Classical kinematics and Finsler structures for nonminimal Lorentz-violating fermions

In the current paper the Lagrangian of a classical, relativistic point particle is obtained whose conjugate momentum satisfies the dispersion relation of a quantum wave packet that is subject to Lorentz violation based on a particular coefficient of the nonminimal Standard-Model Extension (SME). The properties of this Lagrangian are analyzed and two corresponding Finsler structures are obtained. One structure describes a scaled Euclidean geometry, whereas the other is neither a Riemann nor a Randers or Kropina structure. The results of the article provide some initial understanding of classical Lagrangians of the nonminimal SME fermion sector.


I. INTRODUCTION
In his Ph.D. thesis Finsler investigated a geometry whose path length functional was a generalized version of the Riemannian one. In this case the geometric properties of manifolds such as curvature not only depend on the point chosen on the manifold but they are also functions of the angle, which a chosen line element encloses with a given direction in the tangent space of the manifold [1]. Subsequently these types of spaces were called Finsler spaces by Cartan [2,3]. According to Chern [4] it should be avoided saying that Finsler spaces are a generalization of Riemannian ones. Instead, it is better to denote them as Riemannian spaces without the quadratic restriction.
The monographs [5,6] deliver a mathematical introduction of Finsler geometry including various applications. A Finsler space is, indeed, not a point space but a set of line elements. Each is endowed with an underlying Riemannian metric [7], which determines the vector magnitudes and angles between vectors. Besides, a real-valued function on its tangent bundle is introduced, which has certain properties and is often denoted as a Finsler structure. One basic example for a Finsler structure is provided by the time that a salesman needs to travel between different locations on a hillside (see p. 46 in [5], [8]). The solution of the Zermelo navigation problem, which asks the question of finding a geodesic between two points on the surface of the Earth in the presence of wind, leads to a further example of a Finsler structure.
One essential application of Finsler geometry lies in the field of Lorentz symmetry violation, which was initiated by the seminal articles [9][10][11][12]. Since the development of the (minimal) Standard-Model Extension (SME) [13] the investigation of Lorentz violation has become more and more prominent. The SME is a framework of all power-counting renormalizable, Lorentz-violating operators compatible with the Standard Model of elementary particle physics. The minimal SME was extended by the nonminimal SME [14][15][16], which comprises all Lorentz-violating terms with arbitrary operator dimension.
In this context the interest lies in finding a correspondence between the dispersion relation of a quantum wave packet, which follows from the underlying, Lorentz-violating field theory and the kinematics of a classical, relativistic point particle. The classical Lagrangians obtained are then closely linked to Finsler spaces and they help to understand the classical limit of the SME. Furthermore once a Finsler geometry is known for a special set of coefficients, it can be used to describe Lorentz violation in curved backgrounds. In [17,18] classical Lagrangians were derived for certain sets of Lorentz-violating coefficients of the minimal SME fermion sector. In [18][19][20] these Lagrangians were promoted to Finsler structures and their properties were discussed.
In the current article the classical Lagrangian will be obtained for a framework based on a particular Lorentz-violating coefficient of the nonminimal fermion sector. The properties of this Lagrangian will be investigated with the result that it can be promoted to two different Finsler structures.
In general, a playground for Finsler geometry in physics is the investigation of modifications of relativity. One of the first applications was delivered by Randers in [21]. The Finsler structure introduced by him carries his name and is of great importance in science. For example, the structure being a solution of the Zermelo navigation problem is of Randers type. Questions of spacetime causality in relation to Finsler structures were addressed in [22,23] where Randers structures play an essential role here as well. In [24] a Randers structure is used to determine speed limits in quantum information processing.
Further applications of Finsler geometry include but are not restricted to geometrical optics in anisotropic media [25], electron optics under the influence of magnetic fields, thermodynamics, and biology (see [5] for the latter topics), psychometry [8], dynamical systems [26,27], and imaging [28].
Note that Finsler spacetimes have recently been examined in the literature more profoundly. Due to the pseudo-Riemannian signature, the definition of Finsler spacetimes is more involved than that of Finsler geometries with a Riemannian signature. In [29][30][31][32][33] Finsler spacetimes are constructed such that they have a light cone structure and allow for the notion of timelike and lightlike vectors. Implications of a Finsler spacetime geometry on a scalar quantum field theory were investigated in [34,35] and references therein. The concept of Finsler spacetimes is also applied, e.g., in the context of very special relativity [36]. In a certain sense, Finsler spacetimes generalize Lorentz invariance instead of violating it [32].
The paper is organized as follows. Sections II, III give brief introductions to the nonminimal SME fermion sector and tell us how to obtain the classical point-particle Lagrangian from the fermion dispersion relation. In Sec. IV the classical Lagrangian is derived for the sector considered plus its characteristics are investigated. Section V briefly reviews the mathematics of Finsler structures and demonstrates how such structures can be obtained from the classical Lagrangian computed. Finally the results of the paper are summarized in Sec. VI. Calculational details are relegated to the appendix and natural units with = c = 1 are used throughout the article.

