# Classical kinematics for isotropic, minimal

Lorentz-violating fermion operators

###### Abstract

In this article a particular classical, relativistic Lagrangian based on the isotropic fermion sector of the Lorentz-violating (minimal) Standard-Model Extension is considered. The motion of the associated classical particle in an external electromagnetic field is studied and the evolution of its spin, which is introduced by hand, is investigated. It is shown that the particle travels along trajectories that are scaled versions of the standard ones. Furthermore there is no spin precession due to Lorentz violation, but the rate is modified at which the longitudinal and transverse spin components transform into each other. This demonstrates that it is practical to consider classical physics within such an isotropic Lorentz-violating framework and it opens the pathway to study a curved background in that context.

###### pacs:

11.30.Cp, 14.60.Cd, 45.20.Jj, 02.40.-k## I Introduction

Since violations of CPT symmetry and Lorentz invariance were shown to appear in the context of string theory Kostelecky:1988zi ; Kostelecky:1991ak ; Kostelecky:1994rn ; oai:arXiv.org:hep-th/9605088 , the interest in exploring a possible violation of this fundamental symmetry in nature has grown steadily. Subsequently such a violation was also found to occur in loop quantum gravity Gambini:1998it ; Bojowald:2004bb , models of noncommutative spacetimes Carroll:2001ws , spacetime foams models Klinkhamer:2003ec ; Bernadotte:2006ya , and in spacetimes endowed with a nontrivial topology Klinkhamer:1998fa ; Klinkhamer:1999zh . Therefore it can be considered as a window to physics at the Planck scale. A further boom creating a new field of research took place when the minimal Standard-Model Extension (SME) was established Colladay:1998fq . The latter provides a powerful effective framework for describing Lorentz violation for energies much smaller than the Planck scale.

Since then the field has been developing largely in both experimental searches for Lorentz violation and the study of theoretical aspects. There has been a broad experimental search for Lorentz violation (see the data tables Kostelecky:2008ts and references therein) and there are ongoing studies on the properties of quantum field theories based on the SME Kostelecky:2000mm ; oai:arXiv.org:hep-ph/0101087 ; Casana-etal2009 ; Casana-etal2010 ; Klinkhamer:2010zs ; Schreck:2011ai ; Cambiaso:2012vb ; Colladay:2014dua ; Maniatis:2014xja ; Schreck:2013gma ; Schreck:2013kja ; Cambiaso:2014eba ; Schreck:2014qka . Recently, the nonminimal versions of the SME including all higher-dimensional operators of the photon, fermion, and neutrino sector have been constructed as well Kostelecky:2009zp ; Kostelecky:2011gq ; Kostelecky:2013rta .

Although the SME seems to work very well in flat spacetime, certain issues arise when it is coupled to gravitational fields. Around ten years ago
a no-go theorem was proven stating that an explicitly Lorentz-violating field theory cannot be coupled to gravity consistently, because this leads to
incompatibilities with the Bianchi identities Kostelecky:2003fs .^{1}^{1}1Besides, note that certain tensions with the generalized second law
of black-hole thermodynamics may occur when particular Lorentz-violating theories are coupled to a black-hole gravitational background. The reason
is the multiple-horizon structure, e.g., for photons that arises in such frameworks Dubovsky:2006vk ; Eling:2007qd ; Betschart:2008yi ; Kant:2009pm . A
coupling is only possible if Lorentz invariance is violated spontaneously, e.g., in a Bumblebee model Kostelecky:1988zi ; Kostelecky:1989jp ; Kostelecky:1989jw ; Kostelecky:2000mm ; Kostelecky:2003fs ; Bailey:2006fd ; Bluhm:2008yt .

Note that the incompatibilities mentioned were found in the context of Riemann-Cartan spacetimes, i.e., spacetimes endowed with the Riemannian concept of curvature including torsion. An alternative approach to considering Lorentz violation in gravitational backgrounds is to change the fundamental geometrical concept. Hence instead of Riemann-Cartan geometry one might be tempted to use Finsler geometry Finsler:1918 ; Cartan:1933 ; Matsumoto:1986 ; Antonelli:1993 ; Bao:2000 ; Kozma:2003 ; Bao:2004 ; Bucataru:2007 as the basis of a theory of gravity. Geometrical quantities in Finsler spaces such as curvature do not only depend on the particular point considered in the space but also on the angle that a given line element encloses with an inherent direction in this space. Finsler spaces rest on more general length functionals, whereby they can be considered as Riemannian spaces without the quadratic restriction Chern:1996 .

