Conformal Anisotropic Mechanics And Horavas Particle


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Conformal Anisotropic Mechanics And Horavas Particle

  1. 1. ICN-UNAM/123 Conformal Anisotropic Mechanics and Hoˇava’s particle r Juan M. Romero∗ Departamento de Matem´ticas Aplicadas y Sistemas a Universidad Aut´noma Metropolitana-Cuajimalpa, M´xico 01120 DF, M´xico o e e Vladimir Cuesta,† J. Antonio Garcia,‡ and J. David Vergara§ Instituto de Ciencias Nucleares, Universidad Nacional Aut´noma de M´xico, o e Apartado Postal 70-543, M´xico 04510 DF, M´xico e e (Dated: January 7, 2010) In this paper we implement scale anisotropic transformations between space and time in classical mechanics. The resulting system is consistent with the dispersion relation of Hoˇava gravity [1]. Also, r we show that our model is a generalization of the conformal mechanics of Alfaro, Fubini and Furlan. For arbitrary dynamical exponent we construct the dynamical symmetries that correspond to the Schr¨dinger algebra. Furthermore, we obtain the Boltzman distribution for a gas of free particles o compatible with anisotropic scaling transformations and compare our result with the corresponding thermodynamics of the recent anisotropic black branes proposed in the literature. PACS numbers: Valid PACS appear here I. INTRODUCTION quadratic in P0 while the spatial momentum scale as z the models are in principle renormalizable by power Scaling space-time symmetries are very useful tools to counting arguments at least for z = 3. It is claimed also study non linear physical systems and critical phenomena that these models are ghost free. On the other hand due [2, 3]. At the level of classical mechanics an interesting to the non relativistic character of the Hoˇava gravity, r example is the conformal mechanics [4] that is relevant this theory must be considered as an effective version of for several physical systems from molecular physics to a more fundamental theory, compatible with the symme- black holes [5, 6]. Recently a new class of anisotropic tries that we observe in our universe. The consideration scaling transformations between spatial and time coordi- of these models could be useful to understand the quan- nates has been considered in the context of critical phe- tum effects of gravity. nomena [7], string theory [8–10] and quantum gravity [1]: In order to gain some understanding of the content of generalized dispersion relations of the form (2) we will t → bz t, x → bx, (1) construct a reparametrization invariant classical mechan- ics compatible with the scaling laws (1). In particular where z plays the role of a dynamical critical exponent. this implementation of the scaling transformations (1) In particular the result of Hoˇava [1] is quite interesting r will lead to dispersion relations of the form (2). Another since it produces a theory that is similar to general rel- interesting characteristic of our implementation is that it ativity at large scales and may provide a candidate for contains several systems for different limits. In the case a UV completion of general relativity. In principle this of z = 2 the conformal mechanics of Fubini, et al [4] is re- model for Gravity is renormalizable in the sense that the covered. In the limit z → ∞ we get a relativistic particle effective coupling constant is dimensionless in the UV with Euclidean signature and in the limit z = 1 together limit. A fundamental property of these gravity models with m → 0 we obtain a massless relativistic particle. is that the scaling laws (1) are not compatible with the usual relativity. In the IR these systems approach the We also study the symmetries of the anisotropic me- usual General Relativity with local Lorentz invariance chanics and compare our results with the Schr¨dinger o but in the UV the formulation admit generalized disper- algebra for any value of z. Finally, by analyzing the ther- sion relations of the form modynamic properties of a free gas compatible with the scaling transformation (1) we will obtain the same ther- P0 − G(P 2 )z = 0, 2 G = const. (2) modynamic relations for black branes recently proposed in [8]. Here z takes also the role of a dynamical exponent. The organization of the paper is as follows: Section As the dispersion relations used by these models are 2 concerns with the action of the system. In Sec. 3 we study the canonical formalism of the system and its gauge symmetries. Sec. 4 focuses on the equations of motion. ∗ Electronic address: The global symmetries and the Schr¨dinger algebra are o † Electronic address: analyzed in Section 5. In Sec. 6 we consider the ther- ‡ Electronic address: modynamic properties of the system and we conclude in § Electronic address: Section 7.
