Path Integral Formula Research Articles

PERIODIC COHERENT STATES AND A PATH INTEGRAL FOR SPINS

A path integral formalism for quantum spins is discussed with the aid of a new coherent state. Under the study, we can see the reason why the path integral formula proposed recently by Nielsen and Rohrlich works so well and the origin of the parameter in their formula. A generalization to the case where spin magnitude is unfixed is also presented.

QUANTUM MECHANICS AND PATH INTEGRALS IN SPACES WITH CURVATURE AND TORSION

We point out that there is a natural geometric procedure for constructing the quantum theory of a particle in a general metric-affine space with curvature and torsion. Quantization rules are presented and expressed in the form of a simple path integral formula which specifies compactly a new combined equivalence and correspondence principle. The associated Schrödinger equation has no extra curvature nor torsion terms that have plagued earlier attempts. Several well-known physical systems are invoked to suggest the correctness of the proposed theory.

Stochastic mechanics and quantum interference

It is shown that in the framework of the stochastic interpretation of quantum mechanics it is always possible to determine correct real positive transition probabilities that can be added together to obtain the quantum interference patterns. The implication of this fact on the possibility that quantum micro-objects follow trajectories in space and time is discussed. A derivation is given of a path integral formula yielding these transition probabilities.

Path collapse in Feynman formula. Stable path integral formula from local time reparametrization invariant amplitude

The Feynman formula, which expresses the time displacement amplitude 〉 x b | exp (−t H ̌ ) | x a 〈 in terms of a path integral Π 1 N (∫ d n) Π 1 N+1 ( ∫ dp n 2π ) exp{Σ 1 N [ ip n(x n−x n−1) − ϵH (p n, x n)]} with large N, does not exist for systems with Coulomb -1 r potential and gives incorrect threshold behaviours near centrifugal 1 r 2 or angular 1 sin 2θ barriers. We discuss the physical origin of this failure and propose an alternative well-defined path integral formula based on a family of amplitudes that is invariant under arbitrary local time reparametrizations. The time slicing with finite N breaks this invariance. For appropriate choices of the reparametrization function the fluctuations are stabilized and the new formula is applicable to all the above systems.

Canonical transformations in the path integral and the operator formalism

A general framework of canonical transformations in the path integral is presented under the guidance of classical mechanics. The master equation is given in terms of the difference instead of the differential in classical mechanics. In the case of infinitesimal transformations the operator counterpart is found to be a PQ-ordered form of the classical generator. A path integral formula for polar coordinates is also discussed to be understood as a faithful representation of the Schrödinger equation.

A path integral formula for certain fourth-order elliptic operators

We define a discretized path integral formula for the operator −Δ2−V. This formula is the generalization of the Feynman-Kac formula for +Δ−V.

The Z i t t e r b e w e g u n g of a Dirac particle in two-dimensional space-time

The path space measures that have been constructed in the present authors’ previous papers to give path integral formulas in quantum mechanics for a Dirac particle in two-dimensional space-time are shown to be concentrated on those paths that have differential coefficients equal to plus or minus the light velocity in every finite time interval except at finitely many instants of time.

Gravitational Anomaly and Path Integral Jacobian of Anti-Symmetric Tensor Gauge Field

A path integral formula is presented for the covariant gravitational anomaly of the self-dual anti-symmetric tensor gauge field. With the help of this formula, it is shown that the path integral Jacobian of the self-dual field strength under the general coordinate transformation is twice as large as the gravitational anomaly. It is also shown perturbatively that a chiral bi-spinor field accompanied with appropriate Faddeev-Popov ghosts describes a system of two self-dual anti-symmetric tensor gauge fields.

Fermionic path integration and Grassmann Brownian motion

A Grassmann probability theory, with anticommuting random variables and stochastic processes, is developed using an extension of Berezin integration to infinite dimensional spaces. A Kolmogorov-type consistency condition allows integration on spaces of paths in anticommuting space. One particular stochastic process, Grassmann Brownian motion, is described and the associated measure used to give a path-integral formula for the kernel of the evolution operator in fermionic quantum mechanics. The Fourier mode expansion of Grassmann Brownian motion is derived.

Path integration on spheres: Hamiltonian operators from the Faddeev-Senjanovic path integral formula

A study is made of the path integral quantization for a D-dimensional sphere as an example of constrained systems by means of the Faddeev-Senjanovic formula, although no one has checked its validity with paying attentions to a global structure of integration regions. By adopting a well-defined path measure obtained through the time discretization and applying formulas of spherical harmonics, operator Hamiltonians can be picked out from their formula without recourse to any approximation such as a semi-classical one ( h → 0). The analysis is made with a model, H=C p 2+M 2C 2 . We might thus regard the path integral formula of Faddeev-Senjanovic as a powerful breakthrough to “quantum” constrained systems where ordinary canonical approaches sometimes encounter difficulties in giving well-defined canonical momenta because of the uncertainty principle.