For this reason Finsler geometry may be a natural framework to describe preferred directions in a curved spacetime, i.e., Lorentz violation in the presence of gravity. Lately plenty of work has been done to identify Finsler spaces linked to certain cases of the SME fermion sector, which includes studies of the minimal Kostelecky:2010hs ; Kostelecky:2011qz ; Kostelecky:2012ac ; Colladay:2012rv ; Russell:2015gwa and also the nonminimal sector Schreck:2014hga . In the current article isotropic subsets of the minimal fermion sector will be investigated. We will obtain the corresponding Finsler structure and address certain physical problems such as the propagation of a classical, relativistic, pointlike particle in the Lorentz-violating background and the time evolution of particle spin.

The paper is organized as follows. In Sec. II all isotropic coefficients of the minimal SME fermion sector are identified and the corresponding dispersion relations are computed. In Sec. III a generic isotropic dispersion relation is considered and its associated classical, relativistic Lagrangian is derived, which is then promoted to a Finsler structure. Section IV is dedicated to studying the physics of the classical Lagrangian obtained. First of all the motion of the classical particle in an electromagnetic field will be analyzed. Besides, the interest also lies in the behavior of particle spin, which is introduced by hand and treated with the Bargmann-Michel-Telegdi (BMT) equation Bargmann:1959gz . Finally the results are summarized and discussed in Sec. V. Throughout the paper natural units with are used unless otherwise stated.

## Ii Isotropic dispersion laws in the minimal fermion sector

The intention of the current section is to find all isotropic dispersion relations of the minimal SME fermion sector. The full action including both minimal and nonminimal contributions reads as Kostelecky:2013rta

(1a) | ||||

(1b) |

Here is a Dirac spinor field, its Dirac conjugate, and is the fermion mass. The for are the standard Dirac matrices obeying the Clifford algebra and is the unit matrix in spinor space. The operator is a collection of all minimal and nonminimal Lorentz-violating composite operators in the pure fermion sector. All fields and operators are defined in Minkowski spacetime with the metric .

The terminology to denoting Lorentz-violating operators is chosen according to Table 1 of Kostelecky:2013rta . Each operator is characterized by a couple of free Lorentz indices whose number ranges from 0 to 2. These indices control the spin behavior of the corresponding operator. Upon decomposition of into the Dirac bilinears according to Eq. (2) in the latter reference the operators are grouped into scalars, vectors, and second-rank tensors. Additionally, in momentum space these are split into momenta and Lorentz-violating component coefficients, cf. Eqs. (5), (6) in Kostelecky:2013rta . Their transformation properties with respect to (proper and improper) observer Lorentz transformations and charge conjugation are stated in Table 1 of Kostelecky:2013rta as well. Both the scalar and the pseudoscalar operator only appear in the nonminimal sector, i.e., the analysis will be restricted to the vector operators , , , , the scalar operators , , and the tensor operators , . The following calculations will be based on Eq. (39) of Kostelecky:2013rta , which gives the general dispersion relation of the SME fermion sector including all minimal and nonminimal contributions. The dispersion relation involves the operators , , , , defined by Eqs. (2), (7) and , , given by Eq. (35) in the latter reference.

First of all the vector operators and shall be considered. They are contained in and , respectively, and they contribute to . For , , and the modified fermion dispersion relation results in

(2) |

with the fermion four-momentum . Setting the second term on the left-hand side of the latter equation cannot be isotropic for any choice of besides . The corresponding dispersion relation is then given by

(3) |

where is the particle three-momentum. Here denotes the positive-energy dispersion law. This result is encoded in Eq. (94) of Kostelecky:2013rta . Note that a nonzero coefficient just leads to an unobservable shift of the particle energy, which reminds us of the fact that the coefficients can be removed by a phase redefinition Kostelecky:2013rta . As a next step the operator is considered. From , , and we obtain:

(4) |

For the term can only be isotropic, if . Then there are two different dispersion relations that read as

(5) |

The expansion here and all subsequent ones are understood to be valid for a sufficiently small Lorentz-violating coefficient. Due to Lorentz violation the energies of fermion states with different spin projections are no longer degenerate. This behavior resembles a birefringent vacuum for the photon sector.

