Gravitational Quantum Dynamics: A Geometrical Perspective

Abstract

We present a gravitational quantum dynamics theory that combines quantum field theory for particle dynamics in space-time with classical Einstein’s general relativity in a non-Riemannian Finsler space. This approach is based on the geometrization of quantum mechanics proposed in Tavernelli (Ann. Phys. 371:239, 2016)  and combines quantum and gravitational effects into a global curvature of the Finsler space induced by the quantum potential associated to the matter quantum fields. In order to make this theory compatible with general relativity, the quantum effects are described in the framework of quantum field theory, where a covariant definition of ‘simultaneity’ for many-body systems is introduced through the definition of a suited foliation of space-time. As in Einstein’s gravitation theory, the particle dynamics is finally described by means of a geodesic equation in a curved space-time manifold.

Appendices

Appendix A

For completeness, in this appendix we introduce the main concepts of Finsler geometry. A more detailed description of this topic can be found in the literature (see for instance [24, 25, 44, 52]). The dynamics takes place in the tangent bundle TM (a special case of a fiber bundle) with base manifold M of dimension $n=4N$ where N is the number of particles in the system, each one described by 4-coordinates $x_i=(x^{0}_i, x^1_i,x^{2}_i,x^3_i)$. Note that the first coordinate $x^{0}_i$ (with associated velocity $y^{0}_i$) corresponds the time variable associated to the particle i and parametrized by a global time parameter s (see main text). The same formalism can be also applied to the single particle formalism with $N=1$ where all many-body effects are included in the potentials and their effects on the geometry of the space-time manifold.

The non-locality of the quantum potential imposes the extension of the configuration space from the one of a single particle to the full configuration space of all particles considered, which defines a 4N-dimensional space (the manifold M) and the corresponding tangent bundle $TM \equiv P$. P has basis $(e_1, \dots , e_N, {\hat{e}}_1, \dots , {\hat{e}}_N)\equiv (e,{\hat{e}})$ and corresponding coordinates $(x_1,\dots ,x_{{n}}, y_1, \dots , y_{{n}})\equiv ({x},{y})$. Finally, the tangent space to the tangent bundle P ($\hbox {T}_u$P) in a point $u \in P$ is associated to the coordinate basis $(\frac{\partial }{\partial x^1}=\partial _1, \dots , \frac{\partial }{\partial x^{n}}=\partial _{n}, \frac{\partial }{\partial y^1}={{\bar{\partial }}}_1, \dots , \frac{\partial }{\partial y^{n}}=\bar{\partial }_{n})\equiv (\partial ,{\bar{\partial }})$. When dealing with the dynamic of an arbitrary particle, we look at the 4-dimensional sector of the full configuration space (M) that deals with that specific particle. For this subspace, we use the same notation as for the full, with the restriction that the coefficients are within the range $a,b,c=0,\dots , 3$.

The Finsler function F(x, y) defines a (0,2)-d metric tensor field

$$\begin{aligned} g_{ab}(x,y)=\frac{1}{2} {{\bar{\partial }}}_a {{\bar{\partial }}}_{b} F^{2}(x,y), \end{aligned}$$

(A.1)

and the (0,3)-d Cartan tensor

$$\begin{aligned} C_{abc}(x,y)=\frac{1}{4} {{\bar{\partial }}}_a {{\bar{\partial }}}_{b} {{\bar{\partial }}}_c F^{2}(x,y) \, . \end{aligned}$$

(A.2)

Note that for $C_{abc}(x,y)=0$ we recover a Riemannian metric space with the metric tensor $g_{ab}(x)$ independent from y. The corresponding non-linear Cartan connection is given by

$$\begin{aligned} \, {N^{a}_{}}_b \, (x,y)= \, {\varGamma ^{a}_{}}_{bc} \, (x,y) y^c - \, {C^{a}_{}}_{bc} \, (x,y) \, {\varGamma ^c_{}}_{pq} \, (x,y) \, y^p y^q \end{aligned}$$

(A.3)

