Kaushik Ghosh
email: kaushikhit10@gmail.com

Affine

Affine Connections in Quantum gravity and New Fields

Keywords: affine connections, dark energy, inflation, anomaly, little group MSC: 53Z05, 83D05, 81T20, 81T50, 83C45, 20A10 PACS: 02.40.-k, 98.80.Qc, 98.80.Cq, 95.36.+x, 04.50.Kd

come

I. Introduction

A lot of works have been done during the last few decades on two periods of cosmic accelerations: inflation and dark energy [START_REF] Starobinisky | A New Type of Isotropic Cosmological Models Without Singularity[END_REF][START_REF] Guth | Inflationary universe: A possible solution to the horizon and flatness problems[END_REF][START_REF] Spergel | First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters[END_REF][START_REF] Spergel | Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Cosmology[END_REF][START_REF] Komatsu | Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation[END_REF][START_REF] Amendola | Dark Energy[END_REF]. Inflation is driven by the scalar field inflation. We also have multi-scalar inflation and higher spin driven inflation models [START_REF] Dodelson | Modern Cosmology[END_REF][START_REF] Kaloper | Lorentz Chern-Simons terms in Bianchi cosmologies and the cosmic no hair conjecture[END_REF][START_REF] Golovnev | Vector Inflation[END_REF]. Dark energy is not yet detected directly. It is not localized like ordinary matter and is assumed to have negative pressure to explain present cosmic acceleration. We also need dark matter which are observed by their gravitational effects [START_REF] Amendola | Dark Energy[END_REF][START_REF] Zwicky | Die Rotverschieb ung von extragalaktischen Nebeln[END_REF][START_REF] De Felice | f(R) theories[END_REF][START_REF] Vollick | 1/R curvature corrections as the source of the cosmological acceleration[END_REF]. Data from the Cosmic Microwave Background Radiation (CMBR) and supernovae surveys predict that the energy composition of the universe consists of about 20% dark matter, about 70% dark energy and the rest ordinary matter like hadrons and radiation. The first candidate for dark energy is the cosmological constant Λ [START_REF] Amendola | Dark Energy[END_REF][START_REF] Weinberg | The Cosmological Constant Problem[END_REF][START_REF] Kobayashi | Foundation of Differential Geometry[END_REF]. There are two principal alternate approaches. The first adds stress-tensors with negative pressure in the source part of Einstein equation. Cosmons, quintessence, k-essence and perfect fluid models are most well-known among various theories [START_REF] Fujii | Origin of the gravitational constant and particle masses in a scale-invariant scalar-tensor theory[END_REF][START_REF] Chiba | Kinetically driven quintessence[END_REF][START_REF] Kamenshchik | An alternative to quintessence[END_REF]. The other approach modifies the Einstein-Hilbert action. Examples are f (R) gravity [START_REF] Vollick | 1/R curvature corrections as the source of the cosmological acceleration[END_REF][START_REF] Capozzillo | Curvature Quintessence[END_REF], scalar-tensor theories [START_REF] Amendola | Scaling solutions in general nonminimal coupling theories[END_REF][START_REF] Uzan | Cosmological scaling solutions of nonminimally coupled scalar fields[END_REF] and braneworld models [START_REF] Dvali | Porrati: 4-D gravity on a brane in 5-D Minkowski space[END_REF][START_REF] Sahni | Braneworld models of dark energy[END_REF]. Despite a lot of efforts, the origins of the inflation, dark energy and dark matter are yet to completely fit within an extended version of the standard model.

In a few recent manuscripts [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF], we have discussed if a quantum theory of gravity can introduce additional fields besides metric that can be useful to explain inflation and dark energy. The natural candidates are affine connections used to define covariant derivative in a curved spacetime [START_REF] Hocking | Topology[END_REF][START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF][START_REF] Landau | The Classical Theory of Fields[END_REF][START_REF] Wald | General Relativity[END_REF][START_REF] Misner | Gravitation[END_REF]. We found it is more appropriate to use affine connections more general than the metric compatible Christoffel/Levi-Civita connections in a quantum theory of gravity [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF]. This is related to the consistency of canonical commutators and various metric-metric commutators used to quantize the theory. A theory involving curvature is not primarily a theory of metric but in general is a theory of non-trivial connections [START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF]. Hence, it is not bizarre that we can have non-metricity [START_REF] Hehl | Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance[END_REF][START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF] in a quantum theory of gravity. Regarding variational principles, the simplest theory to introduce connections more general than the Levi-Civita connections is to consider the Einstein-Palatini action [START_REF] Wald | General Relativity[END_REF]. Here, the connections are expressed as: Θ α µν = Γ α µν + C α µν , where Γ α µν are the metric compatible Levi-Civita connections and C α µν is a third rank tensor field that can introduce non-metricity. The Lagrangian density is given by √ -gR , where R is the scalar curvature evaluated using the connections Θ α µν . In [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF], we have used a potential-like formalism to express C α µν as various covariant derivatives of lower rank tensor fields. This is similar to the Levi-Civita connections which are constructed from metric. We have introduced two scalar fields that obey homogeneous wave equation in the free theory and can be non-localized. These scalar fields are related to two traces of C α µν , one w.r.t (µ, ν) and the other w.r.t (α, µ). They provide positive and negative stress-tensors in Einstein's equation and thereby can play important role in different cosmological accelerations. These fields give non-metricity although their effects are geometrically different. One of these fields can give us departure from global splitting of spacetime into space and time. In the present article, we will elaborate this in the next section. These fields are similar to the cosmological constant Λ that give deviation from the Newtonian theory of gravity in the weak field slow motion limit [START_REF] Wald | General Relativity[END_REF]. Various astrophysical observations indicate that presently non-metricity produced by these scalar fields is small. This is similar to the cosmological constant in the ΛCDM theory. In this article, we will mostly consider affine connections that are symmetric in the lower indices and give symmetric Ricci tensor. In Sect.II, we will first give an improved discussion of the above mentioned scalar fields and their relevance in cosmology. We will first discuss the scalar field φ that arises when we construct a set of affine connections compatible with a metric conformal to the actual solution. This introduces a massless scalar field similar to dilaton. We will also introduce a massless scalar field ψ that is more significant as far as non-metricity is concerned.

In Sect.III, we will discuss possible coupling between φ, ψ and ordinary fields that are various representations of SL(2, c). In this context, a careful application of the Stokes' theorem reveals that we may need to introduce the right-handed neutrinos as free fields in the electroweak theory in curved spacetime. This remains valid even when we use only the Levi-Civita connections and is required to define anomaly-free vector currents for the neutrinos. The right-handed neutrinos can be important in dark matter research. There are other advantages related with neutrino oscillations to be discussed later. Stokes' theorem also indicates that the axial vector currents for various Dirac fields including that of neutrinos can become anomalous in curved spacetime as well as in flat spacetime when we use the curvilinear coordinates. This is not rare in quantum field theory in flat spacetime. We will demonstrate that in a mathematically consistent Lagrangian description that preserves the principle of general covariance and principle of equivalence, it is sufficient to use Γ α µν as connections to construct actions and conserved stress tensors of ordinary matter and radiation fields. We will find that for fields relevant to the standard model with the addition of the right handed neutrinos, C α µν (φ, ψ) can at most contribute boundary terms and do not appear in the equations of motion. Thus, φ and ψ need not to be included in the actions and stress tensors of ordinary matter and radiation. We will find in Sect.IV that this may also be required to construct a stable theory. This agrees with experimental observations regarding dark energy which does not couple with ordinary matter and radiation. The resulting equations are consistent with the contracted Bianchi identity in Γ α µν . Unless stated otherwise, in this article contracted Bianchi identity will mean twice contracted Bianchi identity. We have found previously [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF] that φ contributes a positive stress tensor and ψ contributes a negative stress tensor in Einstein's equation. Thus, a theory that includes φ and ψ can be useful to explain inflation and dark energy.

