Frédéric Patras

Dynkin operators and renormalization group actions in pQFT

Renormalization techniques in perturbative quantum field theory were known, from their inception, to have a strong combinatorial content emphasized, among others, by Zimmermann's celebrated forest formula. The present article reports on recent advances on the subject, featuring the role played by the Dynkin operators (actually their extension to the Hopf algebraic setting) at two crucial levels of renormalization, namely the Bogolioubov recursion and the renormalization group (RG) equations. For that purpose, an iterated integrals toy model is introduced to emphasize how the operators appear naturally in the setting of renormalization group analysis. The toy model, in spite of its simplicity, captures many key features of recent approaches to RG equations in pQFT, including the construction of a universal Galois group for quantum field theories.

Introduction

Renormalization techniques in perturbative quantum field theory (pQFT) were known, from their inception, to have a strong combinatorial content emphasized, among others, by the celebrated Zimmermann's forest formula. A. Connes and D. Kreimer and, more recently, A. Connes and M. Marcolli, have brought forward that one should try to understand the physics underlying these phenomena by means of tools of noncommutative geometry. Their understanding that a Hopf algebra structure on Feynman graphs underlies the forest formula, their extension to prounipotent groups of the classical Birkhoff decomposition of loops and their discovery of a universal Galois group underlying renormalization are three milestones for our current mathematical understanding of the theory [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem I. The Hopf algebra structure of graphs and the main theorem[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group[END_REF][START_REF] Connes | Renormalization and motivic Galois theory[END_REF][START_REF] Connes | From Physics to Number Theory via Noncommutative Geometry II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory[END_REF][START_REF] Connes | Quantum Fields and Motives[END_REF].

However useful, deep and possibly leading to a new understanding of fundamental interactions, this noncommutative picture has certain drawbacks. For example, a key ingredient of the noncommutative approach is the hypothesis that the local geometry of the fundamental interactions around the spacetime dimension, as encoded in the dimensional regularization (DR) process, governs the physics. How far this assumption is valid is unclear (at least to the author of the present article). Indeed, although DR has particularly nice properties (e.g. with respect to the various gauge symmetries of quantum field theories), all the other usual regularization processes and renormalization schemes that have apparently nothing to do with the geometrical phenomena arising around the space-time dimension (Pauli-Villars, zero-momentum subtraction, cut-offs, and so on) lead ultimately to the same results [START_REF] Itzykson | Quantum field theory[END_REF]. This suggests that, besides geometry, classical combinatorics and algebra should also contribute in an essential way to the renewal of the mathematical foundations of pQFT.

The purpose of the present article is to report on advances related to this research program, emphasizing the ubiquity of the Dynkin operator, especially from the point of view of the renormalization group (RG) analysis of regularized and renormalized amplitudes. To make clear the relevance of the operator for the RG analysis, we introduce a simple toy model for RG actions, namely by studying how solutions of linear differential equations are changed when a rescaling of the equation (reminiscent of the rescaling of energy scales in the usual RG analysis of pQFT) is performed. As we shall see, the corresponding RG equations encode exactly our Dynkin operator analysis of the beta function in pQFT [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalization[END_REF]. We strongly believe that, because of this phenomenon, because of the ubiquity of the Dynkin operator, and because of the relative simplicity and tractability of the underlying algebra, this approach can be meaningful beyond the classical setting of pQFT.

The article is organized as follows. The first section shows how the classical Dynkin operator shows up naturally when performing a RG analysis of solutions of linear differential equations. The second one recalls briefly some general facts about Hopf algebras and renormalization in pQFT, emphasizing the Bogoliubov recursion. The purpose of the third one is twofold. We show how the constructions in the first section lift to the algebraic setting of graded connected Hopf algebras. In the process we recall the construction of a universal Galois group acting on arbitrary graded Hopf algebras. Letting this group act on the particular class of Hopf algebras of Feynman graphs, one recovers the Connes-Marcolli universal Galois group of pQFT. The fourth section emphasizes the role of the Dynkin operator as a universal operator for RG analysis. As an example, we show how the Bogoliubov recursion can be understood that way. The last one focuses on the RG analysis of regularized and renormalized amplitudes in pQFT. Here, the striking result is that the beta function (the generator of the RG equation, where the RG action is obtained by a rescaling of the mass scale) is best understood by a RG analysis of the counterterm (where the RG action on the counterterm is driven now by the number of loops in Feynman diagrams and not any more by a rescaling of mass scales).

This article is largely based on joint works with Ch. Reutenauer [START_REF] Patras | Higher Lie idempotents[END_REF][START_REF] Patras | On Dynkin and Klyachko idempotents in graded bialgebras[END_REF][START_REF] Patras | Lie representations and an algebra containing Solomon's[END_REF] and K. Ebrahimi-Fard, J. Gracia-Bondìa and D. Manchon [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalization[END_REF][START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new identities[END_REF][START_REF] Ebrahimi-Fard | A Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogolioubov's counterterm recursion[END_REF][START_REF] Ebrahimi-Fard | New identities in dendriform algebras[END_REF]. It has benefited from many discussions with J.-Y. Thibon and owes a tribute to the theory of noncommutative symmetric functions as developed in [START_REF] Gelfand | Noncommutative symmetric functions[END_REF]. Precise references to these works will be given in the article. A warm thanks goes to them for fruitful ongoing collaborations and particularly to K. Ebrahimi-Fard and J. Gracia-Bondìa for several comments on a first version of this article.

RG analysis of differential equations

Let us consider the evolution equation:

A ′ (t) = A(t)H(t)
with initial condition A(0) = 1, where A(t) and H(t) belong to a matrix algebra or, more generally, to an operator algebra. The solution is given by the time-ordered exponential series:

T (exp t 0 H(x)dx) := 1 + t 0 dt 1 H(t 1 ) + ... + ∆ n t dt 1 . . . dt n H(t 1 ) . . . H(t n ) + . . . where ∆ n t := {0 ≤ t 1 ≤ • • • ≤ t n ≤ t}.
The general idea behind RG analysis is to study the dependency of a physical system on changes of scales. To put the emphasis on this dependency, we will call systematically (and most probably slightly abusively) the corresponding differential equation the RG equation. In the present situation, one of the simplest possible changes of energy scales consists in rescaling the operator

H(t) to H λ (t) := λH(t), with λ ∈ R * . Writing A λ (t) for the solution of A λ ′ (t) = A λ (t)H λ (t), we get A λ (t) = T (exp t 0 H λ (x)dx).
The problem becomes algebraically interesting and nontrivial when we try to derive a renormalization group equation, that is a differential equation solved by A λ (t) with respect to the perturbation parameter λ. As we shall see later, the computation parallels exactly the one for the RG equations solved by regularized and renormalized amplitudes in pQFT.

In order to derive this differential equation, let us recall a few elementary facts on permutations and iterated integrals. For further details on descents in symmetric groups, we refer to [START_REF] Reutenauer | Free Lie algebras[END_REF] and [START_REF] Blessenohl | Noncommutative character theory of the symmetric group[END_REF]. The more algebraically oriented reader may also refer to the Hopf algebraic approach in [START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF] that will underly most of our forthcoming constructions.

Definition 1.1. The descent set of a permutation σ ∈ S n is the subset of [n -1] defined by:

Desc(σ) := {i < n, σ(i) > σ(i + 1)}.
With the notation σ = (σ(1), . . . , σ(n)), we get, for example: Desc((42315)) = {1, 3}. In general, to each subset S of [n -1] are associated two corresponding "Solomon elements" in Q[S n ]:

D (n) =S := σ∈Sn,Desc(σ)=S σ, D (n) S := σ∈Sn,Desc(σ)⊆S σ.
When no confusion can arise, we write simply D =S and D S for D

D S = T ⊆S D =T , D =S = T ⊆S (-1) |S|-|T | D T .
With the notation

H σ := [ ∆ n t dt 1 . . . dt n H(t 1 ) . . . H(t n )] • σ := ∆ n t dt 1 . . . dt n H(t σ(1) ) . . . H(t σ(n) )
and the extension of the notation to elements

β ∈ C[S n ], β = σ∈Sn µ σ • σ, H β := [ ∆ n t dt 1 . . . dt n H(t 1 ) . . . H(t n )] • β := σ∈Sn µ σ [ ∆ n t dt 1 . . . dt n H(t 1 ) . . . H(t n )] • σ,
the product of k iterated integrals reads:

H n1 H n2 ...H n k = H n1+...+n k • D (n1+...+n k ) {n1,n1+n2,...,n1+...+n k-1 } = H D (n 1 +...+n k ) {n 1 ,n 1 +n 2 ,...,n 1 +...+n k-1 }
where we have written H p for H Idp , Id p = (1...p) ∈ S p . We omit the proof that follows easily by splitting the integration domain into simplices. The construction underlies the classical Mielnik-Plebański [START_REF] Mielnik | Combinatorial approach to Baker-Campbell-Hausdorff exponents[END_REF] explicit solutions to the continuous and discrete Baker-Campbell-Hausdorff formulas by means of Solomon's eulerian idempotent (the logarithm of the identity in the convolution algebra of the symmetric groups group algebras, see [START_REF] Solomon | On the Poincaré-Birkhoff-Witt theorem[END_REF][START_REF] Reutenauer | Theorem of Poincaré-Birkhoff-Witt, logarithm and representations of the symmetric group whose orders are the Stirling numbers[END_REF][START_REF] Patras | Homothéties simpliciales[END_REF][START_REF] Patras | La décomposition en poids des algèbres de Hopf[END_REF][START_REF] Reutenauer | Free Lie algebras[END_REF]). More generally, we have: .

H D (n 1 ) S 1 H D (n 2 ) S 2 = H D (n 1 +n 2 ) S 1 {n 1 } S 2
Let us recall now the definition of the Dynkin operator. From the point of view of Lie theory, this is the element D n of Z[S n ] such that, for an arbitrary sequence of elements x 1 , ..., x n in an associative algebra A:

[. . . [x 1 , x 2 ], ..., x n ] = x 1 ....x n • D n
where [x, y] = xyyx is the Lie bracket on A and where the symmetric group acts on the right:

x 1 ....x n • σ := x σ(1) ...x σ(n) ∀σ ∈ S n . For example, D 2 = (12) -(21) = D ∅ -D ={1} ∈ Z[S 2 ] and D 3 = (123) -(213) -(312) + (321) = D ∅ -D ={1} + D ={12} since [x 1 , x 2 ] = x 1 x 2 -x 2 x 1 , [[x 1 , x 2 ], x 3 ] = x 1 x 2 x 3 -x 2 x 1 x 3 -x 3 x 1 x 2 + x 3 x 2 x 1 .
In general, an easy recursion shows that D n = n-1 i=0 (-1) i D ={1,...,i} .

