A Free-Field Lagrangian for a Gauge Theory of the CPT Symmetry

TL;DR

This paper develops a refined free-field Lagrangian for a gauge theory of CPT symmetry, addressing previous shortcomings and ensuring all theoretical requirements are met.

Contribution

It introduces a modified Lagrangian that satisfies an overlooked requirement, advancing the theoretical framework of gauge theories of CPT symmetry.

Findings

The new Lagrangian satisfies all necessary theoretical conditions.

It improves upon previous models by addressing prior oversights.

The approach simplifies the mathematical formulation of the gauge theory.

Abstract

A simplified mathematical approach is presented and used to find a suitable free-field Lagrangian to complete previous work on constructing a gauge theory of CPT transformations. The new Lagrangian is a slight but important modification of the previous candidate. The new version satisfies an additional requirement which had been overlooked and not satisfied by the previous candidate.

Equations

$$e_a^{\mu }$$

$$\psi$$

$$U$$

$$U^{-1}$$

$${\partial }_{\mu }U$$

$$\left({\partial }_{\mu }U\right)U^{-1}$$

$$X_{\mu }$$

$$Y_{\mu }$$

$${\tilde{Y}}_{\mu }$$

$${\omega }_{\mu ab}\to \Theta \left[-f\right]{\omega }_{\mu ab}+\Theta \left[f\right]{\tilde{\omega }}_{\mu ab}+\Theta \left[-f\right]\delta \left[f\right]{\varsigma }_{\mu ab}+\Theta \left[f\right]\delta \left[f\right]{\tilde{\varsigma }}_{\mu ab}\text{,}$$

$$\left[D_{\mu },D_{\nu }\right]$$

$$S$$

$$\left[D_{\mu },D_{\nu }\right]$$

$$\left[D_{\mu },D_{\nu }\right]-\beta \left\{\left({\partial }_{\mu }{x}_{\nu I}-{\partial }_{\nu }{x}_{\mu I}\right)I+\left({\partial }_{\mu }{x}_{\nu 5}-{\partial }_{\nu }{x}_{\mu 5}\right){\gamma }^5\right\}.$$

$$S$$

Full text

INTRODUCTION

Aside from curiosity, why consider gauging the CPT symmetry?  We assert that gauging the CPT symmetry is necessary to further our understanding of gravitational physics.  The fact that the metric spin connection formulation of general relativity can be derived from gauging the global, proper, continuous Lorentz transformations ($\Lambda$) should be sufficient motivation to consider gauging CPT.  This is because PT is also a proper Lorentz transformation, and it would seem logical to include PT in gauging the *entire *proper Lorentz group of transformations, $PT\Lambda$.  It is necessary to include the C operation because PT is not a universal symmetry, whereas CPT is a universal symmetry.  In effect, we are gauging the $CPT\Lambda$ transformation of the Dirac field to induce the gauging of the full group of proper spacetime Lorentz transformations.

Another reason to gauge the CPT symmetry is because some gravitational issues can be interpreted as requiring an additional force, and the gauging of experimentally verified, global symmetries has been successful in determining the known forces.  Given the historical success from gauging symmetries, we do not have much (if any) choice if one is to continue down that path because CPT is a universal, experimentally verified symmetry that does not require any extra dimensions.

For example, we argue that gauge CPT is a logical alternative to the galactic dark matter hypothesis used in the context of spiral galaxies [1, 2].  The dark matter is invoked to explain galactic rotation curves for two reasons, really.  The primary reason is that the gravity produced by the observed galactic matter does not account for the motion of material (stars, HI gas) as the distance from the galactic center increases.  However, another reason is buried within - a mass independent acceleration.  This acceleration is what steers a hypothetical explanation towards dark matter rather than a missing force - no forces other than gravity produce a mass independent acceleration.  Thus, if gravity is to explain this strange motion, then there must be missing, unseen matter.  However, another possible explanation could logically exist in the form of a missing force or an extension of general relativity because, after all, accelerations are produced by forces.

