Feynman Path Integral Formulation Research Articles

Perturbative analysis of the QCD 4 maps

The Claudson-Halpern stochastic processes for QCD 4 are verified to all orders of perturbation theory resulting in a number of new finite-time, infrared finite, perturbation expansions of the theory. The map viewpoint shows that the superficial quadratic divergences of QCD 4, at least in the temporal gauge, are related to the choice of any particular final state configuration in the Feynman path integral formulation.

Hydrated electron revisited via the feynman path integral route

The Feynman path integral formulation of quantum statistical mechanics is used to study the hydrated electron. The results show that no shell structure is apparent at room temperature and the electron aligns the water molecules by the O-H bond dipole and not the molecular dipole.

Quantum action principle and path integrals for long-range interactions

Schwinger’s action (dynamical) principle for transition amplitudes, in quantum mechanics and field theory, is derived for long-range interactions by using the unescapable fact that a scattering out-(in-) state when translated forward (backward) in time for large positive (negative) times may stilldepend on the quantum-mechanical coupling parameter. For such interactions, the dynamical principle is different from the conventional one used extensively historically in the literature. As a corollary to our main result we show that the integrand in the Feynman path integral formulation is not simply given by exp [i[Action]]. Applications of the derived results are then given to the specific cases for the Coulomb interaction and to quantum electrodynamics.

Path-integral quantum cosmology. II. Bianchi type I with volume-dependent source.

The Feynman path-integral formalism is applied to the degrees of freedom of the fluid-filled Kasner (and related) cosmological models. The measure constructed for the vacuum analysis is used to effect the Arnowitt-Deser-Misner reduction to dynamical degrees of freedom within the functional integral. The Hankel transform of the minisuperspace transition amplitude is obtained. A perturbation expansion about the classical vacuum solution is presented.

Path-integral quantum cosmology. I. Vacuum Bianchi type I.

The Feynman path-integral formalism is applied to study the quantum mechanics of the vacuum Bianchi type-I cosmological model. A measure is obtained for the functional integral representing the minisuperspace transition amplitude which allows the performance of the Arnowitt-Deser-Misner (ADM) reduction within the path integral. This requires a careful analysis of the roles played by the original coordinate time and the intrinsic minisuperspace time. The role of the imposed Feynman boundary conditions is discussed. The integrals are all evaluated to obtain the transition amplitude in closed form. An alternative scheme for evaluation of the transition amplitude is presented to avoid problems related to the nonpolynomial nature of the ADM Hamiltonian.

Analytical Treatment of a Periodic $\delta $-Function Potential in the Path-Integral Formalism

A method for evaluating the density matrix, written as a path-integral, is proposed and applied to a periodic delta function potential. It is the first time that a model which gives rise to a band structure is analytically treated in the framework of the Feynman path-integral formalism. It is shown that the results for the energy spectrum and the wave functions completely coincide with those deduced in the Schrödinger formalism.

Complex trajectories and Feynman path integrals

We show how the notion of complex trajectories can be consistently introduced into the Feynman path integral formalism. WKB solutions are derived which are free of the divergences usually associated with such formulations.

Feynman path integral and the Hückel model

The propagator of the Hückel model is obtained through the Feynman path integral formulation and the corresponding secular equations are generated via the saddle point approximation of the Gaussian integrals appearing in the path integral. The present formalism appears as a practical tool for the evaluation of thermodynamic and magnetic properties of molecular systems and constitutes an alternative method to those given previously using the Green’s function and Dyson equation to get self-energies and reactivity indices.

Magnetic field detrapping of polaronic electrons on films of liquid helium

We have applied an extension of the Feynman path-integral formalism to the problem of noninteracting polaronic electrons on the surface of a liquid-helium film in a perpendicularly applied magnetic field. The ground-state energy, the Feynman-model mass, the magnetization, and the susceptibility of the system are calculated. We find that at a certain magnetic field value, the polaronic electron undergoes a transition from a heavy-mass, self-trapped state to a quasifree Landau state. This detrapping mechanism would have a dramatic effect in a cyclotron-resonance experiment.

N-body saddle-point approximation

By applying the saddle-point approximation to the N-body Feynman path integral formulation, the classical Hartree-Fock Molecular Orbital (M.O.) equations of quantum chemistry are obtained.

Numerical study of Holstein's molecular-crystal model: Adiabatic limit and influence of phonon dispersion

We report on simulations for the molecular-crystal model without dispersion in the adiabatic regime for all lattice dimensionalities. A discrete version of the Feynman path-integral formalism is used to calculate the electron properties. We compare these results with approximate theories and conclude that continuum versions of the molecular-crystal model have little relationship to the discrete lattice model. In addition, the influence of the phonon dispersion on the transition between the "untrapped" and "self-trapped" state of an electron on a one-dimensional lattice is studied within the context of Holstein's molecular-crystal model. We show that the electron-phonon interaction has a critical value below which self-trapping does not occur. We find that this critical value goes to zero if the optical phonon becomes soft.