Theorem 1.1. The RG equation for A λ (t) reads:

λ • ∂ ∂λ A λ (t) = A λ (t) • β(t)
where the beta function (the infinitesimal generator of the RG equation) is given by

β(t) := ∞ n=1 H λ n • D n .
Indeed, let us expand the left and right-hand sides of the equation into homogeneous components with respect to the powers of λ. The component of degree n in the left hand side is simply nH λ n (t). In the right-hand side, we get:

n-1 i=0 H λ i • (H λ n-i • D n-i ).
Expanding this sum according to the product rule for iterated integrals, we get:

n-1 i=0 n-i-1 j=0 (-1) j H λ i • H λ D ={1,...,j} = n-1 j=0 (-1) j H D ={1,...,j} + n-1 i=1 n-i-1 j=0 (-1) j H λ D ={i,i+1,...,i+j} + n-1 i=1 n-i-1 j=0 (-1) j H λ D ={i+1,...,i+j} = H D λ n + n-1 j=1 (-1) j H D ={1,...,j} + n-2 j=0 (-1) j H D ={1,...,1+j} + n-1 i=2 n-i-1 j=0 (-1) j H λ D ={i,i+1,...,i+j} + n-1 i=1 H λ Dn + n-2 i=1 n-i-1 j=1 (-1) j H λ D ={i+1,...,i+j} = n • H λ n
from which the Theorem follows.

A striking corollary of the RG equation for A λ (t) is that A λ (t) and the corresponding beta function β(t) satisfy a formula similar to the one for the universal singular frame in the noncommutative geometrical and motivic Galoisian approach to pQFT [START_REF] Connes | Renormalization and motivic Galois theory[END_REF]. A similar identity also holds in the setting of noncommutative symmetric functions [START_REF] Gelfand | Noncommutative symmetric functions[END_REF]. This is not a mere coincidence, since identities involving Solomon elements in descent algebras translate into properties of Hopf algebras [START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF] and noncommutative symmetric functions [START_REF] Gelfand | Noncommutative symmetric functions[END_REF], and since Hopf algebras provide the right algebraic framework to understand the combinatorial structures of pQFT [START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem I. The Hopf algebra structure of graphs and the main theorem[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group[END_REF] -we postpone an explanation of these phenomena to the forthcoming sections of the article.

Corollary 1.2. Let us write

A λ (t) = n A λ n (t) and β(t) = n β n (t)
, where A λ n (t) and β n (t) refer to the homogeneous component of degree n (with respect to the parameter λ) in A λ (t) and β(t). We have:

A λ n (t) = l≤n k1+...+k l =n,ki>0 β k1 (t)...β k l (t) k 1 (k 1 + k 2 )...(k 1 + ... + k l ) Indeed, since kA λ k (t) = k i=1 A λ k-i (t)β i (t), we have A λ n (t) = βn(t) n + n-1 i=1 1≤j≤n-1 A λ n-i-j (t)βj (t)βi(t) (n-i)n
. Substituting iteratively the value of A λ k (t) in the right hand side of the identity, we arrive at the required identity:

A λ n (t) = l≤n k1+...+k l =n,ki>0 β k1 (t)...β k l (t) k 1 (k 1 + k 2 )...(k 1 + ... + k l )
.

Hopf algebras and RG analysis in QFT

The present, short, section will serve as a reminder of the algebraic structures underlying the renormalization process and as a motivation for the constructions in the next sections. There are several good accounts of the subject, both classical (such as Collins' [START_REF] Collins | Renormalization[END_REF]) and recent (such as Manchon's [START_REF] Manchon | Bogota lectures on Hopf algebras, from basics to applications to renormalization[END_REF] and Figueroa and J. M. Gracia-Bondía's [START_REF] Figueroa | Combinatorial Hopf algebras in quantum field theory I[END_REF]). For that reason, we omit details and limit our account to the leading ideas. Readers familiar with the subject should skip it and proceed directly to the next section. The others should hopefully get insights into the logic behind the renormalization process and some intuition why the algebraic constructions that will be developed later make sense for pQFT.

The main problem of renormalization is to associate a finite quantity to the divergent integrals associated by certain rules (the Feynman rules) to Feynman diagrams (precisely one-particle-irreducible, superficially divergent Feynman diagrams). Using these quantities, physically relevant quantities (such as scattering amplitudes) may be computed through the LSZ reduction formulas [START_REF] Itzykson | Quantum field theory[END_REF]. Concretely, the Feynman diagrams are build out of vertices with prescribed numbers of edges. The vertices represent the fundamental interactions of the theory, whereas the edges stand essentially for the propagation of particles between two interactions. The diagrams arise from a perturbative expansion, and the orders of the expansion correspond naturally to the number of loops in the diagrams.

At first order these quantities (the amplitudes) look like:

φ(m) = R 4 d 4 q (q 2 + m 2 ) 2
and divergences show up. Various techniques have been deviced to remove this kind of divergences. The process is usually a twofold one. First, one introduces one (or several) non physical regularization parameter(s) (regularization step), and the integrals are expanded according to this parameter. An asymptotic study then reveals the structure of the divergences; their removal produces finite quantities (subtraction step). Renormalization group equations arise from the study of the dependence of the solutions in this/these extra parameter(s). Together with the derivation of finite quantities for a given value of the parameter, the RG equations are at the core of the renormalization process.

In practice, there are several techniques (cut-off, Pauli-Villars, Taylor subtraction, dimensional regularization (DR),...). Most of them give satisfactory results (see e.g. [START_REF] Itzykson | Quantum field theory[END_REF]) but (excepted maybe for the cut-off techniques), their physical grounds are still not plainly understood. One of the hopes raised by the noncommutative geometrical approach to renormalization was precisely to pave the way for a new understanding of these phenomena. Although some ideas presented below hold for other schemes than DR (this was proved explicitly for Taylor subtraction in [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalization[END_REF]), we will focus on DR which is certainly the theory most easily dealt with from the mathematical point of view, besides presenting many nice features such as compatibility with gauge invariance.

In the DR process, a regularization parameter ǫ is introduced that should be understood as an infinitesimal deformation of the space-time dimension:

4 D = 4 -ε so that d 4 q is changed to d D q.
A typical dimensional regularization computation then reads, for φ(m) as above [START_REF] Zeidler | Quantum Field Theory I[END_REF],

φ reg (m; ǫ, µ) := iµ ǫ R 4-ǫ d 4-ǫ q (q 2 + m 2 ) 2 = iπ 2 Γ( ǫ 2 )( mπ 1 2 µ ) -ǫ = iπ 2 ( 2 ǫ -C -2 ln( mπ 1 2 µ )) + o(1)
where C stands for Euler's constant and o(1) refers to ǫ -→ 0. A simple subtraction step (or "minimal subtraction", another example is treated in [START_REF] Zeidler | Quantum Field Theory I[END_REF]) consists in removing the divergence. The substracted divergent part φ -(m; ǫ, µ) := -iπ 2 ( 2 ǫ ) is called the counterterm. We will refer to the remaining quantity 1) and to its constant part φ ren (m; ǫ, µ) := φ + (m; 0, µ) as the renormalized amplitudes. A concrete example (the Ginzburg-Landau φ 4 4 scalar model in Euclidean field theory) is treated explicitly in [START_REF] Figueroa | Combinatorial Hopf algebras in quantum field theory I[END_REF], to which we refer, also for precise explanations on notions such as superficial degree of divergence or subdivergences of a Feynman diagram, which we omit to explain but are essential to the understanding of these phenomena. An important point to notice is that, due to the particular structure of DR, an extra "mass scale" parameter µ is needed in the regularization process to insure that the physical dimension of the regularized amplitude remains correct. In DR, the RG analysis is conducted according to this mass scale parameter.

φ + (m; ǫ, µ) := iπ 2 (-C -2 ln( mπ 1 2 µ )) + o(
The process we considered (regularization, asymptotic expansion, removal of the divergence by minimal subtraction) describes correctly what happens at the first order of the expansion (the one-loop order). At higher orders, the regularization principle remains unchanged, but the removal of the singular part (arising from the removal of divergences in the asymptotic expansion) is made more complex. Roughly, some subdiagrams γ of a given Feynman Γ diagram may contribute directly to make the overall integral associated to Γ divergent. These subdivergences have to be removed recursively in order to make the overall renormalization process physically consistent. The recursion is conducted according to the partial order of (sub)diagrams inclusions. This is the purpose of the Bogoliubov recursion, which is best explained in terms of a Hopf algebra structure on diagrams.

This Hopf algebra H is defined as the polynomial algebra generated by the empty set ∅ and the connected proper (or 1PI) Feynman graphs that are (superficially) divergent and/or have (superficially) divergent subdiagrams, with set union as the product operation. The empty set is therefore the unit. The product is written π. Since H is freely generated as a (graded) commutative algebra by connected proper diagrams (a diagram with n loops is attributed the degree n), the coproduct is well defined once it is defined on connected proper diagrams Γ. We get:

∆(Γ) := ∅⊆γ⊆Γ γ ⊗ Γ/γ.
The sum is over all divergent, proper, not necessarily connected subdiagrams of Γ. The map ∆ is coassociative and an algebra map from H to H ⊗ H, making H a true Hopf algebra (the existence of a unit, of a counit and of an antipode follows immediately from the graded structure of the product and the coproduct on H). The counit (which maps ∅ to 1 and the nontrivial graphs to 0 is written η). We refer to [START_REF] Figueroa | Combinatorial Hopf algebras in quantum field theory I[END_REF] for the necessary refinements (e.g. how tadpoles should be handled).

We are now in the position to introduce the setting of the group-theoretic approach to renormalization. Since bare Feynman rules lead to divergences, we have to deal with regularized Feynman rules φ reg . Such a rule defines a map from the set F of (connected, proper) Feynman diagrams to the algebra of Laurent series A := C[ǫ -1 , ǫ]] and extends uniquely, since H is a polynomial algebra over F , to a A-valued character of H, that is, a multiplicative map from

H to C[ǫ -1 , ǫ]]. Equivalently, φ reg it is an element of the prounipotent affine group G C[ǫ -1 ,ǫ]] (H) of A-valued characters on H,
where the group structure is induced by the convolution of linear maps from

H to C[ǫ -1 , ǫ]]: f * g : H ∆ → (H ⊗ H) f ⊗g → (C[ǫ -1 , ǫ]] ⊗ C[ǫ -1 , ǫ]]) π → C[ǫ -1 , ǫ]],
where π stands for the product map.