The situation is slightly similar for the vector operators and being comprised of second-rank tensor coefficients that are contracted with one additional four-momentum. We consider at first. To end up with an isotropic dispersion relation, the coefficients must be chosen such that in Eq. (2) is isotropic. This is only the case if all off-diagonal components vanish and . Since is traceless, that heavily restricts the possibilities of choices for the coefficients, with only one remaining:

(6) |

Then even , which makes the dispersion relation manifestly isotropic. The coefficients behave in a similar manner. Setting , the expression in Eq. (4) must be isotropic. With an analogous argument this leads to

(7) |

For the choice of Eq. (7) it can be checked that resulting in an isotropic dispersion relation. Now the modified dispersion laws in case of nonvanishing coefficients and read as follows:

(8) |

For there is a single dispersion relation for both spin projections of the fermion. At first order in (and for ) this modification corresponds to the modification appearing in the isotropic, CPT-even extension of the photon sector. That is reasonable, since both sectors are related by a coordinate transformation (see Altschul:2006zz and references therein). The result is confirmed by Eq. (95) in Kostelecky:2013rta . For there exist two distinct isotropic dispersion relations.

The next step is to consider the scalar operators and . For the operator it holds that , , , and , which is subsequently inserted in Eq. (39) of Kostelecky:2013rta to give

(9) |

The latter can only be isotropic for leading to the dispersion relation

(10) |

The result corresponds to the observation that is isotropic (see Eq. (97) in Kostelecky:2013rta ) where this effective dimension-5 coefficient also contains according to the first of Eqs. (27) in Kostelecky:2013rta . A similar investigation can be carried out for where , which gives

(11) |

The latter result can only be isotropic for , whereby one obtains

(12) |

Note that by a spinor redefinition the coefficients can be transferred to the operator Altschul:2006ts . In addition, the dispersion relation is only affected at second order for this particular type of coefficients.

Last but not least the tensor coefficients and will be investigated. They are both contained in the tensor operator . The special case derived from the general dispersion relation of Eq. (39) in Kostelecky:2013rta by setting , is given by:

(13a) | ||||

with the convenient definition | ||||

(13b) |

The latter involves the dual of , which is denoted with an additional tilde and formed by contraction of with the four-dimensional Levi-Civita symbol where . Now Eq. (13) can be further simplified by using the properties of . When adding the operators defined in Eq. (13b) the dual is eliminated. Furthermore the square of corresponds to the square of its dual with an additional minus sign. Eventually, contracted with two four-momenta vanishes:

(14) |

The second and third relationship follow from the antisymmetry of . By using these relations, Eq. (13) can be further simplified:

(15) |

For the tensor operator the only term that may lead to anisotropy is the last one on the left-hand side of the latter equation. By explicitly inserting , it can be demonstrated that no choice of the coefficients of leads to an isotropic expression. In Kostelecky:2013rta it was shown that only the dimension-5 coefficients produce an isotropic dispersion law (see Eq. (97) in Kostelecky:2013rta ). According to the fourth of Eqs. (27) in Kostelecky:2013rta these effective coefficients contain where are the dual coefficients of . Furthermore, by symmetry arguments they also comprise . This explains the isotropic dispersion laws of Eq. (II) following from a nonzero coefficient .

Hence an isotropic dispersion relation does not exist for any of the dimension-3 component coefficients . For the tensor operator the situation is different. With it can be checked that there is an isotropic dispersion relation for two different choices of coefficients. The first choice is

(16) |

and all others set to zero, which results in two modified dispersion relations:

(17) |

The nonzero coefficients of Eq. (16) are contained in of Eq. (95) in Kostelecky:2013rta where denotes the dual of . According to the third of Eqs. (27) in Kostelecky:2013rta these effective coefficients also contain , which explains the isotropic dispersion relation of Eq. (5). For this particular choice of the last term on the left-hand side of Eq. (15) is isotropic. The second choice of coefficients, which fulfills that condition, is

(18) |

and all remaining ones set to zero. This case gives rise to a single modified dispersion relation:

(19) |

Note that Eq. (17) comprises a modification at first order in the Lorentz-violating coefficients, whereas the modification in Eq. (19) is of second order in Lorentz violation. The term in Eq. (15) differs for both sets of component coefficients leading to distinct dispersion relations.

To summarize, in the minimal fermion sector of the SME an isotropic dispersion relation exists for a particular choice of , , , , , , and component coefficients. Some of these dispersion relations depend on the spin projection of the fermion, which is the analogy of a birefringent vacuum in the photon sector.

