Riemann-Finsler geometry and Lorentz-violating kinematics

Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a 1-form coefficient and has Finsler structure complementary to the Randers structure.


I. INTRODUCTION
The study of Riemann-Finsler geometry, which has roots in Riemann's 1854 Habilitationsvortrag and Finsler's 1918 dissertation [1,2], is now an established mathematical field with a variety of physical applications. A well-known example intimately linked to physics is Randers geometry [3], in which the Riemann metric at each point is augmented by a contribution from a 1-form. For instance, a pseudo-Randers metric on (3+1)-dimensional spacetime can be identified with the effective metric experienced by a relativistic charged massive particle minimally coupled to a background electromagnetic 1-form potential.
The present work concerns the relationship between a large class of Riemann-Finsler geometries and theories with explicit Lorentz violation. Tiny Lorentz violation offers a promising prospect for experimental detection of new physics from the Planck scale and could arise in an underlying unified theory such as strings [4]. At attainable energies, effective field theory provides a useful tool for describing observable signals of Lorentz and CPT violation [5,6], with explicit Lorentz violation characterized by background coefficients.
However, explicit Lorentz violation is generically incompatible with the Bianchi identities of pseudo-Riemann geometry [7] and so presents an obstacle to recovering the usual geometry of General Relativity. This problem can be avoided via spontaneous Lorentz breaking as, for example, in cardinal gravity [8]. An alternative might be to subsume the usual Riemann geometry into a more general geometrical structure. Here, this is taken to be Riemann-Finsler geometry, and one method is provided to connect it with Lorentz-violating effective field theories. The notion of distance in Riemann-Finsler spaces and pseudo-Riemann-Finsler spacetimes is controlled by additional quantities beyond the Riemann metric (for textbook treatments see, e.g., Refs. [9][10][11][12][13]). Intuitively, the role of these quantities can be played by the background coefficients for explicit Lorentz violation.
The comprehensive realistic effective field theory with Lorentz violation that incorporates both the Standard Model and General Relativity is known as the Standard-Model Extension (SME) [7,14]. To relate the SME to pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces, this work adopts as a starting point the single-fermion renormalizable restriction of the SME in Minkowski spacetime, which is a comparatively simple quantum field theory with explicit Lorentz violation. In the presence of fermion self-interactions, a connection between SME coefficients and pseudo-Riemann-Finsler geometry has been pro-posed by Bogoslovsky [15]. Here, attention is focused on a free SME fermion, which has a wave packet propagating with a dispersion relation modified by Lorentz violation. The dispersion relation is a quartic in the plane-wave 4-momentum p µ and is exactly known [16].
Some of its properties have been discussed by Lehnert [17] and by Altschul and Colladay [18]. It can be associated with an action for a relativistic point particle [19], which encodes in a classical description part of the key physical content of the free quantum field theory while avoiding some of the complications associated with spin.
The basic observation underlying the present work is that the SME-based Lorentzviolating classical lagrangian plays the role of a pseudo-Finsler structure, which leads to some interesting geometrical consequences. Pseudo-Finsler structures play a role in the context of modified particle dispersion relations [20]. They are also relevant for modified photon dispersion relations [21], for which the general photon dispersion relation arising from operators of arbitrary dimension is known [22]. A substantial recent literature links Lorentz violation with pseudo-Riemann-Finsler geometries in the contexts of spacetime, gravity, and field theory [23][24][25][26][27][28][29][30][31][32]. Note, however, that at present no compelling experimental evidence exists for Lorentz violation in nature [33], although SME-based models provide simple explanations for certain unconfirmed experimental results including anomalous neutrino oscillations [34] and anomalous meson oscillations [35].
Interpreting the classical SME lagrangian as a pseudo-Finsler structure implies that the Lorentz-violating trajectories of the relativistic particles are governed by pseudo-Riemann-Finsler metrics and hence define pseudo-Riemann-Finsler geometries in (3+1) dimensions.
