1. Anisotropic Mechanics J.M. Romero, V. Cuesta, J.A. Garcia, and J. D. Vergara Instituto de Ciencias Nucleares, UNAM, Mexico.

2. Introduction Action Principle Canonical Formalism Equations of Motion Dynamical Symmetries Massless case, nonlinear realization of the generators Thermodynamic Properties Conclusions

3. Introduction In recent years, a new class of anisotropic scaling transformations has been considered in the context of critical phenomena {Henkel}, string theory {Son, Barbon, Peet, …} and quantum gravity{Horava,…} where plays the role of a dynamical critical exponent. In particular the result of Horava is quite interesting since it produces a theory that perhaps reduces to general relativity at large scales and may provide a candidate for a UV completion of general relativity.

4. In the IR these systems approach the usual General Relativity with local Lorentz invariance but in the UV the formulation admit generalized dispersion relations of the form As the dispersion relations used by these models are quadratic in while the spatial momentum scale as the models are in principle renormalizable by power counting arguments at least for.

5. In the Horava theory, one hopes to obtain a higher spatial derivative theory of gravity. The idea of the renormalizability of this theory is the following {Germani, et al.}. Consider the linearized gravity In the case of the Einstein-Hilbert action, this can be expanded as Where the second term is a typical self-interaction term and is the gravitational coupling constant, with mass dimension

6. For perturbative renormalization, one defines Expanding around the fixed point we get Since has dimensions of length one expects that. Quantum mechanically the gravitational interaction is irrelevant and this implies that the self-coupling spoils the renormalizability at the Gaussian fixed point.

7. In the Horava theory, one hopes to obtain a higher spatial derivative theory of gravitation, but only quadratic in time The coupling constant for the higher spatial derivative part has negative mass units Now, using the redefinitions

8. We get, with These parameters have now positive mass dimension. Then, the dangerous irrelevant operator of GR has now been turned into a relevant operator at the Lifshitz point.

9. Action Principle Let us consider a d+1-dimensional space-time, for a parametrized non-relativistic particle, These system has the following constraint:

10. To obtain a dispersion relation of the Horava type, we can start from the action where: This action is also invariant under reparametrization, i.e.: With the coordinates and the time are scalars under the reparametrization.

11. Furthermore, if we consider that the potential: Now, for and in the gauge we obtain For arbitrary results: The action (3) will be invariant under the anisotropic scale transformations (1).

12. Introducing an einbein the action (3) has the form Now, in the limit,

13. Canonical momenta This theory has a constraint Canonical Formalism

14. Which can be rewritten, for V(x)=0, in the form In this way, we get a dispersion relation that is very similar to Horava’s gravity model. Notice that in the limit Following Dirac, we propose at the quantum level that the physical sates of the theory are invariant under the same transformations,

15. In infinitesimal form we get In the coordinates representation we obtain So, we get a generalization of the Schroedinger Equation.

16. Gauge of freedom Because of the action (3) is invariant under reparametrizations, there is a local symmetry generated by the constraint. So the hamiltonian action is invariant under the following gauge transformations:

17. Equations of motion with the metric is and the inverse Then, we can rewrite these equations in the form

18. Dynamical Symmetries Dynamical symmetries of the Schroedinger equation

19. For the Dilatations we have the algebra: and the Special Conformal Transformations: It is interesting to notice that close themselves in a subalgebra. So, the Schroedinger algebra is the Galilean algebra plus dilatation and spatial conformal transformations. We are interested in the construction of the explicit generators of the relative Poisson-Lie Schroedinger algebra for any z.

20. Notice that M does not play the role of the center anymore unless z=2 and that C, the generator of the Special Conformal transformations is not in the algebra. For z=2 the phase space realization reads We will denote the generalization of the Schroedinger symmetry algebra for any z as in d dimensions. The algebra is given by the Galilean algebra (29) plus

21. Inspired by this construction we will display a set of dynamical symmetries of the Schroedinger equation for any z. The crux of the argument is to allow M to depend on the magnitude of p squared where A is a constant. Then taking as our Schroedinger equation for any z we have we will ask for the phase space quantities that commutes (in Poisson sense) with S

22. is fixed in terms of the dynamical exponent z by relating our definition (10) with the first class constraint (5) is fixed in terms of the dynamical exponent z by relating our definition (10) with the first class constraint (5) To find what are the analogs of D, C and K we propose and using (11) we obtain

23. So we have the non trivial dynamical symmetries of the anisotropic Schroedinger equation generated by and d Galilean boosts Unfortunately the dynamical symmetries given by (12), (13) and plus J, P, H, M do not close. Nevertheless a subalgebra formed by H,C,D indeed close into a SL(2,R) sector of the full algebra,

24. Massless case. Nonlinear realization Consider the action Consider the action where Then, if it is possible to find a canonical transformation from this action to the relativistic particle action, The symmetries of the Horava particle, will be exactly the same that of the massless relativistic particle.

25. Such, canonical transformation exists and is given by where Then, for the new action we have the massless Klein-Gordon equation and in consequence full conformal symmetry SO(d+1,1),

26. with generators given by and defined by where is the generator of the Spatial conformal transformations. So, the Horava massless particle have the same symmetries than the masless relativistic particle and corresponds only to a nonlinear relization of the generators.

27. Thermodynamic Properties In the following we shall sketch some of the thermodynamic properties of the system. We will consider only the case without potential. In this case the energy of our model is The canonical partition function for the system in a space of d dimensions is Using and the definition of the Gamma function we obtain

28. In this way, if N is the number of particles, the Helmholtz free energy is given by From this expression we may now obtain all the thermodynamic properties of the system, in particular the internal energy is

29. Substituting the number of particles N in terms of the partition function (Z=N in our normalization) the internal energy can be written as, Whereas the entropy is

30. the relationship between energy and entropy in our system is given by This relationship is exactly the obtained in the case of black branes.

31. Concluding remarks We have introduced an action invariant under anisotropic transformations in space and time. This anisotropic mechanical system is consistent with the non-relativistic Horava's dispersion relation. Our system corresponds to the Conformal Mechanics for z=2 and is a generalization of this system for arbitrary z. We show that the full Schroedinger algebra constructed from our generators does not close. However we found that a subalgebra SL(2,R) indeed close. In the massless case the Horava particle is only a nonlinear realization of the relativistic particle.

32. We remark that the thermodynamic properties of our model reproduces the same thermodynamic properties of the recent anisotropic black branes.