Unified treatment of the bound states of the Schioberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schioberg and Eckart potentials using an approximation of the centrifugal term.In order to determine the e-states solutions,we use the Feynman path integral approach to quantum mechanics.We show that by performing nonlinear space-time transformations in the radial path integral,we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system.The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions.We show that the Eckart potential can be derived from the Schioberg potential.The obtained results are compared to those produced by other methods and are found to be consistent.     

Superfield formulation of the phase space path integral

We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.     

Canonical and D-transformations in theories with constraints

We describe a class of transformations in a super phase space (we call them D-transformations) which play the role of ordinary canonical transformations in theories with second-class constraints. Namely, in such theories they preserve the form invariance of equations of motion, their quantum analogs are unitary transformations, and the measure of integration in the corresponding Hamiltonian path integral is invariant under these transformations.     

Space-time transformations in radial path integrals

Nonlinear space-time transformations in the radial path integral are discussed. A transformation formula is derived, which relates the original path integral to the Green function of a new quantum system with an effective potential containing an observable quantum correction ≈?². As an example the formula is applied to spherical brownian motion.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

Of Ghosts, Gauge Volumes, and Gauss's Law

The relationship between the canonical operator and the path integral formulation of quantum electrodynamics is analyzed with a particular focus on the implementation of gauge constraints in the two approaches. The removal of gauge volumes in the path integral is shown to match with the presence of zero-norm ghost states associated with gauge transformations in the canonical operator approach. The path integrals for QED in both the Feynman and the temporal gauges are examined and several ways of implementing the gauge constraint integrations are demonstrated. The upshot is to show that both the Feynman and the temporal gauge path integrals are equivalent to the Coulomb gauge path integral, matching the results developed by Kurt Haller using the canonical formalism. In addition, the Faddeev–Popov form for the Feynman gauge and temporal gauge Lagrangian path integrals are derived from the Hamiltonian form of the path integral.     

Path integrals and the indefiniteness of the gravitational action

The Euclidean action for gravity is not positive definite unlike those of scalar and Yang-Mills fields. Indefiniteness arises because conformal transformations can make the action arbitrarily negative. In order to make the path integral converge one has to take the contour of integration for the conformal factor to be parallel to the imaginary axis. The path integral will then converge at least in the one-loop approximation if a certain positive action conjecture holds. We perform a zeta function regularization of the one-loop term for gravity and obtain a non-trivial scaling behaviour in cases in which the background metric has non-zero curvature tensor, and hence non-trivial topologies.     

Path integrals in polar field variables in QFT

We show how to transform a d-dimensional Euclidean path integral in terms of two (Cartesian) fields to a path integral in terms of polar field variables. First we present a conjecture that states how this transformation should be done. Then we show that this conjecture is correct in the case of two toy models. Finally the conjecture will be proven for a general QFT model with two fields.     

Path integrals and linear canonical transformations

For a system withN bosonic or fermionic degrees of freedom I calculate the coherent state propagator, i.e. the matrix element between coherent states of the evolution operator, for a general quadratic Hamiltonian plus a source term, using the holomorphic form of the path integral. The analysis and the result obtained are used to discuss the transformation properties of the path integral for linear canonical transformations (Bogoliubov-Valatin trfs), a preliminary to the formulation of a geometric theory of path integral quantization.

     

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimensional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions.     

Incompatibility of gauge invariance and nonrelativistic locality in path integral formulation

It is shown that a modification of the usual gauge transformations is essential to the path integral formulation of nonrelativistic quantum mechanics as a consequence of defining the locality condition as follows: The contribution from each path comes entirely from the points on the path. Arguments are based on the similarity between Wiener and Feynman functional (path) integrals.     

Phase Space Path Integral Method for Problem of Obtaining Propagator of a Particle with a Force Quadratic in Velocity

This article uses the phse space path integral method to find the propagator for a particle with a force quadratic in velocity. Two specific canonical transformations has been used for this purpose.     

Operator ordering and supersymmetry

We investigate supersymmetric quantum mechanics in curved space. In particular we discuss, in the framework of the canonical formulation, the role of supersymmetry (SUSY) and of invariance under general coordinate transformations (GCT) in solving the operator ordering ambiguity. It is shown that GCT invariance is instrumental to achieve such result. We also examine the connection with the Weyl ordering of operators and the relevance of our results for the path integral approach.     

Path integral and effective Hamiltonian in loop quantum cosmology

We study the path integral formulation of Friedmann universe filled with a massless scalar field in loop quantum cosmology. All the isotropic models of $k=0,+1,-1$ are considered. To construct the path integrals in the timeless framework, a multiple group-averaging approach is proposed. Meanwhile, since the transition amplitude in the deparameterized framework can be expressed in terms of group-averaging, the path integrals can be formulated for both deparameterized and timeless frameworks. Their relation is clarified. It turns out that the effective Hamiltonian derived from the path integral in deparameterized framework is equivalent to the effective Hamiltonian constraint derived from the path integral in timeless framework, since they lead to same equations of motion. Moreover, the effective Hamiltonian constraints of above models derived in canonical theory are confirmed by the path integral formulation.     

Path integration on Darboux spaces

In this paper, the Feynman path integral technique is applied to two-dimensional spaces of nonconstant curvature: these spaces are called Darboux spaces D _I-D _IV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases; the exceptions being the quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl-Teller potential, and the spheroidal wave functions. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green’s functions and the expansions into the wave functions. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.     

Pure spinor integration from the collating formula

We use the technique developed by Becchi and Imbimbo to construct a well-defined BRST-invariant path integral formulation of pure spinor amplitudes. The space of pure spinors can be viewed from the algebraic geometry point of view as a collection of open sets where the constraints can be solved and a set of free and independent variables can be defined. On the intersections of those open sets, the functional measure jumps and one has to add boundary terms to construct a well-defined path integral. The result is the definition of the pure spinor integration measure constructed in terms of differential forms on each single patch.     

Operator ordering schemes and covariant path integrals of quantum and stochastic processes in Curved space

A method is given for the derivation of covariant path integral solutions of quantum and stochastic processes in curved space.The correspondence between operator ordering schemes and ordinary functions in phase space is studied and applied to the explicit construction of a class of equivalent lattice representations of path integrals. It is shown that this class is uniquely related to a covariant functional integral. Some arguments forR/12 as the quantum mechanical curvature potential are given. Simple rules for nonlinear point transformations are stated. The connection with previous works is discussed.     

Zeta function regularization of path integrals in curved spacetime

This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.