1. Introduction
In the early 20th century, E Cartan first proposed the post-Riemann geometrical structures generated by the microscopic feature of matter fields and considered particularly the coupling of space-time torsion with intrinsic spin [1]. The interest in gravitational theory with torsion and spin had significantly grown in the second half of the 20th century after the gauge theory of gravitation was established [2–6]. The Einstein-Cartan theory (EC) is a degenerate version of Poincaré gauge theory, with a linear gravitational Lagrangian [7–9]. In this model, the torsion does not propagate and couples algebraically to spin, thus one can only feel torsion inside the matter. Although extensive attempts had been made to explore torsion in both theories [10–12] and experiments [13–15], the quest on this fundamental field is far from being satisfactory.
In this paper, we deal with a consistency problem in EC. Recall that in Einstein's General Relativity (GR) the effect of gravity is assessed on a physical system through the Minimal Coupling Procedure (MCP), i.e. first write down the Lagrangian or the equation of motion which holds in Special Relativity (SR), then replace the Minkowski metric ημν with general metric gμν and replace all the ordinary derivatives ∂ with covariant derivatives $\widetilde{{\rm{\nabla }}}$ [16]. For the metric signature (−, +, +, +), let us take the free scalar field Lagrangian ${{ \mathcal I }}_{{SR}}^{\phi }=\int {{\rm{d}}}^{4}x\left[-\tfrac{1}{2}{\partial }_{\mu }\phi {\partial }^{\mu }\phi -\tfrac{1}{2}{m}^{2}{\phi }^{2}\right]$ for example. One obtains the Lagrangian in Riemann space-time by MCP, and subsequently the equation of motion by variation over φ:1.1 ) the equation of motion is derived through the MCP action and coincides with the one derived through MCP from the special-relativistic equation of motion. This is natural because the theory should not depend on whether we regard the Lagrangian or the equation of motion as a starting point of MCP. However, the case becomes subtle in EC. When working in the Riemann-Cartan space-time, the MCP requires replacing ordinary derivatives with covariant derivatives ∇ = ∂ + Γ that contain torsion. Γ is the affine connection in Riemann-Cartan space-time and has a set of 64 components. We define the torsion tensor as ${S}^{\mu }{}_{\nu \rho }=\tfrac{1}{2}({{\rm{\Gamma }}}_{\nu \rho }^{\mu }-{{\rm{\Gamma }}}_{\rho \nu }^{\mu })$, and the contortion tensor as Kλμν = Sλμν + Sμλν + Sνλμ. Following similar steps as in GR one gets:1.2 ) is not the MCP extension of ∂μ∂μφ − m2φ = 0. The situation stands even worse because it is not clear which Lagrangian would correspond to equation ∇μ∇μφ − m2φ = 0; or to say it radically, starting from the minimal coupled equation of motion for φ, one is not able to deduce an analytically expressed Lagrangian.
$\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal I }}_{{GR}}^{\phi }=\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{2}{\widetilde{{\rm{\nabla }}}}_{\mu }\phi {\widetilde{{\rm{\nabla }}}}^{\mu }\phi -\displaystyle \frac{1}{2}{m}^{2}{\phi }^{2}\right],\\ {\widetilde{{\rm{\nabla }}}}_{\mu }{\widetilde{{\rm{\nabla }}}}^{\mu }\phi -{m}^{2}\phi =0.\end{array}\right.\end{eqnarray}$
The covariant derivative is defined as $\widetilde{{\rm{\nabla }}}=\partial +\widetilde{{\rm{\Gamma }}}$, where $\widetilde{{\rm{\Gamma }}}$ is the Christoffel connection. Note that in ( $\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal I }}_{{EC}}^{\phi }=\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{2}{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}^{\mu }\phi -\displaystyle \frac{1}{2}{m}^{2}{\phi }^{2}\right],\\ \mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }{{\rm{\nabla }}}^{\mu }\phi -{m}^{2}\phi =0,\end{array}\right.\end{eqnarray}$
where $\mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }={{\rm{\nabla }}}_{\mu }-{K}_{\mu }$, with Kμ = Kνμν the trace of contortion tensor, and $g=| \det ({g}_{\mu \nu })| $. It shows that the equation of motion in (This is not a problem faced only by the scalar field. For the vector field Aμ (to avoid the gauge problem, we consider massive vector field throughout the whole paper) and Dirac field ψ, one finds1.3 ) and (1.4 ), nor find a concise Lagrangian for the minimal-coupled equations of motion. A detailed definition of the covariant derivatives ∇ acting on matter fields can be found in appendix A .
