# Affine Connection and Quantum Theory

Kaushik Ghosh111E-mail<EMAIL_ADDRESS>Vivekananda College (University
of Calcutta), 269, Diamond Harbour Road, Kolkata - 700063, India

## Abstract

In a few recent manuscripts, we used the affine connection to introduce two
massless scalar fields in the Einstein-Palatini action. These fields lead to
non-metricity. In this article, we will discuss the significance of these
fields in inflation and dark energy. We will construct a Lagrangian formalism
to include these scalar fields in a theory of gravity coupled with ordinary
matter and radiation. We will find that these fields need not to be included
in the actions of interacting gauge theories coupled with conserved fermionic
vector currents as a part of the connection. The same remains valid for
ordinary scalar fields. We can couple the connection-scalars with ordinary
matter by adding suitable interaction terms. In this context, we will find
that Stokes’ theorem leads us to include the right-handed neutrinos in the
electroweak theory in curved spacetime even with the Levi-Civita connection.
This is required to obtain consistent equations of motion and anomaly-free
conserved vector current for the neutrinos. Axial vector currents for
different Dirac fields may remain anomalous. The right-handed neutrinos can be
useful to explain neutrino oscillation and dark matter. We will also discuss
the possibility of introducing massless finite integer spin particles using
second rank symmetric traceless tensors with reference to the corresponding
little group in flat spacetime. We will show that we can use massless $(A,A)$
type fields in Minkowski space to introduce massless finite integer spin
particles.

Key-words: quantum fields in curved spaces, anomaly, right handed neutrino,
inflation, dark energy

MSC: 53Z05, 49S05, 83D05, 81T20, 81T50

PACS: 02.40.-k, 98.80.Qc, 98.80.Cq, 95.36.+x, 04.50.Kd

## I. Introduction

A lot of work has been done during the last few decades on two periods of
cosmic accelerations: inflation and dark energy [1,2,3,4,5,6]. Inflation is
driven by the scalar field inflation. We also have multi-scalar inflation and
higher spin driven inflation models [7,8,9]. Dark energy is not yet detected
directly. It is not localized like ordinary matter and is assumed to have
negative pressure to explain the present cosmic acceleration. We also need
dark matter which is observed by its gravitational effects [6,10,11,12]. 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 $\Lambda$ [6,13]. It is of order $(10^{-3}eV)^{4}$ and is far below
from the value predicted by quantum gravity [14]. This led to construct
alternate theories. There are two principal approaches. The first adds stress-
tensors with negative pressure in the source part of the Einstein equation.
Quintessence, k-essence and perfect fluid models are the most well-known among
various theories [15,16,17]. These fields are expected to originate as matter
fields. There is the possibility that they can come with a negative kinetic
term [5,6]. Corresponding fields are known as ghost fields. Despite a lot of
efforts, the origins of inflation, dark energy and dark matter are yet to
completely fit within an extended version of the standard model. The other
approach modifies the Einstein-Hilbert action. Examples are $f(R)$ gravity
[18,19], scalar-tensor theories [6,20] and braneworld models [21,22]. These
are geometric theory of dark energy [20]. The experimental success of
Einstein’s theory of gravity, linear in the scalar curvature, in explaining
many astrophysical observations led us to search for additional fields that
can be introduced by using a quantum theory of gravity that is linear in the
scalar curvature at the classical level.

In a few recent manuscripts [23,24], we have demonstrated that a quantum
theory of gravity can introduce additional fields besides metric that can be
useful to explain inflation and dark energy. Here, we are considering the free
theory. The natural candidates are the affine connections used to define
covariant derivatives in a curved spacetime [25,26,27,28,29,30]. We found it
is more appropriate to use affine connection more general than the metric
compatible connections in a quantum theory of gravity [23,24]. 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
connection [26]. Hence, it is not bizarre that we can have non-metricity [31]
in a quantum theory of gravity. Regarding variational principle, the simplest
theory to introduce connections more general than the Levi-Civita connection
is to consider the Einstein-Palatini action [28]. Here, the connection is
expressed as:
$\Theta^{\alpha}_{~{}\mu\nu}=\Gamma^{\alpha}_{~{}\mu\nu}+C^{\alpha}_{~{}\mu\nu}$,
where $\Gamma^{\alpha}_{~{}\mu\nu}$ is the metric compatible Levi-Civita
connection and $C^{\alpha}_{~{}\mu\nu}$ is a third rank tensor field that can
introduce non-metricity. The Lagrangian density is given by
$\sqrt{-g}R^{\prime}$, where $R^{\prime}$ is the scalar curvature evaluated
using the connection $\Theta^{\alpha}_{~{}\mu\nu}$. In [23,24], we have used a
potential-like formalism to express $C^{\alpha}_{~{}\mu\nu}$ as various
covariant derivatives of lower rank tensor fields. This is similar to the
Levi-Civita connection which is constructed from metric. We will find in
Sect.III that scalar fields are the most suitable candidates for this purpose.
This is due to the compatibility of the corresponding Einstein equation with
the contracted Bianchi identity in $\Gamma^{\alpha}_{~{}\mu\nu}$. Unless
stated otherwise, in this article, contracted Bianchi identity will mean twice
contracted Bianchi identity. We have previously introduced two scalar fields
that obey homogeneous wave equation in the free theory and can be non-
localized. These scalar fields can be related to two traces of
$C^{\alpha}_{~{}\mu\nu}$, one w.r.t $(\alpha,\mu)$ and the other w.r.t
$(\mu,\nu)$. They provide positive and negative stress-tensors in Einstein’s
equation respectively, and will be useful for constructing new non-trivial
vacuum solutions [29]. Here, we have a scalar-tensor theory directly related
to quantum gravity, that could explain both inflation and dark energy provided
we modify the action by adding suitable potential terms [6]. The scalar fields
mentioned above give non-metricity although their effects are geometrically
different. They are similar to the cosmological constant $\Lambda$ that gives
deviation from the Newtonian theory of gravity in the weak field slow motion
limit [29]. In the present article, we will elaborate on this in the next
section. We will mostly consider affine connection symmetric in the lower
indices that gives symmetric Ricci tensor. In Sect.II, we will first discuss
the scalar field $\omega$ that can be related to an affine connection
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 $\psi$ that is more significant as far as non-metricity is
concerned. $\psi$ contributes a negative stress tensor in Einstein’s equation
and can be useful to explain dark energy.

In Sect.III, we will discuss the possible coupling between $\omega$, $\psi$
and matter fields including radiation that can arise due to the presence of
$C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$ in the covariant derivative. Matter
fields 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 connection
and is required to obtain consistent equations of motion and anomaly-free
conserved vector current for the neutrinos. The right-handed neutrinos can be
important in dark matter research. There are other advantages related to
neutrino oscillations to be mentioned later. Stokes’ theorem also indicates
that the axial vector currents for various Dirac fields, including that of
neutrinos, could become anomalous in curved spacetime as well as in flat
spacetime when we use 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 the principle of equivalence, it is sufficient to use
${\Gamma^{\alpha}_{~{}\mu\nu}}$ as the connection 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, $C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$ can at most contribute
boundary terms to the matter and radiation field actions and do not appear in
the equations of motion. Thus, we may not include
$C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$ in the actions and stress tensors of
ordinary matter and radiation. The resulting equation is consistent with the
contracted Bianchi identity in $\Gamma^{\alpha}_{~{}\mu\nu}$, and merely
contains an additional stress-tensor in the right hand side of Einstein’s
equation coming from $C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$ present in the
curvature scalar. This agrees partly with experimental observations regarding
dark energy which does not couple with ordinary matter and radiation. We will
find in Sect.IV that this may also be required to construct a stable theory
[6]. Deviation from Einstein’s gravity with $\Gamma^{\alpha}_{~{}\mu\nu}$, due
to dark energy, is small. This is apparent from the smallness of the
cosmological constant in the $\Lambda$CDM theory. A similar situation
presently remains valid for the stress-tensor associated with $\omega$ and
$\psi$, if we try to explain dark energy using these fields. This explains why
the Levi-Civita connection and Einstein’s gravity describe the observable
universe to a very good approximation even if $\omega$ and $\psi$ are present.
Non-metricity produced by $\omega$ and $\psi$ is also small.

In Sect.IV, we will discuss possible modifications of Einstein’s theory of
gravity in the presence of $\omega$ and $\psi$. The equations for $\omega$ and
$\psi$ are coupled through derivative terms. We will find that we can separate
the equations by introducing two scalar fields, $\Phi$ and $\Psi$ that are
linear combinations of $\omega$ and $\psi$. The resulting action is a new form
of scalar-tensor theory with two scalar fields. $\Phi$ contributes a positive
stress tensor to Einstein’s equation. When required, we can supplement the
Einstein-Palatini action by $\Phi$ dependent potential term. We can couple
$\Phi$ with ordinary matter by introducing suitable interaction term. $\Phi$
gives finite non-metricity and we can construct a quantum theory where only
$\Phi$ is present. Later, we will consider theories with more than one such
field. $\Psi$ contributes a negative stress-tensor. A scalar matter with
negative kinetic term in Einstein’s equation is known as a phantom/ghost
scalar [6]. We find that a phantom scalar of geometric origin can be present
in quantum gravity. The negative kinetic term of $\Psi$ can lead to
instability if coupled to ordinary matter. We can add $\Psi$ dependent
potential term to get stable theories [6]. We will mention a few words on the
stability of the theory in this section. Another way to generalize Einstein’s
theory is to include $\Phi$, $\Psi$ and the cosmological constant $\Lambda$.
This could lead to a possible geometric theory of inflation together with
dynamic and spatially varying dark energy.

In section:V, we will first discuss the possibility of introducing additional
scalar fields and higher spin 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 connection. 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 stabilizer (little group) for
massless fields in Minkowski space is taken to be: $W=ISO(2)$ [32]. However,
this group is not represented by normal matrices [33] when acting 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 higher spin 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 cannot couple with matter fields in a theory that
preserves local internal gauge invariance. They can be useful for describing
dark energy and dark matter. The massless higher spin particles can also be
useful to construct higher spin driven inflation theories [9,16] in the
absence of spontaneous symmetry breaking. Such a theory need not to be a gauge
theory.

