Phase Space Path Integral Research Articles

Violation of the phase space general covariance as a diffeomorphism anomaly in quantum mechanics

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.

Semiclassical propagation of Wigner functions

We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed. The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrodinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.

Coherent-State Path Integrals and Brownian Motions

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 \({\bf R}^{2}\).

The Klauder–Daubechies construction of the phase-space path integral and the harmonic oscillator

The canonical operator quantization formulation corresponding to the Klauder–Daubechies construction of the phase-space path integral is considered. This formulation is explicitly applied and solved in the case of the harmonic oscillator, thereby illustrating in a manner complementary to Klauder and Daubechies' original work some of the promising features offered by their construction of a quantum dynamics. The Klauder–Daubechies functional integral involves a regularization parameter eventually taken to vanish, which defines a new physical time scale. When extrapolated to the field theory context, besides providing a new regularization of short distance divergences, keeping a finite value for that time scale offers some tantalizing prospects when it comes to strong gravitational quantum systems.

Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations

We give a fairly general class of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of the phase space. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the Riemann integrals, the interchange of the order with a limit and the perturbation expansion formula hold in the phase space path integrals. The use of piecewise bicharacteristic paths naturally leads us to the semiclassical approximation on the phase space.

Multifold phase space path integral synthetic seismograms

SUMMARY We consider a medium consisting of homogeneous layers separated by curved interfaces. In order to evaluate the response of a single generalized ray, the source and receiver wavefields are expanded in a series of plane waves. The coupling of these plane waves at each point of the surfaces of material discontinuity is determined by means of a Kirchhoff integral using generalized reflection and transmission coefficients. The resulting integral, called the multifold phase space path integral (PSPI) consists of a series of integrals over ray parameters and over interfaces touched by the generalized ray on its way from the source to the receiver. This approach is a generalization of the multifold configuration space path integral (CSPI) to which it reduces by successive application of the stationary phase point method over the ray-parameter integrals. The PSPI like the CSPI automatically includes diffractions from corners. In addition classical head waves are included, although for curved interfaces the head waves are only approximate. 2-D synthetic seismograms are converted to equivalent approximate point-source responses by assuming cylindrical symmetry about source and/or receiver. The waveforms and amplitude of PSPI synthetic seismograms compare well with those computed by generalized ray theory for a 1-D model, and with finite difference synthetics for a 2-D model.

Simultaneous histories, path sums, and measurements for noncommuting variables

We analyze von Neumann-like quantum measurements in terms of simultaneous virtual paths constructed for two noncommuting variables. The approach is applied to measurements of operator functions of conjugate variables and to the joint measurements of such variables. The limits of applicability of the restricted phase space path integral are studied. We demonstrate that, for a simple joint measurement, using entangled meter states allows one to manipulate the order in which the measurements are conducted. The effects of 'weakening' a measurement by choosing unsharp meter states are also discussed.

Semiclassical Propagator of the Wigner Function

Propagation of the Wigner function is studied on two levels of semiclassical propagation: one based on the Van Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a pair of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral approach allows for a finer resolution of the quantum spot in terms of Airy functions.

Symmetries in a Constrained System with a Singular Higher-Order Lagrangian

A simple algorithm to construct the generator of gauge transformation for a constrained canonical system with a singular higher-order Lagrangian in field theories is developed. Based on phase-space generating functional of Green function for such a system, the generalized canonical Ward identities under the non-local transformation have been deduced. For the gauge-invariant system, based on configuration-space generating functional, the generalized Ward identities under the non-local transformation have been also derived.The conservation laws are deduced at the quantum level. The applications of the above results to the gauge invariance massive vector field and non-Abelian Chern–Simons(CS) theories with higher-order derivatives are given, a new form of gauge-ghost proper vertices, and Ward–Takahashi identity under BRS transformation and BRS charge at the quantum level are obtained. In the canonical formulation one does not need to carry out the integration over canonical momenta in phase-space path integral as usually performed.

Coherent state path integral for linear systems

We present a computation of the coherent state path integral for a generic linear system using ``functional methods'' (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary -- but not sufficient -- condition for existence of the path integral is that the symplectic transformation generated by the Hamiltonian is (unitarily) implementable on the Fock space for the system.

Path integral for quantum operations

In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive quantum operation satisfied Lindblad equation (quantum Markovian master equation). We consider the path integral for quantum operation with a simple infinitesimal generator.

