17 research outputs found
Quantizations on the circle and coherent states
We present a possible construction of coherent states on the unit circle as
configuration space. Our approach is based on Borel quantizations on S^1
including the Aharonov-Bohm type quantum description. The coherent states are
constructed by Perelomov's method as group related coherent states generated by
Weyl operators on the quantum phase space Z x S^1. Because of the duality of
canonical coordinates and momenta, i.e. the angular variable and the integers,
this formulation can also be interpreted as coherent states over an infinite
periodic chain. For the construction we use the analogy with our quantization
and coherent states over a finite periodic chain where the quantum phase space
was Z_M x Z_M. The coherent states constructed in this work are shown to
satisfy the resolution of unity. To compare them with canonical coherent
states, also some of their further properties are studied demonstrating
similarities as well as substantial differences.Comment: 15 pages, 4 figures, accepted in J. Phys. A: Math. Theor. 45 (2012)
for the Special issue on coherent states: mathematical and physical aspect
Coherent states on the circle
We present a possible construction of coherent states on the unit circle as
configuration space. In our approach the phase space is the product Z x S^1.
Because of the duality of canonical coordinates and momenta, i.e. the angular
variable and the integers, this formulation can also be interpreted as coherent
states over an infinite periodic chain. For the construction we use the analogy
with our quantization over a finite periodic chain where the phase space was
Z_M x Z_M. Properties of the coherent states constructed in this way are
studied and the coherent states are shown to satisfy the resolution of unity.Comment: 7 pages, presented at GROUP28 - "28th International Colloquium on
Group Theoretical Methods in Physics", Newcastle upon Tyne, July 2010.
Accepted in Journal of Physics Conference Serie
Feynman's path integral and mutually unbiased bases
Our previous work on quantum mechanics in Hilbert spaces of finite dimensions
N is applied to elucidate the deep meaning of Feynman's path integral pointed
out by G. Svetlichny. He speculated that the secret of the Feynman path
integral may lie in the property of mutual unbiasedness of temporally proximal
bases. We confirm the corresponding property of the short-time propagator by
using a specially devised N x N -approximation of quantum mechanics in L^2(R)
applied to our finite-dimensional analogue of a free quantum particle.Comment: 12 pages, submitted to Journal of Physics A: Math. Theor., minor
correction
Dihedral symmetry of periodic chain: quantization and coherent states
Our previous work on quantum kinematics and coherent states over finite
configuration spaces is extended: the configuration space is, as before, the
cyclic group Z_n of arbitrary order n=2,3,..., but a larger group - the
non-Abelian dihedral group D_n - is taken as its symmetry group. The
corresponding group related coherent states are constructed and their
overcompleteness proved. Our approach based on geometric symmetry can be used
as a kinematic framework for matrix methods in quantum chemistry of ring
molecules.Comment: 13 pages; minor changes of the tex
Enhanced quantization on the circle
We apply the quantization scheme introduced in [arXiv:1204.2870] to a
particle on a circle. We find that the quantum action functional restricted to
appropriate coherent states can be expressed as the classical action plus
-corrections. This result extends the examples presented in the cited
paper.Comment: 7 page