From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 233

Gaussian Measures in Hilbert Spaces

Gaussian probability measures can be also defined in infinite dimensional topo-
logical vector spaces. The simplest caseis the Hilbert space case. Let us consider
a positive, selfadjoint, trace class operatorCdefined in a Hilbert spaceH:

• (x, Cy)=(Cx,y) for anyx, y∈H;


• (x, Cx)>0 for anyx∈H;


• trC<∞.

Positivity implies the non-degenerate character ofC, which guarantees the ex-
istence of the inverse operatorC−^1 , that again is positive. Let us assume for
concreteness thatH=L^2 (Rn,C)and

Cs=(−Δ+m^2 )−s

withs>n 2. It easy to check thatCsis positive, selfadjoint, trace class operator
inL^2 (Rn,C).
The Minlos theorem establishes that the measure defined byCsis a Bore-
lian probability measure inH=L^2 (Rn,C)[ 12 ]. The momenta of this Gaussian
measure are again obtained in terms of the covariance matrixC,

•〈 1 〉C=1;

•〈(g,f 1 )(g,f 2 )...(g,f 2 m− 1 )〉C=0;

•〈(g,f 1 )(g,f 2 )...(g,f 2 m− 1 〉C

=^1

2 mm!

∑

σ∈S 2 m

(fσ(1),Cfσ(2))(fσ(3),Cfσ(4))...(fσ(2m−1),Cfσ(2m)),

where now we denote byfi∈L^2 (Rn,C) instead ofxthe vectors of the Hilbert
spaceH=L^2 (Rn,C).
The last formula is the infinite dimensional version of Wick’s theorem. And
again the functional derivatives of the generating functional

GC(f)=〈ei(f,g)〉C=

∫

L^2 (Rn,C)

dμC(g)ei(f,g)=e−

(^12) (f,Cf)
,
generate all momenta of the measure,
〈(g,f 1 )(g,f 2 )...(g,fm)〉C=(−i)m
∂mGC
∂f 1 ∂f 2 ...∂fm

∣∣

∣f=0.

However, in field theory the natural covariance is not in general of trace class,
thus, one needs to make appeal to another version of Minlos theorem which applies
to covariances which are not of trace class[ 12 ]. In this case the space of test