  2. 2. 2 II. THE ACTION PRINCIPLE Another interesting limit is in the case z = 1. Even though this limit is not well defined for the action (3) Scale transformation can be realized in non-relativistic it can be implemented using an equivalent form of the theories in different ways. From one hand we can start action (3), in the case of V (x) = 0 results from an action in curved space where some of its isome- z m (x2 ) 2 ˙ tries are the scale transformations. Another way to im- S= dτ + (z − 2)e . (9) 2(z − 1) ez−2 t ˙ plement scale transformations is to start from an action in flat space but constructed in such way that the in- Here, we have introduced an einbein e in a similar way variance under scale transformations is manifest. In this as the standard trick to remove the square root in the work we will take this second point of view. Let us con- action of the free relativistic particle. Using this action sider a d+1−dimensional space-time, with an action prin- (9) we can now take the limit z → 1 if at the same time ciple given by we take the limit m → 0. Assuming that m → 0 at the  z  m same rate as z − 1 in such way that z−1 → µ, where µ m x2 2(z−1) ˙ plays the role of a mass parameter. In consequence we S = dτ L = dτ  1 − V (x) t , (3) ˙ 2 ˙ z−1 t obtain the action √ dt dxi µe x2 ˙ ˙ t= , xi = ˙ , x2 = xi xi , ˙ ˙ ˙ i = 1, 2, · · · , d. S = dτ −1 . (10) dτ dτ 2 ˙ t This action is invariant under reparametrizations of τ , i.e. This action is still invariant under reparametrizations due under the transformations τ → τ = τ (f ), if we require df to the transformation rule of the einbein e → dτ e and it that t and the xi are scalars under reparametrizations. corresponds to a relativistic massless particle whose usual Furthermore, if we consider that the potential satisfies covariant action is V (bx) = b−z V (x), for example, xµ xµ ˙ ˙ a S= dτ . (11) V (x) = , a = const, (4) 2e |x|z the action (3) is invariant under the scaling laws Eq.(1). III. CANONICAL FORMALISM Note that, when z = 2 and τ = t we obtain the action for a non-relativistic particle, furthermore taking into ac- count (4), we obtain Let us consider the canonical formalism of the system (3). The canonical momenta are m 2 a S= dt x − 2 ˙ , (5) 2−z 2 |x| ∂L mz (x2 ) 2(z−1) ˙ pi = = 1 xi , ˙ (12) ∂ xi ˙ 2(z − 1) t z−1 ˙ that corresponds to the action of the conformal mechan- ics [4]. In the case of arbitrary z and for a gauge condition ∂L m (x2 ) 2(z−1) ˙ z τ = t we get pt = =− z + V (x) =−H,(13) ˙ ∂t 2(z − 1) t z−1 ˙ m 2 z a S= dt x ˙ 2(z−1) − , (6) and as expected (by the reparametrization invariance) 2 |x|z the canonical Hamiltonian vanishes that again is invariant under the scaling laws (1). In this sense Eq.(3) represents a generalized conformal ˙ ˙ Hc = pi xi + pt t − L = 0. (14) mechanics. From (12) we get Another interesting property of the action (3) appears  2 1 in the limit z → ∞. In this case for an arbitrary potential mz (x2 ) 2(z−1)  ˙ p2 = pi pi =  1 , (15) we obtain 2(z − 1) t z−1 ˙ m√ 2 ˙ S = dτ x − V (x) t , ˙ (7) that implies in turn 2 −z this action is invariant under reparametrizations. In the m mz z φ = pt + p2 2 + V (x) ≈ 0.(16) particular case of the potential (4) the limit is singular 2(z − 1) 2(z − 1) in the region 0 < |x| < 1, then leaving V (x) = 0 we obtain the action for a relativistic particle with Euclidean If we enforce the constraint, and considering the case signature, when V (x) = 0 we obtain −z m√ 2 m mz z S= dτ x . ˙ (8) pt = − p2 2 , (17) 2 2(z − 1) 2(z − 1)