(with $\, {C^{a}_{}}_{bc} \, (x,y)=g^{ad}(x,y) \, C_{dbc} \, (x,y)$) where $g^{ab}(x,y)$ is the inverse of $g_{ab}(x,y)$ and $\, {\varGamma ^{a}_{}}_{bc} \, = g^{aq}(\partial _{b} g_{qc} + \partial _c g_{qb} - \partial _q g_{bc})$ (to simplify the notation we omit the dependence on the coordinates). The non-linear curvature is then

$$\begin{aligned} \, {R^{a}_{}}_{bc} \, =\delta _c \, {N^{a}_{}}_b \, - \delta _{b} \, {N^{a}_{}}_c \, . \end{aligned}$$

(A.4)

The connection allows to decompose the tangent space $T_u P$ into the vertical space $V_u P$ tangent to $T_u M$. This induces the transformation $\{\partial _a, {{\bar{\partial }}}_{b}\} \rightarrow \{\delta _a=\partial _a - \, {N^{b}_{}}_a \, {\bar{\partial }}_{b}, {{\bar{\partial }}}_{b} \}$ in the basis coordinates of $T_u P$. At this point, one can define a linear covariant derivative that preserves the horizontal-vertical split of the tangent bundle P without inducing mixing. In the horizontal-vertical basis, the linear covariant derivative becomes

$$\begin{aligned} {{\tilde{\nabla }}}_{\delta _a}{\delta _{b}}= {{\tilde{\varGamma }}^c}\,_{ab}\, \delta _c\end{aligned}$$

(A.5)

$$\begin{aligned} {{\tilde{\nabla }}}_{\delta _a} {\delta _{{\bar{b}}}}= \, {{\tilde{\varGamma }}^{{\bar{c}}}}\,_{a{\bar{b}}}\, {\bar{\partial }}_{c} \end{aligned}$$

(A.6)

$$\begin{aligned} {{\tilde{\nabla }}}_{\delta _{{\bar{a}}}} {\delta _{b}}= \, {{\tilde{Z}}^c}\,_{{\bar{a}}b} \, \delta _c \end{aligned}$$

(A.7)

$$\begin{aligned} {{\tilde{\nabla }}}_{\delta _{{\bar{a}}}} {\delta _{{\bar{b}}}}= \, {{\tilde{Z}}^{\bar{c}}}\,_{{\bar{a}}{\bar{b}}} \, {\bar{\partial }}_{c} \end{aligned}$$

(A.8)

where $a,b,c=0,\dots , 3$; ${\bar{a}}, {\bar{b}}=4,\dots ,7$ and

$$\begin{aligned} \, {{\tilde{\varGamma }}^c}\,_{ab} \,= \frac{1}{2} g^{cq} (\delta _a g_{bq} + \delta _{b} g_{aq} - \delta _q g_{ab})\end{aligned}$$

(A.9)

$$\begin{aligned} \, {{\tilde{Z}}^c}\,_{ab} \,= g^{cq} C_{abq}. \end{aligned}$$

(A.10)

In the basis $\{\delta _a,{{\bar{\partial }}}_{b}\}$ the linear curvature (1,3)-tensor on the tangent bundle (TM) is given by ($\alpha ,\beta , \delta ,\gamma ,\phi =0,\dots , 7)$)

$$\begin{aligned} \, {{{\mathbb {R}}}^{\alpha }_{}}_{\beta \gamma \delta } \, = &X_{\delta } \, {\varGamma ^{\alpha }_{}}_{\beta \gamma } \, - X_{\gamma } \, {\varGamma ^{\alpha }_{}}_{\beta \delta } \, + \, {\varGamma ^{\phi }_{}}_{\beta \gamma } \, \, {\varGamma ^{\alpha }_{}}_{\phi \delta } \, - \, {\varGamma ^{\phi }_{}}_{\beta \delta } \, \, {\varGamma ^{\alpha }_{}}_{\phi \gamma } \, \\ \, &+ \, {\varGamma ^{\alpha }_{}}_{\beta \phi } \, \, {W^{\phi }_{}}_{\gamma \delta }\, , \end{aligned}$$

(A.11)

where

$$\begin{aligned} X_\alpha = (\delta _a, {\bar{\partial }}_{{{\bar{a}}}}) \, , \, \, \, {W^{{\bar{a}}}_{}}_{bc} \, = \, {R^{a}_{}}_{bc} \, \, , \, \,{W^{{\bar{a}}}_{}}_{{\bar{b}}c} \, =-\frac{\partial \, {N^{a}_{}}_c \, }{\partial y^b} \, , \, {W^{{\bar{a}}}_{}}_{b{\bar{c}}} \, =\frac{\partial \, {N^{a}_{}}_b \, }{\partial y^c} \, \end{aligned}$$

