Lifshitz field theories, Snyder noncomutative 5 space-time and momentum dependent metric 1 0 2 Juan M. Romero∗ n u J Departamento de Matem´aticas Aplicadas y Sistemas, 2 2 Universidad Auto´noma Metropolitana-Cuajimalpa, ] M´exico, D.F 05300, M´exico h t - p e h J. David Vergara† [ 2 Instituto de Ciencias Nucleares, v 4 Universidad Nacional Autonoma de M´exico, 7 4 A. Postal 70-543 , M´exico D.F., M´exico 6 0 . 1 0 5 Abstract 1 : In this work, we propose three different modified relativistic parti- v i cles. Inthefirstcase, weproposeaparticle withmetrics dependingon X the momenta and we show that the quantum version of these systems r a includes different field theories, as Lifshitz field theories. As a second caseweproposeaparticlethatimpliesamodifiedsymplecticstructure and we show that the quantum version of this system gives different noncommutative space-times, for example the Snyder space-time. In the third case, we combine both structures before mentioned, namely noncommutative space-times and momentum dependent metrics. In this last case, we show that anisotropic field theories can be seen as a limit of noncommutative field theory. ∗[email protected] †[email protected] 1 1 Introduction Recently, different approaches have been developed to obtain a quantum ver- sion of gravity. Some of these approaches are string theory [1], loop quantum gravity [2], noncommutative geometry [3], etc. In (2 + 1) dimensions there are important progress [4, 5], but in (3+1) dimensions we do not know how this theory is, we only have some signs about it. For instance, using the Ehrenfest principle, Bekenstein proposed that in a quantum gravity the area of the event horizon has discrete spectrum [6, 7] 2 2 A = 4πr = γl n, n = 1,2,···. (1) n p In addition, G. ‘t Hooft showed that in (2+1) dimensions the quantum grav- ity implies a discrete space-time in an effective approximation [8]. For those reasons, we can conjecture that in the quantum gravity there are geomet- ric quantities with discrete spectrum. Remarkably, the discrete space-time obtained by G. ‘t Hooft is the so-called Snyder space-time, which is dis- crete, noncommutative and compatible with the Lorentz symmetry [9]. In fact, in this noncommutative space-time the surface area of a sphere is quan- tized [10]. It is worth mentioning that it is possible construct field theories in some noncommutative space-times [11, 12, 13], but to build a gravity or field theory in a noncommutative space-time, as Snyder space-time, is a very difficult task. Some work about Snyder space-time can be seen in [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] and works about non- commutative space-times which imply discrete geometric quantities can be seen in [28, 29, 30, 31, 32]. A major problem to obtain a quantum gravity theory is that the usual gravity is unrenormalizable. Nevertheless, recently Hoˇrava formulated a modified gravity which seems to be free ghosts and power counting renor- malizable [33]. This gravity is invariant under anisotropic scaling ~x → b~x, t → bzt, z,b = constants, (2) with z = 3. The original Hoˇrava gravity has dynamical inconsistencies [34], but were found healthy extensions of it [35, 36]. Hoˇrava