In Sect.IV, we will discuss possible modifications of Einstein's theory of gravity in presence of φ and ψ. The equations for φ and ψ are coupled through derivative terms. We will find that we can separate the equations by introducing two scalar fields Φ and Ψ that are linear combinations of ω = ln(1 + φ) and ψ. Φ contributes a positive stress tensor to Einstein's equation and is a possible candidate for inflation. Ψ contributes a negative stress tensor and is a possible candidate for dark energy. When required, we can supplement the Einstein-Palatini action by Φ and Ψ dependent potential terms. We can couple Φ and Ψ by adding suitable potential terms. We can also couple Φ with ordinary matter and radiation fields. This gives a general version of scalar-tensor theories. Another way to generalize Einstein's theory is to include Φ, Ψ and the cosmological constant Λ. This will lead to a possible theory of inflation together with dynamic and spatially varying dark energy.

In section:V, we will first discuss the possibility of introducing higher spin fields and additional scalar fields. We will find that we need to modify the Einstein-Hilbert-Palatini action if we want to introduce higher spin fields using the affine connections. In flat spacetime, second rank symmetric traceless tensors give (A, A) type representations of higher spin fields like j = 1, 2. The most general candidate to give the little group (stabilizer) for massless fields in Minkowski space is taken to be: W = ISO(2) [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. However, this group is not represented by normal matrices when act on four vectors/tensors. We will find that we have to use a one parameter subgroup of SO(3) as the little group for massless fields in Minkowski space. This allows us to construct a (A, A) type quantum field theory for massless vector fields using symmetric traceless derivatives. Such derivatives are not gauge invariant and can not couple with matter fields in a theory that preserves local internal gauge invariance. They can be useful to describe dark energy and dark matter. The massless higher spin particles can also be useful to construct higher spin driven inflation theories [START_REF] Dodelson | Modern Cosmology[END_REF][START_REF] Kaloper | Lorentz Chern-Simons terms in Bianchi cosmologies and the cosmic no hair conjecture[END_REF][START_REF] Golovnev | Vector Inflation[END_REF] in absence of spontaneous symmetry breaking that need not to be gauge theories.

II. Affine Connections and Scalar Fields

In this section, we will consider a potential-like formalism of the Palatini theory developed in [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF]. This formalism introduces finite on shell non-metricity [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF]. The connections are expressed as: Θ α µν = Γ α µν +C α µν and C α µν are symmetric in the lower indices. Here, we will mostly consider symmetric Ricci tensor to compare with the semi-classical limit of quantum gravity. The later is locally Lorentz invariant quantum field theory in curved spaces. We will use the geometrized units in the following where, h = G = c = 1. The flat spacetime metric is taken to be of signature +2.

In [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF], we have mentioned a few simple ways to introduce non-metricity. We first consider a set of affine connections compatible with a non-singular symmetric covariant field: b µν = g µν + a µν ; a µν = kg µν , where g µν is the metric and k is a constant. These connections give symmetric Ricci tensor. We will return to this aspect later. We first break a µν into a trace and a traceless part:

a µν (x) = φ(x)g µν + āµν (x); φ(x) = a(x) 4 (1)
where, φ is a scalar field, a(x) is the trace of a µν and āµν is trace-free. Corresponding connections can be expressed in the following way [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF][START_REF] Wald | General Relativity[END_REF][START_REF] De Felice | Relativity on Curved Manifolds[END_REF]:

∇ µ A ν = ∂ µ A ν -Θ α µν A α (2) Θ α µν = Γ α µν + C α µν = Γ α µν + δ α (µ ∇ ν) [ln(1 + φ)] - 1 2 g µν ∇ α [ln(1 + φ)] + D α µν
Here Θ α µν satisfy the compatibility conditions: b µν|α = 0, where the bar denotes covariant derivative with connections Θ α µν . Γ α µν are the Levi-Civita connections and ∇ µ are evaluated using Γ α µν . Θ α µν and C α µν are symmetric in the lower indices. C α µν can be expressed in terms of g µν , a µν and their derivatives. The first three terms in the r.h.s of the last line give the contribution of the trace part of b µν given by: g µν + φ(x)g µν . D α µν involve āµν (x). We find:

Θ α αν = ∂ ν (ln |b|) (3) = Γ α αν + C α αν = ∂ ν (ln |g|) + C α αν
The determinants b and g are relative scalars of weight two and transform similarly. Thus, C α αν can only be the gradient of scalar fields. This gives us a symmetric Ricci tensor which, in the present context, is given by [START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF][START_REF] Wald | General Relativity[END_REF][START_REF] De Felice | Relativity on Curved Manifolds[END_REF]:

(∇ µ ∇ ν -∇ ν ∇ µ )A α β = -R α µνκ A κ β + R κ µνβ A α κ (4) R κ µνα = R κ µνα + 2∇ [ν C κ µ]α + 2[C λ [µ|α| C κ ν|λ|] ] R µα = R µα + 2∇ [κ C κ µ]α + 2[C λ [µ|α| C κ κ|λ|] ]
where ∇ µ is the covariant derivative evaluated using the complete connection Θ α µν , ∇ µ is the covariant derivative evaluated using Γ α µν , R κ µνα is the curvature tensor for Γ α µν and R µα is the Ricci tensor associated with the Levi-Civita connections. In flat spacetime, a second rank symmetric traceless tensor give (1, 1) representation of the homogeneous Lorentz group and can describe a j = 0, j = 1 and j = 2 field, [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. We will discuss this aspect further in Sect.V with reference to the contracted Bianchi identities. Thus, we can reduce D α µν further to obtain the following expression for Θ α µν :

Θ α µν = Γ α µν + δ α (µ ∇ ν) [ln(1 + φ)] - 1 2 g µν ∇ α [ln(1 + φ)] (5) +δ α (µ ∇ ν) ξ + U α µν .
It can be shown that ξ contributes a negative stress-tensor in the Einstein's equation. The same remains valid if we make δ α (µ ∇ ν) ξ traceless in (α, µ). ξ and U α αν are associated with āµν (x). Another simple way to introduce non-metricity in the spacetime manifold is to consider the connections to be given by:

Θ α µν = Γ α µν + ∇ α q µν , q µν = q νµ (6) = Γ α µν + g µν ∇ α ψ + ∇ α qµν ; ψ = 1 4 q µ µ , q µ µ = 0 = Γ α µν + g µν ∇ α ψ + δ α (µ ∇ ν) η - 1 4 g µν ∇ α η + V α µν
where the we have extracted η from ∇ µ qµν . The corresponding term is traceless in the lower indices. ψ contributes a negative stress-tensor while η contributes a positive stress-tensor to Einstein's equation. It is possible to introduce a spin 1 field by using traceless symmetric derivatives in qµν , [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. However, the Ricci tensor will not remain symmetric for general q µν . In the following, we will first consider an important special case where C α µν contains only φ and ψ. φ and ψ are related to two possible traces of C α µν . We will later consider the more general cases including the possibility of introducing new vector fields using the present formalism.

