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Lecture XVI 46

Lecture XVI: Applications and Connections

Partition Function of ideal (Non-Interacting) Gas of Quantum Particles

Useful for normalisation of interacting theories

e.g. Non-interacting fermions: H =

aa

As a warm-up exercise, let us first use coherent state representation:

Quantum Partition function

Z0 = tr e(HN) =n

n|e(HN)|n

In coherent state basis:

Z0 =

d[, ]e

P|e(HN)|

Using ident. N.B. n2 = n

e(HN) = eP()a

a =

e()n =

1 +

e() 1

n

Z0 =

d[, ]e

P

| e

1 +

e() 1

()

=

dd

1 2 e2

1 +

e() 1

()

=

dd 1 2 e() 1

=

dd

(1 + e

())

=

1 + e()

i.e. Fermi Dirac distribution

Exercise: show (using CS) that in Bosonic case

Z0 =

n=0en() =

1 e()

1

Bose-Einstein distribution

Lecture Notes October 2005

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Lecture XVI 47

Connection of CSPI with FPI

e.g. Quantum Harmonic oscillator: H =p2

2m+

1

2m2q2

In second quantised form, H = (aa + 1/2), [a, a] = 1, i.e. bosons!

Z= tr(eH) =

()=(0)()=(0)

D[, ]exp

0

+

e/2 in D[, ], () complex scalar field

Parameterise complex field in terms of two real scalar fields

() =m

2

1/2 q() +

ip()

m

Substituting (e.g. = m2

2(q2 + p

2

(m)2)) and noting

0

d qp =

0

d pq

Z=

()=(0)()=(0)

D[p, q]exp

0

d

p2

2m+

1

2m2q2

ipq

cf. (Euclidean time) FPI = it/, = it

/,i

q =

qt

Z=

D[p, q]exp

i

t0

dt (pq H(p, q))

Partition Function of Harmonic Oscillator from CSPI

(i) Bosonic oscillator:

ZB =

Jdet( + )1

D[, ]exp

0

d ( + )

=

(n

dndn)ePnn(in+)n

= Jn

[in + ]1 =

J

n=1

()2 +

2n

21J

n=1

1 +

2n

21

=J

2sinh( /2)

n=1[1 + (x/n)2] = (sinh x)/x

Normalisation: T 0, Z dominated by g.s. limZB = e/2

(= ZF)

i.e. J = ZB =1

2 sinh(/2)

Lecture Notes October 2005

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Lecture XVI 48

(ii) Fermionic oscillator: Gaussian Grassmann integration

ZF = Jdet( + ) = Jn

[in + ] = J

n=0

()2 +

(2n + 1)

2

= Jn=1

1 +

(2n + 1)

2= J cosh( /2)

n=1[1 + (x/(2n + 1))

2] = cosh(x/2)

Using normalisation: limZF = e/2

J = 2e ZF = 2e

cosh(/2).

cf. direct computation:

ZB = e/2

n=0

en, ZF = e/2

1n=0

en.

Note that normalising prefactor J involves only a constant offset of free energy,

F = kBT lnZ

statistical correlations encoded in content of functional integral

In notes, two case studies:

(i) Plasma Theory of the weakly interacting electron gas

(ii) BCS theory of superconductivity a prototype for gauge theories

We will deal with project (ii)

Lecture Notes October 2005