q-path integrals

We construct path integral representations for the evolution operator ofq-oscillators using Bargmann-Fock representations with commuting and non-commuting variables, the differential calculi beingq-deformed for both types of variables. The cases of real and root of unity values ofq-parameter are considered. Forq ²=–1 case we obtain a new form of Grassmann-like path integral.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.On leave of absence from Nuclear Physics Institute, Moscow State University, 119899, Moscow, Russia.     

Relativistic Feynman-type integrals

It is shown that within the framework of the Kershaw stochastic model generalized by the author to the relativistic case a Feynman-type process may be constructed which can formally be understood as a diffusion phenomenon in Euclidean space. This makes it possible to introduce a real probability measure in the scheme of quantum mechanics proposed by Feynman.     

Supersymmetric path integrals

The supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the Wiener integral. It is then pushed forward to a compact Riemannian manifold by means of a Malliavin-type construction. The relation to index theory is discussed.Research supported by an NSF postdoctoral fellowship     

Feynman path integrals

Path integrals techniques are derived from a new definition [1] of Feynman path integrals. These techniques are used to establish that Feynman-Green functions for a given physical system are covariances of pseudomeasures suitable for its path integrals. The variance of a pseudomeasure is a more versatile tool than the Feynman-Green function it defines.     

On surface integrals for triple collision integrals in kinetic theory of gases

The triple collision integrals in kinetic theory of gases are re-examined in terms of mass-normalized coordinate systems and new surface integrals are obtained from them. The five-dimensional surface integrals obtained are more symmetric than those known up to know and appear suggestive of further generalization to more complex problems of many-body systems.     

Path integrals and parastatistics

The propagator and corresponding path integral for a system of identical particles obeying parastatistics are derived. It is found that the statistical weights of topological sectors of the path integral for parafermions and parabosons are simply related through multiplication by the parity of the permutation of the final positions of the particles. Appropriate generalizations of statistics are proposed obeying unitarity and factorizability (strong cluster decomposition). The realization of simple maximal occupancy (Gentile) statistics is shown to require ghost states.     

One loop integrals reduction

By further examining the symmetry of external momenta and masses in Feynman integrals, we fulfilled the method proposed by Battistel and Dallabona, and showed that recursion relations in this method can be applied to simplify Feynman integrals directly.     

One loop integrals revisited

We present a new calculation of the well-known one-loop two-point scalar and tensor functions. We also present a systematic reduction to a certain class of functions which minimizes the effort for calculating tensor integrals drastically. We avoid standard techniques such as Feynman parametrization and Wick rotation.     

One loop integrals reduction

By further examining the symmetry of external momenta and masses in Feynman integrals, we fulfilled the method proposed by Battistel and Dallabona, and showed that recursion relations in this method can be applied to simplify Feynman integrals directly.     

Path integrals on half-line

In the framework of path integrals we present a solution to the Schrödinger equation for a free particle confined to the half-linex > 0. A solution in question corresponds to the boundary condition (/x) (0,t)= (0,t) where is a real constant.     

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

In perturbative calculations of quantum-mechanical path integrals in curvilinear coordinates, one encounters Feynman diagrams involving multiple temporal integrals over products of distributions which are mathematically undefined. In addition, there are terms proportional to powers of Dirac -functions at the origin coming from the measure of path integration. We derive simple rules for dealing with such singular terms from the natural requirement of coordinate independence of the path integrals. Received: 12 December 2000 / Revised version: 16 January 2001 / Published online: 6 April 2001     

Ordinary superconductivity and path integrals

Summary Functional methods are very powerful in dealing with ordinary superconductivity. The ring geometry is discussed in mean field by means of a Higgs-type Ginzburg-Landau Lagrangian. The presence of a junction in the ring, as in SQUIDs, leads to a θ-vacuum as the ground state. The variable θ is related to the phase difference of the order parameter at the junction and Josephson relations are obtained semi-classically. The system is, to any purpose, in two space dimensions, what can imply exotic statistics. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

New relationships between Feynman integrals

New types of relationships between Feynman integrals are presented. It is shown that Feynman integrals satisfy functional equations connecting integrals with different values of scalar invariants and masses. A method is proposed for obtaining such relations. The derivation of functional equations for one-loop propagator- and vertex-type integrals is given. It is shown that a propagator-type integral can be written as a sum of two integrals with modified scalar invariants and one propagator massless. The vertex-type integral can be written as a sum over vertex integrals with all but one propagator massless and one external momenta squared equal to zero. It is demonstrated that the functional equations can be used for the analytic continuation of Feynman integrals to different kinematic domains.     

Path integrals in Riemannian spaces

The covariant path integral for a free particle in curved space will be evaluated by means of a spectral analysis of smooth paths. No discretization rule will be required to put the action on a lattice. The connection between the resulting quantum hamiltonian and the Onsager-Machlup lagrangian for diffusion processes willbe discussed. The present treatment corrects an earlier version.