Computational Physics

Exercises 533

(b) Now consider the integral

I 2 =

∫

dx 1 dx 1 ∗...dxNdxN∗e−x

†Ax

wherexis now a complex vector. Show that

I 2 =

( 2 π)N

det(A)

.

15.2 In this problem and the next we take a closer look at the free field theory. Consider
the one-dimensional, periodic chain of particles with harmonic coupling between
nearest neighbours, and moving in a harmonic potential with coupling constantm^2.
The Lagrangian is given by

L=

1

2

∑∞

n=−∞

[φ ̇n^2 −(φn−φn+ 1 )^2 −m^2 φn^2 ].

We want to find the HamiltonianHsuch that

∫

[Dφn]e−S=〈i|e−(tf−ti)H|f〉

where

S=

∫tf

ti

L[φn(t)]dt

and the path integral

∫

[Dφn]is over all field configurations{φn}compatible withi

attiandfattf.

We use the Fourier transforms

φk=

∑

n

φneikn; φn=

∫ 2 π

0

dk

2 π

φke−ikn.

(a) Show that from the fact thatφnis real, it follows thatφk=φ∗−k, and that the

Lagrangian can be written as

L=

1

2

∫ 2 π

0

dk

2 π

{|φ ̇k|^2 −φ−k[m^2 + 2 ( 1 −cosk)]φk}.

This can be viewed as a set of uncoupled harmonic oscillators with coupling

constantω^2 k=m^2 + 2 ( 1 −cosk).

(b) In Section 12.4 we have evaluated the Hamiltonian for a harmonic oscillator. Use

the result obtained there to find

H=

1

2

∫ 2 π

0

dk

2 π

{ˆπ(k)π(ˆ −k)+φ(ˆ−k)[m^2 + 2 ( 1 −cosk)]φ(ˆk)},

where the hats denote operators;π(ˆ k)is the momentum operator conjugate to

φ(ˆk)– they satisfy the commutation relation

[ˆπ(k),φ(ˆ−k′)]=i

∑

n

eik(k−k′)n= 2 πδ(k−k′),

where the argument of the delta-function should be taken modulo 2π.