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Mathematics of Quantization and Quantum Fields by Christian Gérard, Jan Derezinski

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17

Quasi-free states

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”

(17.1)

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

(17.2)

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

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