### II. Scalar Fields from Affine Connection

The simplest theory to introduce connections more general than the Levi-Civita
connection is to consider the Einstein-Palatini action [29]. Here, the
connection is expressed as:
$\Theta^{\alpha}_{~{}\mu\nu}=\Gamma^{\alpha}_{~{}\mu\nu}+C^{\alpha}_{~{}\mu\nu}$,
where $\Gamma^{\alpha}_{~{}\mu\nu}$ is the metric compatible Levi-Civita
connection and $C^{\alpha}_{~{}\mu\nu}$ is a third rank tensor field that can
introduce non-metricity. In this section we will consider
$C^{\alpha}_{~{}\mu\nu}$ that is symmetric in the lower indices. The curvature
and Ricci tensors are given by the following expressions:

$\displaystyle{{\nabla^{\prime}}_{\mu}}{A_{\nu}}$ $\displaystyle=$
$\displaystyle{\partial}_{\mu}{A_{\nu}}-{{\Theta}^{\alpha}_{~{}\mu\nu}}{A_{\alpha}}$
(1)
$\displaystyle({{\nabla^{\prime}_{\mu}}{\nabla^{\prime}_{\nu}}}-{{\nabla^{\prime}_{\nu}}{\nabla^{\prime}_{\mu}}}){A^{\alpha}_{~{}\beta}}$
$\displaystyle=$
$\displaystyle-{{R^{\prime}}_{\mu\nu\kappa}^{~{}~{}~{}~{}\alpha}}{A^{\kappa}_{~{}\beta}}+{{R^{\prime}}_{\mu\nu\beta}^{~{}~{}~{}~{}\kappa}}{A^{\alpha}_{~{}\kappa}}$
$\displaystyle{R^{\prime}}_{\mu\nu\alpha}^{~{}~{}~{}~{}\kappa}$
$\displaystyle=$
$\displaystyle{{R}_{\mu\nu\alpha}^{~{}~{}~{}~{}\kappa}}+2{{\nabla}_{[\nu}}{C^{\kappa}_{~{}\mu]\alpha}}+2[{C^{\lambda}_{~{}[\mu|\alpha|}}{C^{\kappa}_{~{}\nu|\lambda|]}}]$
$\displaystyle{R^{\prime}}_{\mu\alpha}$ $\displaystyle=$
$\displaystyle{{R}_{\mu\alpha}}+2{{\nabla}_{[\kappa}}{C^{\kappa}_{~{}\mu]\alpha}}+2[{C^{\lambda}_{~{}[\mu|\alpha|}}{C^{\kappa}_{~{}\kappa|\lambda|]}}]$
$\displaystyle R^{\prime}$ $\displaystyle=$
$\displaystyle{g^{\mu\alpha}}[{{R}_{\mu\alpha}}+2{{\nabla}_{[\kappa}}{C^{\kappa}_{~{}\mu]\alpha}}+2[{C^{\lambda}_{~{}[\mu|\alpha|}}{C^{\kappa}_{~{}\kappa|\lambda|]}}]$

where ${{\nabla^{\prime}}_{\mu}}$ is the covariant derivative evaluated using
the complete connection ${{\Theta}^{\alpha}_{~{}\mu\nu}}$, ${\nabla_{\mu}}$ is
the covariant derivative evaluated using ${\Gamma^{\alpha}_{~{}\mu\nu}}$,
${{R}_{\mu\nu\alpha}^{~{}~{}~{}~{}\kappa}}$ is the curvature tensor for
${\Gamma^{\alpha}_{~{}\mu\nu}}$ and ${{R}_{\mu\alpha}}$ is the Ricci tensor
associated with the Levi-Civita connection. The Lagrangian density is given by
$\sqrt{-g}R^{\prime}$, where $R^{\prime}$ is the scalar curvature evaluated
using the connection $\Theta^{\alpha}_{~{}\mu\nu}$. When we apply the
principle of least action to obtain the classical configurations, we obtain
algebraic constraints on $C^{\alpha}_{~{}\mu\nu}$ that lead to the null
solution [29]. In [23,24,25], we have used a potential-like formalism to
express $C^{\alpha}_{~{}\mu\nu}$ as various covariant derivatives of lower
rank tensor fields. This is similar to the Levi-Civita connection which are
constructed from metric. We will find in the following section after Eq.(6) at
point (1) that scalar fields are the most suitable candidates for this
purpose. This is due to the compatibility of the corresponding Einstein
equation with the contracted Bianchi identity in
$\Gamma^{\alpha}_{~{}\mu\nu}$. Here, we express $C^{\alpha}_{~{}\mu\nu}$ as:

$\displaystyle{{\Theta}^{\alpha}_{~{}\mu\nu}}$ $\displaystyle=$
$\displaystyle{\Gamma^{\alpha}_{~{}\mu\nu}}+{{C}^{\alpha}_{~{}\mu\nu}}$ (2)
$\displaystyle{{C}^{\alpha}_{~{}\mu\nu}}$ $\displaystyle=$
$\displaystyle{{\delta}^{\alpha}_{~{}(\mu}}{{\nabla}_{\nu)}}{\omega}-{1\over
2}{g_{\mu\nu}}{{{\nabla}^{\alpha}}{\omega}}+{g_{\mu\nu}}{{\nabla}^{\alpha}{\psi}}.$

The reason for this choice is that the first two terms of
${{C}^{\alpha}_{~{}\mu\nu}}$ gives the contribution of the conformal scalar
field when we construct the connection to be compatible with:

${\tilde{g}_{\mu\nu}}=(1+\phi(x)){g_{\mu\nu}},~{}~{}\omega=ln(1+\phi).$ (3)

We can associate $\omega/\phi$ and $\psi$ with two traces of
$C^{\alpha}_{~{}\mu\nu}$, one in the mixed indices and the other in the lower
indices. We find that:
$C^{\alpha}_{~{}\alpha\nu}={\nabla}_{\nu}(2\omega-\psi)$ and
$C^{\alpha\mu}_{~{}~{}~{}\mu}={\nabla}^{\alpha}(4\psi-\omega)$. We can add
additional scalar fields depending on observations. We will discuss this in
Sect. V. In the present section we will confine our attention to $\omega$ and
$\psi$ only. We will briefly discuss the geometric significance of $\omega$
and $\psi$ in Sect. V [25]. Eq.(2) gives us a symmetric Ricci tensor. The
Einstein-Palatini action is now given by the following expression:

$S={\int}{\sqrt{-g}}{R^{\prime}}{\bf
e}+{\sum}{\kappa_{M}}{S_{M}}({\chi_{a}},g_{\mu\nu})={\int}{\sqrt{-g}}{\mathcal{R^{\prime}}}{\bf
e}+{\sum}{\kappa_{M}}{S_{M}}({\chi_{a}},g_{\mu\nu})$ (4)

where ${\bf e}$ is the coordinate volume element
${dx^{0}}\wedge{dx^{1}}\wedge{dx^{2}}\wedge{dx^{3}}$ [29], $\chi_{a}$
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,
$2{{\nabla}_{[\kappa}}{C^{\kappa}_{~{}\mu]\alpha}}$, present in the expression
of ${R^{\prime}}_{\mu\alpha}$ given by Eq.(1). This requires fixing the normal
derivatives of $\omega$ and $\psi$ on the boundary. Later, we will also
require to fix $\omega$ and $\psi$ on the boundary to obtain corresponding
equations of motion. Thus, we need to remove ambiguities (if any in curved
spacetime) present in solutions with either the Dirichlet boundary conditions
or the Neumann boundary conditions. It is possible to use the Cauchy boundary
conditions with open boundaries in flat spacetime. When possible, we can use
periodic boundary conditions. We can also modify the action by introducing a
suitable boundary term with the boundary integrand given by:
$-C^{\alpha~{}\mu}_{~{}\mu}+C^{\mu~{}\alpha}_{~{}\mu}$, [25]. This is similar
to that done for metric [29] and Chern-Simons theory. Thus, we introduce the
modified Ricci tensor ${\mathcal{R^{\prime}}_{\mu\nu}}$ and modified curvature
scalar $\mathcal{R^{\prime}}$ given by the following expressions:

$\displaystyle{\mathcal{R^{\prime}}}$ $\displaystyle=$
$\displaystyle{g^{\mu\alpha}}{\mathcal{R^{\prime}}_{\mu\alpha}}={g^{\mu\alpha}}\left\\{{{R}_{\mu\alpha}}+2[{C^{\lambda}_{~{}[\mu|\alpha|}}{C^{\kappa}_{~{}\kappa|\lambda|]}}]\right\\}$
$\displaystyle=$ $\displaystyle{R}-{3\over
2}{({\nabla}{\omega})^{2}}+3{({\nabla}{\psi})^{2}}+3[({{\nabla}_{\kappa}{\omega}})({{\nabla}^{\kappa}{\psi}})]$

where $R_{\mu\alpha}$ and $R$ are the Ricci tensor and curvature scalar
associated with the Levi-Civita connection respectively. We obtain the field
equations by considering various variational derivatives. Here, we assume that
${C^{\alpha}_{~{}\mu\nu}}(\omega,\psi)$ is not present in $S_{M}$. This is
consistent with observations. We will illustrate this later in this section.
We have the following modification of the r.h.s of Einstein’s equation
[23,24,25]:

$\displaystyle{{G}_{\mu\alpha}}$ $\displaystyle=$
$\displaystyle{8\pi}[{{P}_{\mu\alpha}(\chi_{a})}+{3\over{16\pi}}{{P}_{\mu\alpha}(\omega)}-{3\over{8\pi}}{{P}_{\mu\alpha}(\psi)}-{3\over{8\pi}}{{P}_{\mu\alpha}(\omega,\psi)}]$
(6) $\displaystyle{{P}_{\mu\alpha}(\omega,\psi)}$ $\displaystyle=$
$\displaystyle({{\nabla}_{(\mu}}{\psi})({{\nabla}_{\alpha)}}{\omega})-{1\over
2}{g_{\mu\alpha}}({{\nabla}_{\kappa}{\psi})({{\nabla}^{\kappa}}{\omega}})$
$\displaystyle=$
$\displaystyle{1\over{(1+\phi)}}[({{\nabla}_{(\mu}}{\psi})({{\nabla}_{\alpha)}}{\phi})-{1\over
2}{g_{\mu\alpha}}({{\nabla}_{\kappa}{\psi})({{\nabla}^{\kappa}}{\phi}})]$
$\displaystyle{{{\nabla}_{\kappa}}{{\nabla}^{\kappa}}{\omega}}$
$\displaystyle=$
$\displaystyle{1\over{1+\phi}}{{{\nabla}_{\kappa}}{{\nabla}^{\kappa}}{\phi}}-{({1\over{1+\phi}})^{2}}{({\nabla}{\phi})^{2}}=0$
$\displaystyle{{{\nabla}_{\kappa}}{{\nabla}^{\kappa}}{\psi}}$ $\displaystyle=$
$\displaystyle 0$

where ${{P}_{\mu\alpha}(\chi_{a})}$ is the stress tensor of ordinary matter
and radiation. $\omega$ and $\psi$ 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
${\Gamma^{\alpha}_{~{}\mu\nu}}$: ${{\nabla}^{\alpha}}{{G}_{\mu\alpha}}=0$. We
will discuss this aspect further in the following section.
${{P}_{\mu\alpha}}(\omega)$ and ${{P}_{\mu\alpha}}(\psi)$ are the stress
tensors of ordinary massless scalar fields. $\psi$ contributes a negative
stress tensor to the Einstein equation. $\omega$ and $\psi$ together with
suitable additional interaction terms give an alternate way to explain
inflation and dark energy [6,35]. Further significance of negative stress
tensor can be found in [36,37]. We find that the coupling of $\omega$ with
$\psi$ in ${\mathcal{R^{\prime}}}$ gives another contribution to the source
stress tensor. We will later show that the equations can be separated in the
scalar fields. ${{C}^{\alpha}_{~{}\mu\nu}}$ in Eq.(2) is now given by the
solutions of the above equations. ${{P}_{\mu\alpha}(\chi_{a})}$ will be
multiplied by appropriate factors when we transform to non-geometrized
ordinary units.