## Iii Construction of the classical Lagrangian and Finsler structure

As of now we intend to consider an isotropic modified dispersion relation of the generic form

(20) |

with a dimensionless parameter where in the standard case . Such a dispersion relation emerges from a particular choice of the coefficients, cf. Eq. (19), or for a nonvanishing (see Eq. (II) by setting ) when absorbing the global modification before the square root into the fermion mass. This dispersion relation is based on the fermion Lagrangian of the SME, i.e., it is a field theory result.

In what follows, for the particular isotropic dispersion relation of Eq. (20) the Lagrangian shall be derived, which describes a classical, relativistic, pointlike particle whose conjugate momentum satisfies the dispersion relation mentioned. It was shown in Kostelecky:2010hs that such a Lagrangian can, in principle, be obtained from five equations involving the four-momentum components and the four-velocity components of the classical particle. One of these equations is the modified dispersion relation. Furthermore, due to parameterization invariance of the classical action along a path the Lagrangian must be positively homogeneous of first degree in the velocity. Then it has to be of the following shape, which forms the second equation:

(21) |

Here is the conjugate momentum of the particle. Note the minus sign in the definition of the latter. If we construct a quantum-mechanical wave packet from the quantum-theoretic free-field equations, its group velocity shall correspond to the velocity of the classical pointlike particle:

(22) |

Because of the assumed isotropy of the Lagrangian the original three conditions, which hold for the spatial momentum components, result in only one equation here for the magnitude of the spatial momentum and the magnitude of the three-velocity. This single equation can be solved with respect to :

(23) |

Then the zeroth four-momentum component can be expressed via the velocity as well:

(24) |

According to Eq. (22) for the sign of has to be chosen as negative. For the sign is taken to be positive. However the absolute value of in Eq. (24) produces an additional minus sign in this case. This leads to:

(25) |

with the preferred timelike direction . If only the positive-energy solution in Eq. (24) is considered, the Lagrangian with a global minus sign must be taken into account as well. It can be checked that Eqs. (20) – (22) are fulfilled by the positive Lagrangian for and by the negative Lagrangian for . Since in the remainder of the paper will be chosen anyhow, the Lagrangian with a global minus sign will be considered from now on. For one obtains the standard result . The Lagrangian itself has an intrinsic metric associated to it, which is used to define scalar products, e.g., . This intrinsic metric corresponds to the Minkowski metric, i.e., .

The following section, which ought to be understood as an interlude, is dedicated to identifying the Finsler structure associated to the Lagrangian with a global minus sign, i.e., (see Bao:2000 for the properties of such a structure). As outlined in the introduction, the basic goal of the community is to understand how Lorentz-violating theories can be coupled to gravitational backgrounds in a consistent manner. A reasonable assumption is that this is possible using the concept of Finsler geometry, since the latter incorporates intrinsic preferred directions into the description of geometrical quantities in a natural way. Readers who are not interested in Finsler geometry can skip the rest of the current section, since the results will not be directly employed in the remainder of the article.

There are two different possibilities of proceeding Kostelecky:2011qz . The first is to set , which results in a three-dimensional Finsler structure describing a Euclidean geometry with a global scaling factor:

(26) |

where is the tangent bundle of the Finsler space. The scalar product of two vectors , in the tangent space is given by with the intrinsic metric . This structure describes a Euclidean space with all dimensions scaled by . A similar space results from applying the same procedure to the Finsler structure of the nonminimal coefficient considered in Schreck:2014hga .

The alternative is to perform a Wick rotation leading to the four-dimensional Finsler structure

(27) |

where . The intrinsic metric here is and is a preferred direction where for and with the used in Eq. (III). The following considerations will be concentrated on the second Finsler structure . The Finsler metric can be computed via

(28) |

and the particular result is independent of . The Finsler structure describes a Euclidean geometry as well. To check this, the Cartan torsion Bao:2004

(29) |

is needed where its mean is defined as

(30) |

with the inverse Finsler metric . For the special Finsler metric in Eq. (28) the mean Cartan torsion
vanishes, which according to Deicke’s theorem Deicke:1953 shows that the corresponding space is Riemannian.^{2}^{2}2In
Kostelecky:2011qz Lagrangians were considered with their intrinsic metric being promoted to a general pseudo-Riemannian
metric. By doing so, the Lagrangian can describe the motion of a relativistic particle on a curved spacetime manifold. Performing the
generalization here would lead to the Finsler structure of Eq. (27) with its scalar products being defined by
an intrinsic metric , which is not necessarily flat. In this case according to Eq. (28) the Finsler metric
would be associated to the structure. Note
that , , and are then understood to be position-dependent functions, in general.
Since does not depend on , its mean Cartan torsion vanishes showing that it still describes a Riemannian
space. In the remainder of the current article the intrinsic metric will be assumed to be flat, though. In this space three
dimensions are scaled by and one dimension remains standard. Therefore the length of a vector in the scaled subspace, which corresponds to
the spatial part of the original spacetime, is scaled where the angle between such vectors stays unmodified. However angles between
vectors change when they have one component pointing along the -axis, which has some influence on, e.g., velocities in the corresponding
spacetime.

