Lorentz Violation and Riemann-Finsler Geometry

TL;DR

This paper explores how Lorentz symmetry violations affect particle motion and geometric structures, deriving a class of Finsler spaces to model these effects in a charge-conserving scalar field theory.

Contribution

It introduces a framework connecting Lorentz violation with Riemann-Finsler geometry, expanding the geometric understanding of Lorentz-violating theories.

Findings

Derived a large class of Finsler spaces from Lorentz violation effects

Analyzed properties of these Finsler spaces

Linked Lorentz violation to modifications in particle dispersion relations

Abstract

The general charge-conserving effective scalar field theory incorporating violations of Lorentz symmetry is presented. The dispersion relation is used to infer the effects of spin-independent Lorentz violation on point-particle motion. A large class of associated Finsler spaces is derived, and the properties of these spaces is explored.

Equations

$${\cal L}(\phi,\phi^{\dagger})=\partial^{\mu}\phi^{\dagger}\partial_{\mu}\phi-m^{2}\phi^{\dagger}\phi+{\textstyle{1\over 2}}\big{[}\partial_{\mu}\phi^{\dagger}(\widehat{k}_{c}){}^{{\mu\nu}}\partial_{\nu}\phi-i\phi^{\dagger}(\widehat{k}_{a}){}^{\mu}\partial_{\mu}\phi+\text{ h.c.}\big{]}.$$

$$p^2-m^2+({\hat{k}}_c){}^{\mu \nu }{p}_{\mu }{p}_{\nu }-({\hat{k}}_a){}^{\mu }{p}_{\mu }=0,$$

$$({\hat{k}}_c){}^{\mu \nu }$$

$$({\hat{k}}_a){}^{\mu }$$

$$R(p)=0,$$

$$\frac{\partial p_0}{\partial P_j}=-\frac{u^j}{u^0},$$

$$L=-u^{\mu }{p}_{\mu },$$

$$L_3^{(d)}$$

$${\tilde{k}}^{(d)}$$

$${\tilde{k}}^{(d)}{}_{{\alpha }_1\cdots {\alpha }_l}$$

$$F^{(d)}=y-\frac{1}{2}y{\tilde{k}}^{(d)}.$$

$$g_{jk}^{(d)}$$

$$\frac{1}{y^2}{G}^j$$

Full text

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1 Introduction

Connections between Riemann-Finsler spaces and theories with Lorentz violation have recently been uncovered.[1] A lack of physical examples is an obstacle on the path toward developing a strong intuition about Finsler spaces. In the first section, the general effective quadratic scalar field theory incorporating violation of Lorentz symmetry will be developed. In the next section, a method to generate the lagrangian describing the motion of an analogue point particle experiencing spin-independent Lorentz violation is derived. The last section explores the properties of these Finsler spaces. These proceedings are based on results in Ref. \refciteBE18.

2 Field theory

For a complex scalar field $\phi$ propagating in an $n$-dimensional Minkowski spacetime with metric $\eta_{\mu\nu}$, the quadratic Lagrange density incorporating Lorentz violation is

[TABLE]

The Lorentz violation is realized by the CPT-odd operator $(\widehat{k}_{a}){}^{\mu}$, and the CPT-even operator $(\widehat{k}_{c}){}^{\mu\nu}$, each of which can include coefficients for Lorentz violation associated with operators of arbitrarily large mass dimension $d$. The hermiticity of ${\cal L}$ implies these operators are themselves hermitian. In the special case of hermitian scalar fields, the term involving $(\widehat{k}_{a}){}^{\mu}$ is proportional to a total derivative. It follows that CPT symmetry is guaranteed when $\phi=\phi^{\dagger}$.

Field redefinitions can eliminate any traces present in the coefficients for Lorentz violation by absorbing them into the terms with lower mass dimension. We can therefore take them to be traceless without loss of generality. The commutativity of derivatives implies that they are totally symmetric in all their indices. From these considerations, it is found that the coefficients contain $(2d-n+2)(d-1)!/(d-n+2)(n-2)!$ independent components.

The dispersion relation for this theory is found to be

[TABLE]

where the operators $(\widehat{k}_{c}){}^{\mu\nu}$ and $(\widehat{k}_{a}){}^{\mu}$ are expressed in momentum space as

[TABLE]

with the sums running over even powers of $p$. For brevity, both types of coefficients will be expressed without the $a$ or $c$ subscripts in what follows, and the appropriate sign difference will be absorbed into the $k^{(d)}$ coefficient where the CPT properties will be determined by the mass dimension $d$.