Phase-space path-integral calculation of the Wigner function

The Wigner function W(q,p) is formulated as a phase-space path integral, whereby its sign oscillations can be seen to follow from interference between the geometrical phases of the paths. The approach has similarities to the path-centroid method in the configuration-space path integral. Paths can be classified by the mid-point of their ends; short paths where the mid-point is close to (q,p) and which lie in regions of low energy (low P function of the Hamiltonian) will dominate, and the enclosed area will determine the sign of the Wigner function. As a demonstration, the method is applied to a sequence of density matrices interpolating between a Poissonian number distribution and a number state, each member of which can be represented exactly by a discretized path integral with a finite number of vertices. Saddle point evaluation of these integrals recovers (up to a constant factor) the WKB approximation to the Wigner function of a number state.

Semiclassical theory of spin-orbit interactions using spin coherent states.

We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start from the path integral in an extended phase space, formulate the classical dynamics of the coupled orbital and spin degrees of freedom, and calculate the ingredients of Gutzwiller's trace formula for the density of states. For a two-dimensional quantum dot with a spin-orbit interaction of Rashba type, we obtain satisfactory agreement with fully quantum-mechanical calculations. The mode-conversion problem, which arose in an earlier semiclassical approach, has hereby been overcome.

PHASE SPACE PATH INTEGRAL IN CURVED SPACE

Phase space path integral is worked out in a Riemannian geometry, by employing a prescription for the infinitesimal propagator that takes Riemann normal coordinates and momenta on an equal footing. The operator ordering induced by this prescription leads to the DeWitt curvature coupling in the Schrödinger equation.

Semiclassical approximations in phase space with coherent states

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

Semiclassical Trace Formulas in Terms of Phase Space Path Integrals

Semiclassical trace formulas are examined using phase space path integrals. Our main concern in this paper is the Maslov index of the periodic orbit, which seems not fully understood in previous works. We show that the calculation of the Maslov index is reduced to a classification of connections on a vector bundle over S1 with structure group Sp(2n, R). We derive a formula for the index of the n-repetition, and show that a Bohr–Sommerfeld type quantization condition including quadratic fluctuation around the orbit is derived using this formula.

THE PHASE SPACE FEYNMAN PATH INTEGRAL WITH GAUGE INVARIANCE AND ITS CONVERGENCE

Convergence of a time-slicing approximation is studied in a general way of the Feynman path integral in phase space with an electromagnetic potential. In the present paper a new approximation of the Feynman path integral in phase space, different from the familiar one in physics, is proposed so that its convergence can be shown independently of the choice of electromagnetic potentials and that gauge invariance of the Feynman path integral defined by its limit can be shown directly. It is also shown that the Feynman path integral in phase space defined above can be expressed in the form of the Feynman path integral in configuration space defined in the familiar way in physics, which Feynman himself stated heuristically.

Exactly solvable path integral for open cavities in terms of quasinormal modes

We evaluate the finite-temperature Euclidean phase-space path integral for the generating functional of a scalar field inside a leaky cavity. Provided the source is confined to the cavity, one can first of all integrate out the fields on the outside to obtain an effective action for the cavity alone. Subsequently, one uses an expansion of the cavity field in terms of its quasinormal modes (QNMs)-the exact, exponentially damped eigenstates of the classical evolution operator, which previously have been shown to be complete for a large class of models. Dissipation causes the effective cavity action to be nondiagonal in the QNM basis. The inversion of this action matrix inherent in the Gaussian path integral to obtain the generating functional is therefore nontrivial, but can be accomplished by invoking a novel QNM sum rule. The results are consistent with those obtained previously using canonical quantization.

Geometrical properties of Maslov indices in periodic-orbit theory

The Maslov indices in periodic-orbit theory are investigated using the phase-space path integral. Based on the observation that the Maslov index is the multi-valued function of the monodromy matrix, we introduce a generalized monodromy matrix in the universal covering space of the symplectic group and show that this index is uniquely determined in this space. The stability of the orbit is shown to determine the parity of the index, and a formula for the index of the n-repetition of the orbit is derived.

Quantal symmetry for a system with a singular higher-order Lagrangian

Based on the phase-space generating functional of the Green function for a constrained Hamiltonian system with a singular higher-order Lagrangian, the canonical Ward identities for such a system under the local and global transformation have been derived, respectively. The quantal conserved charge (QCC) under the global symmetry transformation is also deduced. In general, these QCCs are different from the Noether charge in classical theory. A comparison of these quantal conservation laws with those deriving from the configuration-space path integral for gauge-invariant theories is discussed. We give a preliminary application of our results to Yang-Mills (YM) theory and Chern-Simons (CS) theory with higher-order derivatives. A new form of gauge-ghost proper vertices and new conserved charges at the quantum level are obtained for the YM theory; the quantal Becchi-Rouet-Stora (BRS) conserved charges and conserved angular momentum are also derived for CS theory. The advantage of our canonical formalism is that we do not carry out the integration over the canonical momenta in the phase-space path integral as usually performed.