Download Many-body theory

Many-body theory
Janos Polonyi
Strasbourg University
(Dated: November 24, 2012)
Contents
1
I. Introduction
2
2
5
6
6
8
10
12
12
13
15
16
17
II. Second Quantization
A. Harmonic oscillators
B. Quantum field
C. Observables
1. One-body operators
2. Two-body operators
3. Electron gas
D. Representations of the time evolution
1. Schrödinger representation
2. Heisenberg representation
3. Interaction representation
4. Finite temperature
E. Phonons
III. Green Functions
A. Electrons
1. Definition
2. Relation to expectation values
3. Equation of motion
19
20
20
21
22
IV. Perturbation expansion
A. Interactive Green functions
B. Wick theorem
C. Feynman rules
D. Schwinger-Dyson equation
26
26
27
30
33
V. Examples
A. Collective modes in Coulomb gas
B. Propagation in a disorded medium
C. Linear response formula for the electric conductivity
33
33
38
41
A. Phonons
44
B. Spectral representation
46
I.
INTRODUCTION
These notes represent a brief introduction to the quantum treatment of nonrelativistic many body systems. The
details are kept minimal, students planning to continue in theoretical directions should work through a more exhaustive
presentation of the subject.
The there are three main topics covered, the general description of the second quantized electron-phonon system,
the introduction of Green functions and finally, the theory of perturbation for causal and for retarded Green functions.
2
II.
SECOND QUANTIZATION
The formalism of the usual Quantum Mechanics is not sufficient to consider realistic systems composed of a
large number of particles. The reason is the equivalence of particles, a genuine quantum effect, which requires
the (anti)symmetrization of the wave functions. The resulting wave functions are too complicated for any useful
calculation. Another representation of the many particle system, based on the occupation number offers a simple
mathematical setup where the complications created by the (anti)symmetrization simply disappear.
But there is another, less technical reason of seeking an extension of the usual Quantum Mechanical framework,
related to the fact that energy can be converted into the creation of particles. This phenomenon is well known in high
energy physics where the available kinetic energy can be far higher than the rest mass energy of the particles in the
system. But this does no exhaust the examples of converting energy and particles into each other. Small fluctuations
of a crystalline structure, the phonons which are the normal modes of the leading order, quadratic approximation for
the displacement of the constituents of the ionic crystal can be viewed as the law energy ’elementary particles’ of the
ionic degrees of freedom. They can be created by energies available in the solids and the treatment of such processes
requires the use of some ideas borrowed from high energy physics.
The extension of Quantum Mechanics to treat many body system and processes with changing number of degrees
of freedom is traditionally called ’second quantization’, leaving the name ’first quantization’ for the usual Quantum
Mechanical formalism. These names are misleading as we shall see below but are kept for historical reason.
A.
Harmonic oscillators
Let us now imagine a system of free, equivalent particles characterized by the dispersion relation, the relation
between their energy and momentum, E(~q), q being the wave vector of the particle. Due to the lack of interactions
the total energy is additive,
X
Etot =
E(~qj )
(1)
j
where qj stands for the wave vector of the j-th particle. A better way to write this energy is to regroup particles with
the same momentum by introducing the occupational number, nq giving the number of particles with wave vector q,
and writing
X
Etot =
E(~q)nq
q
=
Z
d3 q
E(~q)n(q).
(2π)3
where the thermodynamical limit has been taken, V = L3 → ∞, ∆q =
the notation
2π
L ,
n(q) = V nq
(2)
in arriving at the second equation where
(3)
is introduced for the occupation number, considered as a function of continuous variable.
To simplify the search for the representation of systems where the number of equivalent particles is not fixed let
us suppose that all particle of our system have the same momentum ~q. The energy spectrum of such a system is
equidistant,
Etot = E(~q)nq .
(4)
The only quantum mechanical system whose spectrum is an arithmetic series is the harmonic oscillator. Therefore,
we introduce a formal harmonic oscillator for each possible value of the wave vector q, together with its Hilbert space
Hq and canonical operator pair, Πq and Qq , satisfying
[Qqj , Πqk ]ξ = i~δj,k
(5)
[Q(q), Π(r)]ξ = i~(2π)3 δ(q − r)
(6)
for finite volume and
3
in the thermodynamical limit where
[A, B]ξ = A · B − ξB · A,
(7)
choosing ξ = 1 at the time being
Q(q) =
√
V Qq ,
√
Π(r) =
V Πr
(8)
and
V
δj,k .
(2π)3
δ(q − r) =
(9)
The Hamilton operator of the oscillator corresponding to the wave vector q is
mq ωq2 2
Π2q
1 Π2 (q)
m(q)ω 2 (q) 2
1
+
Qq =
+
Q (q) = H(q)
Hq =
2mq
2
V 2m(q)
2
V
(10)
with mq = m(q) and ωq = ω(q).
One introduces the creation and destruction operators for particles corresponding to a given wave vector,
a†q =
a† (q) =
mq ωq Qq − iΠq
p
,
2mq ωq ~
aq =
m(q)ω(q)Q(q) − iΠ(q)
p
,
2m(q)ω(q)~
mq ωq Qq + iΠq
p
,
2mq ωq ~
m(q)ω(q)Q(q) + iΠ(q)
p
,
2m(q)ω(q)~
a(q) =
satisfying the canonical commutation relation
[aqj , a†qk ]ξ = δj,k 11,
[aqj , aqk ]ξ = 0,
(11)
(12)
for finite system and
[a(q), a† (r)]ξ = (2π)3 δ(q − r)11,
[a(q), a(r)]ξ = 0
(13)
in the thermodynamical limit. The number operator,
N (q) = a† (q)a(q),
(14)
can be used to express the Hamiltonian as
1
H(q) = ~ω(q) N (q) +
2
.