Corresponding Riemann-Finsler geometries can be generated by Wick rotation or restriction to spatial submanifolds. This construction can be extended to arbitrary dimensional curved spaces in a straightforward way. Following a general discussion of these ideas, this work presents some basic results for a particular Riemann-Finsler space that is constructed using a Riemann metric and a 1-form but differs from Randers space for dimensions n ≥ 3. Its Finsler structure is complementary to the Randers structure in a certain sense described below.
The conventions used here are as follows. Coordinates for (n+1)-dimensional pseudo-Riemann-Finsler spacetime are denoted x µ , µ = 0, 1, . . . , n. The velocities u µ along a curve with path parameter λ are u µ = dx µ /dλ, and a pseudo-Finsler structure is denoted as L = L(x, u). The Minkowski metric in (n+1) dimensions is defined to have positive signature for n > 2. Index raisings or lowerings and contractions are performed with the pseudo-Riemann metric r µν and its inverse r µν ; for example, u µ = r µν u ν and u 2 = u µ r µν u ν . The pseudo-Riemann-Finsler metric is g µν with inverse g µν . To match conventions in mathematics (see, e.g., Ref. [9]), coordinates for n-dimensional Riemann-Finsler space are denoted x j , j = 1, . . . , n, the velocities are y j = dx j /dλ, and a Finsler structure is denoted as F = F (x, y).
The Riemann metric is r jk with inverse r jk , while the Riemann-Finsler metric is g jk with inverse g jk . Index raisings or lowerings and contractions are performed with the Riemann metric; for example, y j = r jk y k , y 2 = y j r jk y k . The norm y of y j is y = y 2 . Partial derivatives with respect to y j are denoted by subscripts; for example, ∂F/∂y j = F y j .

II. SME-BASED FINSLER STRUCTURES
The Lagrange density for the renormalizable single-fermion restriction of the minimal SME in (3+1)-dimensional Minkowski spacetime includes all operators quadratic in the fermion field having mass dimensions three and four [14]. Each Lorentz-violating operator is contracted with a controlling coefficient, so the physics is coordinate independent. The coefficients of mass dimension one are conventionally denoted as a µ , b µ , H µν , while the dimensionless ones are c µν , d µν , e µ , f µ , and g λµν . A constructive procedure has recently been given for generating the classical relativistic point-particle lagrangian L from which the SME plane-wave dispersion relation can be derived [19]. The complete action is involved, but various special cases are tractable and some limits of L have been explicitly obtained.
One example of relevance in what follows is the limiting situation of vanishing coefficients c µν , d µν , e µ , f µ , g λµν , and H µν , for which the particle lagrangian is The two possible signs for the last term reflect the presence of two particle spin projections in the quantum field theory.
This work extends the full L and its limits to include minimal coupling to a background gravitational field given by a pseudo-Riemann metric r µν in (n+1) dimensions and to allow position dependence of all coefficients. For example, this extension affects the contractions in Eq. (1) and permits L ab to describe the motion of a relativistic particle on an (n+1)dimensional curved spacetime manifold in the presence of varying background coefficients a µ (x) and b µ (x). The extended L could be obtained via a suitable Foldy-Wouthuysen transformation [36] of the gravitationally coupled Dirac equation. Position dependence of SME coefficients appears naturally in the gravity context [7,[37][38][39], and Seifert has shown this can result from topologically nontrivial field configurations [40]. Note that comparatively large a µ coefficients could have escaped experimental detection to date [41]. In what follows the fermion mass m is set to unity, m = 1, for simplicity. sufficiently small that the nonsingular pseudo-Riemann metric dominates the background fields, so the effective metric g µν := ∂ 2 (L 2 /2)/∂u µ ∂u ν felt by the relativistic particle is nonsingular. Inspection reveals that the above results are the defining properties of a local pseudo-Riemann-Finsler spacetime with pseudo-Finsler structure L defined on T M\S (for a textbook discussion see, e.g., Ref. [12]). This pseudo-Riemann-Finsler spacetime therefore underlies the motion of a relativistic classical particle experiencing general Lorentz violation.