  3. 3. 3 which can be rewritten in the form with λ a Lagrange multiplier. 2(1−z) 2 z 1 m If we require that is a function of τ , the extended (pt ) − G p2 = 0, G= (18) . z 2z 2(z − 1) action is invariant under the gauge transformations [16] In this way, we get a dispersion relation that is similar √ z−2 to the Horava’s gravity model Eq. (2). δxi = {xi , φ} = Gz(p2 ) 2 pi , ∂V δpi = {pi , φ} = − , Notice that by taking the limits z → 1 and m → 0 in ∂xi (16) we get the constraint δt = {t, φ} = , 1 φ = pt + (p2 ) 2 + V (x), (19) δpt = 0, ∂ that is consistent with the canonical formalism of the ac- δλ = . ∂τ tion (9) and corresponds to a massless relativistic particle with only positive energy. Here {, } are the Poisson brackets. The above trans- The system has one first class constraint (16) and fol- formations, are of the usual type for the parametrized lowing the standard Dirac’s method [11], the “wave equa- particle. However, the rule for the transformation tion” corresponding to the spatial coordinates seems to √ z change drastically. But if we use the definition of the ˆ φ|ψ >= pt + G p2 2 + V (x) |ψ >= 0, ˆ ˆ (20) momenta (12) we will get exactly the usual transfor- i must be implemented on the Hilbert space at quantum mation δxi = dx recovering the full diffeomorphism dt level to obtain the physical states. In other words, the transformations associated with the reparametriza- ”Schr¨dinger equation” corresponding to the anisotropic o tion invariance. This result can be contrasted with the mechanics (3) is same result obtained by Hoˇava’s in his gravity model [1]. r ∂ψ √ z In this way we saw that the action Eq.(3), has very i = G − 2 2 2 + V (x) ψ. (21) ∂t similar properties to the Hoˇava’s gravity namely the r It is interesting to notice that this type of ”Schr¨dinger o same scaling properties and the same invariance un- equation” has been already considered in the literature der reparametrizations. In this sense, we see that our (see for example [12]). The original motivation behind toy model has the same transformation properties than this equation comes from anomalous diffusion equation the Hoˇava’s gravity just as the relativistic particle has r and L´vy flights. For a perspective from Hoˇava gravity e r the same transformation properties as General Relativity see [13] where a relation with spectral dimension and frac- [17]. tals was proposed. Furthermore, a proposal for the free particle quantum-mechanical kernel associated to (21) IV. EQUATIONS OF MOTION was given in terms of the Fox’s H function [14]. On the other hand, in the limit z → ∞, the Schr¨dinger equa- o tion follows from the action (7), for arbitrary potential From the variation of the action (3), and taking into 2 account the definitions of the momenta pt and pi , it fol- we get a primary constraint p2 − m = 0 and several sec- 4 lows that ondary constraints. In the case V = x2 , a similar system has been considered in [15]. For V = 0, we only have d 2 δS = dτ (pi δxi + pt δt) (24) now the first class constraint p2 − m = 0, that at the 4 dτ quantum level corresponds to dpi ∂V dpt −δxi + + δt , 2 2 m2 dτ ∂xi dτ + ψ = 0. (22) 4 where the Euler-Lagrange equations are This equation corresponds to the Klein-Gordon equation with a Euclidean signature metric. dpi ∂V + = 0, (25) dτ ∂xi d Gauge of freedom pt = 0. (26) dτ Notice that for z = 2 and τ = t the equation (25) Because of the action (3) is invariant under reduces to the usual Newton’s second law, and that (26) reparametrizations, there is a local symmetry. The ex- correspond to the energy conservation. tended canonical action for the system is [16] In order to simplify some computations we will use the S= ˙ dτ tpt + xi pi − λφ ˙ (23) gauge condition τ = t. This is a good gauge condition in