$\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal I }}_{{EC}}^{A}=\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\displaystyle \frac{1}{2}{m}^{2}{A}_{\mu }{A}^{\mu }\right],\\ \mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }{F}^{\mu \nu }-{m}^{2}{A}^{\nu }=0,\end{array}\right.\end{eqnarray}$
where Fμν = ∇μAν − ∇νAμ, and $\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal I }}_{{EC}}^{\psi }=\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[\displaystyle \frac{{\rm{i}}}{2}(\bar{\psi }{\gamma }^{\mu }{{\rm{\nabla }}}_{\mu }\psi -{{\rm{\nabla }}}_{\mu }\bar{\psi }{\gamma }^{\mu }\psi )-m\bar{\psi }\psi \right],\\ \displaystyle \frac{{\rm{i}}}{2}{\gamma }^{\mu }{\rm{\nabla }}{}_{\mu }\psi +\displaystyle \frac{{\rm{i}}}{2}{\gamma }^{\mu }\mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }\psi -m\psi =0,\\ \displaystyle \frac{{\rm{i}}}{2}{\rm{\nabla }}{}_{\mu }\bar{\psi }{\gamma }^{\mu }+\displaystyle \frac{{\rm{i}}}{2}\mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }=0.\end{array}\right.\end{eqnarray}$
For both the vector field and Dirac field, neither can we unify all the covariant derivatives in (It is notable that when treating the Dirac field, it is tempting to construct the modified action
$\begin{eqnarray}\begin{array}{rcl}\mathop{{{ \mathcal I }}^{\psi }}\limits^{^\circ } & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[\displaystyle \frac{{\rm{i}}}{2}(\bar{\psi }{\gamma }^{\mu }{{\rm{\nabla }}}_{\mu }\psi -{{\rm{\nabla }}}_{\mu }\bar{\psi }{\gamma }^{\mu }\psi )\right.\\ & & \left.-m\bar{\psi }\psi +\displaystyle \frac{{\rm{i}}}{2}\bar{\psi }{K}_{\mu }{\gamma }^{\mu }\psi \right],\end{array}\end{eqnarray}$
which leads to $\begin{eqnarray}{\rm{i}}{\gamma }^{\mu }{\rm{\nabla }}{}_{\mu }\psi -m\psi =0.\end{eqnarray}$
This is a satisfactory minimal-coupled equation for ψ. But one can promptly calculate its conjugate equation from $\mathop{{{ \mathcal I }}^{\psi }}\limits^{^\circ }$: $\begin{eqnarray}{\rm{i}}\mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }=0,\end{eqnarray}$
which is not minimally coupled. Why is there a break of symmetry between the conjugate equations? The point is that the standard action ${{ \mathcal I }}_{{EC}}^{\psi }$ is a real function, while the quantity $\tfrac{{\rm{i}}}{2}\bar{\psi }{K}_{\mu }{\gamma }^{\mu }\psi $ is a pure imaginary function, so $\mathop{{{ \mathcal I }}^{\psi }}\limits^{^\circ }$ is a complex function. Such a Lagrangian is impossible because it would add redundant constraint to the field. As a consequence, one can no longer deduce the equation of $\bar{\psi }$ from the equation of ψ. The conjugate equation of iγμ∇μψ − mψ = 0 should be ${\rm{i}}{\rm{\nabla }}{}_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }=0$, but not ${\rm{i}}\mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }=0$.So, a subtlety arises in EC that we cannot make both the matter Lagrangian and the equation of motion minimally coupled, and one may seriously ask where to start with the MCP, or whether there is a way to make the theory unique. The key point in this MCP ambiguity is that when manipulating the minimal action principle with a Lagrangian in EC, the quantity $\int {{\rm{d}}}^{4}x\sqrt{g}{\rm{\nabla }}{}_{\mu }{B}^{\mu }=\int {{\rm{d}}}^{4}x\sqrt{g}({\widetilde{{\rm{\nabla }}}}_{\mu }+{K}_{\mu }){B}^{\mu }$ is not a surface integral and does not vanish, where Bμ is an arbitrary vector field.
So far as the authors know, the MCP problem in EC was first observed by Kibble [3] and one can find a legible description by Saa [17]. To solve the problem Saa suggested modifying the volume element as ${{\rm{e}}}^{\theta }\sqrt{g}{{\rm{d}}}^{4}x$, where θ is an introduced scalar field satisfying ∂μθ = Kμ. In this model, the quantity $\int {{\rm{d}}}^{4}x{{\rm{e}}}^{\theta }\sqrt{g}{\rm{\nabla }}{}_{\mu }{B}^{\mu }$ turns out to be a surface term. Another approach provided by Kaźmierczak [18, 19] was to write down a new space-time connection in place of the original one: ${\widehat{{\rm{\Gamma }}}}_{\mu \nu }^{\rho }={{\rm{\Gamma }}}_{\mu \nu }^{\rho }-{\delta }_{\mu }^{\rho }{K}_{\nu }$. This connection guarantees that $\int {{\rm{d}}}^{4}x\sqrt{g}\widehat{{\rm{\nabla }}}{}_{\mu }{B}^{\mu }$ is a surface integral. In summary, there are two main schemes for addressing the problem. One is to alter the integral measure (which will cause departure from generic Riemann-Cartan geometry), and the other is to modify the covariant derivative. In this paper, we propose another possibility of modifying the space-time connection to solve the MCP problem, which we would present in section 2 . In section 3 , we come back to the models of Saa and Kaźmierczak, explain how they work, compare them with our model, and make some further discussion.