The traits of universality and of containing a proper Lorentz transformation that CPT share with $\Lambda$ are the heuristic motivation that a mass independent acceleration (i.e., the equivalence principle is obeyed) would occur if any new forces or extensions of general relativity were to be unveiled by gauging $CPT\Lambda$.  Indeed, the new gauge field $X_{\mu}$ introduced to accommodate local $CPT\Lambda$ transformations is required to contain terms of the form $x_{\mu ab}\sigma^{ab}$ ($\sigma^{ab}=\frac{1}{4}\left[\gamma^{a},\gamma^{b}\right]$) which appear along with the metric spin connection containing term, $\omega_{\mu ab}\sigma^{ab}$, of general relativity.111Greek indices such as $\mu$ denote manifold coordinates.  Latin indices such as $a$ denote the local inertial coordinates.  Otherwise, we use the Bjorken-Drell conventions except for $\sigma^{ab}$. Hence, the spinor fields $\psi$ describing matter will interpret $x_{\mu ab}\sigma^{ab}$ as a gravitational effect.  More detailed arguments regarding gauge CPT as an alternative to the dark matter hypothesis can be found in [1, 2].

Furthermore, gauging CPT should be of interest in the broader problem of reconciling general relativity with quantum theory.  That the CPT symmetry is born neither from special relativity nor quantum theory alone, but rather the phenomenologically successful union of the two, warrants paying closer attention to CPT.  Because general relativity can be obtained by making special relativity (i.e. $\Lambda$) local, it would seem interesting to include CPT when making $\Lambda$ local in order to incorporate quantum theory at a fundamental level.  This is analogous to expanding the early nonrenormalizable weak interaction theories by the renormalizable $SU\left(2\right)\times U\left(1\right)$ electroweak theory - any approach to quantum gravity would need to include the missing spacetime dynamical degrees of freedom unveiled by gauging CPT.

THE GAUGE THEORY OF CPT TRANSFORMATIONS

Gauging $CPT\Lambda$ is a straightforward process analogous to gauging other symmetries.  The original approach ([1, 2]) used variational principles in conjunction with the action integral primarily because of the appearance of Dirac delta functionals when making the discrete, global CPT symmetry a local symmetry transformation.  Dirac delta functionals only have meaning when appearing in a definite integral, and the action integral is the only obvious, natural arena for the Dirac delta functionals which occur.  Also, we view the action integral to be the object of fundamental importance when dealing with symmetries.  For a more in-depth discussion of motivation, experimental possibilities, and derivation of the mathematics used to handle these discrete transformations, see [1, 2].

We begin with the global $CPT\Lambda$ transformation applied to the Dirac spinor $\psi$ and the spacetime vierbein $e_{a}^{\;\mu}$:  $\psi\rightarrow i\gamma^{5}\Lambda_{\psi}\psi$, and $e_{a}^{\;\mu}\rightarrow-e_{b}^{\;\mu}\Lambda_{a}^{\;b}$, where $\Lambda_{\psi}$ and $\Lambda_{a}^{\;b}$ are the spinor and spacetime representations of $\Lambda$.  The $CPT$ transformation appears as the $i\gamma^{5}$ in the spinor transformation and as the factor of $-1$ (due to $PT$) appearing in the vierbein transformation.  We assume there is no non-trivial spacetime analog of the C operation, i.e., there is no such thing as ”anti-spacetime” (an attempt to find a non-trivial spacetime C operation is contained in [2]).  There are two steps in making these transformations local.  First, we make $\Lambda$ a local transformation.  Then, we introduce a real, differentiable function $f$  to be used as the argument of unit step functions $\Theta$, where we define $\Theta\left[f\leq 0\right]=0$, and $\Theta\left[f>0\right]=1$.  In arbitrarily chosen regions where we want to perform the $CPT\Lambda$ operation, we set $f>0$.  In the regions we leave alone, we set $f<0$.  The boundaries between the regions where $CPT\Lambda$ is applied and where it is not are given by $f=0$.  It is to be emphasized that $f$ is not a physical field, just a parameter used to make CPT local; therefore, the function $f$ must disappear from the Lagrangian and ensuing field equations.  One thus obtains the local $CPT\Lambda$ transformations [1]:

[TABLE]

We denote the local $CPT\Lambda$ transformation by the customary $U$ even though the transformation is not unitary.  Unitarity is not necessary in making gauge theories.  For example, $\Lambda$ is not unitary and is used to derive the metric spin connection formulation of general relativity as a gauge theory.  We also note that the presence or absence of a conservation law associated with a global symmetry transformation has no relevance to constructing the ensuing gauge theory.  Again, returning to $\Lambda$, we note that there are no conservation laws associated with global boosts.

