Volume 36, Issue 5, May 1995
Index of content:

Oscillatory integrals on Hilbert spaces and Schrödinger equation with magnetic fields
View Description Hide DescriptionThe theory of oscillatory integrals on Hilbert spaces (the mathematical version of ‘‘Feynman path integrals’’) is extended to cover more general integrable functions, preserving the property of the integrals to have converging finite dimensional approximations. An application to the representation of solutions of the time‐dependent Schrödinger equation with a scalar and a magnetic potential by oscillatory integrals on Hilbert spaces is given. A relation with Ramer’s functional in the corresponding probabilistic setting is found.

Abelian Chern–Simons theory and linking numbers via oscillatory integrals
View Description Hide DescriptionA rigorous mathematical model of Abelian Chern–Simons theory based on the theory of infinite‐dimensional oscillatory integrals developed by Albeverio and Ho/egh‐Krohn is introduced. A gauge‐fixed Chern–Simons path integral is constructed as a Fresnel integral in a certain Hilbert space. Wilson loop variables are defined as Fresnel integrable functions and it is shown in this context that the expectation value of products of Wilson loops with respect to the Chern–Simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops, as conjectured by Witten. Furthermore, a lattice Chern–Simons action is proposed which converges to the continuum limit.

Projective techniques and functional integration for gauge theories
View Description Hide DescriptionA general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out integration over the non‐linear, infinite dimensional spaces of connections modulo gauge transformations. This method of evaluating functional integrals can be used either in the Euclidean path integral approach or the Lorentzian canonical approach. A number of measures discussed are diffeomorphism invariant and therefore of interest to (the connection dynamics version of) quantum general relativity. The account is pedagogical; in particular, prior knowledge of projective techniques is not assumed.

Localization and diagonalization: A review of functional integral techniques for low‐dimensional gauge theories and topological field theories
View Description Hide DescriptionLocalization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low‐dimensional gauge theories are reviewed. These are the functional integral counterparts of the Mathai–Quillen formalism, the Duistermaat–Heckman theorem, and the Weyl integral formula, respectively. In each case, the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups) is introduced, and the finite dimensional integration formulae described. Then some applications to path integrals are discussed and an overview of the relevant literature is given. The applications include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two‐dimensional Yang–Mills theory.

A new perspective on functional integration
View Description Hide DescriptionThe core of this article is a general theorem with a large number of specializations. Given a manifoldN and a finite number of one‐parameter groups of point transformations on N with generatorsY,X _{(1)},...,X _{(d)}, we obtain, via functional integration over spaces of pointed paths on N (paths with one fixed point), a one‐parameter group of functional operators acting on tensor or spinor fields on N. The generator of this group is a quadratic form in the Lie derivatives L_{ X(α)} in the X _{(α)}‐direction plus a term linear in L_{ Y }. The basic functional integral is over L ^{2,1} paths x:T→N (continuous paths with square integrable first derivative). Although the integrator is invariant under time translation, the integral is powerful enough to be used for systems which are not time translation invariant. Seven nontrivial applications of the basic formula are given, and the semiclassical expansion is computed. The methods of proof are rigorous and combine Albeverio–Ho/egh‐Krohn oscillatory integrals with Elworthy’s parametrization of paths in a curved space. Unlike other approaches, Schrödinger type equations are solved directly, rather than solving first diffusionequations and then using analytic continuation.

The functional–analytic versus the functional–integral approach to quantum Hamiltonians: The one‐dimensional hydrogen atom^{a)}
View Description Hide DescriptionThe capabilities of the functional–analytic and of the functional–integral approach for the construction of the Hamiltonian as a self‐adjoint operator on Hilbert space are compared in the context of non‐relativistic quantum mechanics. Differences are worked out by taking the one‐dimensional hydrogen atom as an example, that is, a point mass on the Euclidean line subjected to the inverse–distance potential. This particular choice is made with the intent to clarify a long‐lasting discussion about its spectral properties. In fact, for the four‐parameter family of possible Hamiltonians the corresponding energy‐dependent Green functions are derived in closed form. The multiplicity of Hamiltonians should be kept in mind when modeling certain experimental situations as, for instance, in quantum wires.

Overcoming dissipative distortions by a waveform restorer and designer
View Description Hide DescriptionDissipation causes distortions in wave propagations. A new scheme for restoring such distorted waveforms based on a generalized Kac’s method of solving wave equation with dissipation is proposed. From only the component parts of the wave received over time at a point in space, and without the knowledge of them at other points, the wave that would have been received at the point in the absence of dissipation is then restored. The new scheme can be adapted to function as a waveform designer for designing the to‐be‐transmitted waveform such that the distorted distorted‐waveform to be received becomes the intended one. The details of such a mathematical construction of the waveform restorer and designer are given.

Failure of universality in noncompact lattice field theories
View Description Hide DescriptionThe nonuniversal behavior of two noncompact nonlinear sigma models is described. When these theories are defined on a lattice, the behavior of the order parameter (magnetization) near the critical point is sensitive to the details of the lattice definition. This is counter to experience and to expectations based on the ideas of universality.

How to solve path integrals in quantum mechanics
View Description Hide DescriptionA systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given which will serve as a quick reference for solving path integrals. Explicit formulas for the so‐called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based.