2. Modified coupling of gravitation field and matter fields
We propose a new covariant derivative ${ \mathcal D }$ that slightly alters the standard MCP:
$\begin{eqnarray}{{ \mathcal D }}_{\mu }{B}^{\nu }={\partial }_{\mu }{B}^{\nu }+{{\mathbb{C}}}_{\lambda \mu }^{\nu }{B}^{\lambda }\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal D }}_{\mu }{B}_{\nu }={\partial }_{\mu }{B}_{\nu }-{{\mathbb{C}}}_{\nu \mu }^{\lambda }{B}_{\lambda },\end{eqnarray}$
where ${{\mathbb{C}}}_{\lambda \mu }^{\nu }$ is the modified connection $\begin{eqnarray}{{\mathbb{C}}}_{\lambda \mu }^{\nu }={{\rm{\Gamma }}}_{\lambda \mu }^{\nu }-\displaystyle \frac{1}{3}({\delta }_{\mu }^{\nu }{K}_{\lambda }-{g}_{\lambda \mu }{K}^{\nu }).\end{eqnarray}$
With some algebra one can find: $\begin{eqnarray}\begin{array}{rcl}\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}{{ \mathcal D }}_{\mu }{B}^{\mu } & = & {{\rm{d}}}^{4}x\sqrt{g}[{\widetilde{{\rm{\nabla }}}}_{\mu }+{K}_{\mu }-\displaystyle \frac{1}{3}({\delta }_{\nu }^{\nu }{K}_{\mu }-{g}_{\mu \nu }{K}^{\nu })]{B}^{\mu }\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}{\widetilde{{\rm{\nabla }}}}_{\mu }{B}^{\mu }\\ & = & \displaystyle \int {{\rm{d}}}^{4}x{\partial }_{\mu }(\sqrt{g}{B}^{\mu }).\end{array}\end{eqnarray}$
For a tensor of rank three, the new covariant derivative reads: $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal D }}_{\rho }B{}^{\mu }{}_{\nu }{}^{\lambda } & = & {\partial }_{\rho }B{}^{\mu }{}_{\nu }{}^{\lambda }+{{\mathbb{C}}}_{\epsilon \rho }^{\mu }B{}^{\epsilon }{}_{\nu }{}^{\lambda }-{{\mathbb{C}}}_{\nu \rho }^{\epsilon }{B}^{\mu }{}_{\epsilon }{}^{\lambda }\\ & & +{{\mathbb{C}}}_{\epsilon \rho }^{\mu }B{}^{\mu }{}_{\nu }{}^{\epsilon }.\end{array}\end{eqnarray}$
The above construction can be extended straightforwardly to higher rank tensors.Our definition of ${{ \mathcal D }}_{\mu }$ fits all properties as a covariant derivative:
With the new covariant derivative to hand, we can now build up the theory for the scalar field, massive vector field, and Dirac field respectively.
| a | (a) Linearity $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal D }}_{\lambda }(\alpha {E}^{\mu }{}_{\nu }+\beta {B}^{\mu }{}_{\nu }) & = & {\partial }_{\lambda }(\alpha {E}^{\mu }{}_{\nu }+\beta {B}^{\mu }{}_{\nu })\\ & & +{{\mathbb{C}}}_{\rho \lambda }^{\mu }(\alpha {E}^{\rho }{}_{\nu }+\beta {B}^{\rho }{}_{\nu })\\ & & -{{\mathbb{C}}}_{\nu \lambda }^{\rho }(\alpha {E}^{\mu }{}_{\rho }+\beta {B}^{\mu }{}_{\rho })\\ & = & \alpha ({\partial }_{\lambda }{E}^{\mu }{}_{\nu }+{{\mathbb{C}}}_{\rho \lambda }^{\mu }{E}^{\rho }{}_{\nu }-{{\mathbb{C}}}_{\nu \lambda }^{\rho }{E}^{\mu }{}_{\rho })\\ & & +\beta ({\partial }_{\lambda }{B}^{\mu }{}_{\nu }+{{\mathbb{C}}}_{\rho \lambda }^{\mu }{B}^{\rho }{}_{\nu }-{{\mathbb{C}}}_{\nu \lambda }^{\rho }{B}^{\mu }{}_{\rho })\\ & = & \alpha {{ \mathcal D }}_{\lambda }{E}^{\mu }{}_{\nu }+\beta {{ \mathcal D }}_{\lambda }{B}^{\mu }{}_{\nu },\end{array}\end{eqnarray}$ |
