Quantum Mechanical Path Integral Research Articles

Dimensional regularization of the path integral in curved space on an infinite time interval

We use dimensional regularization to evaluate quantum mechanical path integrals in arbitrary curved spaces on an infinite time interval. We perform 3-loop calculations in Riemann normal coordinates, and 2-loop calculations in general coordinates. It is shown that one only needs a covariant two-loop counterterm ( V DR = ℏ 2 8 R ) to obtain the same results as obtained earlier in other regularization schemes. It is also shown that the mass term needed in order to avoid infrared divergences explicitly breaks general covariance in the final result.

A lattice path integral for supersymmetric quantum mechanics

We report on a study of the supersymmetric anharmonic oscillator computed using a euclidean lattice path integral. Our numerical work utilizes a Fourier accelerated hybrid Monte Carlo scheme to sample the path integral. Using this we are able to measure massgaps and check Ward identities to a precision of better than one percent. We work with a non-standard lattice action which we show has an exact supersymmetry for arbitrary lattice spacing in the limit of zero interaction coupling. For the interacting model we show that supersymmetry is restored in the continuum limit without fine tuning. This is contrasted with the situation in which a `standard' lattice action is employed. In this case supersymmetry is not restored even in the limit of zero lattice spacing. Finally, we show how a minor modification of our action leads to an exact, local lattice supersymmetry even in the presence of interaction.

Minding one’s P’s and Q’s: From the one loop effective action in quantum field theory to classical transport theory

The one loop effective action in quantum field theory can be expressed as a quantum mechanical path integral over world lines, with internal symmetries represented by Grassmanian variables. In this paper, we develop a real time, many body, world line formalism for the one loop effective action. In particular, we study hot QCD and obtain the classical transport equations which, as Litim and Manuel have shown, reduce in the appropriate limit to the non-Abelian Boltzmann-Langevin equation first obtained by B\"{o}deker. In the Vlasov limit, the classical kinetic equations are those that correspond to the hard thermal loop effective action. We also discuss the imaginary time world line formalism for a hot $\phi^4$ theory, and elucidate its relation to classical transport theory.

Quantum path analysis of high-order above-threshold ionization

High-order above-threshold ionization spectra are calculated via an improved Keldysh approximation that takes rescattering into account. An approximate method of evaluating the crucial multidimensional integral proceeds via the saddle point method. The saddle points define complex orbits in position space that depart from the ion and return to it to rescatter. The real parts of these orbits are very closely related to the trajectories of the simple-man model. The spectra are analyzed in terms of these quantum orbits whose constructive and destructive interferences generate the spectrum's intricate structures. In most spectral regions, the six trajectories having the shortest travel times between start and return already provide an excellent approximation to the exact calculation. In exceptional cases, more orbits are required. The quantum orbits provide an illuminating illustration of the quantum mechanical path integral.

Coordinate independence of quantum-mechanical path integrals

We develop simple rules for performing integrals over products of distributions in coordinate space. Such products occur in perturbation expansions of path integrals in curvilinear coordinates, where the interactions contain terms of the form q ̇ 2q n , which give rise to highly singular Feynman integrals. The new rules ensure the invariance of perturbatively defined path integrals under coordinate transformations.

Meshless method for linear elastostatics based on a path integral formulation

A Path Integral (PI) formulation of linear elastostatics is first presented. For this, Navier equations are modified by adding a fictitious ‘time’ derivative of displacements so that equilibrium corresponds to the steady state of the resulting diffusion-like equations. The evolution of displacement is then represented as the propagation, through the fictitious time co-ordinate, of an initial displacement field corresponding to the unloaded state. The resulting procedure somehow mimics the well-known Feynman path integral of quantum mechanics, which is equivalent to the differential formulation embodied in Schrödinger equation. However, the path integral for elastostatics is formulated in terms of infinitesimal propagators of local support. In its simplest form, the formulation can be used as a relaxation method of solution, by updating displacements until convergence. This may be advantageous for problems involving a very large number of unknowns. On the other hand, by equating the updated displacement field to the actual one a direct method of solution is obtained, which leads to non-symmetric (but sparse and banded) discrete equations. Unlike variational principles this formulation does not require integration over the whole domain, effectively eliminating the need of a background mesh for integration. Also, it only requires continuity of the displacement field on the propagator's support. As a consequence, the formulation lends itself to very flexible meshless implementations. To demonstrate this we describe a simple numerical method in which displacements around each node are approximated by quadratic bivariate polynomials, which is the simplest approximation technique. The feasibility of the method is assessed through a number of numerical examples and comparisons with analytical solutions and other meshless methods. Copyright © 2000 John Wiley & Sons, Ltd.