II. FERMION SECTOR OF THE NONMINIMAL STANDARD-MODEL EXTENSION
The SME is a collection of all Lorentz-violating operators of Standard Model fields that are gauge-invariant with respect to SU (3) c × SU (2) L × U (1) Y . The minimal sector comprises all power-counting renormalizable terms, whereas the nonminimal sector includes all contributions up to arbitrary operator dimension. In [16] the operators of the nonminimal SME fermion sector are classified according to their transformation properties under the improper Lorentz transformations P, T, and charge conjugation C plus they are discussed extensively. The action of the nonminimal Lorentz-violating fermion sector reads Here ψ is the standard Dirac field, ψ = ψ † γ 0 the Dirac conjugate field, m ψ is the fermion mass, and 1 4 the unit matrix in spinor space. The gamma matrices γ µ for µ = 1 . . . 4 are standard and satisfy the Clifford algebra {γ µ , γ ν } = 2η µν 1 4 with the flat Minkowski metric η µν with signature (+, −, −, −). The quantity Q comprises any possible Lorentz-violating operator of the fermion sector.
In the recent article [37] certain quantum field theoretic properties of some families of nonminimal operators were investigated. One of the sets of coefficients considered was the dimension-5 part of the Lorentz scalar m, i.e., m = m (5)µν p µ p ν . These coefficients are CPT-even and supposedly the simplest of higher dimension. The considerations within the current paper are further restricted. We assume that all coefficients vanish except of m (5)00 . Then the theory to be considered is characterized by the action of Eq. (2.1) with Q = − m1 4 = −m (5)00 p 2 0 1 4 . The modified off-shell dispersion law reads which is quartic in p 0 . The coefficient m (5)00 has mass dimension −1, which gives the product m (5)00 p 2 0 the suitable mass dimension 1, such that it can be added to the fermion mass m ψ . The solutions of Eq. (2.2) with respect to p 0 are the modified dispersion relations of an on-shell fermion affected by the single coefficient m (5)00 . There are two dispersion relations that are perturbed versions of the standard dispersion law p 0 = p 2 + m 2 ψ with the spatial momentum p. These are stated in [37]. Besides, there are two spurious dispersion laws that do not correspond to the standard limit for a vanishing Lorentz-violating coefficient.

