By G. Velo, A. S. Wightman

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**Extra info for Constructive QFT: Majorana school 1973**

**Example text**

1. 4) is to be interpreted as follows. We carry out the k0 -integration ﬁrst, as a contour integral in the complex k0 -plane, along the contour shown. The contour encloses both the poles of function (k 2 − m2 )−1 , which are at k0 = ±ωk . 3). The commutator ∆(x, y) is a Lorentz-invariant function of the proper distance (x − y)2 . If x0 = y 0 , the commutator vanishes since the ﬁelds commute at equal times. , for all spacelike separations of the points x, y or equivalently for (x − y)2 < 0. 5) φ is a hermitian operator and qualiﬁes as an observable.

21) As an example, consider the 4-point function G(x1 , x2 , x3 , x4 ). In the limit of x01 , x02 , x03 → ∞ and x04 → −∞, we have one-particle in the far past and three particles in the far future, corresponding to the process of a particle decaying into three others. G(x1 , x2 , x3 , x4 ), with these assignments of time-labels, gives the probability amplitude for such a process. Likewise, in the limit of x01 , x02 → ∞ and x03 , x04 → −∞, we have two particles in the far past and two particles in the far future, and the corresponding G(x1 , x2 , x3 , x4 ) gives the amplitude for two-particle scattering.

For Lorentz transformations, δxµ = ω µν xν . 2 Rules of quantization As with any quantum mechanical system, the states are represented by vectors (actually rays) in a Hilbert space H. The scalar product ϕ|α = Ψα [ϕ] is the wave function of the state |α in a ϕ-diagonal representation; it is the probability amplitude for ﬁnding the ﬁeld conﬁguration ϕ(x) in the state |α . Observables are represented by linear hermitian operators on H. Fields are in general linear operators on H, not necessarily always hermitian or observable.