7.6. Invariant functions Consider the functions ()(x) = i (2)d) dd1k C() dk0 eikx k2 m2...

7.6. Invariant functions Consider the functions

(±)(x) = i

(2π)d)

 dd−1k

C(±)

dk0 eik·x k2 −m2 , where the contours of integration are shown in Figure 7.21:

a. Show that the correlation function of the commutator of the field is given by 0| [ϕ(x),ϕ(y) ] |0 =

(x−y),

where (x−y) = (+)(x−y)+
(−)(x−y).

b. Prove that (x) vanishes for equal times (x−y,0) = 0.
Using Lorentz invariance, argue that the relation above implies the vanishing of (x−y) for all space-like intervals.From a physical point of view, the commutativity of the field for space-like intervals is the consequence of the causality principle:
since space-like points cannot be related by light signals, the measurements done at the two points cannot interfere and therefore the operators commute.

c. Prove that (x) and (±)(x) satisfy the homogenous equation (+m2)
(x) = (+m2)
(±)(x) = 0, while the Feynman propagator, which corresponds to an infinite contour of integration, satisfies (+m2)
F(x) = −i δd(x).