III. OBTAINING CLASSICAL LAGRANGIANS FROM DISPERSION RELATIONS
In general, the field equations of the SME lead to modified particle dispersion relations p 0 = p 0 (p i ) with the particle energy p 0 and the spatial momentum components p i ; see, for example, Eq. (2.2) for the nonminimal fermion sector considered. The dispersion relations are necessary conditions for the free field equations to have nontrivial plane wave solutions. By introducing appropriate smearing functions these plane waves can be used to construct quantum wave packets.
Given a dispersion relation p 0 = p 0 (p i ), which is modified by a Lorentz-violating background field, a Finsler structure can be constructed as follows. Consider a classical, relativistic point particle at the spacetime point x ≡ (x 0 , x i ) with a four-velocity u ≡ (u 0 , u i ) (i = 1 . . . 3) whose kinematics is described by a Lagrangian L = L(x, u). The goal is to find a Lagrangian such that its canonical momentum p µ = −∂L(x, u)/∂u µ obeys the dispersion relation p 0 = p 0 (p i ) of the quantum wave packet. Hence one looks for a correspondence between the dispersion relation, which is a quantum field theoretic result, and the Lagrangian of a classical point particle. Note that contrary to most other contexts in field theory the canonical momentum is defined with a minus sign to ensure the kinetic energy in the nonrelativistic limit is nonnegative.
The classical point particle travels along a well-defined trajectory. All physical results, especially the action, should not depend on the choice of its parameterization. This is granted as long as L is positive homogeneous of degree 1 in u: Since the Lagrangian has this property, according to a theorem by Euler [6] it can be written as follows: The latter equation is very helpful. If the momentum can be determined as a function of the velocity, it leads us to the Lagrangian immediately. For most purposes the group velocity of a quantum wave packet can be interpreted as its physical velocity. To have the correspondence to the classical point particle the spatial velocity components of the point particle shall correspond to the group velocity components of the wave packet: The off-shell dispersion relation (for example, Eq. (2.2)) and Eqs. (3.2), (3.3) give five equations of the nine unknowns p µ , u µ , and L. Four of these equations must be used to eliminate p µ in favor of u µ and L. The procedure employed in most cases considered in the literature so far was to use L = −p µ u µ to eliminate p µ , which then led to a polynomial of L. The classical Lagrangian is given by one of the zeros of the latter polynomial with respect to L. In the next section it will become evident that a different procedure will lead to the goal for the case considered here.

IV. CLASSICAL LAGRANGIAN
Now the interest lies in the Lagrangian for a classical point particle reproducing the dispersion relation of a spin-1/2 fermion underlying Lorentz violation with the single nonzero coefficient m (5)00 given by Eq. (2.2). According to Sec. III the group velocity of a quantum-mechanical wave packet ought to be equal to the three-velocity of the corresponding point particle. The group velocity components ∂p 0 /∂p i can be obtained by implicit differentiation of Eq. (2.2) and solving the resulting equation with respect to ∂p 0 /∂p i . This leads to Using Eq. (4.1) the spatial momentum components can be expressed via p 0 . Inserting these relations in the off-shell dispersion relation of Eq. (2.2), the resulting equation can be solved with respect to p 0 . This procedure leads to the following momentum-velocity correspondence: with the definitions Note that for the minimal Lorentz-violating frameworks considered in [17] it was not possible to determine an analogous momentum-velocity correspondence directly from the dispersion relation. It works here as the theory has been restricted to the isotropic sector. Since taking the absolute value of Q 2 complicates many of the analytical calculations, we will restrict the expression above to |m (5)00 | ≤ 1/(2m ψ ). Then Q 2 is nonnegative and the absolute value bars can be omitted. This is in accordance with considering Lorentz violation as a perturbative effect.
Having a momentum-velocity correspondence right from the start is convenient because now the Lagrangian can be constructed via L = −p µ u µ . The result is cast in the form It is considered as a four-dimensional function of the four-velocity components u µ where m ψ and m (5)00 are taken as parameters. In contrast to the cases investigated in [17] the form of the Lagrangian is far from simple and rather unpleasant, since the original equations involve thirdorder polynomials in p 0 . Solving Eqs. (2.2) and (4.1) with a computer algebra system resulted in expressions involving cubic roots of complex quantities, which are themselves multiplied by complex numbers. These expressions are not manifestly real, which is why the Lagrangian was brought to the manifestly real form of Eq. (4.4) by several manipulations. The latter are only valid for real four-velocity components and parameters. Furthermore the Lorentz-violating coefficient m (5)00 must be sufficiently small. As a cross check, the four-momentum can be computed from the Lagrangian via p µ = −∂L/∂u µ . Using this p µ , Eqs. (2.2), (3.3) can be demonstrated to be valid numerically for certain fourvelocities. An analytic proof is prohibitively difficult to perform due to the complicated structure of the Lagrangian.