All Finsler spaces in the context of the minimal SME, which have been considered in other references so far, are related to non-Euclidean spaces. This holds for a-space Kostelecky:2010hs ; Kostelecky:2011qz , b-space Kostelecky:2010hs ; Kostelecky:2011qz , the bipartite spaces Kostelecky:2012ac , and the spaces considered in Colladay:2012rv . A reasonable conjecture is that only single, isotropic dispersion relations such as the one investigated here lead to Euclidean structures.

## Iv Charged relativistic particle in an electromagnetic field

After clarifying the mathematical foundations of the modified Lagrangian in the last section, its physical properties shall be investigated. In what follows, particle trajectories shall be parameterized such that and where is the speed of light and the ordinary three-velocity of the particle. Note that natural coordinates are used with . The Lagrangian then reads as follows:

(31) |

If the particle moves freely, the trajectory will be the same straight line such as in the standard case without any Lorentz violation. Using the metric corresponding to the Lagrangian,

(32) |

the conserved quantities can be computed according to Eqs. (35) and (36) in Girelli:2006fw and they are given by:

(33a) | ||||

(33b) |

with a modified Lorentz factor . For these correspond to the classical energy and spatial momentum (besides a global sign), as expected. The spatial momentum is part of the contravariant four-momentum, whereby the index on has to be raised to produce an additional sign. The quantities and will appear again below.

To understand the modified physics, the classical particle is assigned an electric charge and its propagation in an electromagnetic field will be studied. Therefore a four-potential is introduced and the charged, classical particle is described by the following Lagrangian:

(34) |

with the scalar potential and the vector potential . The equations of motion are obtained from the Euler-Lagrange equations (with the position vector ), which for the particular Lagrangian of Eq. (34) read as follows:

(35a) | ||||

(35b) |

The total time derivative of the vector potential

(36) |

is used to express the right-hand side of Eq. (35b) via the physical electric and magnetic fields , :

(37) |

For the zeroth four-momentum component, i.e., the particle energy, a further equation can be derived directly from the equations of motion for the spatial momentum components:

(38) |

Now a relativistic momentum and energy can be introduced via

(39) |

In fact, with the free Lagrangian it can be cross-checked that

(40) |

which corresponds to Eq. (39) for , and again taking into account that is the spatial momentum of the contravariant momentum four-vector. Therefore raising the index on produces an additional minus sign. With the proper time the equations of motion (37), (38) can be written in a covariant form:

(41) |

where is the electromagnetic field strength tensor. Note that the four-velocity used on the left-hand side of the latter equation involves both modifications in the Lorentz factor and the spatial velocity components, whereas the four-velocity on the right-hand side is standard. The reason for this is that the particle kinematics is modified by the Lorentz-violating background field in contrast to its coupling to the electromagnetic field.

Now the modified equations of motion shall be solved for particular cases to understand how their solutions are affected by Lorentz violation. First, consider the case of a vanishing electric field, , where the particle moves perpendicularly to a magnetic field , i.e., its initial velocity and position shall be given by and , respectively. Here is the constant velocity and the particle distance from the origin at the beginning. The equations of motion in the laboratory frame read

(42) |

where is time-independent, since the magnitude of the velocity does not change in a magnetic field. The latter differential equations with the initial conditions above are satisfied by the following time-dependent particle position and velocity:

(43a) | ||||

(43b) |

This describes a circular motion with radius and angular frequency such as in the standard case where the sign gives the rotational direction. However additional scaling factors appear that can be explained as follows. Kinematic quantities living in tangent space such as and involve a length scale and, therefore, get multiplied by one power of each (cf. Eq. (28)): . In contrast to position and velocity vectors, both the momentum vector and the vector potential live in cotangent space and have an inverse length scale associated to them, which is why they are multiplied by (see the dispersion relation of Eq. (20)): . Taking into account the relationship in momentum space, it follows that . Because of the angular frequency is unaffected by the scaling. Concerning the particle motion, since the magnitude of the velocity is constant, is time-independent. Due to the first of Eq. (41) is satisfied as well, for consistency.