  4. 4. 4 the sense that {φ, τ −t} = 0. Under the above assumption backgrounds and some applications and examples the and using the definition pi we get the equations of motion Schroedinger symmetry was studied in [22]. The genera- tors of the Schr¨dinger algebra include temporal transla- o d mz 2−z ∂V tions H, spatial translations P i , rotations J ij , Galilean x2 ˙ 2(z−1) xi ˙ + = 0, (27) dt 2(z − 1) ∂xi boost K i , dilatation D (where time and space can dilate with different factors), special conformal transformation We thus get C, and the mass operator M .The nonzero commutators z−22(z − 1) ∂V of this algebra are the Galilean algebra gij xj = − x2 ¨ ˙ 2(z−1) , mz ∂xi J ij , J kl = (δ ik J jl + δ jl J ik − δ il J jk − δ jk J il ), 2 − z xi xj ˙ ˙ gij = δij + . (28) J ij , P k = (δ ik P j − δ jk P i ), z − 1 x2˙ J ij , K k = (δ ik K j − δ jk K i ), The matrix gij can be considered as a metric that de- H, K i = P i pends on the velocities. We see that this metric is ho- mogeneous of degree zero in the velocities and in con- P i , K j = −δ ij M, (32) sequence is a Finsler type metric [18]. This metric has three critical points: plus Dilatations z = 1, z = 2, z = ∞. (29) D, P i = P i , D, K i = K i , {D, H} = −2H, (33) For z = ∞, we obtain the inverse metric and Special Conformal transformations xi xj ˙ ˙ {D, C} = −2C, {H, C} = D, C, P i = K i . (34) g ij = δ ij − (2 − z) , (30) x2 ˙ It is interesting to notice that H, C, D close themselves it follows that in an SL(2, R) subalgebra. Thus the Schr¨dinger alge- o bra is the Galilean algebra plus dilatation and Special 2(1 − z) 2 z−2 xi xj ˙ ˙ ∂V Conformal transformations [23]. M is the center of the xi = ¨ x ˙ 2(z−1) δij + (z − 2) . (31) mz x2 ˙ ∂xj algebra. It is also worth noticing that the Schr¨dinger algebra o The limit z = ∞ is particularly interesting, since in in d dimensions, Schr(d), can be embedded into the rela- this case the matrix gij is not invertible. Because of in tivistic conformal algebra in d + 2 space-time dimensions this limit we get the action (7) and in this case is not O(d + 2, 2). This fact is central in the recent literature more valid the constraint (16). Furthermore, from the about the geometric content of the new AdS/NRCFT Hamiltonian analysis of the system appears secondary dualities (for details see [10]). constraints that transform the model in a system with We are interested in the construction of the explicit second class constraints [15]. generator of the relative Poisson-Lie Schr¨dinger algebra o for any z. These will be the symmetries of the corre- sponding anisotropic Schr¨dinger equation (21) associ- o V. DYNAMICAL SYMMETRIES OF THE ated with the non-relativistic anisotropic classical me- ¨ SCHRODINGER EQUATION FOR ANY z chanics (3). We will denote the generalization of the Schr¨dinger o In the same sense as the Galilei group exhaust all the symmetry algebra for any z as Schrz (d) in d dimensions. symmetries of the nonrelativistic free particle in d dimen- The algebra is given by the Galilean algebra (32) plus sions the Schr¨dinger Algebra is the algebra of symme- o tries of the Schr¨dinger equation with V = 0 and can o D, P i = P i , D, K i = (1 − z)K i , be considered as the non relativistic limit of the Con- {D, M } = (2 − z)M, {D, H} = −zH. (35) formal symmetry algebra. Another interesting Galilean Conformal Algebra can also be constructed as the non Notice that M is not playing the role of the center any- relativistic contraction of the full Conformal Relativistic more unless z = 2 and that C, the generator of the Spe- Algebra [19]. In the special case of d = 2 a I¨non¨- o u cial Conformal transformations is not in the algebra [24]. Wigner contraction of the Poincar´ symmetry of a free e We are not aware of any explicit realization of this alge- anyon theory is reduced to a Galilean symmetry with two bra in phase space for arbitrary dynamical exponent z. central charges (see for example [20] and references there For z = 2 the phase space realization reads in). The Schr¨dinger symmetry as a conformal symme- o try of non relativistic particle models was introduced J ij = xi pj − xj pi , H = −pt , P i = pi , some time ago [21] in a holographic context relating a K i = −M xi + tpi system in AdSd+1 with a massless particle in d dimen- 1 sional Minkowski space time. As isometries of certain C = t2 pt + txi pi − M x2 , D = 2tpt + xi pi . 2