Geometro-stochastic quantization of massive fields in curved space-time

It is shown that an earlier introduced concept of quantum geometry, given by a Hilbert bundle over a base space whose elements constitute a generic curved space-time and represent the mean locations of quantum frames, can be generalized to the case of Fock bundles, whose Fock fibres are the carriers of multiparticle states. In this new context, the role of quantum Lorentz frame is taken over by second-quantized frames constructed out of coherent exciton states. The structure groups of the considered bundles contain unitary representations of the Poincare group, with the latter emerging as a gauge group for the internal degrees of freedom of the considered quantum particles. Connections compatible with the Hermitian structure are introduced on bundles of second-quantized frames that are associated to principal bundles of affine Lorentz frames. The corresponding parallel transport is expressed in terms of path integrals for quantum frame propagators. It is shown that the resulting geometro-stochastic quantum field theory in curved space-time does not give rise to the foundational difficulties with the particle concept, with normal ordering and with the definition of the stress-energy tensor, that are inherent in more conventional approaches to quantum field theory in curved space-time. Hence, the derived path integral formulae for the propagation of systems of quantum field particles display in the present framework none of the ambiguities encountered by those approaches.

Vertex functions from the Mandelstam string path integrals

As a pedagogical exercise we derive the vertex functions in the light-cone gauge for an arbitrary string state by direct integration of the Mandelstam path integral formula.

Path integral of constrained fermionic oscillators and its application to the interacting spin-3/2 field

The path-integral formula for the system of constrained fermionic oscillators interacting with ordinary bosons is put into a simpler and practical form. The consequence is the usual naive path integral with the insertion of a certain determinant factor. The factor is rewritten as a one-loop effect of commuting ghost variables of the Faddeev-Popov type. These results are applied to the minimal electromagnetic interaction of massive spin-3/2 field, and unitary Feynman graph rules are obtained.

Extended Feynman formula for damped harmonic oscillator with time-dependent perturbative force

Horváthy's modification of Feynman's path integral formula is extended to the damped harmonic oscillator with time-dependent pertubative force. The propagator at caustics is then obtained by introducing its modified semi-group property. Finally, our new results are confirmed by investigating the classical paths joining two end-point positions.

Extended feynman formula for forced harmonic oscillator

Horvathy's modification of Feynman's original method is generalized to the path integral formula of a forced harmonic oscillator. With this new formula the propagator of a harmonic oscillator with memory is evaluated exactly beyond and at caustics.

Propagation of a Dirac particle. A path integral approach

Rigorous path integral formulas are given which represent, in two space-time dimensions, the fundamental solution of the Cauchy problem for the Dirac equation as well as the retarded and advanced propagators for the Dirac particle. It is also shown that the theory can be applied to a free particle and a particle in a central electric field in four space-time dimensions and reveals some aspects of the path integral. Heuristically discussed is the connection with the phase space path integral or Hamiltonian path integral.

Extended Feynman formula for time-dependent forced harmonic oscillator

Horváthy's modification of Feynman's path integral formula is generalized to the time-dependent forced harmonic oscillator. The propagator at caustics is then obtained by using its modified semi-group property. Finally, with our new formula, the propagator for a charged particle in a time-dependent electromagnetic field is evaluated exactly beyond and at caustics.

Path integral formulation of the propagator for a two-dimensional Dirac particle

The aim of the present note is to give path integral formulas for the solution of the Cauchy problem of the Dirac equation and the relativistic propagator in two space-time dimensions. They have a close analogy with the Feynman-Kac formula for the heat equation, but the countably additive path space measures constructed are other than the Weiner measure. The same method applies to a certain hyperbolic system of the first order.

The simulation of quantum systems with random walks: A new algorithm for charged systems

Random walks with branching have been used to calculate exact properties of the ground state of quantum many-body systems. In this paper, a more general Green's function identity is derived which relates the potential energy, a trial wavefunction, and a trial density matrix to the rules of a branched random walk. It is shown that an efficient algorithm requires a good trial wavefunction, a good trial density matrix, and a good sampling of this density matrix. An accurate density matrix is constructed for Coulomb systems using the path integral formula. The random walks from this new algorithm diffuse through phase space an order of magnitude faster than the previous Green's Function Monte Carlo method. In contrast to the simple diffusion Monte Carlo algorithm, it is an exact method. Representative results are presented for several molecules.

Techniques and Applications of Path Integration

Partial table of contents: Probabilities and Probability Amplitudes for Paths. Correspondence Limit for the Path Integral (Heuristic). Vector Potentials and Another Proof of the Path Integral Formula. Doing the Integral: Free Particle and Quadratic Lagrangians. Brownian Motion and the Wiener Integral Kac's Proof. Perturbation Theory and Feynman Diagrams. SELECTED APPLICATIONS OF THE PATH INTEGRAL. Asymptotic Analysis. The Calculus of Variations. WKB Near Caustics. The Phase of the Semiclassical Amplitude. Scattering Theory. Geometrical Optics. The Polaron. Spin and Related Matters. Quantum Mechanics on Curved Spaces. Relativistic Propagators and Black Holes. Applications to Statistical Mechanics. Critical Droplets. Alias Instantons, and Metastability. Phase Space Path Integral. Omissions, Miscellany, and Prejudices. Indexes.