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# Conformal Anisotropic Mechanics And The HořAva Dispersion Relation

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### Conformal Anisotropic Mechanics And The HořAva Dispersion Relation

2. 2. ROMERO et al. PHYSICAL REVIEW D 81, 065013 (2010) Z 2 z=2 from an action in curved space where some of its isome- m ðx Þ _ tries are the scale transformations. Another way to imple- S¼ d þ ðz À 2Þe : (9) 2ðz À 1Þ ezÀ2 t_ ment scale transformations is to start from an action in ﬂat space but constructed in such a way that the invariance Here, we have introduced an einbein e in a similar way as under scale transformations is manifest. In this work we the standard trick to remove the square root in the action of will take this second point of view. Let us consider a d þ the free relativistic particle. Using this action (9) we can 1-dimensional space-time, with an action principle given now take the limit z ! 1 if at the same time we implement by the limit m ! 0. We assume that m ! 0 at the same rate as z À 1 in such a way that zÀ1 ! , where plays the role m Z Z m ðx2 Þz=2ðzÀ1Þ _ dt S ¼ dL ¼ d À VðxÞt~ _ ; t_ ¼ ; of a mass parameter. In consequence we obtain the action 2 ðtÞ _ 1=zÀ1 d pﬃﬃﬃﬃﬃ Z e x2 _ xi ¼ _ dxi ; x2 ¼ xi xi ; _ _ _ i ¼ 1; 2; Á Á Á ; d: (3) S ¼ d À1 : (10) d 2 t_ This action is invariant under reparametrizations of , i.e., This action is still invariant under reparametrizations due under the transformations ! ¼ ðfÞ, if we require that to the transformation rule of the einbein e ! df e, and it d 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 VðbxÞ ¼ bÀz VðxÞ, for example, ~ ~ Z x x _ _ a S ¼ d : (11) VðxÞ ¼ z ; ~ a ¼ const; (4) 2e jxj ~ the action (3) is invariant under the scaling laws (1). Note III. CANONICAL FORMALISM that, when z ¼ 2 and ¼ t, we obtain the action for a nonrelativistic particle; furthermore taking into account Let us consider the canonical formalism of the system (4), we obtain (3). The canonical momenta are Z m a @L mz ðx2 Þ2Àz=2ðzÀ1Þ _ S ¼ dt x À 2 ; _ 2 (5) pi ¼ ¼ _ xi ; (12) 2 jxj ~ _ @x i 2ðz À 1Þ ðtÞ1=zÀ1 _ which corresponds to the action of the conformal mechan- ics [3]. In the case of arbitrary z and for a gauge condition @L m ðx2 Þz=2ðzÀ1Þ _ pt ¼ ¼À þ VðxÞ ¼ ÀH; ~ ¼ t we get @t_ 2ðz À 1Þ ðtÞz=zÀ1 _ Z m a (13) S ¼ dt ðx2 Þz=2ðzÀ1Þ À z ; _ (6) 2 jxj ~ and as expected (by the reparametrization invariance) the which again is invariant under the scaling laws (1). In this canonical Hamiltonian vanishes, sense Eq. (3) represents a generalized conformal Hc ¼ pi xi þ pt t_ À L ¼ 0: _ (14) mechanics. Another interesting property of the action (3) appears in From (12) we get the limit z ! 1. In this case for an arbitrary potential we mz ðx2 Þ1=2ðzÀ1Þ 2 _ obtain p2 ¼ pi pi ¼ ; (15) 2ðz À 1Þ ðtÞ1=zÀ1 _ Z m pﬃﬃﬃﬃﬃ S ¼ d x2 À VðxÞt_ ; _ ~ (7) which in turn implies 2 Àz m mz this action is invariant under reparametrizations. In the ¼ pt þ ðp2 Þz=2 þ VðxÞ % 0: ~ 2ðz À 1Þ 2ðz À 1Þ particular case of the potential (4) the limit is singular in the region 0 jxj 1; then leaving VðxÞ ¼ 0 we obtain ~ (16) the action for a relativistic particle with Euclidean signa- In the simple case VðxÞ ¼ 0 when we enforce the con- ~ ture, straint we obtain Z m pﬃﬃﬃﬃﬃ Àz S ¼ d x2 : m mz _ (8) pt ¼ À ðp2 Þz=2 ; (17) 2 2ðz À 1Þ 2ðz À 1Þ The case z ! 1 is also very interesting. Even though this which can be rewritten in the form limit is not well deﬁned for the action (3) it can be 2ð1ÀzÞ implemented using an equivalent form of the action (3). 1 m ðpt Þ2 À Gðp2 Þz ¼ 0; G¼ : (18) In the case VðxÞ ¼ 0 we get ~ z2z 2ðz À 1Þ 065013-2