The explicit forms of L and of many of its limits are involved and unknown in detail, so the corresponding pseudo-Finsler geometries may be challenging to explore. However, a variety of special pseudo-Riemann-Finsler spacetimes can be obtained by taking limits in which certain coefficients vanish. One simple example is the structure L a := L ab | b→0 obtained as the limit b µ → 0 of Eq. (1), which takes the familiar pseudo-Randers form.
The 'face' limit L acef of L with coefficients a µ , c µν , e µ , and f µ also yields a pseudo-Randers structure.
An interesting class of comparatively simple limits of L consists of 'bipartite' pseudo-Finsler structures taking the generic form where the symmetric quantity s µν satisfies u µ s µν u ν ≤ 0. Several of the more tractable limits of L fall into this class. For example, the b structure L b := L ab | a→0 is bipartite, with The example with H µν given in Eq. (15) of Ref. [19] is also a bipartite structure L H . The choice s µν = −a µ a ν yields the two structures L |a| = − √ −u 2 ∓ |a · u| jointly spanning L a , so L ab has a tripartite form in this sense. It is likely that other bipartite limits of L remain to be discovered. Note that the quantity s µν is reminiscent of a secondary metric but may lack an inverse. For instance, s µν for the b structure L b is noninvertible because it has a zero eigenvalue for the eigenvector b µ . As a result, the subset S for this example consists of the extended slit u µ = κb µ for real κ. In the more general case, S 1 includes all u µ that are nonzero eigenvectors of s µν with zero eigenvalues.
When S = S 0 , which holds for the face structure and other pseudo-Randers limits of L, global pseudo-Finsler spacetimes arise. However, when S 1 is nonempty, the geometry is only local. An interesting open question is whether it is possible to resolve the geometry at S 1 to yield global pseudo-Finsler spacetimes. The fourth-order polynomial dispersion relation for the wave-packet 4-momentum p µ can be viewed as an algebraic variety R(p µ ).
The structure L is constructed using R, the requirement of homogeneity, and the intrinsic derivatives of L defining the 4-velocity u µ [19]. This construction generates five equations that combine to yield a polynomial P(L), which has physical roots yielding the local pseudo-Finsler structures L of interest and spurious roots corresponding to the set S 1 . The latter arise from the singularities of R, which according to the implicit function theorem are determined by the p µ derivatives of R. Resolving the geometry at S 1 therefore corresponds to resolving the singularities of the variety R.
At the level of quantum field theory, the singularities of R reflect degeneracy of the wavepacket energies, which can be resolved using spin. For example, in the Lorentz-invariant case the two spin projections for the particle modes are degenerate for all momenta because the variety R = (p 2 + 1) 2 is singular everywhere, but considering only one spin projection at a time yields the nonsingular variety R = p 2 + 1 instead. This spin-based resolution also underlies the global nature of the pseudo-Randers face geometry. When any of b µ , d µν , g λµν , or H µν is nonzero, R is generically a nontrivial quartic. Singularities occur for a subset of momenta at which the two spin projections are degenerate in energy, and these generate the set S 1 in T M. It is therefore plausible that the pseudo-Finsler geometry at S 1 for the general structure L could be resolved by the introduction of a spin variable. Note that a resolution for the corresponding variety R is guaranteed by, for example, the Hironaka theorem [42].
The geometry at S 1 might therefore alternatively be resolved using a standard technique for singularities of algebraic varieties such as blowing up. The above comments suggest the existence of a global pseudo-Finsler geometry associated with the general structure L is a reasonable conjecture, but its proof remains open.
An interesting class of SME-based Finsler structures can be obtained from the pseudo-Finsler ones by restriction to the spatial submanifold or by Wick rotation. The full Finsler structure F (x, y) obtained in this way retains much of the complexity of L(x, u), but some limits are amenable to explicit investigation. One comparatively simple example arises by converting to n euclidean dimensions the pseudo-Finsler ab structure L ab given in Eq. (1), yielding the Finsler ab structure Note that F a := F ab | b→0 generates the usual Randers geometry. The Finsler face structure F acef corresponding to L acef also generates a Randers space.