(A.12)

and zero otherwise. ${\mathbb {R}}$ decomposes into the following components labelled by different symbols (capital letters) depending on the addressed (horizontal-vertical) sectors [25]

$$\begin{aligned}&\, {{\mathbb {R}}^{a}_{}}_{bcd} \, = \, {{\mathbb {R}}^{{{\bar{a}}}}_{}}_{{\bar{b}}cd} \, = \, {R^{a}_{}}_{bcd} \, \end{aligned}$$

(A.13)

$$\begin{aligned}&\, {{\mathbb {R}}^{a}_{}}_{bc{\bar{d}}} \, = - \, {{\mathbb {R}}^{a}_{}}_{b{\bar{c}}d} \, = \, {{\mathbb {R}}^{{\bar{a}}}_{}}_{{\bar{b}}c{\bar{d}}} \, = - \, {{\mathbb {R}}^{{\bar{a}}}_{}}_{{\bar{b}}{\bar{c}}d} \, = \, {P^{a}_{}}_{bcd} \, \end{aligned}$$

(A.14)

$$\begin{aligned}&\, {{\mathbb {R}}^{a}_{}}_{b{\bar{c}} {\bar{d}}} \, = \, {{\mathbb {R}}^{\bar a}_{}}_{{\bar{b}}{\bar{c}}{\bar{d}}} \, = \, {S^{a}_{}}_{bcd} \, \end{aligned}$$

(A.15)

while the corresponding Ricci tensor components become

$$\begin{aligned}&{{\mathbb {R}}}_{ab} = R_{ab}, \end{aligned}$$

(A.16)

$$\begin{aligned}&{{\mathbb {R}}}_{{\bar{a}}b} = {^1P}_{ab}, \end{aligned}$$

(A.17)

$$\begin{aligned}&{{\mathbb {R}}}_{a{\bar{b}}} = - {^{2}P}_{ab}, \end{aligned}$$

(A.18)

$$\begin{aligned}&{{\mathbb {R}}}_{{\bar{a}}{\bar{b}}} = S_{ab}. \end{aligned}$$

(A.19)

Note that the linear connection is not uniquely defined and therefore alternative definitions can also be formulated [44]. The horizontal part of the curvature $^lR(\delta _a, \delta _{b})(.)$ can be easily evaluated

$$\begin{aligned} ^l{R^q}_{cab} =\delta _a \, {{\tilde{\varGamma}}^q}\,_{cb} \, - \delta _{b} \, {{\tilde{\varGamma }}^q}\,_{ca} \, + {{\tilde{\varGamma }}^q}\,_{ma} \, \, {{\tilde{\varGamma }}^m}\,_{cb} \, - \, {{\tilde{\varGamma }}^q}\,_{mb} \, \, {{\tilde{\varGamma }}^m}\,_{ca} \, - \, {C^q}_{cm} \, \, {R^m}_{ab} \, , \end{aligned}$$

(A.20)

and is related to the original non-linear curvature through the equation

$$\begin{aligned} \, {R^q_{}}_{ab} \, =- \, {}^{l}{R^q_{}}_{cab} \, y^c. \end{aligned}$$

(A.21)

Finally, the corresponding geodesic curve $s \mapsto \zeta (s)$ defined by

$$\begin{aligned} \ddot{\zeta }^{a} + \, {N^{a}_{}}_b \, (\zeta ,{\dot{\zeta }}) {{\dot{\zeta }}}^b= - {g}^{a b} \partial V(\zeta (s)) /\partial \zeta _{b} \, , \end{aligned}$$

(A.22)

reproduces exactly the dynamics in Eq. (2).