To proceed further, we must take derivatives and make products from the above transformations.  Because of the discrete nature of the transformations, various discontinuities naturally arise.  The action integral, variational methods, and elementary functional analysis lead to the following important rules needed to handle the discontinuities ([1, 2]): $\Theta\left[-f\right]\Theta\left[f\right]=0$, $\left(\Theta\left[\pm f\right]\right)^{n}=\Theta\left[\pm f\right]$ ($n$ is a positive integer $\geq 1$), and $\partial_{\mu}\Theta\left[\pm f\right]=\pm\delta\left[f\right]\partial_{\mu}f$, where $\delta\left[f\right]$ is a Dirac delta functional.  Application of these rules to the above transformations immediately gives us:

[TABLE]

where $\Lambda_{\overline{\psi}}$ is the inverse of $\Lambda_{\psi}$ (also, $\overline{\psi}\rightarrow i\overline{\psi}\gamma^{5}\Lambda_{\overline{\psi}}$ under global $CPT\Lambda$).

Everything follows from these equations.  The Dirac equation is obviously not invariant under $U$, so we introduce a minimally coupled gauge field as part of a covariant derivative acting on $\psi$.  As always, a gauge field $Z_{\mu}$ associated with $U$ must transform as $Z_{\mu}\rightarrow UZ_{\mu}U^{-1}-\frac{1}{\beta}\left(\partial_{\mu}U\right)U^{-1}$, where $\beta$ is the coupling constant.  Because $\left(\partial_{\mu}U\right)U^{-1}$ has terms containing $I$ and $\gamma^{5}$in addition to $\sigma^{ab}$, we immediately see from the linear independence of $I$, $\gamma^{5}$, and $\sigma^{ab}$ that the metric spin connection term, $\omega_{\mu ab}\sigma^{ab}$, cannot compensate for local $CPT\Lambda$ transformations (see [1] for an alternative proof).  Therefore, we introduce a new gauge field $X_{\mu}$ of the form $X_{\mu}=x_{\mu I}I+x_{\mu 5}\gamma^{5}+x_{\mu ab}\sigma^{ab}$ as part of a covariant derivative $D_{\mu}=\partial_{\mu}+\frac{1}{2}\omega_{\mu ab}\sigma^{ab}+\beta X_{\mu}$.  In other words, $Z_{\mu}=\frac{1}{2\beta}\omega_{\mu ab}\sigma^{ab}+X_{\mu}$.  One can easily verify that the following transformation (see [1] for the derivation) for $X_{\mu}$ gives $D_{\mu}\rightarrow UD_{\mu}U^{-1}$:

[TABLE]

The terms $\varsigma_{\mu ab}$, $\widetilde{\varsigma}_{\mu ab}$ come from the differentiation of the transformed vierbein contained in the metric spin connection, $\omega_{\mu ab}$, under local $CPT\Lambda$:

[TABLE]

where $\widetilde{\omega}_{\mu ab}$ satisfies $\widetilde{\omega}_{\mu ab}\sigma^{ab}=\Lambda_{\psi}\omega_{\mu ab}\sigma^{ab}\Lambda_{\overline{\psi}}-2\left(\partial_{\mu}\Lambda_{\psi}\right)\Lambda_{\overline{\psi}}$.  Explicit expressions for $\omega_{\mu ab}$, $\widetilde{\omega}_{\mu ab}$, $\varsigma_{\mu ab}$, and $\widetilde{\varsigma}_{\mu ab}$ can be found in [1].