The Bogoliubov recursion is a process allowing the inductive construction of a decomposition

φ reg = φ -1 - * φ + ,
where φ -∈ G -(A) and φ + ∈ G + (A) are respectively the counterterm character and the renormalized character (they formalize the notion of counterterm and renormalized amplitude, as introduced previously). Here, G -(A) (resp. G + (A)) stands for the group of algebra maps γ from H to C[ǫ -1 ] such that, for any Feynman diagram Γ, γ(Γ) ∈ A -:= ǫ -1 C[ǫ -1 ] (resp. the group of algebra maps from H to

A + := C[[ǫ]]).
Here, the crucial point, first noticed by Ch. Brouder and D. Kreimer [21, Eq.( 55)] is that the projection map R of Laurent series on their divergent part (orthogonally to C[[ǫ]]):

R : A = C[ǫ -1 , ǫ]] -→ A - satisfies the relation: R(xy) + R(x)R(y) = R(R(x)y + xR(y)).
That is, R defines a weight one Rota-Baxter algebra structure on A [START_REF] Rota | Baxter algebras and combinatorial identities. I,II[END_REF][START_REF] Rota | Fluctuation theory and Baxter algebras[END_REF]. This relevance of Rota-Baxter algebra structures for pQFT was first noticed by K. Ebrahimi-Fard and brought into full light in his Thesis (and in a series of papers) [START_REF] Ebrahimi-Fard | RotaBaxter Algebras and the Hopf Algebra of Renormalization[END_REF], stimulating many recent works. Many ideas developed in this article do actually hold for Rota-Baxter algebras of an arbitrary weight θ, that is for algebras provided with an operator R such that the following identity holds:

θR(xy) + R(x)R(y) = R(R(x)y + xR(y)).
We refer e.g. to [START_REF] Ebrahimi-Fard | Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory[END_REF][START_REF] Ebrahimi-Fard | A Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogolioubov's counterterm recursion[END_REF], also for further historical references on the subject.

The Bogolioubov recursion now reads:

φ -= -R(φ), φ + = (1 -R)(φ)
where φ is Bogoliubov's preparation map:

φ = φ - * (φ reg -η)
Equivalently, φ reg = φ -1 - * φ + and φ -solves:

φ -= 1 -R(φ - * (φ reg -η))
The recursion process is by induction on the number of loops in diagrams. The fact that the recursion defines elements of G -(A) and G + (A) is not obvious from the definition and follows from the multiplicativity constraints, as was shown by Kreimer and Connes-Kreimer, see e.g. [START_REF] Kreimer | Chen's Iterated Integral represents the Operator Product Expansion[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem I. The Hopf algebra structure of graphs and the main theorem[END_REF][START_REF] Ebrahimi-Fard | The combinatorics of Bogoliubov's recursion in renormalization[END_REF] and the references therein, also for complementary insights on the Bogoliubov recursion -such as its solution by means of the so-called BCH recursion.

The remaining part of the article will be focused on showing how both the Bogoliubov recursion and the RG equations (the study of the dependency of φ + and φ -on the mass scale parameter) can be understood by means of the Dynkin operator, and how our RG equation for solutions of linear differential equations generalizes in this setting.

A universal Galois group for Hopf algebras and pQFT

The purpose of the present section is to show how to construct a universal Galois group acting on Hopf algebras. The construction goes back to [START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF] and is encoded in the notion of the "descent algebra of a Hopf algebra".

Notice immediately that, since Feynman diagrams are organized into a Hopf algebra, the construction of a universal Galois group acting on Hopf algebras incorporates as a particular case the notion of a universal Galois group for pQFT, as suggested by P. Cartier and constructed by A. Connes and M. Marcolli. This was first noticed in [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalization[END_REF]. This observation has a further interest: indeed, the structure of the universal Galois group of the theory of Hopf algebras carries a very rich structure. Among others, besides a graded Hopf algebra structure on its algebra of coordinates Desc * (the graded dual of the descent algebra, to be introduced below), one can show that each of the graded components of Desc carries another associative product that has nice compatibility properties with respect to the Hopf algebra structure [START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF][START_REF] Gelfand | Noncommutative symmetric functions[END_REF][START_REF] Patras | On Dynkin and Klyachko idempotents in graded bialgebras[END_REF]. One can expect these extra-structure to be meaningful for pQFT.

Let us formalize first the computations on iterated integrals in the first section of the article. For a standard approach to the descent algebra, we refer to [START_REF] Reutenauer | Free Lie algebras[END_REF]. • The product, written * on Desc is given by:

D (n) S * D (m) T := D (n+m) S {n} T +n • The coproduct ∆ is defined by requiring that the D (n) ∅ form a sequence of divided powers, that is, ∆(D (n) ∅ ) := n i=0 D (i) ∅ ⊗ D (n-i) ∅
A fundamental Theorem, due to Solomon, shows that the graded components Desc n ⊂ Q[S n ] of Desc are subalgebras of Q[S n ] (where the product is induced by the product of permutations in S n ). The algebra Desc n is usually referred to as Solomon's algebra of type A n (see [START_REF] Solomon | A Mackey formula in the group ring of a Coxeter group[END_REF]).

The choice of the product * is dictated by the product formulas for iterated integrals. The reasons for the choice of the coproduct are explained in [START_REF] Patras | On Dynkin and Klyachko idempotents in graded bialgebras[END_REF]. They insure that the Hopf algebra structure on Desc is compatible with the action on Hopf algebras, as described in the next Theorem. Definition 3.2. The universal Galois group Gal Hopf of Hopf algebra theory is the set of group-like elements in the Hopf algebra Desc, that is:

Gal Hopf := {x ∈ Desc, ∆(x) = x ⊗ x}
This terminology (which is not standard) is chosen here to emphasize the connexion with the (canonically isomorphic) universal Galois group of renormalization (or motivic Galois group) of [START_REF] Connes | Renormalization and motivic Galois theory[END_REF].

Theorem 3.1.

Let H = n∈N H n be an arbitrary graded connected commutative (resp. cocommutative) Hopf algebra.

The descent algebra Desc and the universal Galois group Gal Hopf act naturally on H. Moreover, the Solomon algebra of type A n (resp. the opposite algebra) acts naturally on H n .

The construction of the action is fairly straightforward: as an obvious corollary of the definition of the product * , the descent algebra is a free associative algebra over the D

m =n H m ).
Here, the definition of the convolution product in End(H) mimics the definition of the convolution of linear morphisms on H as introduced in the previous section and is omitted. Then, since Desc is free associative, there is a unique algebra map from Desc to Desc H ⊂ End(H). This induces a natural action of Desc on H.

The tricky point is the verification that the various structures existing on the descent algebra (such as the product on Desc n or the coproduct) are compatible in a natural way with this action. For these compatibilities, we refer to [START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF][START_REF] Patras | On Dynkin and Klyachko idempotents in graded bialgebras[END_REF].

A nice corollary of this construction, that we mention for the sake of completeness, is that the structure theorems for Hopf algebras (Leray, Cartier-Milnor-Moore) can be recovered in a transparent combinatorial way from this approach. We refer to [START_REF] Patras | Homothéties simpliciales[END_REF][START_REF] Patras | La décomposition en poids des algèbres de Hopf[END_REF][START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF] for details. As could be expected already from Solomon's analysis of the Poincaré-Birkhoff-Witt Theorem, the Solomon's eulerian idempotents play a key role in this construction.

Corollary 3.3. The universal Galois group of Hopf algebra theory is a combinatorial, explicit, realization of the universal Galois group of renormalization theory.

The fact that the universal Galois group of Hopf algebra theory is a universal group for perturbative quantum field theories is a direct consequence of the Thm 3.1 and of the existence of a Hopf algebra structure governing the combinatorics of Feynman diagrams.

The Dynkin operator as a universal operator for the RG analysis

A corollary of the universal properties of the descent algebra is that our toy-model RG equation for iterated integrals (Thm. 1.1) can be lifted to the Hopf algebraic setting. This lift is based on the study of Dynkin operators actions on Hopf algebras in [START_REF] Patras | On Dynkin and Klyachko idempotents in graded bialgebras[END_REF] and is the main purpose of the present section. This is illustrated with the example of the Bogoliubov recursion, following [START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new identities[END_REF][START_REF] Ebrahimi-Fard | A Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogolioubov's counterterm recursion[END_REF]. So, let us turn back for a while to the iterated integral approach to RG analysis. One characteristic feature of the time-ordered exponential expansion of the solution of a first-order linear differential equation is that the n-th order term of the expansion (an n-dimensional iterated integral) rescales as λ n under the action of the Renormalization Group. There is a universal feature to this property: very much as the descent algebra is a universal object for Hopf algebra theory, the Dynkin operator is a universal operator for RG analysis, at least for elementary renormalization groups such as C * .

Let us consider first a formal power series in noncommuting variables and a scalar parameter λ, S(λ) = 1 + n∈N * λ n s n . Mimicking our computation with the time-ordered exponential, we define the series B by: λS ′ (λ) = S(λ)B(λ).

The requirement S(0) = 1 insures that B(λ) is well-defined. When the series S(λ) originates from physics or analysis, e.g. as a solution of a differential equation, it is usually expected to have a group-like behavior.

It is therefore natural to lift the computations to the algebraic setting, view the variables s n as a family of free noncommuting variables, and provide the free associative algebra F generated by the s n with a Hopf algebra structure by requiring that the coproduct ∆ be such that:

∆(S(λ)) = S(λ) ⊗ S(λ).
This equation is enough to define the coproduct and the Hopf algebra structure (since it implies that ∆(s n ) = k≤n s k ⊗ s n-k ). These definitions can be traced back to the seminal work [START_REF] Gelfand | Noncommutative symmetric functions[END_REF]. Indeed, let us notice first that, since the RG equation of Thm. 1.1 holds for an arbitrary timedependent operator H(t), it translates into an equation for the Dynkin operators in the descent algebra, namely:

D := n≥1 D n = ( n∈N D (n) ∅ ) -1 * ( n∈N n • D (n) ∅ ).
This identity, usually derived from the presentation of the Dynkin operator as an iterated bracketing, can be traced back to von Waldenfels, see [START_REF] Patras | On Dynkin and Klyachko idempotents in graded bialgebras[END_REF] for details. The proposition follows, since S(λ) is group-like and since D

∅ (S(λ)) = λ n s n , so that:

( n∈N D (n) ∅ ) -1 * ( n∈N n • D (n) ∅ )(S(λ)) = S(λ) -1 λS ′ (λ).
Let us show now how the Bogoliubov construction of counterterms in pQFT can be accounted for with this construction. Let A be an associative algebra provided with a linear endomorphism L. For x ∈ A, let us consider the series X(λ) solution of the equation:

X(λ) = 1 + L(X(λ) • λx)
with initial condition X(0) = 1, so that the equation can be solved recursively as: X(λ) = 1 + λL(x) + λ 2 L(L(x)x) + .... When X is the solution of the Bogoliubov recursion in pQFT, L is the projection operator R on the divergent part of Laurent series whereas -x is the regularized amplitude φ regη. The RG analysis can then be performed as in F (since any identity satisfied by a formal power series with free variables holds automatically for an arbitrary power series with coefficients in a associative algebra), and we get the RG equation:

λX ′ (λ) = X(λ)Y (λ)
where Y (λ) is the image in A of the corresponding element B(λ) in F (where s 1 is sent to L(x), s 2 to L(L(x)x), and so on).

