We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form
E[exp(AT)] , the (effective) action
AT being a function of particle trajectories up to time T. The estimates in turn yield rigorous lower bounds for ground state energies, via the Feynman–Kac formula. The upper bounds are obtained by writing the action for these functional integrals in terms of stochastic integrals. The method is illustrated in familiar quantum mechanical settings: for the hydrogen atom, for a Schrödinger operator with
1/∣x∣2 potential with small coupling, and, with a modest adaptation of the method, for the harmonic oscillator. We then present our principal applications of the method, in the settings of non-relativistic quantum field theories for particles moving in a quantized Bose field, including the optical polaron and Nelson models.
This book presents selected topics about the path-integral method in Quantum Mechanics and Optics. Starting from an introduction to the grounds of functional integration theory, the main arguments of quantum and statistical mechanics, where the path-integral method works, are exposed. In particular, the partition function, the concept of instanton, tunneling and dissipative phenomena are analysed. The final section is devoted to selected and solved problems. This volume will be useful to beginners as well as more advanced students and researchers in the field.