Lagrangian Path Research Articles

Toroidal orbifold models with a Wess-Zumino term

Closed bosonic string theory on toroidal orbifolds is studied in a Lagrangian path integral formulation. It is shown that a level one twisted WZW action whose field value is restricted to Cartan subgroups of simply-laced Lie groups on a Riemann surface is a natural and nontrivial extension of a first quantized action of string theory on orbifolds with an antisymmetric background field.

Stretching, folding, and alignment along Lagrangian orbits

The stirring and mixing of passive scalars is determined by the geometry of the Lagrangian orbits followed by the advected particles. The efficiency of these processes depends on the ‘‘stretching and folding’’ of a local element (line, area, and volume). This deformation is determined by the nature of the local straining field and the degree of alignment of the advected element with the straining directions. Various ways of quantifying the dynamics of the straining field, that are valid for both three-dimensional flows and dynamical systems in general, are proposed. One of the simplest is a quantity termed the persistence of strain, σ2, which is defined as σ2= 1/2 Tr(A2), where A is the tangent map of the flow. This is shown to provide a compact quantification of the relative strengths of the straining and rotational components of the flow along a Lagrangian path. This, in turn, leads to the concept of a ‘‘strain history’’ of a fluid particle for which we can also compute a ‘‘stretch–fold’’ radio χ, that provides an average measure of these effects along an orbit. The quantity σ2 is closely related to the curvature of an orbit and this suggests a study of the curvature κ and torsion τ of particle trajectories. Further information is provided by extending the definition of Lyapunov exponents ins such a way as to include complex eigenvalue information that is traditionally discarded from the tangent map during the computation of these exponents. This provides additional information about the orientation and alignment of the advected element. These considerations are especially important when considering the fate of small deformable bodies, such as polymers, in flow fields with rapidly fluctuating strain rates.

Equivalence of the Hamiltonian and Lagrangian path integrals for gauge theories

The lagrangian-antifield and the hamiltonian BRST path integrals are shown to be equal for arbitrary gauge theories obeying standard regularity conditions briefly recalled in the text. Various examples are given as an illustration of the results proved in the theoretical part of the paper. One example discusses systems with tertiary and higher-order constraints. One example illustrates the reducible case. And finally one example analyses gauge-fixing conditions involving higher-order time derivatives of the dynamical variables.

Elimination of the auxiliary fields in the antifield formalism

The problem of the elimination of the auxillary fields in the path integral quantization of gauge theories is investigated in the context of the antifield formalism. It is shown that gauge fixing fermions can be found such that: (i) the auxillary fields remain auxiliary after gauge fixing; and (ii) the gauge fixed actions obtained by eliminating the auxillary fields before or after writing down the path integral coincide. The contribution of the auxiliary fields to the integration measure is also discussed. These results are applied (i) to the problem of the equivalence between the lagrangian and hamiltonian path integral formulations; and (ii) to the problem of the elimination of the second class constraints in the hamiltonian formalism.

Geodesic optics via path integral formalism. A modified principle of minimum time

The optical propagator for the Helmholtz equation in geodesic optics is given by the Fourier transform of a path integral in curve spaces. Here we try to express it directly by a path integral in a Riemann space. This optical propagator is obtained as a Lagrangian path integral which let us infer a modified principle of minimum time for geodesic components, that is an effective refractive index is obtained as the usual refractive index plus a correction of order *2.

Canonical quantization of Witten's string field theory in the mid-point time formalism

We carry out canonical quantization of Witten's string field theory in the mid-point time formalism. A divergence which arises in kinetic term can be regularized by discretizing the string. The equal time commutation relation of string fields can be calculated in this manner, and the theory is shown to coincide with the one which is expected in a formal lagrangian path integral quantization.

The relation between operator and path integral covariant quantizations of the Green-Schwarz superstring

By further study of the geometry of the harmonic superspace constraints we make the relation explicit between the operator and path integral approaches to the manifestly covariant harmonic superstring. In particular we find the correct complete set of functionally independent gauge symmetries for the auxiliary variables and identify the ones corresponding to the harmonic superfield postulate in the operator formalism. Then, we deduce in a systematic way the lagrangian path integral from the well defined covariant hamiltonian formulation of the GS superstring.

The canonical formalism and path integrals in curved spaces

The authors discuss the canonical and path integral quantisation of a particle moving on a curved manifold. They advance a simple prescription to quantise the system within the canonical formalism. An effective Hamiltonian Heff is defined as the classical Hamiltonian plus a quantum potential term of order h(cross)2. The Hamilton operator is obtained by Weyl ordering the expression that results when one replaces coordinates and momenta by their corresponding Hermitian operators in the effective Hamiltonian. They construct the phase space path integral representation for the propagator in terms of Heff. When the propagator is expressed as a Lagrangian path integral with an invariant measure an additional correction is introduced.

The product form for path integrals on curved manifolds

A general and simple framework for treating path integrals on curved manifolds is presented. The crucial point will be a product ansatz for the metric tensor and the quantum hamiltonian, i.e. we shall write g αβ = h αγh βγ and H = (1/2m)h αγp αp βh βγ + V + ΔV , respectively, a prescription which we shall call “product form” definition. The p α are hermitian momenta and Δ V is a well-defined quantum correction. We shall show that this ansatz, which looks quite special, is in fact - under reasonable assumptions in quantum mechanics - a very general one. We shall derive the lagrangian path integral in the “product form” definition and shall also prove that the Schro¨dinger equation can be derived from the corresponding short-time kernel. We shall discuss briefly an application of this prescription to the problem of free quantum motion on the Poincare´upper half-plane.

