This book provides self-contained proofs of the existence of ground
states of several interaction models in quantum field theory.
Interaction models discussed here include the spin-boson model, the
Nelson model with and without an ultraviolet cutoff, and the Pauli-Fierz
model with and without dipole approximation in non-relativistic quantum
electrodynamics. These models describe interactions between bose fields
and quantum mechanical matters.A ground state is defined as the
eigenvector associated with the bottom of the spectrum of a self-adjoint
operator describing the Hamiltonian of a model. The bottom of the
spectrum is however embedded in the continuum and then it is non-trivial
to show the existence of ground states in non-perturbative ways. We show
the existence of the ground state of the Pauli-Fierz mode, the Nelson
model, and the spin-boson model, and several kinds of proofs of the
existence of ground states are explicitly provided. Key ingredients are
compact sets and compact operators in Hilbert spaces. For the Nelson
model with an ultraviolet cutoff and the Pauli-Fierz model with dipole
approximation we show not only the existence of ground states but also
enhanced binding. The enhanced binding means that a system for
zero-coupling has no ground state but it has a ground state after
turning on an interaction.The book will be of interest to graduate
students of mathematics as well as to students of the natural sciences
who want to learn quantum field theory from a mathematical point of
view. It begins with abstract compactness arguments in Hilbert spaces
and definitions of fundamental facts of quantum field theory: boson Fock
spaces, creation operators, annihilation operators, and second
quantization. This book quickly takes the reader to a level where a
wider-than-usual range of quantum field theory can be appreciated, and
self-contained proofs of the existence of ground states and enhanced
binding are presented.