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

In this section, we will discuss the possible coupling between the scalars
($\omega$, $\psi$) and ordinary matter fields, including radiation. We have to
consider a few points when we try to construct a Lagrangian description of
gravity coupled with matter fields with a connection more general than the
Levi-Civita connection:

(1) Euler-Lagrange’s equations, Bianchi identity and its contracted forms:

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
${R^{\prime}}_{\mu\nu\alpha}^{~{}~{}~{}~{}\kappa}$ given by Eq.(1) when
derivatives are taken w.r.t the total connection
${\Theta^{\alpha}_{~{}\mu\nu}}$. $\omega$ and $\psi$ in Eq.(2) can be any pair
of regular fields. We can in general have problems when we do away with the
total divergence term in $R^{\prime}$ given by
$2{g^{\mu\alpha}}{{\nabla}_{[\kappa}}{C^{\kappa}_{~{}\mu]\alpha}}$ and apply
the principle of least action to $S$ given by Eq.(4). It is required to have:
${{\nabla}^{\alpha}}{{G}_{\mu\alpha}}=0$, the contracted Bianchi identity
w.r.t ${\Gamma^{\alpha}_{~{}\mu\nu}}$, when we express Einstein’s equation in
the form given by Eq.(6) with the fields associated with
${C^{\alpha}_{~{}\mu\nu}}$ contributing source stress tensor terms in the
right hand side. This remains valid for $\omega$ and $\psi$ individually. The
only scalar quadratic in the first order derivative of a scalar field is the
square of the norm of the covariant derivative and the corresponding term in
$\sqrt{-g}{\mathcal{R^{\prime}}}$ coincides with the Lagrangian density of a
massless scalar field apart from numerical factor. This leads to the familiar
conserved stress tensor of a massless scalar field with a numerical factor in
the r.h.s of Einstein’s equation when we express it in the form given by
Eq.(6). The contracted Bianchi identity w.r.t ${\Gamma^{\alpha}_{~{}\mu\nu}}$
also remains valid for Eq.(6) where we have both $\omega$ and $\psi$ with
derivative coupling. We will show this also in the next section by
diagonalizing the second order expression in ${\nabla_{\mu}}{\omega}$ and
${\nabla_{\nu}}{\psi}$ present in ${\mathcal{R^{\prime}}}$ given by Eq.(5).
This observation allows us to introduce scalar fields in
${C^{\alpha}_{~{}\mu\nu}}$. We can also include various potential terms to be
discussed later. Similar situation may not remain valid when we include higher
spin fields in ${C^{\alpha}_{~{}\mu\nu}}$.

(2) Application of Stokes’ theorem and conservation laws:

The second is the possibility of removing the total divergence term:
${\int}{\sqrt{-g}}{g^{\mu\alpha}}{{\nabla}_{[\kappa}}{C^{\kappa}_{~{}\mu]\alpha}}{d^{4}}x$
by applying the Stokes’ theorem. This is possible when we can express this
term as: ${\int}{d^{4}}x{{\nabla}_{\mu}}({{\sqrt{-g}}V^{\mu}})$, where
$V^{\mu}$ is a contravariant vector field [27,29]. This happens when the
connection in ${{\nabla}_{\kappa}}$ is ${\Gamma^{\alpha}_{~{}\mu\nu}}$
[27,29]. We can add a ${C^{\alpha}_{~{}\mu\nu}}$ only when the total
connection is metric compatible, leading to: ${C^{\alpha}_{~{}\alpha\nu}}=0$,
[27]. The same conditions remain valid when ${C^{\alpha}_{~{}\mu\nu}}$ is not
symmetric in the lower indices. Thus, we have used
${\Gamma^{\alpha}_{~{}\mu\nu}}$ in ${{\nabla}_{\mu}}$. This also agrees with a
local theory based on the principle of equivalence considered below. Same
discussions remain valid with ordinary matter field actions:
$S_{M}=\int{d^{4}x}{{\mathcal{L}_{M}}}$, where ${\sqrt{-g}}$ is included in
${{\mathcal{L}_{M}}}$. While deriving the corresponding Euler-Lagrange’s
equations, we have to eliminate the four-divergence term:
${{\nabla^{\prime}}_{\mu}[{{\partial{\mathcal{L^{\prime}}_{M}}\over{\partial({{{\nabla^{\prime}}_{\mu}}{\chi_{a}}})}}{\delta{\chi_{a}}}}]}$
by applying the Stokes’ theorem. Here, ${\mathcal{L^{\prime}}_{M}}$ is the
matter Lagrangian density evaluated using the total connection
${\Theta^{\alpha}_{~{}\mu\nu}}$. This is not possible until
${{\partial{\mathcal{L^{\prime}}_{M}}\over{\partial({{{\nabla^{\prime}}_{\mu}}{\chi_{a}}})}}{\delta{\chi_{a}}}}$
is a relative contravariant vector field of weight $+1$ [27,29]. Otherwise,
the functional derivative $\frac{\delta S_{M}}{\delta{\chi_{a}}}$ can contain
a non-covariant term. When we can eliminate the total divergence term, the
functional derivative is given by: $\frac{\delta
S_{M}}{\delta\chi_{a}}=-{{\nabla^{\prime}}_{\mu}({{\partial{\mathcal{L^{\prime}}_{M}}\over{\partial({{{\nabla^{\prime}}_{\mu}}{\chi_{a}}})}}})+{\partial{\mathcal{L^{\prime}}_{M}}\over{\partial{\chi_{a}}}}}$.

We now discuss the possible coupling between the connection scalars
$\omega,\psi$ and matter fields that can arise through the equations of motion
obtained by using a variational principal. For scalar fields (including many
non-linear models, [37]), ${C^{\alpha}_{~{}\mu\nu}}$ does not appear in the
corresponding variational derivatives $\frac{\delta S_{M}}{\delta{\chi_{a}}}$
and field equations even if we use ${\Theta^{\alpha}_{~{}\mu\nu}}$ to evaluate
${{\nabla^{\prime}}_{\mu}({{\partial{\mathcal{L^{\prime}}_{M}}\over{\partial({{{\nabla^{\prime}}_{\mu}}{\chi_{a}}})}}})}$.
Thus, we can use ${\Gamma^{\alpha}_{~{}\mu\nu}}$ to construct $S_{M}$,
$\frac{\delta S_{M}}{\delta{\chi_{a}}}$ and field equations. A similar
attribute remains valid for the gauge invariant electromagnetic field tensor
described by the Lagrangian density:
$-\frac{1}{4}{\sqrt{-g}}F_{\mu\nu}F^{\mu\nu}$. We get the following set of
equations for free electromagnetic theory:

$\displaystyle{\nabla_{\mu}}{F^{\mu\nu}}$ $\displaystyle=$
$\displaystyle-{C^{\nu}_{~{}\alpha\beta}}{F^{\alpha\beta}}=0$ (7)
$\displaystyle{\nabla_{[\mu}}{F_{\nu\kappa]}}$ $\displaystyle=$ $\displaystyle
0.$

The right hand side of the first equation is replaced by a current conserved
w.r.t ${{\nabla}_{\mu}}$ when we have sources. The same remains valid with
nonabelian gauge theories. We have, [37]:

$\displaystyle F_{\mu\nu a}$ $\displaystyle=$
$\displaystyle{\nabla_{\mu}}{A_{\nu a}}-{\nabla_{\nu}}{A_{\mu
a}}-{\theta^{bc}_{~{}~{}a}}{A_{\mu b}}{A_{\nu c}}$ (8)
$\displaystyle{\nabla_{\mu}}{F^{\mu\nu}_{~{}~{}a}}$ $\displaystyle-$
$\displaystyle{\theta^{bc}_{~{}~{}a}}{A_{\mu
b}}{F^{\mu\nu}_{~{}~{}c}}=0,~{}{\theta^{bc}_{~{}~{}a}}=-{\theta^{cb}_{~{}~{}a}}$

where ${\theta^{bc}_{~{}~{}a}}$ is the structure constant of the gauge group.
Note that ${C^{\nu}_{~{}\alpha\beta}}(\omega,\psi)$ does not appear explicitly
in ${\mathcal{L^{\prime}}_{M}}$ of all three cases mentioned so far. We now
consider the Dirac fields described by the Lagrangian density [38,39,40]:

$\mathcal{L^{\prime}}=-{\sqrt{-g}}\\{\frac{1}{2}[\bar{\chi}{\gamma^{\mu}}{\nabla^{\prime}_{\mu}}{\chi}-({\nabla^{\prime}_{\mu}}{\bar{\chi}}){\gamma^{\mu}}\chi]+m\bar{\chi}{\chi}\\},~{}{\gamma^{\mu}}={V^{\mu}_{a}}{\gamma^{a}}$
(9)

where $\chi$ are the $SL(2,c)$ spinors, ${V^{\mu}_{a}}$ are the vierbeins
[39,40] and ${\gamma^{a}}$ are the flat spacetime Dirac matrices. Note that
the vierbeins locally satisfy: ${V^{\mu}_{a}}(x){V_{b\mu}}(x)={\eta_{ab}}$,
but ${V^{\mu}_{a}}(x)$ at two different points can not be related by parallel
transport with non-trivial ${C^{\alpha}_{~{}\mu\nu}}$. The covariant
derivative on the $SL(2,c)$ spinors $\chi$ are given by [39,40]:

$\displaystyle{\nabla^{\prime}_{\mu}}$ $\displaystyle=$
$\displaystyle\partial_{\mu}+{\dfrac{1}{2}}{\sigma^{bc}}{V^{\nu}_{b}}({{\nabla^{\prime}}_{\mu}}{V_{c\nu}})$
(10) $\displaystyle=$
$\displaystyle{\nabla_{\mu}}-\frac{1}{2}{\sigma^{bc}}{V_{b}^{\nu}}{V_{c\alpha}}{C^{\alpha}_{~{}\mu\nu}},~{}~{}{\sigma^{bc}}=-\frac{i}{4}[\gamma^{b},\gamma^{c}]$
$\displaystyle=$ $\displaystyle{\nabla_{\mu}}+C_{\mu}.$

For ${C^{\alpha}_{~{}\mu\nu}}$ given by Eq.(2), $\mathcal{L^{\prime}}$
simplifies to:

$\mathcal{L^{\prime}}=-{\sqrt{-g}}\\{\frac{1}{2}[\bar{\chi}{\gamma^{\mu}}{\nabla_{\mu}}{\chi}-({\nabla_{\mu}}{\bar{\chi}}){\gamma^{\mu}}\chi]+m\bar{\chi}{\chi}-\frac{3i}{4}(\bar{\chi}{\gamma^{\mu}}\chi){\nabla_{\mu}}(\omega-\psi)\\}.$
(11)

We can neglect the second term in a theory where the Dirac spinors are coupled
with gauge fields through conserved currents proportional to
$\bar{\chi}{\gamma^{\mu}}\chi$. This includes quantum electrodynamics. Gauge
fields are described as before. Thus, in all four cases mentioned above, we
can replace ${\Theta^{\alpha}_{~{}\mu\nu}}$ by ${\Gamma^{\alpha}_{~{}\mu\nu}}$
in ${\mathcal{L^{\prime}}_{M}}$ and express the latter as ${\mathcal{L}_{M}}$.
We can use ${\mathcal{L}_{M}}$ to obtain corresponding field equations and
construct matter field stress-tensors in ${\nabla_{\mu}}$. Such stress-tensors
lead to Einstein’s equation consistent with the Bianchi identity in
${\Gamma^{\alpha}_{~{}\mu\nu}}$ when we add them in the r.h.s of Eq.(6).