Probabilistic computation of Poiseuille flow velocity fields
View Description Hide DescriptionVelocity fields for Poiseuille flow through tubes having general cross section are calculated using a path integral method involving the first‐passage times of random walks in the interior of the cross sectional domain D of the pipe. This method is applied to a number of examples where exact results are available and to more complicated geometries of practical interest. These examples include a tube with ‘‘fractal’’ cross section and open channel flows. The calculations demonstrate the feasibility of the probabilistic method for pipe flow and other applications having an equivalent mathematical description (e.g., torsional rigidity of rods, membrane deflection). The example of flow through a fractal pipe shows an extended region of diminished flow velocity near the rough boundary which is similar to the suppressed vibration observed near the boundaries of fractal drums.

Functional integration and the theory of knots
View Description Hide DescriptionThis paper examines the role of functional integration in the theory of knots and links. It is shown how it is possible to find many properties and conjectures about link invariants via this point of view and how these considerations lead to ideas about the mathematical status of such forms of integration.

Numerical path integral techniques for long time dynamics of quantum dissipative systems
View Description Hide DescriptionRecent progress in numerical methods for evaluating the real‐time path integral in dissipative harmonic environments is reviewed. Quasi‐adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi‐adiabatic propagator path integral is evaluated by efficient system‐specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunnelingdynamics in the condensed phase are described.

Path integrals on homogeneous manifolds
View Description Hide DescriptionQuantum kinematics are considered for two classes of systems: (i) the coordinate space is a homogeneous Riemannian manifold and the kinetic energy is the Laplace–Beltrami operator, (ii) the phase‐space is a homogeneous Kählerian manifold where the algebra of observables is the universal enveloping algebra of the symmetry group. In both the cases, the path integrals are derived from more fundamental principles. The derivation enables a definite description of the classes of trajectories which constitute the domain of integration in the path integrals. The specification of the classes of functions is important for consistency of the approach. The standard action functional and the variational principle appear in the steepest descent method corresponding to the semiclassical approximation.

Functional integration over geometries
View Description Hide DescriptionThe geometric construction of the functional integral over coset spaces M/G is reviewed. The inner product on the cotangent space of infinitesimal deformations of M defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber G, the functional measure on the coset space M/G is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev–Popov determinant of the more traditional gauge fixed approach in non‐abelian gauge theory. If the general construction is applied to the case where G is the group of coordinate reparameterizations of spacetime, the continuum functional integral over geometries, i.e. metrics modulo coordinate reparameterizations may be defined. The invariant functional integration measure is used to derive the trace anomaly and effective action for the conformal part of the metric in two and four dimensional spacetime. In two dimensions this approach generates the Polyakov–Liouville action of closed bosonic non‐critical string theory. In four dimensions the corresponding effective action leads to novel conclusions on the importance of quantum effects in gravity in the far infrared, and in particular, a dramatic modification of the classical Einstein theory at cosmological distance scales, signaled first by the quantum instability of classical de Sitter spacetime. Finite volume scaling relations for the functional integral of quantum gravity in two and four dimensions are derived, and comparison with the discretized dynamical triangulation approach to the integration over geometries are discussed. Outstanding unsolved problems in both the continuum definition and the simplicial approach to the functional integral over geometries are highlighted.

Kazakov–Migdal model with logarithmic potential and the double Penner matrix model
View Description Hide DescriptionThe Kazakov–Migdal (KM) model is a U(N) lattice gauge theory with a scalar field in the adjoint representation but with no kinetic term for the gauge field. This model is formally soluble in the limit N→∞ though explicit solutions are available for a very limited number of scalar potentials. A ‘‘double Penner’’ model in which the potential has two logarithmic singularities provides an example of an explicitly solublemodel. The formal solution to this double Penner KM Model is reviewed first. Special attention is paid to the relationship of this model to an ordinary (one) matrix model whose potential has two logarithmic singularities (the double Penner model). A detailed analysis is presented of the large N behavior of this double Penner model. The various one cut and two cut solutions are described and cases in which ‘‘eigenvalue condensation’’ occurs at the singular points of the potential are discussed. Then the consequences of our study for the KM model described above are discussed. The phase diagram of the model is presented and its critical regions are described.

Path integration, anticommuting variables, and supersymmetry
View Description Hide DescriptionA geometrictheory of path integration on supermanifolds is developed, and used to establish a path‐integral formula for the super heat kernel exp(−D/^{2} t−D/τ) of the Dirac operator D/ on a Riemannian manifold. The earlier part of the paper describes an extension of Wiener measure, Brownian motion, and stochastic calculus to include paths in spaces parametrized by anticommuting variables; these constructions are then combined with the geometrical data of a Riemannian manifold to construct stochastic differential equations whose solutions provide appropriate superpaths for the study of the Dirac operator.

Time displaced interactions: Classical dynamics and path integral quantization
View Description Hide DescriptionThe path integral was created to quantize systems whose dynamics are nonlocal in time and for which a Hamiltonian formulation could not be found. In this article we quantize a linear oscillator with an interaction that depends on the oscillator’s position at past and future times. The central issue is the classical boundary value problem for such an oscillator. Because two‐time functional boundary conditions are used to implement the quantization, we take up related questions, such as causality in this context, and the general issue of stability for forward and inward data specifications.