| b | (b) Leibniz rule $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal D }}_{\rho }({E}^{\mu }{}_{\nu }{B}^{\lambda }) & = & {\partial }_{\rho }({E}^{\mu }{}_{\nu }{B}^{\lambda })+{{\mathbb{C}}}_{\epsilon \rho }^{\mu }({E}^{\epsilon }{}_{\nu }{B}^{\lambda })\\ & & -{{\mathbb{C}}}_{\nu \rho }^{\epsilon }({E}^{\mu }{}_{\epsilon }{B}^{\lambda })+{{\mathbb{C}}}_{\epsilon \rho }^{\lambda }({E}^{\mu }{}_{\nu }{B}^{\epsilon })\\ & = & {\partial }_{\rho }{E}^{\mu }{}_{\nu }{B}^{\lambda }+{{\mathbb{C}}}_{\epsilon \rho }^{\mu }{E}^{\epsilon }{}_{\nu }{B}^{\lambda }\\ & & -{{\mathbb{C}}}_{\nu \rho }^{\epsilon }{E}^{\mu }{}_{\epsilon }{B}^{\lambda }+{E}^{\mu }{}_{\nu }{\partial }_{\rho }{B}^{\lambda }+{E}^{\mu }{}_{\nu }{{\mathbb{C}}}_{\epsilon \rho }^{\lambda }{B}^{\epsilon }\\ & = & {{ \mathcal D }}_{\rho }{E}^{\mu }{}_{\nu }{B}^{\lambda }+{E}^{\mu }{}_{\nu }{{ \mathcal D }}_{\rho }{B}^{\lambda },\end{array}\end{eqnarray}$ |
| c | (c) Contraction rule $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal D }}_{\rho }{T}^{\mu \lambda }{}_{\lambda } & = & {\partial }_{\rho }{T}^{\mu \lambda }{}_{\lambda }+{{\mathbb{C}}}_{\epsilon \rho }^{\mu }{T}^{\epsilon \lambda }{}_{\lambda }+{{\mathbb{C}}}_{\epsilon \rho }^{\lambda }{T}^{\mu \epsilon }{}_{\lambda }-{{\mathbb{C}}}_{\lambda \rho }^{\epsilon }{T}^{\mu \lambda }{}_{\epsilon }\\ & = & {\partial }_{\rho }{T}^{\mu \lambda }{}_{\lambda }+{{\mathbb{C}}}_{\epsilon \rho }^{\mu }{T}^{\epsilon \lambda }{}_{\lambda }.\end{array}\end{eqnarray}$ |
For the scalar field, we have ${{ \mathcal D }}_{\mu }\phi ={\partial }_{\mu }\phi $. The Lagrangian and the equation of motion readA , and the calculation of variation is given in appendix B .
$\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal I }}_{{\mathbb{C}}}^{\phi }=\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{2}{{ \mathcal D }}_{\mu }\phi {{ \mathcal D }}^{\mu }\phi -\displaystyle \frac{1}{2}{m}^{2}{\phi }^{2}\right]\\ {{ \mathcal D }}_{\mu }{{ \mathcal D }}^{\mu }\phi -{m}^{2}\phi =0.\end{array}\right.\end{eqnarray}$
For the massive vector field: $\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal I }}_{{\mathbb{C}}}^{A}=\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{4}{{ \mathcal F }}_{\mu \nu }{{ \mathcal F }}^{\mu \nu }-\displaystyle \frac{1}{2}{m}^{2}{A}_{\mu }{A}^{\mu }\right]\\ {{ \mathcal D }}_{\mu }{{ \mathcal F }}^{\mu \nu }-{m}^{2}{A}^{\nu }=0,\end{array}\right.\end{eqnarray}$
where $\begin{eqnarray}{{ \mathcal F }}_{\mu \nu }={{ \mathcal D }}_{\mu }{A}_{\nu }-{{ \mathcal D }}_{\nu }{A}_{\mu }.\end{eqnarray}$
For the Dirac field, $\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal I }}_{{\mathbb{C}}}^{\psi }=\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[\displaystyle \frac{{\rm{i}}}{2}(\bar{\psi }{\gamma }^{\mu }{{ \mathcal D }}_{\mu }\psi -{{ \mathcal D }}_{\mu }\bar{\psi }{\gamma }^{\mu }\psi )-m\bar{\psi }\psi \right]\\ {\rm{i}}{\gamma }^{\mu }{{ \mathcal D }}_{\mu }\psi -m\psi =0\\ {\rm{i}}{{ \mathcal D }}_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }=0.\end{array}\right.\end{eqnarray}$
The expression of ${ \mathcal D }$ acting on the spinor field is given in appendix Each of the above equations of motion coincides with the corresponding one that is extended from SR by replacing ∂ with ${ \mathcal D }$. The structure of ${ \mathcal D }$ removes the MCP ambiguity, though it turns out to be a non-minimal coupling in Riemann-Cartan space-time.
