From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 227

The proof of reflection positivity for higher order Schwinger functions follows from
Wick theorem in a similar way.
The cluster property of the two-point Schwinger formula follows from the fact
that the kernel (−∇^2 +m^2 )−^1 (x, y) vanishes in the limit‖x−y‖→∞.
In this formalism the functional integral of the interacting theory can be un-
derstood as a Riemann-Stieltjes measure with respect to the Gaussian measure of
thefreetheoryδμm, i.e.

Sn(f ̃ 1 ,f ̃ 2 ,...,f ̃n)=

∫

S′(R^4 )

δμm(φ)e−V(φ)φ(f ̃ 1 )φ(f ̃ 2 ),...,φ(f ̃n).

Perturbation theory is defined just by the Taylor expansion ofe−V(φ)in power
series and the formal commutation of the Gaussian integration with the Taylor
sum. In theλφ^4 case the perturbation theory is defined by

Sn(f ̃ 1 ,f ̃ 2 ,...,f ̃n)=

∑∞

n=0

1

n!

λn

4!n

∫

S′(R^4 )

δμm(φ)‖φ^2 ‖^2 nφ(f ̃ 1 )φ(f ̃ 2 ),...,φ(f ̃n).

(3.56)

In this formalism UV divergences appear when computing the different terms of
Eq. (3.56) by using Wick’s theorem. But the main advantage of the covariant for-
malism is that in it the preservation of Poincar ́e symmetries under renormalization
is more transparent.

3.9 WhatisBeyond?...........................

From the Euclidean formulation it follows that the functional integral approach
provides us with a constructive method of quantizing a field theory. The pertur-
bative expansion Eq. (3.56) gives us a very explicit way of computing Schwinger
functions. The ultraviolet divergences that arise there, in some cases can be renor-
malized by absorbing them in the bare parameters of the theory.
From this point of view the quantum field theories are classified in two classes:

(i) Theories where a finite set of parameters in the Lagrangian is enough to

absorb all UV divergences; and

(ii) Theories that require an infinite set of independent parameters.

Theories of the first family are called renormalizable whereas those of the sec-
ond class are unrenormalizable. Of course, only theories of the first type are
sensible since from a finite number of parameters they can predict the behavior of
all quantum states.