Ramin Zahedi
email: zahedi@let.hokudai.ac.jp

On the Logical Origin of the Laws Governing the Fundamental Forces of Nature: A New Algebraic-Axiomatic (Matrix) Approach * by

The main idea and arguments of this article are based on my earlier publications (Refs. [1]-[4],

Introduction and Summary

Why do the fundamental forces of nature (i.e., the forces that appear to cause all the movements and interactions in the universe) manifest in the way, shape, and form that they do? This is one of the greatest ontological questions that science can investigate. In this article, we'll consider this basic and and crucial question (and a number of relevant issues) via a new axiomatic mathematical formalism. By definition, a basic law of physics (or a scientific law in general) is: -A theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present‖ [START_REF]Oxford English Dictionary[END_REF]. Eugene Wigner's foundational paper, -On the Unreasonable Effectiveness of Mathematics in the Natural Sciences‖, famously observed that purely mathematical structures and formalisms often lead to deep physical insights, in turn serving as the basis of highly successful physical theories [START_REF] Wigner | The Unreasonable Effectiveness of Mathematics in the Natural Sciences‖[END_REF]. However, all the known fundamental laws of physics (and corresponding mathematical formalisms which are used for their representations), are generally the conclusions of a number of repeated experiments and observations over years and have become accepted universally within the scientific communities [START_REF] Swartz | The Concept of Physical Law[END_REF][START_REF] Barrow | Theories of Everything: The Quest for Ultimate Explanations[END_REF].

This article is based on my earlier publications (Refs. [START_REF] Zahedi | Linearization Method in the Ring Theory‖, Russian Academy of Sciences (RAS)[END_REF]- [START_REF] Zahedi | The Linearization Method Based on the Ring Theory and its Applications in Physics‖, Russian Academy of Sciences (RAS) Publs[END_REF], Springer, 1996Springer, -1998)). In this article, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Sec.2), "it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms"; where in agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values.. Concerning the basic assumption of rationality of relativistic energy-momentum, it is necessary to note that the rational Lorentz symmetry group is not only dense in the general form of Lorentz group, but also is compatible with the necessary conditions required basically for the formalism of a consistent relativistic quantum theory [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF]. In essence, the main scheme of this new mathematical axiomatic approach to fundamental laws of nature is as follows. First based on the assumption of rationality of D-momentum, by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) is derived. Then by first quantization (followed by a basic procedure of minimal coupling to spacetime geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well. Each class of the derived general covariant field equations also includes a definite form of torsion field appeared as generator of the corresponding field' invariant mass. In addition, it is shown that the (1+3)-dimensional cases of two classes of derived field equations represent a new general covariant massive formalism of bispinor fields of spin-2, and spin-1 particles, respectively. In fact, these uniquely determined bispinor fields represent a unique set of new generalized massive forms of the laws governing the fundamental forces of nature, including the Einstein (gravitational), Maxwell (electromagnetic) and Yang-Mills (nuclear) field equations. Moreover, it is also shown that the (1+2)-dimensional cases of two classes of these field equations represent (asymptotically) a new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, respectively, corresponding to the Dirac and Rarita-Schwinger equations.

" , , " Z W W    Z  Z
As a particular consequence, it is shown that a certain massive formalism of general relativitywith a definite form of torsion field appeared originally as the generator of gravitational field's invariant massis obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a certain set of special relativistic algebraic matrix equations. It has been also proved that Lagrangian densities specified for the originally derived new massive forms of the Maxwell, Yang-Mills and Dirac field equations, are also gauge invariant, where the invariant mass of each field is generated solely by the corresponding torsion field. In addition, in agreement with recent astronomical data, a new particular form of massive boson is identified (corresponding to U(1) gauge 50 group) with invariant mass: m γ ≈ 4.90571×10 -kg, generated by a coupled torsion field of the background space-time geometry.

Moreover, based on the definite mathematical formalism of this axiomatic approach, along with the C, P and T symmetries (represented basically by the corresponding quantum operators) of the fundamentally derived field equations, it has been concluded that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times (where this conclusion, in particular, is based on the T-symmetry). It is proved that 'CPT' is the only (unique) combination of C, P, and T symmetries that could be defined as a symmetry for interacting fields. In addition, on the basis of these discrete symmetries of derived field equations, it has been also shown that only left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. Furthermore, it has been shown that the metric of background space-time is diagonalized for the uniquely derived fermion field equations (defined and expressed solely in (1+2)-dimensional space-time), where this property generates a certain set of additional symmetries corresponding uniquely to the SU(2) L U(2) R symmetry group for spin-1/2 fermion fields (representing -1+3‖ generations of four fermions, including a group of eight leptons and a group of eight quarks), and also the SU(2) L U(2) R and SU(3) gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Hence, along with the known elementary particles, eight new elementary particles, including: four new charge-less right-handed spin-1/2 fermions (two leptons and two quarks, represented by -z e , z n and z u , z d ‖), a spin-3/2 fermion, and also three new spin-1 massive bosons (represented by , where in particular, the new boson is complementary right-handed particle of ordinary boson), have been predicted uniquely and expressly by this new mathematical axiomatic approach.

As a particular result, in Sec. 3-4-2, based on the definite and unique formulation of the derived Maxwell's equations (and also determined Yang-Mills equations, represented uniquely with two specific forms of gauge symmetries, in 3-6-3-2), it has been also concluded generally that magnetic monopoles could not exist in nature.

1-1.

The main results obtained in this article are based on the following three basic assumptions (as postulates):

(1)-"A new definite axiomatic generalization of the axiom of -no zero divisors‖ of integral domains (including the ring of integers ℤ);" This algebraic postulate (as a new mathematical concept) is formulated as follows:

"Let ] [ ij a A 
be a n n  matrix with entries expressed by the following linear homogeneous polynomials in s variables over the integral domain ℤ:

) 0 , 0 ( ) 0 (      M M A A r ( 1
)
where M is a non-zero arbitrary (over the integral domain ℤ). In addition, according to the Ref. [START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF], since 0 ) ,..., , , ( homogeneous equation over ℤ, it is also concluded that homogeneous equations defined over the field of rational numbers ℚ, obey the axiomatic relations ( 1) and (1-1) as well. As particular outcome of this new mathematical axiomatic formalism (based on the axiomatic relations ( 1) and (1-1), including their basic algebraic properties), in Sec. 3-4, it is shown that using, a unique set of general covariant massive (tensor) field equations (with new matrix formalism compatible with Clifford, and Weyl algebras), corresponding to the fundamental field equations of physics, are derivedwhere, in agreement with the rational Lorentz symmetry group, it has been basically assumed that the components of relativistic energy-momentum can only take the rational values. In Sections 3-2 -3-6, we present in detail the main applications of this basic algebraic assumption (along with the following basic assumptions (2) and ( 3)) in fundamental physics.

(2)-"In agreement with the rational Lorentz symmetry group, we assume basically that the components of relativistic energy-momentum (D-momentum) can only take the rational values;"

Concerning this assumption, it is necessary to note that the rational Lorentz symmetry group is not only dense in the general form of Lorentz group, but also is compatible with the necessary conditions required basically for the formalism of a consistent relativistic quantum theory [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF]. Moreover, this assumption is clearly also compatible with any quantum circumstance in which the energymomentum of a relativistic particle is transferred as integer multiples of the quantum of action -h‖ (Planck constant).

Before defining the next basic assumption, it should be noted that from the basic assumptions (1) and [START_REF] Zahedi | On Applied Aspects of the Ring Theory‖, Russian Academy of Sciences (RAS)[END_REF], it follows directly that the Lorentz invariant energy-momentum quadratic relation (represented by formula [START_REF] Kaye | Models of Peano arithmetic‖[END_REF], in Sec. 3-1-1) is a particular form of homogeneous quadratic equation (represented by formula in Sec. 2-2). Hence, using the set of systems of linear equations that are determined uniquely as equivalent algebraic representations of the corresponding set of quadratic homogeneous equations (given by equation in various number of unknown variables, respectively), a unique set of the Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) are also determined, representing equivalent algebraic forms of the energy-momentum quadratic relation in various space-time dimensions, respectively. Subsequently, we've shown that by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a unique set of two definite classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is also derived, corresponding to various spacetime dimensions, respectively. In addition, it is also shown that this derived set of two classes of general covariant field equations represent new tensor massive (matrix) formalism of the fundamental field equations of physics, corresponding to fundamental laws of nature (including the laws governing the fundamental forces of nature). Following these essential results, in addition to the basic assumptions [START_REF] Zahedi | Linearization Method in the Ring Theory‖, Russian Academy of Sciences (RAS)[END_REF] and [START_REF] Zahedi | On Applied Aspects of the Ring Theory‖, Russian Academy of Sciences (RAS)[END_REF], it would be also basically assumed that:

(3)-"We assume that the mathematical formalism of the fundamental laws of nature, are defined solely by the axiomatic matrix constitution formulated uniquely on the basis of postulates (1) and ( 2)".

In addition to this basic assumption, in Sec. 3-5, the C, P and T symmetries of the uniquely derived general covariant field equations (that are field equations ( 3) and (4) in Sec. 1-2-1), would be represented basically by their corresponding quantum matrix operators.

1-2.

In the following, we present a summary description of the main consequences of basic assumptions (1) -(3) (mentioned in Sec. 1-1) in fundamental physics. In this article, the metric signature (+ -... -), the geometrized units [START_REF] Guidry | Gauge Field Theories: An Introduction with Applications‖[END_REF] and also the following sign conventions have been used in the representations of the Riemann curvature tensor

(                                                  G R R R R (2)
1-2-1. On the basis of assumptions ( 1) - [START_REF] Zahedi | On the Connection Between Methods of the Ring Theory and the Group Approach‖, Russian Academy of Sciences (RAS)[END_REF], two sets of the general covariant field equations (compatible with the Clifford algebras) are derived solely as follows: 0 ) (

) ( 0     R R k m i        (3) 0 ) ( ) ( 0    F F k m D i        ( 4
)
where  is the general covariant velocity in stationary reference frame (that is a time-like covariant vector),   and    are two contravariant square matrices (given by formulas ( 6) and ( 7)), R  is a column matrix given by formulas ( 6) and [START_REF] Mordell | Diophantine Equations‖[END_REF], which contains the components of field strength tensor   R (equivalent to the Riemann curvature tensor), and also the components of a covariant quantity which defines the corresponding source current (by relations ( 6) and ( 7)), F  is also a column matrix given by formulas ( 6) and [START_REF] Mordell | Diophantine Equations‖[END_REF], which contains the components of tensor field  F (defined as the gauge field strength tensor), and also the components of a covariant quantity which defines the corresponding source current (by relations [START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF] and [START_REF] Mordell | Diophantine Equations‖[END_REF]). In Sec. 3-5, based on a basic class of discrete symmetries of general covariant field equations (3) and (4), it would be concluded that these equations could be defined solely in (1+2) and (1+3) space-time dimensions, where the (1+2) and (1+3)-dimensional cases these field equations are given uniquely as follows (in terms of the above mentioned quantities), respectively:

                  , (5)
-For (1+2)-dimensional space-time we have: , 0 0 , 0 0 , 0 0 0 , ) ( 0

0 0 3 3 1 2 2 1 1 0 0 1 0 0                                                , 0 1 0 0 , 0 0 1 0 , 1 0 0 0 , 0 0 0 1 , 0 0 , 0 0 3 2 1 0 1 0 2 0 1 2                                                           ; ) ( , ) ( , 0 , 0 ) ( ) ( 0 ) ( ) ( ) ( 0 ) ( ) ( 21 10 ) ( 21 10 F F F R R R F F R R k im D J k im J F F R R                                                          (6)
-For (1+3)-dimensional space-time of we get: , 0 0 , 0 [START_REF] Mordell | Diophantine Equations‖[END_REF] In formulas ( 6) and ( 7),  (as initially given quantities). Moreover, in Sections 3-4 -3-6, it has been also shown that the field equations in (1+2) dimensions, are compatible with the matrix representation of Clifford algebra Cℓ 1,2 , and represent (asymptotically) new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, respectively. It has been also shown that these field equations in (1+3) dimensions are compatible with the matrix representation of Clifford algebra Cℓ 1,3 , and represent solely new general covariant massive formalism of bispinor fields of spin-2 and spin-1 particles, respectively.

0 , 0 0 0 ) ( , ) ( 0 0 0 2 3 1 3 2 1 1 0 0 1 0 0                                               
                                             
1-2-2. In addition, from the field equations ( 3) and (4), the following field equations (with ordinary tensor formulations) could be also obtained, respectively:

,                 R T R T R T R R R            (3-1)         ) ( ) ( 0 ) ( R R J R k im R       ; (3-2)
), ( ) (

                            R ) ( ) ( 0 ) ( ) ( R R R k im J            , ). ( 2
) ( 0       k g k g im T R    (3-3) and , 0          F D F D F D    (4-1) ) ( F J F D       ; (4-2)      A D A D F     , ) ( ) ( 0 ) ( ) ( F F F k im D J          , ). ( 2
) ( 0       k g k g im Z F    (4-3)
where in equations (3-1) -(3-2),    is the affine connection given by:

      K     ,  
 is the Christoffel symbol (or the torsion-free connection),   K is a contorsion tensor defined by:

   k g im K R ) 2 ( ) ( 0  
(that is anti-symmetric in the first and last indices),  T is its corresponding torsion tensor given by:

    K K T  
(as the generator of the gravitational field's invariant mass), 1-2-3. In Sec. 3-5, on the basis of definite mathematical formalism of this axiomatic approach, along with the C, P and T symmetries (represented basically by the corresponding quantum operators, in Sec. 3-5) of the fundamentally derived field equations, it has been concluded that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times (where this conclusion, in particular, is based on the T-symmetry). It is proved that 'CPT' is the only (unique) combination of C, P, and T symmetries that could be defined as a symmetry for interacting fields. In addition, on the basis of these discrete symmetries of derived field equations, it has been also shown that only left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. Furthermore, it has been shown that the metric of background space-time is diagonalized for the uniquely derived fermion field equations (defined and expressed solely in (1+2)-dimensional space-time), where this property generates a certain set of additional symmetries corresponding uniquely to the SU(2) L U(2) R symmetry group for spin-1/2 fermion fields (representing -1+3‖ generations of four fermions, including a group of eight leptons and a group of eight quarks), and also the SU(2) L U(2) R and SU(3) gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Hence, along with the known elementary particles, eight new elementary particles, including: four new charge-less right-handed spin-1/2 fermions (two leptons and two quarks, represented by -z e , z n and z u , z d ‖), a spin-3/2 fermion, and also three new spin-1 massive bosons (represented by " , , "

Z W W   
, where in particular, the new boson Z  is complementary right-handed particle of ordinary Z boson), have been predicted uniquely by this new mathematical axiomatic approach (as shown in Sections 3-6-1-2 and 3-6-3-2).

1-2-4.

As a particular consequence, in Sec. 3-4-2, it is shown that a certain massive formalism of the general theory of relativity -with a definite torsion field which generates the gravitational field‗s massis obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a set of special relativistic algebraic matrix relations. In Sec. 3-4-4, it is also proved that Lagrangian densities specified for the derived unique massive forms of Maxwell, Yang-

 R            R k im R k im R k im R R R ) ( ) ( ) ( ) ( 0 ) ( 0 ) ( 0                T
Mills and Dirac equations, are gauge invariant as well, where the invariant mass of each field is generated by the corresponding torsion field. In addition, in Sec. 3-4-5, in agreement with recent astronomical data, a new massive boson is identified (corresponding to U(1) gauge group) with 50 invariant mass: m γ ≈ 4.90571×10 -kg, generated by a coupling torsion field of the backgroundspace-time geometry. Furthermore, in Sec. 3-4-2, based on the definite and unique formulation of thederived Maxwell‗s equations (and also determined Yang-Mills equations, represented unqiely withtwo specific forms of gauge symmetries), it is also concluded that magnetic monopoles could not exist in nature.

1-2-5.

As it would be also shown in Sec. then from this expression for the current in terms of the stress-energy tensor :

] ) ( ) [( 8 ] ) ( ) [( 8
) ( 0 ) ( 0 ) ( 0 ) ( 0 ) (                Tg k im Tg k im B T k im T k im J R R R R R                  (8-2)
where  

T T 

, the gravitational field equations (including a cosmological constant  emerged naturally in the course of derivation process) could be equivalently derived particularly from the massless case of tensor field equations (3-1) -(3-3) in (1+3) space-time dimensions, as follows:

      g Tg T R      4 8 (9)
1-2-6. Let we emphasize again that the results obtained in this article, are direct outcomes of a new algebraic-axiomatic approach 1 which has been presented in Sec. 2. This algebraic approach, in the form of a basic linearization theory, has been constructed on the basis of a new single axiom (that is the axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] in Sec. 2-1) proposed to replace with the ordinary axiom of -no zero divisors‖ of integral domains (that is the axiom [START_REF] Morel | The Manual of Quants: Discrete Physics and Quantitative Analysis‖[END_REF] in Sec. 2). In fact, as noted in Sec. 1-1 and also Sec. 2-1, the new proposed axiom is a definite generalized form of ordinary axiom [START_REF] Morel | The Manual of Quants: Discrete Physics and Quantitative Analysis‖[END_REF], which particularly has been formulated in terms of square matrices (using basically as primary objects for representing the elements of underlying algebra, i.e. integral domains including the ring of integers). In Sec. 3, based on this new algebraic axiomatic formalism, as a new mathematical approach to origin of the laws of nature, "it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms"; where in agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values.

-

--------------------------------------------------------------------------------------------------------1.
Besides, we may argue that our presented axiomatic matrix approach (for a direct derivation and formulating the fundamental laws of nature uniquely) is not subject to the Gödel's incompleteness theorems [START_REF] Lucas | Gödel's Incompleteness Theorems‖[END_REF]. As in our axiomatic approach, firstly, we've basically changed (i.e. replaced and generalized) one of the main Peano axioms (when these axioms algebraically are augmented with the operations of addition and multiplication [START_REF] Kaye | Models of Peano arithmetic‖[END_REF][START_REF] Mendelson | Introduction to Mathematical Logic‖[END_REF][START_REF] Smullyan | Conceptual Roots of Mathematics‖[END_REF]) for integers, which is the algebraic axiom of -no zero divisors‖. Secondly, based on our approach, one of the axiomatic properties of integers (i.e. axiom of -no zero divisors‖) could be accomplished solely by the arbitrary square matrices (with integer components). This axiomatic reformulation of algebraic properties of integers thoroughly has been presented in Sec. 2 of this article.

Theory of Linearization: a New Algebraic-Axiomatic (Matrix) Approach

Based on the Ring Theory

In this Section a new algebraic theory of linearization (including the simultaneous parameterization) of the homogeneous equations has been presented that is formulated on the basis of ring theory and matrix representation of the generalized Clifford algebras (associated with homogeneous forms of degree r ≥ 2 defined over the integral domain ℤ).

Mathematical models of physical processes include certain classes of mathematical objects and relations between these objects. The models of this type, which are most commonly used, are groups, rings, vector spaces, and linear algebras. A group is a set G with a single operation (multiplication)

c b a   ; G c b a  , ,
which obeys the known conditions [START_REF] Durbin | Modern Algebra: An Introduction‖[END_REF]. A ring is a set of elements R, where two binary operations, namely, addition and multiplication, are defined. With respect to addition this set is a group, and multiplication is connected with addition by the distributivity laws:

), ( ) ( ) ( c a b a c b a       ) ( ) ( ) ( a c a b a c b       ; R c b a  , ,
. The rings reflect the structural properties of the set R. As distinct from the group models, those connected with rings are not frequently applied, although in physics various algebras of matrices, algebras of hyper-complex numbers, Grassman and Clifford algebras are widely used. This is due to the intricacy of finding a connection between the binary relations of addition and multiplication and the element of the rings [START_REF] Durbin | Modern Algebra: An Introduction‖[END_REF][START_REF] Zahedi | On Applied Aspects of the Ring Theory‖, Russian Academy of Sciences (RAS)[END_REF]. This Section is devoted to the development of a rather simple approach of establishing such a connection and an analysis of concrete problems on this basis.

I've found out that if the algebraic axiom of -no zero divisors‖ of integral domains is generalized expressing in terms of the square matrices (as it has been formulated by the axiomatic relation ( 17)), fruitful new results hold. In this Section, first on the basis of the matrix representation of the generalized Clifford algebras (associated with homogeneous polynomials of degree r ≥ 2 over the integral domain ℤ),

we've presented a new generalized formulation of the algebraic axiom of -no zero divisors‖ of integral domains. Subsequently, a linearization theory has been constructed axiomatically that implies (necessarily and sufficiently) any homogeneous equation of degree r ≥ 2 over the integral domain ℤ, should be linearized (and parameterized simultaneously), and then its solution be investigated systematically via its equivalent linearized-parameterized formolation (representing as a certain type of system of linear homogeneous equations). In Sections 2-2 and 2-4, by this axiomatic approach a class of homogeneous quadratic equations (in various numbers of variables) over ℤ has been considered explicitly -Closure:

  l k a a ℤ ,   l k a a ℤ (10) -Associativity: , ) ( ) ( p l k p l k a a a a a a      p l k p l k a a a a a a      ) ( ) ( (11) -Commutativity: , k l l k a a a a    k l l k a a a a    (12) -Existence of identity elements: , 0 k k a a   k k a a  1 (13)
-Existence of inverse element (for operator of addition):

0 ) (    k k a a (14) -Distributivity: ), ( ) ( ) ( p k l k p l k a a a a a a a       ) ( ) ( ) ( p l p k p l k a a a a a a a       (15)
-No zero divisors (as a logical bi-conditional for operator of multiplication):

) 0 , 0 ( 0      l l k k a a a a (16)
Axiom [START_REF] Morel | The Manual of Quants: Discrete Physics and Quantitative Analysis‖[END_REF], equivalently, could be also expressed as follows,

0 ) 0 0 (       l k l k a a a a (16-1)
In this article as a new basic algebraic property of the domain of integers, we present the following new axiomatic generalization of the ordinary axiom of -no zero divisors‖ [START_REF] Morel | The Manual of Quants: Discrete Physics and Quantitative Analysis‖[END_REF], which particularly has been formulated on the basis of matrix formalism of Clifford algebras (associated with homogeneous polynomials of degree r ≥ 2, over the integral domain ℤ):

"Let ] [ ij a A 
be a n n  matrix with entries expressed by the following linear homogeneous polynomials in s variables over the integral domain ℤ:

; ) ,..., , , ( 1 3 2 1     s k k ijk s ij ij b H b b b b a a suppose also -  r ℕ: n s I b b b b F A r ) ,..., , , ( 3 2 1
) 0 , 0 ( ) 0 (      M M A A r ( 17
)
where M is a non-zero arbitrary

n s I b b b b F A r ) ,..., , , ( 3 2 1



, the axiomatic biconditional [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] could be also represented as follows:

) 0 , 0 ( ] 0 ) ,..., , , ( [ 3 2 1      M M A b b b b F s (17-1)
where the homogeneous equation 0 ) ,..., , , ( [START_REF] Morel | The Manual of Quants: Discrete Physics and Quantitative Analysis‖[END_REF]. In fact, the axiom [START_REF] Morel | The Manual of Quants: Discrete Physics and Quantitative Analysis‖[END_REF], as a particular case, can be obtained from the axiom (17) (or (17-1), but not vice versa.