Rules for integrals over products of distributions from coordinate independence of path integrals

In perturbative calculations of quantum mechanical path integrals in curvilinear coordinates, Feynman diagrams involve multiple temporal integrals over products of distributions, which are mathematically undefined. We derive simple rules for their evaluation from the requirement of coordinate independence of path integrals.

Quantum mechanical path integrals and thermal radiation in static curved spacetimes

The propagator of a spinless particle is calculated from the quantum mechanical path integral formalism in static curved spacetimes endowed with event-horizons. A toy model, the Gui spacetime, and the 2D and 4D Schwarzschild black holes are considered. The role of the topology of the coordinates configuration space is emphasised in this framework. To cover entirely the above spacetimes with a single set of coordinates, tortoise coordinates are extended to complex values. It is shown that the homotopic properties of the complex tortoise configuration space imply the thermal behaviour of the propagator in these spacetimes. The propagator is calculated when end points are located in identical or distinct spacetime regions separated by one or several event-horizons. Quantum evolution through the event-horizons is shown to be unitary in the fifth variable.

Thermal radiation in nonstatic curved spacetimes: Quantum mechanical path integrals and configuration space topology

A quantum mechanical path integral derivation is given of a thermal propagator in non-static Gui's spacetime. The thermal nature of the propagator is understood in terms of homotopically non-trivial paths in the configuration space appropriate to tortoise coordinates. The connection to thermal emission from collapsing black holes is discussed.

Spinors on manifolds with boundary: APS index theorems with torsion

Index theorems for the Dirac operator allow one to study spinors on manifolds with boundary and torsion. We analyse the modifications of the boundary Chern-Simons correction and APS eta invariant in the presence of torsion. The bulk contribution must also be modified and is computed using a supersymmetric quantum mechanics representation. Here we find agreement with existing results which employed heat kernel and Pauli-Villars techniques. Nonetheless, this computation also provides a stringent check of the Feynman rules of de Boer et al. for the computation of quantum mechanical path integrals. Our results can be verified via a duality relation between manifolds admitting a Killing-Yano tensor and manifolds with torsion. As an explicit example, we compute the indices of Taub-NUT and its dual constructed using this method and find agreement for any finite radius to the boundary. We also suggest a resolution to the problematic appearance of the Nieh-Yan invariant multiplied by the regulator mass^2 in computations of the chiral gravitational anomaly coupled to torsion.

Trace anomalies and the string-inspired definition of quantum-mechanical path integrals in curved space

We consider quantum-mechanical path integrals for non-linear sigma models on a circle defined by the string-inspired method of Strassler, where one considers periodic quantum fluctuations about a center-of-mass coordinate. In this approach one finds incorrect answers for the local trace anomalies of the corresponding n-dimensional field theories in curved space. The quantum field theory approach to the quantum-mechanical path-integral, where fluctuations are not periodic but vanish at the endpoints, yields the correct answers. We explain these results by a detailed analysis of general coordinate invariance in both methods. Both approaches can be derived from the same operator expression and the integrated trace anomalies in both schemes agree. In the string-inspired method the integrands are not invariant under general coordinate transformations and one is therefore not permitted to use Riemann normal coordinates.

Coherent modes in an ensemble of quantum dots generated by dynamic dipole–dipole interactions

The generation and stabilization of coherent modes induced by electric dipole–dipole interactions in a quantum dot ensemble are verified, by means of a standard interaction Hamiltonian scheme, and also by way of a full quantum mechanical path integral on the coherent state basis.

