Abstract The paper is focused on the study of functional integrals defined on discontinuous trajectories. A special type of transformations was proposed for a class of such functional integrals in papers on quantum physics, which makes it possible to reduce integrals of this kind to integrals over continuous trajectories.
The aim of this book is to present and explain the concept of the path integral which is intensively used nowadays in almost all the branches of theoretical physics. The notion of path integral (sometimes also called functional integral or integral over trajectories or integral over histories or continuous integral) was introduced, for the first time, in the 1920s by Norbert Wiener (1921, 1923, 1924, 1930) as a method to solve problems in the theory of diffusion and Brownian motion. This integral, which is now also called the Wiener integral, has played a central role in the further development of the subject of path integration.
Application of the functional integration methods in equilibrium statistical mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. This measure arises in the Bogolyubov T-product approach and is non-Wiener. We consider problems related to integration with respect to the Bogolyubov measure in the space of continuous functions and calculate some functional integrals with respect to this measure. Approximate formulas that are exact for functional polynomials of a given degree and also some formulas that are exact for integrable functionals belonging to a broader class are constructed. We establish the nondifferentiability of the Bogolyubov trajectories in the corresponding function space and prove a theorem on the quadratic variation of trajectories and study the properties implied by this theorem for the scale transformations. We construct some examples of semigroups related to the Bogolyubov measure. Independent increments are found for this measure. We consider the relation between the Bogolyubov measure and parabolic partial differential equations. An inequality for some traces is proved.
The Onsager-Machlup Lagrangian defined by the functional-integral representation of general continuous Markov processes is derived explicitly and concisely. Attention is given both to the mathematics and its physical interpretation. The method is based on (i) the consideration of continuous and differentiable paths connecting fixed end points in the convariant propagator, (ii) the application of a recently developed Fourier-series analysis of these trajectories in locally flat spaces, and (iii) the use of the proper transformations between locally Euclidean and globally Riemannian geometries. The spectral analysis allows for arbitrary paths rather than an a priori straight line even in the short-time propagator and avoids any ad hoc discretization rule. Specialized to globally flat spaces the result agrees with formulas given by Stratonovich, Horsthemke and Bach, Graham, and Dekker. It is demonstrated that a unique covariant path integral is equivalent to a whole class of stochastically equivalent lattice expressions.
Abstract A new type of four-path interferometer is proposed to investigate the nature of the ‘discontinuous trajectories’ of a photon. Our derivation shows that the absence of ‘weak traces’, which makes the trajectories of the photon seem to be discontinuous, is actually due to the multi-path interference, and cannot be a strong criterion to say ‘The photon has not been there.’ The presence of the weak traces is a sufficient but not necessary condition for the fact that the photon has been the corresponding places. Moreover, the propagating paths of the photon must always be continuous.