  5. 5. 5 Inspired by this construction and the embedding of the For the Galilean boost we have Schr¨dinger algebra in d dimensions into the full rela- o K i = tpi + ρM (p2 )xi , tivistic Conformal algebra in d + 2 [10] we will display a set of dynamical symmetries of the Schr¨dinger equation o and the solution for the constant ρ using the condition for any z. (38) is The crux of the argument is to allow M to depend on 1 the magnitude of pi squared M (p2 ). We will make the ρ=− . anzats that M is a homogeneus function of degree α in (1 − α) pi So we have the non trivial dynamical symmetries of the anisotropic Schr¨dinger equation generated by o ∂M i p = 2αM, M (p2 ) = A(p2 )α , (36) ∂pi D = ztpt + xi pi , (40) 2 z z 1 where A is a constant. Then taking as our Schr¨dinger o C = t2 pt + txi pi − M (p2 )x2 , (41) equation (16) with V (x) = 0 and for any z we have 4 2 2 that are dilatation and Special Conformal transforma- p2 tions and d Galilean boosts S = pt + H = pt + , (37) z 2M (p2 ) K i = tpi − M (p2 )xi . (42) 2 we will ask for the phase space quantities that commutes A similar result was obtained previously in [7] for a differ- (in Poisson sense) with the equation (37), i.e., ential representation of the algebra (35) using the concept {O, S} = 0, (38) of fractional derivatives. Unfortunately the dynamical symmetries given by over S = 0. Any such phase space quantity will be a dy- (40), (41) and (42) plus J ij , P i , H, M do not close in namical symmetry of the anisotropic Schr¨dinger equa- o Schrz (d) (35). Nevertheless a subalgebra formed by tion (17). α is fixed in terms of the dynamical exponent H, C, D indeed close into an SL(2, R) sector of the full z by relating our definition (37) with the first class con- algebra, straint (17) z {D, H} = −zH, {D, C} = −zC, {H, C} = D. 2 2−z 1 α= , A= √ , (39) To these generator we can add the angular momentum 2 2 G J ij that commutes with them. From the other hand the anisotropic system (3) is also invariant under the J ij , P i , H are obvious symmetries of the anisotropic Poincar´ symmetry alone. e Schr¨dinger equation. To see what are the analogs of o D, C and K i let us start with D as defined for the z = 2 case but with an anisotropic reescaling between space VI. THERMODYNAMIC PROPERTIES and time D = tpt + βxi pi . In the following we shall sketch some of the thermo- dynamic properties of the system. We will consider only By taking the Poisson bracket with S we have the case without potential. In this case the energy of our model is β(1 − α)p2 √ {D, S} = pt + , H = Gpz . (43) M (p2 ) 1 Let us denote by V the volume and β = kT , with T the then temperature and k the Boltzmann constant, the canon- 1 ical partition function for the system in a space of d di- β= . mensions is 2(1 − α) V √ z V √ z Using the same idea with the Special Conformal genera- Z = N dpe−β Gp = N dΩd dppd−1 e−β Gp tor we propose VΩd ∞ √ z = N dppd−1 e−β Gp . (44) C = t2 pt + γtxi pi + σM (p2 )x2 , 0 √ Using u = β Gpz , and the definition of the Gamma where γ, σ are constants to be determined from the re- function we obtain quirement (38). The solutions are VΩd d 1 2 Z= Γ . (45) √ d z γ=− , σ=− . z N β G z (1 − α) (1 − α)2