As a next example consider a particle moving in a vanishing magnetic field, , where the particle initially moves perpendicularly to the electric field with the initial conditions and . The equations of motion in the laboratory frame then read as follows:

(44) |

An integration with respect to using the initial condition leads to:

(45) |

This is a system of algebraic equations for and (), which can be solved to give

(46a) | |||

where a subsequent integration results in | |||

(46b) |

Here the trajectory again involves additional scalings with . The behavior can be understood when taking into account that each of the velocities , , and positions , gets one power of : . Due to and the scaling , the electric field is subject to . Concerning the physical behavior of the particle, the velocity component in -direction goes to zero starting from its initial value . This is due to the relativistic increase in mass, since there is no force along the -direction compensating for this effect. Therefore the distance traveled in -direction grows logarithmically, i.e., very slowly. The velocity in -direction steadily increases to reach its maximum value as expected. For large times the particle then travels with the practically constant velocity resulting in the uniform motion for . For consistency, the first of Eq. (41) is fulfilled when inserting the electric field vector and the velocity components of Eq. (46a).

### iv.1 Introduction of particle spin

Since spin is a manifestly quantum-mechanical concept, the classical particle studied in the previous sections does not have any spin associated to it, although it shall be based on a Lorentz-violating fermion. However it is possible to introduce spin for a classical particle according to the lines of Bargmann:1959gz . The authors of the latter reference derive a relativistic equation of motion (often denoted as the BMT equation where this abbreviation refers to the authors’ second names) for the spin of a classical particle of electric charge and mass in an electromagnetic field. By doing so, they take the equation of motion for the spin three-vector as a basis:

(47) |

Generalization to arbitrary frames leads to the BMT equation:

(48) |

Here is the Landé factor of the particle, is the particle velocity, the spin
four-vector, and the proper time.
The spin four-vector can be understood as a covariant particle polarization vector. According to Bargmann:1959gz it is the
expectation value of the Pauli-Lubanski (pseudo)vector , which
is a four-vector by construction Lubanski:1942 .^{3}^{3}3The operator stated on the first page of Lubanski:1942 is with
the metric used here. Here is the four-dimensional Levi-Civita symbol with the condition
and are the generators of the Lorentz group. The latter are a covariant generalization
of the angular momentum three-vector. The scalar product of with itself is one of the two Casimir operators of the Poincaré algebra
giving the eigenvalues for a particle at rest. Since is the particle spin eigenvalue, can be
interpreted as a covariant generalization of the nonrelativistic spin operator.

The quantum mechanical treatment of fermion spin in the SME was carried out in Kostelecky:2013rta ; Gomes:2014kaa . In the latter references the time evolution of the spin expectation value was obtained from the expectation value of the commutator of the spin operator and the Lorentz-violating Hamiltonian (cf. Bluhm:1999dx where this approach was introduced). At first order in Lorentz violation a Larmor-like precession of the particle spin is induced by controlling coefficients leading to two distinct fermion dispersion relations. These are subsets of the effective and operators that comprise the operators , , , and . Such a behavior is reminiscent of the standard case when spin precession occurs for the valence electron of a hydrogen atom in an external magnetic field accompanied by a splitting of its energy levels.

Therefore considering dispersion laws of the form of Eq. (20), the only set of isotropic controlling coefficients that may lead to spin precession is given by Eq. (18). However it is evident that their correction to the standard fermion dispersion relation is of quadratic order. Inserting these coefficients into Eq. (62) of Kostelecky:2013rta , which controls the spin part of the Hamiltonian at first order in Lorentz violation, gives zero as expected. Therefore the spin operator commutes with the Hamiltonian at first order in the controlling coefficients for the sets of coefficients studied. Based on the Heisenberg equations no additional time dependence of the spin operator emerges from Lorentz violation at first order for the sets of coefficients leading to the classical Lagrange function of Eq. (III). Note that the set of coefficients given by Eq. (16) results in two distinct isotropic dispersion relations and classical Lagrangians corresponding to operators with these properties are not considered in the current article.

To ensure that Lorentz violation does not give rise to an additional contribution to the zeroth component of the BMT equation, the explicit form of the Pauli-Lubanski vector can be examined. In matrix form the generators of the Lorentz group are written as