Mode regularization, time slicing, Weyl ordering, and phase space path integrals for quantum-mechanical nonlinear sigma models

A simple, often invoked, regularization scheme of quantum mechanical path integrals in curved space is mode regularization: one expands fields into a Fourier series, performs calculations with only the first $M$ modes, and at the end takes the limit $M \to \infty$. This simple scheme does not manifestly preserve reparametrization invariance of the target manifold: particular noncovariant terms of order $\hbar^2$ must be added to the action in order to maintain general coordinate invariance. Regularization by time slicing requires a different set of terms of order $\hbar^2$ which can be derived from Weyl ordering of the Hamiltonian. With these counterterms both schemes give the same answers to all orders of loops. As a check we perform the three-loop calculation of the trace anomaly in four dimensions in both schemes. We also present a diagrammatic proof of Matthews' theorem that phase space and configuration space path integrals are equal.

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.

Thermal Vacuum Polarization Using the Quantum Mechanical Path Integral

It has been shown how the quantum mechanical path integral can be used to do perturbative calculations in both quantum and thermal field theory to any order of the loop expansion. However, it is not readily apparent how gauge invariance is made manifest in this approach; in this paper we demonstrate how the vacuum polarization in electrodynamics at one-loop order is in fact transverse. We employ the one-dimensional Green's function [Formula: see text] in conjunction with an integration-by-parts procedure akin to that used by Strassler and Bern and Kosower. Surface terms in this approach are all zero. We obtain the high temperature expansion for the vacuum polarization in the static limit.

P-Adic Path Integrals for Quadratic Actions

The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude [Formula: see text] for one-dimensional systems with quadratic actions is calculated in an exact form, which is the same as that in ordinary quantum mechanics.

Renormalized path integral in quantum mechanics

We obtain direct, finite, formulations of a renormalized quantum mechanical system with no reference to ultraviolet cutoffs and running coupling constants, in both the Hamiltonian and path integral pictures. The path integral description requires a modification to the Wiener measure on continuous paths that describes an unusual diffusion process wherein colliding particles occasionally stick together for a random interval of time before going their separate ways.

Non-Abelian Bosonization as a Nonholonomic Transformation from a Flat to a Curved Field Space

There exists a simple rule by which path integrals for the motion of a point particle in a flat space can be transformed correctly into those in a curved space. This rule arose from well- established methods in the theory of plastic deformations, where crystals with defects are described mathematically by applying active nonholonomic coordinate transformations to ideal crystals. In the context of time-sliced path integrals, this has given rise to aquantum equivalence principlewhich determines the short-time action and functional integration measure of fluctating orbits in spaces with curvature and torsion. The nonholonomic transformations have a nontrivial Jacobian which in curved spaces produces an additional energy proportional to the curvature scalar, thereby canceling an equal term found earlier by DeWitt in his formulation of Feynman's time-sliced path integral in curved space. The importance of this cancelation has been documented in various systems (H-atom, particle on the surface of a sphere, spinning top). Here we point out its relevance to the bosonization of a non-Abelian one-dimensional quantum field theory, whose fields live in a flat field space. The bosonized version is a quantum-mechanical path integral of a point particle moving in a space with constant curvature. The additional term introduced by the Jacobian is crucial for the identity between original and bosonized theory. A useful bosonization tool is the so-called Hubbard–Stratonovich formula for which we find a nonabelian version.

Barton expansion and the path integral approach in thermal field theory.

It has been shown how on-shell forward scattering amplitudes (the ``Barton expansion'') and quantum mechanical path integral (QMPI) can both be used to compute temperature dependent effects in thermal field theory. We demonstrate the equivalence of these two approaches and then apply the QMPI to compute the high temperature expansion for the four-point function in QED, obtaining results consistent with those previously obtained from the Barton expansion.

A functional approach to loop diagrams

A closed-form expression for the N-loop contribution to the generating functional can be written down using the heat kernel [Formula: see text]. If [Formula: see text] where Aμ and V are functions of the background field, then by using quantum-mechanical techniques, this heat kernel can be expanded in powers of the background field, allowing one to compute Green's functions. We demonstrate that one can also employ to this end a distinct functional approach developed by Onofri, which circumvents both loop-momentum integrals and the quantum-mechanical path integral. We illustrate the technique by computing the two-point function in scalar electrodynamics to one-loop order.