3,896 research outputs found
Constrained Quantum Systems as an Adiabatic Problem
We derive the effective Hamiltonian for a quantum system constrained to a
submanifold (the constraint manifold) of configuration space (the ambient
space) in the asymptotic limit where the restoring forces tend to infinity. In
contrast to earlier works we consider at the same time the effects of
variations in the constraining potential and the effects of interior and
exterior geometry which appear at different energy scales and thus provide, for
the first time, a complete picture ranging over all interesting energy scales.
We show that the leading order contribution to the effective Hamiltonian is the
adiabatic potential given by an eigenvalue of the confining potential
well-known in the context of adiabatic quantum wave guides. At next to leading
order we see effects from the variation of the normal eigenfunctions in form of
a Berry connection. We apply our results to quantum wave guides and provide an
example for the occurrence of a topological phase due to the geometry of a
quantum wave circuit, i.e. a closed quantum wave guide.Comment: 19 pages, 4 figure
Effective Hamiltonians for Thin Dirichlet Tubes with Varying Cross-Section
We show how to translate recent results on effective Hamiltonians for quantum
systems constrained to a submanifold by a sharply peaked potential to quantum
systems on thin Dirichlet tubes. While the structure of the problem and the
form of the effective Hamiltonian stays the same, the difficulties in the
proofs are different.Comment: 6 pages, 1 figur
Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension
We extend a recent classification of three-dimensional spatially isotropic
homogeneous spacetimes to Chern--Simons theories as three-dimensional gravity
theories on these spacetimes. By this we find gravitational theories for all
carrollian, galilean, and aristotelian counterparts of the lorentzian theories.
In order to define a nondegenerate bilinear form for each of the theories, we
introduce (not necessarily central) extensions of the original kinematical
algebras. Using the structure of so-called double extensions, this can be done
systematically. For homogeneous spaces that arise as a limit of (anti-)de
Sitter spacetime, we show that it is possible to take the limit on the level of
the action, after an appropriate extension. We extend our systematic
construction of nondegenerate bilinear forms also to all higher-dimensional
kinematical algebras.Comment: 52 pages, 2 figures, 11 tables; v2: matches published version,
additional references added and incorporated referee suggestion
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