Feynman integral and perturbation theory in quantum tomography

Martin Grothaus, Anna Vogel, Christopher Casenas Bernido +1 more

In this note the concepts of path integrals and techniques how to construct them are presented. Here we concentrate on a White Noise approach. Combining White Noise techniques with a generalized time‐dependent Doss' formula Feynman integrands are constructed as white noise distributions beyond perturbation theory.

Alberto Ibort, Vladimir I. Man’ko, Giuseppe Marmo +2 more

G.M. D'Ariano, Lorenzo Maccone, M.G.A. Paris

M. Asorey, Alberto Ibort, Giuseppe Marmo

The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.

Richard E. Crandall

Combinatorial relations can be used to convert the non-relativistic time-sliced Feynman path integral into perturbation expansions. These methods reveal that when the time interval is sliced into N increments, each order of perturbation theory sustains an error O(1/ square root N). In this way the author provides exact path integral results for the following potentials: delta-function comb, finite well, tunnelling barrier, and a generalized exponential cusp. For the tunnelling barrier it is seen how the celebrated (-1) reflection factor arises in the limit of infinite barrier height. The one-dimensional Coulomb problem is solved as a limiting case of the exponential cusp. In addition, for power potentials the author indicates how this path integral approach yields sometimes divergent, nevertheless asymptotic perturbation expansions.

Jorge Casanova, C. E. López, Juan José García‐Ripoll +2 more