gravity has different interesting properties, some of them are [37, 38, 39, 40, 41, 42, 43, 44, 45]. Field theories invariant under the anisotropic scaling transformations (2) can be seen in [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56], notably these field the- ories improve their high energy behavior. Furthermore, in the usual general 2 relativity it has been found space-times invariant under the anisotropic scal- ing (2), see [57]. Significantly, using the gravity/gauge correspondence, these space-times can be related with some condensed matter systems [58, 59, 60]. Now, as a road to obtain new physics, different authors have been pro- posed that the Minkowski geometry should be changed by the Finsler geom- etry [61, 62, 63, 64, 65, 66]. In this new geometry the metric depends on velocities, notice that in this case the metric can depends on the momenta. Remarkably, since 1938 Max Born proposed a theory with a metric that depends on the momenta as a suggestion for unifying quantum theory and relativity [67]. In this work, we propose three different modified relativistic particles. In the first case, we propose a particle with a metric depending on the mo- menta and we show that the quantum version of this system includes differ- ent field theories, as anisotropic field theories. As a second case we propose a particle that implies a modified symplectic structure and we show that the quantum version of this system gives different noncommutative space- times, for example the Snyder space-time. In the third case, we combine both structures before mentioned, namely noncommutative space-times and momentum-dependent metric. In this last case, we show that anisotropic field theories can be seen as a limit of noncommutative field theory. This paper is organized as follow: in Sec. 2, we study the first modified relativistic particle and show that in this framework different Lifshitz field theories can be obtained; in Sec. 3, we propose a modified particle that its quantum version implies noncommutative space-times; in Sec. 4, we combine theresultsobtainedinSec. 2and3. Finally, inSec. 5,weprovideasummary. 2 Modified actions TheHamiltonianactionforthemassiverelativisticparticlewiththemomenta fixed in the end points is given by λ λ S = dτ −p˙ xµ − p2 −m2 = dτ −ηµνp˙ x − ηµνp p −m2 , (3) µ µ ν µ ν Z (cid:18) 2 (cid:19) Z (cid:18) 2 (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) 3 whereλisaLagrangemultiplier. Now,ifweconsideramomentum-dependent metric Ωµν(p), (4) we can propose the generalized relativistic particle action λ S = dτ −ηµνp˙ x − Ωµν(p)p p −m2 . (5) µ ν µ ν Z (cid:18) 2 (cid:19) (cid:0) (cid:1) In the section 3, we will show that the quantum version of this system in- cludes different field theories, as Lifshitz field theories. In addition, if Λµν(p) (6) is a symmetric matrix we can also introduce the alternative action λ S = dτ −Λµν(p)p˙ x − ηµνp p −m2 . (7) µ ν µ ν Z (cid:18) 2 (cid:19) (cid:0) (cid:1) Notice that in this case the space-time metric is not modified. In the section 4, we will show that this system has a modified symplectic structure and also we will show that the quantum version of this system gives different noncommutative