10.1. Operatorial identities There is a simple example that shows the necessity of considering the operatorial identities only in a weak sense, i.e. true only for the matrix elements. Consider an interacting scalar field φ(x) and suppose that for x0→−∞ its interactions vanish. In this case it seems natural to pose the operatorial identity lim x0→−∞

φ(x) = φin(x)

where φin(x) is a free bosonic field. However, this leads to a contradiction. In fact, it the relation above were true, we would have lim x0→−∞ lim y0→−∞

0 | φ(x)φ(y) | 0 = 0 | φin(x)φin(y) | 0.

Since φin(x) is a free field, the right-hand side is the usual progagator Gfree(x−y) of a scalar free field Gfree(x−y) =  ddp

(2π)d 1

p2 −m2 eip˙(x−y).

a. Use the Lorentz invariance to fix the dependence on the coordinates of the propagator G(x−y) of the interacting field φ(x).

b. Argue that the propagator does not coincide in the limit x0→−∞with Gfree(x−y).