3. 3. CONFORMAL ANISOTROPIC MECHANICS AND THE . . . PHYSICAL REVIEW D 81, 065013 (2010) pﬃﬃﬃﬃ In this way, we get a dispersion relation that is similar to xi ¼ fxi ; g ¼ Gzðp2 ÞzÀ2=2 pi ; ˇ that proposed in Horava gravity [Eq. (2)]. The relation (17) @V implies that the momentum pt is negative for z 1. In pi ¼ fpi ; g ¼ À ; t ¼ ft; g ¼ ; order to preserve the equivalence between (17) and (18) for @xi z 1 we assume that pt takes only negative values in (18). @ Notice that by taking the limits z ! 1 and m ! 0 in (16) pt ¼ 0; ¼ : @ we get the constraint Here f; g are the Poisson brackets. The above transforma- ¼ pt þ ðp2 Þ1=2 þ VðxÞ: ~ (19) tions are of the usual type for the parametrized particle. However, the rule for the transformation corresponding to In the case VðxÞ ¼ 0 this constraint is consistent with the the spatial coordinates seems to change drastically. But if canonical formalism of the action (9) and corresponds to a we use the deﬁnition of the momenta (12) we will get massless relativistic free particle. i exactly the usual transformation xi ¼ dx recovering Now, from the Eq. (16) the system has one ﬁrst class dt constraint and, following the standard Dirac’s method [11], the full diffeomorphism transformations associated with the ‘‘wave equation’’ the reparametrization invariance. This result can be con- pﬃﬃﬃﬃ ˇ trasted with the same result obtained by Horava’s in his j c ¼ ðpt þ Gðp2 Þz=2 þ VðxÞÞj c ¼ 0 ^ ^ ^ ~ (20) gravity model [10]. In this way we saw that the action Eq. (3) has very must be implemented on the Hilbert space at quantum level ˇ similar properties to Horava’s gravity, namely, the same to obtain the physical states. In other words, the scaling properties and the same invariance under repara- ¨ ‘‘Schrodinger equation’’ corresponding to the anisotropic metrizations. In this sense, we see that our toy model has mechanics (3) is ˇ the same transformation properties as Horava’s gravity, @c pﬃﬃﬃﬃ just as the relativistic particle has the same transformation i@ ¼ ð GðÀ@2 r2 Þz=2 þ VðxÞÞ c : ~ (21) properties as general relativity [17]. @t ¨ It is interesting to notice that this type of ‘‘Schrodinger IV. EQUATIONS OF MOTION equation’’ has been already considered in the literature (see, for example, [12]). The original motivation behind From the variation of the action (3), and taking into this equation comes from the anomalous diffusion equation account the deﬁnitions of the momenta pt and pi , it follows ´ ˇ and Levy ﬂights. For a perspective from Horava gravity see that Z d [13] where a relation with spectral dimension and fractals was proposed. Furthermore, a proposal for the free particle dpi @V S ¼ d ðp x þ pt tÞ À xi þ quantum-mechanical kernel associated to (21) was given in d i i d @xi terms of the Fox’s H function [14]. On the other hand, in dp the limit z ! 1, the Schrodinger equation follows from ¨ þ t t ; (24) d the action (7); for arbitrary potential we get a primary constraint p2 À m ¼ 0 and several secondary constraints. 