Suppose that we have a state ψ on the polynomial algebra generated by the fields ϕ(y) satisfying the CCR or CAR relations. For simplicity, assume that ψ is even, that is, vanishes on odd polynomials. Clearly, this state determines a bilinear form on Y given by the “2-point function”

We say that a state ψ is quasi-free if all expectation values

can be expressed in terms of (17.1) by the sum over all pairings.

This chapter is devoted to a study of (even) quasi-free states, both bosonic and fermionic. This is an important class of states, often used in physical applications. Fock vacuum states belong to this class. It also includes Gibbs states of quadratic Hamiltonians.

Representations obtained by the GNS construction from quasi-free states will be called quasi-free representations. They are usually reducible. Many interesting concepts from the theory of von Neumann algebras can be nicely illustrated in terms of quasi-free representations.

Quasi-free states can be easily realized on Fock spaces, using the so-called Araki–Woods, resp. Araki–Wyss representations in the bosonic, resp. fermionic case. Under some additional assumptions, in particular in the case of a finite number of degrees of freedom, these representations can be obtained as follows. First we consider a Fock space equipped with a quadratic Hamiltonian. Then we perform the GNS construction with respect to the corresponding Gibbs state. Finally, we apply an appropriate Bogoliubov rotation.