From Classical Mechanics to Quantum Field Theory

232 From Classical Mechanics to Quantum Field Theory. A Tutorial

gets its main contribution from the interval (−c, c). The main properties of the
Gaussian measure are given by the average of its momenta,

•〈 1 〉c=1;

•〈x^2 m+1〉c=0;

•〈x^2 m〉c=(2m−1)!!〈x^2 〉m=(2m−1)!!c^2 m.

The last formula is known as Wick’s theorem.
All characteristics of Gaussian measures can be derived from the average of a
single special function

gc(y)=〈gc(y)〉c=e−

c 2 y 2

.

In particular, all momenta of the measure can be obtained from the derivatives of
gcat the origin

〈xm〉=(−i)m

dm

dym

gc

∣∣

∣

y=0

.

The multidimensional generalization is straightforward. LetCbe a positive,
symmetric matrix, i.e.

(x, Cy)=(Cx,y), (x, Cx)>0).

Positivity implies the non-degenerate character ofC,detC= 0, which guarantees
the existence of the inverse matrixC−^1.
The Gaussian probability measure is defined by

dμC=

e−^12 (x,C

− (^1) x)
√
2 πdetC
dnx.
The momenta of the multidimensional Gaussian measure are obtained in terms of
the covariance matrixC,
•〈 1 〉C=1;
•〈xi^1 xi^2 ...xi^2 m−^1 〉C=0;
•〈xi^1 xi^2 ...xi^2 m〉C=

1

2 mm!

∑

σ∈S 2 m

Cσ(i 1 )σ(i 2 )Cσ(i 3 )σ(i 4 )...Cσ(i 2 m− 1 σ(i 2 m)).

The last formula is the multidimensional Wick’s theorem. The generating function
ofdμCis

gC(y)=〈ei(x,y)〉C=e−^12 (y,Cy).

All the momenta of the Gaussian measure, can be obtained from the derivatives
ofgCat the origin

〈xi^1 xi^2 ...xim〉C=(−i)m ∂

mgC

∂xi 1 ∂xi 2 ...∂xim

∣∣

∣y=0.