Computational Physics

478 Computational methods for lattice field theories

where

Knl=( 2 d+m^2 )δnl−

∑

μ

δn,l+μ. (15.46)

Defining Fourier-transformed fields as usual:

φk=

∑

n

φneik·n; (15.47a)

φn=

1

Ld

∑

k

φke−ik·n, (15.47b)

wherenandlrun from 0 toL−1, periodic boundary conditions are assumed, and
the components ofkassume the values 2mπ/L,m=0,...,L−1. Then we can
rewrite the free-field action as

SE=

1

2 L^2 d

∑

k

φkKk,−kφ−k, (15.48)

asKk,−kare the only nonzero elements of the Fourier transformKk,k′:

Kk,k′=Ld

[

−

∑

μ

2 cos(kμ)+( 2 d+m^2 )

]

δk,−k′

=Ld

[

4

∑

μ

sin^2

kμ

2

+m^2

]

δk,−k′ (15.49)

where the sum is now only over the positiveμdirections;kμis theμ-component
of the Fourier wave vectork.
The partition function

Z=

∫

[Dφk]exp

(

−

1

2 L^2 d

∑

k

φkKk,−kφ−k

)

=

∫

[Dφk]exp

(

−

1

2 L^2 d

∑

k

|φk|^2 Kk,−k

)

(15.50)

(up to a normalisation factor) is now a product of simple Gaussian integrals, with
the result (N=Ld):

Z=( 2 πN^2 )N/^2 /

∏

k

√

Kk,−k=( 2 πN^2 )N/^2 /

√

detK=( 2 πN^2 )N/^2

√

det(K−^1 ).

(15.51)
The partition function appears as usual in the denominator of expressions for expect-
ation values. We can calculate for example the two-point correlation or Green’s