Things become more complicated with weak interaction that include relative
(axial) vector currents [27,37]. To illustrate, we consider the electroweak
theory [37] without the complex scalar fields and use the Levi-Civita
connection:

$\displaystyle\mathcal{\tilde{L}}$ $\displaystyle=$
$\displaystyle{\sqrt{-g}}[-\frac{1}{4}{\vec{A}_{\mu\nu}}.{\vec{A}^{\mu\nu}}-\frac{1}{4}{{B}_{\mu\nu}}{{B}^{\mu\nu}}$
$\displaystyle-$
$\displaystyle{\bar{e}_{R}}({\gamma^{\mu}}{\nabla_{\mu}}+ig^{\prime}{\gamma^{\mu}}{B_{\mu}}){e_{R}}-{\bar{L}_{e}}({\gamma^{\mu}}{\nabla_{\mu}}+i\frac{g^{\prime}}{2}{\gamma^{\mu}}{B_{\mu}}-i\frac{g}{2}{\sigma_{i}}{\gamma^{\mu}}{A^{i}_{\mu}}){L_{e}}]$
$\displaystyle{L_{e}}$ $\displaystyle=$
$\displaystyle{\begin{pmatrix}\nu_{L}\\\
e_{L}\end{pmatrix}},~{}e_{R/L}=\frac{1}{2}(1\pm\gamma_{5}){\chi_{e}},~{}\nu_{L}=\frac{1}{2}(1-\gamma_{5}){\chi_{\nu}}$

where $\chi_{e}$ and $\chi_{\nu}$ are the Dirac fields for the electrons and
neutrinos respectively. Other symbols have usual meanings [37]. As before,
${\nabla_{\mu}}$ is evaluated using the Levi-Civita connection. 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:

$\displaystyle{\mathcal{\tilde{L}}}_{D}$ $\displaystyle=$
$\displaystyle{\sqrt{-g}}[-{\bar{\chi}_{e}}{\gamma^{\mu}}{\nabla_{\mu}}{\chi_{e}}$
$\displaystyle-\frac{1}{2}{\bar{\chi}_{\nu}}{\gamma^{\mu}}{\nabla_{\mu}}{\chi_{\nu}}+\frac{1}{2}{\bar{\chi}_{\nu}}{\gamma^{\mu}}{\gamma_{5}}{\nabla_{\mu}}{\chi_{\nu}}].$

We do not have any problem with the electron field part. However, the
variational derivative: $\frac{\delta{\tilde{S}}_{D}}{\delta{\chi_{\nu}}(x)}$
is non-covariant. This is because
${\sqrt{-g}}[{\bar{\chi}_{\nu}}{\gamma^{\mu}}{\gamma_{5}}({\delta{\chi_{\nu}}})]$
is a vector and, as mentioned before, we cannot eliminate the total divergence
in $\int{\bf
e}{\nabla_{\mu}}[{\sqrt{-g}}({\bar{\chi}_{\nu}}{\gamma^{\mu}}{\gamma_{5}}{\delta{\chi_{\nu}}})]$
completely by using the Stokes’ theorem [27,29]. We can eliminate total
divergence terms by using the Stokes’ theorem for relative vector fields of
weight $+1$. We find that:
$\frac{\delta{\tilde{S}}_{D}}{\delta{\bar{\chi}_{\nu}}(x)}=-{\sqrt{-g}}{\gamma^{\mu}}{\nabla_{\mu}}{\chi_{\nu
L}}(x)$ and
$\frac{\delta{\tilde{S}}_{D}}{\delta{\chi_{\nu}}(x)}={\sqrt{-g}}[{\bar{\chi}_{\nu
L}}(x){\gamma^{\mu}}{\overleftarrow{\nabla}_{\mu}}+\frac{1}{2}{\Gamma^{\mu}_{~{}\mu\alpha}}({\bar{\chi}_{\nu}}{\gamma^{\alpha}}{\gamma_{5}})]$.
This is expected because of the mixing of scalar density [26] and scalar in
${\mathcal{\tilde{L}}}_{D}$ for ${\chi_{\nu}}$. We also find that we cannot
derive a consistent pair of equations for ${{\chi}_{\nu}}$ and
${\bar{\chi}_{\nu}}$ by applying the principle of least action directly to the
above action when ${\bar{\chi}_{\nu}}$ is taken to be the adjoint of
${{\chi}_{\nu}}$ given by ${\chi_{\nu}^{\dagger}}{\gamma^{0}}$. The latter
choice is required to construct various physical variables like stress-tensor
from ${\mathcal{\tilde{L}}}_{D}$ and will be used in the present article. We
do not have problems when: ${\Gamma^{\mu}_{~{}\mu\nu}}=0$, which is not valid
in general. We solve this problem by adding the kinetic term for $\nu_{R}$.
Thus, we modify ${\mathcal{\tilde{L}}}_{D}$ to:

${\mathcal{L}}_{D}=-\frac{1}{2}{\sqrt{-g}}[{\bar{\chi}_{e}}{\gamma^{\mu}}{\nabla_{\mu}}{\chi_{e}}-({\nabla_{\mu}}{\bar{\chi}_{e}}){\gamma^{\mu}}{\chi_{e}}+{\bar{\chi}_{\nu}}{\gamma^{\mu}}{\nabla_{\mu}}{\chi_{\nu}}-({\nabla_{\mu}}{\bar{\chi}_{\nu}}){\gamma^{\mu}}{\chi_{\nu}}].$
(14)

The equations for $\nu_{L}$ and ${\bar{\nu}}_{L}$ can be obtained from that of
${\chi_{\nu}}$ and ${\bar{\chi}_{\nu}}$. There are other important advantages.
We will find below that ${\mathcal{L}}_{D}$ can be used to construct an
expression for conserved vector current for the neutrino while this is not
valid for ${\mathcal{\tilde{L}}}_{D}$. 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
[41], and can be important in explaining neutrino oscillation [42]. They can
also be relevant in the dark matter research. We now consider possible
conserved currents associated with ${\mathcal{\tilde{L}}}_{D}$,
${\mathcal{L}}_{D}$ and the theory that we obtain when we replace
${\mathcal{\tilde{L}}}_{D}$ by ${\mathcal{L}}_{D}$ in Eq.(12).

(i) We find that the global phase transformation:
${\chi}\rightarrow{e^{-ic}}{\chi}$, where $c$ is a constant, is an exact
symmetry of both ${\mathcal{\tilde{L}}}_{D}$ and ${\mathcal{L}}_{D}$. We
generate a conserved vector current for the Dirac spinors by generalizing to
local phase transformation:
${\chi_{e/\nu}}\rightarrow{e^{-i\alpha(x)}}{\chi_{e/\nu}}$ and define the
current as: ${\sqrt{-g}}{\nabla_{\mu}j^{\mu}}=i\frac{\delta
S_{M}}{\delta\alpha(x)}$, [38]. We have: ${\nabla_{\mu}j^{\mu}}=0$, 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 $\frac{\delta S_{M}}{\delta\alpha(x)}=0$,
[43]. We do not have any problem with electrons. However, with
${\mathcal{\tilde{L}}}_{D}$, we cannot define a covariant functional
derivative $\frac{\delta S_{M}}{\delta\alpha(x)}$ for the neutrinos. This is
because, as before, we cannot apply the Stokes’ theorem to the vector:
${\sqrt{-g}}{\alpha(x)}(\bar{\chi_{\nu}}{\gamma^{\mu}}{\gamma_{5}}\chi_{\nu})$,
where $\alpha(x)$ is any regular scalar field and hence, cannot define a
covariant functional derivative $\frac{\delta{~{}~{}~{}}}{\delta\alpha(x)}$ of
the functional: $\int{\bf
e}{\sqrt{-g}}[{\bar{\chi}_{\nu}}{\gamma^{\mu}}{(1-\gamma_{5})}\chi_{\nu}]{\nabla_{\mu}\alpha}$.
Similar comments can remain valid for curvilinear coordinates in flat
spacetime. We do not have such problems with ${\mathcal{L}}_{D}$. The right-
handed neutrinos have been used before to cancel anomalies [41]. Thus,
${\mathcal{L}}_{D}$ is also more appropriate to define the conserved vector
current for the neutrinos.