space-times, for example the Snyder space-time. Furthermore, we can take the action λ S = dτ −Λµν(p)p˙ x − Ωµν(p)p p −m2 , (8) µ ν µ ν Z (cid:18) 2 (cid:19) (cid:0) (cid:1) which contains the actions (5) and (7). In all these cases, we assume that we use the Minkowski metric for raising and lowering indices. Notice that these three actions are invariant under reparametrization trans- formations dτ˜ τ → τ(τ˜), λ → λ , dτ in fact this symmetry appears in the usual relativistic particle [68]. Due that the reparametrization symmetry is a local symmetry, according to Dirac’s Method [69, 68] these three action have a first class constraint C(x,p) ≈ 0, 4 which generates the reparametrization symmetry (the ”gauge symmetry” for these systems). Now, if A(x,p) is a function in the phase space, according to the Dirac’s Method, an infinitesimal gauge transformation are given by δA = ǫ(x,p){A(x,p),C(x,p)}. Notice that only if A(x,p) is a gauge invariant quantity we have {A(x,p),C(x,p)} = 0. Furthermore, at quantum level Dirac’s method sets that the physical states are invariant under the action of the first class constraints, i.e., exp(ζCˆ)|ψi = |ψi, (9) which implies Cˆ|ψi = 0. (10) Here Cˆ is the quantum version of the constraint C(x,p). In the next sections we will show that these three actions are different and each one of them gives an alternative quantum physics. 3 Lifshitz case First we consider the following action λ S = dτ −ηµνp˙ x − Ωµν(p)p p −m2 , (11) µ ν µ ν Z (cid:18) 2 (cid:19) (cid:0) (cid:1) from this action we obtain the classical constraint C = Ωµν(p)p p −m2 ≈ 0. (12) µ ν Then, using the Dirac’s Method, at quantum level and in the coordinate representation we get the wave equation −Ωµν(−i∂)∂ ∂ −m2 φ = 0, (13) µ ν (cid:0) (cid:1) which is a modified Klein-Gordon equation. 5 3.1 Scalar Field The equation (13) can be obtained from the action 1 S = dxd+1 ∂ φΩµν (−i∂)∂ φ−m2φ2 . (14) µ ν Z 2 (cid:0) (cid:1) In fact, if the matrix Ωµν (−i∂) has only an even number of derivatives, we arrive to δS = − dxd+1δφ Ωµν (−i∂)∂ ∂ φ+m2φ = 0, (15) µ ν Z (cid:0) (cid:1) which implies the equation of motion Ωµν (−i∂)∂ ∂ φ+m2φ = 0, (16) µ ν that is equivalent to the equation (13). Notice, that if Ωµν (−i∂) = ηµν +hµν (−i∂), (17) we have 1 S = dxd+1 ∂ φ∂µφ+(∂ φ)hµν (−i∂)(∂ φ)−m2φ2 µ µ ν Z 2 (cid:0) (cid:1) 1 = dxd+1 ∂ φ∂µφ−φhµν (−i∂)(∂ ∂ φ)−m2φ2 . (18) µ µ ν Z 2 (cid:0) (cid:1) In particular, if we take Ωµν(p) = ηµν +l2pµpν, l = constant, (19) namely Ωµν (−i∂) = ηµν +l2pˆµpˆν = ηµν −l2∂µ∂ν, (20) we arrive to 1 S = dxd+1 ∂ φ∂µφ+l2φ(cid:3)2φ−m2φ2 . (21) µ Z 2 (cid:0) (cid:1) 6 In general, if G(p2) is a smooth function, when Ωµν(p) = ηµν +G p2 pµpν, (22) (cid:0) (cid:1) we obtain Ωµν (−i∂) = ηµν −G(−(cid:3))∂µ∂ν, (23) for this case, the action (18) becomes 1 S = dxd+1 ∂ φ∂µφ+φG(−(cid:3))(cid:3)2φ−m2φ2 . (24) µ Z 2 (cid:0) (cid:1) Thisisaquantum fieldtheorywithhighordertimederivatives, which implies the existence of ghost field solutions [70]. To avoid these kind of solutions we introduce the matrix Ω0µ(p) = η0µ, Ωij(p) = δij +l2pipj, (25) the quantum version of this last equation is given by Ω0ν (−i∂) = η0µ, Ωij(p) = δij −l2∂i∂j, (26) and for the action (18) we get 1 S = dxd+1 ∂ φ∂µφ−l2∂ φ∂i∂j∂ φ−m2φ2 µ i j Z 2 (cid:0) (cid:1) = dxd+11 ∂ φ∂µφ+l2φ ∇2 2φ−m2φ2 . (27) µ Z 2 (cid:16) (cid:17) (cid:0) (cid:1) Thisistheactionforananisotropicscalarfieldwithdynamicexponent z = 2, also this field is named Lifshitz scalar field [52, 56]. Moreover, for the case Ω0ν(p) = η0ν, Ωij(p) = δij +α −p~ 2 z−2pipj, (28) (cid:0) (cid:1) namely Ω0ν(−i∂) = η0ν, Ωij(−i∂) = δij −α ∇2 z−2∂i∂j, (29) (cid:0) (cid:1) 7 the action (18) becomes 1 S = dxd+1 ∂ φ∂µφ+αφ ∇2 zφ−m2φ2 . (30) µ Z 2 (cid:0) (cid:0) (cid:1) (cid:1) Which is the action for an anisotropic scalar field with dynamic exponent z, [52, 56]. Then, a Lifshitz scalar field can be seen as a scalar field in a generalized metric depending on the momenta. In general, if G(p~ 2) is a smooth function, we can propose the matrix Ω0ν(p) = η0ν, Ωij(p) = δij +G p~ 2 pipj, (31) (cid:0) (cid:1) which at quantum level is Ω0ν (−i∂) = η0ν, Ωij(−i∂) = ηij −G −∇2 ∂i∂j, (32) (cid:0) (cid:1) gives the action S = dxd+11 ∂ φ∂µφ+φG −∇2 ∇2 2φ−m2φ2 . (33) µ Z 2 (cid:16) (cid:17) (cid:0) (cid:1)(cid:0) (cid:1) In the next subsections we will study other fields in a momentum dependent metric. 3.2 Dirac Field Now we construct a Dirac equation in a metric that depends on the mo- menta. In this case, we require a tetrad formalism associate to the momenta dependent metric. For this reason, let us introduce the tetrad eµ = eµ(−i∂), (34) a a which satisfies eµ(−i∂)eν (−i∂)ηab = Ωµν (−i∂). (35) a b Then, using the usual Dirac’s matrices, that satisfy the ordinary Clifford algebra γa,γb = 2ηab, (36) + (cid:8) (cid:9) 8 and the tetrad basis introduced in (34) we obtain the following matrices Γµ(−i∂) = eµ(−i∂)γa, (37) a which satisfy {Γµ(−i∂),Γν(−i∂)} = 2Ωµν (−i∂). (38) + With the matrices Γµ we propose the modified Dirac equation −iΓµ∂ ψ +mψ = 0. (39) µ Notice that using this last equation and (38), we arrive to (−iΓν∂ −m)(−iΓµ∂ +m)ψ = ν µ = −ΓµΓν∂ ∂ ψ −m2ψ µ ν ΓµΓν +ΓνΓµ 2 = − ∂ ∂ −m ψ = 0, µ ν (cid:18) 2 (cid:19) namely − Ωµν (−i∂)∂ ∂ +m2 ψ = 0, (40) µ ν (cid:0) (cid:1) which is the modified Klein-Gordon equation (13). Then, the generalized Dirac’s equation can be seen as a Dirac’s equation in a metric depending on the momenta. In particular, if we take Ωµν (−i∂) = ηµν +hµν (−i∂), (41) at first order, the tetrad results 1 eµ(−i∂) = ηµ + hµ(−i∂), (42) a a 2 a which satisfies eµeνηab ≈ ηµν +hµν (−i∂). (43) a b In this approximation we obtain 1 Γµ = γµ + hµγa (44) 2 a 9 and the modified Dirac’s equation is given by 1 (−iγµ∂ +m)ψ −i hµγa∂ ψ = 0. (45) µ 2 a µ For the case hµν (−i∂) = −G(−(cid:3))∂µ∂ν, (46) we arrive to i (−iγµ∂ +m)ψ + G(−(cid:3))∂µ∂ γa∂ ψ = 0, (47) µ a µ 2 namely i (−iγµ∂ +m)ψ + G(−(cid:3))(cid:3)γµ∂ ψ = 0. (48) µ µ 2 While, if we take hµ0(−i∂) = 0, hij(−i∂) = −G −∇2 ∂i∂j, (49) (cid:0) (cid:1) we get i (−iγµ∂ +m)ψ + G −∇2 ∇2γµ∂ ψ = 0. (50) µ µ 2 (cid:0) (cid:1) In particular when 2 2 z−2 G −∇ = α ∇ , α = constant, (51) (cid:0) (cid:1) (cid:0) (cid:1) we obtain (−iγµ∂ +m)ψ + iα −∇2 z−1γµ∂ ψ = 0, (52) µ µ 2 (cid:0) (cid:1) which is the anisotropic Dirac’s equation with dynamic exponent z, [46, 56]. Therefore, the anisotropic Dirac’s equation can be seen as a Dirac’s equation in a metric depending on the momenta. 10