3. Discussion
3.1. Three treatments on the uniqueness problem
Let us now discuss a little bit about Saa's model. The modified volume element ${{\rm{e}}}^{\theta }\sqrt{g}{{\rm{d}}}^{4}x$ makes the integral $\int {{\rm{d}}}^{4}x{{\rm{e}}}^{\theta }\sqrt{g}{\rm{\nabla }}{}_{\mu }{B}^{\mu }$ a surface integral:
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \int {{\rm{d}}}^{4}x{{\rm{e}}}^{\theta }\sqrt{g}{\rm{\nabla }}{}_{\mu }{B}^{\mu } & = & \displaystyle \int {{\rm{d}}}^{4}x{{\rm{e}}}^{\theta }\sqrt{g}[{\widetilde{{\rm{\nabla }}}}_{\mu }{B}^{\mu }+{K}_{\mu }{B}^{\mu }]\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}[-({\widetilde{{\rm{\nabla }}}}_{\mu }{{\rm{e}}}^{\theta }){B}^{\mu }+{K}_{\mu }{B}^{\mu }{{\rm{e}}}^{\theta }]\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}[-{{\rm{e}}}^{\theta }{B}^{\mu }{\partial }_{\mu }\theta +{K}_{\mu }{B}^{\mu }{{\rm{e}}}^{\theta }]\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}[-{{\rm{e}}}^{\theta }{B}^{\mu }{K}_{\mu }+{K}_{\mu }{B}^{\mu }{{\rm{e}}}^{\theta }]\\ & = & 0,\end{array}\end{eqnarray}$
where one uses the condition that eθBμ vanishes to infinity at the second line, and ∂μθ = Kμ at the fourth line. This model with the modified volume element guarantees that a covariant divergence of a vector ∇μBμ turns to be a surface term, thus, one can freely integrate by parts with covariant derivative ∇. The price in this model is to introduce an elementary field θ in order to handle the contortion trace. The model also predicts that part of the torsion tensor propagates outside the matter region and shows inconsistencies to the known experiments [20, 21].Compared with Saa's approach, Kaźmierczak does not introduce a new field, and only slightly alters the space-time connection: ${\widehat{{\rm{\Gamma }}}}_{\mu \nu }^{\rho }\equiv {{\rm{\Gamma }}}_{\mu \nu }^{\rho }-{\delta }_{\mu }^{\rho }{K}_{\nu }$. For this covariant derivative one has
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}{\widehat{{\rm{\nabla }}}}_{\mu }{B}^{\mu } & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}[{\widetilde{{\rm{\nabla }}}}_{\mu }{B}^{\mu }+{K}_{\mu }{B}^{\mu }-{\delta }_{\mu }^{\rho }{K}_{\rho }{B}^{\mu }]\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}{\widetilde{{\rm{\nabla }}}}_{\mu }{B}^{\mu }\\ & = & \displaystyle \int {{\rm{d}}}^{4}x{\partial }_{\mu }(\sqrt{g}{B}^{\mu }).\end{array}\end{eqnarray}$
By writing ${\widehat{{\rm{\nabla }}}}_{\mu }{B}^{\nu }$ = ${\partial }_{\mu }{B}^{\nu }+{{\rm{\Gamma }}}_{\lambda \mu }^{\nu }{B}^{\lambda }-{\delta }_{\lambda }^{\nu }{K}_{\mu }{B}^{\lambda }$ = ${\partial }_{\mu }{B}^{\nu }+{{\mathbb{N}}}_{\lambda \mu }^{\nu }{B}^{\lambda }$, one can prove that the derivative $\widehat{{\rm{\nabla }}}$ retains the linearity, Leibniz rule and contraction rule just as in section 2 . However, one can check that ${\widehat{{\rm{\Gamma }}}}_{\mu \nu }^{\rho }$ is no longer metric:
$\begin{eqnarray}\begin{array}{rcl}{\widehat{{\rm{\nabla }}}}_{\mu }{g}_{\nu \lambda } & = & {\widetilde{{\rm{\nabla }}}}_{\mu }{g}_{\nu \lambda }-{K}^{\rho }{}_{\nu \mu }{g}_{\rho \lambda }+{\delta }_{\nu }^{\rho }{K}_{\mu }{g}_{\rho \lambda }\\ & & -{K}^{\rho }{}_{\lambda \mu }{g}_{\nu \rho }+{\delta }_{\lambda }^{\rho }{K}_{\mu }{g}_{\nu \rho }\\ & = & 2{K}_{\mu }{g}_{\nu \lambda }\ne 0,\end{array}\end{eqnarray}$
where the properties ${\widetilde{{\rm{\nabla }}}}_{\mu }{g}_{\nu \lambda }=0$ (just as in GR) and Kλνμ = − Kνλμ have been used [22].In comparison, for our modified derivative ${ \mathcal D }$, the connection ${{\mathbb{C}}}_{\lambda \mu }^{\nu }$ is still metric:
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal D }}_{\mu }{g}_{\nu \lambda } & = & {\partial }_{\mu }{g}_{\nu \lambda }-{{\mathbb{C}}}_{\nu \mu }^{\rho }{g}_{\rho \lambda }-{{\mathbb{C}}}_{\lambda \mu }^{\rho }{g}_{\nu \rho }\\ & = & {\widetilde{{\rm{\nabla }}}}_{\mu }{g}_{\nu \lambda }-{{\mathbb{K}}}^{\rho }{}_{\nu \mu }{g}_{\rho \lambda }-{{\mathbb{K}}}^{\rho }{}_{\lambda \mu }{g}_{\nu \rho }\\ & = & {\widetilde{{\rm{\nabla }}}}_{\mu }{g}_{\nu \lambda }=0.\end{array}\end{eqnarray}$
The point is that the modified contortion tensor ${{\mathbb{K}}}_{\nu \lambda \mu }={K}_{\nu \lambda \mu }-\tfrac{1}{3}({g}_{\nu \mu }{K}_{\lambda }-{g}_{\lambda \mu }{K}_{\nu })$ is anti-symmetric in the first two indices, and so keeps the covariant derivative of metric to be zero just as in EC.It is worthwhile to remark that one may attempt to simply set the constraint Kμ = 0 to solve the uniqueness problem in EC theory.1() However, one must remember that such a constraint modifies the geometry, and one is no longer in the Riemann-Cartan space-time. To deal with this constraint, a Lagrange multiplier can be introduced into the action, and one would obtain very different gravitational field equations:
$\begin{eqnarray*}{{\rm{G}}}^{\mu }{}_{a}=8\pi G\left({T}^{\mu }{}_{a}-\displaystyle \frac{1}{4}{\tau }^{b}{K}_{{ba}}{}^{\mu }+\displaystyle \frac{1}{2}{e}_{[b| }{}^{\nu }{e}_{| a]}{}^{\mu }{\tilde{{\rm{\nabla }}}}_{\nu }{\tau }^{b}\right)\end{eqnarray*}$
$\begin{eqnarray*}{S}_{a}{}^{b\mu }=-4\pi G\left({\tau }_{a}{}^{b\mu }-\displaystyle \frac{1}{4}{\tau }^{b}{e}_{a}{}^{\mu }\right),\end{eqnarray*}$
where Gμν is the Einstein tensor as in EC, T is energy-momentum tensor, τ the spin tensor of matter field, and ebμ the tetrad.The departure from the Riemann-Cartan space-time would lead to more severe complications if one considers an extension of the EC theory, e.g., with a propagating torsion, or higher-order terms of the torsion. Therefore, in this paper, we choose to stick to the Riemann-Cartan space-time as in the original EC theory. The contortion tensor K is a basic geometric quantity with its inherent definition and is not traceless in the Riemann-Cartan space-time. The ${\mathbb{K}}$ tensor we constructed is not the contortion tensor in the Riemann-Cartan space-time, it is solely introduced for coupling to the matter field. In this way, we keep all the possible forms of constructions within the Riemann-Cartan space-time.