Moreover, according to the Ref. [START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF], since 0 ) ,..., , , (

3 2 1  s b b b b F
is a homogeneous equation over ℤ, it is also concluded that homogeneous equations defined over the field of rational numbers ℚ, obey the axiomatic relations [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] and (17-1) as well.

As a crucial additional issue concerning the axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF], it should be noted that the condition

-   r ℕ: n s I b b b b F A r ) ,..., , , ( 3 2 1



‖ which is assumed initially in the axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF], is also compatible with matrix representation of the generalized Clifford algebras [Refs. 40 -47] . However, in some particular cases and applications, we may also assume some additional conditions for the generators k E , such as the Hermiticity or anti-Hermiticity (see Sections 2-2, 2-4 and 3-1). In Sec. 3, we use these algebraic properties of the square matrix A (corresponding with the homogeneous quadratic equations), where we present explicitly the main applications of the axiomatic relations ( 17) and (17-1) in foundations of physics (where we also use the basic assumptions ( 2) and (3) mentioned in Sec. 1-1).

It is noteworthy that since the axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] has been formulated solely in terms of square matrices, in Ref. [START_REF] Zahedi | On the Logical Origin of the Laws Governing the Fundamental Forces of Nature: A New Axiomatic Matrix Approach‖[END_REF] we have shown that all the ordinary algebraic axioms ( 10) -( 15) of integral domain ℤ (except the axiom of -no zero divisors‖ ( 16)), in addition to the new axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF], could be also reformulated uniformly in terms of the set of square matrices. Hence, we may conclude that the square matrices, logically, are the most elemental algebraic objects for representing the basic properties of set of integers (as the most fundamental set of mathematics).

In the following, based on the axiomatic relation (17-1), we've constructed a corresponding basic algebraic linearization (including a parameterization procedure) approach applicable to the all classes of homogeneous equation. Hence, it could be also shown that for any given homogeneous equation of degree r ≥ 2 over the ring ℤ (or field ℚ), a square matrix A exists that obeys the relation .

In this regard, for various classes of homogeneous equations, their equivalent systems of linear equations would be derived systematically. As a particular crucial case, in Sections 2-2 and 2-4, by derivation of the systems of linear equations equivalent to a class of quadratic homogeneous equations (in various number of unknown variables) over the integral domain ℤ (or field ℚ), these equations have been analyzed (and solved) thoroughly by this axiomatic approach. In the following, the basic schemes of this axiomatic linearization-parameterization approach are described.

First, it should be noted that since the entries

n s I b b b b F A r ) ,..., , , ( 3 2 1



, we may also assume certain additional conditions for the matrix generators k E , such as the Hermiticity (or anti-Hermiticity), and so on (see Sections 2-2, 2-4 and 3- 1). In fact, these conditions could be required, for example, if a homogeneous invariant relation (of physics) be represented by a homogeneous algebraic equation of the type: , 0 ) ,..., , , ( In Sec. 2-2, as one of the main applications of the axiomatic relations ( 17) and (17-1), we derive a unique set the square matrices for s = 0,1,2,3,4,.., respectively. In addition, in Sec. 2-3 using this systematic axiomatic approach, for some particular forms of homogeneous equations of degrees 3, 4 and 5 , their equivalent systems of linear equations have been derived as well. It is noteworthy that using this general axiomatic approach (on the basis of the logical biconditional relations [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] and (17-1)), for any given r th degree homogeneous equation in s unknown variables over the integral domain over ℤ, its equivalent system(s) of linear equations 0   M A is derivable (with a unique size, if in the course of the derivation, we also assume a minimum value for n , i.e. the size of corresponding square matrix n n A  ). Furthermore, for a given homogeneous equation of degree r in s unknown variables, the minimum value for n , i.e. the size of the corresponding square matrix given homogeneous equation of degree r in s unknown variables, on the basis of the axiomatic relations [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] and , see also the preprint versions of this article in Refs. [START_REF] Zahedi | On the Logical Origin of the Laws Governing the Fundamental Forces of Nature: A New Axiomatic Matrix Approach‖[END_REF].

n n A  in its equivalent matrix equation: 0   M A , is: 1 1 min     s s r r n for 2  r

2-2.

In this section, on the basis of axiomatic relation and general methodological notes (mentioned above), for the following general form of homogeneous quadratic equations their equivalent systems of linear equations are derived (uniquely): 0 ) , ,..., , , , ( 0 [START_REF] Kauffman | Knots and Physics[END_REF] Equation ( 18) for s = 0,1,2,3,4,.. is represented by, respectively: , 0 [START_REF] Svozil | Randomness & Undecidability in Physics: Chapter 3: Discrete Physics‖[END_REF] , 0 [START_REF] Amoroso | The Physics of Reality: Space, Time, Matter; Section I: Laws of Forms, Majorana Fermions, and Discrete Physics‖[END_REF] , 0 [START_REF] Zenil | A Computable Universe: Understanding and Exploring Nature as Computation‖[END_REF] . 0 [START_REF] Penrose | Spinors and Space-Time‖[END_REF] It is necessary to note that quadratic equation ( 18) is isomorphic to the following ordinary representations of homogeneous quadratic equations: , 0 0 ,

1 1 0 0     s i i i s s f e f e f e f e Q
0 0 0 0     f e f e i i i
1 1 0 0 1 0      f e f e f e i i i (20) , 0 2 2 1 1 0 0 2 0       f e f e f e f e i i i
3 3 2 2 1 1 0 0 3 0        f e f e f e f e f e i i i
4 4 3 3 2 2 1 1 0 0 4 0         f e f e f e f e f e f e i i i
   s j i j i ij c c G (18-1)      s j i s j i j i ij j i ij d d G c c G 0 , 0 , , (18-2)
using the linear transformations: 2-2-1. As it will be shown in Sec. 2-2-2, the reason for choosing equation [START_REF] Kauffman | Knots and Physics[END_REF] as the standard general form for representing the homogeneous quadratic equations (that could be also transformed to the ordinary representations of homogeneous quadratic equations (18-1) and (18-2), by linear transformations (18-3)) is not only its very simple algebraic structure, but also the simple linear homogeneous forms of the entries of square matrices A (expressed in terms of variables i i f e , ) in the corresponding systems of linear equations 0   M A obtained as the equivalent form of quadratic equation [START_REF] Kauffman | Knots and Physics[END_REF] in various number of unknown variables.

Moreover, as it is shown in the following, we may also assume certain Hermiticity and anti-Hermiticity conditions for the deriving square matrices A (in the corresponding systems of linear equations 0   M A equivalent to the quadratic equation ( 18)), without any restriction in the existence and procedure of derivation of these matrices. By adding these particular conditions, for a specific number of variables in equation [START_REF] Kauffman | Knots and Physics[END_REF], its equivalent matrix equation 0   M A could be determined uniquely. In Sec. 3, where we use the algebraic results obtained in Sections 2-2 and 2-4 on the basis of axiomatic relations [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] and , in fact, the assumption of these Hermiticity and anti-Hermiticity properties is a necessary issue. These Hermiticity and anti-Hermiticity additional conditions are defined as follows:

" First, by supposing:

0 0 f e  and i i f e   (for s i ,..., 2 , 1 
), the quadratic equation ( 18) would be represented as: ) be anti-Hermitian:

* i i E E   ".
2-2-2. As noted and would be also shown below, by assuming the above additional Hermiticity and anti-Hermiticity conditions, the system of linear equations 0   M A corresponding to quadratic equation [START_REF] Kauffman | Knots and Physics[END_REF], is determined uniquely for any specific number of variables i i f e , . Hence, starting from the simplest (or trivial) case of quadratic equation [START_REF] Kauffman | Knots and Physics[END_REF], i.e. equation [START_REF] Svozil | Randomness & Undecidability in Physics: Chapter 3: Discrete Physics‖[END_REF], its equivalent system of linear equations is given uniquely as follows:

0 0 0 2 1 0 0                m m f e M A ( 24
)
where it is assumed 0  M , and in agreement with (17-1) we also have:

2 0 0 0 0 0 0 2 ) ( 0 0 0 0 I f e f e f e A                (24-1)
For equation [START_REF] Ilachinski | Cellular Automata: A Discrete Universe‖[END_REF], the corresponding equivalent system of linear equations is determined as: [START_REF] Freedman | [END_REF] where we have:

0 0 0 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 1 1 0                                                 m m m m e e f f f e f e M M A A M A
                                           (25-1)
Notice that matrix equation ( 25) could be represented by two matrix equations, as follows:

, 0 3 0 1 1 0                   m m f e f e M A (25-2) 0 2 1 0 1 1 0                    
), so we may choose the matrix equation (25-2) as the system of linear equations equivalent to the quadratic equation ( 20)where for simplicity in the indices of parameters i m , we may simply replace arbitrary parameter 3 m with arbitrary parameter 1 m , as follows (for

0 1        m m ): . 0 1 0 1 1 0               m m f e f e ( 26
)
The system of linear equations corresponding to the quadratic equation ( 21) is obtained as: [START_REF] Straumann | General Relativity: With Applications to Astrophysics‖[END_REF] where in agreement with [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] we have:

1 2 0 0 2 1 0 1 2 2 1 0 1 2 0                                                                                          
8 2 2 1 1 0 0 0 2 1 0 1 2 2 1 0 1 2 0 0 2 1 0 1 2 2 1 0 1 2 0 0 2 1 0 1 2 2 1 0 1 2 0 0 2 1 0 1 2 2 1 0 1 2 0 2 ) (
                                                                                     
In addition, similar to equation [START_REF] Freedman | [END_REF], the obtained matrix equation ( 27) is equivalent to a system of two matrix equations, as follows: ), so we may choose the equation (27-2) as the system of linear equations corresponding to the quadratic equation ( 21 where the column parametric matrix M in (29) is non-zero 0  M . In a similar manner, the uniquely obtained system of linear equations corresponding to the quadratic equation [START_REF] Penrose | Spinors and Space-Time‖[END_REF], is given by: 0 where we've assumed the parametric column matrix M in ( 30) is non-zero, 0  M . In a similar manner, the systems of linear equations (written in matrix forms similar to the matrix equations ( 24), ( 26), [START_REF] Wald | General Relativity‖[END_REF], [START_REF] Elton | The Dirac equation without spinors‖[END_REF] and [START_REF] Griesshammer | Introduction to Supersymmetry and Exact Nonperturbative Results in SUSY-QCD‖[END_REF]) with larger sizes are obtained for the quadratic equation [START_REF] Kauffman | Knots and Physics[END_REF] ). Moreover, based on the axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF], in fact, by solving the obtained system of linear equations corresponding to a homogeneous equation of degree r , we may systematically show (and decide) whether this equation has the integral solution.

0 2 1 3 0 1 2 3 3 2 1 0 3 1 2 0 2 1 3 0 1 2 3 0                                                                     
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 14 13 12 11 10 0 4 3 2 1 0 4 3 1 2 0 4 2 1 3 0 4 1 2 3 0 3 2 1 4 0 3 1 2 4 0 2 1 3 4 0 1 2 3 4 4 3 2 1 0 4 3 1 2 0 4 2 1 3 0 4 1 2 3 0 3 2 1 4 0 3 1 2 4 0 2 1 3 4 0 1 2 3 4 0                                                                                                                                            

2-3.

Similar to the uniquely obtained systems of linear equations corresponding to the homogeneous quadratic equations (in Sec. 2-2), in this section in agreement with the axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF], we present the obtained systems of linear equations, i.e.

  M A

(by assuming the minimum value for n , i.e. the size of square matrix n n A  ), corresponding to some homogeneous equations of degrees 3, 4 and 5, respectively. For the homogeneous equation of degree three of the type: [START_REF] Illge | Massive Fields of Arbitrary Spin in Curved Space-Times‖[END_REF] the corresponding system of linear equations is given as follows: [START_REF] Pilkuhn | Relativistic Quantum Mechanics[END_REF] where A is a 27×27 square matrix written in terms of the square 9×9 matrices A , given by: , (

, 0 ) , , , , , ( 1 1 1 2 2 2 2 2 2 2 0 0 0 2 0 2 2 1 1 0 0       g f e f e f e f e f e f e f e f e F
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 2 1 0 2 1 0 2 0 1 2 1 0 0 2 1 0 0 1 2 2 1 0 2 2 1 0 0 2 2 1                                                f f e f f g f f e e e e
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 2 1 0 2 1 0 2 2 0 1 2 2 1 0 0 2 2 1 0 0 1 2 1 0 2 1 0 0 2 2                                                e f e f
                                             2 2 0 0 1 2 2 1 0 2 2 1 0 2 0 1 2 1 0 0 2 1 0 0 1 2 1 0 2 1 0 0 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f e f e f
) 33
The uniquely obtained system of linear equations (i.e.

  M A

, by assuming the minimum size for the square matrix [START_REF] Weinberg | The Quantum Theory of Fields[END_REF] where A is a 27×27 square matrix written in terms of the 9×9 matrices 1 A , 2 A and 3 A given by: (

For the 4 th degree homogeneous equation of the type: 0 ) , , , , , ( [START_REF] Ticciati | Quantum Field Theory for Mathematicians[END_REF] the corresponding system of linear equations is given as, , 0 :::

4 3 2 1 2 3 1 3 2 1 4 3 2 1 2 1      f f f f e e e e f f f f e e F
0 0 0 0 0 0 0 0 0 0 0 0 16 2 1 4 3 2 1                             m m m A A A A M A (38)
where A is a 16×16 square matrix represented in terms of the 4×4 matrices In addition, the system of linear equations corresponding to 5 th degree homogeneous equation of the type, 0 ) , , , , , , ( where A is a 25×25 square matrix expressed in terms of the following 5×5 matrices

5 4 3 2 1 4 2 1 3 2 2 1 2 2 3 1 2 4 1 5 4 3 2 1 2 1       f f f f f e e e e

2-4.

In this Section by solving the derived systems of linear equations ( 26), [START_REF] Wald | General Relativity‖[END_REF], [START_REF] Elton | The Dirac equation without spinors‖[END_REF] and [START_REF] Griesshammer | Introduction to Supersymmetry and Exact Nonperturbative Results in SUSY-QCD‖[END_REF] corresponding to the quadratic homogeneous equations ( 20) - [START_REF] Penrose | Spinors and Space-Time‖[END_REF] in Sec. 2-2, the general parametric solutions of these equations are obtained for unknowns i e and i f . There are the standard methods for obtaining the general solutions of the systems of homogeneous linear equations in integers [START_REF] Mordell | Diophantine Equations‖[END_REF][START_REF] Ireland | A Classical Introduction to Modern Number Theory‖[END_REF]. Using these methods, for the system of linear equations (26) (and consequently, its corresponding quadratic equation ( 20)) we get directly the following general parametric solutions for unknowns 1 0 , e e and 1 0 , f f

: 1 0 1 1 1 1 1 0 0 0 , , , m lk f m lk e m lk f m lk e      (43) where 1 0 ,k k , l m m , , 1
are arbitrary parameters. In the matrix representation the general parametric solution [START_REF] Pappacena | Matrix pencils and a generalized Clifford algebra‖[END_REF] has the following form: where

                                         1 0 1 1 1 0 1 0 1 0 0 0 , 1 0 0 1 k k m m l K lM f f k k lm K lM e e f e
2 mI M e 
, K is a column parametric matrix and f M is also a parametric anti-symmetric matrix.

For the system of linear equations (28) (and, consequently, for its corresponding quadratic homogeneous equation ( 21)), the following general parametric solution is obtained directly:

) ( , ), ( , ), ( , 3 1 2 0 2 2 2 1 0 3 2 1 1 1 2 2 1 1 0 0 0 m k m k l f m lk e m k m k l f m lk e m k m k l f m lk e          (44)
where

l m m m m k k k , , , , , , , 3 2 1 2 1 0
are arbitrary parameters. In matrix representation the general parametric solution (44) could be also written as follows: , K is a column parametric matrix and f M is also a parametric anti-symmetric matrix.

                                                                  
In addition, it could be simply shown that by adding two particular solutions of the types } , {

i i i f e e  
is also obtained, as follows:

0 ) ( 0 ) ( ) 0 , 0 ( 0 0 0 0                    s i i i i s i i i i i s i i i s i i i f e e f e f e f e f e (44-2)
Using the general basic property (44-2) in addition to the general parametric solution (44) of quadratic equation ( 21) (which has been obtained directly from the system of linear equations ( 28) corresponding to quadratic equation ( 21)), exceptionally, the following equivalent general parametric solution is also obtained for quadratic equation ( 21):

) ( ), ( ), ( ), ( ), ( ), ( 3 1 2 0 2 1 2 2 1 0 3 2 1 2 1 1 2 2 1 1 0 3 0 0 m k m k l f km m k l e m k m k l f km m k l e m k m k l f km m k l e             (45)
where

k k k k , , , 2 1 0 , l m m m m , , , ,
h f h e h e h e     03 2 31 1 , h f h f  
, in addition to the replacements of nine arbitrary parameters

3 2 1 0 , , , u u u u , w v v v v , , , , 3 2 1 0
, with new nine parameters of the types

3 2 1 0 , , , u u u u , w v v v v , , , , 3 2 1 0 , given as: , , , , 2 1 2 0 1 3 0 u k u k u k u k     , 2 u k  , 1 1 v m  , 0 2 v m  , 3 3 v m  , 2 v m  w l  ,
exceptionally, could be also represented as follows:

); ( ), ( ), ( ), ( ), ( ), (

3 0 0 3 03 1 2 2 1 21 1 3 3 1 31 0 2 2 0 20 0 1 1 0 10 3 2 2 3 23 v u v u w h v u v u w h v u v u w h v u v u w h v u v u w h v u v u w h             (45-1)
where it could be expressed by a single uniform formula as well (for μ, ν = 0,1,2,3):

) (      v u v u w h   (45-2)
A crucial and important issue concerning the algebraic representation (45-2) (as the differences of products of two parametric variables  u and  v ) for the general parametric solution [START_REF] Coskun | On representations of Clifford algebras of Ternary cubic forms‖[END_REF], is that it generates a symmetric algebra Sym(V) on the vector space

, V where V v u  ) , (
  [START_REF] Bourbaki | Algebra‖[END_REF]. This essential property of the form (45-2) would be used for various purposes in the following and also in Sec. 3 (where we show the applications of this axiomatic linearization-parameterization approach and the results obtained in this Section and Sec. 2-4, in foundations of physics).

In addition, as it is also shown in the following, it should be mentioned again that the algebraic form (45-2) (representing the symmetric algebra Sym(V)), exceptionally, is determined solely from the parametric solution (44-1) (obtained from the system of equations ( 28)) by using the identity (44-2). In fact, from the parametric solutions obtained directly from the subsequent systems of linear equations. i.e. equations ( 29), [START_REF] Griesshammer | Introduction to Supersymmetry and Exact Nonperturbative Results in SUSY-QCD‖[END_REF] and so on (corresponding to the quadratic equations ( 22), ( 23 In the following (also see Ref. [START_REF] Zahedi | On the Logical Origin of the Laws Governing the Fundamental Forces of Nature: A New Axiomatic Matrix Approach‖[END_REF]), we present the parametric solutions that are obtained directly from the systems of linear equations ( 29), [START_REF] Griesshammer | Introduction to Supersymmetry and Exact Nonperturbative Results in SUSY-QCD‖[END_REF] and so on, which also would be the parametric solutions of their corresponding quadratic equations [START_REF] Kauffman | Knots and Physics[END_REF] in various number of unknowns (on the basis of axiom ( 17)). Meanwhile, the following obtained parametric solutions for the systems of linear equations ( 29), [START_REF] Griesshammer | Introduction to Supersymmetry and Exact Nonperturbative Results in SUSY-QCD‖[END_REF] and so on, similar to the parametric solutions ( 43) and ( 44), include one parametric term for each of unknowns i e , and sum of s parametric terms for each of unknowns i f (where

s i ,..., 3 , 2 , 1 , 0  ).
Hence, the following parametric solution is derived directly from the system of linear equations ( 29) (that would be also the solution of its corresponding quadratic equation ( 22)):

).