We thus consider the case when āµν and qµν vanish but both φ and ψ are present. This helps us to avoid the problems associated with quantization and connections discussed in [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF]. The connections are no longer compatible with g µν , b µν and q µν . C α µν is given by the following expression [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF]:

Θ α µν = Γ α µν + C α µν (7)
C α µν = δ α (µ ∇ ν) ω - 1 2 g µν ∇ α ω + g µν ∇ α ψ, ω = ln(1 + φ)
The Einstein-Palatini action is now given by the following expression:

S = √ -gR e + κ M S M (χ a , g µν ) = √ -gR e + κ M S M (χ a , g µν ) ( 8
)
where e is the coordinate volume element dx 0 ∧dx 1 ∧dx 2 ∧dx 3 [START_REF] Wald | General Relativity[END_REF], χ a represents collection of ordinary matter and radiation and S M is the corresponding action. In the last expression we have removed total divergence terms coming from the second term, 2∇ [κ C κ µ]α , present in the expression of R µα given by Eqn.( 4). This requires to fix the normal derivatives of φ and ψ on the boundary or we may use periodic boundary conditions. We can also modify the action by introducing suitable boundary terms with the boundary integrand given by: -C α µ µ + C µ α µ . This is similar to that done for metric [START_REF] Wald | General Relativity[END_REF]. We introduce the modified Ricci tensor R µν and modified curvature scalar R given by the following expressions:

R = g µα R µα = g µα R µα + 2[C λ [µ|α| C κ κ|λ|] ] (9) = R - 3 2 (∇ω) 2 + 3(∇ψ) 2 + 3[(∇ κ ω)(∇ κ ψ)]
where R µα and R are the Ricci tensor and curvature scalar associated with the Levi-Civita connections respectively. We obtain the field equations by considering various variational derivatives. Here, we assume that φ and ψ are not present in S M . This is consistent with observations. We will illustrate this later. We have the following modification of the r.h.s of Einstein's equation [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Wald | General Relativity[END_REF]:

G µα = 8π[P µα (χ a ) + 3 16π P µα (ω) - 3 8π P µα (ψ) - 3 8π P µα (ω, ψ)] (10)
P µα (ψ, φ) = (∇ (µ ψ)(∇ α) ω) - 1 2 g µα (∇ κ ψ)(∇ κ ω) = 1 (1 + φ) [(∇ (µ ψ)(∇ α) φ) - 1 2 g µα (∇ κ ψ)(∇ κ φ)] ∇ κ ∇ κ ω = 1 1 + φ ∇ κ ∇ κ φ -( 1 1 + φ ) 2 (∇φ) 2 = 0 ∇ κ ∇ κ ψ = 0
where P µα (χ a ) is the stress tensor of ordinary matter and radiation. ω and ψ obey the massless Klien-Gordan equation that have non-trivial solutions for most spacetimes. The above equations are consistent with the contracted Bianchi identity associated with Γ α µν : ∇ α G µα = 0. We will discuss this aspect further in the following section. P µα (ω) and P µα (ψ) are the stress tensors of ordinary massless scalar fields. ω together with suitable potential terms can be useful to explain inflation. ψ contributes a negative stress tensor to Einstein equation. This gives an alternate way to explain dark energy. Further significance of negative stress tensor can be found in [START_REF] Narlikar | Mini-bangs in cosmology and astrophysics[END_REF][START_REF] Ford | Negative Energy, Wormholes and Warp Drive[END_REF]. We find that coupling of ω with ψ in R gives another contribution to source stress tensor. We will later show that the equations can be separated in the scalar fields.C α µν in Eq.( 7) are now given by the solutions of the above equations. P µα (χ a ) will be multiplied by appropriate factors when we transform to non-geometrized ordinary units.

We now discuss the geometrical significance of the non-metricities associated with φ and ψ. Non-metricity is defined in terms of the non-metricity tensor Q µαβ [START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF]:

Q µαβ = -∇ µ g αβ ( 11
)
where ∇ µ is evaluated using the complete connection: Γ α µν + C α µν . We have the following expressions for Q µαβ :

Q µαβ = g αβ ∇ µ [ln(1 + φ)]; 2g µ(α ∇ β) ψ (12)
The first expression is valid when only φ is finite while the second one is valid only when ψ is finite. Both φ and ψ are present in the manifold in a theory given by Eq. [START_REF] Zwicky | Die Rotverschieb ung von extragalaktischen Nebeln[END_REF]. They are given by solutions of the homogeneous wave equation in Eq. [START_REF] Zwicky | Die Rotverschieb ung von extragalaktischen Nebeln[END_REF]. Resulting Q µαβ is the sum of the above two expressions. We can express Q µαβ in the following way [START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF]:

Q µαβ = Q µ g αβ + Qµαβ ; ( 13
)
where Qµαβ is traceless in the last two indices. Qµαβ vanishes for φ. Corresponding C α µν preserves the light cone under parallel transport due to the reparameterization invariance of the form of the geodesic equation:

t µ ∇ µ t ν = f t ν ,
where f is a scalar function on the curve [START_REF] Wald | General Relativity[END_REF]. Here, we can use a parameterization so that:

f = 1 2 t µ ∇ µ [ln(1 + φ)].
Examples of non-metric affine connections with vanishing Q are given in [START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF]. These are local gauge theories of the Weyl group that can have torsion. Both trace and traceless parts of Q µαβ are finite for ψ. Depending on the strength of ψ, corresponding connections may not preserve the light cone under parallel transport [START_REF] Hehl | Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance[END_REF][START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF]. Global (3 + 1) -splitting of the complete spacetime manifold into space and time may not exist under this circumstance.

III. Matter Fields in Presence of Non-metricity Scalar Fields

In this section we will discuss possible coupling between φ, ψ and ordinary matter fields including radiation. The fields φ and ψ are introduced to have finite non-metricity Q µαβ . They are required in quantum theory of gravity. This is discussed in detail in [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Ghosh | Non-metric fields from quantum gravity[END_REF]. We can introduce the scalar fields like φ, ψ without reference to āµν and qµν . The last two fields are introduced to have higher spin fields. We will discuss the possibility of introducing higher spin fields using āµν and qµν in Sect.V. We have to consider a few points when we try to construct a Lagrangian description of gravity coupled with matter fields with a set of connections more general than the Levi-Civita connections:

(1) The first is regarding the Euler-Lagrange's equations, Bianchi identity and its contracted forms. We do not have any problem with these identities for R κ µνα given by Eq.( 4) when derivatives are taken w.r.t the total connection Θ α µν . ω and ψ in Eq.( 7) can be any pair of regular fields. We can in general have problems when we do away with the total divergence term in R given by 2g µα ∇ [κ C κ µ]α and apply the principle of least action to S given by Eq.( 8). It is required to have: ∇ α G µα = 0, the contracted Bianchi identity w.r.t Γ α µν , when we express Einstein's equation in the form given by Eq. [START_REF] Zwicky | Die Rotverschieb ung von extragalaktischen Nebeln[END_REF], where the fields associated with C α µν contribute source stress tensor terms in the right hand side. This remains valid for ω and ψ individually. The only scalar quadratic in the first order derivative of a scalar field is square of the norm of covariant derivative and corresponding term in √ -gR coincides with the Lagrangian density of a massless scalar field apart from numerical factors. This leads to familiar conserved stress tensors of scalar fields. The contracted Bianchi identity w.r.t Γ α µν also remains valid for Eq.( 10) where we have derivative coupling between ω and ψ. We can also include various potential terms to be discussed later. This may not be the situation when we include higher spin fields in C α µν .

