Semiclassical approximations in phase space with coherent states

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.

Starting from the phase space path integral, we have derived the Ward identities in canonical formalism for a system with regular and singular Lagrangian. This formulation differs from the traditional discussion based on path integral in configuration space. It is pointed out that the quantum canonical equations for systems with singular Lagrangians are different from the classical ones obtained from Dirac's conjecture. The preliminary applications of Ward identities in phase space to the Yang-Mills theory are given. Some relations among the proper vertices and propagators are deduced, the PCAC, AVV vertices and generalized PCAC expressions are also obtained. We have also pointed out that some authors in their early work had ignored the treatment of the constraints.

The present article is primarily a review of the projection-operator approach to quantize systems with constraints. We study the quantization of systems with general first- and second-class constraints from the point of view of coherentstate, phase-space path integration, and show that all such cases may be treated, within the original classical phase space, by using suitable path-integral measures for the Lagrange multipliers which ensure that the quantum system satisfies the appropriate quantum constraint conditions. Unlike conventional methods, our procedures involve no λ-functionals of the classical constraints, no need for dynamical gauge fixing of first-class constraints nor any average thereover, no need to eliminate second-class constraints, no potentially ambiguous determinants, as well as no need to add auxiliary dynamical variables expanding the phase space beyond its original classical formulation, including no ghosts. Besides several pedagogical examples, we also study: (i) the quantization procedure for reparameterization invariant models, (ii) systems for which the original set of Lagrange multipliers are elevated to the status of dynamical variables and used to define an extended dynamical system which is completed with the addition of suitable conjugates and new sets of constraints and their associated Lagrange multipliers, (iii) special examples of alternative but equivalent formulations of given first-class constraints, as well as (iv) a comparison of both regular and irregular constraints.

The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al, one of which was erroneously termed Herman–Kluk approximation.

We consider a topological quantum mechanics described by a phase space path integral and study the 1-dimensional analog for the path integral representation of the Kontsevich formula. We see that the naive bosonic integral possesses divergences, that it is even naively non-invariant and thus is ill-defined. We then consider a superextension of the theory which eliminates the divergences and makes the theory naively invariant. This super-extension is equivalent to the correct choice of measure and was discussed in the literature. We then investigate the behavior of this extended theory under diffeomorphisms of the extended phase space and despite of its naive invariance find out that the theory possesses anomaly under nonlinear diffeomorphisms. We localize the origin of the anomaly and calculate the lowest nontrivial anomalous contribution.

Within the scope of simple quantum mechanics we present a semiclassical theory which is exact. While the semiclassical theory of canonical phase space path integrals is now well established [1, 2] we examine here the case where the classical phase space is the two-sphere. After summarizing some relevant features of a classical spin, we briefly discuss the localization of classical phase space integrals and then present an extension for a quantum spin. The semiclassical propagator is employed to solve the Jaynes-Cummings model.

We present a stochastic path integral formalism for continuous quantum measurement that enables the analysis of rare events using action methods. By doubling the quantum state space to a canonical phase space, we can write the joint probability density function of measurement outcomes and quantum state trajectories as a phase space path integral. Extremizing this action produces the most-likely paths with boundary conditions defined by preselected and postselected states as solutions to a set of ordinary differential equations. As an application, we analyze continuous qubit measurement in detail and examine the structure of a quantum jump in the Zeno measurement regime.

The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in [1] for the two corresponding path integral forms suggested by Klauder and Skagerstan in [2]. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.

A numerical method for solving optimal control problems

Intuitively, the Feynman path integral corresponds to a weighted sum over classical paths, an interpretation that fails for the phase space path integral. To address the question of whether there exists a path integral expression conforming to a sum over paths in phase space, an examination of the discrete coherent state path integral (CSPI) is undertaken. Via an alternative formulation of the CSPI, it is shown that the coherent state action for a broad class Hamiltonians can be transformed from the variables (q,p) to (q,q̇). For these Hamiltonians, such a transformation along with the inclusion of all terms O(ε(zi−zi−1)) yields an expression which, for finite ε, can be interpreted as a sum over classical paths with Gaussian weight. The numerical evaluation of this expression through importance sampling (Monte Carlo) is demonstrated.

An initial value representation for the coherent state propagator with complex trajectories

A stochastic approach to the rigorous foundation of the coherent-state (phase-space) path integral is given. Stochastic integrals and some generalizations of the Feynman–Kac theorem are used for this purpose. In this approach, quantum mechanics is described in terms of the Fock–Bargmann representation; a classical Hamiltonian is related to the corresponding quantum Hamiltonian on the Fock–Bargmann space, seen as a Hilbert subspace of \(L^{2}({\bf R}^{2})\). The coherent-state path integral is realized as a conditional expectation of a stochastic process defined by the exponential of the Fisk–Stratonovich integral of the fundamental 1-form along a path of Brownian motion on the phase space \({\bf R}^{2}\).

SecDec: A general program for sector decomposition

Initial value representations are constructed to avoid the search for trajectories that are only defined in semiclassical approximations by their boundary conditions. We show how to incorporate these procedures within the full Weyl representation, so that quantum expectation values are given by phase space integrals over the evolving Wigner function. Spurious semiclassical singularities at caustics are cancelled, even though there is no increase in the number of trajectories, as compared to usual semiclassical formulae. The whole construction remains exact in the case of quadratic Hamiltonians. The evolution of (density) operators depends on either a forward and backward trajectory, given by an initial value, or else on a pair of trajectories, propagating backwards from their final value. The latter option reduces numerical errors in the computation of trajectories. The general scheme also leads to analogous algorithms for evolving the quantum fidelity, which can be approximated perturbatively with a single trajectory, reducing to the ‘dephasing representation’ for small times. The theory is developed within a generalized ‘Maslov method’, based on semiclassical Fourier transforms, in order to avoid singularities in the limit of small times.

Superfield formulation of the phase space path integral