Nonrelativistic conformal groups. II. Further developments and physical applications

Abstract

The finite-dimensional conformal groups associated with the Galilei and (oscillating or expanding) Newton–Hooke space–time manifolds was characterized by the present authors in a recent work. Three isomorphic group families, one for each nonrelativistic kinematics, were obtained, whose members are labeled by a half-integer number l. Since the action of these groups on their corresponding space–time manifolds is only local, a linearization is introduced here such that the corresponding action is well defined everywhere. In particular, the (l=1)-conformal cases that can be obtained by contraction from the well-known Minkowskian conformal group are treated in more detail. As an application of physical interest, the conformal invariance of the Galilean electromagnetism is studied. In order to achieve it, the pertinent local representations of the Galilean conformal algebras are derived.

Extended Galilean symmetries of non-relativistic strings

“…We will write the generator of the most general point transformation and will impose on it the condition of Noether symmetry, which gives the associated non-relativistic Killing equations that one must solve. For a massive non-relativistic particle, the maximal set of symmetries is larger than the Galilei group, and it is in fact the Schrödinger group [20,21], which is the group corresponding to the z = 2 case of an infinite set of z-Galilean conformal algebras [22][23][24].…”

“…If we consider the conformal gauge λ = 1, µ = 0 the equations of motion becomė 24) and they should be supplemented with the constraints (3.5), (3.6). 4 We thank Paul Townsend for remarks concerning the boundary conditions.…”

Nonrelativistic conformal groups and their dynamical realizations

“…[15,16]). It consists of rotations J, translations P , boosts B and time translations H which form the Galilean algebra, together with dilatations D, conformal transformations K and, finally, space dilatations D s .…”

“…The Schrödinger group, when supplemented with space dilatation transformations becomes l = 1 2 member of the whole family of nonrelativistic conformal groups [15,16], indexed by halfinteger l. Various structural, geometric and physical aspects of the resulting Lie algebras have been intensively studied [17]- [35]. For l = N 2 , N-odd (N-even in the case of dimension two), the nonrelativistic conformal algebra admits central extension.…”

Nonlocal dynamics and infinite nonrelativistic conformal symmetries

“…In particular, it has been shown (see [30]) that when the relevant eigenfrequencies are proportional to consecutive odd integers then the maximal symmetry group of the PU oscillator is the l-conformal Newton-Hooke group with half-integer l = N − 1 2 , which is isomorphic to the l-conformal Galilei group (for this reason we will simply refer to it as the l-conformal non-relativistic group in what follows). These groups are natural conformal extensions of the NewtonHooke (Galilei) group, (in particular for N = 1 we obtain the Schrödinger one) and originally have been introduced in the context of non-relativistic gravity [31,32]; however, there are many dynamical models, both classical (including also the second order dynamical equations) and quantum, which exhibit such symmetries [33]- [41]. Furthermore, in a number of papers [42]- [49] infinite dimensional extensions of those symmetries, for fixed order l, have been also considered in the context of some more sophisticated models.…”