First-Order Perturbation Theory: Operator Formalism

Omar Morandi, Luigi Barletti, Giovanni Frosali

A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representation is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulation of the Wigner dynamics in the presence of an external magnetic field is investigated.

Bryce Seligman DeWitt

GEORGE G. HALL

Wojciech Jaworski, David M. Wardlaw

We show that a useful concept of the time spent by a quantum-mechanical particle in a given spatial region can be expressed in terms of a Hermitian sojourn time operator. Mean values of this operator give the mean sojourn time (dwell time) of the particle. We show that the sojourn time operators occur in a natural way in first-order perturbation theory for the barrier perturbed by any finite-range potential. The perturbed S operator, and also the effect of the perturbation on the change of observables due to the scattering, are, to a first-order approximation, fully expressible in terms of sojourn time operators. We anticipate that this result can be generalized to the case when the translational motion of the tunneling particle is weakly coupled to additional degrees of freedom. As a specific example of such a system, we explicitly reconsider the Larmor clock, originally proposed as a device measuring interaction times. We show that the change of the spin components, to a first-order approximation, is indeed fully expressible in terms of a sojourn time operator. In particular, when set as a clock, the system measures mean values of the sojourn time operator or real parts of some of its matrix elements.

Luigi E. Picasso, L. Bracci, Emilio d’Emilio

Song He, Yuan Sun, Yu-Xuan Zhang

Abstract In this paper, we investigate the partition functions of conformal field theories (CFTs) with the T $$ \overline{T} $$ T ¯ deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtained by the perturbative QFT approach are consistent with results obtained by the Hamiltonian formalism in literature.