Applied to the bipartite pseudo-Finsler structure (2), the above procedure generates a bipartite Finsler structure F s given by

III. THE b STRUCTURE
As an explicit example of an SME-based Riemann-Finsler geometry, consider the b structure F b := F ab | a→0 obtained as a limit of Eq. (4). In fact this specifies two Finsler structures, one for each choice of the ± sign, originating in the two spin degrees of freedom in the SME.
For notational simplicity, it is convenient to write the ab structure F ab (x, y) as where ρ := y 2 , α := a · y, β : For a Riemann space with metric r jk , the Finsler structure is F r = ρ, for Randers a space it is F a = ρ + α, and for b space it is F b = ρ + β. The dependence on x j arises through r jk (x), a j (x), and b j (x). Constancy of the metric and coefficients would imply that the canonical momentum is conserved and that the Riemann-Finsler space is locally Minkowski, which parallels the treatment of Ref. [19]. The notational pairings (r, ρ), (a, α), (b, β) here match the standard literature on Lorentz violation; the conventional mathematics notation is recovered by the replacements (r, ρ) → (a, α), (a, α) → (b, β), (b, β) → (⋆, ⋆).
A first observation is that the b structure F b offers a kind of complement to the Randers a structure F a . Given a nonzero 1-form a j , F a can be constructed by adding to F r the parallel projection of the velocity y j along a j , where y = a · y/ a is the a-normalized parallel projection. By splitting the Randers structure into two pieces, the last term can be written α = ± a y 2 . However, given another nonzero 1-form b j , a complementary structure can be obtained by combining F r with the perpendicular projection of the velocity y j along b j instead. This gives the b structure F b , where y j ⊥ = y j − (b · y)b j /b 2 is the b-normalized perpendicular projection. One natural formulation of this perpendicular projection uses the Gram determinant or gramian.
Given two vectors b j , y j and the Riemann metric r jk , the gramian gram(b, u) is given by gram(b, u) = b 2 y 2 − (b · y) 2 , so the b structure can be written In euclidean space, the gramian of two vectors represents the square of the area of the parallelogram formed by the vectors. The Cauchy-Schwarz inequality implies the gramian is a nonnegative quantity, gram(b, y) ≥ 0, confirming that the square-root term in Eq. (9) is real, as required.
For low dimensions, the b structure F b generates known geometries. When n = 1 the gramian vanishes, gram(b, y) = 0, so the Riemann-Finsler space reduces to a Riemann curve.
When n = 2, the parallel and perpendicular projections y j and y j ⊥ span vector spaces of the same dimension. This enables the introduction of a vector v j via the identification y j its two signs therefore maps to the two pieces of a Randers structure. An equivalent way to see this result is to identify the corresponding Randers 1-form a j with the dual of b j , a j = ǫ jk b k , which is perpendicular to b j . The contribution to F b from the perpendicular projection to b j is equivalent to a contribution to F b from the parallel projection to the dual ǫ jk b k , so for n = 2 the b structure generates a Randers geometry. However, the duality equivalence is unavailable in higher dimensions, so for n ≥ 3 the b space is expected to be neither a Riemann nor a Randers geometry. This result is proved in the next section by direct construction of the Matsumoto torsion.
To be a Finsler structure, F b must satisfy certain basic criteria [9]. One is nonnegativity on T M. For the positive sign in Eq. (8), F b is always nonnegative because ρ ≥ 0 and β ≥ 0.
For the negative sign, F b ≥ 0 iff b < 1. This can be checked as follows. If b j is zero then F b = ρ, which is nonnegative. If y j is zero then F b = 0, which is also nonnegative. If both b j and y j are nonzero, define the nonzero real angle cos θ = (b · y)/( b y ). Then for the same reason. The nonnegativity of F b is therefore assured for both signs in F b when b < 1. This condition is assumed in what follows.