The way this general pattern specializes to the Bogoliubov recursion is particularly intriguing, since it leads to new formulas and ideas for the construction of the counterterm in pQFT which have an independent mathematical interest. The corresponding analysis was performed in [START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new identities[END_REF][START_REF] Ebrahimi-Fard | A Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogolioubov's counterterm recursion[END_REF] in the more general framework of arbitrary Rota-Baxter algebras and, although we state them in the context of the Bogoliubov recursion for dimensionally regularized amplitudes in pQFT, the following results hold in that general setting.

Recall first the notion of (left) pre-Lie or Vinberg algebra: this is a vector space equipped with a bilinear product • such that:

(a • b) • c -a • (b • c) = (b • a) • c -b • (a • c)
so that, in particular, the bracket [a, b] • satisfies the Jacobi identity. The algebra L of maps from the Connes-Kreimer Hopf algebra of Feynman graphs to the algebra of Laurent series carries a pre-Lie structure, defined by: a

• b := R(a)b -bR(a) + ba,
where R stands as usual for the projection on the divergent part of the Laurent series. We set: for a ∈ L,

l (n) (a) := l (n-1) (a) • a r (n) (a) := R(l (n) (a)).
Proposition 4.2. The solution φ λ -to the generalized Bogolioubov recursion φ λ -= 1 + R(φ λ - * λ(φ regη)) satisfies the RG equation:

λφ λ - ′ = φ λ -• ξ(λ) where ξ(λ) = n≥1 r (n) (λ(φ reg -η)).
The computation of ξ(λ) is far from obvious and depends heavily on the existence of a Rota-Baxter algebra structure on the algebra of Laurent series. We refer to [START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new identities[END_REF][START_REF] Ebrahimi-Fard | A Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogolioubov's counterterm recursion[END_REF] for a proof. The physical relevance of this equation is largely unclear at the moment since the rescaling φ reg -→ λφ reg (which breaks the multiplicative properties of the regularized amplitude φ reg ) does not seem to have any obvious physical meaning. However, as we already observed, the mathematical structures underlying the formula are very rich, interesting on their own, and suggest a relevance of pre-Lie structures in the construction of counterterms and renormalized amplitudes.

RG equations and the beta function in pQFT

Whether or not the previous analysis can be extended to include the classical RG equations in pQFT may seem problematic. Indeed, in that case, the RG analysis has to take into account the phenomenon of renormalization, so that understanding algebraically the RG equations requires the understanding of the interactions between the behavior of divergences in regularized amplitudes and the changes of mass scales. However, surprisingly, the Dynkin operator approach to RG equations still holds in that case, leading to a new, Lie theoretic, understanding of the beta function.

Let H be, once again, the graded connected commutative Hopf algebra of Feynman diagrams of a given quantum field theory treated perturbatively. As usual, A = C[ε -1 , ε]] stands for the algebra of Laurent series. To the (pro-unipotent) group of characters G(A) (the group of algebra morphisms from H to A) is naturally associated a (pro-nilpotent) Lie algebra (its elements are called infinitesimal characters): L(A) := {φ ∈ Hom(H + , A), (H + ) 2 ⊂ Ker(φ)} where H + := n>0 H n . Since G and L are pro-unipotent and pro-nilpotent, they behave essentially as a group of unipotent (resp. a Lie algebra of nilpotent) matrices and the log and the exp maps induce bijections between G and L. However, the logarithm and exponential maps are not suited to the understanding of the RG equations: as could be expected from our previous analysis of RG equations, one has to use instead of the classical logarithm the Dynkin operator: Theorem 5.1. Right composition with D is a bijective map from G(A) to L(A). The inverse map is given by

Γ : α ∈ L(A) -→ n k 1 ,...,k l ∈N * k 1 +•••+k l =n α k1 * • • • * α kn k 1 (k 1 + k 2 ) . . . (k 1 + • • • + k n ) ∈ G(A).
The computation of the inverse follows from the identity in the Cor. 1.2, when suitably translated into an identity in the descent algebra -and, therefore, into a universal identity for Hopf algebras. A purely algebraic proof was first obtained in [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalization[END_REF], to which we refer for details.

Recall that, in dimensional regularization, the regularized amplitude depends on ε and on the mass scale µ. The mass scale appears at the power µ nε in the regularized amplitude of an n-loop graph Γ (we write n = |Γ|). Here, the (mass scale) renormalization group (MRG) is the group of rescalings of µ, and acts therefore on the group of A-valued characters G(A) as a one-parameter action of C * ∋ t: φ t (Γ) := t ε|Γ| φ(Γ), where, for simplicity, we write from now on φ for φ reg (m; ε, µ) and Γ stands for an arbitrary graph (not necessarily connected). We have, for φ t ∈ G(A) the decomposition:

φ t = (φ t ) -1 - * (φ t ) + .
For reasons that will become clear soon, we also introduce a second renormalization group (the "loop" RG, LRG) action:

φ (λ) (Γ) := λ |Γ| φ(Γ),
which is the one we have encountered in the previous section and are now familiar with (according to which the components of degree n in a Hopf algebra, here the graphs with n loops, are rescaled by a factor λ n ). The two RG actions commute and, moreover, the LRG action commutes to the Bogoliubov decomposition in the sense that:

(φ t ) (λ) = ((φ t ) (λ) ) -1 - * ((φ t ) (λ) ) + = ((φ t ) -1 -) (λ) * ((φ t ) + ) (λ) Moreover, t • ∂ ∂t φ t = ǫ • λ ∂ ∂λ (φ t ) (λ)
|λ=1 . The physical constraints associated to the renormalization process translate then into the locality condition: Theorem 5.2 (Collins). Let φ be a dimensionally regularized Feynman rule character. The counterterm character in φ t = (φ t ) -1 - * (φ t ) + satisfies t ∂(φ t ) - ∂t = 0

Or (φ t ) -is equal to φ -, i.e. independent of t. The physical reason for this is that the counterterms can be taken mass-independent; this goes back at least to [START_REF] Collins | Structure of the counterterms in dimensional regularization[END_REF]. For this fact, and more details on the physical significance of the locality property in pQFT, we refer the reader to [START_REF] Collins | Renormalization[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group[END_REF].

We say the A-valued characters with this property are local characters: φ ∈ G loc (A) ⊂ G(A). Notice that, by definition of the MRG action, G + (A) ⊂ G loc (A). We will be mainly interested in local counterterms, that is, elements in G -(A) loc := G -(A) ∩ G loc (A). Notice that G -(A) loc is stable under the LRG action. Performing a RG analysis for this action yields immediately a differential equation for the counterterm: Lemma 5.1. We have, for any ψ ∈ G -(A), the LRG equation λ(ψ (λ) ) ′ = ψ (λ) • (ψ (λ) • D).

Moreover, the map ψ -→ ψ • D is a bijection between G -(A) and L(A -).

We omit the proof since it follows, once again, from the universal properties of the Dynkin operator for RG analysis and from the existence of a Hopf algebra structure on H.

Actually, one can show [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalization[END_REF] that the map ψ → Dyn(ψ) := ε(ψ • D) is a bijection between G loc -(A) and L(C). Interesting phenomena show up when one tries to understand locality from this point of view since, for local characters, the infinitesimal generators of the LRG equation (for the counterterm) and of the MRG equation (for the renormalized character) surprisingly agree: Theorem 5.3. Let φ ∈ G loc (A). For the renormalized character φ ren (t) := (φ t ) + (ε = 0), we have: .

t ∂
Taking the limit when ǫ → 0, we get: t ∂ ∂t φ ren (t) = Dyn(φ -1 -) * φ ren (t).

These equations show in the end that the classical abstract RG equation can be understood as an equation involving the action of the Dynkin operators of the Hopf algebras of Feynman graphs. Moreover, the simple toy model introduced at the begining of the article accounts fairly well for the combinatorial subtelties of the construction. These simple arguments, that do not appeal to the complex structures of noncommutative geometry but rely on the more classical tools of free Lie calculus and symmetric group actions, appear to us in the end as a simple (down-to-earth, but efficient) way to account for the universal Galoisian approach to renormalization. Together with K. Ebrahimi-Fard and J. Gracia-Bondìa (with whom this research program is currently conducted), we hope these ideas to be flexible enough to accomodate for the more complex computations showing up in "real life" pQFT (e.g. with higher dimensional renormalization groups corresponding to several coupling constants; with other renormalization schemes than RG, and so on).

S

  . By inclusion/exclusion (i.e. Möbius inversion), the two are related by the formulas:

Definition 3 . 1 .

 31 Let us write Desc for the linear span of Solomon elements D (n) S , where n runs over N and S over the subsets of [n -1]. The vector space Desc is provided uniquely with a cocommutative Hopf algebra structure by the following requirements:

∅

  . Let us define the descent algebra Desc H of a Hopf algebra H as the convolution subalgebra of End(H) generated by the graded projections p n : H -→ H n (orthogonally to

Proposition 4 . 1 .

 41 We have, in F : B(λ) = D(S(λ)) where D := n≥1 D n stands for the sum of Dynkin operators acting naturally on the Hopf algebra F in view of Thm 3.1.