                   (46)
where , K is a column parametric matrix and f M is also a parametric anti-symmetric matrix.

l k k k k , , , ,                                                                                  
However, in solutions ( 46 are not arbitrary, and in fact, in the course of obtaining the solution (46) from the system of linear equations [START_REF] Elton | The Dirac equation without spinors‖[END_REF], a condition appears for these parameters as follows:

0 7 3 6 2 5 1 4     m m m m m m m m (47)
The condition ( 47) is also a homogeneous quadratic equation that should be solved first, in order to obtain a general parametric solution for the system of linear equations [START_REF] Elton | The Dirac equation without spinors‖[END_REF]. Since the parameter 4 m has not appeared in the solution [START_REF]Clifford Algebras with Numeric and Symbolic Computations‖[END_REF], it could be assumed that 0 4  m , and the condition ( 47) is reduced to the following homogeneous quadratic equation, which is equivalent to the quadratic equation ( 20) (corresponding to the system of linear equations ( 28)):

, 0 4  m 0 7 3 6 2 5 1    m m m m m m , ( 47-1)
where the parameter m would be arbitrary. The condition (47-1) is equivalent to the quadratic equation [START_REF] Amoroso | The Physics of Reality: Space, Time, Matter; Section I: Laws of Forms, Majorana Fermions, and Discrete Physics‖[END_REF]. Hence by using the general parametric solution (45-1) (as the most symmetric solution obtained for quadratic equation ( 21) by solving its corresponding system of linear equations ( 28)), the following general parametric solution for the condition (47-1) is obtained:

) ( ), ( ), ( ), ( ), ( ), ( 2 1 1 2 7 1 3 3 1 6 3 2 2 3 5 0 3 3 0 3 0 2 2 0 2 0 1 1 0 1 v u v u w m v u v u w m v u v u w m v u v u w m v u v u w m v u v u w m             (48) where 3 2 1 0 , , , u u u u , w v v v v , , , , 3 2 1 0
, m are arbitrary parameters. Now by replacing the solutions [START_REF] Witten | Three Lectures On Topological Phases Of Matter‖[END_REF] (obtained for

7 6 5 3 2 1 , , , , , m m m m m m in terms of the new parameters 3 2 1 0 , , , u u u u , w v v v v , , , , 3 2 1 0
) in the relations [START_REF]Clifford Algebras with Numeric and Symbolic Computations‖[END_REF], the general parametric solution of the system of linear equations (29) (and its corresponding quadratic equation ( 22)) is obtained in terms of the arbitrary parameters

3 2 1 0 , , , k k k k , 3 2 1 0 , , , u u u u , , , , , , 3 2 1 0 w v v v v l m, .
For the system of linear equations (30) (and its corresponding quadratic equation ( 23)), the following parametric solution is obtained:

) ( , ), ( , ), ( , ), ( , ), (
1 0 1 1 1 5 4 3 3 2 2 1 1 0 0 0 m k m k m k m k l f m lk e m k m k m k m k l f m lk e m k m k m k m k l f m lk e m k m k m k m k l f m lk e m k m k m k m k l f m lk e                            (49) where l k k k k k , , , , , 4 3 2 1 0
are arbitrary parameters. In the matrix representation the solution (49) could be also written as follows: , K is a column parametric matrix and f M is also a parametric anti-symmetric matrix.

                                                                                                             
However, similar to the system of equations ( 29), in the course of obtaining the solutions ( 49) or (49-1) from the system of linear equations ( 30), the following conditions appear for parameters

                (50)
that is similar to the condition [START_REF] Smith | Decomposition of generalized Clifford algebras‖[END_REF]. Here also by the same approach, since the parameters

     m m m m m
, and the set of conditions ( 50) are reduced to the following system of homogeneous quadratic equations which are similar to the quadratic equation ( 20) (corresponding to the system of linear equations ( 28

                   
The conditions (50-1) are also similar to the quadratic equation [START_REF] Amoroso | The Physics of Reality: Space, Time, Matter; Section I: Laws of Forms, Majorana Fermions, and Discrete Physics‖[END_REF]. Hence using again the general parametric solution (45-1), the following general parametric solutions for the system of homogeneous quadratic equations

4 0 3 3 0 3 2 0 0 2 2 0 1 1 0 1 v u v u w m v u v u w m v u v u w m v u v u w m v u v u w m v u v u w m m m m m v u v u w m m v u v u w m v u v u w m v u v u w m                          (51) where 4 3 2 1 0 4 3 2 1 0 , , , , , , , , , v v v v v u u u u u
and w are arbitrary parameters. Now by replacing the solution (51) (that have been obtained for

w v v v v v u u u u u , , , , , , , , , , 4 3 2 1 0 4 3 2 1 0
) in the relations [START_REF] Sidharth | Frontiers of Fundamental Physics 4‖[END_REF], the general parametric solution of the system of linear equations (30) (and its corresponding quadratic equation ( 23)) is obtained in terms of the arbitrary parameters , , ,

1 0 4 3 2 1 0 4 3 2 1 0 w v v v v v u u u u u k k k k k l m, . , , , , , , , , , , , , , 4 3 2
Meanwhile, similar to the relations ( 48) and ( 51), it should be noted that arbitrary parameter 1 m in the general parametric solution [START_REF] Pappacena | Matrix pencils and a generalized Clifford algebra‖[END_REF] and arbitrary parameters 20) and ( 21), respectively, by solving their equivalent systems of linear equations ( 26) and ( 28)), by keeping their arbitrariness property, could particularly be expressed in terms of new arbitrary parameters

1 0 1 0 , , , v v u u and 2 1 0 2 1 0 , , , , , v v v u u u , as follows, respectively: ); ( 0 1 1 0 1 v u v u w m   (43-2) ). ( ), ( ), ( 0 3 3 0 3 0 2 2 0 2 0 1 1 0 1 v u v u w m v u v u w m v u v u w m       (44-3)
In fact, as a particular common algebraic property of both parametric relations (43-2) and (44-2), it could be shown directly that by choosing appropriate integer values for parameters

w v v u u , , , , 1 0 1 0 in the relation (43-2), the parameter 1 m (defined in terms of arbitrary parameters ) , , , , 1 0 1 0 w v v u u
could take any given integer value, and similarly, by choosing appropriate integer values for parameters , respectively. In addition, for the general quadratic homogeneous equation [START_REF] Kauffman | Knots and Physics[END_REF] with more number of unknowns, the general parametric solutions could be obtained by the same approaches used above for quadratic equations ( 20) -( 23), i.e. by solving their corresponding systems of linear equations (defined on the basis of axiom ( 17)). Moreover, using the isomorphic transformations (18-3) and the above general parametric solutions obtained for quadratic equations ( 20) -( 23),… (via solving their corresponding systems of linear equations ( 26), ( 28), ( 29), [START_REF] Griesshammer | Introduction to Supersymmetry and Exact Nonperturbative Results in SUSY-QCD‖[END_REF],…), the general parametric solutions of quadratic equations of the regular type (18-2) (in various number of unknown) are also obtained straightforwardly. All the parametric solutions that are obtained by this new systematic matrix approach for the homogeneous quadratic equations and also higher degree homogeneous equations of the type 0 ) ,..., , , (

w v v v u u u , , , , , , 2 1
3 2 1  s x x x x F
(defined in the axiom ( 17)), are fully compatible with the solutions and conclusions that have been obtained previously for various homogeneous equations by different and miscellaneous methods and approaches [START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF][START_REF] Mordell | Diophantine Equations‖[END_REF][START_REF] Ireland | A Classical Introduction to Modern Number Theory‖[END_REF]. In Sec. 3, we've used the uniquely specified systems of homogeneous linear equations (and also their general parametric solutions) corresponding to the homogeneous quadratic equationswhere, in particular, it has been assumed basically that the components of the relativistic energy-momentum vector (as one of the most basic physical quantities) in the Lorentz invariant energy-momentum (homogeneous) quadratic relation, can only take the rational values.

A Unique Mathematical Derivation of the Laws Governing the Fundamental

Forces of Nature: Based on a New Algebraic-Axiomatic (Matrix) Approach

In this Section, , as a new mathematical approach to origin of the laws of nature, using the new basic algebraic axiomatic (matrix) formalism (presented in Sec.2), "it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms"; where in agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energymomentum can only take rational values. Concerning the basic assumption of rationality of relativistic energy-momentum, it is necessary to note that the rational Lorentz symmetry group is not only dense in the general form of Lorentz group, but also is compatible with the necessary conditions required basically for the formalism of a consistent relativistic quantum theory [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF]. In essence, the main scheme of this new mathematical axiomatic approach to fundamental laws of nature is as follows. First in Sec. 3-1-1, based on the assumption of rationality of D-momentum, by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) is derived. Then in Sec. 3-4, by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well. Each class of the derived general covariant field equations also includes a definite form of torsion field appeared as generator of the corresponding field' invariant mass. In addition, in Sections 3-4 -3-5, it is shown that the (1+3)-dimensional cases of two classes of derived field equations represent a new general covariant massive formalism of bispinor fields of spin-2, and spin-1 particles, respectively. In fact, these uniquely determined bispinor fields represent a unique set of new generalized massive forms of the laws governing the fundamental forces of nature, including the Einstein (gravitational), Maxwell (electromagnetic) and Yang-Mills (nuclear) field equations. Moreover, it is also shown that the (1+2)-dimensional cases of two classes of these field equations represent (asymptotically) a new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, respectively, corresponding to the Dirac and Rarita-Schwinger equations.

As a particular consequence, in Sec. 3-4-2, it is shown that a certain massive formalism of general relativitywith a definite form of torsion field appeared originally as the generator of gravitational field's invariant massis obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a certain set of special relativistic algebraic matrix equations. In Sec. 3-4-4, it has been also proved that Lagrangian densities specified for the originally derived new massive forms of the Maxwell, Yang-Mills and Dirac field equations, are also gauge invariant, where the invariant mass of each field is generated solely by the corresponding torsion field. In addition, in Sec. 3-4-5, in agreement with recent astronomical data, a particular new form of massive boson is identified (corresponding to U(1) gauge group) with invariant mass: 50 m γ ≈ 4.90571×10 -kg, generated by a coupled torsion field of the background space-time geometry.

Moreover, in Sec. 3-5-2, based on the definite mathematical formalism of this axiomatic approach, along with the C, P and T symmetries (represented basically by the corresponding quantum operators) of the fundamentally derived field equations, it has been concluded that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times (where this conclusion, in particular, is based on the time-reversal symmetry). In Sections 3-5-3 and 3-5-4, it is proved that 'CPT' is the only (unique) combination of C, P, and T symmetries that could be defined as a symmetry for interacting fields. In addition, in Sec. 3-5-4, on the basis of these discrete symmetries of derived field equations, it has been also shown that only left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. Furthermore, in Sec. 3-6, it has been shown that metric of the background space-time is diagonalized for the uniquely derived fermion field equations (defined and expressed solely in (1+2)dimensional space-time), where this property generates a certain set of additional symmetries corresponding uniquely to the SU(2) L U(2) R symmetry group for spin-1/2 fermion fields (representing -1+3‖ generations of four fermions, including a group of eight leptons and a group of eight quarks), and also the SU(2) L U(2) R and SU(3) gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Hence, along with the known elementary particles, eight new elementary particles, including: four new charge-less right-handed spin-1/2 fermions (two leptons and two quarks, represented by -z e , z n and z u , z d ‖), a spin-3/2 fermion, and also three new spin-1 massive bosons (represented by " , , "

Z W W   
, where in particular, the new boson Z  is complementary right-handed particle of ordinary Z boson), are predicted uniquely by this new mathematical axiomatic approach. Furthermore, as a particular result, in Sec. 3-4-2, based on the definite and unique formulation of the derived Maxwell's equations (and also determined Yang-Mills equations, represented uniquely with two specific forms of gauge symmetries, in 3-6-3-2), it is also concluded generally that magnetic monopoles could not exist in nature.

3-1.

As noted in Sec. 1-1, the main results obtained in this article are based on the following three basic assumptions (as postulates):

(1)-"A new definite axiomatic generalization of the axiom of -no zero divisors‖ of integral domains (including the integr ring ℤ) is assumed (represented by formula [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF], in Sec. 2-1);" This basic assumption (as a postulate) is a new mathematical concept. In Sec. 2-1, based on this new axiom, a general algebraic axiomatic (matrix) approach (in the form of a basic linearizationparameterization theory) to homogeneous equations of degree r ≥ 2 (over the integer domain, extendable to field of rational numbers), has been formulated. A summary of the main results obtained from this axiomatic approach have been presented in Sec. 1-1. As particular outcome of this new mathematical axiomatic formalism (based on the axiomatic relations ( 17) and (17-1), including their basic algebraic properties presented in detail, in Sections 2-1 -2-4), in Sec. 3-4, it is shown that using, a unique set of general covariant massive (tensor) field equations (with new matrix formalism compatible with Clifford, and Weyl algebras), corresponding to the fundamental field equations of physics, are derivedwhere, in agreement with the rational Lorentz symmetry group, it has been basically assumed that the components of relativistic energy-momentum can only take the rational values. In Sections 3-2 -3-6, we present in detail the main applications of this basic algebraic assumption (along with the following basic assumptions (2) and ( 3)) in fundamental physics.

(2)-"In agreement with the rational Lorentz symmetry group, we assume basically that the components of relativistic energy-momentum (D-momentum) can only take the rational values;" Concerning this assumption, it is necessary to note that the rational Lorentz symmetry group is not only dense in the general form of Lorentz group, but also is compatible with the necessary conditions required basically for the formalism of a consistent relativistic quantum theory [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF]. Moreover, this assumption is clearly also compatible with any quantum circumstance in which the energy-momentum of a relativistic particle is transferred as integer multiples of the quantum of action -h‖ (Planck constant).

Before defining the next basic assumption, it should be noted that from the basic assumptions ( 1) and ( 2), it follows directly that the Lorentz invariant energy-momentum quadratic relation (represented by formula [START_REF] Kaye | Models of Peano arithmetic‖[END_REF], in Sec. 3-1-1) is a particular form of homogeneous quadratic equation (18-2). Hence, using the set of systems of linear equations that have been determined uniquely as equivalent algebraic representations of the corresponding set of quadratic homogeneous equations (given by equation (18-2) in various number of unknown variables, respectively), a unique set of the Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) are also determined, representing equivalent algebraic forms of the energymomentum quadratic relation in various space-time dimensions, respectively. Subsequently, we've shown that by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a unique set of two definite classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is also derived, corresponding to various space-time dimensions, respectively. In addition, it is also shown that this derived set of two classes of general covariant field equations represent new tensor massive (matrix) formalism of the fundamental field equations of physics, corresponding to fundamental laws of nature (including the laws governing the fundamental forces of nature). Following these essential results, in addition to the basic assumptions (1) and ( 2), it would be also basically assumed that:

(3)-"We assume that the mathematical formalism of the fundamental laws of nature, are defined solely by the axiomatic matrix constitution formulated uniquely on the basis of postulates (1) and ( 2)".

In addition to this basic assumption, in Sec. 3-5, the C, P and T symmetries of uniquely derived general covariant field equations (that are equations ( 71) and ( 72), in Sec. [START_REF] Zahedi | On the Connection Between Methods of the Ring Theory and the Group Approach‖, Russian Academy of Sciences (RAS)[END_REF][START_REF] Zahedi | The Linearization Method Based on the Ring Theory and its Applications in Physics‖, Russian Academy of Sciences (RAS) Publs[END_REF], are also represented basically by their corresponding quantum matrix operators.

3-1-1.

Based on the basic assumption (2), i.e., the assumption of rationality of the relativistic energymomentum, the following Lorentz invariant quadratic relations (expressed in terms of the components of D-momentums  p ,  p of a relativistic massive particle (given in two reference frames), and also components of quantity

  k m p 0 st 
, where m 0 is the invariant mass of particle and  k is its covariant velocity in the stationary reference frame):

, p p g p p g          (51) st st ν μ μν ν μ μν p p g p p g  (52) . ) ( ) ( ) )( ( 2 0 2 0 0 00 0 0 c m k m g k m k m g   
ν μ μν would be particular cases of homogeneous quadratic equation (18-2) in Sec. 2-2, and hence, they would be necessarily subject to the process of linearization (along with a parameterization procedure), using the systematic axiomatic approach presented Sections 2, 2-2 and 2-4 (formulated based on the basic assumption (1)).

The Lorentz invariant relations ( 51) and ( 52) (as the norm of the relativistic energy-momentum) have been defined in the D-dimensional space-time, where m 0 is the invariant mass of the particle,  p and  p are its relativistic energy-momentums (i.e. D-momentums) given respectively in two reference frames,  k is a time-like covariant vector given by: ) 0 ,..., 0 , ( ) 0 ,..., 0 , ( 00 0

g c k k   
, -c‖ is the speed of light, and the components of metric have the constant values. As noted in Sec. 1-2, in this article, the sign conventions (2) (including the metric signature (+ --…-)) and geometric units would be used (where in particular -c = 1‖). However, for the clarity, in some of relativistic formulas (such as the relativistic matrix relations), the speed of light -c‖ is indicated formally.

As a crucial issue here, it should be noted that in the invariant quadratic relations ( 51) and ( 52), the components of metric which have the constant values (as assumed), necessarily, have been written by their general representations  g (and not by the Minkowski metric   , and so on). This follows from the fact that by axiomatic approach of linearization-parameterization (presented in Sections 2-1 -2-4) of quadratic relations ( 51) and ( 52) (as particular forms of homogeneous quadratic equation (18-2) which could be expressed equivalently by quadratic equations of the types (18) via the linear transformations (18-3)), their corresponding algebraic equivalent systems of linear equations could be determined uniquely. In fact, based on the formulations of systems of linear equations obtained uniquely for the quadratic equations [START_REF] Kauffman | Knots and Physics[END_REF] in Sections 2-2 -2-4, it is concluded directly that the algebraic equivalent systems of linear equations corresponding to the relations ( 51) and ( 52), are determined uniquely if and only if these quadratic relations be expressed in terms of the components  g represented by their general forms (and not in terms of any special presentation of the metric's components, such as the Minkowski metric, and so on). However, after the derivation of corresponding systems of linear equations (representing uniquely the equivalent algebraic matrix forms of the quadratic relations ( 51) and ( 52) in various space-time dimensions), the Minkowski metric could be used in these equations (and the subsequent relativistic equations and relations as well).

Hence, using the systems of linear equations ( 24), ( 26), ( 28), ( 29), [START_REF] Griesshammer | Introduction to Supersymmetry and Exact Nonperturbative Results in SUSY-QCD‖[END_REF],…, obtained uniquely on the basis of the axiom (17) by linearization (along with a parameterization procedure) of the homogeneous quadratic equations ( 19) -( 23),… (which could be transformed directly to the general quadratic equation (18-2), by the isomorphic linear transformations (18-3)), and also using the parametric relations (43-2), (44-3), ( 48) and (52) (expressed in terms of the arbitrary parameters  u and  v ), as the result of linearization (along with a parameterization procedure) of the invariant quadratic relations (51) and ( 52), the following systems of linear equations are also derived uniquely corresponding to various space-time dimensions, respectively:

-For (1+0)-dimensional case of the invariant relation [START_REF] Lucas | Gödel's Incompleteness Theorems‖[END_REF], we obtain:

0 ] )][ ( [ 0    s p p g    (53)
where ν = 0 and parameter s is arbitrary;

-For (1+1)-dimensional space-time we have:

0 ) ( ) ( ) ( ) ( 0 1 1 0 0 0 1 1 1 0                        s w v u v u p p p p g p p p p g       ( 54
)
where ν = 0,1 and

, , 1 0 u u , , , 1 0 w v v
s are arbitrary parameters;

-For (1+2)-dimensional space-time we have:

0 ) ( ) ( ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( ) ( 0 ) ( 0 ) ( 1 2 2 1 2 0 0 2 0 1 1 0 0 0 2 1 0 0 1 1 2 2 2 2 1 0 1 1 2 0                                                               s w v u v u w v u v u w v u v
                  ( 55
)
where ν = 0,1,2 and , , ,

2 1 0 u u u , , , , 2 1 0 w v v v
s are arbitrary parameters;

-For (1+3)-dimensional space-time we obtain:

0 ) ( ) ( ) ( 0 ) ( ) ( ) ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 2 1 3 3 1 3 2 2 3 0 3 3 0 0 2 2 0 0 1 1 0 0 3 2 1 0 3 1 2 0 2 1 3 0 1 2 3 3 2 1 0 3 1 2 0 2 1 3 0 1 2 3 0                                                                            s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u f e e e f e f f f e f f f e f f f f f e f e e e
f e e e f e e e [START_REF] Swartz | The Concept of Physical Law[END_REF] where ν = 0,1,2,3 and

3 2 1 0 , , , , u u u u w , 3 2 1 0 , , , v v v v
, s are arbitrary parameters, and we also having:

; ) ( ), ( ), ( ), ( ), ( ), ( , ) ( ), ( 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 0 p p f p p g e p p f p p g e p p f p p g e p p f p p g e                                         (56-1)
-For (1+4)-dimensional case, the system of linear equations corresponding to the invariant quadratic relation ( 51) is specified as follows: 0 [START_REF] Barrow | Theories of Everything: The Quest for Ultimate Explanations[END_REF] where ν = 0,1,2,3,4 ,

) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0 ) ( 0 ) ( ) ( ) ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 1 1 3 3 1 2 3 3 2 1 4 4 1 2 4 4 2 3 4 4 3 4 0 0 4 0 3 3 0 2 0 0 2 0 1 1 0 0 4 3 2 1 0 4 3 1 2 0 4 2 1 3 0 4 1 2 3 0 3 2 1 4 0 3 1 2 4 0 2 1 3 4 0 1 2 3 4 4 3 2 1 0 4 3 1 2 0 4 2 1 3 0 4 1 2 3 0 3 2 1 4 0 3 1 2 4 0 2 1 3 4 0 1 2 3 4 0                                                                                                                                                       s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u f e e e e
4 3 2 1 0 , , , , u u u u u , w v v v v v , , , , , 4 3 2 1 0
, s are arbitrary parameters, and we have:

; ), ( , ), ( , ), ( , ), ( , ), (
                                             (57-1)
The systems of linear equations that are obtained for (1+5) and higher dimensional cases of the invariant quadratic relation [START_REF] Lucas | Gödel's Incompleteness Theorems‖[END_REF], have also the formulations similar to the obtained systems of linear equations ( 53) -( 57), and would be expressed by the matrix product of a N N 2 2  square matrix and a 1 2  N column matrix in (1+ N)-dimensional space-time. For (1+5)-dimensional case of the invariant relation [START_REF] Lucas | Gödel's Incompleteness Theorems‖[END_REF], the column matrix of the corresponding system of linear equations (expressed by the matrix product of a 32 32  square matrix and a 1 32  column matrix) is given by (where

