Hidden symmetries of dynamics in classical and quantum physics

Abstract

This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description of physical systems as varied as non-relativistic, relativistic, with or without gravity, classical or quantum, and are related to the existence of conserved quantities of the dynamics and integrability. In recent years their study has grown intensively, due to the discovery of non-trivial examples that apply to different types of theories and different numbers of dimensions. Applications encompass the study of integrable systems such as spinning tops, the Calogero model, systems described by the Lax equation, the physics of higher dimensional black holes, the Dirac equation, supergravity with and without fluxes, providing a tool to probe the dynamics of non-linear systems.

Hidden symmetry and (super)conformal mechanics in a monopole background

“…Thus, I 1 and I 2 are orthogonal to each other at t = 0, but in general case of α = ν 2 their scalar product is not zero and changes periodically with period π/ω. For the particular choice α = ν 2 , the vectors I 1 and I 2 are orthogonal vector integrals of motion of order 2 in the kinetic momenta, and so, they correspond to the "hidden symmetries" [1] of the system. They are, however, dynamical, explicitly time-dependent integrals of motion (similarly to generators of the conformal Newton-Hook symmetry D and K), d dt I 1,2 = ∂I 1,2 ∂t + {I 1,2 , H} = 0.…”

“…Hidden symmetries are associated with peculiar classical and quantum properties of a system [1]. They are generated by higher order in canonical momenta integrals of motion.…”

The Eisenhart lift: a didactical introduction of modern geometrical concepts from Hamiltonian dynamics

“…If V does not have a global minimum then what follows will only apply in a region where V > 0. We can notice that g n+1,n+1 > 0 implies that we can write 9) or in other words the left hand side defines a positive quantity that we can think of as the square of the length of an infinitesimal piece of trajectory in the space (q i , y). g AB then determines how the length is calculated starting from dq A .…”

Ricci-flat spacetimes admitting higher rank Killing tensors

“…This method was further applied to a variety of integrable systems with the aim to produce other examples of irreducible higher rank Killing tensors [7][8][9][10][11][12]. It is important to stress, however, that none of the Lorentzian spacetimes studied in [5,[7][8][9][10][11][12] solves the vacuum Einstein equations. The conventional Eisenhart lift [6] is an embedding of a dynamical system with n degrees of freedom x 1 , .…”

“…. , x n ) (for a recent review with numerous examples see [12]). The equations of motion of the original system are contained within the null geodesics associated with the metric…”