Appendix B

The generalization of the Dirac field equation to the curved space-time geometry defined by $g_{ab}(x)$ (in the coordinates $x^{a}$ of the curved manifold) requires the notion of the so-called tetrad or vierbein formalism [31], which enables the definition of a local normal (Minkowskian) coordinates ${\tilde{x}}^{\mu }$ at a space-time point X. In terms of these new coordinates ${{\tilde{x}}}^{\mu }$ the metric tensor simplifies to $\eta _{\mu \nu }$ and

$$\begin{aligned} g_{ab}(x)= {e}_{a}^{\mu}(x) {e^{\nu }_{}}_{b}(x) \, \eta _{\mu \nu } \end{aligned}$$

(B.1)

where

$$\begin{aligned} {e}_{a}^{\mu}(x) = \left( \frac{\partial {\tilde{x}}^{\mu }}{\partial x^{a}} \right) _{x=X} \end{aligned}$$

(B.2)

defines the vierbein $(\mu =0,1,2,3)$.

When passing to curve space-time, the covariant derivative of Dirac fields, ${\mathcal {D}}_a$, acquires therefore an additional term, the so-called spin connection $\omega _a(x)$ defined as [31, 32]

$$\begin{aligned} \omega _a = \frac{1}{2} \varOmega ^{\mu \nu } \, {e^b_{\mu}} e_{\nu b,a} \, \end{aligned}$$

(B.3)

with $\varOmega ^{\mu \nu } = \frac{1}{4} [{{\tilde{\gamma }}}^{\mu },\tilde{\gamma }^{\nu }] \,$ and ${{\tilde{\gamma }}}^{\mu } = \, {e^{\mu }_{}}_a \, \gamma ^{a} \, ,$ which leads to (including the electromagnetic coupling term)

$$\begin{aligned} {\mathcal {D}}_{\mu } \psi&= \, {E^{a}_{}}_{\mu} \, \left( \partial _a \psi + \omega _{a} \psi - i e A_a \psi \right) \end{aligned}$$

(B.4)

$$\begin{aligned} {\mathcal {D}}_{\mu } {\bar{\psi }}&= \, {E^{a}_{}}_{\mu} \, \left( \partial _a \psi - {\bar{\psi }} \omega _{a} + i e {\bar{\psi }}A_a \right) \, , \end{aligned}$$

(B.5)

where $\, {E^{a}_{}}_\mu \, (x)$ is the inverse of $\, {e^\mu _{}}_a \, (x)$ ($\, {E^{a}_{}}_\mu \, \, \, {e^\mu _{}}_b \, = \delta ^{a}_{b}$). The corresponding Dirac field action is given by [31]

$$\begin{aligned} {\mathcal {A}}_D \left[ \psi (x), A(x); \, {e^\mu _{}}_a \, (x) \right] = i \int d^{4}x \, \sqrt{ |g|} \, {\bar{\psi }}(x) {{\tilde{\gamma }}}^{\mu } \overset{\leftrightarrow }{{\mathcal {D}}_{\mu }} \psi (x) - m {{\bar{\psi }}}(x) \psi (x) \,, \end{aligned}$$

(B.6)

which upon variation with respect to ${{\bar{\psi }}}$ leads to the Dirac equation (see also Eq. (6))

$$\begin{aligned} (i {{\tilde{\gamma }}}^{\mu } \, {\mathcal {D}}_{\mu } - m) \, {\psi } (x)=0 \, . \end{aligned}$$

(B.7)

Appendix C

In this appendix, we describe the process that led to the definition of the covariant quantum Lagrangian defined in Eq. (16). The goal is to find the simplest possible extension of Eq. (15), which has Eq. (5) as its non-relativistic limit. The quantum potential term in Eq. (5), $-Q(\zeta ;\zeta _i)\, \dot{\zeta _i}^{0}$ (for the particle i), can be obtained by adding

$$\begin{aligned} L^{0{(i)}}_{Q}(\zeta ,{{\dot{\zeta }}})=-\int d^{4}x \, \delta ^{4}(x-\zeta _i(s)) \frac{Q(\zeta ;\zeta _i)}{\sqrt{-g}} \sqrt{-g_{ab}(x) \frac{d \zeta _i^{a}(s)}{ds}\frac{d \zeta _i^b(s)}{ds}} \,. \end{aligned}$$