2 4 where the Euler-Lagrange equations are In the case V ¼ x2 , a similar system has been considered in [15]. For V ¼ 0, we only have now the ﬁrst class constraint dpi @V þ ¼ 0; (25) p2 À m ¼ 0, which at the quantum level corresponds to 2 d @xi 4 2 @2 r2 þ m c ¼ 0: (22) d 4 pt ¼ 0: (26) d This is the expected Helmholtz equation and may be Notice that for z ¼ 2 and ¼ t Eq. (25) reduces to the viewed as the Klein-Gordon equation with a Euclidean usual Newton’s second law, and that (26) corresponds to signature metric. the energy conservation. Because the action (3) is invariant under reparametriza- In order to simplify some computations we will use the tions, there is a local symmetry. The extended canonical gauge condition ¼ t. This is a good gauge condition in action for the system is [16] the sense that f; À tg Þ 0. Under the above assumption Z and using the deﬁnition pi we get the equations of motion S ¼ dðtpt þ xi pi À Þ _ _ (23) d mz @V ðx2 Þð2ÀzÞ=2ðzÀ1Þ xi þ _ _ ¼ 0: (27) with a Lagrange multiplier. dt 2ðz À 1Þ @xi If we require that is a function of , the extended action is invariant under the gauge transformations [16] We thus get 065013-3
4. 4. ROMERO et al. PHYSICAL REVIEW D 81, 065013 (2010) 2ðz À 1Þ @V this algebra are the Galilean algebra gij xj ¼ Àðx2 ÞðzÀ2Þ=2ðzÀ1Þ € _ ; mz @xi fJ ij ; J kl g ¼ ðik J jl þ jl J ik À il J jk À jk J il Þ; (28) 2 À z xi xj _ _ gij ¼ ij þ : fJ ij ; Pk g ¼ ðik Pj À jk Pi Þ; z À 1 x2 _ fJ ij ; K k g ¼ ðik K j À jk Ki Þ; (32) The matrix gij can be considered as a metric that depends on the velocities. We see that this metric is homogeneous fH; Ki g ¼ Pi ; of degree zero in the velocities and in consequence is a fPi ; Kj g ¼ Àij M; Finsler type metric [18]. This metric has three critical points: plus dilatations z ¼ 1; z ¼ 2; z ¼ 1: (29) fD; Pi g ¼ Pi ; fD; Ki g ¼ K i ; fD; Hg ¼ À2H; For z Þ 1, we obtain the inverse metric (33) xi xj _ _ and special conformal transformations gij ¼ ij À ð2 À zÞ 2 ; (30) _ x fD; Cg ¼ À2C; fH; Cg ¼ D; fC; Pi g ¼ Ki : and it follows that (34) 2ð1 À zÞ 2 ðzÀ2Þ=ð2ðzÀ1ÞÞ _ _ xi xj @V xi ¼ € ðx Þ _ ij þ ðz À 2Þ 2 : It is interesting to notice that H, C, D close themselves in mz x @xj _ an SLð2; RÞ subalgebra. Thus the Schrodinger algebra is ¨ (31) the Galilean algebra plus dilatation and special conformal transformations [24]. M is the center of the algebra. The limit z ¼ 1 is particularly interesting, since in this ¨ It is also worth noticing that the Schrodinger algebra in d case the matrix gij is not invertible. In this limit we get the dimensions, SchrðdÞ, can be embedded into the relativistic action (7) and in this case the constraint (16) is not valid. conformal algebra in d þ 2 space-time dimensions Oðd þ Furthermore, from the Hamiltonian analysis of the system 2; 2Þ. This fact is central in the recent literature about the there appear secondary constraints that transform the geometric content of the new anti–de Sitter/nonrelativistic model into a system with second class constraints [15]. conformal ﬁeld theories dualities (for details see [9]). We are interested in the construction of the explicit V. DYNAMICAL SYMMETRIES OF THE ¨ generator of the relative Poisson-Lie Schrodinger algebra SCHRODINGER EQUATION FOR ANY z ¨ for any z. These generators will be the symmetries of the ¨ corresponding anisotropic