A. Properties of the classical Lagrangian
Although the classical Lagrangian given by Eq. (4.4) is rather complicated, it is possible to deduce some of its properties either analytically or numerically.

1) Limit for vanishing Lorentz-violating coefficient:
At first by looking at Eq. (4.4) one may think that the Lagrangian has a pole at m (5)00 = 0. This would indicate a spurious Lagrangian that does not correspond to the standard result for vanishing Lorentz violation. However consider the limit of the term under the square root for vanishing Lorentz-violating parameter: This is the reason why the Lagrangian does not have a pole at m (5)00 = 0. On the contrary, L is regular in this limit and corresponds to the standard case:  It is important to remark that this limit only exists for time-and lightlike u as indicated. Details of how to obtain it can be found in App. A 1. The sign function takes into account that the point-particle velocity u i /u 0 in Eq. (3.3) changes its sign when u 0 changes the sign. Therefore the Lagrangian has a discontinuity on the |u|-axis, i.e., for u 0 = 0.
2) Limit for vanishing velocity components: Equation (4.4) seems to have a pole for both u 0 = 0 and |u| = 0. For this reason these limits shall be investigated. Due to Q 1 (u 0 = 0, |u|) = −8u 6 Q 3 3 and Q 2 (u 0 = 0, |u|) = 2u 2 Q 3 the factor in square brackets after the square root in Eq. (4.4) results in Therefore the Lagrangian does not have a pole for u 0 → 0, but it is not continuous in this limit (see the previous item). As a next step consider |u| = 0, for which Q 1 (u 0 , |u| = 0) = −(u 0 ) 6 and Q 2 (u 0 , |u| = 0) = (u 0 ) 2 . We then obtain for the radicand under the square root of Eq. (4.4): Because of this the Lagrangian does not have a pole in the limit |u| → 0, as well. Furthermore no pole appears for the combined limit u µ → 0 µ .

3) Global sign of the Lagrangian:
Due to the square root in the Lagrangian it is real only for values of the velocity components, fermion mass, and the Lorentz-violating coefficient lying within a domain such that and for the factor behind the square root the following estimate can be obtained: for (u 0 ) 2 ≥ u 2 and m (5)00 ≥ 0. Hence for time-and lightlike four-velocity, u 0 > 0 due to the prefactor, and nonnegative Lorentz-violating coefficient we have that L(u; m ψ , m (5)00 ) ≤ 0. This simple analytical estimate can be refined numerically. The Lagrangian is negative, zero or positive for the four-momentum components lying in certain regimes. Therefore we define the following sets: The values of (u 0 , u) lying in these domains can be determined numerically. For (u 0 , u) ∈ R 1 ∩ R 3 and (u 0 , u) ∈ R 2 ∩ R 4 we have L(u; m ψ , m (5)00 ) ≤ 0, whereas for (u 0 , u) ∈ R 2 ∩ R 3 and (u 0 , u) ∈ R 1 ∩ R 4 it holds that L(u; m ψ , m (5)00 ) ≥ 0. For both cases the equality sign is only valid when the set R 1 is involved. Otherwise the Lagrangian cannot be zero.

4) Symmetries:
The form of the Lagrangian in Eq. (4.4) allows to show that L(u 0 , −u; m ψ , m (5)00 ) = L(u 0 , u; m ψ , m (5)00 ). So it is symmetric with respect to a reflection at the point u = 0. However the second argument of L(u 0 , u; m ψ , m (5)00 ) will always be assumed to be nonnegative.

