833 research outputs found
Quantum conformal mechanics
The quantum mechanics of one degree of freedom exhibiting the exact conformal
SL(2,R) symmetry is presented. The starting point is the classification of the
unitary irreducible representations of the SL(2,R) group (or, to some extent,
its universal covering). The coordinate representation is defined as the basis
diagonalizing the special conformal generator K. It is indicated how the
resulting theory emerges from the canonical/geometric quantization of the
Hamiltonian dynamics on the relevant coadjoint orbits.Comment: 30 page
Nonlocal dynamics and infinite non-relativistic conformal symmetries
We study the symmetry of the class of nonlocal models which includes the
nonlocal extension of the Pais-Uhlenbeck oscillator. As a consequence, we
obtain an infinite dimensional symmetry algebra, containing the Virasoro
algebra, which can be considered as a generalization of the non-relativistic
conformal symmetries to the infinite order. Moreover, this nonlocal extension
resembles to some extent the string model and on the quantum level it leads to
the centrally extended Virasoro algebra.Comment: 21 pages. Minor changes, references modifie
Unitary representations of N-conformal Galilei group
All unitary irreducible representations of centrally extended (N-odd)
N-conformal Galilei group are constructed. The "on-shell" action of the group
is derived and shown to coincide, in special but most important case, with that
obtained in: J. Gomis, K. Kamimura, Phys. Rev. {\bf D85} (2012), 045023.Comment: References update
On the geometry of conformal mechanics
A geometric picture of conformally invariant mechanics is presented. Although
the standard form of the model is recovered, the careful analysis of global
geometry of phase space leads to the conclusion that, in the attractive case,
the singularity related to the phenomenon of "falling on the center" is
spurious. This opens new possibilities concerning both the interpretation and
quantization of the model. Moreover, similar modification seem to be relevant
in supersymmetric and multidimensional generalization of conformal mechanics.Comment: 8 pages, 4 figures, two references adde
Heisenberg algebra for restricted Landau problem
Algebraic derivation of modified Heisenberg commutation rules for restricted
Landau problem is given.Comment: 6 pages, no figures,we added two references and corrected three typo
Niederer's transformation, time-dependent oscillators and polarized gravitational waves
It is noted that the Niederer transformation can be used to find the explicit
relation between time-dependent linear oscillators, including the most
interesting case when one of them is harmonic. A geometric interpretation of
this correspondence is provided by certain subclasses of pp-waves; in
particular the ones strictly related to the proper conformal transformations.
This observation allows us to show that the pulses of plane gravitational wave
exhibiting the maximal conformal symmetry are analytically solvable.
Particularly interesting is the circularly polarized family for which some
aspects (such as the classical cross section, velocity memory effect and
impulsive limit) are discussed in more detail. The role of the additional
integrals of motion, associated with the conformal generators, is clarified by
means of Ermakov-Lewis invariants. Possible applications to the description of
interaction of electromagnetic beams with matter are also indicated.Comment: 26 pages, Substantially revised according to the suggestions of the
referees (the role of the integrals of motion, associated with the conformal
generators, is clarified by means of Ermakov-Lewis invariants). Accepted for
publication in CQ
N-Galilean conformal algebras and higher derivatives Lagrangians
It is shown that the N-Galilean conformal algebra, with N-odd, is the maximal
symmetry algebra of the free Lagrangian involving (N+1)/2-th order time
derivative
Nonrelativistic conformal groups and their dynamical realizations
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 Schroedinger group and the
standard Newton equations are obtained for N=1 (l=1/2).Comment: 18 pages, no figures; few references adde
N-Galilean conformal algebras and quantum theory with higher order time derivatives
It is shown that centrally extended N-Galilean conformal algebra, with N-odd,
is the maximal symmetry algebra of the Schrodinger equation corresponding to
the free Lagrangian involving (N+1)/2-th order time derivatives.Comment: references update
Free Particle Wave Function and Niederer's Transformation
The solutions to the free Schroedinger equation discussed by P. Strange
(arXiv: 1309.6753) and A. Aiello (arXiv: 1309.7899) are analyzed. It is shown
that their properties can be explained with the help of Niederer's
transformation
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