  6. 6. 6 In this way, if N is the number of particles, the Helmholtz temporal coordinates. This anisotropic mechanical sys- free energy is given by tem is consistent with the non-relativistic Hoˇava’s dis- r   persion relation. Furthermore, it was shown that for our particle model the system has the same local symmetries  VΩd d  as Hoˇava Gravity. Another interesting point is that the r F = −kT N ln Z = −kT N ln  Γ (46) . √ d z z anisotropic mechanics proposed here corresponds to the z N β G Conformal Mechanics of [4] for z = 2 and is a general- ization of it for arbitrary z. Also our system includes, in From this expression we may now obtain all the ther- the limit z → ∞, a relativistic particle with Euclidean modynamic properties of the system, in particular the signature and in the double scaling limit z → 1, m → 0 internal energy is m with z−1 constant a massless relativistic particle. From ∂ ln Z d the equations of motion we show that naturally emerges U = N kT 2 = N kT. (47) a Finsler type metric that could implies that the Hoˇava’s r ∂T z gravity can be related to Finsler geometry. Also, we stud- where N is the total number of particles in the gas. Sub- ied the dynamical symmetries of our model and we found stituting the number of particles N in terms of the parti- all dynamical symmetries of the anisotropic Schr¨dinger o tion function (Z = N in our normalization) the internal equation for arbitrary z that correspond to the genera- energy can be written as tors of the Schr¨dinger algebra Schrz (d). We show that o the full Schr¨dinger algebra constructed from our genera- o d VΩd Γ z d d+z tors does not close. However we found that a subalgebra U= (kT ) z . (48) √ d z z SL(2, R) indeed close. An interesting point is that the z N G explicit realization of the generators is not linear and are analogous to the realization given in [7]. As a final Whereas the entropy is point, we remark that the thermodynamic properties of d+z our model reproduces the same thermodynamic proper- d 1 ∂U VΩd Γ z (k) z d d d + z ties of the recently proposed anisotropic black branes. S= dT = (T ) z (49) . T ∂T V √ d z z d z N G Note added. While this article was in the peer re- viewing process at PRD several papers addressed issues We conclude from (48) and (49), that the relationship related to our work [25–29]. between energy and entropy in our system is given by ST d U= . (50) d+z This relationship is exactly the obtained in the case of black branes [8]. Acknowledgments VII. CONCLUSIONS This work was partially supported under grants In this work we have introduced an action invariant CONACyT-SEP 55310, CONACyT 50-155I, DGAPA- under anisotropic transformations between spatial and UNAM IN109107 and DGAPA-UNAM IN116408. [1] P. Hoˇava, Phys. Rev. D, 79 084008 (2009). r (1999), hep-th/9911066. arXiv:0901.3775. [6] H. E. Camblong, C. R. Ordonez, Phys. Rev. D 68 125013 [2] P.C. Hohenberg, B.I. Halperin, Theory of Dynamic Crit- (2003), hep-th/0303166. ical Phenomena, Rev. Mod. Phys. 49 435 (1977). [7] M. Henkel, Nucl. Phys. B 641, 405 (2002) [arXiv:hep- [3] J. L. Cardy, Scaling And Renormalization In Statistical th/0205256]. Physics, Cambridge, UK: Univ. Pr. (1996). [8] G. Bertoldi, B. A. Burrington, A. Peet, arXiv:0907.4755. [4] V. Alfaro, S. Fubini, G. Furlan, Nuovo Cim. A 34, 569 [9] J.L.F. Barbon, C.A. Fuertes JHEP 0809:030 (2008), (1976). arXiv:0806.3244. [5] R. Britto-Pacumio, J. Michelson, A. Strominger, A. [10] D. T. Son, Phys. Rev. D 78, 046003 (2008) Volovich, Lectures on superconformal quantum mechanics [arXiv:0804.3972 [hep-th]]. and multiblack hole moduli spaces, Contributed to NATO [11] P. A. M. Dirac, Lectures on Quantum Mechanics, Dover, Advanced Study Institute on Quantum Geometry. Pub- New York, (2001). lished in Progress in string theory and M-theory, 235-264 [12] N. Laskin, Phys. Lett. A 268, 298 (2000) [arXiv:hep-
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