Estimates on Functional Integrals of Non-Relativistic Quantum Field Theory, with Applications to the Nelson and Polaron Models

Abstract

In this dissertation we present a computational tool that allows one to provide lower bounds for the ground state energy of several quantum-mechanical Hamiltonians. Given a Hamiltonian H, its ground state energy can be expressed as a Feynman-Kac formulawhere A T is negative the time integral of the effective potential of the Schrödinger operator H, evaluated at a Brownian path on the time interval [0, T ]. It is shown how a Clark-Ocone expansion for the action A T , namely an expansion as its deterministic expectation plus a random oscillatory part, represented as a stochastic integral, can lead to an upper bound for the exponential moment of A T , E e A T , of the form e BT +o(T ) , for some constant B. This leads, in particular, to a lower bound for H given by −B. The method is illustrated in two canonical models in non-relativistic quantum field theory: the Nelson and polaron models.ii

A Governança No Combate À Desertificação Em Comunidades Rurais

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