aa r X i v : . [ h e p - t h ] F e b Hidden Symmetries in Deformed Very Special Relativity
N. Dimakis ∗ Center for Theoretical Physics, College of Physics Sichuan University, Chengdu 610064, China
We study particle dynamics in a space-time invariant under the
DISIM b (2) group - the deforma-tion of the ISIM (2) symmetry group of very special relativity. We find that the Lorentz violationleads to the creation of higher order (hidden) symmetries which are connected to those broken atthe space-time level. Through the perspective of the conserved quantities of the special relativisticcase the Lorentz violation is linked to specific non-commutative relations in phase-space.
I. INTRODUCTION
The subject of Lorentz violation has a long history in theoretical physics and is motivated by different arguments ina broad spectrum of theories (string theory, loop quantum gravity, non-commutative geometry etc. [1–4]). In this workwe are interested in the symmetry structure of the particle dynamics in the deformed version of very special relativity(VSR), which incorporates such a violation. In this context, we reveal the existence of higher order symmetries whichare connected to breaking Lorentz invariance.VSR was introduced by Cohen and Glashow in [5]. The basic idea is that Lorentz symmetry is not a fundamentalsymmetry of nature but rather this role is reserved for one of its proper subgroups. One such realization consistsof taking the four-parameter similitude
SIM (2) subgroup of the Lorentz group, together with the translations, inorder to form the eight dimensional
ISIM (2) group. In [6], it was demonstrated that the
ISIM (2) group admitsa physically acceptable deformation which was called
DISIM b (2), with b being a nonzero dimensionless parameterwhose value needs to be very small ( b < − ). This deformed algebra is compatible with a line-element belongingto a Finsler type of geometry which was initially introduced by Bogoslovksy [7, 8]. Relativistic particle dynamics inFinsler geometry is often used to model dispersion relations emanating from Lorentz violation in field theory [9–12].In our case the geometry is characterized by the Bogoslovsky-Finsler line-element ds = g µν dx µ dx ν " ( ℓ µ dx µ ) − g µν dx µ dx ν b , (1)with g µν being the flat space-time metric and ℓ a covariantly constant, null, future directed Killing vector of g µν . The ds of (1) is a homogeneous function of degree 2 in the dx µ , which places it in the class of Finsler geometries. Ofcourse, when the parameter b is zero the typical pseudo-Riemannian line-element of special relativity is recovered. The b → ℓ in (1) persists in those relations (see [13]).In what follows we work in light-cone coordinates, x µ = ( v, u, x , y), in which the pseudo-Riemannian metric g involved in (1), is g = g µν dx µ dx ν = 2 dudv + δ ij dx i dx j , (2)while the null vector becomes ℓ = ℓ µ ∂ µ = ∂ v . Of course the context of the theory can be generalized by adoptingcurved space-time metrics (for studies in pp-wave geometries see [14, 15]).Throughout this work, Greek indices cover the whole spectrum of space-time variables, the i, j are reserved for thex − y plane, while the u, v subscripts are used to denote the components in the corresponding null direction. Greekindices are raised and lowered with the four-dimensional metric (2), while for i, j the δ ij is used for this purpose.Finally, in our conventions, ds < g < iso (3 ,
1) algebra, theBogoslovsky-Finsler line-element is invariant under the transformations generated by T µ = ∂ µ , B ij = x i ∂ j − x j ∂ i , B vi = u∂ i − x i ∂ v (3a)and N = (1 + b ) v∂ v + ( b − u∂ u + bx i ∂ i . (3b) ∗ [email protected], [email protected] These vectors are the generators of the
DISIM b (2) group and elements of the corresponding disim b (2) algebra.The vectors (3a) are of course also part of iso (3 , T µ and B µν = x µ ∂ ν − x ν ∂ µ , which are the isometriesof the pseudo-Riemannian metric g . The three remaining vectors of the Poincar´e algebra which leave invariant g butfail to be symmetries of ds are represented by: B uv = v∂ v − u∂ u , B ui = v∂ i − x i ∂ u . (4)The vector N is the one carrying the deformation parameter b . When the latter is zero, the N | b =0 is identified withthe B uv vector which together with the seven vectors from (3a) form the generators of the ISIM (2) group.We have thus the following setting: i) The eight-dimensional disim b (2) algebra spanned by (3) which