The neutrino in the Friedmann universe: Non-zero rest-mass effects

We have considered the problem of the propagation of nonzero rest-mass neutrinos in the Friedmann dust universes of three types: open, flat, and closed, as well as in the radiation-dominated epoch of thehot universe. The total Lagrangian path of the particle has been calculated, and this is shown to be finite for all the three universes-contrary to the total path of the photon, which is infinite in the open and flat universes.

Exact path-integration for the two-dimensional Coulomb problem

The lagrangian path integral for the Coulomb problem in two dimensions is explicitly calculated in the parabolic coordinates with the new “time” parameter of Duru and Kleinert.

Exact-Path-Integral Treatment of the Hydrogen Atom

The Lagrangian path integral for the hydrogen atom is calculated exactly by rescaling paths and performing the Kustaanheimo-Stiefel transformation in each short time integral.

Path integrals and constraints: Particle in a box

Equivalence between the lagrangian path integral and the hamiltonian path integral with constraints is shown for a particle in a one-dimensional box.

Multilegged propagators in strong-coupling expansions

This paper is a continuation of a previous paper on strong-coupling expansions in quantum field theory. We are concerned here with one-dimensional quantum field theories (quantum-mechanical models). Our general approach is to derive graphical rules for constructing the strong-coupling expansion from a Lagrangian path integral in the presence of external sources. After reviewing the normalization of one-dimensional path integrals, we examine in detail the model Hamiltonian H = vertical-barpvertical-bar + vertical-barqvertical-bar. We show that in the strong-coupling expansion the graphs are constructed from multilegged propagators attached to multilegged vertices. We use these graphical rules to calculate the ground-state energy for this Hamiltonian. One motivation for examining expansions involving multilegged propagators is provided by the Lagrangian for quantum chromodynamics whose strong-coupling expansion also involves multilegged propagators.

Path integral formulation of reggeon quantum mechanics

The Hamiltonian path integral for reggeon quantum mechanics with cubic and quartic interactions is unambiguously defined. On the basis of a recursion formula on the time lattice, a generalized Trotter formula is given and proven to be equivalent to a Lagrangian path integral for a Schroedinger problem in an interval. The potential occuring in this formula is the same found in previous papers, and the corresponding path integral exists also in the limit of vanishing quartic coupling. This provides a regularization procedure for the purely cubic case for both α(0) less or greater than one. All previous results are confirmed, in particular the tunnel-like energy gap.

Canonical transformations and path integrals

A limited class of canonical transformations is introduced into the Lagrangian path integral method of quantization. Path integral quantization in different representations is discussed and a simple example is given.

Stochastic E-L Wave Theory: Perturbation Techniques

Although stochastic E-L (Eulerian-Langrangian) wave theory does not require random perturbation techniques in principle to provide solutions, in practice, ensemble expectations are usually resolved by perturbing the random medium. The fundamental concepts for the evaluation of wave functionals are first treated in one dimension, then a central limit theorem for Lagrangian functionals permits these concepts to be treated in three dimensional analysis via stochastic perturbation. The resultant relation for calculating the pertinent Eulerian ensemble expectation corresponding to the Lagrangian functional in question is shown to depend in a natural manner upon the Lagrangian path spreading and, in the case of a statistically isotropic medium, to reduce to the proper nonstochastic limit. Thence, stochastic E-L wave theory can be demonstrated via stochastic perturbation techniques applied to secular Lagrangian relations. In the stationary Markovian limit, the comparable Weiner integral results.

An approach to the Eulerian-Lagrangian Problem for Sound Propagation in Continuous Stochastic Media

The formal solution of the Eulerian-Lagrangian problem for sound propagation in continuous stochastic media is reframed so that the emphasis on the need for complete knowledge of the statistical nature of the Lagrangian functional of interest is shifted to the need for knowledge of the asymptotic behavior of certain stochastic Lagrangian integrals which result from the application of a central limit theorem for stochastic functionals. The resultant practical relation for calculating the Eulerian ensemble expectation corresponding to the Lagrangian functional in question is shown to depend in a natural manner upon the Lagrangian path spreading and, in the case of a statistically isotropic medium, to reduce to the proper nonstochastic limit.

Approach to the Eulerian-Lagrangian Problem for Sound Propagation in Continuous Stochastic Media

The formal solution of the Eulerian-Lagrangian problem for sound propagation in continuous stochastic media is reframed so that the emphasis on the need for complete knowledge of the statistical nature of the Lagrangian functional of interest is shifted to the need for knowledge of the asymptotic behavior of certain stochastic Lagrangian integrals which result from the application of a central limit theorem for stochastic functionals. The resultant practical relation for calculating the Eulerian ensemble expectation corresponding to the Lagrangian functional in question is shown to depend in a natural manner upon the Lagrangian path spreading and, in the case of a statistically isotropic medium, to reduce to the proper nonstochastic limit.

Summation over Feynman Histories in Polar Coordinates

Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central-force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse-square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.