Schrodinger equation (21) asso- In the same sense that the Galilei group with no central ciated with the nonrelativistic anisotropic classical me- charges [19] exhausts all the symmetries of the nonrelativ- chanics (3). ¨ istic free particle in d dimensions, the Schrodinger algebra We will denote the generalization of the Schrodinger¨ ¨ is the algebra of symmetries of the Schrodinger equation symmetry algebra for any z as Schrz ðdÞ in d dimensions. with V ¼ 0 and can be considered as the nonrelativistic The algebra is given by the Galilean algebra (32) plus limit of the conformal symmetry algebra. Another interest- ing Galilean conformal algebra can also be constructed as fD; Pi g ¼ Pi ; fD; Ki g ¼ ð1 À zÞKi ; the nonrelativistic contraction of the full conformal rela- (35) tivistic algebra [21]. In the special case of d ¼ 2 a Iononu- ¨ ¨ fD; Mg ¼ ð2 À zÞM; fD; Hg ¼ ÀzH: ´ Wigner contraction of the Poincare symmetry of a free Notice that M is not playing the role of the center anymore anyon theory is reduced to a Galilean symmetry with two unless z ¼ 2 and that C, the generator of the special central charges (see, for example, [22] and references conformal transformations, is not in the algebra [25]. We ¨ therein). The Schrodinger symmetry as a conformal sym- are not aware of any explicit realization of this algebra in metry of nonrelativistic particle models was introduced phase space for arbitrary dynamical exponent z. For z ¼ 2 some time ago [20] in a holographic context relating a the phase space realization reads system in AdSdþ1 with a massless particle in d-dimensional Minkowski space-time. As isometries of J ij ¼ xi pj À xj pi ; H ¼ Àpt ; Pi ¼ pi ; certain backgrounds and some applications and examples, ¨ the Schrodinger symmetry was studied in [23]. The gen- K i ¼ ÀMxi þ tpi C ¼ t2 pt þ txi pi À 1Mx2 ; 2 ¨ erators of the Schrodinger algebra include temporal trans- D ¼ 2tpt þ xi pi : lations H, spatial translations Pi , rotations J ij , Galilean boost Ki , dilatation D (where time and space can dilate Inspired by this construction and the embedding of the with different factors), special conformal transformation ¨ Schrodinger algebra in d dimensions into the full relativ- C, and the central charge M. The nonzero commutators of istic conformal algebra in d þ 2 [9], we will display a set of 065013-4
5. 5. CONFORMAL ANISOTROPIC MECHANICS AND THE . . . PHYSICAL REVIEW D 81, 065013 (2010) ¨ dynamical symmetries of the Schrodinger equation for any 1 ¼À : z. ð1 À Þ The crux of the argument is to allow M to depend on the magnitude of pi squared Mðp2 Þ. We will make the ansatz So we have the nontrivial dynamical symmetries of the that M is a homogeneous function of degree in pi ¨ anisotropic Schrodinger equation generated by @M i D ¼ ztpt þ xi pi ; (40) p ¼ 2M; Mðp2 Þ ¼ Aðp2 Þ ; (36) @pi z2 2 z 1 ¨ where A is a constant. Then taking as our Schrodinger C¼ t pt þ txi pi À Mðp2 Þx2 ; (41) 4 2 2 equation (16) with VðxÞ ¼ 0 and for any z, we have ~ that are dilatation and special conformal transformations p2 and d Galilean boosts S ¼ pt þ H ¼ pt þ : (37) 2Mðp2 Þ z Ki ¼ tpi À Mðp2 Þxi : (42) We will ask for the phase space quantities that commute (in 2 the Poisson sense) with Eq. (37), i.e., A similar result was obtained previously in [6] for a fO; Sg ¼ 0; (38) differential representation