5) Differentiability:
First of all, the argument inside the inverse trigonometric functions shall be investigated: is ∈ C ∞ . The sine and cosine functions are C ∞ and the square root is C ∞ as long as its argument is larger than zero (see the third item).
Finally, the dimensionless quantity L ψ /m ψ is plotted in Fig. 1. Some of its properties such as the discontinuity for u 0 = 0 are directly visible.

V. FINSLER STRUCTURES AND MANIFOLDS
After understanding the properties of the Lagrangian it shall be promoted to a Finsler structure. In general, a Lagrangian depends on n+1 velocity components u 0 , u i where u i for i = 1 . . . n are the spatial components. The underlying metric of the Lagrangian is called r µν and it is used to lower and raise indices, e.g, u µ = r µν u ν . It is a pseudo-Riemannian metric with signature (+, −, . . . , −), whereas a Finsler structure in n dimensions is characterized by a Riemannian metric with signature (+, +, . . . , +). First, the defining properties of a Finsler structure shall be reviewed.
Consider an n-dimensional manifold M with its tangent bundle T M where x i ∈ M , y i ∈ T M for i = 1 . . . n. The underlying Riemannian metric will be denoted as r ij (x). M is promoted to a Finsler manifold by introducing a function F : T M → [0, ∞) with F = F (x, y) where the following properties hold: 3) F (x, κy) = κF (x, y) for all κ > 0 (positive homogeneity of first degree for y),

4) and the Hessian matrix
is positive definite for y = 0. Some texts, e.g., [5] include the first property, whereas it is omitted in [6]. The function F is called the fundamental function, metric function, Lagrangian [5] or simply a Finsler structure [6] and g ij is named the derived metric, fundamental Finsler tensor or just metric Finsler tensor [5].

