From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 221

fields and the corresponding Wightman functions which after analytic continua-
tion become Schwinger functions. Indeed if one considers a Euclidean timeτ=it
theτ-evolution of the field operators becomes

φE(x,τ)=eτHφ(x,0)e−τH.

The smearing ofφE(x,t) by a test functionf ̃defines the Euclidean field operators

φE(f ̃)=

∫

R^4

d^4 xφE(x,t)f ̃(x,t).

Now the vacuum expectation values of products of field operatorsφE(f ̃)isnot
always well defined because the Euclidean time evolution is given by hermitian
operatorsUE(τ)=U(it) which define a semigroup instead of a group unlike the
case of real time evolution. The hermitian operatorsUE(τ) are only bounded
for positive values of the Euclidean timeτ<0. For such a reason the vacuum
expectation values of products of field operatorsφE(f ̃) require some time-ordering
of the domains of the test functions. If the support of the family of functions
{f ̃i}i=1, 2 ,···,nofS(R^4 ) are time ordered, i.e. τ 1 >τ 2 >···>τnfor any point
x=(x 1 ,x 2 ,···,xn) withfi(xi)=0fori=1, 2 ,···,n,then

Sn(f ̃ 1 ,f ̃ 2 ,...,f ̃n)=〈 0 |φE(f ̃ 1 )φE(f ̃ 2 )...φE(f ̃n)| 0 〉 (3.49)

is a well defined function and does coincide in that case with the analytic extension
of the corresponding Minkowskian vacuum expectation values.
Moreover, it can be extended for multivariable test functions ̃fn ∈S(R^4 n)
defined inR^4 nby

Sn( ̃fn)=

∫

R^4

d^4 x 1

∫

R^4

d^4 x 2 ...

∫

R^4

d^4 xnφE(x 1 )φE(x 2 )...φE(xn)f ̃n(x 1 ,x 2 ,...,xn)
(3.50)
when ̃fnhas support in a time ordered subset ofR^4 n, i.e. f ̃n(x 1 ,x 2 ,...,xn)=0
ifx∈R^4 ndoes not satisfy any of the inequalitiesτ 1 >τ 2 >···>τn.Inthe
particular case of ̃fn=f ̃ 1 f ̃ 2 ...f ̃nthe expectation value Eq. (3.50) reduces to
Eq. (3.49). ButSn( ̃fn) can be extended to multivariable functions with more
general support by analytic extension from the Mikowskian definition.
The interesting point is that the analytic extension also provides a finite value
for the case where the supports are not time-ordered. These analytically extended
functions are known as Schwinger functions. Although they can only be expressed
as vacuum expectation values of products of Euclidean fields Eq. (3.49) when
the supports of the test functions are time-ordered, in practice, because of their
symmetry under permutations, they can always be calculated in that way.
The relevance of Schwinger functions is that the quantum field theory can be
completely formulated in terms of them and the fundamental principles reformu-
lated in the following way.