5 4 3 2 1 0 , , , , , u u u u u u , s w v v v v v v , , , , , , ,
                                                                                                                                  s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u S w v u v u w v u v u w v u v u w v u v u w v u v u S S S S (57-2)
In a similar manner, using the axiomatic approach presented in Sec. 2, the systems of linear equations corresponding to the energy-momentum invariant relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF] in various space-time dimensions are obtained uniquely as follows, respectively (note that by using the geometric units, we would take

1  c ):
-For (1+0)-dimensional space-time we obtain:

  0 00 0 00 0           s g c m g p g   (58)
where ν = 0 and parameter s is arbitrary;

-For (1+1)-dimensional space-time we have:

0 ) ( )) ( ( ) ( ) ( 0 1 1 0 00 0 0 00 0 10 1 1 00 0 00 0                         s w v u v u g c m p g c m g p g p g c m g p g     (59)
where ν = 0,1 and

, , 1 0 u u , , , 1 0 w v v
s are arbitrary parameters;

-For (1+2) dimensions we have (where ν = 0,1,2 and , , ,

2 1 0 u u u , , , , 2 1 0 w v v v s are arbitrary parameters): 0 ) ( ) ( ) ( )) ( ( 0 ) ( ) ( 0 )) ( ( ) ( ) ( 0 ) ( 0 ) (
                                                        s w v u v u w v u v u w v u v u g c m p g c m g p g g c m g p g g c m p p p p g c m g p g g c m g p g p g c m g p g g c m g p g             (60)
-For (1+3)-dimensional space-time we obtain:

0 ) ( ) ( ) ( 0 ) ( ) ( ) ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 2 1 3 3 1 3 2 2 3 0 3 3 0 0 2 2 0 0 1 1 0 0 3 2 1 0 3 1 2 0 2 1 3 0 1 2 3 3 2 1 0 3 1 2 0 2 1 3 0 1 2 3 0                                                                            s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u f e e e f e f f f e f f f e f f f f f e f e e e
f e e e f e e e (61) where ν = 0,1,2,3 and

3 2 1 0 , , , u u u u , w v v v v , , , , 3 2 1 0
, s are arbitrary parameters, and we also having:

; ), ( , ), ( , ), ( ), ( ), (
p f g c m g p g e p f g c m g p g e p f g c m g p g e g c m p f g c m g p g e                      (61-1)
-For (1+4)-dimensional space-time, the system of linear equations corresponding to the invariant quadratic relation ( 52) is derived as follows: 0 [START_REF] Hehl | How Does the Electromagnetic Field Couple to Gravity, in Particular to Metric, Nonmetricity, Torsion, and Curvature? in Gyros, Clocks, Interferometers[END_REF] where ν = 0,1,2,3,4 and

) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0 ) ( 0 ) ( ) ( ) ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 1 1 3 3 1 2 3 3 2 1 4 4 1 2 4 4 2 3 4 4 3 4 0 0 4 0 3 3 0 2 0 0 2 0 1 1 0 0 4 3 2 0 4 3 1 2 0 4 2 1 3 0 4 1 2 3 0 3 2 1 4 0 3 1 2 4 0 2 1 3 4 0 1 2 3 4 4 3 2 1 0 4 3 1 2 0 4 2 1 3 0 4 1 2 3 0 3 2 1 4 0 3 1 2 4 0 2 1 3 4 0 1 2 3 4                                                                                                                                                       s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u f e
4 3 2 1 0 , , , , u u u u u , w v v v v v , , , , ,
                          (62-1)
The systems of linear equations that are obtained for (1+5) and higher dimensional cases of the energymomentum quadratic relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF], have also the formulations similar to the obtained systems of linear equations ( 58) -( 62), and would be expressed by the matrix product of a

N N 2 2 
square matrix and a 1 2  N column matrix in (1+N)-dimensional space-time. For the (1+5)-dimensional case of energymomentum relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF], the column matrix of the corresponding system of linear equations (expressed by the matrix product of a 32 32  square matrix and a 1 32  column matrix, similar to (57-2)) is given by: . ) (

) ( ) ( ) ( ) ( ) ( 0 ) ( ) ( ) ( 0 )
0 2 2 0 0 1 1 0                                                                                                                                   s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u S w v u v u w v u v u w v u v u w v u v u w v u v u S S S S (62-2) where 5 4 3 2 1 0 , , , , , u u u u u u , s w v v v v v v , , , , , , , 5 4 3

3-2.

From the derived systems of linear equations ( 54) -( 57) corresponding to the (1+1) -(1+4)dimensional cases of the invariant relation [START_REF] Lucas | Gödel's Incompleteness Theorems‖[END_REF], and also using the general parametric solutions ( 43) -(51) (obtained for systems of linear equations ( 26) -( 30)), the rational Lorentz transformations (which are completely dense in the standard group of Lorentz transformations [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF], as noted in Sec. 3-1) are derived for momentums  p and  p . For instance, assuming the Minkowski metric, from the system of linear equations ( 55), a parametric form of rational Lorentz transformations for three-momentums  p and  p in (1+2)dimensional space-time, is derived as follows:

                                                                                      2 1 0 2 1 0 2 2 2 1 2 0 2 2 2 1 2 0 2 2 2 1 2 0 1 0 2 2 2 2 1 2 0 2 0 1 2 2 2 1 2 0 1 0 2 2 2 2 1 2 0 2 2 2 1 2 0 2 2 2 1 2 0 2 1 0 2 2 2 1 2 0 2 0 1 2 2 2 1 2 0 2 1 0 2 2 2 1 2 0 2 2 2 1 2 0 1 1 1 ) ( 2 1 ) ( 2 1 ) ( 2 1 1 1 ) ( 2 1 ) ( 2 1 ) ( 2 1 1 p p p p p p z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z (63)
where the parameters  s in (63) are given by the formulas:

w v u v u z ) ( 0 1 1 0 0   , w v u v u z ) ( 2 0 0 2 2   , , ) ( 1 2 2 1 3 w v u v u z  
that are expressed in terms of the arbitrary parameters

w v v v u u u , , , , , , 2 1 0 2 1 0
. These parameters would be also determined and expressed in terms of the initially given physical variables (such as the relative velocity between the reference frames). However, as it has been also noted in Sec. 2-4 concerning a particular common algebraic property of parametric relations (43-2) and (44-3) which are equivalent to the above expressions, by choosing appropriate integer values for parameters

w v v v u u u , , , , , , 2 1 0 2 1 0
, the parameters 2 1 0 , , z z z could take any given integer values. Thus, we may directly determine the relevant expressions for parameters  s in terms of the initially given physical values and variables. Hence, as a particular case, from the isomorphic transformations (63), in addition to these determined expressions for the parameters  s (in terms of the relative velocity between the reference frames in x-direction and the speed of light): 0 , ) 1 (

2 1 0      z z z   , 2 1 1     , , c v  
we obtain the equivalent form of Lorentz transformations in the standard configuration [START_REF] Forshaw | Dynamics and Relativity‖, Manchester Physics Series[END_REF]: Similar to the derived transformations (63-1), the Lorentz transformations (in standard configuration) are derived by the same approach for higher-dimensional space-times.

                                               

3-3.

The Lorentz invariant systems of linear equations ( 59) -( 62),…, (obtained on the basis of the axiom [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] and relevant general results obtained in Sections 2-2 and 2-4 for homogeneous quadratic equations) as equivalent forms of the Lorentz invariant energy-momentum quadratic relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF], could be expressed generally by the following matrix formulation in (1+N)-dimensional space-time:

, 0 ) ( 0   S k m p       (64)
where -For (1+1)-dimensional case we get:

                  , , (65)
; ) ( , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 , 1 0 0 0 0 1 1 0 1 1 0 0                                        s w v u v u S     (66) where , , 1 0 u u , , , 1 0 w v v
s are arbitrary parameters.

-For (1+2)-dimensional case we obtain where , , ,

2 1 0 u u u , , , , 2 1 0 w v v v s are arbitrary parameters): , 0 0 , 0 0 , 0 0 0 , ) ( 0 0 0 3 3 1 2 2 1 1 0 0 1 0 0                                                , 0 0 , 0 0 1 0 2 0 1 2                          , ) ( ) ( ) ( 1 2 2 1 2 0 0 2 0 1 1 0                 s w v u v u w v u v u w v u v u S                              0 1 0 0 , 0 0 1 0 , 1 0 0 0 , 0 0 0 1 3 2 1 0     (67)
-For (1+3)-dimensional case we obtain:

, 0 0 , 0 0 , 0 0 0 ) ( , ) ( 0 0 0 2 3 1 3 2 1 1 0 0 1 0 0                                                , 0 0 , 0 0 , 0 0 , 0 0 6 7 3 7 6 3 4 5 2 5 4 2                                               , ) ( ) ( ) ( 0 ) ( ) ( ) ( 2 1 1 2 1 3 3 1 3 2 2 3 0 3 3 0 0 2 2 0 0 1 1 0                                  s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u S . 0 0 0 0 0 0 0 0 1 0 1 0 , 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 , 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 , 0 0 0 0 1 0 0 1 0 0 0 0 0 , 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 , 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 7 6 5 4 3 2 1 0                                                                                                                       (68) where 3 2 1 0 , , , u u u u , w v v v v , , , , 3 2 1 0
, s are arbitrary parameters. Moreover, the 4×4 matrices i  (68) generate the Lorentz Lie algebra in (1+3) dimensions.

-For (1+4)-dimensional case we have:

, ) ( 0 0 0 1 0 0             , 0 0 0 ) ( 1 0 0             , 0 0 2 2 1           , 0 0 3 3 1            , 0 0 5 4 2           , 0 0 4 5 2              , 0 0 7 6 3            , 0 0 6 7 3             , 0 0 9 8 4           , 0 8 9 4              , ) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0 ) ( 0 ) ( ) ( ) ( 1 2 2 1 1 3 3 1 2 3 3 2 1 4 4 1 2 4 4 2 3 4 4 3 4 0 0 4 0 3 3 0 2 0 0 2 0 1 1 0                                                              s w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u S , 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 , 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 2 1 0                                                                                                                     , 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 , 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 4                                                                                           . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 , 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 9 8 7                                                                                           (69) where 4 3 2 1 0 , , , , u u u u u , s w v v v v v , , , , , , 4 3 2 1
0 are arbitrary parameters. Furthermore, similar to the 4×4 i  matrices in [START_REF] Bialinicki | The Maxwell wave function of the photon[END_REF], the 8×8 matrices i  (69) generate the Lorentz Lie algebra in (1+4) dimensions.

For (1+5)-dimensional case the size of matrices   and    is 32 32  . S is also a 1 32  column matrix given by: . ) (

) ( ) ( ) ( ) ( ) ( 0 ) ( ) (
                                                                                                                                  s w v u v
of a one-to-one correspondence between these (non-zero) entries of matrix S and the entries 

h (for μ > ν) of a 1 1 2 2    N N square matrix ] [  h H defined in (1+N) dimensions, by: w v u v u h ) (        (70-1)
where μ, ν = 0,1,2,…, N, and

0   h for μ = ν.
Note that the algebraic form (70-1) is equivalent to form (45-2) which, as it has been noted in Sec. 2-4, generates a symmetric algebra Sym(V) on the vector space V , where

V v u  ) , (  
" [START_REF] Bourbaki | Algebra‖[END_REF].

Hence, as a basic algebraic property of the form (70-1), a natural unique isomorphism is defined between the underlying vector space V of the symmetric algebra Sym(V) (which is generated by algebraic form (70-1)) and the Weyl algebra W(V). Moreover, based on this isomorphism, the Weyl algebra W(V) could be defined as a (first) quantization of the symmetric algebra Sym(V), where the generators of the Weyl algebra W(V) would be represented by the corresponding (covariant) differential operators (such as    i , as per quantum mechanics usage).

In Sec. 3-4, we use these general and basic algebraic properties of the column matrix S , in particular, in the procedure of quantization of the algebraic matrix equation ( 64).

3-3-2.

In addition to the above algebraic properties of the parametric entries of column matrix S , that are represented uniformly by the algebraic formula (70-1), in terms of the arbitrary parameters:

N u u u u u ,..., , , , 3 2 1 0 , w v v v v v N , ,..., , , , 3 2 1 0
, the following basic properties hold as well:

Displaying the column matrix S by two half-sized 1 2 1   N column matrices S and S   (containing respectively the upper and lower entries of S , similar to the formulas (57-2) and (62-2) representing the (1+5)-dimensional case of matrix S ) such that:

          S S S
, then we have:

(1). The number of entries of the column matrix S that are zero, is exactly:

) N ( N  1 2
, and the other N entries are represented solely either by the formulation:

, w ) v u v u ( h 0 0 0     
or by its negative form, i.e.:

w ) v u v u ( h h     0 0 0 0    
, where μ = 1,2,…, N, and 0  h denote the N entries (except the first entry 00 h that is zero) of the first column of square matrix ] [  h H (defined by the formula (70-1));

(2). The number of entries of the column matrix S   that are zero, is exactly:

) 1 2 ) 1 ( 2 ( 1     N N N ,
, w ) v u v u ( h       
or by its negative form, i.e.:

w ) v u v u ( h h          
, where μ > ν , μ, ν = 1,2,…, N, and  h denote the components of square matrix ] [  h H , and the last entry of column matrix S   is also represented by the arbitrary parameter s .

(3). If we exchange S and S   in the column matrix

          S S S
, that could be shown by,

                                 S S S S I I S I I S Ch 0 0 0 0 ) ( (70-2)
then based on the general formulation of matrix S (defined uniquely by formulas (66) -( 70) for various space-time dimensions) and its algebraic properties (1) and (2) (mentioned above), it is concluded directly that the matrix equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF] given with the new column matrix

) (Ch

S

(70-2), i.e. equation:

0 ) ( ) ( 0   Ch S k m p      
, is which could be defined solely in (1+2) space-time dimensions for 0 , 0 , 0

1 1    v u s
, in (1+3) space-time dimensions for 0  s , and in (1+4) space-time dimensions for 0 , 0 , 0

1 1    v u s
(which is reduced and be equivalent to the (1+3)-dimensional case of matrix equation ( 64)). In (1+1) and (1+5) and higher space-time dimensions, the matrix equation 0 ) (

) ( 0   Ch S k m p      
would be which are defined if and only if all the entries of column matrix

) (Ch

S

are zero. This means that the matrix equation (64

): 0 ) ( 0   S k m p       , is symmetric in the exchange of S and S   (in the column matrix           S S S
), solely in (1+2)-dimensional space-time for 0 , 0 , 0

1 1    v u s
, and in (1+3)-dimensional space-time for 0  s . In Sec. 3-5-2, this particular algebraic property of the column matrix S would be used for concluding a new crucial and essential issue in fundamental physics.

In the following Section, the natural isomorphism between the symmetric algebra Sym(V) (generated uniquely by the algebraic form (70-1)) and the Weyl algebra W(V) mentioned in Sec. 3-3-1, in addition to the general algebraic properties of column matrix S presented in Sec. 3-3-2, would be used and applied directly in the procedure of first quantization of the Lorentz invariant system of linear equations (64).

3-4. A new unique mathematical derivation of the fundamental (massive) field equations of physics (representing the laws governing the fundamental forces of nature):

By first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of the Lorentz invariant system of linear equation ( 64) (representing uniquely the equivalent form of energy-momentum quadratic relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF], see Sec. 3-3) expressed in terms of the Clifford algebraic matrices ( 65) -( 70),... , two classes of general covariant field equations are derived uniquely as follows (given in (1+N) space-time dimensions):

0 ) ( ) ( 0     R R k m i        , ( 71
) 0 ) ( ) ( 0    F F k m D i        ( 72
) ( ) ( 0 ) ( ) ( R R R k im J           
(which appears in the course of the derivation of field equation ( 71), see Sec.

3-4-2 for details). In a similar manner, in the tensor field equation ( 72), Ψ

F is also a column matrix as a (first) quantized form of the algebraic column matrix S (defined by relations ( 64) -(70-1) in Sections 3-3, 3-3-1 and 3-3-2), determined and represented uniquely by formulas ( 73) -( 77),…, in to various space-time dimensions. The column matrix Ψ F contains both the components of tensor field  F (defined as the gauge field strength tensor), and also the components of covariant quantity ) ( F  that defines the corresponding source current vector by relation: relation:

) ( ) ( 0 ) ( ) ( F F F k im D J         
(which appears in the course of the derivation of field equation ( 72), see Sec. 3-4-2 for details). Moreover, the general covariance formalism of the field equations ( 71) and ( 72), would be also shown in Sec. 3-4-1.

In addition, in Sec. 3-5, based on a basic class of discrete symmetries for the field equations ( 71) and ( 72), along with definite mathematical axiomatic formalism of the derivation of these equations, it is shown that these equations could be defined solely in (1+2) and (1+3) space-time dimensions. It is shown that (1+3) dimensional cases of these equations represent uniquely a new formalism of bispinor fields of spin-2 and spin-1 particles, respectively. It is also shown that the (1+2)-dimensional cases of these equations, represent asymptotically new massive forms of bispinor fields of spin-3/2 and spin-1/2 particles, respectively.

Moreover, in Sec. 3-5-2, based on the definite mathematical formalism of this axiomatic derivation approach, the basic assumption (3) in Sec. 3-1, along with the C, P and T symmetries (represented basically by their corresponding quantum matrix operators) of the fundamentally derived general covariant field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], it is concluded that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times (where this conclusion, in particular, is based on the Tsymmetry). In Sections 3-5-3 and 3-5-4, it is proved that 'CPT' is the only (unique) combination of C, P, and T symmetries that could be defined as a symmetry for interacting fields. In addition, in Sec. 3-5-4, on the basis of these discrete symmetries of the field equations ( 71) and ( 72), it is shown that only lefthanded particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents.

Furthermore, in Sec. 3-6, it is argued that the metric of background curved space-time is diagonalized for the spin-1/2 fermion field equations (defined by the field equation (110) as a generalized form of (1+2)dimensional case of equation ( 72 , where in particular Z  is the complementary right-handed particle of ordinary Z boson), are predicted by this new mathematical axiomaticapproach.

As a particular result, in Sec. 3-4-2, based on the definite and unique formulation of the derived Maxwell‗s equations (and also Yang-Mills equations, defined by the (1+3)-dimensional case of the field equation ( 72), compatible with specific gauge symmetry groups as shown in Sec. 3-6-1-2 and 3-6-3-2), it is also concluded that magnetic monopoles could not exist in nature.

3-4-1. Axiomatic Derivation of General Covariant Massive Field Equations (71) and (72):

First it should be noted that via first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of the algebraic systems of linear equations (64) (as a matrix equation given by the Clifford algebraic matrices ( 65) -( 70),…, in various space-time dimensions), two categories of general covariant field equations (with a definite matrix formalism compatible with the Clifford algebras and their generalizations, see Sec 3-3 and also Appendix A) are derived solely, representing by the tensor equations ( 71) and ( 72) in terms two tensor fields   R and ,  F respectively. In fact, as it has been mentioned in Sections 3-3-1 and 3-3-2, there is a natural isomorphism between the Weyl algebra and the symmetric algebra generated by the algebraic form (70-1) which represents the general formulation of the entries of algebraic column matrix S in the matrix equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF]. In addition, the procedure of minimal coupling to space-time geometry would be simply defined as a procedure which, starting from a theory in flat spacetime, substitutes all partial derivatives by corresponding covariant derivatives and the flat space-time metric by the curved space-time (pseudo-Riemannian) metric. Moreover, as mentioned in Sec. 3-3-1, on the basis of this natural isomorphism, the Weyl algebra could be also represented as a quantization of the symmetric algebra generated by the algebraic form (70-1)). Hence, using this natural isomorphism, by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of matrix equation ( 64), two definite classes of general covariant massive (tensor) field equations are determined uniquely, expressed in terms of two basic connection forms (denoting by two derivatives   and  D corresponding respectively to the diffeomorphism (or metric) invariance and gauge invariance), along with their corresponding curvature forms, denoting respectively by   R (as the gravitational field strength tensor, equivalent to Riemann curvature tensor) and  F (as the gauge field strength tensor). This natural isomorphism could be represented by the following mappings (corresponding to the curvature forms   R and  F , respectively):

             R R w v u v u R R ) , ( ) ( ) (          , (71-1) μν F F F ig D D D D w v u v u ) ( ) ( ) (              . (72-1)
where ), ( ) (

                            R      A D A D F  
, and F g ,  A are respectively the corresponding coupling constant and gauge field (that is defined generally as a Lie algebra-valued 1-form representing by a unique vector field [START_REF] Frampton | Gauge Field Theories‖[END_REF]). Based on this natural unique isomorphism represented by the mappings (71-1) and (72-1), the column matrices Ψ R and Ψ F (in the expressions of field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], respectively) would be determined uniquely various dimensional space-times, represented by formulas ( 73) -( 77),… .