Another criterion for a Finsler structure is C ∞ regularity. Since the Riemann metric is positive definite, the component ρ of F b is C ∞ on the usual slit bundle T M\S 0 for which y j = 0. In contrast, the component β vanishes on the slit extension S 1 for which y j ⊥ = 0 and y j = 0, so on T M\S 0 only C 0 continuity of F b is assured in the general case. However, β is positive definite outside the set S = S 0 + S 1 for which gram(b, y) = 0. This implies that F b is C ∞ on T M\S. Where necessary, the restriction of F b to T M\S is assumed in what follows. As discussed in the previous section, when S 1 is nonempty this restriction implies the geometry associated with F b is typically singular on S and hence is y local.
Exceptions are the case n = 1, which generates a Riemann curve and is y global, and the case n = 2, which can be mapped to a y-global Randers geometry as described above.  [44]. In projection, the degenerate energies appear as cusps on the energy-momentum plot [16]. Resolving these singularities and generating the corresponding y-global Riemann-Finsler geometries for F b is an interesting open problem.
The two remaining criteria for F b to be a Finsler structure are positive homogeneity of degree one in y j , F b (x, κy) = κF b (x, y) for κ > 0, and positive definiteness of the symmetric Finsler metric g jk := (F b 2 /2) y j y k associated with F b . The former holds by inspection, but to demonstrate the latter some explicit results are useful.
A short calculation shows g jk can be expressed compactly as where B := β + b 2 ρ and where κ j represents the convenient combination involving the y j derivatives of ρ and β. The latter are ρ y j = y j /ρ and β y j = s jk y k /β, where s jk = b 2 r jk −b j b k for the b structure. One way to investigate positive definiteness of g jk is via the determinant det g. For n = 1 the determinant is det g = det r, matching expectations for a Riemann curve. For arbitrary n ≥ 2, some calculation gives the pleasantly simple formula For n = 2, the first factor reduces to the identity and the remaining factors match the wellknown determinant of the Randers metric, as might be expected from the n = 2 mapping between the a and b structures. Also, in the limit b → 0 the formula produces det g = det r, as required.
Given the result (12), a standard argument [9] verifies positive definiteness of g jk . Introducing F ǫb = ρ + ǫβ, it follows from (12) that det g ǫ is positive and so g ǫ jk has no vanishing eigenvalues. At ǫ = 0 the eigenvalues of g ǫ jk are those of r jk and hence are all positive, while as ǫ increases to 1 no eigenvalue can change sign because none vanishes. This ensures positive definiteness and also invertibility of g jk .

IV. SOME PROPERTIES OF b SPACE
For any n > 1, the Finsler b space with metric (10) cannot be a Riemann geometry. One way to see this is to construct the the Cartan torsion C jkl := (g jk ) y l /2, which measures the non-euclidean nature of a Finsler structure viewed as a Minkowski norm on any tangent space T M x . For b space, the Cartan torsion takes the simple form where the sum is over cyclic permutations of j, k, l. Here, κ jk is the combination of the second y j derivatives of ρ and β, which are ρ y j y k = (r jk − ρ y j ρ y k )/ρ and β y j y k = (s jk − β y j β y k )/β. Note that β y j y k vanishes for n = 2. Since C jkl is nonzero, Diecke's theorem [45] implies that F b is non-euclidean as a Minkowski norm, so b space cannot be a Riemann geometry. The mean Cartan torsion I j := (ln(det g)) y j /2 is found to be which is also nonvanishing for n > 1.
For any n > 2, the b space also differs from Randers space. This can be seen by calculating the Matsumoto torsion M jkl , which separates Randers and non-Randers metrics when n > 2.
