Maxwell Field with Torsion

Abstract

We propose a gauge-invariant model of propagating torsion which couples to the Maxwell field and to charged particles. As a result, we have an Abelian gauge-invariant action leading to a theory with nonzero torsion consistent with available experimental data, which can be used to establish a lower bound for our new coupling parameter.

Appendices

Appendix A: Semi-Minimal Coupling

In the following, we consider the general gauge transformation and modified Maxwell field tensor given by (6) and (11), respectively. For arbitrary α, we show that a minimal coupling on the level of differential forms can be achieved which circumvents issues related to restrictions on the connection coefficients and the failure of MCP when applied to the homogeneous equations encountered in [8].

The Maxwell field tensor (11) can be written invariantly in terms of the differential two-form

$$F=dA-\alpha T\wedge A, $$

where A=A _μ d x ^μ and T=T _μ d x ^μ are the vector potential and torsion trace one-forms. Consider the map D:Λ^p→Λ^p+1 defined in the space of p-forms ω such that

$$D\omega=d\omega-\alpha T\wedge\omega. $$

We note that D is not a graded derivation, i.e., one does not have D(ω _p ∧ω _q )=D ω _p ∧ω _q +(−1)^p ω _p ∧D ω _q , where ω _p is a p -form and ω _q is a q-form.

One can show that for an arbitrary p-form ω

$$D^{2}\omega=0 $$

since T is exact, T=d φ. From the nilpotency of the map D, it follows that

$$DF\equiv0, $$

which is the analog of the homogeneous Maxwell equations d f≡0, where f=d A. The homogeneous equations in a local coordinate map are

$$\left(\partial_{\mu}-\alpha T_{\mu}\right)F_{\nu\sigma}+\left(\partial_{\sigma}-\alpha T_{\sigma}\right)F_{\mu\nu}+\left(\partial_{\nu}-\alpha T_{\nu}\right)F_{\sigma\mu}\equiv0. $$

Thus, the homogeneous equations are identically zero, and no restriction on either F or T arise.

The analog of the nonhomogeneous Maxwell equations without sources, ∗d∗f=0 is

$$\ast D\ast F=\left(-\bar{\nabla}_{\mu}F^{\mu\nu}+\alpha T_{\mu}F^{\mu\nu}\right)dx_{\nu}=0~, $$

which coincides with the equations of motion (12).

We have thus shown that the semi-minimal coupling given by the substitution of the de-Rahm exterior product d by the map D provides the Maxwell equations coupled to the trace part of the torsion tensor:

$$\begin{array}{@{}rcl@{}} f=dA&\rightarrow& F=DA,\\ df\equiv0&\rightarrow& DF\equiv0,\\ \ast d\ast f=0&\rightarrow&\ast D\ast F=0. \end{array} $$

Appendix B: Maxwell Equations with Sources

In this section, we apply the formalism presented in the previous section in order to calculate conserved currents.

Let us introduce the current density three-form

$$j=\frac{1}{3!}j^{\mu}\varepsilon_{\mu\nu\kappa\sigma}dx^{\nu}\wedge dx^{\kappa}\wedge dx^{\sigma}, $$

such that the inhomogeneous Maxwell equations become

$$ D\ast F=j. $$

(B1)

Since D ²=0, one has the condition D j=0 to ensure consistency of the Maxwell equations, which in a local chart reads

$$ \left(\bar{\nabla}_{\mu}-\alpha T_{\mu}\right)j^{\mu}=0. $$

(B2)

The interaction term in the action is gauge-invariant provided the current satisfies the conservation (B2):

$$\delta_{\epsilon}\int d^{4}x\sqrt{-g}j^{\mu}A_{\mu}=-\frac{1}{\alpha}\int d^{4}x\sqrt{-g}\left(\bar{\nabla}_{\mu}-\alpha T_{\mu}\right)j^{\mu}~. $$

Now, consider the three-form τ _T =i _T F, where i _T is the interior derivative along the vector field T, which in coordinates is given by

$$\tau_{T}=\frac{\alpha}{3!}\left(T_{\mu}f_{\nu\sigma}+T_{\sigma}f_{\mu \nu}+T_{\nu}f_{\sigma\mu}\right)~ dx^{\mu}\wedge dx^{\nu}\wedge dx^{\sigma}~. $$

This three-form is covariantly conserved:

$$D\tau_{T}=d\tau_{T}-\alpha T\wedge\tau_{T}\equiv0~, $$

which is consistent with D F≡0. Since T∧τ _T vanishes, it follows that τ _T is a closed form,

$$ d\tau_{T}=0. $$

(B3)

Thus, one can construct a conserved quantity, the gauge invariant one-form j _T =∗τ _T , which has the local expression

$$j_{T\mu}=-\frac{\alpha}{2}\varepsilon_{\mu\nu\rho\kappa}T^{\nu}F^{\rho \kappa}\equiv-\frac{\alpha}{2}\varepsilon_{\mu\nu\rho\kappa}T^{\nu}f^{\rho \kappa}~. $$

Following (B3), one has d∗j _T =0:

$$\partial_{\mu}j_{T}^{\mu}\equiv\nabla_{\mu}j_{T}^{\mu}\equiv0~. $$

Thus, if Σ is space-like hypersurface, one has the conserved quantity

$$Q=\alpha{\int}_{\Sigma}d^{3}x\sqrt{\tilde{g}}\mathbf{T}\cdot\mathbf{B}~, $$

where $\tilde {g}_{\mu \nu }$ is the induced metric on Σ, and T and B are torsion and magnetic field vectors. One can show that the total charge Q is a boundary term and vanishes at infinity in case the fields have vanishing boundary values at infinity, as in the case of propagating torsion theory. From j _T , one can construct a dual torsionic source for electromagnetism, called “electric current” in [5], j _E =i _T F, which in coordinates reads $j_{E}^{\mu }=-\alpha F^{\mu \nu }T_{\nu }$.