180 research outputs found
Weak Coherent State Path Integrals
Weak coherent states share many properties of the usual coherent states, but
do not admit a resolution of unity expressed in terms of a local integral. They
arise e.g. in the case that a group acts on an inadmissible fiducial vector.
Motivated by the recent Affine Quantum Gravity Program, the present article
studies the path integral representation of the affine weak coherent state
matrix elements of the unitary time-evolution operator. Since weak coherent
states do not admit a resolution of unity, it is clear that the standard way of
constructing a path integral, by time slicing, is predestined to fail. Instead
a well-defined path integral with Wiener measure, based on a continuous-time
regularization, is used to approach this problem. The dynamics is rigorously
established for linear Hamiltonians, and the difficulties presented by more
general Hamiltonians are addressed.Comment: 21 pages, no figures, accepted by J. Math. Phy
Path Integral Quantization and Riemannian-Symplectic Manifolds
We develop a mathematically well-defined path integral formalism for general
symplectic manifolds. We argue that in order to make a path integral
quantization covariant under general coordinate transformations on the phase
space and involve a genuine functional measure that is both finite and
countably additive, the phase space manifold should be equipped with a
Riemannian structure (metric). A suitable method to calculate the metric is
also proposed.Comment: plain Latex, 9 pages, no figure
The Feynman Path Integral: An Historical Slice
Efforts to give an improved mathematical meaning to Feynman's path integral
formulation of quantum mechanics started soon after its introduction and
continue to this day. In the present paper, one common thread of development is
followed over many years, with contributions made by various authors. The
present version of this line of development involves a continuous-time
regularization for a general phase space path integral and provides, in the
author's opinion at least, perhaps the optimal formulation of the path
integral.Comment: LaTeX, 24 pages, no figure
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