General Theory for Quantum Statistics in Two Dimensions

Because of complicated topology of the configuration space for indistinguishable particles in two dimensions, Feynman's path-integral formulation allows exotic statistics. All possible quantum statistics in two-space are characterized by an angle parameter $\ensuremath{\theta}$ which interpolates between bosons and fermions. The current formalisms in terms of topological action of multivalued wave functions can be derived in a model-independent way.

Semiclassical collision theory within the Feynman path-integral formalism: The perturbed stationary state formulation

Classical equations are derived for the relative motion of two molecules as the internal degrees of freedom make a given quantum transition by extending the semiclassical theory of Peckhuks into the adiabatic representation. Using a stationary phase approximation to Feynman’s path integral representation of quantum mechanics as a starting point, classical trajectories are obtained which fulfill angular momentum and energy conservation. The features of the adiabatic treatment are analyzed through the numerical integration of the collinear harmonic oscillator-atom collision. The calculations show that the use of a perturbed basis appreciably reduces the computational time required by the iterative procedure of integration. However no solution can be obtained as in the diabatic approach for some classically forbidden transitions.

Exact propagator for reflectionless potentials

The exact space-time propagation for a general reflectionless potential is obtained in closed form involving error functions. It is indicated how bound states and transmission coefficients may be recovered from asymptotic behaviour of the propagator. A sum rule is derived that shows how the leading terms of the short-time form of the propagator can be used rigorously in a Feynman path integral formalism.

NewU-matrix theory in quantum mechanics

We have analyzed Dyson's $U$-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the $U$ matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our $U$ matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our $U$ matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our ${U}^{(s)}$ matrix, which is the $U$ matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.

Finite-bandwidth effects in one-dimensional Wigner lattices

The Feynman path-integral formalism is used to calculate the thermodynamic properties of a one-dimensional (1D) spinless fermion model with nearest- and next-nearest-neighbor interaction. The mapping of this 1D quantum model on a two-dimensional classical model allows the use of a Monte Carlo method to calculate the properties of long chains. We present results for typical values of the interaction parameters, density, and temperature. We find that a finite bandwidth smears out the details of the classical ground-state configurations discussed by Hubbard. If the density $\ensuremath{\rho}\ensuremath{\ne}\frac{1}{2}$, we find that the static structure factor and the static susceptibility display a maximum at $2{k}_{F}$ only when the next-nearest-neighbor interaction is nonzero.

Galvanomagnetic phenomena of a polaron for arbitrary coupling constants

On the basis of the Feynman path-integral formalism with the use of an optimised trial action which has an infinite number of variational parameters, the magnetoconductivity tensor for arbitrary magnetic fields, temperatures and electron-phonon coupling constants is derived. A double-time thermal Green function is utilised which is calculated from the free-energy expression subject to a driving field. This simple and direct derivation is consistent with the conventional approximation to evaluate the free energy. Analytical expressions are given of the high-field magnetoresistance of a Frohlich polaron for the limiting cases of weak and strong couplings and of low and high temperatures.

Statistical properties of polarons in a magnetic field. I. Analytic results

The free energy of an ideal polaron gas in a static magnetic field is calculated using Feynman's path-integral formalism. The trial action, used in Feynman's polaron theory, is extended to take into account the anisotropy of the effective electron-phonon interaction. This results in a free-energy expression with four variational parameters. According to Feynman's conjecture, the resulting free energy is an upper bound to the exact result. The approximate free energy is expected to provide accurate results for arbitrary electron-phonon coupling strength ($\ensuremath{\alpha}$), temperature ($T$), and magnetic field strength ($\mathcal{H}$). The free energy per polaron is evaluated for limiting values of $\ensuremath{\alpha}$, $T$, and $\mathcal{H}$.

Boundary conditions, symmetry breakdown and Feynman path integral

The analysis of boundary conditions in variational formulation of classical field theory is performed. It is used in the Feynman path integral formulation of quantum theory including non-trivial cases of dynamical symmetry breakdown.

Quantum electrodynamic theory of voltage carrying states in a current biased Josephson weak link

The Feynman path integral formulation of the Josephson effect is applied to the problem of the influence of quantum electrodynamic fluxoid tunnelling on the characteristics of a current biased, singly connected, Josephson weak link. The well known Josephson voltage biased frequency effect, 4GwV=(2eV/h(cross)), has a natural duality extension to a current bias frequency effect omega I=( pi I/e).