From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 201

The subspace spanned by the vectorsφ(f 1 )φ(f 2 )···φ(fn)| 0 〉for arbitrary

test functionsf 1 ,f 2 ,···fn∈S(R^3 ) is a dense subspace ofH.

• P6Poincar ́e covariance: Letf ̃∈S(R^4 ) be a test function defined in
  Minkowski space-time and

φ(f ̃)=

∫

R^4

d^4 xφ(x)f ̃(x),

whereφ(x)=φ(x,t)=eitP^0 φ(x)e−itP^0 .Then^4 ,

U(Λ,a)φ(f ̃)U(Λ,a)†=φ(f ̃(Λ,a)),

with

f ̃(Λ,a)(x)=f ̃(Λ−^1 (x−a)).

• P7(Bosonic) local causality:For anyf,g∈S(R^3 ) the corresponding field
  operatorsφ(f),φ(g)commute^5 ,

[φ(f),φ(g)] = 0. (3.12)

3.3.1 Canonicalquantization

As in the case of quantum mechanics, there are special cases where the quantum
field theory arises from the quantization of a classical field theory^6.
Let us consider a scalar real fieldφin Minkowski space-timeR^4. The classical
field theory is defined according to the variational principle by stationary field
configurationsφ(x) of the classical action functional

S[φ(x)]≡

∫

d^4 xL(φ, ∂μφ)=

∫

d^4 x

( 1

2 ∂μφ∂

μφ−V(φ)). (3.13)

The equations of motion are obtained, thus, from the Euler-Lagrange equations

∂μ

[

δL

δ(∂μφ)

]

−δL

δφ

=0 =⇒ φ+δV

δφ

=0, (3.14)

where=∂μ∂μ. Notice that the Poincar ́e invariance of the action implies the
Poincar ́e invariance of the equations of motion.

(^4) For higher spin fields, the Poincar ́e representation satisfies the covariant transfomation law
U(Λ,a)φ(f)U(Λ,a)†=S(Λ)−^1 φ(f(Λ,a)), whereSis a linear n-dimensional representation of
Lorentz group andφis a field with n-components.
(^5) In the fermionic case, the commutator [·,·] is replaced by an anticommutator{·,·}.
(^6) See section 1.3.