(ii) ${\chi}\rightarrow{e^{-ic\gamma_{5}}}{\chi}$ is another symmetry
transformation [43] of both ${\mathcal{\tilde{L}}}_{D}$ and
${\mathcal{L}}_{D}$, although we cannot define axial vector currents using
$\frac{\delta S_{M}}{\delta\alpha(x)}$ when we generalize $c$ to $\alpha(x)$.
This remains valid for both ${\mathcal{\tilde{L}}}_{D}$ and
${\mathcal{L}}_{D}$ and is due to the reasons discussed before regarding the
Stokes’ theorem in curved spacetime. Axial vector currents like
$\bar{\chi}{\gamma^{\mu}}{\gamma_{5}}\chi$ remain conserved for the classical
solutions by the equations of motion obtained from ${\mathcal{L}}_{D}$.
Various axial vector currents may remain approximately conserved in the
quantum theory in the 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 [37,43]. This
includes pion decay. We thus replace Eq.(12) by the following expression:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle{\sqrt{-g}}\\{-\frac{1}{4}{\vec{A}_{\mu\nu}}.{\vec{A}^{\mu\nu}}-\frac{1}{4}{{B}_{\mu\nu}}{{B}^{\mu\nu}}$
$\displaystyle-$
$\displaystyle\frac{1}{2}[{\bar{\chi}_{e}}{\gamma^{\mu}}{\nabla_{\mu}}{\chi_{e}}-({\nabla_{\mu}}{\bar{\chi}_{e}}){\gamma^{\mu}}{\chi_{e}}+{\bar{\chi}_{\nu}}{\gamma^{\mu}}{\nabla_{\mu}}{\chi_{\nu}}-({\nabla_{\mu}}{\bar{\chi}_{\nu}}){\gamma^{\mu}}{\chi_{\nu}}]$
$\displaystyle+$ $\displaystyle
ig^{\prime}{\bar{e}_{R}}{\gamma^{\mu}}{B_{\mu}}{e_{R}}+i\frac{g^{\prime}}{2}{\bar{L}_{e}}{\gamma^{\mu}}{B_{\mu}}{L_{e}}-i\frac{g}{2}{\bar{L}_{e}}{\sigma_{i}}{\gamma^{\mu}}{A^{i}_{\mu}}{L_{e}}\\}.$

(iii) ${\mathcal{\tilde{L}}}$ 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 $\mathcal{L}$ with $\nu_{R}$ remaining unchanged under such
transformations. However, $S_{M}$ obtained from ${\mathcal{\tilde{L}}}$ is not
invariant under pure gauge transformations: $A_{\mu}\rightarrow
A_{\mu}+{\nabla_{\mu}}{\lambda(x)}$, without any change in the phases of
various Dirac fields. This is consistent with (i) and (ii) and indicates
possible axial current anomaly. The same remains valid for $\mathcal{L}$. 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:
$\bar{\chi}{\gamma^{\mu}}M\chi,~{}M=1,{\gamma_{5}}$. As before, Stokes’
theorem implies that $\frac{\delta{S_{M}}}{\delta\lambda(x)}$ is not tensorial
in general under pure gauge transformations even if we impose
$\nabla_{\mu}(\bar{\chi}{\gamma^{\mu}}\chi)=0$. Thus, with $\mathcal{L}$ we
cannot define conserved currents by using: $\frac{\delta
S_{M}}{\delta\lambda(x)}$. This is the reason that we have started with
invariance under phase transformations of Dirac fields to define possible
conserved currents [43] in the present theory rather than the gauge
transformations. A 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 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
that depend on $\mathcal{L}$, regularization schemes and possibly boundary
conditions [43,44,45]. (i), (ii) and (iii) can be important in the early
universe.

We now replace the Levi-Civita connection in Eq.(15) by
${\Theta^{\alpha}_{~{}\mu\nu}}$. As in the case of Eq.(9), it is easy to show
that $\omega$ and $\psi$ can be eliminated from the action obtained from
$\mathcal{L}$. This helps us to take
$\mathcal{L^{\prime}}({\Theta^{\alpha}_{~{}\mu\nu}})=\mathcal{L}({\Gamma^{\alpha}_{~{}\mu\nu}})$
for the Dirac fields in the electroweak theory with $\mathcal{L}$ suitably
generalized from Eq.(15) by adding the complex scalar fields [37]. Thus, we
can construct the standard model using ${\nabla_{\mu}}$ only.
${C^{\alpha}_{~{}\mu\nu}}(\omega,\psi)$ given by Eq.(2) does not appear
explicitly in action for scalar fields, abelian and nonabelian gauge fields.
It also does not appear in the equations of motion for these fields even if we
include it in the corresponding covariant derivatives. For spin half spinors,
${C^{\alpha}_{~{}\mu\nu}}(\omega,\psi)$ only contributes boundary terms when
we use Eq.(14). ${C^{\alpha}_{~{}\mu\nu}}(\omega,\psi)$ will not be present in
matter field stress-tensors also. This partly justifies our efforts to explain
dark energy using $\psi$. 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 connection ${\Theta^{\alpha}_{~{}\mu\nu}}$ provided
we modify the action so that we do not have any problem with
${\Gamma^{\alpha}_{~{}\mu\nu}}$. The right-handed neutrino, $\nu_{R}$, remains
a free field and can be important for various purposes mentioned before,
including QFT in flat spacetime. We can follow (3) below when we use Eq.(12)
to describe electroweak theory. Otherwise, we have coupling between $\omega$,
$\psi$ and neutrinos that can give novel features in the semiclassical theory,
quantum fields in curved spacetime. In this case, we have to consider a
particular combination of $\omega$ and $\psi$ to have a stable theory. This is
discussed in the next section. We can not define conserved vector current for
the neutrinos even when $\omega,\psi=0$. We can have coupling between
$(\omega,\psi)$ and general tensor fields that are not antisymmetric. In this
case, we can again follow (3) below. It is unexpected that $\omega$ and $\psi$
will couple with classical sources if they do not couple with quantum fields.
The stress tensor of such sources, when added to the source of Einstein’s
equation given by Eq.(6), gives equation of motion consistent with the Bianchi
identity [29]. Torsion will be discussed later.

(3) Restricting ${{\Theta}^{\alpha}_{~{}\mu\nu}}$ to
${\Gamma^{\alpha}_{~{}\mu\nu}}$ in matter and radiation actions:

According to the principle of general covariance, we can restrict
${{\Theta}^{\alpha}_{~{}\mu\nu}}$ to ${\Gamma^{\alpha}_{~{}\mu\nu}}$, when we
need to replace the partial derivatives acting on matter and radiation fields
by covariant derivatives. When we try to construct a theory of gravity with
sources, of the form given by Eq.(6), we supplement the r.h.s of Eq.(6) with
stress tensors that are symmetric and conserved w.r.t ${\nabla_{\mu}}$. The
stress tensor of ideal fluid does not contain the connection and is conserved
w.r.t ${\Gamma^{\alpha}_{~{}\mu\nu}}$. 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 connection
[31]. Partial derivatives acting on $SL(2,c)$ spinors are generalized from
flat spacetime to curved spacetime by replacing them with suitable
combinations of covariant derivatives given by ${\nabla^{\prime}_{\mu}}$,
${V^{\mu}_{a}}{\partial_{\mu}}$, ${V^{\mu}_{a}}{\nabla^{\prime}_{\mu}}$ and
${{\nabla^{\prime}}_{\nu}}{V^{\mu}_{a}}$ where ${V^{\mu}_{a}}$ are the
vierbeins. The choice of connection depends on the contracted Bianchi
identity. In general, stress tensor for $SL(2,c)$ spinors are always conserved
w.r.t ${{\nabla^{\prime}}_{\mu}}$ when the connection in
${{\nabla^{\prime}}_{\mu}}$ is ${\Gamma^{\alpha}_{~{}\mu\nu}}$, [40].
${\Gamma^{\alpha}_{~{}\mu\nu}}$ is also used in the Newman-Penrose formalism.
Thus, a theory where we do not include ${C^{\alpha}_{~{}\mu\nu}}(\omega,\psi)$
in the covariant derivatives when acting on matter and radiation from the
beginning, is a possible theory.

We find from the above three points that it is nontrivial to extend the
standard model to curved spacetime even with the Levi-Civita connection. 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. Observations in (2) and (3) imply that for fields relevant to the
standard model, fields that are various representations of $SL(2,c)$ with the
addition of the right handed neutrinos, we can use only
${\Gamma^{\alpha}_{~{}\mu\nu}}$ as the connection present in the covariant
derivatives acting on them even when $C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$ is
finite. Variables like stress-tensors of ordinary matter and radiation fields
also contain only ${\Gamma^{\alpha}_{~{}\mu\nu}}$ as the connection. 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 do the
same to defining the gauge invariant electromagnetic field tensor in the
presence of torsion [31]. This explains why Levi-Civita connection and
Einstein’s gravity explain the observable universe to a very good
approximation. Under these circumstances,
$C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$ only contributes a stress-tensor in the
r.h.s of the modified Einstein’s equation given by Eq.(6) that presently has
small effect on spacetime.

Depending on $\psi$, we may not have: ${V^{\mu}_{a}}{V_{b\mu}}={\eta_{ab}}$,
where ${\eta_{ab}}$ is the flat spacetime metric, when we parallel transport
${V^{\mu}_{a}}$ from one point to a distant point in the spacetime manifold
[23,24,25]. Thus, we do not have any problem with the principle of
equivalence, but the global splitting of spacetime into space and time may not
remain exact. In the Einstein-Palatini formalism considered here, $\omega$ and
${\psi}$ 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 is observed through large scale cosmological observations. $\psi$
contributes a negative source stress tensor to Einstein’s equation and the
preceding discussions indicate that $\omega$ and ${\psi}$ need not be present
in the covariant derivatives acting on matter and radiation fields. Thus,
$\omega/\phi$ and $\psi$ need not to have ordinary matter and radiation as
their sources [31]. All these make $\omega$ and $\psi$ as possible candidates
to explain inflation and 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. This may be required to construct a
stable theory. We will discuss this issue in the next section. We find that a
part of dark matter can be the right-handed neutrinos. ${\phi}$ is similar to
the dilaton [46]. Cosmological observations indicate that non-metricities
produced by $\omega$ and ${\psi}$ are presently very small. This is consistent
with the smallness of the cosmological constant in the $\Lambda$CDM model.
$\omega$ and $\psi$ manifest themselves through long range geodesics and the
global structure of spacetime. However, local inhomogeneities and anisotropies
in $\omega$ and $\psi$ can cause significant effects on geodesic motions in
the corresponding regions. $\omega$ and $\psi$ are useful to discuss geometry.
In the next section, we will find that it is more useful to consider linear
combinations of $\omega$ and $\psi$ in discussing stability and 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 ${\omega}$ and ${\psi}$. This will be useful in cosmic
epoch. We now compare Eqs.(4,5) with the action of scalar-tensor theories [6]:

$S={\int{\bf
e}}{\sqrt{-g}}{\big{[}}f(\phi,R)-\zeta(\phi){({\nabla}{\phi})^{2}}{\big{]}}+{S_{m}(\chi_{a},\nu_{R},{g_{\mu\nu}})}$
(16)

where $\chi_{a}$ is the collection of ordinary matter and radiation. Eq.(5)
indicates that: $\zeta(\omega)={3\over 2},~{}~{}\zeta(\psi)=-3$. $f(\phi,R)=R$
for both fields. Thus, the scalars mentioned at the beginning of the first
section like quintessence and k-essence can be related to quantum gravity and
can have geometrical origin. When required, we can modify the theory by adding
suitable $\omega$ and $\psi$ dependent terms in Eq.(5). We get:

$\mathcal{L}={\sqrt{-g}}(\mathcal{R^{\prime}}-\mathcal{V})$ (17)

where $\mathcal{V}$ contains the added terms which should be consistent with:
${{\nabla}^{\alpha}}{{G}_{\mu\alpha}}=0,$ if we express the equations in the
form similar to Eq.(6). In general, the above Lagrangian density will give
coupled equations for $\omega$ and $\psi$. We obtain a decoupled theory by
diagonalizing the symmetric bilinear form in ${{\nabla}^{\mu}{\omega}}$ and
${{\nabla}^{\nu}{\psi}}$ given by Eq.(5), [47]. This bilinear form is similar
to: ${a^{2}}-2ab-2{b^{2}}$ and the corresponding symmetric matrix has
eigenvalues: ${\lambda_{\pm}}=(-1\pm\sqrt{13})/2$ . $\mathcal{R^{\prime}}$ is
now given by the following expression:

$\displaystyle{\mathcal{R^{\prime}}}$ $\displaystyle=$
$\displaystyle{R}-{3\over
2}[{({\nabla}{\omega})^{2}}-2{({\nabla}{\psi})^{2}}-2({{\nabla}_{\kappa}{\omega}})({{\nabla}^{\kappa}{\psi}})]$
$\displaystyle=$ $\displaystyle{R}-{3\over
2}{g^{\mu\nu}}[{\lambda_{+}}{({{\nabla}_{\mu}{\Phi^{\prime}}})({{\nabla}_{\nu}{\Phi^{\prime}}})}+{\lambda_{-}}{({{\nabla}_{\mu}{\Psi^{\prime}}})({{\nabla}_{\nu}{\Psi^{\prime}}})}]$

where $\Phi^{\prime}$ and $\Psi^{\prime}$ are related to $\omega$ and $\psi$
by the following expression:

${\begin{pmatrix}\Phi^{\prime}\\\
\Psi^{\prime}\end{pmatrix}}={S^{T}}{\begin{pmatrix}\omega\\\
\psi\end{pmatrix}},~{}~{}~{}S={\begin{pmatrix}{c^{+}_{1}}&{c^{-}_{1}}\\\
{c^{+}_{2}}&{c^{-}_{2}}\end{pmatrix}}.$ (19)

Here ${c^{\pm}_{i}}$ are the components of two orthonormal eigenvectors. $S$
is given by the following expression:

$S={\begin{pmatrix}{\sqrt{4\over{26-6\sqrt{13}}}}&{\sqrt{4\over{26+6\sqrt{13}}}}\\\
{{3-\sqrt{13}\over 2}}\sqrt{4\over{26-6\sqrt{13}}}&{{3+\sqrt{13}\over
2}}\sqrt{4\over{26+6\sqrt{13}}}\end{pmatrix}}.$ (20)

We can absorb the positive numerical factors into
$\Phi^{\prime},\Psi^{\prime}$ and the modified curvature scalar is given by:

${\mathcal{R^{\prime}}}={R}-{1\over
2}[{({{\nabla}{\Phi}})^{2}}-{({{\nabla}{\Psi}})^{2}}].$ (21)

The above expression gives a more convenient way to discuss the physical
effects of the non-metricity scalar fields $\omega$ and $\psi$. We can express
$C^{\alpha}_{~{}\mu\nu}$ in terms of $\Phi$ and $\Psi$ using the inverse of
Eq.(19). We find that $\Psi$ contributes a negative stress tensor to
Einstein’s equation and is a phantom scalar [6] of geometric origin. We always
get a scalar field with a negative stress tensor in Einstein’s equation,
irrespective of the initial linear combinations of $\nabla^{\mu}\omega$ and
$\nabla^{\nu}\psi$, provided they give finite $C^{\alpha~{}\mu}_{~{}\mu}$.
$\Psi$ vanishes when $C^{\alpha}_{~{}\mu\nu}$ is traceless in the lower
indices. We can introduce various potential terms like
${{\mathcal{V}}_{1}}(\Phi)$, ${{\mathcal{V}}_{2}}(\Psi)$ and as per
requirement. ${{\mathcal{V}}_{1}}(\Phi)$ and ${{\mathcal{V}}_{2}}(\Psi)$ can
give us a theory more general than that given by Eq.(16) and new effects in
the scalar-tensor theories. The contracted Bianchi identity will remain valid
for this theory. Thus, transforming back to $(\omega,\psi)$ we will get a
theory with $(\omega,\psi)$ that preserves the contracted Bianchi identity.

We can consider a theory containing $\Phi$, $\Psi$ and the cosmological
constant $\Lambda$. This will lead to a possible theory of inflation together
with dynamic and spatially varying dark energy. Einstein’s equation will be
generalized to:

$G_{\mu\nu}+{\Lambda}{g_{\mu\nu}}-{\Lambda_{\mu\nu}(\Phi,\Psi)}=8{\pi}{P_{\mu\nu}(\chi_{a})}$
(22)

where ${\Lambda_{\mu\nu}(\Phi,\Psi)}$ includes
${P_{\mu\nu}(\Phi)},{P_{\mu\nu}(\Psi)}$ and potential terms for $\Phi$ and
$\Psi$. The left hand side gives a generalization of the $\Lambda$CDM theory
linear in curvature and consistent with the Bianchi identities. This is along
the line of introducing the cosmological constant in Einstein’s equation.
${{P}_{\mu\nu}(\chi_{a})}$ is the stress tensor of ordinary matter and
radiation including the right handed neutrinos. We can construct new non-
trivial solutions of the vacuum Einstein’s equation [29,48,49], i.e, for
${P_{\mu\nu}(\chi_{a})}=0$, with different choices of
${\Lambda_{\mu\nu}(\Phi,\Psi)}$. An important issue is the stability of the
theory. $\Psi$ provides a negative kinetic term, thus behaving like a phantom
scalar [5,50,51,52,53]. Phantom field is a possible candidate to explain dark
energy [5,6]. A phantom field, interacting with ordinary matter and radiation,
can lead to instability both in classical and quantum theory. Stable classical
theories are discussed by a few authors by adding suitable potential terms for
the phantom fields [52,53]. There remains the issue of unstable vacuum
[54,55]. In the present theory, $\Psi$ does not couple with ordinary matter
and scattering between the two sectors causing instability of the vacuum is
not possible. Processes of the form: $Vacuum\rightarrow 2\gamma+2\psi$, where
$\gamma$ is a photon and $\psi$ gives a phantom quanta, need careful analysis.
The coupling between $g_{\mu\nu}$ and $\Psi$, given by
${({{\nabla}{\Psi}})^{2}}$, can be put in the form ${\Psi}{\nabla^{2}}{\Psi}$,
vanishes for the classical solutions, thus prohibiting the decay of vacuum to
a detectable on-shell spectrum. Total derivatives in the action have no effect
on the classical states. ${\Psi}{\nabla^{2}}{\Psi}$ becomes a polynomial in
$\Psi$ with a ${{\mathcal{V}}(\Psi)}$ for the on-shell spectrum in a
nonperturbative theory, and there is no coupling with the graviton. Thus,
classically stable self-interacting theory of $\Psi$ can give a stable quantum
theory. A theory with static or nearly static $\Psi$ can be useful for
discussing dark energy. A scalar field with negative energy density is also
required to construct the steady state model of the expanding universe
[48,56,57,58,59].

$\Phi$ contributes a positive stress tensor and gives a stable theory. Both
${Q_{\mu}}$ and ${{\bar{Q}}_{\mu\alpha\beta}}$ are finite for $\Phi$. Thus, we
can consider a quantum theory with only $\Phi$ present as the non-metricity
field. In the next section we will consider theories that can contain more
than one $\Phi$ fields. Such a theory can be useful to explain both inflation
and dark energy [20]. We can introduce quadratic self-interaction term in
$\Phi$. We can couple $\Phi$ with ordinary matter by adding suitable
interaction terms. Being of geometric origin, the coupling between
$g_{\mu\nu}$ and $\Phi$ does not involve $G$. The same remains valid for
$\Psi$. $\Phi$ and $\Psi$ are quantum gravitational in origin [23,24], and it
is expected that they can play important roles in the very early universe.
$g_{\mu\nu},\Phi,\Psi$ should be treated similarly in discussing quantum
effects like vacuum fluctuations. Various free field configurations of $\Phi$
and $\Psi$, their vacuum energies, quantum fluctuations and classical
configurations with possible interactions can be useful to explain inflation
and dark energy [6,35,50].

Equation (22) leads to a generalization of the $\Lambda$CDM theory. We do not
obtain the Newtonian theory in the weak field and slow motion limit when
$\Lambda\neq 0$, [29]. This implies that $\Lambda$ is small. This is similar
to the non-metricities produced by the pair ($\omega$, $\psi$) or ($\Psi$,
$\Phi$) as discussed in the previous section. In this context, $\Phi$ and
$\Psi$ have the advantage of being dynamical and hence can produce significant
classical and quantum effects even in a free theory. This is along the line of
perturbative quantum field theory and Casimir effect. Various non-trivial
vacuum solutions of Einstein’s equation, with and without $\Lambda$ [48,49],
imply that affine connection need not to have ordinary matter and radiation as
its sources even in the classical theory. The same can remain valid for $\Phi$
and $\Psi$. This agrees with the comments in the previous paragraph and the
observed energy composition of our universe if these fields are useful to
explain inflation and dark energy. It is unexpected that $10\%$ ordinary
matter can produce $70\%$ dark energy. 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 [60,61,62]. We will discuss applications of Eq.(22) in cosmology
in a forthcoming article. ${P_{\mu\nu}(\chi_{a})}$ 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.(22), will contain higher order curvature
invariants evaluated using the Levi-Civita connection together with quantum
corrections coming for various matter fields and possibly for non-metricity
fields $\Phi$ and $\Psi$. The last set will depend on the requirement of
potential terms for $\Phi$ and $\Psi$. This is the case with some low energy
effective actions of string theory that contain ghost field [6]. Lastly, a
possible approach to explaining dark energy could be to consider $\Psi$ as the
possible source of $\Lambda$. Discussions on possible sources of $\Lambda$ for
constant curvature spaces can be found in [48]. These are similar to $\Psi$.
We cannot 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

In Sect.II, we found that we can introduce scalar fields through
${C^{\alpha}_{~{}\mu\nu}}$ without reference to any symmetric second rank
tensors like ${\tilde{g}_{\mu\nu}}$ discussed in this article or $a_{\mu\nu}$
and $q_{\mu\nu}$ discussed previously [24,25]. It is thus possible to
introduce additional scalar fields depending on observations, i.e, when such
scalars couple differently with other fields including $\omega,\psi$. We can
take the corresponding part of ${C^{\alpha}_{~{}\mu\nu}}$ to be traceless in
the mixed indices or in the lower indices. Considering various models to
explain inflation and dark energy discussed in [6,32,33], we can generalize
Eq.(2) to the following expression:

