On the origin of the integral representations in quantum optics

Claudio Ccapa Ttira, César Daniel Fosco, Adolfo P. C. Malbouisson +1 more

We impose the periodicity conditions corresponding to the Matsubara formalism for thermal field theory as constraints in the imaginary-time path integral. These constraints are introduced by means of time-independent auxiliary fields which, by integration of the original variables, become dynamical fields in the resulting ``dual'' representation for the theory. This alternative representation has the appealing property of involving fields that live in one dimension less than the original ones, with a quantum partition function whose integration measure is identical to the one of its classical counterpart, albeit with a different (spatially nonlocal) action.

József Janszky, Петер Адам, I. Földesi +1 more

Alexander S. Lukyanenko, Inna Lukyanenko

A path integral representation of the evolution operator for the four-dimensional Dirac equation is proposed. A quadratic form of the canonical momenta regularizes the original representation of the path integral in the electron phase space. This regularization allows one to obtain a representation of the path integral over trajectories in the configuration space. This form of the path integral is useful for the formulation of perturbation theory in an external electromagnetic field.

Helman Le Pas de Sécheval, C. Bendjaballah

S. Twareque Ali, L. V. Denisov

A formula for a composite Fourier-wavelet transform is suggested, from which the noncommutative Fourier and wavelet transforms can be obtained as particular examples, as well as several other Fourier-type transforms. It is shown how this composite transform can be used to extend certain results, originally obtained for square-integrable group representations, to more general measures, arising in quantum probability theory.

Ravinder Rupchand Puri