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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
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