  ∂t φ ren (t) = Dyn(φ -1 -) * φ ren (t),the abstract RG equation Indeed, since φ is local, we have:φ t = (φ t ) -1 - * (φ t ) + = φ -1 - * (φ t ) + .

	and get:	t	∂(φ t ) + ∂t	= φ - * t	∂ ∂t	φ t = φ - * ǫλ	∂ ∂λ	(φ t ) (λ)	|λ=1
		= φ - * ǫλ(	∂ ∂λ	(φ -1 -) (λ) * (φ t + ) (λ) + (φ -1 -) (λ) *	∂ ∂λ	(φ t + ) (λ) ) |λ=1
				= Dyn(φ -1 -) * φ t + + ǫλ	∂ ∂λ	(φ t + ) (λ)	|λ=1


Emmanouil Markoulakis

John Chatzakis

Anthony Konstantaras

Emmanuel Antonidakis

A Synthetic Macroscopic Magnetic Unipole

Keywords: macroscopic quantum effect, synthetic magnetic unipole, ferromagnetism, permanent magnets, magnetic unipolar array, vortex unipole magnetization, ferrolens

We demonstrate experimentally with a prototype for the first time, that an artificial emergent magnetic unipole hedgehog field in a simply connected three-dimensional domain is possible, emulating effectively the Dirac model by simply using a novel permanent magnets' topology and geometrical arrangement in a ring array. Although similar effects were demonstrated by others over the last decade, with these effects usually observed and lasting for a small fraction in time and applied at the quantum or microscopic scale using primary BECs, Skyrmions, Spin Ice and recently Chiral Magnets this was never shown until now and performed at the macro scale and by using normal magnets. The synthetic magnetic unipole ring array prototype progressively twists and steers the magnetic flux into vanishing curl field (i.e. vortex) geometry towards the centre of the ring and its air cap. This mere act alone proves enough to create the desired effect and an apparent isolated magnetic unipole region is observed at the centre of this magnetic ring array, as we have mapped with a 3-axis magnetometer and show with the quantum magnetic-optic device flux viewer, the ferrolens. This apparent unipolar magnetic array exhibits anomalous behavior at its centre, air cap and registers with a magnetometer or a normal compass needle as having the same magnetic polarity on both opposite faces of the ring on the same axis passing through the centre of the ring. Also the other opposite polarity pole is dispersed and confined along the periphery of the ring array with a 90° angle to the unipole axis and none existing in the far field and nowhere to be found. Even in the near field, counting the total number of discrete poles of the magnets of the array including its emergent unipole at the centre, counts to an odd number. The results were analyzed and software simulated and some interesting conclusions are drawn about the ultimate quantum vortex nature of ferromagnetism.

Introduction

First of all we have to clarify here that we consider that the term "magnetic monopole" regardless of what adjective word you put in front, is strictly reserved to describe the elusive hypothesized to exist natural quantum elementary particle. Therefore in order to avoid confusion we prefer instead to use in our research the term "synthetic magnetic unipole (i.e. uni-one)" to refer to the artificial human made implementation attempt to physically, loosely emulate the theoretic Dirac model for magnetic monopoles which is a dipole where the two opposite magnetic charge monopoles are separated and connected by a semi-infinite line thus infinite long and infinite thin curve in space, the Dirac String. So even Dirac [START_REF] Dirac | Quantised singularities in the electromagnetic field[END_REF] tells us in his theory that a totally isolated magnetic monopole is not possible to exist in any way as Maxwell [START_REF] Maxwell | A dynamical theory of the electromagnetic field[END_REF] does, therefore essentially and exclusively magnetism is a dipole phenomenon unless proven otherwise in the future. It is our understanding and belief that a completely isolated hypothesized magnetic monopole in the rare case it can exist will have nothing magnetic about it and therefore is an impossibility and the whole term in this specific interpretation of it, is a contradiction.

However, with that being said there are still potential significant scientific and technological merits today from theoretical and experimental research on this field and specially the last years from loosely synthetic implementations [START_REF] Bender | Synthetic versus Dirac monopoles[END_REF] of Dirac theorized monopoles usually artificially induced via an external magnetic field or laser steering on Bose-Einstein Condensates [START_REF] Ray | Observation of Dirac monopoles in a synthetic magnetic field[END_REF] [START_REF] Ollikainen | Experimental realization of a Dirac monopole through the decay of an isolated monopole[END_REF] but sometimes also natural occurring and observed when brought at very low temperatures as topological texture defects, thus skyrmions lattices in a condensed or soft matter medium like Spin-Ice materials [START_REF] Castelnovo | Magnetic monopoles in spin ice[END_REF] [START_REF] Ladak | Direct observation of magnetic monopole defects in an artificial spin-ice system[END_REF] or recently observed and manipulated in chiral magnets [START_REF] Milde | Unwinding of a Skyrmion Lattice by Magnetic Monopoles[END_REF][START_REF] Kish | Skyrmions in Chiral Magnets[END_REF][START_REF] Okumura | Tracing Monopoles and Anti-monopoles in a Magnetic Hedgehog Lattice[END_REF][START_REF] Okumura | Magnetic hedgehog lattices in noncentrosymmetric metals[END_REF][START_REF] Kanazawa | Critical phenomena of emergent magnetic monopoles in a chiral magnet[END_REF]. All above described are created or existing at the quantum or microscopic size scale as non-isolated magnetic unipole quasi-particles [START_REF] Pushkarov | Quasiparticle Theory Of Defects In Solids[END_REF] . Parenthetical, although a bit outside the context of our research presented herein but not entirely, as we will show later on, we must reference that it is theorized and with some indirect experimental reasoning that Dirac magnetic monopoles are hidden inside ferromagnetic materials and are correlated closely to the Anomalous Hall Effect (AHE) [START_REF] Fang | The anomalous Hall effect and magnetic monopoles in momentum space[END_REF]. What in general these quasiparticles magnetic unipoles have in common is that they are topological defects emergent phenomena, in the form of an energy vortex in the medium that resembles as an effect and therefore emulates the theorized Dirac magnetic monopole. Also, their monopole field is not inherent from any applied external field but merely intrinsic due to the angular momentum of their vortex motion and in most cases these vortices are short lived. The vortex nature of the theorized magnetic monopoles is actually what Dirac geometrically is describing in his work. A Dirac monopole is a point-like source of a possibly artificial magnetic field that forms at the endpoint of a quantum whirlpool. In quantum mechanics, an electron is described by a diffuse wave-like object rather than a pointlike particle. Paul Dirac was the first person who understood the importance of studying the end points of quantummechanical whirlpools within these electron waves. He noticed that when an electron has such a terminating vortex, a magnetic monopole inevitably forms at the end point (i.e. apex of the vortex). A terminating vortex is the defining characteristic of the Dirac monopole magnetic charge.

In our experiments we adopted and implemented the same theoretical approach but applied it at the macroscale by using normal neodymium permanent magnets in a novel ring array formation which progressively twists the magnetic flux of the individual magnets in the array to an end point at the center of the ring and air cap of the ring array and therefore creating an artificial magnetic flux whirlpool. This magnetic flux whirl opposite to other approaches for creating synthetic magnetic unipoles which are usually short lived, is permanent thus stable. Also the synthetic magnetic unipole field in our case is not created by the induced vortex motion topological defect in a medium caused by an external applied magnetic field or other method of steering but instead is intrinsic and originating from the Net Quantum Magnet Field (QMF) [START_REF] Markoulakis | Real time observation of a stationary magneton Results[END_REF][16] existing inside our unipolar magnetic array material and then projected to the outside, at the center of the air cap of the ring array in a small region a couple of millimeters. This array of individual magnets acts like a magnetic flux twister focusing and vortexing the flux coming from all the like poles of its individual magnets facing to the same angular direction in the circle (i.e. clockwise or counterclockwise) to an end point, apex tip of the vortex located at the center of the hole of the ring array.

In other words, the net magnetic field of our array prototype contains itself the synthetic magnetic unipole at the tip, end point of its magnetic flux vortex field formation and no other medium is needed in order to generate the unipole by angular momentum.

Considering also what is theorized by Dirac about magnetic vortices end points and also the possibility of Dirac magnetic monopoles hiding inside ferromagnetic materials responsible to one extend for the Anomalous Hall Effect as concluded previously in the literature [START_REF] Fang | The anomalous Hall effect and magnetic monopoles in momentum space[END_REF] and inferred also by some experimental deduction, there is a fair chance our prototype array to act as a Dirac magnetic monopole trap in the best case scenario or inducing a magnetic unipole quasiparticle in the worst case.

We did not use any static magnetic shielding on our prototype and it is not to be confused with the well known Halbach magnetic array configurations [START_REF] Halbach | Design of permanent multipole magnets with oriented rare earth cobalt material[END_REF][18] which suppress the one net pole of the array (i.e. weak pole but you can still measure a field) relative to the other net pole polarity (i.e. strong pole) by using spatial partial field cancelation technique in one side of the array and by forcing unlike poles magnetization axes in close proximity together whereas our synthetic magnetic unipole prototype ring array is doing something entirely different. It is fussing the magnetic flux coming from all the same sides of the discrete magnets in the array to the center of the ring and diffusing and spreading out along the perimeter of the ring, the flux coming from all the other pole sides of the magnets. End result in our prototype, both faces of the ring array have uniaxially the same magnetic polarity and along the perimeter of the array no field is present or can be measured for the far field (i.e. for distances larger than the diameter of the ring array).

Next, we will describe the materials, devices and methods we used in our experiments with the unipolar prototype and also its design and implementation details.

Materials and Methods

Ferrolens

As with our previous research we use the quantum magnetic-optic flux viewer ferrolens to probe inside the ferromagnets material and projects to the outside their net Quantum Magnet Field (QMF) [15][16][19]. This superparamagnetic 25μm thin film encapsulated inside a flat optical lens and physical device, displays a real time 2D imprint with some holographic depth, of the vorticity and curl of the magnetic flux inside the permanent magnet's bulk material. In fig. 1(a) the polar view of the field of a typical neodymium ring magnet is shown by the ferrolens. The vorticity and curl of its magnetic flux lines is displayed in its air cap. The field shown depicts see-through both polar fields of the magnet (i.e. N and S) therefore the total field of the ring magnet is shown which is torus shaped. This is a unique property of the ring magnets when displayed with the ferrolens since they are able to confine the total field inside their air capped hole as shown by the lens. Fig. 1 (b) shows the inner magnetic field of an air capped without a ferromagnetic core, current solenoid, of which its theoretical model for its magnetic field is also illustrated in fig. 1 (c). The inner field of an air capped electric solenoid in contrast to the field of a ferromanget, has no vorticity and nearly zero curl on its magnetic flux lines and is very homogeneous. This is nicely shown by the ferrolens in this particular snapshot, fig. 1(b), from the experiments. You are looking in this snapshot, see-through the field on both magnetic poles N and S of the solenoid located at its two ends. The magnetic flux lines closer to the inner walls of the solenoid have larger curl than the lines at its center region which are straight with zero curl and vertical to the ferrolens surface and cannot therefore scatter light thus they are not made visible on the lens. The lines however close to the inner surface of the solenoid which are more curled are nicely picked up by the ferrolens and displayed with holographic depth. The magnetic array's frame shown in fig. 2 (a) is designed in such a way that the axis of each magnetic N42 plate (see fig. 2b) when inserted vertically into a slot is not exact concentric to the circle's center as shown in fig. 2(c) and therefore not aligned with the imaginary radius line but instead intentionally eccentrically skewed 12° so that each magnetic plate axis does not point the ring's exact center but is instead tangent to a virtual 2.08mm calculated circle shown with blue in fig. 2(c). We have to stress out here how much important it is that the magnetic plates used in the design of the unipolar array to be as thin as possible else the magnetic flux of the array will not be steered and twisted precisely at the center of the ring array and the desired unipole effect will not be achieved. The angular separation of the twelve magnets used in the design is 30°. Each magnetic plate has 10mmx10mmx1mm dimensions and is a N42 grade neodymium axial magnetized magnet.

