Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of mathematical aspects of quantization and quantum field theory. The authors present both basic and more advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, like tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance). Applications of these topics to physical problems are discussed in later chapters. Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: diagrammatics and the Euclidean approach to constructive quantum field theory. With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics.
We consider in this paper the problem of the existence of a ground state for a class of Hamiltonians used in physics to describe a confined quantum system ("matter") interacting with a massless bosonic field. These Hamiltonians were called Pauli-Fierz Hamiltonians in [DG]. Examples, like the spin-boson model or a simplified model of a confined atom interacting with a bosonic field are given in [DG, Sect.
3.3].Pauli-Fierz Hamiltonians can be described as follows: Let K and K be respectively the Hilbert space and the Hamiltonian describing the matter. The assumption that the matter is confined is expressed mathematically by the fact thatThe bosonic field is described by the Fock space Γ(h) with the one-particle space h = L 2 (IR d , dk), where IR d is the momentum space, and the free Hamiltonian dΓ(ω(k)) = ω(k)a * (k)a(k)dk.The positive function ω(k) is called the dispersion relation. The constant m := inf ω can be called the mass of the bosons, and we will consider here the case of massless bosons , ie we assume that m = 0.The interaction of the "matter" and the bosons is described by the operatoris a function with values in operators on K. Thus, the system is described by the Hilbert space H := K ⊗ Γ(h) and the Hamiltoniang being a coupling constant.If K = C, the Hamiltonian H is solvable (see eg [Be, Sect. 7]) and H is defined as a selfadjoint operator if 1 ω(k) |v(k)| 2 dk < ∞,
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