(C.1)

to Eq. (15). In the Newton limit we can set $\sqrt{\sum _{k=1}^3 ({{\dot{\zeta }}}_i)^{2}} \ll c$ and therefore

$$\begin{aligned} L^{cl (i)}_{Q}(\zeta ,{{\dot{\zeta }}})= -\int d^{4}x \, \delta ^{4}(x-\zeta _i(s)) \frac{Q(\zeta ;\zeta _i)}{\sqrt{-g}} \sqrt{-g_{00}(x)} \, {{\dot{\zeta }}}_i^{0} \,. \end{aligned}$$

(C.2)

Finally, taking the classical limit of the metric tensor (see [43]),

$$\begin{aligned} g_{ab}(\zeta _i)= \begin{pmatrix} -(1+V^g(\zeta _i)) &{} 0 &{} 0 &{} 0 \\ 0 &{} 1&{} 0&{} 0 \\ 0 &{} 0 &{} 1&{} 0 \\ 0 &{} 0&{} 0&{} 1 \end{pmatrix} \end{aligned}$$

(C.3)

for which $g_{00}=\det (g_{ab})\equiv g$ we get the desired non-relativistic limit

$$\begin{aligned} L^{cl (i)}_{Q}(\zeta ,{{\dot{\zeta }}}) = -Q(\zeta ;\zeta _i) \, {\dot{\zeta }}_i^{0}. \end{aligned}$$

(C.4)

Appendix D

The matter field component of the energy-momentum tensor is obtained through the variation of

$$\begin{aligned} {\mathcal {A}}^{(p)}_m =\sum _i m_i \int d^{4} x \, ds \, \delta ^{4}(x-\zeta _i(s)) \sqrt{{\tilde{g}}_{ab}(x) \frac{d\zeta ^{a}_i}{ds} \frac{d\zeta ^b_i}{ds}} \end{aligned}$$

(D.1)

that leads to

$$\begin{aligned} T^{ab}&= \frac{2}{\sqrt{|{{\tilde{g}}}|}} \frac{\delta }{\delta {{\tilde{g}}}_{ab}} {\mathcal {A}}^{(p)}_m \end{aligned}$$

(D.2)

$$\begin{aligned}&= \frac{2}{\sqrt{| {{\tilde{g}}}|}} \sum _i m_i \int ds \, \frac{\delta ^{4}(x-\zeta _i(s)) }{\sqrt{{{\tilde{g}}}_{cd}(\zeta _i) \frac{d \zeta _i^{c}}{ds} \frac{d \zeta _i^{d}}{ds} }} \frac{d \zeta _i^{a}}{ds} \frac{d \zeta _i^{b}}{ds} \end{aligned}$$

(D.3)

$$\begin{aligned}&= \frac{2}{\sqrt{|{{\tilde{g}}}|}} \sum _i m_i \int ds \, \delta ^{4}(x-\zeta _i(s)) \frac{d \zeta _i^{a}}{ds} \frac{d \zeta _i^{b}}{ds} \end{aligned}$$

(D.4)

where the last step is possible by setting s such that $\sqrt{{{\tilde{g}}}_{cd}(\zeta _i) \frac{d \zeta _i^{c}}{ds} \frac{d \zeta _i^{d}}{ds} } =1$.

Appendix E

For completeness, in this appendix we summarize the main steps that lead to the energy momentum tensor, $T_{ab}$, given in Eqs. (30) and (31). To this end, we follow the development in Ref. [55] (see Theorem 4.2 therein), which starts with the definition of $T_{ab}$ as the variation of the field action with respect to the metric tensor (in a generic n dimensional manifold M):

$$\begin{aligned} \delta _g {\mathcal {A}} = \delta _g \int d^n x \, {{\mathcal {L}}} = -\frac{1}{2} \int d^n x \, \sqrt{|g|} T^{ab} \delta g_{ab} \, , \end{aligned}$$

(E.1)

with $g= \det g$. Using the variation identities

$$\begin{aligned} \delta g^{ab} = - g^{ac} g^{bd} \delta g_{cd} \end{aligned}$$

(E.2)

for the inverse metric tensor and

$$\begin{aligned} \delta \sqrt{|g|} =\frac{1}{2} \sqrt{|g|} \, g^{ab} \delta g_{ab} \end{aligned}$$