leaves invariantthe ds and ii) the ten-dimensional Poincar´e algebra of (3a) and (4) which are isometries of g . We shall refer to thelatter as the b = 0 case throughout the manuscript due to g = ds | b =0 .Apart from this fundamental difference at the symmetry level though, there exists a striking resemblance betweenline-elements (1) and (2). According to a theorem proven by Roxburgh [16], certain classes of Finslerian spaceshave the property of reproducing the exact same geodesics as Riemannian metrics. The Bogoslovsky-Finsler line-element (1) belongs to this class and, even though it has a distinct symmetry structure compared to the g of (2), ithas its extrema exactly on the same trajectories as the latter. This very interesting property is the motive of thiswork. We seek to find the deeper connection between the two systems and what happens to the symmetries and thecorresponding integrals of motion that are broken in the Bogoslovsky-Finsler case by the Lorentz symmetry violation. II. THE HIDDEN SYMMETRIES
In order to dig deeper into the symmetry structure of the problem we study the motion of a particle of mass m in the Bogoslovsky-Finsler spacetime characterized by line-element (1). The corresponding action can be writtenas S = − m R √− ds = R ˜ Ldτ (we work in units c = 1), where ˜ L = − m ( ℓ µ ˙ x µ ) b ( − ˙ x µ ˙ x µ ) − b . The dot denotesdifferentiation with respect to the parameter along the curve which we symbolize with τ . It is more convenienthowever to use instead of ˜ L the equivalent Lagrangian L = − e ˙ u b ( − ˙ x µ ˙ x µ ) − b − e m , (5)where e = e ( τ ) is an auxiliary degree of freedom called the einbein [17] and in which ℓ µ dx µ = du was used. Lagrangian L corresponds to a well defined Hamiltonian which can be obtained through the Dirac-Bergmann algorithm [18, 19];in contrast to ˜ L which is a function homogeneous of degree 1 in the velocities and thus its Hamiltonian is bound tobe identically zero [20].The equivalence of the two Lagrangians is straightforward upon calculation of the equations of motion. The Euler-Lagrange equation for the degree of freedom e , i.e. ∂L∂e = 0, leads to e = 1 m ˙ u b ( − ˙ x µ ˙ x µ ) − b , (6)which is the constraint relation of the system. Substitution of e from (6) into L maps the latter to ± ˜ L , depending onthe sign in front of the square root that one takes when solving (6). The use of (6) inside the second order equationsof motion, reduces them to ¨ u = ˙ u ¨ v ˙ v , ¨ x i = ˙ x i ¨ v ˙ v (7)with v ( τ ) remaining an arbitrary function through which the parametrization invariance of the system is expressed.This last set (7) is equivalent to the one obtained from the Euler-Lagrange equations of ˜ L , which - as a constrainedsystem of equations - can be solved algebraically with respect to just three accelerations; the solution being (7).More important than the equivalence at the level of the equations is that ˜ L and L admit the same symmetries.The transformations induced by (3) leave invariant both Lagrangians. The situation is similar to what happensin the Riemannian case, where b = 0, and L | b =0 becomes the quadratic equivalent of the square root Lagrangian˜ L | b =0 = − m p − ˙ x µ ˙ x µ ; either one can be used to study a geodesic problem.The “time” gauge choice v ( τ ) = τ leads to e =constant, which corresponds to the typical affine parametrization.As expected by the theorem proven by Roxburgh [16], eqs. (7) are the same as those for the motion in Minkowskispace-time where b = 0. The difference of the two systems rests in the association of the constants of integration withthe physical parameters m and the now nonzero b through the constraint equation (6).As we already mentioned, the symmetry structure of the Bogoslovsky-Finsler line-element and consequently ofLagrangians ˜ L and L is quite different from that of the b = 0 case. However, the fact that the same set of secondorder equations provides solutions in both cases signifies that there are conservation laws to be accounted for in the b = 0 case since it admits a smaller symmetry group.Our study on conserved charges can be better expressed in the Hamiltonian formulation. To this end the Dirac-Bergmann algorithm for constrained systems [18, 19] is applied. We refrain from presenting details on the theory ofconstrained systems and we refer to relevant textbooks [21, 22]. The resulting Hamiltonian constraint reads H = − (1 − b ) b − (1 + b ) b p − bv ( − p µ p µ ) b + m = 0 , (8)where p µ = ∂L∂ ˙ x µ . Equation (8) leads to the dispersion relation: p µ p µ = 2 p u p v + p i p i = − m (1 − b ) (cid:16) p v m (1 − b ) (cid:17) b b ,which was first presented in [6]. The equality to zero in (8) holds on mass shell and in the formalism of constrainedsystems it is referred to as a weak equality [18].As is expected by Noether’s theorem, the symmetries (3) of the Bogoslovsky-Finsler line-element generate linear inthe momenta integrals of motion which are I µ = p µ , I ij = x i p j − x j p i , I vi = up i − x i p v ,I N = (1 + b ) vp v + ( b − up u + bx i p i , (9)and of course have the property of commuting with H . Apart from the above conserved charges, we may notice thatthe following quantities, which are rational functions in the momenta, are also conserved: I uv = vp v − up u + b b u p µ p µ p v (10a) I ui = vp i − x i p u + b b x p µ p µ p v . (10b)It is straightforward to check that truly { I uv , H } = 0 = { I ui , H } , where { , } are the usual Poisson brackets. Aninteresting point about this new charges is that they look like b -distorted “boosts”, since for b = 0 they fall to thelinear Minkowski space charges generated by vectors (4).The fact that the quantities I uv , I ui possess a nonlinear dependence on the momenta means that they are notgenerated by space-time vectors like (4), but of what is called higher order or hidden symmetries of the Lagrangian.The components of such symmetry generators depend also on derivatives of the coordinates. For more informationon these types of symmetries we refer to [23, 24] (maybe the most famous hidden symmetry in physics is the oneassociated with the Carter constant for the geodesic motion in Kerr space-time [25]). The symmetry generators of I uv and I ui are X uv = − u∂ u + (cid:18) v + b − b ˙ x µ ˙ x µ ˙ u u (cid:19) ∂ v (11a) X ui = − x i ∂ u + b − b ˙ x µ ˙ x µ ˙ u x i ∂ v + v∂ i . (11b)It is easy to check that if we take into account the relation p µ p µ p v = (1 + b ) ˙ x µ ˙ x µ (1 − b ) ˙ u , (12)then we directly obtain the I uv and I ui of (10) through calculating the inner product of the vectors with the momentum,i.e. I uv = ( X uv ) µ p µ , I ui = ( X ui ) µ p µ .The transformations which the X uv and X ui induce in the space of ( x µ , ˙ x ν ) leave invariant both Lagrangians L and˜ L . The ensuing transformations are recovered by extending the symmetry vectors X = X µ ∂ µ in the space of the firstderivatives, i.e. X [1] = X + ˙ X µ ∂∂ ˙ x µ . The X [1] are called the first prolongations of the vectors X [23]. These extendedvectors have a wide application in the study of symmetries in dynamical systems, many prominent examples being incosmology [26, 27]. The components ˙ X µ of course contain now accelerations, but these can be substituted throughthe use of the second order Euler-Lagrange equations (7) and thus obtain a transformation rule confined in the space( x µ , ˙ x ν ). The invariance of the Lagrangians can be simply shown by noting that X [1] uv ( L ) = 0 = X [1] ui ( L ) modulo eqs.(7).The X uv and X ui of (11) are not space-time vectors. Nevertheless they can be linked to such upon the solutionspace. First we note that the ratio, R = ˙ x µ ˙ x µ ˙ u , appearing in (11) is itself a constant of motion: ˙ R = 0, by virtue of theEuler-Lagrange equations (7). If we use the constraint equation (6), together with the first integral of motion fromeq. (9), which reads in the velocity phase-space I v = p v = ∂L∂ ˙ v = 1 e (1 − b ) ˙ u b +1 ( − ˙ x µ ˙ x µ ) b = π v , (13)then we may write the on mass shell value of R in terms of the parameters m and π v (we use π v as the on mass shellconstant value of the momentum p v ) R = ˙ x µ ˙ x µ ˙ u = − (cid:20) (1 − b ) m π v (cid:21) b . (14)In view of the last expression, we produce a reduced (on mass shell) version of vectors (11) as ξ uv = − u∂ u + " v − b (1 − b ) − b b (cid:18) m π v (cid:19) b u ∂ v (15a) ξ ui = − x i ∂ u − b (1 − b ) − b b (cid:18) m π v (cid:19) b x i ∂ v + v∂ i . (15b)These ξ uv , ξ ui are indeed space-time vectors and when we set b = 0 in (15), we recover the missing Minkowski spacesymmetries B uv and B ui appearing in (4). We can even use them to write simplified, linear in the momenta, on massshell equivalent expressions for the (10). Those are Q uv = ( ξ uv ) µ p µ and Q ui = ( ξ ui ) µ p µ , which commute with theHamiltonian when the relations H = 0 and p v = π v are enforced.The ξ uv , ξ ui close an algebra together with the seven vectors (3a). It is however a trivial deformation of the Poincar´ealgebra, i.e. it is the same algebra expressed in different coordinates. This can be directly seen by noticing that theset of these ten vectors are isometries of the following flat space metric¯ g = g − b (1 − b ) − b b (cid:18) m π v (cid:19) b ℓ µ ℓ ν dx µ dx ν , (16)where g is given by (2). The space-time transformation for which ¯ g g serves as an algebra automorphism and makesthe corresponding vectors (up to linear combinations) assume the usual expressions leading to the typical Poincar´ealgebra. Equation (16) implies that the metric ¯ g is disformally related to g . Disformal transformations were initiallydefined by Bekenstein as generalizations to conformal transformations [28] (for applications see also [29, 30]).We need to stress here that the ξ uv , ξ ui of (15) are not themselves symmetry vectors but the reduced expressionsof the higher order symmetries given by the X uv , X ui in (11). The symmetry group of the system (when referringto space-time vectors) is the DISIM b (2), it is interesting to see however that the effect of the parameter b and theviolation of Lorentz invariance does not result in a complete elimination of the broken symmetries from the system.They are converted into the higher order symmetries, X uv , X ui , which reduce on mass shell to expressions given bythe original Killing vectors distorted appropriately by b , the ξ uv and the ξ ui respectively. Similar distortions of brokensymmetries have emerged in a different setting involving proper conformal Killing vectors in the case of the motionof a massive particle in Riemannian pp-wave space-times [31]: For example, it is well known that in the case of nullgeodesics the proper Conformal Killing vectors (CKVs) generate integrals of motion. However this property is lostwhen one considers a massive particle (in a sense the presence of the mass m breaks these symmetries). In [31] itwas shown that the proper CKVs still contribute by producing conservation laws under a similar mass dependentdistortion. Here, in a Finsler geometry, we see it happening at the level of Killing vectors whose symmetry propertyis broken by the introduction of the nonzero parameter b .We can actually extend the hidden symmetries we encountered here by also using proper CKVs together with thenecessary distortions. However this goes outside the scope of this letter where we mainly want to focus to the effect ofthe distortion on the Lorentz symmetry vectors. Just as an example though, we mention that by taking the followinglinear combination of the symmetry N given in (3b) together with the vector ξ uv of (15a) ξ h = 1 b N − − bb ξ uv = " v + (cid:18) (1 − b ) m π (cid:19) b u ∂ v + x i ∂ i , (17)we obtain the distortion of the homothecy, the ξ h | ( b =0= m ) is the homothetic vector of g . We can now use ξ h towrite the (reduced) linear conserved quantity Q h = ( ξ h ) µ p µ and even go backwards with the substitution of (14) inconjuction with (12) to finally express the original integral of motion which is generated by a higher order Noethersymmetry and which reads I h = 2 vp v + x i p i − − b b p µ p µ p v u. (18)A direct calculation shows that { I h , H } = 0. III. NON-CANONICAL COORDINATES
As we mentioned, the mapping ¯ g ( u, v, x i ) g ( U, V, x i ) allows us to obtain the Poincar´e algebra generators out oflinear combinations of the ten vectors of (3a) and (15). In order to avoid any confusion we use the ( u, v, x i ) for theoriginal coordinates and ( U, V, x i ) for those after the transformation. The mapping ( u, v, x i ) ( U, V, x i ) we need is v = V + b (1 − b ) − b b (cid:18) m π v (cid:19) b U, (19)while the rest of the coordinates remain unchanged, i.e. u = U, x i = x i . Under the aforementioned coordinate changewe have T u + b (1 − b ) − b b (cid:18) m π v (cid:19) b T v T U := ∂ U ,ξ ui − b (1 − b ) − b b (cid:18) m π v (cid:19) b B vi B Ui := V ∂ i − x i ∂ U , (20)while for the rest we get: ξ uv B UV , B vi B V i , B ij B ij , T v T V and T i T i . Thus, the Poincar´e algebracorresponding to the symmetries of the flat metric g ( U, V, x i ) = 2 dU dV + dx i dx i is completely recovered.What would be of interest though, is to acquire a mapping that connects the full symmetry generated charges (10)of the Bogoslovsky-Finsler line-element to the conserved quantities emerging from the Poincar´e