In addition, as mentioned in Sec. 3-4 in detail, the last entry of algebraic column matrix S in matrix equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF] (as it has been shown in the relations ( 64) -( 70)), is represented by the arbitrary algebraic parameter s . In the course of the derivation of field equations ( 71) and ( 72) (via the first quantization procedure mentioned above, and the mappings (71-1) and (72-1)), the arbitrary parameter s could be substituted solely by two covariant quantities 71) and ( 72), respectively) by the conditional relations:

) ( ) ( 0 ) ( ) ( R R R k im J           and ) ( ) ( 0 ) ( ) ( F F F k im D J         .
In addition, as another basic issue concerning the general covariance formulation of tensor field equations ( 71) and ( 72), we should note that each of these equations (as a system of equations) includes also an equation corresponding to the 2 nd Bianchi identity, as follows, respectively:

0 ) ( ) ( ) ( ) ( 0 ) ( 0 ) ( 0                    R k im R k im R k im R R R    , (71-2) 0
) ( ) ( ) ( ) ( 0 ) ( 0 ) ( 0                F k im D F k im D F k im D F F F    (72-2)
However, the tensor field   R as the Riemann curvature tensor, obeys the relation (71-2) tensor, if and only if a torsion tensor is defined in as:

) ( ) 2 ( ) ( 0       k g k g im T R   
, and subsequently the relation (71-2) be equivalent to the 2 nd Bianchi identity of the Riemann tensor. Consequently, the covariant derivative   should be also defined with this torsion, that we may show it by   

. Moreover, as it has been also shown in Sec. 3-4-2, concerning the relation (72-2), we may also define a torsion field as:

) ( ) 2 ( ) ( 0      k g k g im Z F   
, and write the relations (71-2) and (72-2) (representing the 2 nd Bianchi identities) as follows:

, of the field equations ( 71) and (72) in Sec. 3-4-2. In addition, based on the formulations of torsions   T and   Z (that have appeared naturally in the course of derivation of the field equations ( 71) and ( 72)) and general properties of torsion tensors (in particular, this property that a torsion tensor can always be

                R T R T R T R R R            (71-2-a) 0          F D F D F D   
F F F R R R F F R R k im D J k im J F R                                         (73)
-For (1+2)-dimensional space-time we obtain:

; ) ( , ) ( , , ) ( ) ( 0 ) ( ) ( ) ( 0 ) ( ) (
F F F R R R F F R R k im D J k im J F F F R R R                                                           (74)
-For (1+3)-dimensional space-time we have:

; ) ( , ) ( , 0 , 0 ) ( ) ( 0 ) ( ) ( ) ( 0 ) ( ) (
F F F R R R F F R R k im D J k im J F F F F F F R R R R R R                                                                                      (75)
-For (1+4)-dimensional space-time we get,

F F F R R R F F R R k im D J k im J F F F F F F F F F F R R R R R R R R R R                                                                                                                                          (76)
-For (1+5)-dimensional space-time we obtain: [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF] where in the relations ( 73) -( 77

) ( ) ( 0 ) ( ) ( ) ( 0 ) ( ) (
F F F R R R F F R R k im D J k im J F F F F F F F F F F F F F F F R R R R R R R R R R R R R R R                                                                                                                                                                                                                                                  

3-4-2.

From the field equations ( 71) and (72) (derived uniquely with certain matrix formalisms compatible with the Clifford and Weyl algebras), the following general covariant field equations, with ordinary tensor formalisms, are obtained (but not vice versa), respectively:

,                 R T R T R T R R R            (78-1)         ) ( ) ( 0 ) ( R R J R k im R       ; (78-2)
), ( ) (

                            R ) ( ) ( 0 ) ( ) ( R R R k im J            , ). ( 2
) ( 0       k g k g im T R    (78-3) and , 0          F D F D F D    (79-1) ) ( F J F D       ; (79-2)      A D A D F     , ) ( ) ( 0 ) ( ) ( F F F k im D J          , ). ( 2
) ( 0       k g k g im Z F    (79-3)
where    is the affine connection:

      K     , 
  is the Christoffel symbol (or the torsion- free connection),   K is the contorsion tensor defined by:

   k g im K R ) 2 ( ) ( 0  
(that is antisymmetric in the first and last indices),  T is the torsion given by: It should be emphasized again that the general covariant (tensor) field equations (78-1) -(78-3) and (79-1) -(79-3), merely show the general covariance property of the axiomatically determined field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF]. However, a crucial issue here is that tensor equations (78-1) -(78-3) and (79-1) -(79-3) (which in fact could be obtained from the original equations ( 71) and ( 72), but not vice versa), don't hold the definite matrix formalisms (compatible with certain Clifford and Weyl algebras) which are represented fundamental field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF]. Hence, based on this mathematical axiomatic approach, it would be concluded that the fundamental force fields of physics, including the gravitational, electromagnetic and nuclear fields, cannot be represented described, completely and sufficiently, via the ordinary tensor field equations of the type (78-1) -(78-3) and (79-1) -(79-3). In fact, as shown in Sections 3-3 -3-4-1, on the basis of this axiomatic approach, these fundamental physical fields should be represented (and described) soley by the axiomatically determined field equations ( 71) and ( 72) holding definite matrix formalisms (compatible with certain Clifford and Weyl algebras).

    K K T   (that

3-4-2-1. Derivation of the Einstein field Equations

Along with the massive gravitational field equations (78-1) -(78-3) (obtained uniquely from the originally derived field equations ( 71)) that are expressed solely in terms of   R as the field strength tensor and also torsion's depended terms, we also assume the following relation as basic definition for the Ricci tensor (where the Riemann curvature tensor and Ricci tensor don't obey the interchange symmetries:

    R R  ,   R R 
, because of the torsion [START_REF] Wald | General Relativity‖[END_REF]):

           R k im R k im R k im R R R ) ( ) ( ) ( ) ( 0 ) ( 0 ) ( 0               (78-4)
where the relation particularly remains unchanged by the transformation:

   g R R    (78-5)
(  is equivalent to the cosmological constant). It should be noted that by taking 0   , from the 2 nd Bianchi identity of the Riemann curvature tensor and relation it could be shown that the Ricci tensor is also the contraction of the Riemann tensor, i.e.

    R R 
(which is equivalent to the ordinary definition of the Ricci tensor). However, this ordinary definition for the Ricci tensor, necessarily, doesn't imply the above transformation. In fact, in the following, we show that this basic transformation is necessary for having the cosmological constant in the gravitational field equations (including the Einstein field equations which could be derived from the above equations and relations) expressed in terms of the Ricci and stress-energy tensors. As a direct result, a unique equivalent expression of gravitational field equations, in terms of the Ricci tensor  R and stress-energy tensor  T , could be also determined from the basic definition (78-4) (for Ricci curvature tensor, based on this axiomatic formalism), and field equations (78-1) -(78-3), along with the following expression for current

) ( R J  (defined in terms of the stress-energy tensor  T , ) (   T T  , and metric  g , in D-dimensional space-time): ] ) ( ) [( 8 ] ) ( ) [( 8
) ( 0 ) ( 0 ) ( 0 ) ( 0 ) (                Tg k im Tg k im B T k im T k im J R R R R R                  , (78-6)
where

  T T  for 0 ) ( 0  R m , 0  B for 2 , 1  D
, and

) 2 /( 1   D B for 3  D
. , the Einstein field equations (as the massless case) are determined directly as follows:

     g BTg T R      ) ( 8 (78-7)
3-4-3. Showing that magnetic monopoles could not exist in nature. As a direct consequence of the uniquely derived general covariant field equations ( 72) that are specified by the matrices ( 73) -( 77) and ( 65) -( 70) (or the general covariant field equations (79-1) -(79-3) obtained from the original equation ( 72)), which , in fact, represent the electromagnetic fields equivalent to a generalized massive form of the Maxwell's equations (as well as a generalized massive form of the Yang-Mills fields corresponding to certain gauge symmetry groups, see Sec. [START_REF] Zahedi | On the Connection Between Methods of the Ring Theory and the Group Approach‖, Russian Academy of Sciences (RAS)[END_REF][START_REF] Zahedi | The Linearization Method Based on the Ring Theory and its Applications in Physics‖, Russian Academy of Sciences (RAS) Publs[END_REF][START_REF] Durbin | Modern Algebra: An Introduction‖[END_REF][START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF], it is concluded straightforwardly that magnetic monopoles could not exist in nature.

3-4-4. On the local gauge invariance of uniquely derived new general covariant massive (matrix) forms of the Maxwell's (and Yang-Mills) and Dirac equations.

The Lagrangian density specified for the tensor field  F in the field equations (79-1) -(79-3) is

(supposing 0 ) (  F J 
) [START_REF] Frampton | Gauge Field Theories‖[END_REF]:

  F F g L F * ) ( ) 4 1 (    (80)
where g is the metric's determinant. Moreover, the trace part of torsion field   Z in (79-3) is obtained as:

     k N k ) im ( N Z Z ) F (     2 0 ( 81
)
where (1+N) is the number of space-time dimensions and  2

) ( 0 F im  
. Now based on the definition of covariant vector  k (as a time-like covariant vector), we simply get:

       k :
. This basic property, along with and formula (81), imply the general covariant massive field equations (79-1) -(79-3) (formulated originally with the torsion field (79-3) generating the invariant mass

) ( 0 F m of field  F )
, and the corresponding Lagrangian density (80), be invariant under the U(1) Abelian gauge group [START_REF] Guidry | Gauge Field Theories: An Introduction with Applications‖[END_REF][START_REF] Frampton | Gauge Field Theories‖[END_REF][START_REF] Fresneda | Maxwell Field with Torsion[END_REF][61][START_REF] Hehl | How Does the Electromagnetic Field Couple to Gravity, in Particular to Metric, Nonmetricity, Torsion, and Curvature? in Gyros, Clocks, Interferometers[END_REF][63]. However, in Sec. 3-6, we show that assuming the spin-1/2 fermion fields (describing generally by the field equation (110-9) compatible with specific gauge symmetry group (110-12), as shown in Sec. 3-6-1-2) and their compositions as the source currents of the (1+3)-dimensional cases of general covariant massive field equation ( 72) (describing the spin-1 boson field), then this field equation would be invariant under two types of gauge symmetry groups, including: SU(2) L U(2) R and SU(3), corresponding with a group of seven bosons and a groups of eight bosons (as shown in Sec. 3-6-3-2).

3-4-5. Identifying a new particular massive gauge boson.

According to Refs. [60 -63], in agreement with the recent astronomical data, we can directly establish a lower bound for a constant quantity which is equivalent to the constant that is generated by a coupling torsion field of the type (79-3) of the background curved space-time.

In addition, it should be noted that, in general, based on the covariant massive field equations ( 71) and ( 72) derived by our axiomatic approach (or field equations (78-1) -(78-3) and (79-1) -(79-3) obtained from ( 71) and ( 72)), the invariant masses of the elementary particles are generated by torsion fields of the types (78-3) (for spin-3/2 and spin-2 particles) and (79-3) (for spin-1/2 and spin-1 particles, see Sec. [START_REF] Zahedi | On the Connection Between Methods of the Ring Theory and the Group Approach‖, Russian Academy of Sciences (RAS)[END_REF][START_REF] Zahedi | The Linearization Method Based on the Ring Theory and its Applications in Physics‖, Russian Academy of Sciences (RAS) Publs[END_REF][START_REF] Durbin | Modern Algebra: An Introduction‖[END_REF][START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF].

Hence, this approach could be also applied for massive neutrinos concluding that their masses are generated by the coupling torsion fields (of the type (79-3)). Such massive particle fields coupled to the torsions (of the type (79-3)) of the background space-time geometry could be completely responsible for the mysteries of dark energy and dark matter [START_REF] Kouwn | Massive Photon and Dark Energy[END_REF].

3-5. Quantum Representations of C, P and T Symmetries of the Axiomatically Derived

General Covariant Massive (tensor) Field Equations ( 71) and ( 72):

As it has been shown in Sections 3-3, 3-3-1, 3-3-2, 3-4 and 3-4-1, the general covariant massive (tensor) field equations ( 71) and ( 72) as the unique axiomatically determined equations (representing the fundamental field of physics, as assumed in Sec. 3-1), are represented originally with definite matrix formalisms constructed from the combination of two specific matrix classes including the column matrices ( 73) -( 77),… compatible with the Weyl algebras (based on the isomorphism (71-1) -(72-1)), and the square matrices ( 65) -( 70),… that are compatible with the Clifford algebras and their generalizations; see Sections 3-3, 3-3-1, 3-3-2 and 3-4-1 and also Appendix A for detail).

In agreement with the principles of relativistic quantum theory [START_REF] Weinberg | The Quantum Theory of Fields[END_REF], and also as another primary assumption in addition to the basic assumption (3) defined in Sec. 3-1, we basically represent the C, P and T symmetries of the source-free cases of by the following quantum matrix operators (with the same forms in both flat and curved space-time), respectively:

(Note: In Sec. 3-5-3, we show that only a certain simultaneous combination of the C, P and T transformations could be defined for the field equations ( 71) and ( 72) with non-zero source currents.)

(1)-Parity Symmetry (P-Symmetry):

         I I P ˆP 0 0 ( 83
)
where I is the identity matrix, and the size of matrix P  in (1+N)-dimensional space-time is

N N 2 2  .
The operator P ˆobeys the relations:

T P P P P P P , 1 , 1 ) det( 1 2         (83-1)
(2)-Time-Reversal Symmetry (T-Symmetry):

K î K T T ˆCh P     0 (84)
where the operator K ˆ denotes complex conjugation, the operator P  defined by formula (83) and the operator Ch  in (1+1) and (1+2) space-time dimensions, is given by:

, I I Ch         0 0 (84-1)
and in (1+3) and higher space-time dimensions, Ch  is denoted by:

        0 0 iI iI Ch  (84-2)
where the size of matrix Ch  in (1+N)-dimensional space-time is:

N N 2 2  . Moreover, in (1+1) and (1+2) space-time dimensions, the time reversal operator T ˆ(84) and the Hermitian operator

Ch P i T    0 ˆ
(specified in the formula (84)) obey the relations:

, , 1 ˆ0 0 1 0 0 2 T T T T T T         (84-3)
and in (1+3) and higher space-time dimensions, T ˆand 0 T obey the relations:

T T T T T T 0 0 1 0 0 2 , 1 ˆ      (84-4)
Concerning the time reversal symmetry, it should be noted that the relations (84-3) are solely compatible with the fermionic fields, and relations (84-4) are solely compatible with the bosonic fields. In addition, it should be noted that these basic quantum mechanical properties (i.e. the relations (84-3) and (84-4)) of the time reversal symmetry (84), are fully compatible with corresponding properties of the field tensors  (3)-Charge Conjugation Symmetry (C-Symmetry):

F F C F R R C R K iI C K iI C           ) ( , ) ( (85)
where

K iI C



, I is the identity matrix, the operator K ˆ denotes complex conjugation, and the charge conjugation operator C ˆ defined by:

T ) ( ) ( R C R C    , T ) ( ) ( F C F C   
. The charge conjugation operator C ˆ obeys the following relations:

T C C C C C C , 1 ˆ1          (85-1)
As a basic additional issue, it is worth to note that the time-reversal operator (84) could be also expressed basically in terms of the parity matrix operator P  (83), matrix operator K iI C  given in the definition of charge-conjugated transformation (85), and matrix Ch  defined by formulas (84-1) and (84-2), as follows:

T C Ch P    (85-2)
where we have:

P Ch Ch P       .

3-5-1. Basic Properties of Matrix (operator) Ch

 (defined by formulas (84-1) and (84-2)):

In this Section, the main properties of matrix operator Ch  (defined by unitary matrices (84-1) and (84-2)) hav been presented. Each of general covariant (tensor) field equations ( 71) and ( 72) (including their source-free and non source-free cases), as a system of differential equations, is symmetric and has the same spectrum by multiplying by matrix Ch  . The multiplied column matrices

R Ch Ch R     ) ( and F Ch Ch F     ) (
then obey the equations ( 71) and ( 72), respectively, but with opposite sign in mass term such that: 0

) ( ) ( ) ( 0     Ch R R k m i        , 0 ) ( ) ( ) ( 0     Ch F F k m i        .
As a general additional issue concerning the column matrices

R Ch Ch R     ) ( and F Ch Ch F     ) (
, should be also added that the sign change of the mass terms introduced in the field equations ( 71) and ( 72) is immaterial (the same property also hold for the ordinary formulation of Dirac equation, and so on [START_REF] Pilkuhn | Relativistic Quantum Mechanics[END_REF]). In other words, the field equations ( 71) of the form 0 ) (

) ( 0     R R k m i        are
equivalent, and similarly the field equations ( 72) of the form 0 ) (

) ( 0     F F k m i      



would be equivalent as well. However, since the algebraic column matrix S in the matrix equation ( 64) (derived and represented uniquely in terms of the matrices (66) -( 70),… corresponding to various space-time dimensions), is not symmetric by multiplying by matrix Ch  (84-1) and (84-2) (except for (1+2) and (1+3)-dimensional cases of column matrix S , based on the definite algebraic properties of matrix S presented in Sections 3-3, 3-3-1, 3-3-2), it is concluded that except the (1+2) and (1+3)- dimensional cases of the fundamental field equations ( 71) and ( 72), these field equations could not be defined with the column matrices of the types

) ( ) ( R Ch Ch R     and ) ( ) ( F Ch Ch F     (if assuming
that the column matrices R  and F  are defined with field equations ( 71) and ( 72), i.e. they have the formulations similar to the formulations of originally derived column matrices ( 73) -( 77),… corresponding to various space-time dimensions). This conclusion follows from this fact that the filed equations ( 71) and ( 72) have been derived (and defined) uniquely from the matrix equation ( 64) via the axiomatic derivation approach (including the first quantization procedure) presented in Sections 3-4, 3-4-1. In Sec. 3-5-2, using this property (i.e. multiplication of column matrices R  and F  defined in the fundamental field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], by matrix Ch  (84-1) and (84-2) from the left), this crucial and essential issue would be concluded directly that by assuming the timereversal invariance of the general covariant filed equations ( 71) and ( 72 ), these fundamental field equations could be defined solely in (1+2) and (1+3) space-time dimensions (with the column matrices of the forms (96-1) and (98-2), respectively). Subsequently, in Sec. 3-5-3, , it would be also shown that only a definite simultaneous combination of all the transformations , Ĉ P ˆ, T ˆ and also matrix Ch  (given by quantum operators ( 83) -( 87)) could be defined for the field equations ( 71) and ( 72) with non-zero source currents. In addition, the matrix operator Ch  in (1+1) and (1+2) space-time dimensions obeys the relations:

T Ch Ch Ch Ch Ch ) ( ) ( ) ( , 1 ) ( 1 2            , ( 86
)
and in the (1+3) and higher dimensions obeys the following relations as well:

T Ch Ch Ch Ch Ch ) ( ) ( ) ( , 1 ) ( 1 2             (87)
Furthermore, in Sec. 3-5-4, the matrix Ch  would be also used basically for defining and representing the left-handed and right handed components of the column field matrices R  and F  defined originally in the field equations ( 71) and ( 72).

3-5-2. Showing that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times:

The proof of this essential property of nature within the new mathematical axiomatic formalism presented in this article, is mainly based on the T-symmetry (represented basically by quantum matrix operators (84)) of the fundamentally derive general covariant field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF]. As shown in Sec. 3-5, the source-free cases (as basic cases) of field equations ( 71) and ( 72) are invariant under the time-reversal transformation defined by matrix operator (84). Moreover, in Sec. 3-5-3, it would be also shown that these field equations with non-zero source currents are solely invariant under the simultaneous transformations of all the , Ĉ P ˆ, and T ˆ ( 83) -( 85), multiplied by matrix Ch  (given by formulas (84-1) and (84-2)). Now, following the definite mathematical formalism of the axiomatic derivation approach of fundamental field equations ( 71) and ( 72), assuming that any column matrix R X , or F Y , expressible in the tensor formulation of general covariant field equation ( 71) or [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], is basically definable, if and only if, it could be also derived originally as a column matrix via the axiomatic derivation approach presented in Sections 3-4 and 3-4-1.

On this basis, it could be shown that the time-reversal transformed forms of the column matrices R  and F  given in the expressions of source-free cases of field equations ( 71) and ( 72), are definable solely in (1+2) and (1+3)-dimensional space-times. Based on this result, along with the basic assumption (3) in Sec. 3-1, it is concluded directly that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times. We show this in the following in detail.

As noted, in fact, the above conclusion follows directly from the formulations of uniquely determined time-reversal transformed forms of column matrices R  and F  given in the expressions of sourcefree cases of field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF]. Denoting these column matrices by , where the time-reversal operator (84) is defined by:

K î K T T ˆCh P     0
, they would be determined as follows in various dimensions:

-For (1+1)-dimensional space-time we have:

; ) , ( 0 ) , ( ) , ( ) , ( Ж , ) , ( 0 ) , ( ) , ( ) , ( Ж 10 0 10 0                           t iF t T t T t t iR t T t T t F F F R R R x x x x x x x x  (88)
-For (1+2)-dimensional space-time it is obtained: 92), although they could be expressed merely in the tensor formulations of field equations ( 71) and ( 72), however, except the (1+2) and (1+3)-dimensional cases of these transformed column matrices, all the other cases cannot be derived originally as a column matrix via the axiomatic derivation approach presented in Sections 3-4 and 3-4-1 (following the formulation of originally derived column matrices ( 73) -( 77 In addition, it is also worth to note that on the basis of our derivation approach, since there are not the corresponding isomorphism (that could be represented by the unique mappings (71-1) and (72-1), in Sec. 3-4-1) between the entries of column matrices R Ж and F Ж (88), (91), (92),… and the entries (with the exactly same indices) of column matrix S (in the algebraic matrix equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF], where its last entry, i.e. arbitrary parameter -s‖ is zero compatible with the source-free cases of the field equations ( 71) and ( 72)) that are given uniquely as follows in (1+1) and (1+4), (1+5),... and higher space-time dimensions, respectively, using the definitions (66) -( 70),… (in Sec. , and also the algebraic properties of column matrix S (presented in Sections 3-3-1 and 3-3-2) representing in terms of two half-sized 1 2 1   N column matrices

                                              ) t
                                                                                  ) t
                                                                                                                                                                                                                                                                      ) t

S and

S   such that:

          S S S
(where

N u u u u u ,..., , , , 3 2 1 0
,

w v v v v v N , , ..., , , , 3 2 1 0
are arbitrary parameters):

-For (1+1)-dimensional space-time we have:

                  0 ) ( 0 1 1 0 w v u v u S S S ; (93)
-For (1+4), (1+5)-dimensional space-times we get, respectively: ;

1 3 3 1 2 3 3 2 1 4 4 1 2 4 4 2 3 4 4 3 4 0 0 4 0 3 3 0 2 0 0 2 0 1 1 0                                                                              w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u S w v u v u w v u v u w v u v u w v u v u S S S S , 0 ) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0 ) ( 0 ) ( ) ( ) ( , 1 2 2 1
...                                                                                                                                   w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u S w v u v u w v u v u w v u v u w v u v u w v u v u S S S S (94)
it would be directly concluded that in (1+1) and (1+4), (1+5),... and higher space-time dimensions, the column matrices R Ж and F Ж could not be defined as the column matrices in unique formulations of the axiomatically derived general covariant field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF]. In other words, for the (1+2)dimensional cases of the transformed column matrices R Ж and F Ж (89), the corresponding isomorphism (represented uniquely by the mappings (71-1) and (72-1)) could be defined between the components of these matrices and the entries of column matrix S (67), for s = 0 (compatible with , 0

) (  R   0 ) (  F  ), if and only if: 0 ) , ( 02   t iR x  and 0 ) , ( 02   t iF x
. This could be shown as follows:

. F in (1+2)-dimensional space-time (without vanishing these tensor fields), based on their basic definitions given by formulas (71-1) and (72-1).