of the algebra (35) using the concept of fractional derivatives. over S ¼ 0. Any such phase space quantity will be a Unfortunately the dynamical symmetries given by (40)– ¨ dynamical symmetry of the anisotropic Schrodinger equa- (42), plus J ij , Pi , H, M, do not close in Schrz ðdÞ (35). tion (17). is ﬁxed in terms of the dynamical exponent z Nevertheless a subalgebra formed by H, C, D indeed by relating our deﬁnition (37) with the ﬁrst class constraint closes into an SLð2; RÞ sector of the full algebra, (17) z fD; Hg ¼ ÀzH; fD; Cg ¼ ÀzC; fH; Cg ¼ D: 2Àz 1 2 ¼ ; A ¼ pﬃﬃﬃﬃ : (39) 2 2 G To these generators we can add the angular momentum J ij that commutes with them. J ij , Pi , H are obvious symmetries of the anisotropic ¨ Schrodinger equation. To see what are the analogs of D, C, and Ki , let us start with D as deﬁned for the z ¼ 2 case VI. THERMODYNAMIC PROPERTIES but with an anisotropic rescaling between space and time In the following we shall sketch some of the thermody- D ¼ tpt þ
6. 6. xi pi : namic properties of the system. We will consider only the case without potential. In this case the energy of our model By taking the Poisson bracket with S we have is
7. 7. ð1 À Þp2 pﬃﬃﬃﬃ fD; Sg ¼ pt þ ; H ¼ Gpz : (43) Mðp2 Þ Let us denote by V the volume and
8. 8. ¼ kT , with T the 1 then temperature and k the Boltzmann constant, that the canoni- 1 cal partition function for the system in a space of d dimen-
9. 9. ¼ : sions is 2ð1 À Þ V Z pﬃﬃﬃ z V Z pﬃﬃﬃ z Using the same idea with the special conformal generator Z ¼ N dpeÀ
10. 10. Gp ¼ N dd dppdÀ1 eÀ
11. 11. Gp ~ we propose @ @ Vd Z 1 pﬃﬃﬃ z C ¼ t2 pt þ txi pi þ Mðp2 Þx2 ; ¼ N dppdÀ1 eÀ
12. 12. Gp : (44) @ 0 pﬃﬃﬃﬃ where , are constants to be determined from the re- Using u ¼
13. 13. Gpz , and the deﬁnition of the gamma func- quirement (38). The solutions are tion, we obtain 1 2 ¼À ¼À Vd d ð1 À Þ ; ð1 À Þ2 : Z ¼ N pﬃﬃﬃﬃ d=z À : (45) z@ ð
14. 14. GÞ z For the Galilean boost we have In this way, if N is the number of particles, the Helmholtz Ki ¼ tpi þ Mðp2 Þxi ; free energy is given by Vd d and the solution for the constant using the condition (38) F ¼ ÀkTN lnZ ¼ ÀkTN ln N pﬃﬃﬃﬃ d=z À : (46) is z@ ð
15. 15. GÞ z 065013-5
16. 16. ROMERO et al. PHYSICAL REVIEW D 81, 065013 (2010) From this expression we may now obtain all the thermody- Furthermore, it was shown that for our particle model the namic properties of the system. In particular, the internal ˇ system has the same local symmetries of Horava gravity. energy is Another interesting point is that the anisotropic mechanics @ lnZ d proposed here corresponds to the conformal mechanics of U ¼ NkT 2 ¼ NkT; (47) [3] for z ¼ 2 and is a generalization of it for arbitrary z. @T z Also our system includes, in the limit z ! 1, a relativistic where N is the total number of particles in the gas. particle with Euclidean signature and, in the double scaling Substituting the number of particles N in terms of the limit z ! 1, m ! 