From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 219

which smeared with test functionsf ̃∈S(R^4 )

φ(f ̃)=

∫

R^4

d^4 xφ(x,t)f ̃(x, t)

satisfy

U(t)φ(f ̃)U(t)†=φ(f ̃(I,t)),

where

f ̃(I,a)(x, t)=f ̃(x, t−a).

In terms of the new covariant field operatorsφ(f ̃), the principles of quantum field
theory are similar to the ones introduced in section 3. The only changes affect the
last three principles which now read:

• PC5Field theory (real boson):For anyf ̃∈S(R^4 ) there is field operator
  φ(f ̃)inHwhich satisfiesφ(f ̃)=φ(f ̃)∗. The subspace spanned by the
  vectorsφ(f ̃ 1 )φ(f ̃ 2 )···φ(f ̃n)| 0 〉for arbitrary test functionsf ̃ 1 ,f ̃ 2 ,···f ̃n∈
  S(R^4 ) is a dense subspace ofH.


• PC6Poincar ́e covariance: For any Poincar ́e transformation (Λ,a)and
  classical field test function definedinMinkowskispace-timef ̃∈S(R^4 )
  U(Λ,a)φ(f ̃)U(Λ,a)†=φ(f ̃(Λ,a)),
  where
  f ̃(Λ,a)(x)=f ̃(Λ−^1 (x−a)).


• PC7(Bosonic) local causality: For anyf, ̃ ̃g∈S(R^4 ) whose domains are
  space-like separated^10 the corresponding field operatorsφ(f ̃),φ( ̃g)com-
  mute^11
  [φ(f ̃),φ( ̃g)] = 0. (3.48)

It is not difficult to show that these principles are satisfied by the free field the-
ory. The only non-trivial test is the calculation of the commutator of free fields
[φ(f),φ(g)]. After some simple algebra, it can be shown that it is an operator
proportional to the identity operator times a real function offandgwhich can
be estimated from the vacuum expectation value of the operator

[φ(f ̃),φ( ̃g)] =I

∫

R^4

d^4 x

∫

R^4

d^4 yf ̃(x)Δ(x−y) ̃g(y)=I〈 0 |[φ(f ̃),φ( ̃g)]| 0 〉,

(^10) f, ̃g ̃are space-like separated if for anyx, y∈R (^4) such thatf ̃(x)=0and ̃g(y)=0,d(x, y)<0.
(^11) In the fermionic case the commutator [·,·] is replaced by an anticommutator{·,·}.