(E.3)

for the metric determinant, we obtain

$$\begin{aligned} \delta _g \int d^n x \, {\mathcal {L}} = \int d^n x\, \left( \frac{1}{2} g^{ab} {\mathcal {L}} \, \delta g_{ab} + \frac{\partial {\mathcal {L}}}{\partial g_{ab}} \delta g_{ab} \right) \end{aligned}$$

(E.4)

from which (comparing with Eq. (E.1)) we get

$$\begin{aligned} \sqrt{|g|} \, T^{ab} = -2 \frac{\partial {\mathcal {L}}}{\partial g_{ab}} - g^{ab}{\mathcal {L}} \end{aligned}$$

(E.5)

or equivalently

$$\begin{aligned} \sqrt{|g|} \, T_{ab} = 2 \frac{\partial {\mathcal {L}}}{\partial g^{ab}} - g_{ab}{\mathcal {L}}\, , \end{aligned}$$

(E.6)

where we used

$$\begin{aligned} \frac{\partial g^{ab}}{\partial g_{cd}}= -\frac{1}{2} \left( g^{ac} g^{bd} + g^{ad} g^{bc}\right) . \end{aligned}$$

(E.7)

For the Dirac field contribution to the energy momentum tensor in a flat space-time geometry, one gets (with $g_{ab}={{\tilde{g}}}_{ab}$, and $D_a=\partial _a-i e A_a$)

$$\begin{aligned} {\mathcal {L}}_D&= \frac{i}{2} \sqrt{|{{\tilde{g}}}|} {{\bar{\psi }}}(x) \gamma ^b \overset{\leftrightarrow }{D}_{b} \psi (x) \nonumber \\&= \frac{i}{2} \sqrt{|{{\tilde{g}}}|} {{\bar{\psi }}}(x) \gamma _a {{\tilde{g}}}^{ab} \overset{\leftrightarrow }{D}_{b} \psi (x) \nonumber \\&= \frac{i}{2} \sqrt{|{{\tilde{g}}}|} {{\bar{\psi }}}(x) \gamma _a \tilde{g}^{ba} \overset{\leftrightarrow }{D}_{b} \psi (x) \end{aligned}$$

(E.8)

or equivalently

$$\begin{aligned} {\mathcal {L}}_D = \frac{i}{2} \sqrt{|{{\tilde{g}}}|} {{\bar{\psi }}}(x) \gamma ^b \overset{\leftrightarrow }{D}_{b} \psi (x) = \frac{i}{2} \sqrt{|{{\tilde{g}}}|} {{\bar{\psi }}}(x) \gamma _{b} {{\tilde{g}}}^{ba} \overset{\leftrightarrow }{D}_a \psi (x) \, , \end{aligned}$$

(E.9)

which implies that

$$\begin{aligned} 2{\mathcal {L}}_D = \frac{i}{2} \sqrt{|{{\tilde{g}}}|} \left( {{\bar{\psi }}}(x) \gamma _a {{\tilde{g}}}^{ba} \overset{\leftrightarrow }{D}_{b} \psi (x) + {{\bar{\psi }}}(x) \gamma _{b} {{\tilde{g}}}^{ba} \overset{\leftrightarrow }{D}_a \psi (x) \right) \end{aligned}$$

(E.10)

and therefore

$$\begin{aligned} {\mathcal {L}}_D = \frac{i}{4} \sqrt{|{{\tilde{g}}}|} \left( {{\bar{\psi }}}(x) \gamma _a {{\tilde{g}}}^{ab} \overset{\leftrightarrow }{D}_{b} \psi (x) + {{\bar{\psi }}}(x) \gamma _{b} {{\tilde{g}}}^{ab} \overset{\leftrightarrow }{D}_a \psi (x) \right) . \end{aligned}$$

(E.11)

Finally, inserting into equation Eq. (E.6) one gets

$$\begin{aligned} T_{ab} = \frac{i}{2} \left( {{\bar{\psi }}}(x) \gamma _a \overset{\leftrightarrow }{D}_{b} \psi (x) + {{\bar{\psi }}}(x) \gamma _{b} \overset{\leftrightarrow }{D}_a \psi (x) \right) - \frac{\tilde{g}_{ab}}{\sqrt{|{{\tilde{g}}}|}} {\mathcal {L}}_D. \end{aligned}$$

(E.12)

The extension to curved space-time requires the introduction of the vielbein fields $\, {e^\mu _{}}_a \, (x)$ (with inverse $\, {E^{\mu }_{}}_a \,$, such that $\, {E^{a}_{}}_\mu \, \, {e^{\nu }_{}}_a \, =\delta ^{\nu }_{\mu }$) related to the metric by the relation $g_{ab}=\eta _{\mu \nu } \, {e^{\mu }_{}}_a \, \, {e^{\nu }_{}}_b \,$. In the following, we only briefly summarize the main results without giving any derivation. For more details see [31, 32].