3.2. A new observation on EC, comparison of EC and our approach
We now examine how different our theory is from EC. For the scalar field:
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal I }}_{{\mathbb{C}}}^{\phi } & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{2}{g}^{\mu \nu }{{ \mathcal D }}_{\mu }\phi {{ \mathcal D }}_{\nu }\phi -\displaystyle \frac{1}{2}{m}^{2}{\phi }^{2}\right]\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{2}{g}^{\mu \nu }{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}_{\nu }\phi -\displaystyle \frac{1}{2}{m}^{2}{\phi }^{2}\right]\\ & = & {{ \mathcal I }}_{{EC}}^{\phi }.\end{array}\end{eqnarray}$
Our model shows no differences from the EC action, and so is the equation of motion. Furthermore, both ${{ \mathcal I }}_{{\mathbb{C}}}^{\phi }$ and ${{ \mathcal I }}_{{EC}}^{\phi }$ share the same expression with ${{ \mathcal I }}_{{GR}}^{\phi }\,=\int {{\rm{d}}}^{4}x\sqrt{g}\left[-\tfrac{1}{2}{\widetilde{{\rm{\nabla }}}}_{\mu }\phi {\widetilde{{\rm{\nabla }}}}^{\mu }\phi -\tfrac{1}{2}{m}^{2}{\phi }^{2}\right]$, thus they all lead to the same equation of motion: ${\widetilde{{\rm{\nabla }}}}_{\mu }{\widetilde{{\rm{\nabla }}}}^{\mu }\phi -{m}^{2}\phi =0$. This equation differs from ∇μ∇μφ − m2φ = 0 which is built up with the complete covariant derivative in EC, by a term Kμ∇μφ. This fact indicates that the scalar field does not couple to torsion in the above-mentioned theories, and the uniqueness problem would not arise in EC if we present the scalar theory with $\widetilde{{\rm{\nabla }}}$(and also with ${ \mathcal D }$). However, the $\widetilde{{\rm{\nabla }}}$ should be regarded as a modified derivative in EC just as ${ \mathcal D };$ and more importantly, we can not unify vector and Dirac fields with $\widetilde{{\rm{\nabla }}}$ since it would simply reduce to GR.For the Dirac field, we notice a very interesting property that our modified action actually equals the EC action:
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal I }}_{{\mathbb{C}}}^{\psi } & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[\displaystyle \frac{{\rm{i}}}{2}(\bar{\psi }{\gamma }^{\mu }{{ \mathcal D }}_{\mu }\psi -{{ \mathcal D }}_{\mu }\bar{\psi }{\gamma }^{\mu }\psi )-m\bar{\psi }\psi \right]\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[\displaystyle \frac{{\rm{i}}}{2}(\bar{\psi }{\gamma }^{\mu }{{\rm{\nabla }}}_{\mu }\psi -{{\rm{\nabla }}}_{\mu }\bar{\psi }{\gamma }^{\mu }\psi )-m\bar{\psi }\psi \right.\\ & & -\displaystyle \frac{{\rm{i}}}{24}\bar{\psi }({e}_{\mu a}{K}_{b}-{e}_{\mu b}{K}_{a}){\gamma }^{\mu }{\gamma }^{a}{\gamma }^{b}\psi \\ & & \left.-\displaystyle \frac{{\rm{i}}}{24}\bar{\psi }({e}_{\mu a}{K}_{b}-{e}_{\mu b}{K}_{a}){\gamma }^{a}{\gamma }^{b}{\gamma }^{\mu }\psi \right]\\ & = & {{ \mathcal I }}_{{EC}}^{\psi }.\end{array}\end{eqnarray}$
This means that, as for the case of the scalar field, one can re-arrange ${{ \mathcal I }}_{{EC}}^{\psi }$ in a different form to avoid the ambiguity problem.Indeed, the property ${{ \mathcal I }}_{{\mathbb{C}}}^{\psi }={{ \mathcal I }}_{{EC}}^{\psi }$ tells us that both theories should yield the same equations of motion, as can be verified below:
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{i}}}{2}{\gamma }^{\mu }{\rm{\nabla }}{}_{\mu }\psi +\displaystyle \frac{{\rm{i}}}{2}{\gamma }^{\mu }\mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }\psi -m\psi \\ \quad =\,\displaystyle \frac{{\rm{i}}}{2}{\gamma }^{\mu }{\rm{\nabla }}{}_{\mu }\psi +\displaystyle \frac{{\rm{i}}}{2}{\gamma }^{\mu }({\rm{\nabla }}{}_{\mu }-{K}_{\mu })\psi -m\psi \\ \quad =\,{\rm{i}}{\gamma }^{\mu }{\rm{\nabla }}{}_{\mu }\psi -\displaystyle \frac{{\rm{i}}}{2}{\gamma }^{\mu }{K}_{\mu }\psi -m\psi \\ \quad =\,{\rm{i}}{\gamma }^{\mu }{\rm{\nabla }}{}_{\mu }\psi -\displaystyle \frac{{\rm{i}}}{12}(4{\gamma }^{b}{K}_{b}+2{\gamma }^{a}{K}_{a})-m\psi \\ \quad =\,{\rm{i}}{\gamma }^{\mu }\left[{\partial }_{\mu }\psi +\displaystyle \frac{1}{4}{W}_{{ab}\mu }{\gamma }^{a}{\gamma }^{b}\psi \right]\\ \quad -\,\displaystyle \frac{1}{4}\displaystyle \frac{{\rm{i}}}{3}({e}_{\mu a}{K}_{b}-{e}_{\mu b}{K}_{a}){\gamma }^{\mu }{\gamma }^{a}{\gamma }^{b}\psi -m\psi \\ \quad =\,{\rm{i}}{\gamma }^{\mu }\left[{\partial }_{\mu }\psi +\displaystyle \frac{1}{4}\left({W}_{{ab}\mu }-\displaystyle \frac{1}{3}\left({e}_{\mu a}{K}_{b}-{e}_{\mu b}{K}_{a}\right)\right){\gamma }^{a}{\gamma }^{b}\psi \right]\\ \quad -\,m\psi \\ \quad =\,{\rm{i}}{\gamma }^{\mu }{{ \mathcal D }}_{\mu }\psi -m\psi \\ \quad =\,0.\end{array}\end{eqnarray}$