3 Classical kinematics

A method has been developed to extract point particle lagrangians from a given field theory.[3] Using the three equations

[TABLE]

the idea is to identify the centroid of a localized wave packet with the point particle. The method starts with the dispersion relation $R(p)$. Next, the components of the classical velocity $u^{\mu}$ of the particle are related to the group velocity of the corresponding wave packet. The last equation is required by translation invariance of $L$, along with the requirement that $L$ be one-homogeneous in the velocity, $L(\lambda u)=\lambda L(u)$. The first two relations can then be used to eliminate the components of the $n$-momentum to write $L$ only as a function of the velocity $u^{\mu}$.

These equations can be combined to produce a quadratic polynomial in $L$. For the case $d=n$, solving this quadratic leads to the exact lagrangian. For the nonminimal cases $d\geq 5$, corrections to the usual lagrangian can be determined through an iterative procedure. The process begins with an expansion in $(k^{(d)}){}$ of the roots of the quadratic. Call this root $L=L(u^{\mu},p_{\mu},(k^{(d)}){})$. Define $L_{0}=L(u^{\mu},p_{\mu},0)=-\sqrt{u^{\mu}\eta_{\mu\nu}u^{\nu}}$, and then $L_{j}=L(u^{\mu},p^{(j-1)}_{\mu}(u^{\mu}),(k^{(d)}){})$ where $p^{(j)}_{\mu}=-\partial L_{j}/\partial u^{\mu}$. This leads to

[TABLE]

where the

[TABLE]

contain the directional dependence $\hat{u}^{\mu}=u^{\mu}/u$, $u=\sqrt{u^{\mu}\eta_{\mu\nu}u^{\nu}}$. This Lagrangian matches the first order correction found by Reis and Schreck for the nonminimal fermion sector using an ansatz-based technique.[4]

4 Finsler geometry

The Finsler structure (or Finsler norm) plays a central role in the study of Finsler spaces. Classical lagrangians satisfy many of the requirements in the definition of Finsler structures. Though there are important differences preventing the lagrangians derived above from fulfilling the definition of a Finsler structure, a method exists to generate associated Finsler structures from a given lagrangian.[5]

For the lagrangians developed from the scalar field theories discussed above, the prescription to generate a Finsler structure in this case is given by $p_{x}(u)\rightarrow(-i)^{n}p_{x}(y)$, $k^{(d)x}\rightarrow i^{n}k^{(d)x}$, $L\rightarrow-F=-y\cdot p$, $u^{x}\rightarrow(i)^{n}y^{x}$.

As a demonstration of the kinds of geometrical quantities one can calculate in these spaces, we use the Finsler space associated with the first order limit of the lagrangian given in Eq. (7). The Finsler structure associated with this lagrangian is

[TABLE]

The fundamental tensor of a Finsler space determines the metric and therefore also the geodesics. The definition of this tensor is $g_{jk}={\textstyle{1\over 2}}\partial^{2}F^{2}/\partial y^{j}\partial y^{k}$. For the limit under consideration, the fundamental tensor is given by

[TABLE]

Inspection shows $g_{jk}$ reduces to a purely riemannian one for the cases $d=n$ and $d=n-2$. This is consistent with the fact that the $d=n$ coefficient can be absorbed into the metric at the level of the field theory, while a $d=n-2$ coefficient would correspond to a rescaling of the mass term.

The situation is not as straightforward for other values of mass dimension. It has been demonstrated that the nonvanishing of the Cartan torsion implies non-riemmannian nature of the underlying space.[6] Calculation of this tensor shows it vanishes for $d=n$ and $d=n-2$, and also in the case of $n=1$, which represents a Riemann curve, but is nonvanishing in other cases. Calculation of the Matsumoto torsion[7] shows only $d=n-1$ reduces to a Randers metric.

The geodesics are obtained from the geodesic equation $F\tfrac{d}{d\lambda}(y^{j}/F)+G^{j}=0$. A calculation shows the geodesic spray coefficients $G^{j}$ are

[TABLE]

where a $\bullet$ subscript denotes contraction of $\hat{y}^{j}$ with a lower $j$ index, with all contractions exterior to any derivatives.

It is clear from this expression that if the background field is covariantly constant with respect to the riemannian metric, $\widetilde{D}_{j}\widetilde{k}^{(d)}{}_{l}=0$, then the geodesics are unaffected. This situation was conjectured to hold in general and is confirmed here, but remains unproved.