From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 225

In this case the Schwinger functions can be derived from (3.53). For any two
functionsf ̃ 1 ,f ̃ 1 ∈SR^4 with ordered supports, i.e. for anyx 1 ,x 2 withf ̃ 1 (x 1 )=
f ̃ 2 (x 2 )=0,τ 1 >τ 2

S 2 (f ̃ 1 ,f ̃ 2 )=〈 0 |Φ∗E(f ̃ 1 )ΦE(f ̃ 2 )| 0 〉=

∫

dx 1 dx 2 f ̃ 1 (x 1 )

a^2 π

|x 1 −x 2 |^2 π

f ̃ 2 (x 2 ).

This model corresponds to the spin wave regime of theO(2) sigma model defined
by the classical action

S[Φ(x)] =

1

2

∫

d^2 x∂μΦ∗∂μΦ=

1

2

∫

d^2 x∂μφ∂μφ.

which in that regime does coincide with (3.13) in 1+1 dimensional theories.

3.8.1 Functional integral approach

The major advantage of the Euclidean approach is that the Schwinger functions are
better behaved than the corresponding Wightman distributions and what is more
important they can be derived in most of the cases from functional integration
with respect to a probability measure. This also allows by introducing a suitable
regularization with a systematic numerical approach.
The result which was first suggested by Symanzik[ 26 ]and Nelson[ 16 ]is that
formally speaking the Schwinger functions can be considered as the momentum
operators of a functional measure defined in the space of distributionsS(R^4 )by
the exponential of the Euclidean classical actionSE, i.e.

Sn(f ̃ 1 ,f ̃ 2 ,...,f ̃n)=

∫

S′(R^4 )

δφe−SE(φ)φ(f ̃ 1 )φ(f ̃ 2 )...φ(f ̃n).

Inthecaseoffreefieldtheory

SE(φ)=

1

2

‖∇φ‖^2 +

m^2

2

‖φ‖^2 =

1

2

(φ,(−∇^2 +m^2 )φ)

and we have that

S 2 (f, ̃ ̃g)=

∫

S′(R^4 )

δφ e−SE(φ)φ(f ̃)φ( ̃g)=

∫

S′(R^4 )

δμmφ(f ̃)φ( ̃g)

=

1

2

(f, ̃(−∇^2 +m^2 )−^1 ̃g) (3.54)

whereδμmis the Gaussian measure defined onS′(R^4 ) with vanishing mean and
covariance operator (−∇^2 +m^2 )−^1.
It is obvious that Schwinger function (3.54) does coincide with the analytic ex-
tension of the Wightman function of the free theory. In fact, from the functional
integral formulation it is easy to check that the Schwinger functions satisfy the