From Classical Mechanics to Quantum Field Theory

220 From Classical Mechanics to Quantum Field Theory. A Tutorial

where

Δ(x−y)=

∫

d^3 k

(2π)^3

1

2

√

k^2 +m^2

(

eik·(x−y)−e−ik·(x−y)

)

,

andk·(x−y)=k·(x−y)−

√

k+m^2 (x 0 −y 0 ). The local causality property
Eq. (3.48) follows from the fact that the causal propagator kernel Δ(x−y) vanishes
for equal timesx 0 =y 0 , since the two terms in

Δ(x−y,0) =

∫

d^3 k

(2π)^3

1

2

√

k^2 +m^2

(

eik·(x−y)−e−ik·(x−y)

)

=0,

give the same contributions, as can be shown by flipping the sign ofkin one of
them. Although the expression of causal propagator kernel Δ(x−y) is relativistic
invariant, it seems to be non-covariant. However, it can be written in an explicitly
covariant form

Δ(x−y)=D(x−y)−D(y−x),

where

D(x−y)=

∫

d^4 k

(2π)^4

θ(k 0 )δ(k^2 +m^2 )eik·(x−y).

This result shows that the covariant quantization approach could also be derived
from the Peierls covariant classical approach to field theory [ 17 ], by replacing
Peierls brackets by commutators (see appendix C).
To check that fundamental principles are satisfied in an interacting theory is
more difficult, but it can be shown that in perturbation theory they are satisfied
even after renormalization.

3.7.1 Euclideanapproach

Working with field operators in the Fock space is hard because they are unbounded
operators. For such a reason it is more convenient to consider their expectation
values on the different states. Since the full Fock space is generated by the com-
pleteness principle by the field operators, it is enough to consider the expectation
values of the products of field operators on the vacuum state.
These expectation values are known as Wightman functions

W(f ̃ 1 ,f ̃ 2 ,...,f ̃n)=〈 0 |φ(f ̃ 1 )φ(f ̃ 2 )...φ(f ̃n)| 0 〉.

However the unbounded character of the field operatorsφ(f ̃) reflects in the os-
cillating behavior of the Wightman functions. For such a reason it is much more
convenient to introduce the Euclidean time analytic extensions of the quantum