General Very Special Relativity is Finsler Geometry

We ask whether Cohen and Glashow's Very Special Relativity model for Lorentz violation might be modified, perhaps by quantum corrections, possibly producing a curved spacetime with a cosmological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous deformations, but none of these give rise to non-commutative translations analogous to those of the de Sitter deformation of the Poincar\'e group: spacetime remains flat. Only a 1-parameter family DISIM_b(2) of deformations of SIM(2) is physically acceptable. Since this could arise through quantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be taken into account. The Lorentz-violating point particle action invariant under DISIM_b(2) is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive DISIM_b(2)-invariant wave equations for particles of spins 0, 1/2 and 1. The experimental bound, $|b|<10^{-26}$, raises the question ``Why is the dimensionless constant $b$ so small in Very Special Relativity?''

Local Lorentz and CPT invariance are fundamental assumptions in almost all current physical theories. It is important to test these assumptions experimentally, lest evidence of new physics beyond the standard model be overlooked. Current experimental limits on violations of local Lorentz and CPT invariance are extremely stringent. Thus what is required are novel alternative non-Lorentz invariant theories, capable of circumventing these tight limits. Recently, Cohen and Glashow [1] have made the ingenious proposal that the local laws of physics need not be invariant under the full Lorentz group, generated by M µν , but rather, under a SIM(2) subgroup, whose Lie algebra is generated by (M +i , M ij , M +− = M 03 , ) (with i and j ranging over the values 1 and 2) [2]. This they referred to as Very Special Relativity. Taking the semi-direct product with the translations (P + , P − , P i ) gives an 8-dimensional subgroup of the Poincaré group called ISIM(2) [3].
The great merits of Cohen and Glashow's suggestion are that CPT symmetry is preserved and that ISIM(2) leaves invariant no vector or tensor fields, known as "spurion fields." For example, a spurionic vector field may be thought of as the 4-velocity of the aether [4]. In fact SIM(2) consists of those Lorentz transformations Λ µ ν leaving invariant the null direction n µ = δ µ + , i.e. such that Λ µ ν n ν = λn µ for some λ which depends on Λ µ ν . The generator M +− acts by sending n µ → λn µ . This scaling symmetry implies that one cannot take n µ to define the actual 4-velocity of the aethereal motion, but only its direction, thus rendering the presence of such an aether more difficult to detect. A theory of this kind appears to be compatible with all current experimental limits on violations of Lorentz invariance and spatial isotropy [1,5].
Subsequently, Cohen and Freedman [6], and later Lind-ström and Roček [7], showed that ISIM is compatible with supersymmetry. There are several ways one might try to incorporate gravity. One is where we make local the ISIM(2) algebra [8]. Another is to consider a global space-time and to see if it is compatible with very special relativity ideas. In this case there appear to be difficulties [9]. For example, the maximal subgroup of SO(4, 1), the isometry group of de Sitter spacetime, is SIM(3), which is 7-dimensional [10]. SIM(3) contains SIM(2) as a subgroup, but the stabilizer, or tangent space group, is SO(3), not SIM(2). Alternatively, we recall that that Poincaré group admits a unique deformation into the de Sitter (anti-de Sitter) group [11], with [P µ , P ν ] = 1 3 ΛM µν , where the parameter Λ is the cosmological constant. One may ask whether ISIM(2) admits a similar deformation, such that the translations P µ become non-commutative. If so, the coset of the deformed group divided by SIM(2) (or its deformation) could be thought of as a curved spacetime.
Here we show that there are indeed continuous deformations of ISIM (2), but in all of them the translations remain commutative. Among them is a 1-parameter family of deformations, which we denote by DISIM b (2). For any values of b this is an 8-dimensional subgroup of the 11-dimensional Weyl group, i.e. the semi-direct product of dilatations with the Poincaré group. Subgroups with different values of b are not isomorphic. Interestingly, if one constructs a point-particle action for the deformed groups DISIM b (2), using the methods of non-linear realisations [12], one arrives at Lagrangians of Finsler form, first proposed by Bogoslovsky (see [13] and references therein). Therefore the deformation of very special relativity leads in a natural way to Finsler geometry. In the remainder of this letter we shall outline the derivation of these results, and comment on their physical significance.