(2) The second is the possibility of removing the total divergence term:

√ -gg µα ∇ [κ C κ µ]α d 4
x by applying the Stokes' theorem. This is possible when the corresponding connections in ∇ κ are Γ α µν [START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF][START_REF] Wald | General Relativity[END_REF]. We can add a set of C α µν only when the total connection is metric compatible leading to: C α αν = 0, [START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF]. The same conditions remain valid when C α µν is not symmetric in the lower indices. Thus, we have used Γ α µν in ∇ µ . This also agrees with a local theory based on the principle of equivalence considered below. Same discussions remain valid with ordinary matter field actions: S M = d 4 xL M . While deriving the corresponding Euler-Lagrange's equations, we have to eliminate the four-divergence term:

∇ µ [ ∂L M ∂(∇ µχa )
δχ a ] by applying the Stokes' theorem. Here, L M is the matter Lagrangian density. This is not possible until the conditions mentioned above are imposed on the connections in ∇ µ . Otherwise, the functional derivative δS M δχa will contain the additional term:

-C α αν [ ∂L M ∂(∇ ν χa) ],
where the connections are given by Θ α µν . This will in general not lead to a conserved stress tensor w.r.t Γ α µν in the r.h.s of Einstein's equation given by Eq.( 10). This problem does not arise for scalar fields (including many non-linear models, [START_REF] Itzykson | Quantum Field Theory[END_REF]) and gauge invariant stress tensor of electromagnetic fields described by the Lagrangian density: -1 4

√

-gF µν F µν . We get the following set of equations for free electromagnetic theory:

∇ µ F µν = -C ν αβ F αβ = 0 (14) ∇ [µ F νκ] = 0
The right hand side of the first equation is replaced by a current conserved w.r.t ∇ µ when we have sources. Similar situation remains valid with nonabelian gauge theories. We have, [START_REF] Itzykson | Quantum Field Theory[END_REF]:

F µνa = ∇ µ A νa -∇ ν A µa -θ bc a A µb A νc (15)
∇ µ F µν a -θ bc a A µb F µν c = 0, θ bc a = -θ cb a
where θ bc a are the structure constants of the gauge group. We now consider the Dirac fields described by the Lagrangian density [START_REF] Weinberg | The Quantum Theory of Fields[END_REF][START_REF] Birrell | Quantum Fields in Curved Spaces[END_REF][START_REF] Weinberg | Gravitation and Cosmology[END_REF]:

L = - √ -g{ 1 2 [ χγ µ ∇ µ χ -(∇ µ χ)γ µ χ] + m χχ}, γ µ = V µ a γ a (16)
where χ are the SL(2, c) spinors, V µ a are the vierbeins [START_REF] Birrell | Quantum Fields in Curved Spaces[END_REF][START_REF] Weinberg | Gravitation and Cosmology[END_REF] and γ a are the flat spacetime Dirac matrices. The covariant derivative on the SL(2, c) spinors χ are given by [START_REF] Birrell | Quantum Fields in Curved Spaces[END_REF][START_REF] Weinberg | Gravitation and Cosmology[END_REF]:

∇ µ = ∂ µ + 1 2 σ bc V ν b (∇ µ V cν ) (17) = ∇ µ - 1 2 σ bc V ν b V cα C α µν , σ bc = - i 4 [γ b , γ c ] = ∇ µ + C µ
For C α µν given by Eq.( 7), L simplifies to:

L = - √ -g{ 1 2 [ χγ µ ∇ µ χ -(∇ µ χ)γ µ χ] + m χχ - 3i 4 ( χγ µ χ)∇ µ (ω -ψ)} (18)
We can neglect the second term in a theory where the Dirac spinors are coupled with gauge fields through conserved currents proportional to χγ µ χ. This includes quantum electrodynamics. Gauge fields are described as before. Thus, in all four cases mentioned above we can replace Θ α µν by Γ α µν in L M and express the later as L M . We can use L M to construct corresponding matter field stress-tensors in ∇ µ . Such stress-tensors lead to Einstein's equation consistent with the Bianchi identity when we add them in the r.h.s of Eq. [START_REF] Zwicky | Die Rotverschieb ung von extragalaktischen Nebeln[END_REF].

Things become more complicated with weak interaction that include relative (axial) vector currents [START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF][START_REF] Itzykson | Quantum Field Theory[END_REF]. To illustrate, we consider a simple model of electroweak theory [START_REF] Itzykson | Quantum Field Theory[END_REF] without the complex scalar fields and use the Levi-Civita connections:

L = √ -g[- 1 4 A µν . A µν - 1 4 B µν B µν (19)
-ēR (γ µ ∇ µ + ig γ µ B µ )e R -Le (γ µ ∇ µ + i g 2 γ µ B µ -i g 2 σ i γ µ A i µ )L e ] L e = ν L e L , e R/L = 1 2 (1 ± γ 5 )χ e , ν L = 1 2 (1 -γ 5 )χ ν
where χ e and χ ν are the Dirac fields for the electrons and neutrinos respectively. Other symbols have usual meanings [START_REF] Itzykson | Quantum Field Theory[END_REF]. As before, ∇ µ is evaluated using the Levi-Civita connections. We have discussed the free gauge field terms before. Here, we first consider the Dirac field terms that are only coupled with gravity. This part can be expressed as:

LD = √ -g[-χe γ µ ∇ µ χ e ( 20
)
- 1 2 χν γ µ ∇ µ χ ν + 1 2 χν γ µ γ 5 ∇ µ χ ν ]
We do not have any problem with the electron field part. However, the variational derivative:

δ SD δχν (x) is non-covariant. This is because √ -g[ χν γ µ γ 5 (δχ ν )
] is a vector and we can not eliminate the total divergence in e∇ µ [ √ -g( χν γ µ γ 5 δχ ν )] completely by using the Stokes' theorem [START_REF] Lovelock | Tensors, Differential Forms, and Variational Principals[END_REF][START_REF] Wald | General Relativity[END_REF]. We can eliminate total divergence terms by using the Stokes' theorem for relative vector fields of weight +1. We find that:

δ SD δ χν (x) = - √ -gγ µ ∇ µ χ νL (x) and δ SD δχν (x) = √ -g[∇ µ χνL (x)γ µ + 1 2 Γ µ µν ( χν γ ν γ 5 )].
We also find that we can not derive a consistent pair of equations for χ ν and χν by applying the principle of least action directly to the above action when χν is taken to be the adjoint of χ ν given by χ † ν γ 0 . The later choice is required to construct various physical variables like stress-tensor from LD and will be used in the present article. We do not have problems when: Γ µ µν = 0, which is not valid in general. We solve this problem by adding the kinetic term for ν R . Thus, we modify and LD to:

L D = - 1 2 [ χe γ µ ∇ µ χ e -(∇ µ χe )γ µ χ e + χν γ µ ∇ µ χ ν -(∇ µ χν )γ µ χ ν ] (21)
The equations for ν L and νL can be obtained from that of χ ν and χν . There are other important advantages.