The Magnetic Unipolar Array

Type N or S, of the emergent unipole at the center of the ring as also stated at the legend of fig. 2 depends on angular direction of the magnetic moments of the magnetic plates, clockwise (CW) or counterclockwise (CCW), cascaded in series around the ring and in combination with their skew 12° angle direction left or right. The various combinations and the emergent each time unipole type is shown below:

N

As a rule of thumb to the above Table . 1, if the leaning of the magnetic plates is in the same direction as the angular direction of the magnetic moments of the individual magnets around the ring then the emergent unipole at the center of the ring is always a North (N) unipole. Likewise, when leaning is at opposite direction of angular momentum then this results always to a South (S) unipole. As an example the angular magnetic moment in fig. 2 c) show the front side of the unipolar ring array finished prototype PVC frame with the assembled in 12 magnetic plates. We call this front side of the prototype from now on as Active Side to distinguish it more rigorously from its backside shown in fig. 3(d).

Importantly, notice here that there is actually no difference except a small difference in field strength we will discuss later on, between the field at the front side Active Side of the unipolar array and its backside. Both sides of the prototype four register as a South type S unipole. This is because according to Table .1, when the ring array is viewed from its front, the so called active side, the angular magnetic moment of the ring array turns clockwise around (CW) with the magnet plates leaning to the left therefore an S unipole at the center of the ring. Similarly, when the ring array is viewed from its backside the situation of Table .1 reverses thus a counterclockwise (CCW) rotation of the magnetic moment with the magnets now leaning to the right therefore the end result as shown in Table .1 is the same as previously thus a S unipole.

In fig. 3(e), interestingly we see a 2D imprint of the resulting magnetic field geometry of the array using a commercial magnetic field viewing film. We see a teeth like structure is formed like arrows pointing to the center air capped hole of the ring array. We can count here the discrete 24 poles of the array on the perimeter and also observe the emergent unipole vortex forming at the center of the hole similar to a screw and apparently isolated from the rest of the field (see here a singled out larger size version, https://tinyurl.com/ybtof269). This vortex unipole magnetization is also shown in a close-up here in fig. 4. Notice in fig. 5 that a same circular magnetization array but without skewed angles on the magnets position introduced and instead all equatorial axes of the magnets being entirely concentric to the centre of the ring will produce no magnetic field at all at the air capped hole region of the ring or elsewhere outside the ring bulk material. Its field is similar to a toroidal solenoid where no field exists outside the solenoid.

Other Instrumentation

A sensitive ±70μΤ range, 3-axis magnetometer with resolution 143mV/μT was used in conjunction with a digital oscilloscope for making the field strength measurements for the surface mapping of the field of the Unipolar Array prototype and by using also a xy-coordinates digitizer.

Results and Discussion

Decay of the Unipolar Array Field

It is well known that any point-charge electric or hypothetically magnetic, which has a spherical symmetry, would decay with distance from the source with the inverse square law 1/r 2 contrary to dipole fields, of two opposite polarity separated charges which decay with the inverse cube law 1/r 3 . For non-point charges the above described are still true and hold for distances r>>D (i.e. far field) where D is the dimension of the non-point charge for monopolar fields or the line distance between the centers of the two charges for dipole fields. Also, the field of a hypothetical magnetic charge monopole would have a non-zero divergence and zero curl magnetic flux lines which would extent radially out in space thus the hedgehog effect.

Measuring the decay of the field strength, magnetic flux density B with distance on the z-axis, of our prototype unipolar array on the far field to see if it complies more to the inverse square law which is true for monopoles than the inverse cube which holds for dipoles, would possible provide important data about the actual nature of our prototype under testing and its characterization. We see in fig. 6(b) that the far field measured values for the unipolar prototype draw in a straight line and therefore comply with the inverse square law. Only at the last most distanced measurement point thus at 10cm the values started to deviate from the inverse square law. This deviation of the normalized value of this last plot point can be contributed to the fact that at this point the measured value of the field strength of the array dropped at 17μΤ (see fig. 6a) which is actually much smaller value than the Earth's magnetic field (i.e. 40 to 50μΤ) and the curl of the Earth's magnetic flux at that point started to significantly interfere with the measurements. Observing the plot of the normalized values to the inverse cube law with distance of the far field at fig. 6(c), we see that in contrast to fig. 6(b) of the normalized inverse square law plot, the values here start to deviate early on from a straight line and continue to do so as a whole therefore the measured field strength values of the unipolar prototype do not decay with the inverse cube law therefore inferring to the anomalous characteristics of our prototype's field.

As a reference for comparison we repeated the experiment for a normal axial magnetized ring magnet and draw the corresponding plots for the measured field decay values as shown in fig. 7. We used a ferrite magnet which is much less strong than neodymium magnets so that we get comparable field strength values at the same distances, see fig. 7(a) https://tinyurl.com/yayggfrb . As expected we see in fig. 7(b) https://tinyurl.com/yaqnh4ks of the normalized inverse cube values, an almost perfectly straight line, therefore a normal dipole magnet's field decay complies with the inverse cube law with distance as measured thus the field decays exponentially with distance. Furthermore, in fig. 7(c) https://tinyurl.com/yaz7m86p of the inverse square normalized plot, we see that the values deviate clearly and curve out from a straight line and therefore do not comply with an inverse square with distance decay.

As a final observation here we cannot fail to notice in fig. 6 the close matching up of the prototype's measured field values from its front side (i.e. active side) and backside which infers to the symmetry of the unipole field. Distance was always measured from the plane of the magnetic plates to the magnetometer sensor as shown in fig. 6(d). The extra plastic frame material thickness existing at the backside of the array was also taken into consideration so it does not mess up the distance measurements.

Next, we will continue the measurements for the prototype's field characterization by graphing its magnetic field surface maps.

Surface mapping of the Unipolar Array's Field

The discrete measurements for the surface mapping of the field of the array were taken at 30° intervals around the ring faces both front and backside of the prototype, at the exact positions where the magnetic plates are located inside the

with the magnetometer's resolution given by the manufacturer at 143mV/μΤ. Negative values on the z-axis of the maps indicate a South Pole whereas positive values a North Pole. The formed yellow spikes on the rim of the map of fig. 8(a) are due our discrete measurements points taken at regular intervals and the actual field of the ferrite ring magnet will have the same strength uniformly around the rim yellow region shown, with no spikes. We see however that the highest field strength at the far field of the ferrite ring magnet is measured at its exact center air capped hole.

In fig. 8(b)&(c) we can see the surface mapped field of our unipolar ring array prototype v4 of its front side face and backside accordingly. By observing these two maps we quickly realize that it is the same unipole field observed by two opposite spatial regions although the unipole field emanating from the backside measures relatively at an average 2 μT less. Also, the field registers on both sides of the prototype axially as a South S Unipole with the highest field strength concentrated at the center air capped hole of the prototype ring array at the far field. Interestingly, we discover that the field around the frame of the unipolar array where the magnetic plates are located seems to dissipate progressively around the ring with a slope function which is a similar characteristic with the angular velocity function of an irrotational free vortex therefore strongly suggesting that a magnetic flux vortex is present in our synthetic unipole. This will become more evident next were we map the field of the prototype ring array along its periphery at the far field, which is 90° perpendicular to its unipole axis, as shown in fig. 9 of the experimental setup. The distance from the magnetometer sensor is about 2 cm out to the periphery of the prototype and is hold fixed. The small humps and dips along the black dashed close loop line ellipsoid shown in fig. 9 of the surface map are the measurement points and where the magnetic plates are located on the periphery of the prototype.

Something extraordinary is happening here emphasizing the anomalous characteristics of our unipole array prototype. As we can see, the field on the periphery at the first half 0° to 180° section of the ring array, registers as a South Pole, and progressively diminishes in strength from about -30μΤ (i.e. minus sign indicates a S Pole, -4400/143 see fig. 9) to Fig. 9 Surface map of the field on the periphery of the prototype unipolar array. 0μT accordingly and then for the second half of the ring from 180° point, back to 0° it registers as a North Pole with an progressively increasing strength from 0μΤ to about +3μΤ (plus sign indicates a North Pole, 400/143 see fig. 9). We notice that the ring array prototype has on half of its perimeter the same polarity S Pole as its unipole at the center of the ring and on the other half, a North Pole but greatly suppressed in strength relative to its South Pole on the periphery (see fig. 9 illustration). As a net result the generated poles of our prototype including the emergent unipole at the center are three thus an odd number and therefore an anomalous result.

Ferrolens Observations of the Unipole Vortex

Although the prototype v4 of the magnetic unipolar array with a one axis Hall sensor directly upon the center of the front side or backside ring hole, at zero distance measures 22mT S pole field strength for the front and 17.6mT S pole for the backside of the prototype, (i.e. the difference between back and front is actually less than the one measured because the added 2mm plastic frame thickness on the backside of the ring array) and barely registers on a typical ferrolens which has a minimum sensitivity for an adequate display of the field at least 40mT, we actually succeeded on displaying in real time the formed apparent isolated induced unipolar vortex on the lens thin film from our prototype array. This was achieved by using the latest development generation of ferrolenses commercially available (i.e. Ferrocell commercial name) which have a more sensitive diluted thin film mixture and less thickness of the film (i.e. 509 type Ferrocell) [20].