We are now ready to find the Lagrangian using the usual machinery of gauge theories.  We begin by deriving the candidate found in the previous versions of this paper222This Lagrangian was first proposed in [1] as the ”peel-off” Lagrangian.  However, gauge covariance was not shown. and then introduce the modification leading to the new candidate.  First, we know that $X_{\mu}$ must be massless (see [1] for an alternative proof).  We also know that $\left[D_{\mu},D_{\nu}\right]$ transforms gauge covariantly.  Unfortunately, using $Tr\left\{\left[D_{\mu},D_{\nu}\right]\left[D^{\mu},D^{\nu}\right]^{{\dagger}}\right\}$ as a free-field term introduces $R^{2}$ into the Lagrangian instead of the required $R$ of general relativity, where $R$ is the Einstein-Hilbert scalar curvature term formed from just the metric spin connection and a couple of vierbeins.  Use of just $\left[D_{\mu},D_{\nu}\right]$ is problematic because it is not clear what to contract it with, courtesy of the absence of the Latin (i.e. inertial) indices in the $x_{\mu I}$ and $x_{\mu 5}$ terms.  Also, at some point in the total Lagrangian, the $\omega_{\mu ab}$ and $x_{\mu ab}$ terms must appear on an unequal footing outside of the $\frac{1}{2}\omega_{\mu ab}\sigma^{ab}+\beta x_{\mu ab}\sigma^{ab}$ combination - otherwise, the sole purpose of $x_{\mu ab}$ is just to be a ”fudge factor” introduced to get rid of the $\varsigma_{\mu ab}$, $\widetilde{\varsigma}_{\mu ab}$ terms appearing in the transformation of $\omega_{\mu ab}$.

The key to constructing the previous Lagrangian was that $R$ is invariant [1] and covariant (shown below) under local $CPT\Lambda$ transformations even though $\omega_{\mu ab}$ is not.  (Briefly [1], the $R$ formed from the metric spin connection, $\omega_{\mu ab}$, is the same as the familiar $R$ formed from the metric tensor $g^{\mu\nu}$, and the $R$ from $g^{\mu\nu}$ is gauge invariant.  Gauge invariance follows because $g^{\mu\nu}=\eta^{ab}e_{a}^{\;\mu}e_{b}^{\;\nu}\rightarrow\Theta\left[-f\right]g^{\mu\nu}+\Theta\left[f\right]g^{\mu\nu}$, where $\eta^{ab}$ is the Minkowski metric tensor.  From this we immediately see that derivatives of the metric tensor also transform in the same manner, $\partial_{\rho}g^{\mu\nu}\rightarrow\Theta\left[-f\right]\partial_{\rho}g^{\mu\nu}+\Theta\left[f\right]\partial_{\rho}g^{\mu\nu}$.  So, the metric tensor and $R$ and any functions of these only pick up removable singularities under local $CPT\Lambda$ transformations.  These singularities have no effect on the action integral and are therefore dropped.  Again, the action integral and variational approach ([1, 2]) are of fundamental importance in dealing with singularities.)

We now show that the Einstein-Hilbert $R$ transforms gauge covariantly.  Or, equivalently, we need to show that $R_{\mu\nu ab}\sigma^{ab}\rightarrow UR_{\mu\nu ab}\sigma^{ab}U^{-1}$ under local $CPT\Lambda$ transformations ( the transformation of the vierbeins used in the contraction of $R_{\mu\nu ab}$ to obtain $R$ cause no complications because they do not introduce any delta functionals or gamma matrices).  We begin with re-expressing $R_{\mu\nu ab}$ in terms of the more familiar form of the Riemann curvature tensor:  $R_{\mu\nu ab}=R_{\mu\nu\rho\sigma}e_{a}^{\;\rho}e_{b}^{\;\sigma}$.  We recall that $R_{\mu\nu\rho\sigma}$ is invariant under local $CPT\Lambda$ transformations because $R_{\mu\nu\rho\sigma}$ is comprised of the metric tensor and its derivatives.  So, we have:

$R_{\mu\nu ab}\sigma^{ab}=$ $R_{\mu\nu\rho\sigma}e_{a}^{\;\rho}e_{b}^{\;\sigma}\sigma^{ab}$