We will find below that L D can be used to construct an expression for conserved vector current for the neutrino while this is not valid for LD . These comments remain valid in flat spacetime with curvilinear coordinates. In flat spacetime, the right-handed neutrinos can be useful to remove triangle anomalies in the lepton sector [START_REF] Yanagida | Horizontal symmetry and mass of the t quark[END_REF], and can be important to explain neutrino oscillation [START_REF] Yanagida | Horizontal Symmetry and Masses of Neutrinos[END_REF]. They can also be relevant in the dark matter research. We now consider possible conserved currents associated with LD , L D and the theory that we obtain when we replace LD by L D in Eq.( 19). (i) We find that the global phase transformation: χ → e -ic χ, where c is a constant, is an exact symmetry of both LD and L D . We generate a conserved vector current for the Dirac spinors by generalizing to local phase transformation: χ e/ν → e -iα(x) χ e/ν and define the current as: √ -g∇ µ j µ = i δS M δα(x) , [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. We have: ∇ µ j µ = 0, for the classical fields. Current conservation also follows from the equations of motion directly. In an anomaly free gauge theory minimally coupled with the Dirac fields, the current remains conserved and the action remains invariant under local phase transformation even for the non-classical configurations relevant to the quantum theory. Thus, we can define the conserved current using the above expression with δS M δα(x) = 0, [START_REF] Preskin | An Introduction to Quantum Field Theory[END_REF]. We do not have any problem with the electrons. However, with LD , we can not define a covariant functional derivative δS M δα(x) for the neutrinos. This is because, as before, we can not apply the Stokes' theorem to the vector: √ -gα(x)( χν γ µ γ 5 χ ν ), where α(x) is any regular scalar field and hence, can not define a covariant functional derivative δ δα(x) of the functional: e √ -g[ χν γ µ (1 -γ 5 )χ ν ]∇ µ α. Similar comments can remain valid for curvilinear coordinates in flat spacetime. We do not have such problems with L D . The right-handed neutrinos had been used before to cancel anomalies [START_REF] Yanagida | Horizontal symmetry and mass of the t quark[END_REF]. Thus, L D is also more appropriate to define observed conserved vector current for the neutrinos. (ii) χ → e -icγ5 χ is another symmetry transformation [START_REF] Preskin | An Introduction to Quantum Field Theory[END_REF] of both LD and L D , although we can not define axial vector currents using δS M δα(x) when we generalize c to α(x). This remains valid for both LD and L D and is due to the reasons discussed before regarding the Stokes' theorem in curved spacetime. Axial vector currents like χγ µ γ 5 χ remain conserved for the classical solutions by the equations of motion obtained from L D . Various axial vector currents may remain approximately conserved in the quantum theory in presence of gravity. However, the comments regarding Stokes' theorem in curved spacetime remain valid for curvilinear coordinates in flat spacetime. Thus, we expect anomalies for various axial vector currents in a naive interacting theory even in flat spacetime. Anomalies in the axial vector currents are common in flat spacetime when the fermions are coupled with various gauge fields through the axial vector currents [START_REF] Itzykson | Quantum Field Theory[END_REF][START_REF] Preskin | An Introduction to Quantum Field Theory[END_REF]. This includes pion decay. We thus replace Eq.( 19) by the following expression:

L = √ -g{- 1 4 A µν . A µν - 1 4 B µν B µν (22) - 1 2 [ χe γ µ ∇ µ χ e -(∇ µ χe )γ µ χ e + χν γ µ ∇ µ χ ν -(∇ µ χν )γ µ χ ν ] + ig ēR γ µ B µ e R + i g 2 Le γ µ B µ L e -i g 2 Le σ i γ µ A i µ L e }
(iii) L is invariant under simultaneous gauge and local phase transformations of various Dirac fields with e L and e R transforming differently due to parity violating effects. The same remains valid for L with ν R remaining unchanged under such transformations. However, S M obtained from L is not invariant under pure gauge transformations: A µ → A µ + ∇ µ λ(x), without any change in the phases of various Dirac fields. This is consistent with (i) and (ii) and indicate possible axial current anomaly. The same remains valid for L. Note that for a pure gauge transformation without any change in the phases of Dirac fields, the variation of the action comes from the coupling terms between the potentials and fermionic bilinear forms: χγ µ M χ, M = 1, γ 5 . As before, Stokes' theorem implies that δS M δλ(x) is not tensorial in general under pure gauge transformations even if we impose ∇ µ ( χγ µ χ) = 0. Thus, with L we can not define conserved currents by using: δS M δλ(x) . This is the reason that we have started with invariance under phase transformations of Dirac fields to define possible conserved currents in the present theory rather than the gauge transformations [START_REF] Preskin | An Introduction to Quantum Field Theory[END_REF]. Similar aspect remains valid with curvilinear coordinates in flat spacetime and we can have axial vector current anomaly in flat spacetime as mentioned before. The above discussions indicate that various regularization schemes used in the quantum theory need to preserve pure gauge transformation and corresponding local phase transformations in the Dirac fields mentioned before. We can still have anomalies in axial vector currents depending on L, regularization schemes and possibly boundary conditions [START_REF] Preskin | An Introduction to Quantum Field Theory[END_REF][START_REF] Weinberg | The Quantum Theory of Fields[END_REF][START_REF] Ramond | Field Theory: A Modern Primer[END_REF]. (i), (ii) and (iii) can be important in the early universe.

We now replace the Levi-Civita connections in Eq.( 22) by Θ α µν . As in the case of Eq.( 16), it is easy to show that ω and ψ can be eliminated from the action obtained from L. This helps us to take L (Θ α µν ) = L(Γ α µν ) for the Dirac fields in the electroweak theory with L suitably generalized from Eq.( 22) by adding the complex scalar fields [START_REF] Itzykson | Quantum Field Theory[END_REF]. Thus, we can construct the standard model using ∇ µ only. C α µν given by Eq.( 7) do not appear in the equations of motion when we express them in terms of ∇ µ and also in the corresponding stress tensors. This partly justifies our efforts to explain dark energy using ψ. We conclude that we do not have any problem with the contracted Bianchi identity when we generalize the standard model to curved spacetime described by the connections Θ α µν provided we modify the action so that we do not have any problem with Γ α µν . The right-handed neutrino, ν R , remains a free field and can be important for various purposes mentioned before including QFT in flat spacetime. We can follow (3) below not to have contradiction with the contracted Bianchi identity when we use Eq.( 19) to describe electroweak theory. Otherwise, we have coupling between ω, ψ and neutrinos that can give deviations from the semiclassical theory, quantum fields in curved spacetime. The coupling term is of the form:

√ -g[ χν γ µ (1 -γ 5 )χ ν ]∇ µ (ω -ψ).
We can have coupling between (φ, ψ) and general tensor fields that are not antisymmetric. In this case also we can follow (3) below. It is unexpected that ω and ψ will couple with classical sources if they do not couple with quantum fields. Torsion will be discussed later.