0 ) , ( , 0 ) , ( , 0 ) , ( ) , ( 0 ) , ( Ж , ) , ( ) , ( 0 ) , ( Ж 0 ) ( ) ( )
                                                                         t iF t iR v u v u t iF t iF t iF t iR t iR t iR w v u v u w v u v u w v u v u S F R on) Quantizati (First Procedure Derivation
Hence, definite mathematical framework of our axiomatic derivation approach (presented in Sec. 3-4), in addition to the time-reversal invariance (represented by the quantum operator (84)) of source-free cases of general covariant field equations ( 71) and ( 72), imply the (1+2)-dimensional case of column matrices Ψ R and Ψ F given by relations (74) (where we assumed , 0

) (  R   0 ) (  F 
), could be given solely as follows, to be compatible with the above assumed conditions (i.e. being compatible with the mathematical framework of axiomatic derivation of field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], and also the time-reversal invariance defined by quantum operator (84)), and consequently, as the column matrices could be defined in the formulations of the fundamental tensor field equations ( 71) and ( 72), respectively:

                            0 0 , 0 0 21 10 21 10 F F R R F R   (96)
The formulations (96) that are represented the column matrices Ψ R and Ψ F in the field equations ( 71) and ( 72) compatible with the above basic conditions, are also represented these matrices in the field equations ( 71) and ( 72) with non-zero source currents compatible with two basic conditions (similar to above conditions) including a unique combination of the C, P and T symmetries (that have been represented by quantum operators (83) -( 85)) for these cases of field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], and also the mathematical framework of axiomatic derivation of equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF]. In fact, as it has been shown in Sec. 3-5-1, the field equations ( 71) and ( 72) with non-zero source currents could have solely a certain combination (given by formulas (86) and (87)) of the C, P and T symmetries (that are represented by the operators (83), ( 84) and ( 85)). This unique combined symmetry in addition to the unique formulations (96) of source-free cases of column matrices Ψ R and Ψ

F in (1+2)-dimensional space-time, implies these matrices could take solely the following forms to be defined in the formulations of the fundamental tensor field equations ( 71) and ( 72) (with non-zero source currents):

) ( ) ( 0 ) ( ) ( ) ( 0 ) ( ) ( 21 10 ) ( 21 10 )
( , ) ( , 0 , 0 F F F R R R F F R R k im D J k im J F F R R                                                        (96-1)
In the same manner, concerning the (1+4)-dimensional cases of column matrices R Ж and F Ж (91), there would be a mapping between the entries of these matrices and entries (with the same indices) of algebraic column matrix S [START_REF] Mohr | Solutions of the Maxwell equations and photon wave functions[END_REF], where s = 0 (compatible with , 0

) (  R   0 ) (  F 
), if and only if: ,

 t iR x  , 0 ) , ( 41   t iR x  , 0 ) , ( 31   t iR x  0 ) , ( 21   t iR x  , , 0 ) , ( 10 
                                                                                                                                                                                                                                                                          t iF
v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u w v u v u S F R on) Quantizati (First Procedure Derivation
This means that in (1+4) space-time dimensions, the mathematical framework of our axiomatic derivation approach (described in Sec. 3-4) in addition to the time reversal invariance (defined by the quantum operator (84)) of the source-free case of the derived general covariant fundamental field equations [START_REF] Sushkov | New Experimental Limits on Non-Newtonian Forces in the Micrometer Range[END_REF] and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF] imply the column matrices Ψ R and Ψ F (76) (for , 0

) (  R   0 ) (  F 
) could take solely the following forms (in general) to be defined in the formulations of the field equations ( 71) and ( 72): In addition, similar to the formulations (96-1), as it has been shown in Sec. 3-5-1, the field equations ( 71) and ( 72) with non-zero source currents have a certain (and unique) combination of the C, P and T symmetries (that have been defined by the operators (83), ( 84) and ( 85)). This combined symmetry in addition to the forms (98), imply also the (1+4)-dimensional cases of column matrices Ψ R and Ψ F represented by formula (76) could take solely the following forms (in general) to be defined in the formulations of fundamental field equations ( 71) and (72):

                                                                                                       
) ( ) ( 0 ) ( ) ( ) ( 0 ) ( ) (
F F F R R R F F R R k im D J k im J F F F F F F R R R R R R                                                                                                                                    (98-1)
Consequently, the (1+4)-dimensional cases of column matrices Ψ R and Ψ F that are originally given by formulations [START_REF] Zahedi | On the Logical Origin of the Laws Governing the Fundamental Forces of Nature: A New Axiomatic Matrix Approach‖[END_REF], are reduced to formulas (98-1) which are equivalent to the (1+3)-dimensional cases of these matrices (given originally by column matrices of the forms (75)), i.e.: . )

F F F R R R F F R R k im D J k im J F F F F F F R R R R R R                                                                                    (98-2)
Moreover, as it would be also noted in Sec. 3-6, it is noteworthy to add that the tensor field   R in column matrix Ψ R (98-2) (expressing the general representation of column matrices definable in the formulation of (1+3)-dimensional case of general covariant field equation ( 71)), in fact, equivalently represents a massive bispinor field of spin-2 particles in (1+3) space-time dimensions (which could be identified as a definite generalized massive matrix formulation of the Einstein gravitational field, as it has been also shown in Sec. 3-4-2), and the tensor field  F in the column matrix Ψ F (98-2) (expressing the general representation of column matrices definable in the formulation of (1+3)-dimensional case of general covariant field equation ( 72)), in fact, equivalently represents a massive bispinor field of spin-1 particles in (1+3) space-time dimensions (which could be identified as definite generalized massive formulation of the Maxwell electromagnetic field, as it has been also shown in Sections 3-4-2 and 3-4-4; and also Yang-Mills fields compatible with specific gauge groups, as it would be shown in Sec. [START_REF] Zahedi | On the Connection Between Methods of the Ring Theory and the Group Approach‖, Russian Academy of Sciences (RAS)[END_REF][START_REF] Zahedi | The Linearization Method Based on the Ring Theory and its Applications in Physics‖, Russian Academy of Sciences (RAS) Publs[END_REF][START_REF] Durbin | Modern Algebra: An Introduction‖[END_REF][START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF].

Summing up, in this Section (Sec. 3-5-2) we showed that the axiomatic approach of derivation of the field equations ( 71) and ( 72) (described in Sections 3-1, 3-3 and 3-4) in addition to their time reversal invariance (represented basically by the quantum operator (84)), imply these fundamentally derived equations could be solely defined in (1+2) and (1+3) space-time dimensions. -Hence, based on the later conclusion and also the basic assumption (3) (defined in Sec. 3-1), we may conclude directly that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times, and cannot have more than four space-time dimensions.‖ Based on the axiomatic arguments and relevant results presented and obtained in this Section, in the following Sections we consider solely the (1+2) and (1+3)-dimensional cases of general covariant field equations ( 71) and ( 72) that are defined solely with the column matrices of the forms (96-1) and (98-2).

3-5-2-1. Equivalent (asymptotically) representations of the bispinor fields of spin-3/2 and spin-1/2 particles, respectively, by general covariant field equations (71) and (72) (formulated solely with column matrices of the types (96-1)) in (1+2) space-time dimensions:

It is noteworthy that according to the Ref. [START_REF] Elton | The Dirac equation without spinors‖[END_REF] and also based on the basic properties of the Riemann curvature tensor   R in (1+2) space-time dimensions [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF] (in particular the identity:

        G R 
, where   G is the Einstein tensor and ), it would be concluded that   R which is defined by (1+2)-dimensional case of the general covariant massive field equation ( 71) (which could be defined solely with a column matrix of the type Ψ R (96-1)), represents asymptotically a general covariant bispinor field of spin-3/2 particles (that would be asymptotically equivalent to the Rarita-Schwinger equation). In a similar manner, according to the Ref. [START_REF] Elton | The Dirac equation without spinors‖[END_REF], and also following the basic properties of field strength tensor  F in (1+2) space-time dimensions (that as a rank two anti-symmetric with three independent components holding, in particular, the identities:

     T F  ,     F T ) 2 1 (  ,
showing that  F could be represented equivalently by a vector  T with three independent components as well) it would be concluded that  F which is defined by (1+2)-dimensional case of the general covariant massive (tensor) field equation ( 72) (which could be defined solely with a column matrix of the type Ψ F (96-1)), represents asymptotically a general covariant bispinor field of spin-1/2 particles (that would be asymptotically equivalent to the Dirac equation [START_REF] Elton | The Dirac equation without spinors‖[END_REF]). Furthermore, as it would be shown in Sec. 3-6, the general covariant field equations (72) (representing asymptotically the spin-1/2 fermion fields) is also compatible with the SU(2) L U(2) R symmetry group (representing -1+3‖ generations for both lepton and quark fields including a new charge-less fermion).

3-5-2-2. Equivalent representations of the bispinor fields of spin-2 and spin-1 particles, respectively, by general covariant field equations (71) and (72) (defined solely with column matrices of the types (98-2)) in (1+3) space-time dimensions:

It should be also note that according to the Refs. [31 -36], the basic properties of the Riemann curvature tensor including the relevant results presented in Sec. 3-4-2 , it would be concluded that the field strength tensor   R (i.e. the Riemann tensor) the in (1+3) space-time dimensions by general covariant massive (tensor) field equation ( 71) (formulated solely with a column matrix of the type Ψ R (98-2)), represents a general covariant bispinor field of spin-2 particles (as a generalized massive formulation of the Einstein gravitational field equation). In a similar manner, according to the Refs. [31 -36], the field strength tensor  F which is defined in (1+3) space-time dimensions by the general covariant massive (tensor) field equation ( 72) (formulated solely with a column matrix of the type Ψ F (98-2)), represents a general covariant bispinor field of spin-1 particles (representing new generalized massive formulations of the Maxwell's equations, and also Yang-Mills field equations). Furthermore, as it would be shown in Sec. 3-6, the general covariant field equations ( 72 As it has been shown in Sections 3-5-1 and 3-5-2, since the algebraic column matrix S in the matrix equation ( 64) (derived and represented uniquely in terms of the matrices (66) -( 70),… corresponding to various space-time dimensions), is not symmetric by multiplying by matrix Ch  (84- 1) and (84-2) (except for (1+2) and (1+3)-dimensional cases of column matrix S , based on the definite algebraic properties of matrix S presented in Sections 3-3, 3-3-1, 3-3-2), it is concluded that except the (1+2) and (1+3)-dimensional cases of the fundamental field equations ( 71) and ( 72), these field equations could not be defined with column matrices of the type 71) and (72), i.e. they have the formulations similar to the formulations of originally derived column matrices ( 73) - [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF],… corresponding to various space-time dimensions). This conclusion follows from this fact that the filed equations ( 71) and ( 72) have been derived (and defined) uniquely from the matrix equation ( 64) via the axiomatic derivation approach (including the first quantization procedure) presented in Sections 3-4, 3-4-1. As it has been shown in Sec. 3-5-2, using this property (i.e. multiplication of column matrices R  and F  , defined in the unique expressions of fundamental field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], by matrix Ch  from the left), this crucial and essential issue is concluded directly that by assuming the time-reversal invariance of the general covariant filed equations ( 71) and ( 72 ), these fundamental field equations could be defined solely in (1+2) and (1+3) space-time dimensions (with the column matrices of the forms (96-1) and (98-2), respectively).

Hence, the definite mathematical formalism of the axiomatic approach of derivation of fundamental field equations ( 71) and ( 72), along with the C, P and T symmetries (represented by the quantum matrix operators ( 83) -( 87), in Sec. 3-5) of source-free cases (as basic cases) of these equations, in fact, imply these equations with non-zero source currents, would be invariant solely under the simultaneous combination of all the transformations , Ĉ P ˆ, and

F Ch F R Ch R COMB COMB                   (99) ). , ( ) , ( ˆ), , ( ) , ( ˆt r C P T t r Z t r C P T t r Z F Ch F R Ch R COMB COMB                 (100)
The unique combined form of transformation COMB Z ˆ(99) (and also COMB Z ˆ(100), where

COMB COMB Z Z  
) is based on the following two basic issues: Firstly, it follows from the definite formulations of uniquely determined column matrices ( 73) - [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF],… (corresponding to various space-time dimensions, however, as noted above, based on the arguments presented in Sec. 3-5-2, the only definable column matrices in the formulations of field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], are of the types Ψ R and Ψ F represented by formulas (96-1) and (98-2), in (1+2) and (1+3) space-time dimensions, respectively), where the source currents

) ( ) ( 0 ) ( ) ( R R R k im J            , ) ( ) ( 0 ) ( ) ( F F F k im J          
. In other words, the unique formulation of derived combined symmetries On the basis of the 'CPT' theorem [START_REF] Weinberg | The Quantum Theory of Fields[END_REF][START_REF] Pal | An Introductory Course of Particle Physics[END_REF], it would be concluded directly that the unique combined forms of transformations (99) and (100) (representing the only combination of , Ĉ P ˆ, and T ˆ transformations multiplied by matrix Ch  , that could be defined as a symmetry for general covariant field equations (71) and ( 72) with non-zero source currents), should be equivalent only to simultaneous combination of , Ĉ P ˆ, and T ˆ transformations (that have been defined uniquely by formulas ( 83) -( 85)). Moreover, based on the 'CPT' theorem, the simultaneous combination of transformations , Ĉ P ˆ, and T ˆ should: -interchange the particle field and its corresponding antiparticle field; inverts the spatial coordinates r r     ; reverse the spin of all particle fields; leave the direction of the momentum invariant; and, therefore, should interchange the left-handed and right-handed components of both particle field and its corresponding antiparticle field‖. Hence, we should have:

; )] ( [ )] ( [ )] ( [ )] ( [ ˆ, )] ( [ )] ( [ )] ( [ )] ( [ ˆ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( Right F Right F Ch Left F Ch Left F Right R Right R Ch Left R Ch Left R r r r C P T r Z r r r C P T r Z COMB COMB                                   (101) ; )] ( [ )] ( [ )] ( [ )] ( [ ˆ, )] ( [ )] ( [ )] ( [ )] ( [ ˆ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( Left F Left F Ch Right F Ch Right F Left R Left R Ch Right R Ch Right R r r r C P T r Z r r r C P T r Z COMB COMB                                   (102) ; )] ( [ )] ( [ )] ( [ )] ( [ ˆ, )] ( [ )] ( [ )] ( [ )] ( [ ˆ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( Right F Right F Ch Left F Ch Left F Right R Right R Ch Left R Ch Left R r r r C P T r Z r r r C P T r Z COMB COMB                               (103) . )] ( [ )] ( [ )] ( [ )] ( [ ˆ, )] ( [ )] ( [ )] ( [ )] ( [ ˆ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( Left F Left F Ch Right F Ch Right F Left R Left R Ch Right R Ch Right R r r r C P T r Z r r r C P T r Z COMB COMB                               (104)
where the column matrices

)], ( ) ( [ 2 1 )] ( [ )], ( ) ( [ 2 1 )] ( [ ) ( ) ( r γ r r r γ r r F Ch F Left F R Ch R Left R                       )]; ( ) ( [ 2 1 )] ( [ )], ( ) ( [ 2 1 )] ( [ ) ( ) ( r γ r r r γ r r F Ch F Right F R Ch R Right R                       (105) )], ( ) ( [ 2 1 )] ( [ )], ( ) ( [ 2 1 )] ( [ ) ( ) ( r γ r r r γ r r F Ch F Left F R Ch R Left R                 )]. ( ) ( [ 2 1 )] ( [ )], ( ) ( [ 2 1 )] ( [ ) ( ) ( r γ r r r γ r r F Ch F Right F R Ch R Right R                 (106)
where we have:

, )] ( [ )] ( [ ) ( , )] ( [ )] ( [ ) ( ) ( ) ( ) ( ) ( Right F Left F F Right R Left R R r r r r r r                       . )] ( [ )] ( [ ) ( , )] ( [ )] ( [ ) ( ) ( ) ( ) ( ) ( Right F Left F F Right R Left R R r r r r r r                       (107)
Now using the definitions (105) and (106) in the formulas (101) -(104), we obtain:

; )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ , )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ ) ( ) ( ) ( ) ( Left F F F Ch F Ch F Ch Left F Ch Left R R R Ch R Ch R Ch Left R Ch r r r γ r γ r r r r r γ r γ r r                                                   (101-1) ; )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ , )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ ) ( ) ( ) ( ) ( Right F F F Ch F Ch F Ch Right F Ch Right R R R Ch R Ch R Ch Right R Ch r r r γ r γ r r r r r γ r γ r r                                                   (102-1) ; )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ , )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ ) ( ) ( ) ( ) ( Left F F F Ch F Ch F Ch Left F Ch Left R R R Ch R Ch R Ch Left R Ch r r r γ r γ r r r r r γ r γ r r                                             (103-1) . )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ , )] ( [ )] ( ) ( [ 2 1 )] ( ) ( [ 2 1 )] ( [ ) ( ) ( ) ( ) ( Right F F F Ch F Ch F Ch Right F Ch Right R R R Ch R Ch R Ch Right R Ch r r r γ r γ r r r r r γ r γ r r                                             (104-1)
Based on the relations (101-1) -(104-1), it would be concluded directly that only the left-handed components of particle fields representing by

) ( )] ( [ Left R r    , ) ( )] ( [ Left F r   
, and the right-handed components of their corresponding antiparticle fields representing by 103)). Hence (and also following the basic assumption (3) defined in Sec. , it is concluded directly that only the left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. This means that only left-handed bosonic fields (along with their complementary right-handed fields) could be coupled to the corresponding fermionic source currents; which also means that only lefthanded fermions (along with their complementary right-handed fermions) can participate in any interaction with the bosons (which consequently would be only left-handed bosons or their complementary right-handed bosons).

3-6. Showing the invariance of axiomatically derived general covariant (tensor) field

equations ( 72) in (1+2)-dimensional space-time (defined generally with column matrices of the type Ψ F (96-1), representing spin-1/2 fermion fields) under the SU(2) L U(2) R symmetry group, and also their invariance in (1+3) dimensions (defined generally with column matrices of the type Ψ F (98-2) representing the spin-1 boson fields which assumed that are coupled to the spin-1/2 fermionic source currents) under the SU(2) L U(2) R and SU(3) symmetry groups (distinctly):

One of the natural and basic properties of the (1+2)-dimensional space-time geometry is that the metric tensor can be -diagonalized‖ [START_REF] Rindler | Relativity: SPECIAL, GENERAL[END_REF]. Using this basic property, the invariant energy-momentum quadratic relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF] (in Sec. 3-1-1) would be expressed as follows: . This set includes seven different types of the sign inversions (in total), which could be represented simply by the following symmetric group of transformations (based on the formalism of the corresponding Lorentz symmetry group of invariant relation (108)), respectively: then, along with the set seven real-valued transformations (108-1) -(108-7), this relation would be also invariant under these corresponding sets of complex transformations:

0 ) ( ) ( ) ( ) (
) , , , ( ) , , , ( ) , , , ( ) 1 ( 2 ) 1 ( 1
   (108-1) ) , , , ( ) , , , ( ) , , , ( ) 2 ( 2 ) 2 ( 1
   (108-2) ) , , , ( ) , , , ( ) , , , ( ) 3 ( 2 ) 3 ( 1
      (108-4) ) , , , ( ) , , , ( ) , , , ( ) 5 ( 2 ) 5 ( 1
     (108-5) ) , , , ( ) , , , ( ) , , , ( ) 6 ( 2 ) 6 ( 1
     (108-6) ) , , , ( ) , , , ( ) , , , ( ) 7 ( 2 ) 7 ( 1
) , , , ( ) , , , ( ) ( 2 ) ( 1 ) ( 0 )
( 0 2 1 0 0          a a a st a st ip ip ip ip p p p p , ) , , , ( ) , , , ( ) ( 2 ) ( 1 ) ( 0 )
( 0 2 1 0 0 a a a st a st ip ip ip ip p p p p          .
(108-9) for 7 ,...,

In Sec. 3-6-1-1, using the transformations (108-1) -(108-7) (along with their corresponding complex forms (108-9)), a certain set of seven simultaneous (different) general covariant field equations (corresponding to a group of seven bispinor fields of spin-1/2 particles) would be determined as particular cases of the (1+2)-dimensional form of general covariant field equation (72) (defined with a column matrix of the type (96-1)).

3-6-1.