This torsion is defined as M jkl := C jkl − 1 (n+1) (jkl) I j h kl , where the angular metric h jk is h jk := g jk − F y j F y k . For F b , the Matsumoto torsion can be written as Since this is nonzero for n > 2, the Matsumoto-Hōjō theorem [46] shows that the b structure F b cannot correspond to a Randers structure for n > 2, despite being constructed from a 1-form b j and despite its comparative simplicity and calculability.
One way to explore features of a Riemann-Finsler space is to study its geodesics (for a textbook treatment see, e.g., Ref. [47]). The Finsler geodesics for b space are solutions of the equation where the spray coefficients G j := g jm Γ mkl y k y l are defined in terms of the Christoffel symbol Γ jkl for the Riemann-Finsler metric g jk , The geodesics solving Eq. (17) are valid for any choice of diffeomorphism gauge or, equivalently, for any choice of geodesic speed.
The spray coefficients G j for b space can be calculated explicitly by first deriving G j := Γ jkl y k y l and then contracting with the inverse Riemann-Finsler metric to get G j := g jk G k .
Some calculation reveals the compact result Here, a lower index m contracted with r mk ρ y k is denoted by a bullet •, with contractions understood to be external to any derivatives. Also, the Christoffel symbol γ jkl for the Riemann metric r jk takes the usual form while the symbol γ jkl is defined analogously as using the form of s jk for F b .
To proceed, the inverse Riemann-Finsler metric is required. This can be determined to be where Contracting with G j gives the spray coefficients G j as where D j is the Riemann covariant derivative and contractions are understood to be external to derivatives, as before. It follows that if D j b k = 0 then γ j •• = 0 and so also γ ••• = γ ••• = 0. The Finsler spray coefficients (24) therefore become G j = ρ 2 γ j •• = γ j kl y k y l , which are the usual Riemann spray coefficients for the metric r jk . For constant Finsler speed or, equivalently, the gauge choice F b = 1 fixing the curve parameter λ to a definite time λ = t, the geodesics then become solutions of the usual Riemann geodesic equationẍ j + γ j klẋ kẋl = 0.
Remarkably, this result shows an r-parallel b j coefficient has no effect on the motion. Intuitively, local conditions along the geodesics appear uniform, so local geodesic observations cannot unambiguously detect nonzero b j . This suggests a suitable transformation or coordinate redefinition could be found to remove a parallel b j from F b , in analogy to the removal of certain unphysical coefficients in suitable limits of the SME [7,14,19,22,41,48,49]. For example, at least one component of the Randers coefficient a µ can be removed by a phase redefinition of the fermion [7]. At the relativistic quantum level, the b µ coefficients cannot generically be removed due to the entanglement of the spin components, which is absent at the classical level away from the set S. However, for constant b µ in Minkowski spacetime, a chiral phase transformation can eliminate b µ in the massless limit [14], and Lehnert has exhibited a nonlocal field redefinition that simultaneously removes b µ from both spin components [49].
The expression (24) for the spray coefficients permits in principle the direct derivation of various geometric quantities characterizing b space, including the nonlinear connection N j k := (G j ) y j /2, the Berwald connection B Γ j kl := (G j ) y j y k /2, and the Berwald h-v curvature B P k j lm := −F b (G j ) y j y k y l /2. The Cartan, Chern (Rund), and Hashiguchi connections and the various associated curvatures and torsions can also in principle be obtained. However, the explicit formulae appear lengthy and are omitted here.
One result of interest pertaining to Berwald curvature is that any b space having b j parallel with respect to r jk is a Berwald space. Since D j b k = 0 implies γ j •• = 0 and hence G j = γ j kl y k y l , and since γ j kl is independent of y j , three y derivatives of G j vanish. The Berwald h-v curvature is therefore zero, and so any r-parallel b space is a Berwald space.
The converse statement that any Berwald b space is necessarily an r-parallel space appears plausible but is left open here.
Note that the analogous results for Randers space, which in present terminology state that any a space is a Berwald space iff it is an r-parallel space, are well established [50][51][52][53].