$\rightarrow R_{\mu\nu\rho\sigma}\left(\Theta\left[-f\right]e_{a}^{\;\rho}-\Theta\left[f\right]e_{c}^{\;\rho}\Lambda_{a}^{\;c}\right)\left(\Theta\left[-f\right]e_{b}^{\;\sigma}-\Theta\left[f\right]e_{d}^{\;\sigma}\Lambda_{b}^{\;d}\right)\sigma^{ab}$

$=$ $R_{\mu\nu\rho\sigma}\left(\Theta\left[-f\right]e_{a}^{\;\rho}e_{b}^{\;\sigma}+\Theta\left[f\right]\Lambda_{a}^{\;c}\Lambda_{b}^{\;d}e_{c}^{\;\rho}e_{d}^{\;\sigma}\right)\sigma^{ab}$.

We compare this with $UR_{\mu\nu ab}\sigma^{ab}U^{-1}$:

$UR_{\mu\nu ab}\sigma^{ab}U^{-1}$

$=\left(\Theta\left[-f\right]I+\Theta\left[f\right]i\gamma^{5}\Lambda_{\psi}\right)R_{\mu\nu ab}\sigma^{ab}\left(\Theta\left[-f\right]I-\Theta\left[f\right]i\gamma^{5}\Lambda_{\overline{\psi}}\right)$

$=\left(\Theta\left[-f\right]R_{\mu\nu ab}\sigma^{ab}+\Theta\left[f\right]iR_{\mu\nu ab}\gamma^{5}\Lambda_{\psi}\sigma^{ab}\right)\left(\Theta\left[-f\right]I-\Theta\left[f\right]i\gamma^{5}\Lambda_{\overline{\psi}}\right)$

$=\Theta\left[-f\right]R_{\mu\nu ab}\sigma^{ab}+\Theta\left[f\right]R_{\mu\nu ab}\gamma^{5}\Lambda_{\psi}\sigma^{ab}\gamma^{5}\Lambda_{\overline{\psi}}$

$=\Theta\left[-f\right]R_{\mu\nu ab}\sigma^{ab}+\Theta\left[f\right]R_{\mu\nu ab}\Lambda_{\psi}\sigma^{ab}\Lambda_{\overline{\psi}}$

$=\Theta\left[-f\right]R_{\mu\nu ab}\sigma^{ab}+\Theta\left[f\right]R_{\mu\nu ab}\Lambda_{c}^{\;a}\Lambda_{d}^{\;b}\sigma^{cd}$

$=\Theta\left[-f\right]R_{\mu\nu\rho\sigma}e_{a}^{\;\rho}e_{b}^{\;\sigma}\sigma^{ab}+\Theta\left[f\right]R_{\mu\nu\rho\sigma}e_{a}^{\;\rho}e_{b}^{\;\sigma}\Lambda_{c}^{\;a}\Lambda_{d}^{\;b}\sigma^{cd}$

$=R_{\mu\nu\rho\sigma}\left(\Theta\left[-f\right]e_{a}^{\;\rho}e_{b}^{\;\sigma}+\Theta\left[f\right]\Lambda_{a}^{\;c}\Lambda_{b}^{\;d}e_{c}^{\;\rho}e_{d}^{\;\sigma}\right)\sigma^{ab}$.

From the gauge covariance of $\left[D_{\mu},D_{\nu}\right]$ and $R$, we can find a gauge covariant free-field term for $X_{\mu}$.  We begin with the expansion of $\left[D_{\mu},D_{\nu}\right]$:

[TABLE]

The terms in $\left\{...\right\}$ give $R$ upon contraction with a couple of vierbein.  Because both $\left[D_{\mu},D_{\nu}\right]$ and the $\left\{...\right\}$ terms transform gauge covariantly, we see that the remaining terms - denoted as $H_{\mu\nu}$ - must also transform gauge covariantly.  We can now form a gauge covariant Lagrangian term from $H_{\mu\nu}$ as $Tr\left\{H_{\mu\nu}H^{\mu\nu{\dagger}}\right\}$ - this is the previous free-field Lagrangian for $X_{\mu}$.  The Hermitian action we obtain from minimal coupling of $\omega_{\mu ab}$, $X_{\mu}$, and $\psi$ is thus [1]:

[TABLE]

$e=\det\left(e_{a}^{\;\mu}\right)$ and $H_{\mu\nu}=\frac{\beta}{2}\left(\omega_{\mu ab}\left[\sigma^{ab},X_{\nu}\right]-\omega_{\nu ab}\left[\sigma^{ab},X_{\mu}\right]\right)+\beta^{2}\left[X_{\mu},X_{\nu}\right]+\beta\left(\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu}\right)$.  The resulting equations of motion are found in [1].