Following the definite formulation of (1+2)-dimensional case of system of linear equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF] (formulated in terms of the matrices (67)), for the energy-momentum relation (108) (along with the transformations (108-1) -(108-7)), the following set of seven systems of linear equations (with different parametric formalisms) is determined uniquely. The general parametric solution of each of these systems of linear equations, obeys also the quadratic relation (108) (representing a set of seven forms, with different parametric formulations, of the general parametric solutions of quadratic relation (108)). This set of the seven systems of linear equations could be represented uniformly by a matrix equation as follows: S is also given uniquely as follows (formulated on the basis of definite parametric formulation of column matrix S (67) in (1+2) space-time dimensions):

0 ) ( ) ( ) ( 0 ) (  
                  s w v u v u w v u v u w v u v u S a a a a a a a a a a a a a ) ( ) ( ) ( ) ( 1 ) ( 2 ) ( 2 ) ( 1 ) ( 2 ) ( 0 ) ( 0 ) ( 2 ) ( 0 ) ( 1 ) ( 1 ) ( 0 ) ( (109-1)
which includes seven cases with specific parametric formulations expressed respectively in terms of seven groups of independent arbitrary parameters: 109), it is necessary to add that this parametric expression has been determined specifically by assuming (as a basic assumption in addition to the systematic natural approach of formulating the matrix equation ( 109), based on the definite formulation of axiomatically determined matrix equation ( 64)) the minimum value for total number of the arbitrary parameters in all of seven cases of column matrix ) (a S , which implies equivalently the minimum value for total number of the arbitrary parameters in all of seven simultaneous (different) cases of matrix equation (109) (necessarily with seven independent parametric solutions representing a certain set of seven different equivalent forms of the general parametric solution of quadratic relation (108), based on the general conditions of basic definition of the systems of linear equations corresponding to homogeneous quadratic and higher degree equations, presented in Sec. 2, and Sections 2-2 -2-4, 3-1-1 concerning the homogenous quadratic equations).

) ( 2 ) ( 1 ) ( 0 ) ( 2 ) ( 1 ) ( 0 , , , , ,
In the following, in the derivation of the corresponding field equations (from matrix equation ( 109)), we will also use the above particular algebraic property of parameters s which has been expressed commonly in the expressions of all of seven simultaneous cases of matrix equation (109) (and also in Sec. 3-6-2, concerning the (1+3)-dimensional corresponding form of matrix equation (109), which holds the similar property).

3-6-1-1.

In addition, along with the transformations (108-1) -(108-7) and algebraic matrix equation (109), using the corresponding complex transformations (108-9), we may also formally have the following equivalent matrix equation (with the complex expression): 0 ), in the following, using this basic property of the momentum operator, we derive a certain set of seven different simultaneous general covariant field equations from the matrix equations ( 109) and (109-2) (based on the general axiomatic approach of derivation of general covariant massive field equation [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF], presented in Sections 3-4 -3-4-5, in addition to certain forms of quantum representations of the C, P and T symmetries of this field equation, presented in Sections 3-5 -3-5-4). Furthermore, in Sec. 3-6-1-2, it would be also shown that the uniform representation of this determined set of seven simultaneous field equations, describe a certain group of seven simultaneous bispinor fields of spin-1/2 particles (corresponding, respectively, to a new righthanded charge-less fermion in addition to three right-handed anti-fermions, along with their three complementary left-handed fermions). Furthermore, concerning the gravitational field equation ( 71), it should be noted that following from the fact that the general covariant field equation ( 71) should describe, uniquely and uniformly, the background space-time geometry via a certain form of the Riemann curvature tensor (which should be determined from the tensor field equation ( 71)), the matrix equation (109) could not be used for the derivation of a set of simultaneous different spin-3/2 fermion fields in (1+2) dimensions (there would be the same condition for the field equation ( 71) in higher-dimensional space-times).

) ( ) ( ) ( 0 )
Hence, based on the axiomatic approach of derivation of (1+2)-dimensional case of field equation [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF] (defined solely by a column matrix of the form (96-1) in (1+2) space-time dimensions, as shown in Sec. 3-5-2), from the matrix equation ( 109) and (109-2) (defined solely by column matrix (109-1)), and also taking into account the momentum operator's property:

    p p
ˆ, the following group of seven simultaneous (different) general covariant field equations could be determined:

0 ) ( ) ( ) ( 0 ) (    f F f f k m D i        (110)
specifying by the following group of transformations (based on the corresponding group of transformations (108-1) -(108-7) and (108-9)), for 7

,..., 3 , 2 , 1  f , respectively ) , , , ( ) , , , (
2 1 0 0 ) 1 ( 2 ) 1 ( 1 ) 1 ( 0 ) 1 ( 0 D D m D D D m D   (110-1) ) , , , ( ) , , , ( 2 1 0 0 ) 2 ( 2 ) 2 ( 1 ) 2 ( 0
) 2 ( 0 D D m D D D m D   (110-2) ) , , , ( ) , , , ( 2 1 0 0 ) 3 ( 2 ) 3 ( 1 ) 3 ( 0
) 3 ( 0 D D m D D D m D    (110-3) ) , , , ( ) , , , ( 2 1 0 0 ) 4 ( 2 ) 4 ( 1 ) 4 ( 0 ) 4 ( 0         iD iD im iD D D m D (110-4) ) , , , ( ) , , , ( 2 1 0 0 ) 5 ( 2 ) 5 ( 1 ) 5 ( 0 ) 5 ( 0        iD iD im iD D D m D (110-5) ) , , , ( ) , , , ( 2 1 0 0
) 6 ( 2 ) 6 ( 1 ) 6 ( 0 ) 6 ( 0        iD iD im iD D D m D (110-6) ) , , , ( ) , , , ( 2 1 0 0
) 7 ( 2 ) 7 ( 1 ) 7 ( 0 ) 7 ( 0       iD iD im iD D D m D (110-7)
where the column matrix ) ( f F  would be also given as follows (based on the definite formulation of column matrix F  (96-1) in Sec. 3-5-2, expressing the general representation of column matrices definable in the formulation of (1+2)-dimensional case of general covariant field equation ( 72)):

F f f f F f f f F k im D J F F      ) ( , 0 ) ( 0 ) ( ) ( ) ( 21
) ( 10
) (                     (110-8)
where in all of the seven simultaneous cases of field equation (110) defined respectively by the column matrices

) ( f F  (110-8) (for 7 ,..., 3 , 2 , 1  f
), the scalar quantity F  (that as a given initial quantity, defines the source currents ) ( f J  (110-8)), necessarily, has the same value, based on the definite parametric formulation of the algebraic column matrix (109-1) (in particular, the common form of the corresponding arbitrary parameter s in the expressions of all of the seven simultaneous cases of matrix equation ( 109)).

3-6-1-2.

Following the definite formulations of set of seven general covariant (massive) field equations (110) (specified, respectively, by the group of seven transformations (110-1) -(110-7)), the set of these could be represented uniformly by the following general covariant field equation as well (defined solely in (1+2) space-time dimensions):

0 ) ( 0    F k m D i        (110-9)
where the column matrix F  given by:

F F F k im D J F F         ) ( , 0 0 21 10                    (110-10)
and the field strength tensor  F , scalar F   , along with the source current  J are defined as follows:

   7 1 )
( f f f F F    , . ) ( ) ( 7 1 ) ( 7 1 )
( 0 ) ( 0 2          J J k im D k im D I f f f f F f f F               (110-11)
where I is the 2 2  identity matrix, and

2 f f l   (for 7 ,..., 3 , 2 , 1  f ) are a set of seven 2 2  complex matrices given by, 2 f f l   ,            7 6 5 4 3 2 1 , , , , , , l l l l l l l                                                                                      1 0 0 1 , 0 0 , 0 1 1 0 , 1 0 0 1 , 0 0 , 0 1 1 0 , 0 0 i i i i i i   (110-12)
which as would be shown in the following, represents uniformly a combined gauge symmetry group of the form: SU(2) L U(2) R , where the sub-set of three matrices -

3-6-2.

Assuming the spin-1/2 fermion fields describing by general covariant massive field equations (110-9) (defined by column matrix (110-10) in (1+2) space-time dimensions with a digonalized metric), as the coupling source currents of spin-1 boson fields (describing generally by (1+3)-dimensional case of general covariant field equation ( 72) formulated with a column matrix of the type Ψ F (98-2)), it is concluded that the (1+3)-dimensional metric could be also diagonalized for corresponding spin-1 boson fields. This conclusion follows directly from the above assumption that the (1+3)-dimensional metric of spin-1 boson fields (coupled to the corresponding fermionic source currents) would be also partially diagonalized such that: then, along with the set fifteen real-valued transformations (111-1) -(111-15), this relation would be also invariant under these corresponding sets of complex transformations: In Sec. 3-6-3, using the transformations (111-1) -(111-15) (along with their corresponding complex forms (111-17)), a set of fifteen different general covariant field equations would be determined, including two certain groups of simultaneous field equations (corresponding, respectively, to a group of seven bispinor fields and a group of eight bispinor fields of spin-1 particles), as particular cases of the (1+3)-dimensional form of general covariant field equation ( 72) (defined with a column matrix of the type (98-2)).

0   μ g (for μ = 0,
      (111-13) ) , , , , ( ) , , , , ( ) , , , , ( ) 14 ( 3
      (111-14) ) , , , , ( ) , , , , ( ) , , , , ( ) 15 ( 3
) , , , , ( ) , , , , ( ) ( 3 )
( 2 ) ( 1
) ( 0 ) ( 0 3 2 1 0 0            b b b b st b st ip ip ip ip ip p p p p p , ) , , , , ( ) , , , , ( ) ( 3 )
( 2 ) ( 1
) ( 0 ) ( 0 3 2 1 0 0 b b b b st b st ip ip ip ip ip p p p p p            . ( 111

3-6-3.

Similar to the set of seven algebraic matrix equations (109) (determined uniquely as the algebraic equivalent matrix representation of the energy-momentum relation (108)), based on the definite formulation of the system of linear equation ( 64) in (1+3) space-time dimensions (formulated in terms of the matrices (68)), for the energy-momentum relation (111) (along with the transformations (111-1) -(111-15)) the following two sets of systems of linear equations are also determined uniquely, including respectively a set of seven and a set of eight systems of equations (with different parametric formalisms). The general parametric solution of each of these systems of linear equations, obeys also the quadratic relation (111) (representing a set of fifteen forms, with different parametric formulations, of the general parametric solutions of quadratic relation (111)). Each of these sets of the systems of linear equations could be represented uniformly by a matrix equation as follows, respectively: 0 S are also given uniquely as follows by two distinct expressions (formulated on the basis of definite parametric formulation of column matrix S (68) in (1+3) space-time dimensions): of the seven cases of algebraic matrix equation (112-1), and also each of the eight cases of algebraic matrix equation (112-2), definitely, is not a necessary issue at the present stage. However, since the corresponding momentum operator  p ˆ has a complex value (where ), ˆ    p p in the following, using this basic property of momentum operator  p ˆ, we derive, distinctly, two certain groups of the general covariant field equations, including a group of seven different simultaneous field equations from the matrix equations (112-1) and (112-3), and a group of eight different simultaneous field equations from the matrix equations (112-2) and (112-4) (based on the general axiomatic approach of derivation of general covariant massive field equations (72) presented in Sections 3-4 -3-4-5, and the quantum representations of C, P and T symmetries of this equation, presented in Sections 3-5 -3-5-4). Furthermore, in Sec. 3-6-3-2, it would be also shown that each of these determined two sets of seven and eight simultaneous field equations describe, respectively, a uniform group of seven spin-1boson fields (corresponding to two lefthanded massive charged bosons, along with their two complementary right-handed bosons; a left-handed massive charge-less boson, along with its complementary right-handed boson; and a single right-handed massless and charge-less boson), and a uniform group of eight spin-1 boson field (corresponding to eight massless charged bosons).

) ( ) ( ) ( 0 ) ( 1 1 1   b b b S k m p       , (112-1) 0 ) ( ) ( ) ( 0 ) ( 2 2 2   b b b S k m p       (112-2) where 7 ,..., 3 , 2 , 1 1  b , 15 ,..., 9 , 8 2  b ,   k m p b b st ) ( 0 ) ( 1 1  ,   k m p b b st ) ( 0 ) (
Hence, similar to the (1+2)-dimensional general covariant field equation (114) derived in Sec. 3-6-1-1, based on the axiomatic approach of derivation of the (1+3)-dimensional case of field equation ( 72) (defined solely by a column matrix of the form (98-2) in (1+3) space-time dimensions, as shown in Sec. 3-5-2), from the matrix equations (112-1), (112-3) and (112-2), (112-4) (defined solely by column matrices (113)), also taking into account the momentum operator's property:

  p p  
, the following two distinct groups of seven and eight simultaneous (different) general covariant field equations could be determined, respectively:

0 ) ( ) ( ) ( 0 ) ( 1 1 1    b Z b b k m D i        (114-1) 0 ) ( ) ( ) ( 0 ) ( 2 2 2      b G b b k m D i        (114-2)
specifying by the following two groups of transformations (based on the corresponding groups of transformations (111-

1 0 0 ) 1 ( 3 ) 1 ( 2 ) 1 ( 1 ) 1 ( 0 ) 1 ( 0 D D D m D D D D m D    (114-1-1) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 2 ( 3 ) 2 ( 2 ) 2 ( 1 ) 2 ( 0 ) 2 ( 0 D D D m D D D D m D    (114-1-2) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 3 ( 3 ) 3 ( 2 ) 3 ( 1 ) 3 ( 0 ) 3 ( 0 D D D m D D D D m D    (114-1-3) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 4 ( 3 ) 4 ( 2 ) 4 ( 1 ) 4 ( 0 ) 4 ( 0           iD iD iD im iD D D D m D (114-1-4) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 5 ( 3 ) 5 ( 2 ) 5 ( 1 ) 5 ( 0 ) 5 ( 0         iD iD iD im iD D D D m D (114-1-5) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 6 ( 3 ) 6 ( 2 ) 6 ( 1 ) 6 ( 0 ) 6 ( 0         iD iD iD im iD D D D m D (114-1-6) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 7 ( 3 ) 7 ( 2 ) 7 ( 1 ) 7 ( 0 ) 7 ( 0         iD iD iD im iD D D D m D (114-1-7) , ( ) , , , , ( 3 2
and the second group is given as follows, respectively:

) , , , , ( ) , , , , ( 3 2
1 0 0 ) 1 ( 3 ) 1 ( 2 ) 1 ( 1 ) 1 ( 0 ) 1 ( 0 D D D m D D D D m D        (114-2-1) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 2 ( 3 ) 2 ( 2 ) 2 ( 1 ) 2 ( 0 ) 2 ( 0 D D D m D D D D m D        (114-2-2) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 3 ( 3 ) 3 ( 2 ) 3 ( 1 ) 3 ( 0 ) 3 ( 0 D D D m D D D D m D        (114-2-3) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 4 ( 3 ) 4 ( 2 ) 4 ( 1 ) 4 ( 0 ) 4 ( 0 D D D m D D D D m D          (114-2-4) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 5 ( 3 ) 5 ( 2 ) 5 ( 1 ) 5 ( 0 ) 5 ( 0             iD iD iD im iD D D D m D (114-2-5) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 6 ( 3 ) 6 ( 2 ) 6 ( 1 ) 6 ( 0 ) 6 ( 0               iD iD iD im iD D D D m D (114-2-6) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 7 ( 3 ) 7 ( 2 ) 7 ( 1 ) 7 ( 0 ) 7 ( 0               iD iD iD im iD D D D m D (114-2-7) ) , , , , ( ) , , , , ( 3 2
1 0 0 ) 8 ( 3 ) 8 ( 2 ) 8 ( 1 ) 8 ( 0 ) 8 ( 0               iD iD iD im iD D D D m D (114-2-8)
where the column matrices  are also given as follows, written on the basis of definite formulations of algebraic column matrices (113), in addition to the unique formulation of column matrix (98-2) (expressing the general representation of column matrices definable in the formulation of (1+3)dimensional case of general covariant field equation ( 72)):

, ) ( , 0 ) ( 0 ) ( ) ( ) ( 31
) ( 31
) ( 23
) ( 30
) ( 20
) ( 10
) ( 1 1 1 1 1 1 1 1 1 1 Z b b b Z b b b b b b b Z k im D J Z Z Z Z Z Z                                      G b b b G b b b b b b b G k m i D J G G G G G G      ) ( , 0 ) ( 0 ) ( ) ( ) ( 31
) ( 31
) ( 23
) ( 30
) ( 20
) ( 10
) ( 2 2 2 2 2 2 2 2 2 2                                    (114-3)
where in all of the seven simultaneous (different) field equations (112-1) formulated with column matrix (described respectively by general covariant massive field equations (114-4) and (114-5)), and on the basis of C, P and T symmetries of these field equations (as two particular forms of the (1+3)-dimensional case of tensor field equation ( 72)), represented by their corresponding quantum operators (defined in Sections 3-5 -3-5-4), it would be concluded that the field equation (114-4) describes uniformly a definite group of seven simultaneous bispinor fields of spin-1 particles (corresponding to seven matrices

Conclusion

The main results obtained in this article, are mainly, the outcomes of the new algebraic axiom (17) (along with the basic assumptions (2) -(3) defined in Sec. . This new axiom as a definite generalized form of the ordinary axiom of -no zero divisors‖ of integral domains (including the domain of integers), has been formulated soley in terms of square matrices (with integer entries, appeared as primary objects for representing the integer elements in their corresponding algebraic axiomatic formalism). In Sec. 3 of this article, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Sec.2), "it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms"; where as a basic additional assumption (that is the assumption (2) in Sec. 3-1), in agreement with the rational Lorentz symmetry group, it has been also assumed that the components of relativistic energy-momentum (D-momentum) can only take the rational values. Concerning the basic assumption of rationality of relativistic energy-momentum, it is necessary to add (as mentioned in Sec. 3-1) that the rational Lorentz symmetry group is not only dense in the general form of Lorentz group, but also is compatible with the necessary conditions required basically for the formalism of a consistent relativistic quantum theory [START_REF] Hill | Relativistic Theory of Discrete Momentum Space and Discrete Space-Time‖[END_REF]. In essence, the main scheme of the new mathematical axiomatic approach to fundamental laws of nature presented in Sec. 3, is as follows. First in Sec. 3-1-1, based on the assumption of rationality of D-momentum, by linearization (along with a parameterization procedure) of the Lorentz invariant energymomentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) has been derived. Then in Sec. 3-4, by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) has been derived uniquely as well. Each class of the derived general covariant field equations also includes a definite form of torsion field appeared as generator of the corresponding field' invariant mass. In addition, in Sections 3-4 -3-5, it has been shown that the (1+3)-dimensional cases of two classes of derived field equations represent a new general covariant massive formalism of bispinor fields of spin-2, and spin-1 particles, respectively. In fact, these uniquely determined bispinor fields represent a unique set of new generalized massive forms of the laws governing the fundamental forces of nature, including the Einstein (gravitational), Maxwell (electromagnetic) and Yang-Mills (nuclear) field equations. Moreover, it has been also shown that the (1+2)-dimensional cases of two classes of these field equations represent (asymptotically) a new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, respectively, corresponding to the Dirac and Rarita-Schwinger equations.

As a particular consequence, in Sec. 3-4-2, it has been shown that a certain massive formalism of general relativitywith a definite form of torsion field appeared originally as the generator of gravitational field's invariant massis obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a certain set of special relativistic algebraic matrix equations. In Sec. 3-4-4, it has been also proved that Lagrangian densities specified for the originally derived new massive forms of the Maxwell, Yang-Mills and Dirac field equations, are also gauge invariant, where the invariant mass of each field is generated solely by the corresponding torsion field. In addition, in Sec. 3-4-5, in agreement with recent astronomical data, a new particular form of massive boson has been identified 50 (corresponding to U(1) gauge group) with invariant mass: m γ ≈ 4.90571×10 -kg, which isgenerated by a coupled torsion field of the background space-time geometry.

Moreover, in Sec. 3-5-2, based on the definite mathematical formalism of this new axiomatic approach, along with the C, P and T symmetries (represented basically by the corresponding quantum matrix operators) of uniquely derived two fundamental classes of general covariant field equations, it has been concluded that the universe could be realized solely with the (1+2) and (1+3)dimensional space-times (where this conclusion, in particular, is based on the time-reversal symmetry). In Sections 3-5-3 and 3-5-4, it has been proved that 'CPT' is the only (unique) combination of C, P, and T symmetries that could be defined as a symmetry for interacting fields.. In addition, in Sec. 3-5-4, on the basis of these discrete symmetries of derived field equations, it has been also shown that only left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. Furthermore, in Sec. 3-6, it has been shown that metric of the background space-time is diagonalized for the uniquely derived fermion field equations (defined and expressed solely in (1+2)-dimensional space-time), where this property generates a certain set of additional symmetries corresponding uniquely to the SU(2) L U(2) R symmetry group for spin-1/2 fermion fields (representing -1+3‖ generations of four fermions, including a group of eight leptons and a group of eight quarks), and also the SU(2) L U(2) R and SU(3) gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Hence, along with the known elementary particles, eight new elementary particles, including: four new charge-less right-handed spin-1/2 fermions (two leptons and two quarks, represented by -z e , z n and z u , z d ‖), a spin-3/2 fermion, and also three new spin-1 massive bosons (represented by " , , "

Z W W   
, where in particular, the new boson Z  is complementary righthanded particle of ordinary Z boson), have been predicted uniquely by this fundamental axiomatic approach. As a particular result, in Sec. 3-4-2, based on the definite and unique formulation of the derived Maxwell's equations (and also determined Yang-Mills equations, represented uniquely with two specific forms of gauge symmetries, in 3-6-3-2), it has been also concluded generally that magnetic monopoles could not exist in nature.

The new results obtained in this article, which are connecting with a number of longstanding essential issues in science and philosophy, demonstrate the wide efficiency of a new fundamental algebraic-axiomatic formalism presented in Sec. 2 of this article.

  matrix. Then the following axiom is assumed (as a new axiomatic generalization of the ordinary axiom of -no zero divisors‖ of integral domain ℤ):

  and consequent of this biconditional. In addition, based on the initial assumption   r ℕ:

  antecedent and consequent of biconditional (1-1). The axiomatic biconditional (1-1), defines a system of linear equations of the type 0

  relativistic forms of energy-momentum quantum operator (where   is the general covariant derivative and  D is gauge covariant derivative, defining in Sections 3

  is general covariant derivative defined with torsion  T . In equations (4-1) -(4-3),  D  is the general relativistic form of gauge covariant derivative defined with torsion field   Z (which generates the gauge field's invariant mass), and  A denotes the corresponding gauge (potential) field.

