Remarks on Multi-Dimensional Conformal Mechanics

Conformal invariance takes a new meaning in two dimensions and requires special attention. Indeed, there exists in two dimensions an infinite variety of coordinate transformations which although not everywhere well-defined are locally conformal: they are the holomorphic mappings from the complex plane (or a part it) onto itself. Amongst this infinite set of mappings one must distinguish the 6-parameter global conformal group made of one-to-one mappings of the complex plane into itself. A local field theory should be sensitive to local symmetries and it is the local invariance that enables exact solutions of two-dimensional conformal field theories.

Physical Review D, 2012

Nonrelativistic conformal groups, indexed by l = N 2 , are analyzed. Under the assumption that the mass parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown that the corresponding dynamical system describes, within Ostrogradski framework, the nonrelativistic particle obeying (N + 1)-th order equation of motion. As a special case, the Schrödinger group and the standard Newton equations are obtained for N = 1 (l = 1 2).

2016

The conformal transformations corresponding to N-Galilean conformal symmetries, previously defined as canonical symmetry transformations on phase space, are constructed as point transformations in coordinate space.

Annals of Physics, 1969

International Journal of Modern Physics D

Conformal transformations of the following kinds are compared: (1) conformal coordinate transformations, (2) conformal transformations of Lagrangian models for a D-dimensional geometry, given by a Riemannian manifold M with metric g of arbitrary signature, and (3) conformal transformations of (mini-)superspace geometry. For conformal invariance under these transformations the following applications are given respectively: (1) Natural time gauges for multidimensional geometry, (2) conformally equivalent Lagrangian models for geometry coupled to a spatially homogeneous scalar field, and (3) the conformal Laplace operator on the n-dimensional manifold ℳ of minisuperspace for multidimensional geometry and the Wheeler-de Witt equation. The conformal coupling constant ξc is critically distinguished among arbitrary couplings ξ, for both, the equivalence of Lagrangian models with D-dimensional geometry and the conformal geometry on n-dimensional minisuperspace. For dimension D=3, 4, 6 or 10...

Annals of Physics, 2004

We consider the Lagrangian particle model introduced in [1] for zero mass but nonvanishing second central charge of the planar Galilei group. Extended by a magnetic vortex or a Coulomb potential the model exibits conformal symmetry. In the former case we observe an additional SO(2, 1) hidden symmetry. By either a canonical transformation with constraints or by freezing scale and special conformal transformations at t = 0 we reduce the six-dimensional phasespace to the physically required four dimensions. Then we discuss bound states (bounded solutions) in quantum dynamics (classical mechanics). We show that the Schrödinger equation for the pure vortex case may be transformed into the Morse potential problem thus providing us with an explanation of the hidden SO(2, 1) symmetry.

Physical Review D

We study deformations of the quantum conformal mechanics of De Alfaro-Fubini-Furlan with rational additional potential term generated by applying the generalized Darboux-Crum-Krein-Adler transformations to the quantum harmonic oscillator and by using the method of dual schemes and mirror diagrams. In this way we obtain infinite families of isospectral and non-isospectral deformations of the conformal mechanics model with special values of the coupling constant g = m(m + 1), m ∈ N, in the inverse square potential term, and for each completely isospectral or gapped deformation given by a mirror diagram, we identify the sets of the spectrum-generating ladder operators which encode and coherently reflect its fine spectral structure. Each pair of these operators generates a nonlinear deformation of the conformal symmetry, and their complete sets pave the way for investigation of the associated superconformal symmetry deformations.

Journal of Mathematical Physics, 2010

A review for the 50th anniversary of the Journal of Mathematical Physics.

Nuclear Physics B, 2015

031701], representation theory of the centrally extended l-conformal Galilei algebra has been applied so as to construct second order differential equations exhibiting the l-conformal Galilei group as kinematical symmetry. It was suggested to treat them as the Schrödinger equations which involve Hamiltonians describing dynamical systems without higher derivatives. The Hamiltonians possess two unusual features, however. First, they involve the standard kinetic term only for one degree of freedom, while the remaining variables provide contributions linear in momenta. This is typical for Ostrogradski's canonical approach to the description of higher derivative systems. Second, the Hamiltonian in the second paper is not Hermitian in the conventional sense. In this work, we study the classical limit of the quantum Hamiltonians and demonstrate that the first of them is equivalent to the Hamiltonian describing free higher derivative nonrelativistic particles, while the second can be linked to the Pais-Uhlenbeck oscillator whose frequencies form the arithmetic sequence ω k = (2k − 1), k = 1, . . . , n. We also confront the higher derivative models with a genuine second order system constructed in our recent work [A. Galajinsky, I. Masterov, Nucl. Phys. B 866 (2013) 212] which is discussed in detail for l = 3 2 .