(3) When we try to construct a theory of gravity with sources we supplement the r.h.s of Eq.( 10) with stress tensors that are symmetric and conserved w.r.t ∇ µ . Stress tensor of ideal fluid does not contain the connections and is observed to be conserved w.r.t Γ α µν . Quantum fields are introduced in flat spacetime as different irreducible representations of SL(2, c), parity and various internal symmetries. Corresponding actions are generalized to local Lorentz invariance in curved spacetime to couple with gravity. This is according to the principle of equivalence. Gauge theories are described by gauge covariant derivatives of potentials that do not contain the connections [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF]. Different covariant derivatives acting on SL(2, c) spinors are generalized from the flat spacetime to curved spacetime by replacing partial derivatives with suitable combinations of ∇ µ , V µ a ∂ µ , V µ a ∇ µ and ∇ ν V µ a where V µ a are the vierbeins. The choice of connections depends on the contracted Bianchi identity. In general, stress tensor for SL(2, c) spinors are always conserved w.r.t ∇ µ when the connections in ∇ µ are Γ α µν , [START_REF] Weinberg | Gravitation and Cosmology[END_REF]. Γ α µν are also used in the Newman-Penrose formalism. We find that it is nontrivial to extend the standard model to curved spacetime even with the Levi-Civita connections. We have to introduce the right-handed neutrino as a free field singlet in the standard model to have a theory with conserved vector current for the neutrinos. φ and ψ do not appear in the equations of motion and also in the stress tensors even if we start with Θ α µν in the corresponding covariant derivatives. ( 2) and ( 3) imply that it is sufficient but not necessary to use Γ α µν as connections to construct S M . Thus, we assume that only Γ α µν are present in the actions of ordinary matter and radiation fields, i.e, fields that are representations of SL(2, c). Other variables derived from the actions like stress-tensors of ordinary matter and radiation fields also contain only Γ α µν as the connections. We find that this agrees with: (i) a principle of least action, (ii) principle of general covariance, (iii) Stokes' theorem, (iv) contracted Bianchi identity in presence of finite non-metricity and (v) principle of equivalence. We will find in the following section that this may also be required to construct a stable theory with finite φ and ψ. We do the same to define the gauge invariant electromagnetic field tensor in presence of torsion [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF]. In general, we will have to modify Einstein's equation given by Eq.( 10) when C α µν appears in the r.h.s. If not so, i.e, when there is no contradiction with the contracted Bianchi identity, ω and ψ can couple with the corresponding matter fields classically.

Depending on ψ, we may not have: V µ a V bµ = η ab , where η ab is the flat spacetime metric, when we parallel transport V µ a from one point to a distant point in the spacetime manifold. This can be discussed using Eq.( 12). Thus, we do not have any problem with the principle of equivalence locally but global splitting of spacetime into space and time may not remain exact. In the Einstein-Palatini formalism considered here, ω and ψ obey the homogeneous wave equation and can be finite everywhere i.e they are non-localized. We can add suitable potential terms for themselves. We will discuss this later. Dark energy does not interact with ordinary matter directly and are observed through large scale cosmological observations. ψ contributes a negative source stress tensor to Einstein's equation and the preceding discussions indicate that ω and ψ need not be present in the covariant derivatives acting on matter and radiation fields. Thus, φ/ω and ψ need not to have ordinary matter and radiation as their sources [START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF]. All these make ψ a possible candidate for dark energy. This qualitatively agrees with the energy budget mentioned in section:I. It is unexpected that 10% ordinary matter can produce 70% dark energy. We find that a part of dark matter can be the right-handed neutrinos. φ is a possible candidate for inflation. φ is similar to the dilaton [START_REF] Gasperini | The Pre-Big Bang Scenario in String Cosmology[END_REF]. Cosmological observations indicate that non-metricities produced by φ and ψ are presently very small. This is consistent with the smallness of the cosmological constant in the ΛCDM model. φ and ψ manifest themselves through long range geodesics and global structure of spacetime. However, local inhomogeneities and anisotropies in φ and ψ can cause significant effects on geodesic motions in the corresponding regions. In the next section, we will find that it is more useful to consider linear combinations of ω and ψ to explain their effects like inflation and dark energy.

IV. Extension of Einstein's Theory of Gravity

We can use classical theories and quantum field theory in curved spaces to find the effects of (φ, ψ) when the full quantum effects like vacuum fluctuations are not much significant. φ, ψ, g µν should be treated similarly to discuss such effects. This will be useful in cosmic epoch. We now compare Eqs. [START_REF] Kaloper | Lorentz Chern-Simons terms in Bianchi cosmologies and the cosmic no hair conjecture[END_REF][START_REF] Golovnev | Vector Inflation[END_REF] with the action of scalar-tensor theories [START_REF] Amendola | Dark Energy[END_REF]:

S = e √ -g f (φ, R) -ζ(φ)(∇φ) 2 + S m (χ a , ν R , g µν ) (23)
where χ a is the collection of ordinary matter and radiation. Eqs. [START_REF] Golovnev | Vector Inflation[END_REF] indicate that:

ζ(ω) = 3 2 , ζ(ψ) = -3. f (φ, R) = R
for both fields. Thus, the scalars mentioned at the beginning of the first section like quintessence and k-essence can have geometrical origin. When required, we can modify the theory by adding suitable φ and ψ dependent terms in Eq.( 9). We get:

L = √ -g(R -V) (24)
where V contains the added terms which should be consistent with: ∇ α G µα = 0, if we express the equations in the form similar to Eqn. [START_REF] Zwicky | Die Rotverschieb ung von extragalaktischen Nebeln[END_REF]. In general, the above Lagrangian density will give coupled equations for ω and ψ. We can construct a simple theory by choosing

V = V 1 (ω) + V 2 (ψ) + 3[(∇ κ ω)(∇ κ ψ)].
This gives a set of two decoupled equations for ω and ψ. A more useful approach towards this direction is to consider the modified curvature scalar given by Eq.( 9) and diagonalize the symmetric bilinear form in ∇ µ ω and ∇ ν ψ, [START_REF] Gilmore | Lie Groups, Lie Algebras, and Some of Their Applications[END_REF]. This bilinear form is similar to: a 2 -2ab -2b 2 and the corresponding symmetric matrix has eigenvalues:

λ ± = (-1 ± √ 13)/2 .
R is now given by the following expression:

R = R - 3 2 [(∇ω) 2 -2(∇ψ) 2 -2(∇ κ ω)(∇ κ ψ)] (25) = R - 3 2 g µν [λ + (∇ µ Φ )(∇ ν Φ ) + λ -(∇ µ Ψ )(∇ ν Ψ )]
where Φ and Ψ are related to ω and ψ by the following expression:

Φ Ψ = S T ω ψ , S = c + 1 c - 1 c + 2 c - 2 (26)
Here c ± i are the components of two orthonormal eigenvectors. S is given by the following expression:

S =   4 26-6 √ 13 4 26+6 √ 13 3- √ 13 2 4 26-6 √ 13 3+ √ 13 2 4 26+6 √ 13   (27)
We can absorb the positive numerical factors into Φ , Ψ and the modified curvature scalar is given by:

R = R - 1 2 [(∇Φ) 2 -(∇Ψ) 2 ] (28)
The above expression gives a more convenient way to discuss the physical effects of the non-metricity scalar fields φ and ψ. We find that Ψ contributes a negative stress tensor to Einstein's equation. We can introduce various potential terms like V 1 (Φ), V 2 (Ψ) and V(Φ, Ψ) as per requirement. The last term gives coupling between Φ and Ψ. V 1 (Φ), V 2 (Ψ) and V(Φ, Ψ) can give us a theory more general than that given by Eq.( 23) and new effects in the scalar-tensor theories. The contracted Bianchi identity will remain to hold for this theory. Thus, transforming back to (ω, ψ) we will get a theory with (ω, ψ) that preserves the contracted Bianchi identity. We will discuss more on V(Φ, Ψ) in the following. We can consider a theory containing Φ, Ψ and the cosmological constant Λ. This will lead to a possible theory of inflation together with dynamic and spatially varying dark energy. Einstein's equation will be generalized to:

G µν + Λg µν + Λ µν (Φ, Ψ) = 8πP µν (χ a ) (29)
where Λ µν (Φ, Ψ) includes P µν (Φ), P µν (Ψ), potential terms for Φ and Ψ as well as possible coupling terms between the two fields. P µν (χ a ) is the stress tensor of ordinary matter and radiation including the right handed neutrinos. Φ can be useful to explain inflation. We can include possible interaction terms between Φ and χ a . In this context, we note that a theory where Ψ is coupled to the matter fields, radiation and Φ may be unstable depending on the theory. This is due to the negative stress-tensor associated with Ψ. Thus, Ψ can remain as a free field similar to Λ to give relatively stable cosmological solutions. This justifies its explanation as dark energy and absence of φ and ψ in various matter and radiation field stress-tensors. An interesting case will be to consider a self-interacting theory for Ψ. The same can remain valid with finite V(Φ, Ψ). The above equation leads to a generalization of the ΛCDM theory. We do not obtain the Newtonian theory in the weak field and slow motion limit when Λ = 0 [START_REF] Wald | General Relativity[END_REF]. This indicates that Λ is presently small. This is similar to the non-metricities produced by the pair (ψ, φ) or (Ψ, Φ) as discussed in the previous section. We find that inflation and dark energy possibly indicate that we have to generalize the classical structures of spacetime in the quantum domain. This is not completely unexpected. A complete theory of quantum mechanics can be partly spacetime independent algebraic theory as suggested by Einstein and apparent in various quantum entanglement experiments [START_REF] Bub | Interpreting the Quantum World[END_REF][START_REF] Fine | The Shaky Game[END_REF][START_REF]Closing the Door on Einstein and Bohr's Quantum Debate[END_REF]]. We will discuss applications of Eq.( 29) in cosmology in a forthcoming article. P µν (χ a ) will be multiplied by appropriate factors when we transform to non-geometrized ordinary units. Quantum corrections to the effective actions of various theories mentioned above like that given by Eq.( 29) will contain higher order curvature invariants evaluated using the Levi-Civita connections together with quantum corrections coming for various matter fields and possibly for non-metricity fields (Φ, Ψ) or (ω, ψ). The last set will depend on the requirement of potential and coupling terms for (Φ, Ψ) or (ω, ψ).

Lastly, a possible approach to explain dark energy can be to consider Ψ as possible source of Λ. Discussions on possible sources of Λ for constant curvature spaces can be found in [START_REF] Hawking | The Large Scale Structure of Space-Time[END_REF]. These are similar to Ψ. We can not generate the cosmological constant with perfect fluids for which both energy density and pressure are positive definite. We will discuss this aspect later.

V. Additional Fields and Little Group

In the theory considered in Sect.II containing a µν and q µν , we can try to introduce two additional scalar fields. One of them is ξ in Eq.( 5) and the other is η in Eq. [START_REF] Amendola | Dark Energy[END_REF]. It is not possible to have a set of āµν and qµν with finite ξ and η but vanishing U α µν and V α µν respectively. We can not introduce qµν without modifying the action. Same comments will remain valid for āµν . This is due to the contracted Bianchi identity. However, Θ α µν is symmetric in the lower indices and can have forty independent components. We have so far used three second rank symmetric tensors g µν , a µν and q µν to construct Θ α µν . It is thus possible to introduce another scalar not associated with g µν , a µν and q µν that contributes only a first order derivative term to C α µν similar to φ and ψ. We can take the corresponding part of C α µν to be traceless in the upper indices or in the lower indices. This term will be either of the form given for ξ in Eq.( 5) or that for η in Eq.( 6) respectively. The resulting theory is consistent with the contracted Bianchi identity. ξ contributes a negative stress tensor and η contributes a positive stress tensor to Einstein's equation. However, unless the additional scalar couples with any other field including ω and ψ differently or have distinct potential terms including or excluding quadratic terms, we can express the resulting C α µν in the form given by Eq.( 7) and need not to get beyond the two-scalar model considered here. These comments can be relevant to multi-scalar inflation theories [START_REF] Dodelson | Modern Cosmology[END_REF][START_REF] Liddle | Cosmological Inflation and Large Scale Structures[END_REF] and dark energy.

We now consider the case of higher spin (j > 0) fields that can be introduced using C α µν symmetric in the lower indices and expressed as derivatives of lower rank tensors. These fields can be important for theories like higher spin driven inflation [START_REF] Kaloper | Lorentz Chern-Simons terms in Bianchi cosmologies and the cosmic no hair conjecture[END_REF][START_REF] Golovnev | Vector Inflation[END_REF]. We consider the case that includes qµν :

C α µν = δ α (µ ∇ ν) ω - 1 2 g µν ∇ α ω + g µν ∇ α ψ + ∇ α qµν (30)
Regarding dimensions, we have: [q µν ] = [g µν ]. We have discussed in Sect.III that after we have removed the total divergence terms in R and expressed the Einstein's equation in the form given by Eq. [START_REF] Zwicky | Die Rotverschieb ung von extragalaktischen Nebeln[END_REF], where the contribution of the non-metricity fields appear as source terms, the resulting equations have to be consistent with the Bianchi identity in Γ α µν . This together with the equations of motion impose severe restrictions on qµν . Thus, we have to modify the Einstein-Palatini action by adding suitable qµν dependent terms so that the resulting theory is consistent with the contracted Bianchi identity. We can also use the metric-affine f (R ) theories. This may indicate that the resulting theory is described by the low energy effective action of a more fundamental theory. We can replace qµν by the traceless derivative: t µν = ∇ (µ B ν) of a vector field B α with ∇ α B α = 0, to have a spin one field from geometric sector. We then have:

C α µν = δ α (µ ∇ ν) ω - 1 2 g µν ∇ α ω + g µν ∇ α ψ + ∇ α t µν (31)
where t µ µ = ∇.B = 0. Note that we have used a symmetric field t µν to introduce a spin one field and this field will not give a symmetric R µν although corresponding R µα will be symmetric. We again have problem with the contracted Bianchi identity similar to that mentioned before. We will later address this problem. We also note that we can not eliminate B α from the action of the Dirac field spinors and the later can couple with B α . Such coupling is yet to be observed. Thus, the two scalar model considered in this article are more consistent with the observations. Torsion can be introduced using torsion potentials [START_REF] Hammond | Torsion gravity[END_REF].