$\displaystyle{C^{\alpha}_{~{}\mu\nu}}$ $\displaystyle=$
$\displaystyle{{\delta}^{\alpha}_{~{}(\mu}}{{\nabla}_{\nu)}}{\omega}-{1\over
2}{g_{\mu\nu}}{{{\nabla}^{\alpha}}{\omega}}$ $\displaystyle+$
$\displaystyle{g_{\mu\nu}}{\nabla^{\alpha}}{\psi}+{{\delta}^{\alpha}_{~{}(\mu}}{{\nabla}_{\nu)}}{\eta}-{1\over
4}{g_{\mu\nu}}{{{\nabla}^{\alpha}}{\eta}}$
$\displaystyle{\mathcal{R^{\prime}}}$ $\displaystyle=$
$\displaystyle{R(\Gamma^{\alpha}_{~{}\mu\nu})}-{1\over
2}[{\sum_{1}^{3}}(-1)^{s_{i}}{(\nabla\Phi_{i})}^{2}]$

here $s_{i}=0/1$. Two of $s_{i}$ are zero and two are $1$, i.e, corresponding
$\Phi_{i}$ are phantom fields. As before, we do not need to include
${C^{\alpha}_{~{}\mu\nu}}({\Phi}_{i})$ in the matter and radiation field
actions when we include the right handed neutrinos. We can introduce self
interactions depending on observations. We can also couple those $\Phi_{i}$
with positive kinetic terms to ordinary matter. The above model resemblances
various models to address inflation and dark energy, including the multi-
scalar theory of inflation [7,35,50], k-essence and phantom fields to explain
dark energy, steady-state model of the expanding universe [48,56,57] and re-
bouncing model of the universe [48], as special cases. The theory with only
$\Phi$ and $\Psi$ associated with two traces of ${C^{\alpha}_{~{}\mu\nu}}$
appears to be mathematically most straightforward.

We now consider the case of higher spin ($j>0$) fields that can be introduced
using ${C^{\alpha}_{~{}\mu\nu}}$ symmetric in the lower indices by expressing
the latter as derivatives of lower rank tensors. In flat spacetime, second
rank symmetric traceless tensors give $(A,A)$ type representations of higher
spin fields like $j=1,2$ [38]. We will show in the next section that we can
construct a $(A,A)$ type quantum field theory for massless vector fields using
symmetric traceless derivatives. Such derivatives are not gauge invariant and
cannot couple with matter fields in a theory that preserves local internal
gauge invariance. They can be useful for describing dark energy and dark
matter. The massless higher spin particles can also be useful to construct
higher spin driven inflation theories [7,8,9] in the absence of spontaneous
symmetry breaking that need not to be gauge theories. We consider the case
that includes a second rank symmetric traceless field ${{\bar{q}_{\mu\nu}}}$
[23,24,25]:

${{C}^{\alpha}_{~{}\mu\nu}}={{\delta}^{\alpha}_{~{}(\mu}}{{\nabla}_{\nu)}}{\omega}-{1\over
2}{g_{\mu\nu}}{{{\nabla}^{\alpha}}{\omega}}+{g_{\mu\nu}}{{\nabla}^{\alpha}{\psi}}+{\nabla^{\alpha}}{{\bar{q}_{\mu\nu}}}.$
(24)

Regarding dimensions, we have: $[{{\bar{q}_{\mu\nu}}}]=[g_{\mu\nu}]$. We have
discussed in Sect.II that after we have removed the total divergence terms in
$R^{\prime}$ and expressed the Einstein’s equation in the form given by
Eq.(6), where the contribution of the non-metricity fields appears as source
terms, the resulting equations have to be consistent with the contracted
Bianchi identity in ${\Gamma^{\alpha}_{~{}\mu\nu}}$. This, together with the
equations of motion, imposes severe restrictions on ${{\bar{q}_{\mu\nu}}}$.
The exception being the scalar fields mentioned before. Thus, we have to
modify the Einstein-Palatini action by adding suitable ${{\bar{q}_{\mu\nu}}}$
dependent terms so that the resulting theory is consistent with the contracted
Bianchi identity. We can also use the metric-affine $f(R^{\prime})$ theories.
This may indicate that the resulting theory is described by the low energy
effective action of a more fundamental theory. We can replace
${\bar{q}_{\mu\nu}}$ with the traceless derivative:
$t_{\mu\nu}={{\nabla_{(\mu}}{{B}_{\nu)}}}$ of a vector field $B_{\alpha}$ with
$\nabla_{\alpha}B^{\alpha}=0$, to have a spin one field from geometric sector.
We then have:

${{C}^{\alpha}_{~{}\mu\nu}}={{\delta}^{\alpha}_{~{}(\mu}}{{\nabla}_{\nu)}}{\omega}-{1\over
2}{g_{\mu\nu}}{{{\nabla}^{\alpha}}{\omega}}+{g_{\mu\nu}}{{\nabla}^{\alpha}{\psi}}+{\nabla^{\alpha}}{t_{\mu\nu}}$
(25)

where ${t_{\mu}^{~{}\mu}}=\nabla.B=0$. Note that we have used a symmetric
field $t_{\mu\nu}$ to introduce a spin one field and this field will not give
a symmetric ${{R^{\prime}}_{\mu\nu}}$ although the corresponding
${\mathcal{R^{\prime}}_{\mu\alpha}}$ will be symmetric. We again have a
problem with the contracted Bianchi identity similar to that mentioned before.
We will later address this problem. We also note that we cannot eliminate
$B^{\alpha}$ from the action of the Dirac field spinors and the latter can
couple with $B^{\alpha}$. Such coupling is yet to be observed. Thus, the two-
scalar model considered in this article is more consistent with the
observations.

We can introduce another scalar field when ${C^{\alpha}_{~{}\mu\nu}}$ is not
symmetric in the lower indices. This can be associated with the trace of
corresponding torsion tensor. We generalize Eq.(2) to the following
expression:

$\displaystyle{{\Theta}^{\alpha}_{~{}\mu\nu}}$ $\displaystyle=$
$\displaystyle{\Gamma^{\alpha}_{~{}\mu\nu}}+{{C}^{\alpha}_{~{}\mu\nu}}$ (26)
$\displaystyle{{C}^{\alpha}_{~{}\mu\nu}}$ $\displaystyle=$
$\displaystyle{{\delta}^{\alpha}_{~{}(\mu}}{{\nabla}_{\nu)}}{\omega}-{1\over
2}{g_{\mu\nu}}{{{\nabla}^{\alpha}}{\omega}}+{g_{\mu\nu}}{{\nabla}^{\alpha}{\psi}}+{{\delta}^{\alpha}_{~{}[\mu}}{{\nabla}_{\nu]}}{\rho},~{}~{}\omega=ln(1+\phi).$

To discuss the geometric significance of the above connection we use the non-
metricity tensor ${Q_{\mu\alpha\beta}}$ [31]:

${Q_{\mu\alpha\beta}}=-{{\nabla^{\prime}}_{\mu}}{g_{\alpha\beta}}$ (27)

where ${\nabla^{\prime}}_{\mu}$ is evaluated using the complete connection:
${\Gamma^{\alpha}_{~{}\mu\nu}}+{{C}^{\alpha}_{~{}\mu\nu}}$. We have the
following expressions for ${Q_{\mu\alpha\beta}}$:

${Q_{\mu\alpha\beta}}={g_{\alpha\beta}}{{\nabla}_{\mu}{\omega}}+2{g_{\mu(\alpha}}{{\nabla}_{\beta)}{\psi}}+{g_{\mu(\alpha}}{{\nabla}_{\beta)}{\rho}}-{g_{\alpha\beta}}{{\nabla}_{\mu}{\rho}}.$
(28)

It is convenient to express ${Q_{\mu\alpha\beta}}$ as the sum of a trace and
trace-free part in the last two indices [31]:

${Q_{\mu\alpha\beta}}={Q_{\mu}}{g_{\alpha\beta}}+{{\bar{Q}}_{\mu\alpha\beta}}.$
(29)

${{\bar{Q}}_{\mu\alpha\beta}}$ is traceless in the last two indices.
${{\bar{Q}}_{\mu\alpha\beta}}$ vanishes when $\psi,\rho=0$. Corresponding
${{C}^{\alpha}_{~{}\mu\nu}}$ preserves the light cone under parallel transport
due to the reparameterization invariance of the form of the geodesic equation:
$t^{\mu}{{\nabla}_{\mu}t^{\nu}}={f}t^{\nu}$, where $f$ is a scalar function on
the curve [25,29]. Both trace and traceless parts of ${Q_{\mu\alpha\beta}}$
are finite for $\psi$ and $\rho$. Depending on $\psi$ and $\rho$,
corresponding connection may not preserve the light cone under parallel
transport [31]. Global $(3+1)$ -splitting of the complete spacetime manifold
into space and time may not exist under these circumstances. The modified
curvature scalar is now given by the following expression:

$\displaystyle{\mathcal{R^{\prime}}}$ $\displaystyle=$
$\displaystyle{g^{\mu\alpha}}{\mathcal{R^{\prime}}_{\mu\alpha}}={g^{\mu\alpha}}\left\\{{{R}_{\mu\alpha}}+2[{C^{\lambda}_{~{}[\mu|\alpha|}}{C^{\kappa}_{~{}\kappa|\lambda|]}}]\right\\}$
$\displaystyle=$ $\displaystyle{R}-{3\over
2}{({\nabla}{\omega})^{2}}+3{({\nabla}{\psi})^{2}}+{3\over
4}{({\nabla}{\rho})^{2}}+3({{\nabla}_{\kappa}{\omega}})({{\nabla}^{\kappa}{\psi}})+{1\over
2}({{\nabla}_{\kappa}{\omega}})({{\nabla}^{\kappa}{\rho}}).$

Note that ${\mathcal{R^{\prime}}_{\mu\alpha}}$ is no longer symmetric. We can
again diagonalize and rescale various scalar fields to obtain:

${\mathcal{R^{\prime}}}={R}-{1\over
2}[{({{\nabla}{\Phi}})^{2}}-{({{\nabla}{\Psi}})^{2}}-{({{\nabla}P})^{2}}]$
(31)

We find that $\rho$ contributes another phantom field. $\Phi,\Psi$ and $P$ are
given by linear combinations with different strengths of all three fields:
$\omega$, $\psi$ and $\rho$. $\rho$ does not couple with scalar fields and
Dirac fields when the Lagrangian of the latter is given by expressions similar
to Eq.(14). However, $\rho$ couples with the gauge fields. This is evident
from Eq.(7) and is expected in presence of torsion [31]. The field tensor is
given by:
${F^{\alpha\beta}}={g^{\alpha\mu}}{g^{\beta\nu}}{F_{\mu\nu}}={g^{\alpha\mu}}{g^{\beta\nu}}[{{\nabla_{\mu}}{A_{\nu}}}-{{\nabla_{\nu}}{A_{\mu}}}+{T^{\alpha}_{~{}\mu\nu}}{A_{\alpha}}]={g^{\alpha\mu}}{g^{\beta\nu}}[{{\partial}_{\mu}}A_{\nu}-{{\partial}_{\nu}}{A_{\mu}}]$.
Here:
${T^{\alpha}_{~{}\mu\nu}}=2{{\delta}^{\alpha}_{~{}[\mu}}{{\nabla}_{\nu]}}{\rho}$
and
$-{C^{\nu}_{~{}\alpha\beta}}{F^{\alpha\beta}}=-{F^{\nu\beta}}{\nabla_{\beta}}{(\rho)}$.
The coupling of ${F^{\alpha\beta}}$ with torsion violates the current
conservation in an interacting theory. There is no problem in a theory with
free gauge fields. Vacuum effects can again be useful to explain inflation,
dark energy and related problems [6,32,33,63,64,65]. Higher spin fields can
also be introduced using torsion potentials [66].

