Suppose that we have a state ψ on the polynomial algebra generated by the fields (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 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. ...