As a consistency check, the conjugate equation is:
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{i}}}{2}{\rm{\nabla }}{}_{\mu }\bar{\psi }{\gamma }^{\mu }+\displaystyle \frac{{\rm{i}}}{2}\mathop{{\rm{\nabla }}}\limits^{\star }{}_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }\\ \quad =\displaystyle \frac{{\rm{i}}}{2}{\rm{\nabla }}{}_{\mu }\bar{\psi }{\gamma }^{\mu }+\displaystyle \frac{{\rm{i}}}{2}({\rm{\nabla }}{}_{\mu }-{K}_{\mu })\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }\\ \quad ={\rm{i}}{\rm{\nabla }}{}_{\mu }\bar{\psi }{\gamma }^{\mu }-\displaystyle \frac{{\rm{i}}}{2}\bar{\psi }{K}_{\mu }{\gamma }^{\mu }+m\bar{\psi }\\ \quad ={\rm{i}}{\rm{\nabla }}{}_{\mu }\bar{\psi }{\gamma }^{\mu }+\displaystyle \frac{{\rm{i}}}{12}\bar{\psi }(-2{K}_{b}{\gamma }^{b}-4{K}_{a}{\gamma }^{a})+m\bar{\psi }\\ \quad ={\rm{i}}{\rm{\nabla }}{}_{\mu }\bar{\psi }{\gamma }^{\mu }+\displaystyle \frac{{\rm{i}}}{12}\bar{\psi }({e}_{\mu a}{K}_{b}-{e}_{\mu b}{K}_{a}){\gamma }^{a}{\gamma }^{b}{\gamma }^{\mu }+m\bar{\psi }\\ \quad ={\rm{i}}{{ \mathcal D }}_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }\\ \quad =0.\end{array}\end{eqnarray}$
For the vector field, however, our theory shows a concrete difference from EC:
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal I }}_{{\mathbb{C}}}^{A} & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{4}{{ \mathcal F }}_{\mu \nu }{{ \mathcal F }}^{\mu \nu }-\displaystyle \frac{1}{2}{m}^{2}{A}_{\mu }{A}^{\mu }\right]\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\left[-\displaystyle \frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\displaystyle \frac{1}{2}{m}^{2}{A}_{\mu }{A}^{\mu }\right.\\ & & \left.-\displaystyle \frac{1}{6}({A}_{\mu }{K}_{\nu }-{A}_{\nu }{K}_{\mu })({F}^{\mu \nu }+\displaystyle \frac{1}{3}{A}^{\mu }{K}^{\nu })\right]\\ & = & {{ \mathcal I }}_{{EC}}^{A}-\displaystyle \int {{\rm{d}}}^{4}x\sqrt{g}\\ & & \times \,\left[\displaystyle \frac{1}{6}({A}_{\mu }{K}_{\nu }-{A}_{\nu }{K}_{\mu })({F}^{\mu \nu }+\displaystyle \frac{1}{3}{A}^{\mu }{K}^{\nu })\right].\end{array}\end{eqnarray}$
By the above discussion, we conclude that although the uniqueness problem can be solved partially within EC (what we meant by partially is that we are able to unify the covariant derivatives for the scalar field and Dirac field, but not for the vector field), it is somehow accidental and one still needs a satisfactory way to make the whole theory consistent. The model which we propose presents another possibility to solve the problem, and the new coupling between matter and gravity differs from EC only for the vector field.
3.3. The gravitational action
The effort that we have made in this paper is aiming to give a hint on the guiding principle of how matter couples with a gravitational field in a more general space-time than Riemann, especially when torsion plays an important role. It is worth mentioning that the pure gravity part of action may remain untouched compared to EC. The Hilbert-Einstein action reads:
$\begin{eqnarray}{{ \mathcal I }}_{g}=\int {{\rm{d}}}^{4}x\sqrt{g}\left[\displaystyle \frac{1}{4\pi {\rm{G}}}{\rm{R}}\right],\end{eqnarray}$
where R = Rμμ is the curvature scalar. Rμν is the Ricci tensor defined as Rααμν, with the Riemann tensor ${{\rm{R}}}^{\rho }{}_{\alpha \mu \nu }\,=2({\partial }_{[\mu | }{{\rm{\Gamma }}}_{\alpha | \nu ]}^{\rho }+{{\rm{\Gamma }}}_{\beta [\mu | }^{\rho }{{\rm{\Gamma }}}_{\alpha | \nu ]}^{\beta })$. As a further attempt, one may build up the Riemann tensor through the commutator $[{{ \mathcal D }}_{\mu },{{ \mathcal D }}_{\nu }]$ instead of [∇μ, ∇ν]. The resulting Riemann tensor and Ricci tensor show some extra contortion terms than Rμν and Rραμν. We leave this option for future study.