algebra of the flatspace metric g ( U, V, x i ). For this we use as a guide transformation (19) and substitute in it the constants m , π v withrespect to their phase-space dynamical equivalents from (14) and (12). After this process, we obtain the followingnon-canonical transformation from U, V to u, v variables U = u, V = v + b b ) p µ p µ p v u,p U = p u − b b ) p µ p µ p v , p V = p v , (21)with the x i , p i remaining unchanged. It can be seen that this transformation maps the linear conserved chargesgenerated by the Poincar´e symmetry algebra of g ( U, V, x i ) to the higher order ones of the Bogoslovsky-Finsler metric.To make it specific, with the use of (21), we obtain the correspondence I U := p U = I u − b b ) p µ p µ p v I v (22) I Ui := V p i − x i p U = I ui + b b ) p µ p µ p v I vi . (23)The rest are mapped directly to the corresponding quantities, i.e. I V := p V = I v , I UV := V p V − U p U = I uv , I V i = I vi while the I i , I ij are obviously not affected by the transformation (21).Due to (21) being a non-canonical transformation, the relevant I in the ( U, V ) variables do not in general commutewith the resulting Hamiltonian obtained by using (21) in (8). This can be achieved by introducing the fundamentalbracket relations which are implied by transformation (21). We thus write a new bracket [ , ] whose nonzero valuesin the ( U, V ) coordinates are [
U, V ] = b U (1 + b ) p V , [ U, p U ] = 11 + b , [ V, x i ] = − b U p i (1 + b ) p V , [ V, p U ] = b (cid:0) p U p V + p i p i (cid:1) (1 + b ) p V , [ V, P V ] = 1 , [ x i , p U ] = − bp i (1 + b ) p V , [ x i , p j ] = δ ij . (24)These are calculated with the help of the Poisson brackets in the ( u, v ) coordinates, e.g. [ U, V ] = { U ( u, v, p ) , P U ( u, v, p ) } .It is easy to verify that the Jacobi identity is satisfied by (24). We notice the space-time non-commutativity introducedsince V does not commute with either U or x i .With relations (24) the transformed Hamiltonian (eq. (8) under use of (21)) commutes with the Poincar´e chargesin the U, V variables and it reproduces the correct equations of motion in these coordinates. In addition, the structureof the disim b (2) algebra is not affected. For example, if we take the transformed I N conserved charge which reads inthe U, V coordinates I N = (1 + b ) ( V p V − U p U ) + bx i p i − bU p i p i p V , (25)we immediately calculate [ p U , I N ] = (1 − b ) p U and [ p V , I N ] = − (1 + b ) p V ; exactly as we have { p u , I N } = (1 − b ) p u and { p v , I N } = − (1 + b ) p v with the I N of (9) in the original u, v variables.We thus see that from the point of view of the symmetries of special relativity, the effect of the Lorentz violation withthe introduction of a Finslerian line-element can be simulated by the introduction of non-commutative coordinates inspace-time and in particular ones that satisfy relations (24). The parameter b in this context signifies the deviationfrom the Poisson bracket formalism.The compatibility of non-commutative space-times with the original notion of VSR and the symmetry groups thatit involves has been explored previously in [32] for non-commutative matrices θ µν that depend solely on space-timecoordinates, i.e. [ x µ , x ν ] = θ µν ( x ). In our case the correspondence we make involves non-commutativity in all ofphase-space. Non-commutative expressions including momenta through a different approach have been used before[33] in order to reproduce the disim b (2) algebra from B µν and p µ . However, the algebra given there, introducesa nonzero bracket between the null momenta, { p u , p v } 6 = 0, which is not in agreement with the original disim b (2)algebra. IV. CONCLUSION
We demonstrated that the broken symmetries due to Lorentz violation in the scenario of the deformed VSR takepart in the creation of higher order symmetries from which integrals of motion that are rational functions in themomenta emerge. With a non-canonical transformation we mapped this enhanced set of conserved quantities to theusual integrals of motion of the free particle in special relativity.We used this correspondence to define new brackets that make the charges of the Minkowskian motion commutewith the Hamiltonian of the Bogoslovsky-Finsler case in the new variables. The brackets give rise to non-commutativerelations in space-time. It is interesting to note that in (24) the bracket, [
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