0 with zÀ1 constant, a massless relativ- m partition function (Z ¼ N in our normalization) the inter- istic particle. From the equations of motion we showed that nal energy can be written as there naturally emerges a Finsler type metric that could Vd ÀðdÞ d ˇ imply that Horava’s gravity can be related to Finsler ge- U¼ pﬃﬃﬃﬃ z ðkTÞðdþzÞ=z : (48) ometry. Also, we studied the dynamical symmetries of our z@N ð GÞd=z z model and we found all dynamical symmetries of the Whereas the entropy is ¨ anisotropic Schrodinger equation for arbitrary z that cor- ¨ respond to the generators of the Schrodinger algebra Z 1 @U Vd ÀðdÞðkÞðdþzÞ=z d d=z d þ z Schrz ðdÞ. We showed that the full Schrodinger algebra ¨ S ¼ dT ¼ pﬃﬃﬃﬃ z ðTÞ : T @T V z@N ð GÞd=z z d constructed from our generators does not close. However, (49) we found that a subalgebra SLð2; RÞ indeed closes. An interesting point is that the explicit realization of the gen- We conclude from (48) and (49) that the relationship erators is not linear and is analogous to the realization between energy and entropy in our system is given by given in [6]. As a ﬁnal point, we remarked that the ther- STd modynamic properties of our model reproduces the same U¼ : (50) thermodynamic properties of the recently proposed aniso- dþz tropic black branes. This relationship is exactly that one obtained in the case of black branes [7]. ACKNOWLEDGMENTS This work was partially supported under grants VII. CONCLUSIONS CONACyT-SEP 55310, CONACyT 50-155I, DGAPA- In this work we introduced an action invariant under UNAM IN109107, and DGAPA-UNAM IN116408. anisotropic transformations between spatial and temporal Note added.—While this article was in the peer review- coordinates. This anisotropic mechanical system is consis- ing process at Physical Review D, several papers addressed ˇ tent with the nonrelativistic Horava’s dispersion relation. issues related to our work [26–30]. [1] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, ˇ [10] P. Horava, Phys. Rev. D 79, 084008 (2009). 435 (1977). [11] P. A. M. Dirac, Lectures on Quantum Mechanics (Dover, [2] J. L. Cardy, Scaling and Renormalization in Statistical New York, 2001). Physics (Cambridge University Press, Cambridge, [12] N. Laskin, Phys. Lett. A 268, 298 (2000); Phys. Rev. E 62, England, 1996). 3135 (2000); 66, 056108 (2002). [3] V. Alfaro, S. Fubini, and G. Furlan, Nuovo Cimento Soc. [13] P. Horava, Phys. Rev. Lett. 102, 161301 (2009). Ital. Fis. 34A, 569 (1976). [14] A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H- [4] R. Britto-Pacumio, J. Michelson, A. Strominger, and A. Function: Theory and Applications (Springer, New York, Volovich, in Progress in String Theory and M-theory, 2010). edited by Laurent Baulieu, Michael Green, Marco Picco, [15] M. S. Plyushchay, Mod. Phys. Lett. A 4, 837 (1989); M. S. and Paul Windey (Kluwer, The Netherlands, 2001), p. 235. Plyushchay and A. V. Razumov, Int. J. Mod. Phys. A 11, [5] H. E. Camblong and C. R. Ordonez, Phys. Rev. D 68, 1427 (1996). 125013 (2003). [16] M. Henneaux and C. Teitelboim, Quantization of Gauge [6] M. Henkel, Nucl. Phys. B641, 405 (2002). System (Princeton University Press, Princeton, NJ, 1992). [7] G. Bertoldi, B. A. Burrington, and A. Peet, Phys. Rev. D [17] C. Teitelboim, Phys. Rev. D 25, 3159 (1982). 80, 126004 (2009). [18] F. Girelli, S. Liberati, and L. Sindoni, Phys. Rev. D 75, [8] J. L. F. Barbon and C. A. Fuertes, J. High Energy Phys. 09 064015 (2007). (2008) 030. [19] The Galilei symmetry of the free particle action can be [9] D. T. Son, Phys. Rev. D 78, 046003 (2008). ¨ central extended to Schrodinger symmetry by the inclu- 065013-6
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