New quantum states and the dynamics : coherent state path integrals based on arbitrary fiducial vectors

Masao Matsumoto

We develop the formulation of the spin(SU(2)) coherent state path integrals based on arbitrary fiducial vectors. The resultant action in the path integral expression extensively depends on the vector; It differs from the conventional one in that it has a generalized form having some additional terms. We also study, as physical applications, the geometric phases associated with the coherent state path integrals to find that new effects of the terms may appear in experiments. We see that the formalism gives a clear insight into geometric phases.

Enrico Onofri

Ioannis Bakas, H. La Roche

Generalised coherent states are used in the quantum mechanical study of physical systems with homogeneous phase spaces, for which there is a group theoretical approach to quantisation. New sets of coherent states are introduced based on dynamical subalgebras and also with fiducial vectors in the rigged Hilbert space. The path integral expression for coherent state transition amplitudes is examined from a group theoretical point of view.

MASAO MATSUMOTO

We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term à la Balachandran et al. as well as some additional terms, both of which depend on fiducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.

Chon‐Fai Kam, Wei-Min Zhang, D.H. Feng