A. Construction of a Finsler structure
According to [19] a Finsler structure can be constructed from a Lagrangian by either restricting L(u; m ψ , m (5)00 ) to the spatial domain or by performing a Wick rotation. The fermion mass is then often set to 1 in this procedure. However it will be kept in what follows such that the mass dimensions of the various terms are not spoilt. Pursuing the first possibility, the expansions with (r ij ) = diag (1, 1, 1). The result corresponds to the Finsler structure of the Euclidean threedimensional space being scaled with a dimensionless factor A. This Finsler structure is only defined for nonzero m (5)00 although according to the first item in the previous section the Lagrangian corresponds to the standard result for m (5)00 → 0. However note that this limit only exists for time-and lightlike u: (u 0 ) 2 − u 2 ≥ 0. The divergence in m (5)00 can be explained as follows. Considering m = m (5)00 p 2 0 it is evident that the Lorentz-violating coefficient is directly coupled to the zeroth component of the four-momentum. This translates from the wave packet to the velocity of the classical point particle, which forbids the combined limit u 0 = 0 and m (5)00 = 0.
The Finsler structure of Eq. (5.3) describes a Euclidean geometry where distances between two points are scaled by the factor A in comparison to the conventional Euclidean geometry with the structure F (y) = r ij y i y j . Angles are not affected by the scaling. Note that such a geometry is described by the spatial part of the Friedmann-Lemaître-Robertson-Walker metric (with zero curvature), which has a wide application in cosmological models. In the latter metric there appears a time-dependent scale factor.
The second possibility, i.e., a Wick rotation of the Lagrangian with u 0 = iy 4 fails to produce a Finsler structure. It can be demonstrated numerically that F (y, y 4 ) ≡ L(iy 4 , y; m ψ , m (5)00 )/m ψ does not have a positive definite metric g ij according to Eq. (5.1). 1 The reason why this is the case will be explained as follows. The sets R 1 and R 2 defined in Eqs. (4.10a), (4.10b) separate L(u 0 , u; m ψ , m (5)00 ) into two parts with completely different properties. For the limit of vanishing Lorentz-violating coefficient these sets are given by lim m (5)00 →0 According to Eq. (4.6a) the Lagrangian corresponds to the standard result in the limit of zero m (5)00 if (u 0 ) 2 − u 2 ≥ 0. Hence the Lagrangian describes the physics of a classical, relativistic point particle in the presence of nonminimal Lorentz violation caused by the coefficient m (5)00 only if u ∈ R 1 . Performing a Wick rotation of (u 0 ) 2 − u 2 > 0 would lead to (u 0 ) 2 − u 2 → (iy 4 ) 2 − y 2 = −(y 4 ) 2 − y 2 < 0 and then u ∈ R 2 . However for u lying in the latter domain the Lagrangian is not supposed to describe the physics of the same classical point particle. What is then the meaning of L(u 0 , u; m ψ , m (5)00 ) in that regime? The answer to this question will be examined as follows.
First of all, for simplicity the further analysis will be restricted to L = L(u 0 , |u|; m ψ , m (5)00 ) as a function of (u 0 , |u|) ∈ R 2 due to the isotropy of the Lagrangian. Then we define F (2) (y) ≡ F (2) (y 1 , y 2 ) ≡ 1 m ψ L(y 2 , y 1 ; m ψ , m (5)00 ) , y ≡ (y 1 , y 2 ) ∈ R 2 ∩ R 3 , (5.5a) Here the index "(2)" of F (2) (y) indicates that this is a two-dimensional function of y where m ψ , m (5)00 are considered as parameters. The sets R 2 , R 3 are the two-dimensional restrictions of R 2 , R 3 to (u 0 , |u|). Note that R 2 of Eq. (4.10b) is the set for which the Lagrangian does not describe the physics of a classical point particle moving in a Lorentz-violating background. It can be determined by computing the zeros of h(u 0 , u). Due to the homogeneity of the function h, to obtain the zeros the ansätze u 0 = ±α|u| are made where α is calculated numerically (see Fig. 2 for a certain range of the Lorentz-violating coefficient). As a next step the metric corresponding to F (2) shall be investigated. It is computed according to Eq. (5.1): (5.6) This is an extremely complicated (2 × 2)-matrix, which will not be stated explicitly. The computation is straightforward to be done with a computer algebra system as only derivatives have to be performed. Now the definiteness of this metric shall be checked by calculating its eigenvalues and looking at their signs. Choosing special values for m ψ and the Lorentz-violating coefficient m (5)00 , the eigenvalues are plotted in Fig. 3. One can see that the first eigenvalue is larger than zero for the shown range of y, whereas the second eigenvalue is negative most probably for y / ∈ R 2 . 2 Hence there exist strong indications that the metric is positive definite for y ∈ R 2 .
Furthermore F (2) (y) > 0 for y ∈ R 2 ∩ R 3 and the Lagrangian is C ∞ for y ∈ R 2 ∩ R 3 (cf. Eq. (4.13)) plus it is positive homogeneous of first degree according to Eq. (4.14). Therefore the performed numerical investigations indicate that the Lagrangian L(y 2 , y 1 ; m ψ , m (5)00 ) itself is a two-dimensional Finsler structure without any Wick rotation at all as long as y lies in the domain R 2 ∩ R 3 . That is why it is reasonable to classify it. A first step is to compute the Cartan torsion C ijk , which is given by [38] C ijk ≡ 1 2 Note that some authors define C ijk with an additional prefactor F (see, e.g., [6]). The mean Cartan torsion is defined as with the inverse Finsler metric g ij . According to a theorem by Deicke a Finsler space is a Riemann space if and only if I vanishes [39]. Both C ijk and I can be obtained for Eq. (5.5a). The computation is again straightforward but the result is very lengthy, which is why it will not be given. However it can be demonstrated that I does not vanish for certain numerical parameters. For example, with y 2 = 1/2, y 1 = 1, and m ψ m (5)00 = 1/10 one obtains Hence the Finsler space, which is defined by the Lagrangian for a certain subset of velocities, is definitely not Riemannian. A further important quantity, which helps to classify Finsler spaces, is the Matsumoto torsion: where n is the dimension of the Finsler space to be considered [38]. The Matsumoto-Hōjō theorem tells us that a Finsler space with dimension ≥ 3 is a Randers space if and only if the Matsumoto torsion vanishes [40]. However note that for a two-dimensional Finsler space M ijk = 0, which is why the Matsumoto-Hōjō theorem cannot be applied. This concludes the analysis of the twodimensional Finsler structure of Eq. (5.5a). Finally the following four-dimensional function is considered where y ≡ (y 1 , y 2 , y 3 ): Then analogously to Eq. (5.6) a metric tensor g (4) (y) can be constructed again, which is now a (4 × 4)-matrix. Its eigenvalues behave similarly to the eigenvalues of the two-dimensional Finsler metric g (2) . There are strong numerical indications that all four eigenvalues are positive as long as y ∈ R 2 and therefore the metric is probably positive definite for y lying in this domain. Besides, F (4) (y) > 0 and F (4) (y) ∈ C ∞ for y ∈ R 2 ∩ R 3 plus it is positive homogeneous of first degree. This is what makes F (4) a Finsler structure as long as y ∈ R 2 ∩ R 3 . Also for this case the mean Cartan torsion, Eq.
where b µ are the CPT-odd, pseudovector fermion coefficients of the minimal SME. The latter is given in Eqs. (5), (6) in [19] by setting the CPT-odd vector coefficients a µ equal to zero.