In fig. 10(a) we see the prototype placed on top of the ferrolens. Fig. 10(b) is a black & white photograph for enhanced contrast, of the unipole magnetic flux vortex field of the protototype array as shown by the lens. The cross section of the black vortex formed at the center is no more than 3mm. This photograph was taken with the prototype placed under the ferrolens. Fig. 10(c)&(d) show the vortex when viewed from the front side and backside of the prototype accordingly, through the prototype's ring array 8mm hole. In this case the prototype was placed on top of the ferrolens. This vortexing magnetic flux and apparent isolated vortex shown by the ferrolens is not the classical magnetic macroscopic field but the net Quantum Magnet Field (QMF) [START_REF] Markoulakis | Real time observation of a stationary magneton Results[END_REF][16] inside the prototype ring array's material projected to the outside and shown by the ferrolens quantum magnetic optic physical sensor. It displays the vorticity and curl of the field inside the magnetic material of the prototype array which are different than the classical magnetic field formed outside the magnet on air.

If there is a unipole charge or net unipole charge, formed inside this vortex at its apex as predicted by Dirac theory [START_REF] Dirac | Quantised singularities in the electromagnetic field[END_REF], it would emanate magnetic flux radially with almost zero curl in the form of an hedgehog field to all directions and this would be the same for both QMF and classical magnetic field. Previously, in section 2.1 and fig. 1(b) we have shown experimentally with the field inside an electric solenoid, that the ferrolens is capable of sensing and displaying straight without curl or very little, magnetic flux lines as long these are not perpendicular to the lens surface. It might be possible to see these straight flux lines with the ferrolens coming out radially from the vortex mouth to all directions as a hedgehog field, given a large relative optical zoom and a sensitive enough ferrolens. This is shown in fig. 11. Notice that the flux lines appear divergent coming from a center point, the vortex field.

Next we will simulate these results from our observations with the ferrolens of our Unipolar Magnetic Ring Array and from our analysis and data presented so far of all the experiments we have performed, thus a possible macroscopic synthetic magnetic unipole quasiparticle field device or even possible according the literature [START_REF] Fang | The anomalous Hall effect and magnetic monopoles in momentum space[END_REF], a ferromagnetic Dirac magnetic monopole trap. Unfortunately, for direct observation of these particle charges, since the ferrolens observations cannot provide us with the submicron optical resolution and magnification needed, SQUID Scanning Magnetic Microscopy would be the best choice here, however we do not have this option yet although we are planning for it in the future. Therefore, we have to deduce the existence of these synthetic unipole charges indirectly by observing, measuring and analyzing their resulting field in our novel unipolar magnetic array.

Simulations of the Unipolar Magnetic Array

The simulations of the ferrolens display of the prototype unipolar magnetic array were executed with the commercially available software simulator program of Michael Snyder, pic2mag v1.0 Multithreading. It is the only scientific program currently as far as we know, that can simulate successfully the ferrolens induced magnetic field of magnetic arrays and its display. We will use pic2mag s/w to further verify our experimental results and observations and also see details of the total field of the prototype unipolar array which were not possible previously entirely to observe with the actual physical device ferrolens because the sensitivity limitations of the device we have described in section 3.3. The near field, at the bulk magnetic material on the face and periphery of the ring array which is greatly confined inside the ring as we have measured and will show later on with more detail in the simulations, even at distances no more than 5mm falls off quickly at the z-axis to values varying from 7mT as we have measured with a Hall sensor to as low as a few μT and therefore not adequate field strength prototype. How the simulation works, you have to input first the exact magnetic array's geometry, dimensions. We have used a xy coordinate's digitizer to input the array's topology into the simulator. Then using RGB color coding on each discrete magnetic plate we inputted the exact spatial orientation of the magnetic moment of each individual magnet plate with reference to a 360° circle. The program then executes and simulates the ferrolens display of the field Fig. 12 Simulations of the prototype v4 unipolar magnetic array in various configurations listed on Table .1. Both field views of the prototype unipolar ring array are shown when observed from its front side or backside.

for displaying the magnetic flux in the array by the actual ferrolens. When a magnet is placed in contact with the ferrolens there is always a separation distance from the encapsulated thin film inside the optical lens to the magnet. This is because the thickness of the optical glass used by the lens. In our case we used ferrolenses with an optical glass thickness in between the magnet and the lenses' thin film of about 2.5mm.

The simulation has no such limitations and can display the magnetic flux vorticity and curl inside the magnetic array inside the prototype unipole array. In fig. 12 we see the results of these simulations for various configurations of the prototype listed in Table .1 section 2.2 with the field viewed on both sides, front and back of the prototype unipole array. Thus, direction of skew angle of magnet plates left ( \ ) or right ( / ) and angular rotation direction of the magnetic moment of the ring array, clockwise (NP North Pole rotation CW) or counterclockwise (NP North Pole rotation CCW).

In all cases we see the vorticity and curl of the magnetic flux inside the array thus the Quantum Magnet Field (QMF) of the unipolar array as displayed by the simulation. The software simulates the display of the ferrolens assuming a magnet's side is put under the lens and in contact. We clearly can see in all the simulations the emergent isolated unipole quantum magnet flux vortex displayed as an oval, egg shaped purple loop at the center of each array air capped hole. This actually is more or less what we have observed with the actual, physical ferrolens device and also measured in the other experiments with the magnetometers. Apart of the QMF of the prototype shown by the simulator, the simulator can also calculate the classical macroscopic on air, N-S vector magnetic field of the unipole array shown in fig. 13. The software simulator pic2mag v1.0 for illustrating the vector field uses small bicolored magnetic rods which act as tiny compass needles shown in fig. 13. Unconventionally, it uses red for the South Pole of the rods and blue indicating their North Pole. In fig. 13(a) we see a close-up zoomed in version of our South S unipole simulation at the center of the array's air hole. We observe that the vector field diverges to all direction from the center of the hole where the vortex is formed as Dirac theory predicts like a hedgehog. Also, the rods North poles (blue) point inwards therefore a South Unipole at the center of the prototype ring array. Fig. 13(b) shows the total classical vector field.

Detection of magnetic unipole with a solenoid

Apart of using SQUID scanning magnetic microscopy for submicron mapping the synthetic unipole field at the center of the magnetic flux vortex, located at the center of the prototype ring array hole, see previously fig10, we can use also a SQUID magnetometer to simply detect the presence of magnetic monopole(s):

"Passing a magnetic monopole through a coil of wire induces a net current in the coil. This is not the case for a magnetic dipole or higher order magnetic pole, for which the net induced current is zero, and hence the effect can be used as an unambiguous test for the presence of magnetic monopoles. In a wire with finite resistance, the induced current quickly dissipates its energy as heat, but in a superconducting loop the induced current is long-lived. By using a highly sensitive "superconducting quantum interference device" (SQUID) one can, in principle, detect even a single magnetic monopole." [START_REF] Cunningham | High Energy Physics -Glenn Cunningham -Google Books High Energy Physics (Scientific e-Resources[END_REF].

The SQUID method of detection shown in fig. 14 is very sensitive and most suitable for detection of magnetic monopoles or related quasiparticles at the microscopic scale like Spin-Ice materials and beyond and could theoretically even detect a single magnetic monopole inside matter. However, in our case of the synthetic magnetic mono (uni) pole prototype array (i.e. smmpa) and due its macroscopic ferromagnetic nature the above described experimental setup for detection of magnetic monopoles could be replicated but without necessarily the use of a SQUID magnetometer.

Instead, as shown in fig. 15(a) a relative long electric solenoid was special made and wire coiled around a 3D printed plastic pipe and its two wire ends were connected with an R=1Ω shunt resistor. The induced electric current waveform in the solenoid was displayed and measured with a digital oscilloscope connected across the shunt resistor. Various types of dipole ring magnets including our unipole smmpa array were allowed to free fall vertically using a wooden stick guide, to ensure that the magnets fall vertical with their pole and do not flip inside the solenoid, throughout the total length of the solenoid and the induced signal was recorded on the oscilloscope for comparison. The inductance of the solenoid used in the experiments was measured with a RLC bridge to be L=1.35 mH with an ohmic wire resistance of RL=10 Ω. In fig. 16 we see the recorded signals correspondingly of the induced current inside the solenoid by dropping vertically the magnets inside the solenoid. The center segment of each waveform corresponds to the signal produced when the whole length of the magnet is inserted inside the solenoid and the magnet is free falling along the solenoid's total length. Surprisingly in fig. 16(a), we observe that for the long ferrite dipole ring magnet of fig. 15(b) when dropped inside the solenoid and as long its total length is inserted and travelling inside the solenoid, the induced current segment shown is not zero but a variable slope current and therefore does not initially comply with the predicted in theory [START_REF] Cunningham | High Energy Physics -Glenn Cunningham -Google Books High Energy Physics (Scientific e-Resources[END_REF] zero current in this case. However, by further experimentation we discovered that this was due the fact that the length of the dipole magnet lm, was comparable to the length of the solenoid ls and therefore producing a ratio l s /l m closer to one. In practice a much larger length solenoid or a much shorter dipole magnet will produce the desired effect and comply with the theoretical prediction. We decided to repeat the experiment with a much shorter dipole ferrite ring magnet shown in fig. 15(c) and its induced current signal in the solenoid is shown in fig. 16(b). As expected the net current segment at the center of the waveform shown in fig. 16(b) is this time almost zero and fixed as theory ideally predicts validating therefore our previous observation and conclusion. The total waveform shown of the signal during entry and exit of the magnet from the solenoid appears asymmetric due the free fall gravitational acceleration of the magnet. Additionally in fig. 16(c), by applying magnetic braking we succeeded in extending the period where the magnet is inside the solenoid and therefore further demonstrating the net zero current flattening effect predicted by theory and also reducing the free fall acceleration therefore the total waveform appears more symmetric.

Next, we dropped our South Unipole prototype v4 ring array inside the solenoid at free fall. The produced induced current signal is shown in fig. 17(a). It shows unambiguously that the middle current segment of the waveform where the whole length of the unipole is introduced inside the solenoid and traveling along its length, has a non-zero net current value. Therefore complying with the theoretical prediction about magnetic monopoles induced current in a solenoid [START_REF] Cunningham | High Energy Physics -Glenn Cunningham -Google Books High Energy Physics (Scientific e-Resources[END_REF]. Continuing, we dropped our smmpa prototype v3 in a North Unipole configuration through the solenoid and the recorded current signal is shown in fig. 17(b). As expected the total signal waveform appears now inverted compared to the South Unipole signal shown in fig. 17(a). As a side note, it is hard not to notice although irrelevant to our discussion here that the total waveform resembles strongly to the letter M. In all smmpa drop experiments the shunt resistor R of 1Ω was replaced with a 10Ω resistor value to increase the signal output measured by the oscilloscope since the measured magnetic strength of the prototypes smmpa are less than that of the typical dipole magnets we used in the experiments. Also, notice that the smmpa prototypes have about the same length as the short ferrite dipole ring magnet of fig. 15(c).