Continuous deformations of Lie algebras have been extensively explored, by both mathematicians and physicists, under the rubric of Lie-algebra cohomology [11].
Here we give an elementary account based on the Cartan-Maurer equations, which provides a simple and easily automated scheme for determining the deformations of a given Lie algebra g with structure constants C a b c . We suppose there exists a family of deformed Lie algebras g t parameterised by a continuous variable t, with structure constantŝ (1) We are only interested in deformations which do not arise merely from a (t-dependent) change of basis: Introducing a basis λ a of left-invariant 1-forms of the original algebra, such that Defining D ≡ d + C∧ , the first-order deformation equations (3) may then be written as where the second equation expresses the requirement of non-triviality of the deformation. Because D 2 = 0 as a consequence of the Jacobi identities of the undeformed algebra, dC + C∧ C = 0, the differential D may be regarded as a co-boundary operator acting on g-valued forms. The non-trivial linearised deformations are therefore in 1-1 correspondence with the second cohomology group H 2 (g; g). If a non-trivial linear deformation A is found, the next step is to investigate the Jacobi identities at order t 2 . These read and can be re-expressed in terms of the vector and matrix valued forms as This equation can only be solved if D(A • A) = 0, which implies that A•A must be both D-closed and exact. Thus there is a potential obstruction to finding a deformation at quadratic order: A • A should not have a projection in the third cohomology group H 3 (g; g). There are analogous equations to (7) at higher orders in t. If H 3 (g; g) vanishes, then the equations may be solved at all nonlinear orders. If H 3 (g; g) is non-zero, then the higherorder analogues of (A • A) should have no projections into it.
We start with the ISIM(2) Cartan-Maurer relations where g −1 dg = λ + P + + λ − P − + λ i P i + λ +i M +i + λ +− M +− + λ 12 M 12 . Defining N ≡ M +− and J ≡ M 12 , the corresponding non-trivial Lie brackets are therefore Expanding the vector-valued 2-form A a on a basis of 2forms and solving the resultant linear equations in (5) reveals that there is a 2-parameter family of non-trivial solutions, i.e., H 2 (isim(2); isim (2)) is 2-dimensional. Substituting this linearised solution into the full Jacobi identities, we find that it gives a 2-parameter family of exact Lie algebras of the leading-order form (8), with additional terms as follows: where µ = (+, −, i). Here a and b are arbitrary constant parameters.
As in the undeformed case, the algebra here has the structure of a semi-direct sum of sim(2) and the translations R 4 , i.e. sim(2)⋉R 4 . While the M +i act on the translations as in the undeformed case, the adjoint action of the generators N and J is given by [N, P µ ] = P ν C N ν µ and [J, P µ ] = P ν C J ν µ , where the matrices C N and C J are given respectively by Minkowski spacetime may be thought of as the symmetric space E(3, 1)/SO(3, 1) with SO(3, 1) playing the role of the tangent-space group, and the tangent space being spanned by the translations P µ . In our case we wish to replace SO(3, 1) by SIM (2). However, it follows by exponentiating C J that J does not generate a compact SO(2) subgroup unless the deformation parameter a vanishes.
From now on we shall restrict attention to this a = 0 case, for which we denote the deformed algebra by disim b (2). The non-trivial Lie brackets for disim b (2) are given by The deformed group DISIM b (2) is a subgroup of the Weyl group with an action on Minkowski spacetime given by translations, and boosts in the +i directions, together with a combination of a boost in the +− direction and a dilatation. Specifically, the deformed generator N acts as The group DISIM b (2) does not leave invariant the standard Minkowski line element ds = (η µν dx µ dx ν ) 1/2 , but rather, the Finslerian line element This is of the form first suggested by Bogoslovsky [13].
We shall now construct, using the theory of nonlinear realisations, a DISIM b (2)-invariant Lagrangian for a point particle.