## VI. Higher Spin Massless Bosons and Little Group

Here, we consider the issue of constructing a semi-classical limit of the
theories considered in this article. One of which can be a locally Lorentz
invariant quantum field theory in curved spaces including $t_{\mu\nu}$
discussed in Sect.III. We discuss this with reference to little groups in
Minkowski space and $(A,A)$ type fields [38]. We do not have any problem for
the massless scalar fields for which $A=0$, [38]. Massless integer spin
particles are usually described by $(A,0)\oplus(0,A)$ type fields [38,67,68].
Here, $A$ is an integer. The most general candidate to give the little group
(stabilizer) for massless particles with spin greater than zero in Minkowski
space is taken to be: $W=ISO(2)$, [38]. This forces us to use
$(1,0)\oplus(0,1)$ type gauge invariant antisymmetric derivatives to describe
massless spin-one particles [38]. A similar situation remains valid with other
massless integral spin particles. However, $ISO(2)$ is not represented by
normal matrices [69] when acting on four vectors/tensors. In the following, we
will find that we have to use a one parameter subgroup of $SO(3)$ as the
little group for massless higher spin fields in Minkowski space. This allows
us to construct a $(1,1)$ type quantum field theory for massless vector fields
using symmetric traceless derivatives in addition to $(1,0)\oplus(0,1)$ type
antisymmetric derivatives. Thus, we can use the symmetric traceless
derivative: ${t_{\mu\nu}}={\partial_{(\mu}B_{\nu)}}$ with ${\partial}.B=0$ to
introduce a massless spin-one particle.

In flat spacetime, the standard momentum $k^{\mu}$ 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 $+2$, [38]. The
little group that keeps the standard momentum unchanged is given by $ISO(2)$
and for spacetime tensors, it is taken to be
$W(\theta,\alpha,\beta)=S(\alpha,\beta)R_{z}(\theta)$, [38]. Where
$R_{z}(\theta)$ is a spatial rotation about the Z -axis by angle $\theta$ and
$S(\alpha,\beta)$ is given by:

${S^{\mu}_{~{}\nu}(\alpha,\beta)}=\begin{pmatrix}1&0&-\alpha&\alpha\\\
0&1&-\beta&\beta\\\ \alpha&\beta&1-\gamma&\gamma\\\
\alpha&\beta&-\gamma&1+\gamma\end{pmatrix}.$ (32)

Here, $\gamma=(\alpha^{2}+\beta^{2})/2$. 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 [38]:

${W^{\mu}_{~{}\nu}}{e^{\nu}(k,\sigma)}=\exp{(-i\sigma\theta(k,W))}{e^{\mu}(k,\sigma)}.$
(33)

However, the matrix representing ${S^{\mu}_{~{}\nu}}$ is not normal
$(S^{T}S\neq SS^{T})$. This is also valid for $W=SR$ and the above eigenvalue
equation for ${e^{\mu}(k,\sigma)}$ does not exist [66]. Thus, we restrict $W$
to $R_{z}(\theta)$ 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 $S(\alpha,\beta)$ with the
charges continuously dependent on $\theta$, [38].

We now consider ${e^{\mu}(k,\sigma)}$ that satisfy the following equation
instead of Eq.(30):

$[R_{z}(\theta)]^{\mu}_{~{}\nu}{e^{\nu}(k,\sigma)}=\exp{(-i\sigma\theta)}{e^{\mu}(k,\sigma)}$
(34)

where, $\sigma=0,\pm{1}$. A convenient choice for $e^{\mu}(k,0;\pm 1)$ is:
$e^{\mu}(k,\pm 1)={1\over\sqrt{2}}(1,\pm i,0,0)$ and $e^{\mu}(k,0)=(0,0,1,1)$.
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: $({{\beta
k}\over{\sqrt{1-\beta^{2}}}},0,k,{{k}\over{\sqrt{1-\beta^{2}}}})$. Firstly, we
can apply a boost $\beta$ along the $x^{1}$ axis. Alternatively, we can rotate
$(0,0,k,k)$ to $(k\beta,0,k\sqrt{1-\beta^{2}},k)$ and thereafter apply a boost
along the direction of the spatial momentum by $\alpha$ where:

$1-\beta^{2}={{1-\alpha}\over{1+\alpha}},~{}~{}\alpha=|\vec{\alpha}|.$ (35)

The above two Lorentz transformations do not lead to the same
${e^{\mu}(\vec{p},\pm 1)}$. This is related to the four -dimensional geometry.
However, we can always have:
${k_{\mu}}{e^{\mu}(k,\sigma)}={p_{\mu}}{e^{\mu}(\vec{p},\sigma)}=0={\partial_{\mu}}{B^{\mu}}(x)$,
where ${B^{\mu}}(x)$ is the Fourier transform of ${e^{\mu}(\vec{p},\sigma)}$
given by, [38]:

$B^{\mu}(x)=\int{{d^{3}}p\over\sqrt{2p^{0}}}{(2\pi)^{-3/2}}{\sum_{\sigma}}[{e^{\mu}(\vec{p},\sigma)}{\exp(ip.x)}a(\vec{p},\sigma)+c.c]$
(36)

where, $a(\vec{p},\sigma)$ is an operator. Under a Lorentz transformation $L$,
${B^{\mu}}$ transform as:

$U(L){B_{\mu}(x)}{U^{-1}}(L)={L^{\nu}_{~{}\mu}}{B_{\nu}(Lx)}$ (37)

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 $B_{\nu}$ through the $(1,1)$ type symmetric
traceless field: ${{\partial_{(\mu}}{{B}_{\nu)}}}$ (here we have
$\partial.B=0$) or through the ${(1,0)\oplus(0,1)}$ type antisymmetric field
${{\partial_{[\mu}}{{B}_{\nu]}}}$. The latter is used to describe QED to
preserve the $U(1)$ internal symmetry present in the corresponding sources
[37]. In this context, we note that the Coulomb gauge $\vec{\nabla}.\vec{B}=0$
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
[37]. 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 [68]. The above conclusions are also consistent with the Gupta-
Bleuler quantization of the electromagnetic field [37].
${{\nabla_{(\mu}}{{B}_{\nu)}}}$ is not gauge invariant and $B_{\alpha}$, with
the kinetic term obtained using ${{\nabla_{(\mu}}{{B}_{\nu)}}}$, cannot
minimally couple with matter fields that possess internal 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 $(\omega,\psi)$ and can be relevant to explaining 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
[8,9,16] that need not to be gauge theories.

## V. Conclusion

To conclude, in this article we have proposed new geometric fields from the
gravity sector as possible candidates for inflation and dark energy.
Previously, Einstein-Palatini action was used to introduce two massless scalar
fields $\omega$ and $\psi$, that give non-metricity. It was found to be more
consistent with a quantum theory of gravity. The connection is expressed as:
$\Gamma^{\alpha}_{~{}\mu\nu}+C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$. The
Einstein-Palatini formalism is particularly suitable for introducing scalar
fields through $C^{\alpha}_{~{}\mu\nu}$. We found that for fields relevant to
the standard model with the addition of the right handed neutrinos,
$C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$ can at most contribute boundary terms to
the matter and radiation field actions and do not appear in the equations of
motion. Thus, we do not need to include $C^{\alpha}_{~{}\mu\nu}(\omega,\psi)$
in the actions and stress tensors of ordinary matter and radiation that are
various representations of $SL(2,c)$, through the covariant derivatives. This
scheme, developed in Sect.III, can also be required to construct a stable
theory with dark energy. The issue of stability is discussed in Sect.IV, where
we have performed linear transformation on $\omega$ and $\psi$, to obtain a
theory of two decoupled scalars $\Phi$ and $\Psi$. $\Psi$ gives a negative
stress tensor. We can introduce potential terms in $\Phi$ and $\Psi$. The
negative kinetic term in $\Psi$ prohibits any coupling between itself and
other fields, including $\Phi$. We can have a self-interacting theory of
$\Psi$. We can couple $\Phi$ to ordinary matter. All these give a versatile
model where vacuum expectation values and quantum fluctuations can play
important roles. We can construct various models by adding suitable
interaction terms. We can generalize the $\Lambda$CDM theory by including
$\Phi$ and $\Psi$ to give dynamic and spatially varying dark energy and
inflation. $\Lambda$, $\Phi$ and $\Psi$ are similar in many respects.
Together, they give the most general theory linear in scalar curvature that is
consistent with the Bianchi identity in $\Gamma^{\alpha}_{~{}\mu\nu}$. They
need not to have ordinary matters as their sources and can be non-localized.
They contribute positive and negative stress tensors in Einstein’s equation
that are not coupled to the metric through the gravitational constant.
$\Lambda$, $\Phi$ and $\Psi$ give us departures from the classical theory and
the 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. Eq.(29) can be used to
construct new non-trivial solutions of the vacuum Einstein’s equation. Lastly,
we can consider a theory that contains more than one $\Phi$ fields but no
$\Psi$ field.

We found that we may have to introduce the right-handed neutrinos into the
standard model to define the equation of motion and conserved vector currents
for neutrinos in curved spacetime. This is related to the existence of
required variational derivatives in curved spacetime. The same remains valid
for curvilinear coordinates in flat spacetime. The right handed neutrinos have
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 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 neutrinos couple with $\Phi$
and $\Psi$ when we do not include the right handed neutrinos in the action. In
this case, we can not define conserved neutrino vector currents in curved
spacetime even if we use the Levi-Civita connection. The above observations
can be important in the early universe.

In this article, we have used $(A,A)$ type representations to introduce higher
integral 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. Such fields do not couple with ordinary
matter through gauge theories. These fields and the right handed neutrinos can
be useful in dark matter, dark energy and higher spin driven inflation models.
We can modify the action or use $f(R^{\prime})$ theories to include higher
spin fields. Lastly, we have introduced a phantom scalar using torsion that
gives non-metricity. It couples with gauge fields and can violate current
conservation laws. Its vacuum effects can be useful in cosmology.

## Acknowledgement

I am thankful to a few reviewers for some improvements, [24].

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