The $x_{\mu ab}$ terms are the obvious terms to examine as possible explanations for current gravitational issues because they couple to the matter fields in the same manner as the spin connection.  Experimental speculations and crude predictions can be found in [1, 2].  Also, the $x_{\mu ab}$ field equations should explain the Baryonic Tully-Fisher law and the Faber-Jackson law.  However, difficulties encountered by the author in trying to explain the slope appearing in the Tully-Fisher law using a gauge theory of the CPT symmetry - or *any *force obeying an inverse square law - have led to consideration of a slightly different free-field Lagrangian of the gauge theory of the CPT symmetry.  The physical implications of the ”chiral” terms, $x_{\mu I}$ and $x_{\mu 5}$, and their associated anomalies are not developed enough to warrant a discussion.

THE NEW LAGRANGIAN

To proceed further, we recall that the CPT symmetry is born from the union of special relativity with quantum theory; therefore, we intuitively expect $X_{\mu}$ to be a ”bridge” connecting the two [1,2].  The free-field Lagrangians of the quantum field theories appearing in the Standard Model are based on (using the applicable covariant derivatives) $Tr\left\{\left[D_{\mu},D_{\nu}\right]\left[D^{\mu},D^{\nu}\right]^{{\dagger}}\right\}$, and the free-field Lagrangian $R$ of general relativity is of the form (obtained from $\omega_{\mu ab}$) $\left[D_{\mu},D_{\nu}\right]$ appropriately contracted.  Therefore, we expect $X_{\mu}$ to appear in both types of free-field Lagrangians.  The previous free-field Lagrangian did not satisfy this criterion.

A simple, intuitive candidate is to retain the Lagrangian discussed above, but replace the Einstein-Hilbert $R$ by a similar term formed by substituting $\frac{1}{2}\omega_{\mu ab}$ with $\frac{1}{2}\omega_{\mu ab}+\beta x_{\mu ab}$ instead.  We denote this modified curvature scalar by $R_{X\omega}$.  This new free-field Lagrangian formed from $Tr\left\{H_{\mu\nu}H^{\mu\nu{\dagger}}\right\}$ and $R_{X\omega}$ satisfies all the criteria we require - provided that the new candidate transforms gauge covariantly, too.  We now turn our attention to this task.  We begin by again expanding $\left[D_{\mu},D_{\nu}\right]$:

[TABLE]

The term $R_{X\omega}$ is formed from the contraction (with a couple of vierbein) of:

[TABLE]

So, if we can show that $\beta\left\{\left(\partial_{\mu}x_{\nu I}-\partial_{\nu}x_{\mu I}\right)I+\left(\partial_{\mu}x_{\nu 5}-\partial_{\nu}x_{\mu 5}\right)\gamma^{5}\right\}$ transforms gauge covariantly, then so does $R_{X\omega}$.