In order to construct a DISIM b (2)-invariant Lagrangian with worldline reparameterisation invariance, we allow the Goldstone coordinates (w, w i , x µ ) to depend on the worldline coordinate τ (see, for example, [14]). We shall restrict our attention to Lagrangians that are linear in the left-invariant 1-forms pulled back to the particle's worldline. Requiring invariance under SO(2) then implies that we must discard λ +i and λ i , and thus we consider the Lagrangian where α, β and γ are arbitrary constants. Since the last term is a total derivative, we can discard it. Eliminating the non-dynamical Goldstone coordinates w and w i , one obtains, in physical units, the Lagrangian Calculating th canonical momenta from (17), we obtain the DISIM b (2)-invariant dispersion relation or Hamiltonian constraint . (18) Note that that for b = 0 we recover the ordinary free relativistic particle, which does not see the light-like direction n µ . The cases b = ±1 are special, and are best investigated directly from equation (16). The conclusion is that for b = 1 we obtain the massless equation η µν p µ p ν = 0, whilst for b = −1 we haveẋ i = 0 andẋ − = 0, and the dynamics is trivial in this case.
Upon quantisation, p µ → −i∂ µ , we obtain a generalised Klein-Gordon equation of the form This is in general a non-local equation, since it involves fractional derivatives. Although the special case b = 1 appears to give a local modification of the usual Klein-Gordon equation involving a term linear in n µ ∂ µ this is really equivalent to the standard massless Klein-Gordon equation (as discussed earlier for this special value of b). Specifically, the first-order term can be removed by making the phase transformation φ → φ e −imnµx µ /2 .
The free Maxwell equations are also invariant under the action of the Weyl group and so they too are clearly invariant under DISIM b (2). The invariance of A µ dx µ , together with (12), implies that ( x any invariant action must have L → λ 4b L. Thus we can add a mass term, giving If b = 1 we can further include a non-Lorentz invariant Chern-Simons term [16], L cs = 1 2 ℓ −1 ǫ µνρσ n µ A ν F ρσ , where ℓ is an arbitrary length scale. Since it is a subgroup of the Weyl group, DISIM b (2) leaves invariant the massless Dirac Lagrangian. Bogoslovsky and Goenner [15] have pointed out that adding to the massless Dirac Lagrangian a term of the form m in µψ γ µ ψ ψψ gives a non-linear DISIM b (2)-invariant generalisation of the massive Dirac equation. This follows from the scalings ψ → λ 3b/2 ψ, γ µ ∂ µ → λ b γ µ ∂ µ under the action of the generator M +− . As with the generalised Klein-Gordon equation (19), the case b = 1 is special: The additional term may then be removed by a phase transformation of the form ψ → ψ e imnµx µ . Note however that, as discussed below, experimental bounds constrain |b| to be very much less than 1. CPT will be preserved if an operator exists in the complexification of DISIM b (2) that reverses x µ . As discussed for ISIM(2) in [1], a candidate CPT operator is e iφJ e iαN . This has the following action on the momenta: (P + , P − , P 1 + iP 2 ) → e −bα (e −α P + , e α P − , e iφ (P 1 + iP 2 )) .
(22) Requiring that P µ → −P µ implies that for integers n + , n − and n 3 . Although b = 1 + p/q is rational, with p odd, one may always choose n + and n − so that b is arbitrarily close to any given real number.
We have shown that ISIM(2) admits no deformations with de Sitter-like non-commutative translations. However, it is interesting to note that ISIM(2), unlike the Poincaré group but like the Galilei group, admits a central extension: the cohomology group H 2 (isim(2), R) is non-trivial. We find that it is generated by λ +− ∧ λ 12 , and so may adjoin to isim(2) a central element Z, whose only non-trivial Lie bracket is Thus unlike the translations, the boosts and rotations can be rendered non-commutative. This also works for the full 2-parameter family of deformations of ISIM (2). Including the extra generator Z, appending e θZ , and proceeding with the comstruction of an invariant particle Lagrangian leads to unmodified equations of motion, since the only effect is to add a total derivativeθ to the Lagrangian. It was argued in [13] that aether-drift experiments imply |b| < 10 −10 . However, it follows from (17) that every particle has a mass tensor m ij = (1 − b)m(δ ij + bn i n j ). Hughes-Drever type limits [17] on the anisotropy of inertia then potentially imply that |b| < 10 −26 . However, this depends on the precise form of the interactions [18,19]. Since a non-vanishing b could arise through quantum corrections, Very Special Relativity faces the question, analogous to the puzzle posed by the cosmological constant in traditional relativity: "Why is b so small?"