It is natural to conjecture that any SME-based Riemann-Finsler space is a Berwald space iff it has r-parallel coefficients for Lorentz violation. This attractive conjecture is amenable to direct investigation in various special cases, while a general proof is likely to offer valuable

insights.
Another open challenge is to identify physical interpretations of SME-based pseudo-Riemann-Finsler and Riemann-Finsler structures, including the b structure. Examples for the a structure are well known. As mentioned in the introduction, the dynamics of a relativistic charged particle moving in an electromagnetic potential is governed by a pseudo-Randers a structure L a , while the Randers a structure F a has applications in several physical contexts including Zermelo navigation, optical metrics, and magnetic flow (see, e.g., Refs. [54][55][56][57][58]). Physical applications of the b structure would also be interesting from both physical and mathematical perspectives. By construction, the SME-based pseudo-Riemann-Finsler b structure L b controls the motion of a relativistic particle in the presence of Lorentz violation involving the b µ coefficients. However, identifying an application of the Riemann-Finsler b structure F b appears challenging.
Some insight can be obtained by converting the variational problem associated with F b into a form with similarities to the Randers structure F a . This can be accomplished by introducing two additional coordinate variables, a 2-form Σ jk = −Σ kj and a scalar κ, and defining (F b ) Σκ := ρ + b j Σ jk y k + κρ( 1 2 trΣ 2 + 1).
Note that the factor of ρ in the last term is included to maintain explicit homogeneity of degree one in y j but has no essential effect on the argument to follow. Note also that the conjugate velocities for Σ jk and κ are absent from (F b ) Σκ , so an effective metric defined in the enlarged space with coordinates (x j , Σ jk , κ) would have zero eigenvalues.
In the variational problem (26), the scalar κ plays the role of a Lagrange multiplier, enforcing the norm constraint Σ jk Σ jk = 2. Variation with respect to the 2-form Σ jk imposes the condition b j y k − y j b k = 2κρΣ jk . These equations can be solved to yield κ = ±β/2ρ and Σ jk = ±(b j y k − b k y j )/β, which in turn can be used to show that the canonical momentum p j associated with x j in F b coincides on shell with that in (F b ) Σκ , i.e., p j := ∂F b /∂y j = ∂(F b ) Σκ /∂y j . It follows that (F b ) Σκ and F b have the same geodesics. The two signs in The expression (26) reveals that for the b structure the combination b k Σ kj plays a role analogous in certain respects to that of the Randers a j coefficient. Since Σ jk is a dynamical variable, this suggests b space can be viewed in terms of a Randers space with a dynamical coefficient a j . Shen [54] has shown that the usual Randers geodesics can be identified with solutions to the Zermelo problem of navigation control in an external wind related to the coefficient a j (for a detailed exposition, see the treatment by Bao and Robles [56]). The dynamical coefficient b k Σ jk therefore suggests a related interpretation for b space in which the effect of the external flow b j is adjustable, in analogy to the change of effective wind direction arising from the combination of a boat's sail and keel. A direct application to the Zermelo problem falls short because in the Randers case the external flow is related not only to the Zermelo wind but also to the Riemann metric of the navigation space, whereas the term ρ in the expression (26) is independent of Σ jk . However, an interpretation of F b along these lines may be achievable for a system described by the more general theory of optimal control.
Another approach is to seek a physical system in which the notion of distance is intrinsically quartic rather than quadratic. In the optical-metric interpretation, for example, the Randers structure F a generates geodesics matching the spatial trajectories of null geodesics in a stationary spacetime, which are determined by a quadratic spacetime interval ds 2 = 0 [57]. In contrast, geodesics of the b structure F b match geodesics defined by a null quartic interval ds 4 = 0 in a certain class of spacetimes. The quartic nature of b space is directly reflected in its close ties to the motion of a massive Dirac fermion, which for nonzero b µ generically has four distinct modes corresponding to the two spin degrees of freedom for particles and antiparticles, whereas the spin-independent Randers case involves only two distinct modes.