We now consider the issue of constructing a semi-classical limit of the above possible theories one of which can be a locally Lorentz invariant quantum field theory in curved spaces including t µν . We discuss this with reference to little groups in Minkowski space. We will find that it is possible to use symmetric traceless tensors that give (A, A) type representations of the Lorentz group to describe massless higher spin particles. We do not have any problem with massless scalar fields for which A = 0. t µν give (A, A) type representation symmetries like U (1) in a theory that preserves such symmetries. Thus, the effect of these fields would be purely gravitational to influence the metric and they appear only in Einstein's equations. This is similar to the discussions given in Sec.I regarding (φ, ψ) and can be relevant to explain dark energy and dark matter. On the other hand, coupling between these fields and matter fields can lead to gauge invariance breaking effects like particle-antiparticle asymmetry. The present section demonstrates that we can use massless higher spin bosons to construct higher spin driven inflation theories [START_REF] Dodelson | Modern Cosmology[END_REF][START_REF] Kaloper | Lorentz Chern-Simons terms in Bianchi cosmologies and the cosmic no hair conjecture[END_REF][START_REF] Golovnev | Vector Inflation[END_REF] that need not to be gauge theories. Lastly,, we can always use a set of C α µν antisymmetric in the lower indices to introduce a spin one field in the spacetime manifold [START_REF] Ghosh | Affine connections in quantum gravity and new scalar fields[END_REF][START_REF] Hehl | General relativity with spin and torsion: Foundations and prospects[END_REF].

VI. Conclusion

To conclude, in this article we have illustrated the possibility of explaining a few cosmological observations using new fields from the gravity sector. We have extended general relativity, the classical theory of gravity, to include non-metricity within the physical framework of a variational principle. We had earlier found this to be more consistent with a quantum theory of gravity. We have used a potential-like formalism that introduces symmetric second rank tensors. We have found that these tensors can give massless scalar fields even in the simplest case where the action is given by the curvature scalar. We have considered a theory that includes two such scalar fields φ and ψ. We found that in a Lagrangian formalism that preserves a few general principles like the principle of equivalence, these fields need not to be present in the expressions of covariant derivatives acting on ordinary matter and radiation that are various representations of SL(2, c). Exception can arise with the neutrinos. However, coupling between (φ, ψ) and neutrinos are not relevant when we include the right-handed neutrinos in the standard model. This scheme, developed in Sect.III, can also be required to construct a stable theory. This is discussed in Sect.IV where we have performed linear transformation on ω = ln(1 + φ) and ψ to obtain a theory of two decoupled scalars Φ and Ψ. Ψ gives a negative stress tensor and is a possible candidate for dark energy. Φ is a possible candidate for inflation. We can introduce potential terms in Φ and Ψ that include coupling between the two. We can also consider the possibility of a theory with three such scalar fields. We can generalize the ΛCDM theory by including Φ and Ψ to give dynamic and spatially varying dark energy and inflation. Λ, Φ and Ψ are similar in many respects. They need not to have ordinary matter as their sources and can be non-localized. They can also contribute positive and negative stress tensors in Einstein's equation. Λ, Φ and Ψ give us departures from the classical theory and global splitting of spacetime into space and time, thus modifying the global structure of spacetime. However, present cosmological observations suggest that such effects are finite but small.

We found that we may have to introduce the right-handed neutrinos in the standard model to have anomaly free conserved vector currents for neutrinos in curved spacetime. This is related with the existence of required variational derivatives in curved spacetime as well as curvilinear coordinates in flat spacetime. The right handed neutrinos had been used before to eliminate anomalies. The axial vector currents for various Dirac fields including the neutrinos can still remain anomalous. Anomalies in axial vector currents are common in flat spacetime. We have considered the right-handed neutrinos to be free field singlets of the SU (2) gauge theory in the present article. They can be useful in the dark matter research. They can also be useful to explain neutrino oscillation. The above observations can be important in the early universe.

In this article, we have used (A, A) type representations to introduce higher spin fields. We have found that for massless fields, we have to use a one parameter subgroup of SO(3) as the little group in Minkowski space. This allows us to use (A, A) type fields to introduce massless integral spin fields with spin greater than zero. This can be useful in dark energy and higher spin driven inflation models. We can modify the action or use f (R ) theories to include higher spin fields.

VII. Acknowledgement

I am thankful to a few reviewers for a few improvements.

of j = 1 massless particles. In flat spacetime, the standard momentum k µ used to construct the little group for massless fields is taken to be (0, 0, k, k), where the fourth component represents time and the metric is of signature +2, [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. The little group that keep the standard momentum unchanged is given by ISO(2) and for spacetime tensors, it is taken to be W (θ, α, β) = S(α, β)R z (θ), [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. Where R z (θ) is a spatial rotation about the Z -axis by angle θ and S(α, β) is given by:

Here, γ = (α 2 + β 2 )/2. To construct a Lorentz covariant quantum field theory of a massless four vector field, the momentum space four vector with the standard momentum (0, 0, k, k) have to obey [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]:

However, the matrix representing S µ ν is not normal (S T S = SS T ). This is also valid for W = SR and the above eigenvalue equation for e µ (k, σ) does not exist [START_REF] Cullen | Matrices and Linear Transformations[END_REF]. Thus, we restrict W to R z (θ) only. This is consistent with the fact that massless particle states in flat spacetime are characterized by helicity and there are no charges associated with the generators of S(α, β) with the charges continuously dependent on θ, [START_REF] Weinberg | The Quantum Theory of Fields[END_REF].

We now consider e µ (k, σ) that satisfy the following equation instead of Eq.( 33):

where, σ = 0, ±1. A convenient choice for e µ (k, 0; ±1) is:

) and e µ (k, 0) = (0, 0, 1, 1). We note that for null vectors like (0, 0, k, k) in Minkowski space, there can be more than one way to obtain a null vector like:

. Firstly, we can apply a boost β along the x 1 axis. Alternatively, we can rotate (0, 0, k, k) to (kβ, 0, k 1 -β 2 , k) and thereafter apply a boost along the direction of the spatial momentum by α where:

The above two Lorentz transformations do not lead to the same e µ ( p, ±1). This is related to the fourdimensional geometry. However, we can always have:

is the Fourier transform of e µ ( p, σ) given by, [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]:

where, a( p, σ) is an operator. Under a Lorentz transformation L, B µ transform as:

where U (L) are the unitary transformations constructed from a suitable Lagrangian density. To construct a local theory invariant under parity, we can introduce a massless vector field B ν through the (1, 1) type symmetric traceless field: ∂ (µ B ν) (here we have ∂.B = 0) or through the (1, 0) ⊕ (0, 1) type antisymmetric field

. The later is used to describe QED to preserve the U (1) internal symmetry present in the corresponding sources [START_REF] Itzykson | Quantum Field Theory[END_REF]. In this context, we note that the Coulomb gauge ∇. B = 0 together with B 0 = 0 is a non-covariant choice of gauge and a Lorentz transformation on one such field will in general not yield another such field [START_REF] Itzykson | Quantum Field Theory[END_REF]. Things change in QED when we modify the canonical commutators to eliminate the unphysical degrees of freedom. In this case, the right hand side of the above equation is appended by pure gauge terms that ensure Lorentz covariance [START_REF] Bjorken | Relativistic Quantum Fields[END_REF]. The above conclusions are also consistent with the Gupta-Bleuler quantization of the electromagnetic field [START_REF] Itzykson | Quantum Field Theory[END_REF]. ∇ (µ B ν) is not gauge invariant and B α , with the kinetic term obtained using ∇ (µ B ν) , can not minimally couple with matter fields that possess internal