Last but not least, we wanted to see if the length ratio of the solenoid over the unipole prototype length l s /l smmpa , affects in any way the generated current signal waveform. This is demonstrated at fig. 17(c) where we constructed a relative large length pseudo-south unipole magnet by cutting the dipole ferrite magnet of fig. 15(b) in half and forcing together by epoxy gluing the two formed North Pole ends at the middle forming therefore a single S-S pseudo unipole magnet. The recorded current waveform generated by dropping the pseudo-unipole magnet inside the solenoid is shown in fig. 17(c). By comparing fig. 16 with fig. 17 waveforms we can see clearly, in contrast to dipole magnets that size of the unipole magnet has no effect on the general shape and symmetry of the induced current waveform, therefore further complying with the predictions deduced in theory.

Conclusions

By refining our research and observations [START_REF] Markoulakis | Real time observation of a stationary magneton Results[END_REF][16] [START_REF] Markoulakis | Real time visualization of dynamic magnetic fields with a nanomagnetic ferrolens[END_REF] over the last years, we have come to the conclusion that ferromagnetism is a matter net dipole energy vortex macroscopic quantum phenomenon taking place inside the ferromagnet's material thus the net Quantum Magnet Field (QMF) which is different in curl and vorticity from the field outside the magnet on air, its classical macroscopic magnetic N-S vector field. With the net Quantum Magnet dipole vortex field inside the material being the cause and the classical field outside, the effect.

The primary drive for this research presented herein, is for answering the question what would happen if we artificially emulated what an actual Quantum Magnet is doing thus twisting magnetic flux energy into a free vortex whirlpool formation as predicted by Dirac P.A.M. in 1931. To our surprise, but not really, a magnetic pole was created out of the blue, validating our research about the ultimate intrinsic vortex irrotational nature of ferromagnetism, matter phenomenon.

We achieved our goal by using our research and experience in conceiving a novel ferromagnetic array and prototype introducing a unique as far as we known, vortex unipole magnetiztion which twists progressively the net Quantum Magnet flux into a vortex formation within the array's ring air hole at its center. Differentiating from most other research in the literature reported so far, apart that this is a macroscopic research and device for emulating synthetic Dirac unipoles at room temperature and not performed at the microscopic (e.g. Spin Ice) or quantum size level (e.g. BECs), in our case the emergent synthetic magnetic unipole at the center is permanent and is not due and created by the reaction response of another medium but intrinsic to the array itself and its projected Quantum Magnet field. It is possible this to be the first ever reported ferromagnetic Dirac monopole trap that isolates permanently a synthetic magnetic charge taken out within its own ferromagnetic matter. The existence of synthetic magnetic monopoles at room temperature inside ferromagnetic matter was theorized with some indirect experimental data, previously in the literature [START_REF] Fang | The anomalous Hall effect and magnetic monopoles in momentum space[END_REF].

We have herein described in detail and characterized the device prototype, its magnetic field and operation, analyzed the results and experimental data numerically and by simulation and supplement related material [20]. For direct observation of the field we used the quantum magnetic optic flux viewer ferrolens we have used also in our previous research. We regret however that we could not have access for this research to SQUID scanning magnetic microscopy which has a submicron spatial resolution. We reserve this for future research and invite also other researchers to replicate our research. However, we have demonstrated by using experimentally an adapted simplified version of the solenoid induction current detection method [START_REF] Cunningham | High Energy Physics -Glenn Cunningham -Google Books High Energy Physics (Scientific e-Resources[END_REF] for magnetic monopoles in section 3.5 without the use of a SQUID magnetometer, that our macroscopic prototype synthetic magnetic unipole array unambiguously exhibits magnetic monopole behavior and therefore successfully emulates Dirac magnetic monopoles.

Although there are some other proposals and experiments done macroscopically to create or trap synthetic magnetic quasiparticles [START_REF] Béché | Magnetic monopole field exposed by electrons[END_REF], our research as far as we know is the only presently pure magnetic by using normal permanent ferromagnets. The unique signal signature patterns of magnetic monopoles synthetic or natural, passing through a non-ideal ohmic solenoid we recorded for the first time and shown in fig.17 can be used by researches to aid them in the detection of magnetic monopoles.

Fig. 1 (

 1 Fig. 1 (a) Polar view of the field of a neodymium typical ring magnet shown on the ferrolens. The ring magnet has an axial magnetization by thickness (i.e. one face of the ring is the North Pole and the other the South Pole). Ferrolens shows the vorticity and curl of the magnetic flux inside the magnet's bulk material and projects it to the outside. The field displayed is confined inside the ring air cap. (b)&(c) The homogeneous field inside an electrical d.c. solenoid shown in real time by the ferrolens. Magnetic flux is going in straight parallel lines through the solenoid's length from one end (pole) of the solenoid to the other as theory predicts and also shown here by the ferrolens. The see-through property of the lens is demonstrated with holographic depth thus looking the one end of the solenoid we can see the flux lines terminating at the other end. (d) Neodymium cylindrical magnet put on its side under the ferrolens. Its two net Quantum Magnet Field (QMF) polar vortices are shown. Black holes left and right are the positions where the North and South Pole centers are physically located on the magnet. (e) Close-up magnified view of the vortex field inside the pole of a neodymium cube magnet.

Fig. 2 (

 2 Fig. 2 (a) Illustrations of the Top and Top-side views of the 3D printed unipolar magnetic array's prototype v3, frame. (b) N42 neodymium magnetic plates used in the implementation of the unipolar array. Twelve magnetic plates are inserted vertically each into the corresponding frame slots shown. The polarity of the individual magnets inside the array follows a heads to tails pattern N-S-N-S... etc around the ring. (c) Drawing of the unipolar array's final prototype design v4. The type North or South of the emergent unipole formed inside the blue circle area, depends on the angular direction of the of the magnetic moments of the individual twelve magnets in the ring array, clockwise cw or counterclockwise ccw in combination with the 12° skew angle direction of each magnet plate in relation to the concentric radius line each time. Thus, if the magnetic plates at the ring array's perimeter lean to the left or to the right.

Fig. 3 (

 3 Fig. 3 (a) CNC Blue print of final prototype 4. (b)&(c) Top (Active side) and top-side views of prototype version 4. (d) Backside of unipolar array prototype. (e) Field 2D imprint of prototype 4 front active side, shown with a magnetic field viewing film.printer files for prototype version 3 can be found in this link, https://tinyurl.com/ycw9uf5p. Final prototype version 4 was not 3D printed but CNC machined from hard PVC material for extra stiffness. The CNC blue print and final constructed

Fig. 4

 4 Fig. 4 Screw like formation of the unipole vortex at the centre as shown with a magnetic field viewing film.

Fig. 5

 5 Fig. 5 Circular aluminum frame concentric magnetization ring array. No field (i.e. light green color) exists outside the ring bulk material.

Fig. 6 (

 6 Fig. 6 (a) (b) &(c) Plots of z-axis field strength decay with distance measurements of the unipolar array prototype v4. (d) The experimental setup with the prototype and the 3-axis magnetometer. The magnetometer here is measuring the field only on the z-axis. We see in fig.6(a) https://tinyurl.com/y7kb7kjc the linear xy axes plot of the decay of the prototype's z-axis field with distance. Fig.6(b) https://tinyurl.com/yde5bf42 shows the normalized values version plot to the inverse square distance. Ideally if the unipolar prototype complies with the inverse square law with distance, the plot will be a straight line. Similar fig.6(c) https://tinyurl.com/y7hcp2cl shows the normalized values to the inverse cube with distance. Both plots of the front, active side of the array and its backside (blue plot) are shown. Fig.6(d) shows the experimental setup.The z-axis position in space was chosen in such a way that the magnetometer does little as possible register the magnetic N-S vector field of the Earth although a total control experiment is very difficult because the curl of the magnetic flux of Earth's magnetic field and it would require

Fig. 8 (

 8 Fig. 8 (a) Normal ring magnet. (b)&(c) Prototype Unipole.

Fig. 10 (

 10 Fig. 10 (a) Prototype unipolar array placed on top of the ferrolens. (b) Black and white photograph of the isolated unipolar magnetic vortex field of the prototype array as shown by the ferrolens. (c)&(d) color photographs of the unipolar magnetic flux vortex formed and displayed by the ferrolens. Photographs were taken through the 8mm diameter hole, front and backside of the prototype ring array's plastic frame.

Fig. 11

 11 Fig. 11 Hedgehog field lines displayed by the ferrolens, diverging out of the unipolar magnetic ring array's vortex field formed at the centre of the ring air capped hole.

Fig. 13 (

 13 Fig. 13 (a) Simulated vector hedgehog field of the South unipole. (b) Simulated vector field of the whole unipole array.

Fig. 14 SQUID

 14 Fig. 14 SQUID Magnetic Monopole Detector Experimental Setup.

Fig. 15 (

 15 Fig. 15 (a) Experimental setup, a 15cm length solenoid about 750 turns of 0.2mm copper insulated wire coiled on a 25cm long 3D printed plastic pipe. Both wire ends of the solenoid are connected via a 1Ω shunt resistor and measured by a digital oscilloscope. The smmpa unipole ring array prototype and a large length ferrite dipole ring magnet are shown and used in the experiments free falling inside the solenoid vertically and the induced current signal on the solenoid was recorded. (b) Large length ferrite dipole magnet used in the experiments. (c) Short length ferrite dipole magnet used in the experiments.

Fig. 16

 16 Fig. 16 Recorded induced current signals from free fall dipole magnet experiments through a solenoid.

Fig. 17

 17 Fig. 17(a) South Unipole prototype solenoid current signal. (b) North Unipole prototype solenoid current signal. (c) South unipole magnet solenoid current signal.

  Fig. 17(a) South Unipole prototype solenoid current signal. (b) North Unipole prototype solenoid current signal. (c) South unipole magnet solenoid current signal.

Table . 1

 . Unipolar Array Combinations

	Magnetic moment	Leaning of	Resulting unipole
	angular direction	magnetic plates
	CW	Right	N
	CW	Left	S
	CCW	Right	S
	CCW	Left

Acknowledgements

The authors are grateful to I. Rigakis, L. Frantzeskakis and Prof. E. Maravelakis for their assistance in the prototype versions construction and solenoid as well to B. Kerr, M. Snyder and T. Vanderelli for their support.