  3-4-3, if the Ricci curvature tensor is defined basically by the following relation in terms of Riemann curvature tensor (determined by field equations (3-1) -(3-3)): , (8-1)

  matrix. Then the following axiom is assumed (as a new axiomatic generalization of the ordinary axiom of -no zero divisors‖ of integral domain ℤ):

  antecedent and consequent of biconditional. The axiomatic biconditional (17-1), defines a system of linear equations of the type 0 integral domain ℤ). The axiom (17) (or (17-1)) for 1  n , is equivalent to the ordinary axiom of -no zero divisors‖

.

  In fact, we may represent uniquely the square matrix A (with assumed properties in the axiom[START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF]) by this homogeneous linear form: independent from the variables k b ) would be generators of the corresponding generalized Clifford algebra associated with the r th degree homogeneous polynomial )

ija 1 ,(

 1 of square matrix A are linear homogeneous forms expressed in terms of the integral variables p b , i.e. we may also represent the square matrix A by this linear matrix form: implies that the square matrices k E (which their entries are independent from the variables k b ) would be generators of the corresponding generalized Clifford algebra associated with the r th degree homogeneous polynomial) for r ≥ 2, such as the standard quadratic forms defined in the quadratic equation[START_REF] Kauffman | Knots and Physics[END_REF] in Sec. 2-2), without any restriction in the existence and procedure of derivation of their corresponding square matrices assuming in axiom (17)) obeying the Clifford algebraic relation:

  as assumed in the axiom[START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF], where the variables k b denote the components of corresponding physical quantity (such as the relativistic energy-momentum, as it has been assumed in Sec. 3-1 of this article based on the basic assumption (2) noted in Sections 1-1 and 3-1).

  n n A  (by assuming a minimum value for n , i.e. the size of the corresponding matrix n n A  ) corresponding to the quadratic homogeneous equations of the type: 1,2,3,4,.., respectively. Subsequently, in Sec. 2-4, by solving the corresponding systems of

G

  is a symmetric and invertible square matrix, i.e.:

  represent the quadratic equation[START_REF] Kauffman | Knots and Physics[END_REF], based on the axiomatic relation (17-1)) could be also expressed by the homogeneous linear matrix form: the relevant Hermiticity and anti-Hermiticity conditions, we assume that any square matrix A in the deriving matrix equation: 0   M A (as the equivalent representation of quadratic equation (18)), should also has this additional property that by supposing:

  matrix equations (25-2) and (25-3) are equivalent (due to the assumption of arbitrariness of parameters

  matrix equations and are equivalent (due to the assumption of arbitrariness of parameters

  )where for simplicity in the indices of parameters i m , we may simply replace the arbitrary parameters 5 the quadratic equations[START_REF] Zenil | A Computable Universe: Understanding and Exploring Nature as Computation‖[END_REF] the corresponding system of linear equations is obtained uniquely as follows:

  n n A  ) corresponding to the well-known homogeneous equation of degree three: form (in compatible with the new axiom (17)):

  parametric solution (45) by the direct bijective replacements of six unknown variables ) , ( i i f e(where i = 0,1,2) with the six new variables of the type  h , given by:

  expanded parametric solutions of the type (45) (equivalent to the algebraic form (45-2)) are not derived.

  parametric solution[START_REF] Chapman | On the generalized Clifford algebra of a monic polynomial‖[END_REF] (which have been obtained as the solutions of quadratic equations (

  any given integer values. Therefore, using this common algebraic property of the parametric relations (43-2) and (44-2), the arbitrary parameter 1

0 m is the invariant mass of a relativistic particle2 2 N

 2 velocity(that is a time-like covariant vector) in the stationary reference frame,   and   ~are two contravariant N N square matrices (corresponding to the matrix representations of Clifford algebras Cℓ 1,2 , Cℓ 1,3 , Cℓ 1,4 ,…, Cℓ 1,N (for N ≥ 2) and their generalizations[START_REF] Zahedi | Linearization Method in the Ring Theory‖, Russian Academy of Sciences (RAS)[END_REF][START_REF] Tesser | Generalized Clifford algebras and their representations‖[END_REF][START_REF]Clifford Algebras with Numeric and Symbolic Computations‖[END_REF], see also Appendix A) that by the isomorphic linear relations[START_REF] Jackiw | Chern-Simons modification of general relativity[END_REF] are expressed in terms of two corresponding contravariant parametric column matrix. These matrices in (1+1), (1+2), (1+3), (1+4) and (1+5) space-time dimensions are given uniquely as follows, respectively:

  and except the last entry (represented by arbitrary parameter s ), all the other

  are the general relativistic forms of energy-momentum quantum operator (where   is the general covariant derivative, and  D is gauge covariant derivative, for detail see the ordinary tensor formalisms of these equations, representing by formulas (78-1) -(79-3), in Sec. 3-4-1), covariant velocity in stationary reference frame (that is a time-like covariant vector), square matrices (compatible with the matrix representations of certain Clifford algebras, see Sec. 3-3 and also Appendix A) defined by formulas (65) -(70) in Sec 3-3. In the field equation (72), Ψ R is a column matrix as a (first) quantized form of the algebraic column matrix S (defined by relations (64) -(70-1) in Sections 3-3, 3-3-1 and 3-3-2), determined and represented uniquely by formulas (73) -(77),…, in various space-time dimensions. The column matrix Ψ R contains the components of field strength tensor   R (equivalent to the Riemann curvature tensor), and also the components corresponding source current tensor by relation:

   (given in the field equations (

  tensor representations (i.e. the formulas (78-1) -(79-3))

J(

   are the source currents expressed, necessarily, in terms of the covariant quantities ) as the initially given quantities), respectively. For higher-dimensional space-times, the column matrices R  and F  (with similar formulations) are determined uniquely as well.

  generates the invariant mass of the gravitational field), and  A is the vector potential. Moreover, in equations (79-1) -(79-3) the general covariant derivative  D  has been defined specifically with the torsion field   Z (that generates the invariant mass of the field  F ).

.

  Hence, a new massive particle (corresponding to the U(1) symmetry group) would be identified with the invariant mass:

F

  and   R presented in Sec.[START_REF] Zahedi | On the Connection Between Methods of the Ring Theory and the Group Approach‖, Russian Academy of Sciences (RAS)[END_REF][START_REF] Zahedi | The Linearization Method Based on the Ring Theory and its Applications in Physics‖, Russian Academy of Sciences (RAS) Publs[END_REF][START_REF] Durbin | Modern Algebra: An Introduction‖[END_REF][START_REF] Dickson | History of the Theory of Numbers‖, Volume II: Diophantine Analysis[END_REF], where the tensor field  F (describing by general covariant field equation (72)) represents (asymptotically) solely a massive bispinor field of spin-1/2 particles (as a new general covariant massive formulation of the Dirac equation) in (1+2) space-time dimensions, and also represents a massive bispinor field of spin-1as new massive general covariant (matrix) formulations of both Maxwell and Yang-Mills field equations compatible with specified gauge symmetry groupsin (1+3) space-time dimensions; and tensor field   R (describing by general covariant field equation (71)) represents (asymptotically) solely a bispinor field of spin-3/2 particles (as a new massive general covariant form of the Rarita-Schwinger equation) in (1+2) space-time dimensions , and also represents a massive bispinor field spin-2 particles (equivalent to a generalized massive form of the Einstein equations) in (1+3) space-time dimensions.

  where the quantum operator T ˆis given uniquely by formula (84), i.e.:

  1+4)-dimensional space-time we have:

  the formulations of the derived time-reversal transformed column matrices R

  )). Below this conclusion (and subsequent remarkable results) is discussed in more detail. **********************************

  be assumed for the field strength tensors   R and 

  to the (1+3)-dimensional source-free cases of column matrices Ψ R and Ψ F (represented uniquely by formulas (75))

3 - 5 - 3 .

 353 ) (representing the spin-1 boson fields coupling to the spin-1/2 fermionic currents) is also compatible with the SU(2) L U(2) R and SU(3) symmetry groups. Moreover, based on these determined gauge symmetries for the derived fermion and boson field equations, four new charge-less spin-1/2 fermions (representing by -z e , z n ; z u , z d ‖, where two righthanded charhe-less quarks z u and z d emerge specifically in two subgroups with anti-quarks such that: right-handed particle of ordinary Z boson), are predicted uniquely by this new mathematical axiomatic approach. Showing that only a definite simultaneous combination of the quantum mechanical transformations C ˆ, P ˆ, T ˆ and Ch  (given uniquely by the matrix operators (83) -(87)) could be defined for the general covariant massive (tensor) field equations (71) and (72) with non-zero source currents:

(

  if assuming that the column matrices R  and F  are defined with field equations (

  ) (represented by the transformations R T ˆ and F T ˆ, where the quantum operator T ˆis given uniquely by formula (84), i.e.:

3 - 5 - 4 .

 354 Showing that only the left-handed particle (along with their complementary righthanded antiparticle) fields could be coupled to the corresponding source currents:

  transformations , Ĉ P ˆ, and T ˆ(83) -(85). Furthermore, in agreement and based on the definitions and properties of quantum operators C ˆ, P ˆ, T ˆ and matrix Ch  given by formulas (83) -(87), the left-handed and right-handed components of column matrices of the types (96-1) and (98-2) (representing the unique formulations of column matrices that could be defined in the field equations (71) and (72), as mentioned in Sec. 3-5-2) are defined solely as follows for the column matrices )

  obey the transformations (101-1) and (104-1) (as the necessary conditions given respectively by relations (101) and (104)). On the other hand, the right-handed components of particle fields respectively), don't obey the transformations (102-1) and (103-1) (as the necessary conditions given respectively by relations (102) and (

  as defined in Sec. 3-1-1) 0 m and  p are the particle's rest mass and momentum (3-momentum), form of the 3-velocity of particle in stationary reference frame. As it would be shown in the following, a crucial and essential property of the quadratic relation (108) is its invariance under a certain set of sign inversions of the components of particle's momentum:

  of the Clifford algebra Cℓ 1,2 ) defined solely by formulas (65) and (67), and parametric column matrix) (a

  w (i.e. having the same forms in all of the seven cases of column matrix ) (a S ). In addition, concerning the specific parametric expression (109-1) of column matrix ) (a S in the formulation of matrix equation (

  expression of each of the seven cases of algebraic matrix equation (109), definitely, is not a necessary issue at the present stage. However, since the corresponding momentum operator

‖

  corresponds to U(2) R group.

  matrices (compatible with matrix representations of the Clifford algebra Cℓ 1,3 ) defined solely by formulas[START_REF] Jackiw | Chern-Simons modification of general relativity[END_REF] and[START_REF] Bialinicki | The Maxwell wave function of the photon[END_REF], and parametric column matrices

2 .

 2 also in all of the eight simultaneous (different) field equations (112Following the definite formulations of set of seven field equations (114-1), and set of eight field equations (114-2 ) specified, respectively, by the transformations (114-1-1) -(114-1-7) and (114-2-1) -(114-2-8), these two sets of the field equations could be represented uniformly by the following general covariant field equations as well (defined solely in (1+3) space-time dimensions), respectively:

  the set of matrices (110-12) in Sec. 3-6-1-1, represents uniformly a combined gauge symmetry group of the form: SU(2) L U(2) R , where the subset of three matricesequivalent to the Gell-Mann matrices (representing the SU(3) gauge symmetry group):

9 )

 9 Now based on the definite matrix formalisms of the field strength tensors  Z and  G (114-7)

  photon (determined as a right-handed charge-less boson, compatible with the positive-frequency corresponding to the right-handed circular polarization state of photon), and Z W W , ,  

  The basic properties of the integral domain ℤ with binary operations

										(		,	) 	are represented as
	follows, respectively [5] (	 a i	,	a	j	,	a	k	,...	 ℤ):

2-1.

  ,

																4	12	3	14	2	15
	2	2	2	0	2	4	11	3	13	1	15	3	3	3	0	3	4	10	2	13	1	14
	4	4	4	0	5	3	10	2	11	1	12

  )), where this property generates a certain set of additional symmetries corresponding uniquely to the SU(2) L U(2) R symmetry group for spin-1/2 fermion fields (represented by two main groups of -1+3‖ generations, corresponding respectively to two subgroups of leptons and two subgroups of quarks), in addition to the SU(2) L U(2) R and SU(3) gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Moreover, based on these uniquely determined gauge symmetries, four new charge-less spin-1/2 fermions (representing by -z e , z n ; z u , z d ‖, where two right-handed charhe-less quarks z u and z d emerge specifically in two subgroups with anti-

	quarks such that: ( by " , , " Z W W   	s	,	, u	b	, z u ) and (	c	,	d	,	t	, z d )), and also three new massive spin-1 bosons (representing

1 0 are arbitrary parameters. In the matrix representation, the parametric solution[START_REF]Clifford Algebras with Numeric and Symbolic Computations‖[END_REF] is represented as follows:

1 0) given on the basis

Acknowledgment

Special thanks are extended to Prof. and Academician Vitaly L. Ginzburg (Russia), Prof. and Academician Dmitry V. Shirkov (Russia), Prof. Leonid A . Shelepin (Russia), Prof. Vladimir Ya. Fainberg (Russia), Prof. Wolfgang Rindler (USA), Prof. Roman W. Jackiw (USA), Prof. Roger Penrose (UK), Prof. Steven Weinberg (USA), Prof. Ezra T. Newman (USA), Prof. Graham Jameson (UK), Prof. Sergey A. Reshetnjak (Russia), Prof. Sir Michael Atiyah (UK) (who, in particular, kindly encouraged me to continue this work as a new unorthodox mathematical approach to fundamental physics), and many others for their support and valuable guidance during my studies and research. ************************

--Forces.of.Nature.Laws-Jan.2015-signed.pdf , https://eprints.lib.hokudai.ac.jp/dspace/handle/2115/59279, https://cds.cern.ch/record/1980381, https://ui.adsabs.harvard.edu/#abs/2015arXiv150101373Z, https://inspirehep.net/record/1387680 ,

S S [START_REF] Jaffe | Quantum Yang-Mills theory[END_REF] where

are arbitrary parameters.

Similar to the formulations (66) - [START_REF] Jaffe | Quantum Yang-Mills theory[END_REF], for the higher dimensional cases of invariant quadratic relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF], the column matrix S and square matrices   and    (defining the square matrices   and    that correspond to the matrix representations of Clifford algebras and their generalization, see Sec. 3-3 and also Appendix A) are obtained with similar algebraic structures, where in (1+N) space-time dimensions the size of square matrices   and    is N N 2 2  and the size of column matrix S is 1 2  N .

3-3-1. General algebraic formulation of the column matrix S given in the matrix equation (64)

As noted in Sec. 3-3, the matrix equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF] represents uniquely the equivalent form of the Lorentz invariant energy-momentum quadratic relation [START_REF] Kaye | Models of Peano arithmetic‖[END_REF] (as the norm of the D-momentum), based on the axiomatic relations [START_REF] Noyes | Some thoughts on discrete physics and the reconstruction of quantum mechanics‖[END_REF] and and relevant general results obtained in Sections 2-2 and 2-4 for homogeneous quadratic equations over the integral domain over ℤ. Hence (as it has been also mentioned in Sec. , the general algebraic formulation of the entries of column matrices S obtaining in subsequent higher space-time dimensions, are similar to formulations of the obtained matrices S (66) - [START_REF] Jaffe | Quantum Yang-Mills theory[END_REF] corresponding, respectively, to the (1+0), (1+1), (1+2), (1+3), (1+4) and (1+5)-dimensional cases of Lorentz invariant matrix equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF]. Hence, the algebraic formulation of column matrix S in (1+N) space-time dimensions would be generally defined as follows: the last entry of S is represented solely by the arbitrary parameter s , entries of S could be represented uniformly by the following unique algebraic formulation (expressing in terms of the arbitrary parameters:

,..., , , , treated as an independent tensor field, or equivalently, as part of the space-time geometry [72 -74]), it could be concluded directly that torsion field   T generates the invariant mass of corresponding gravitational field, and torsion field   Z generates the invariant mass of corresponding gauge field, respectively. Hence, based on our axiomatic derivation approach including the mappings (71-1) and (72-1) (mentioned above), the (1+1), (1+2), (1+3), (1+4), (1+5),…, dimensional cases of column matrices Ψ

R

and Ψ

F in the specific expressions of general covariant massive (tensor) field equations ( 71) and ( 72), are determined uniquely as follows, respectively; For (1+1)-dimensional space-time we have: (100), in particular, is a direct consequent of the above conditional expressions for source currents

As noted in Sec. 3-4-1, these relations appear as necessary conditions in the course of the axiomatic derivation of general covariant field equations ( 71) and [START_REF] Hehl | Four lectures in Poincare gauge field theory[END_REF]. In fact, in the field equations [START_REF] Sushkov | New Experimental Limits on Non-Newtonian Forces in the Micrometer Range[END_REF] ) (

Now based on these conditional expressions in addition to this natural and basic circumstance that the source currents ) ( R J  and ) ( F J  should be also transferred respectively as a rank three tensor and a vector, under the parity, time-reversal and charge conjugation transformations (defined by formulas (83) -( 85)) of the field equations ( 71) and ( 72), it would be concluded directly that the transformations (99) and (100) are the only simultaneous combinations of transformations , Ĉ P ˆ, T ˆ (also including the matrix

Ch



, necessarily, as it would be shown in the following paragraph), which could be defined for the field equations ( 71) and ( 72) with -non-zero‖ source currents. 99) and (100), follows simply from the basic arguments presented in Sec. 3-5-2. In fact, in these uniquely determined combinations, the simultaneous multiplication by matrix Ch  (from the left) is a necessary condition for that the transformed column matrices:

Secondly, appearing the matrix operator

)

(given in the transformations (99) and (100)) could be also defined in the field equations ( 71) and ( 72), based on the formulations of column matrices (96-1) and (98-2), as mentioned in Sec. 3-5-2) (however, it is worth to note that this argument is not merely limited to the definability of column matrices of the types (96-1) and (98-2), and it could be also represented on the basis of unique formulations of all the originally derived column matrices ( 73) -( 77),… corresponding to various space-time dimensions).

In the next Section, we show how the 'CPT' theorem in addition to the unique formulations of the combined transformations (99) and (100) (representing the only definable transformation forms, including C, P and T quantum mechanical transformations, for the field equations ( 71) and ( 72) with -non-zero‖ source currents), imply only the left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents.

Now based on the matrix formulation of field strength tensor  F (defined by the general covariant field equation (110-9)), and on the basis of C, P and T symmetries of this field equation (as a particular form of the (1+2)-dimensional case of field equation ( 72)) that have been represented basically by their corresponding quantum operators (in Sections 3-5 -3-5-4), it would be concluded that the general covariant field equation (110-9) describes uniformly a group of seven spin-1/2 fermion fields corresponding to, respectively: -three left-handed fermions (for Furthermore, as it would be shown in Sections 3-6-3 -3-6-3-2, as a natural assumption, by assuming the seven types of spin-1/2 fermion fields that are described by general covariant field equation (110-9), as the source currents of spin-1 boson fields (that will be represented by two determined unique groups describing respectively by general covariant field equations (114-4) and (114-5), in Sec. 3-6-3-2), it would be concluded that there should be, in total, four specific groups of seven spin-1/2 fermion fields (each) with certain properties, corresponding to -1+3‖ generations of four fermions, including two groups of four leptons each, and two groups of four quarks each. Moreover, based on this basic circumstances, two groups of leptons would be represented uniquely by: -[( , z d ), could explain the baryon asymmetry, and subsequently, the asymmetry between matter and that is equivalent to:

, where (similar to the (1+2)dimensional case in Sec. 3-1-1) 0 m and  p are the particle's rest mass and momentum (4-momentum), The first group includes,

and the second group is given by, respectively: S also includes eight cases with specific parametric formulations expressed respectively in terms of eight groups of independent arbitrary parameters:

, and two common arbitrary parameters s and w (with the same forms in all of seven cases of the column matrix S in the formulation of matrix equations (112-1) and (112-2), have been determined specifically by assuming (as a basic assumption in addition to the systematic natural approach of formulating the matrix equations (112-1) and (112-2), based on the definite formulation of axiomatically determined matrix equation ( 64)) the minimum value for total number of arbitrary parameters in both column matrices S , which implies equivalently the minimum value for total number of arbitrary parameters in all of the fifteen simultaneous (different) cases of matrix equations (112-1) and (112-2) (necessarily with fifteen independent parametric solutions representing totally a certain set of fifteen different equivalent forms of the general parametric solution of quadratic relation (111), based on the general conditions of basic definition of the systems of linear equations corresponding to homogeneous quadratic and higher degree equations, presented in Sec. 2, and Sections 2-2 -2-4, 3-1-1 concerning the homogenous quadratic equations).In the following, similar to the fundamental general covariant field equation (109-2) derived in Sec. 3-6-1-1, in the derivation of the corresponding field equations (from matrix equations (112-1) and (112-2), respectively), we will also use the above particular algebraic properties of parameters s and s which, respectively, have been expressed commonly in the expressions of all of seven simultaneous cases of matrix equation (112-1), and in the expressions of all of eight cases of matrix equation (112-1).

3-6-3-1. Moreover, similar to the invariant relation (108) and derived matrix equation (109), along with the transformations (111-1) -(111-15) and algebraic matrix equations (112-1) and (112-2), using the corresponding complex transformations (111-17), we may also formally have the following equivalent matrix equations (with the complex expression), respectively: and photon, and also eight gluon fields (with their known definite properties, including the color charges represented by ‗color octet' [START_REF] Weinberg | The Quantum Theory of Fields[END_REF][START_REF] Pal | An Introductory Course of Particle Physics[END_REF]). In addition, based on the group representation of three additional new bosons that are predicted uniquely by this new mathematical axiomatic approach, denoting by: Moreover, following the axiomatic approach of derivation of matrix equation [START_REF] Deser | Three-Dimensional Massive Gauge Theories[END_REF], matrices