VI. CONCLUSIONS AND OUTLOOK
To summarize, in this paper the Lagrangian was obtained for a classical, relativistic point particle whose conjugate momentum fulfills the modified dispersion relation of a quantum wave packet underlying Lorentz violation caused by an isotropic dimension-5 operator. The speciality is that it was possible to obtain the momentum-velocity correspondence and, therefore, the Lagrangian directly without having to solve a polynomial equation of high degree. This behavior is different from all cases of the minimal SME fermion sector that have been investigated in the literature so far. It is traced back to the isotropic nature of the coefficient considered and it is expected to be possible in general as long as an isotropic Lorentz-violating framework is considered.
Having obtained the Lagrangian its properties were discussed. The Lagrangian can be positive, zero or negative in certain domains of the four-velocity components plus it is C ∞ apart from the region u 0 = 0 that has to be excluded. It corresponds to the standard Lagrangian for a vanishing Lorentz-violating coefficient as long as the four-velocity is time-or lightlike. For this reason it describes the physics of a classical, relativistic point particle only for a certain domain of fourvelocities.
The final goal was to promote the Lagrangian to a Finsler structure and to understand its characteristics. Restricting the Lagrangian to the spatial domain results in a Finsler structure describing a scaled Euclidean geometry. Performing a Wick rotation fails to produce a Finsler structure. However, interestingly the Lagrangian itself is a Finsler structure for the four-velocity components lying in a subset of the domain where it does not describe the physics of a point particle. It was demonstrated that this Finsler structure is neither a Riemann nor a Randers structure.
One last comment shall note a possible connection to [41]. In the latter reference it is shown that certain complex Riemannian manifolds have real slices of all possible signatures. The Lagrangian L(u 0 , u; m ψ , m (5)00 ) considered might provide such an example for Finsler spaces, if it can be embedded into a complex Finsler manifold. Then both the Lagrangian of Eq. (4.4) restricted to the domain R 1 and the Finsler structure of Eq. (5.11) restricted to the domain R 2 ∩ R 3 may be real slices of the complex Finsler manifold with different signatures. This is an interesting open problem for future studies.
The outlook is to obtain and study the classical Lagrangians plus the Finsler structures for alternative sets of nonminimal Lorentz-violating coefficients. A special interest lies in Lagrangians having a simpler form compared to the one considered in the current paper. This will help to better understand the properties of the nonminimal SME.

VII. ACKNOWLEDGMENTS
It is a pleasure to thank V. A. Kostelecký for reading the manuscript and giving helpful suggestions. Furthermore the author is indebted to N. Russell for helpful discussions. This work was performed with financial support from the Deutsche Akademie der Naturforscher Leopoldina within Grant No. LPDS 2012-17.