Contents 6.2 The Nelson model in Fock space 6.2.1 Definition 6.2.2 Infrared and ultraviolet divergences 6.2.3 Embedded eigenvalues 6.3 The Nelson model in function space 6.4 Existence and uniqueness of the ground state 6.5 Ground state expectations 6.5.1 General theorems 6.5.2 Spatial decay of the ground state 6.5.3 Ground state expectation for second quantized operators. .. 6.5.4 Ground state expectation for field operators 6.6 The translation invariant Nelson model 6.7 Infrared divergence 6.8 Ultraviolet divergence 6.8.1 Energy renormalization 6.8.2 Regularized interaction 6.8.3 Removal of the ultraviolet cutoff 6.8.4 Weak coupling limit and removal of ultraviolet cutoff ....

The Pauli-Fierz Hamiltonian describes a system of N electrons minimally coupled to a quantized radiation field. The electrons have spin and an ultraviolet cutoff is imposed on the quantized radiation field. For arbitrary values of coupling constants, self-adjointness and essential self-adjointness of the Pauli-Fierz Hamiltonian are proven by means of a functional integral.

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman-Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L p -L q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

This paper presents functional integral representations for heat semigroups with infinitesimal generators given by self-adjoint Hamiltonians (Pauli–Fierz Hamiltonians) describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. By the functional integral representations, some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schrödinger operators with classical vector potentials.

The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.