We begin by defining $F_{\mu\nu}\equiv\left[D_{\mu},D_{\nu}\right]=\sum\limits_{i}y_{i}\Gamma^{i}$, where the $\Gamma^{i}$ are the eight linearly independent matrices $I$, $\gamma^{5}$, and $\sigma^{ab}$; and the $y_{i}$ are their coefficients (we will not be using the Einstein summation convention for this proof).  We define the transformation of $F_{\mu\nu}$ under gauge $CPT\Lambda$ (i.e., $U$ from above) as $F_{\mu\nu}^{{}^{\prime}}=\sum\limits_{i}y_{i}^{{}^{\prime}}\Gamma^{i}$.  We also know that $F_{\mu\nu}^{{}^{\prime}}=UF_{\mu\nu}U^{-1}$, therefore, $\sum\limits_{i}y_{i}^{{}^{\prime}}\Gamma^{i}=\sum\limits_{i}y_{i}U\Gamma^{i}U^{-1}$.  Let $U\Gamma^{i}U^{-1}=\sum\limits_{j}a_{j}^{i}\Gamma^{j}$, then $\sum\limits_{i}y_{i}^{{}^{\prime}}\Gamma^{i}=\sum\limits_{i}\sum\limits_{j}y_{i}a_{j}^{i}\Gamma^{j}=\sum\limits_{i}\sum\limits_{j}y_{j}a_{i}^{j}\Gamma^{i}$, so $y_{i}^{{}^{\prime}}=\sum\limits_{j}y_{j}a_{i}^{j}$.  We note that $UIU^{-1}=I$, $U\gamma^{5}U^{-1}=\gamma^{5}$, and $U\sigma^{ab}U^{-1}=\sum\limits_{cd}r_{cd}\sigma^{cd}$, where the $r_{cd}$ are just the coefficients appearing in the transformation of $\sigma^{ab}$.  Therefore, the transformation of the terms containing $I$ or $\gamma^{5}$ are not mixed up with the $\sigma^{ab}$ terms.  We now want to know what is required to have a single $y_{i}\Gamma^{i}$ term transform gauge covariantly.  In other words, what is required to have $y_{i}^{{}^{\prime}}\Gamma^{i}=y_{i}U\Gamma^{i}U^{-1}$?  Or, equivalently, what is required for $\sum\limits_{j}y_{j}a_{i}^{j}\Gamma^{i}=y_{i}U\Gamma^{i}U^{-1}$?  Hitting both sides of this equation by $\left(\Gamma^{i}\right)^{-1}$ gives $\sum\limits_{j}y_{j}a_{i}^{j}I=y_{i}\left(\Gamma^{i}\right)^{-1}U\Gamma^{i}U^{-1}$.  So, for this equation to be true, the following two conditions must be met:  $I=\left(\Gamma^{i}\right)^{-1}U\Gamma^{i}U^{-1}$, and $\sum\limits_{j}y_{j}a_{i}^{j}=y_{i}$.  The first condition is met for $\Gamma^{i}=I$ and $\Gamma^{i}=\gamma^{5}$.  The second condition is met for the $U$ of gauge $CPT\Lambda$ because $a_{i}^{j}=\delta_{i}^{j}$ for $\Gamma^{i}=I$ and $\Gamma^{i}=\gamma^{5}$.  Therefore, the entire term $\beta\left\{\left(\partial_{\mu}x_{\nu I}-\partial_{\nu}x_{\mu I}\right)I+\left(\partial_{\mu}x_{\nu 5}-\partial_{\nu}x_{\mu 5}\right)\gamma^{5}\right\}$ transforms gauge covariantly - and so does $R_{X\omega}$.  Therefore, without consideration of the appearance of the chiral anomaly or any other anomalies which might appear, the Hermitian action we obtain from minimal coupling of $\omega_{\mu ab}$, $X_{\mu}$, and $\psi$ is thus:

[TABLE]

The appearance of $X_{\mu}$ in both $R_{X\omega}$ and $H_{\mu\nu}$ leads to interesting equations of motion for the $x_{\mu ab}$ terms.  Also, the replacement of $R$ by $R_{X\omega}$ provides the intuition that the gravitational lensing effects attributed to the hypothetical dark matter are explained by gauge $CPT\Lambda$ instead.  Finally, we note that in the limit as $X_{\mu}\rightarrow 0$, we obviously recover general relativity.

This paper is dedicated to the memory of the author’s Mother.

REFERENCES

[1] K. Koltko, ”Gauge CPT as a Possible Alternative to the Dark Matter Hypothesis,” arXiv:  1308.6341 [physics.gen-ph] and cross-listed as [astro-ph.GA] (2013).

[2] K. Koltko, Attempts to Find Additional Dynamical Degrees of Freedom in Spacetime Using Topological and Geometric Methods, 2000 Ph.D. Thesis, University of South Carolina.