The variation of the action functional (without mass term)

$$\begin{aligned} {\mathcal {A}}_D \left[ \psi , A; \, {e^\mu _{}}_a \, (x) \right] = i \int d^{4}x \, \sqrt{|g|} {\bar{\psi }} {{\tilde{\gamma }}}^{\mu } \overset{\leftrightarrow }{{\mathcal {D}}}_{\mu } \psi \end{aligned}$$

(E.13)

with respect to the frame fields $\, {E^{a}_{}}_{\mu } \,$ yields [32]

$$\begin{aligned} T^{ab} (x) =\frac{i}{2} \left( {\bar{\psi }} \gamma ^{a} \overset{\leftrightarrow }{{\mathcal {D}}^b} \psi + {\bar{\psi }} \gamma ^b \overset{\leftrightarrow }{{\mathcal {D}}^{a}} \psi \right) \, , \end{aligned}$$

(E.14)

which corresponds to Eq. (33), when transformed to the operator space.

Appendix F

For completeness, we summarize the main steps that lead to the energy momentum tensor for the electromagnetic field in the Finsler space. The derivation follows closely the one of N. Voicu in [52].

For the Lagrangian density (with $c=1$) corresponding to Eq. (36),

$$\begin{aligned} {\mathcal {L}}= -\frac{1}{16 \pi } \, \sqrt{{\mathcal {G}}} \, F_{AB} F^{AB} \, ,\end{aligned}$$

(F.1)

the partial derivatives

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial A_{a,b}}&=-\frac{1}{4 \pi } F^{ba} \sqrt{{\mathcal {G}}} \end{aligned}$$

(F.2)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial A_{a . {\bar{b}}}}&=-\frac{1}{4 \pi } F^{{\bar{b}} a} \sqrt{{\mathcal {G}}} \end{aligned}$$

(F.3)

give

$$\begin{aligned} \, {{\tilde{T}}^b\,_c} \,=\frac{1}{4 \pi } \left( -F^{ba} A_{a,c} + \frac{1}{4} \delta ^b_c F_{AB} F^{AB}\right) \end{aligned}$$

(F.4)

$$\begin{aligned} \, {{\tilde{T}}^{{{\bar{b}}}}}\,_c \,=- \frac{1}{4 \pi } F^{{\bar{b}} a} A_{a,c} \end{aligned}$$

(F.5)

where we use the notation $X_{a,c} = \frac{\partial X_a}{\partial x^{{c}}}$ and $X_{a.c} = \frac{\partial X_a}{\partial y^{{\bar{c}}}}$. To symmetrize these expressions, we can add terms derived from divergences of generic tensors without changing the conserved charges [52]. Using

$$\begin{aligned} \frac{1}{4 \pi } \left( F^{ba} A_{a,c} + F^{b {\bar{a}}} A_{c . {\bar{a}}}\right) \sqrt{{\mathcal {G}}}&= \frac{1}{4 \pi } \left( F^{ba} A_{c} \sqrt{{\mathcal {G}}}\right) _{,a} + \frac{1}{4 \pi } \left( F^{b {\bar{a}}} A_c \sqrt{{\mathcal {G}}}\right) _{. {\bar{a}}} \end{aligned}$$

(F.6)

$$\begin{aligned} \frac{1}{4 \pi } F^{{\bar{b}} a} A_{c,a} \sqrt{{\mathcal {G}}}&= \frac{1}{4 \pi } \left( F^{{\bar{b}}a} A_c \sqrt{{\mathcal {G}}}\right) _{,a} \end{aligned}$$

(F.7)

in Eqs. (F.4) and (F.5), we finally obtain the energy momentum tensor in Eq. (38).