(15)
The energy of the n-th excited state |niq ,
N (q)|niq = n|niq ,
(16)
is
1
En (q) = ~ω(q) n +
2
(17)
which leads to the condition ~ω(q) = E(~q), leaving m(q) arbitrary. The relations
√
a† (q)|niq = n + 1|n + 1iq
√
a(q)|niq = n|n − 1iq
|niq =
will be useful later.
(a† (q))n
√
|0iq ,
n!
a(q)|0iq = 0
(18)
4
The Hilbert space of the system of all harmonic oscillator, HF , called Fock-space is HF = ⊗
Hamilton operator acting on it is
2
Z
d3 q
Π (q)
m(q)ω 2 (q) 2
Htot =
+
Q (q) .
(2π)3 2m(q)
2
Q
k
The total number of particles is represented by the operator
Z
d3 q †
a (q)a(q)
Ntot =
(2π)3
Let us consider some simple states for bosons. The ground state is clearly that of with no particles,
Y
|0i = ⊗
|0iq .
Hk and the
(19)
(20)
(21)
q
Let us denote the state with a single particle of wave vector q by
|qi = a† (q)|0i.
The one particle state whose wave function is Ψ1 (~q) in the momentum representation is
Z
d3 q
|Ψ1 i =
Ψ1 (~q)a† (q)|0i.
(2π)3
(22)
(23)
The state with two particles of wave vectors qj , j = 1, 2 is
|q1 , q2 i = a† (q1 )a† (q2 )|0i.
(24)
Notice that due to the second commutation relation in (13) the two particle state is symmetrical with respect to the
exchange of the particles,
|q1 , q2 i = |q2 , q1 i
(25)
which requires the choice ξ = 1 for bosons. The two particle state characterized by the wave function Ψ2 (~q1 , ~q2 ) in
momentum representation is
Z 3
d q1 d 3 q2
1
Ψ2 (~q1 , ~q2 )a† (q1 )a† (q2 )|0i.
(26)
|Ψ2 i =
2
(2π)6
Finally, a general state of the Fock-space has an ill defined particle number and can be written as
Z 3
Z
d q1 d 3 q2
d3 q
1
†
†
†
|Ψi = Ψ0 +
Ψ1 (~q)a (q) +
Ψ2 (~q1 , ~q2 )a (q1 )a (q2 )|0i + · · · |0i.
(2π)3
2
(2π)6
(27)
Ground states with ill defined particle number are the hallmark of the Bose-Einstein condensates and broken symmetries.
The relation
N |p1 , . . . , pn i = n|p1 , . . . , pn i
(28)
is a simple result of applying the commutation relation (13).
One expects some complications for fermions due to the Pauli exclusion principle. The impossibility of placing two
fermions in the same state can be expressed by the equation
(a† (q))2 =
1 †
[a (q), a† (q)]− = 0.
2
(29)
In fact, applying both sides of this equation on an arbitrary state Ψi on has a† (q)a† (q)|Ψi = 0, indicating the absence
of vectors in the Fock-space with two fermions in the same state. The Pauli principle restricts the dimensions of
Hq = {c0 |0iq + c1 |1iq } to 2. This suggest that the canonical commutation relations will be imposed with ξ = −1 for
fermions.
5
The relations (21)-(28) can be re-derived for fermions except the modification of Eq. (25),
|q1 , q2 i = −|q2 , q1 i
which results from Eq. (13) applied
Z
N |Ψ1 i =
Z
=
with ξ = −1. As a simple example let us consider
Z
d3 q †
d3 r
a
(q)a(q)
Ψ1 (r)a† (r)|0i
(2π)3
(2π)3
d3 qd3 r
Ψ1 (r)a† (q)[ξa† (r)a(q) + (2π)3 δ(q − r)]|0i = |Ψ1 i.
(2π)6
(30)
(31)
The spin can be taken into account in a trivial manner, by the introduction of a harmonic oscillator for each pair
(q, σ) which amounts to the replacements a(q) → aσ (q), [aσ (q), a†κ (r)]− = (2π)3 δ(q − r)δσ,κ .
B.
Quantum field
It is well known that the operator algebra of a harmonic oscillator is the simplest in terms of the creation and
destruction operators. This observation gives the idea of considering
a(p) =
m(p)ω(p)X(p) + iπ(p)
p
,
2m(p)ω(p)~
(32)
(an operator valued function, acting on the Fock-space and defined in the space of wave vectors) as the fundamental
variable for the description of the system of infinitely many harmonic oscillators. Our intuition is better developed in
real (coordinate) space due to a very restrictive constrain on physical laws, the locality. Thus we seek the corresponding
variable constructed in real space what can formally be defined by means of a Fourier transformation,
Z
d3 q iqx
ψσ (x) =
e aσ (q),
(33)
(2π)3
whose hermitian conjugate is
ψσ† (x)
=
Z
d3 q −iqx †
e
aσ (q).
(2π)3
(34)
The inverse Fourier transformations are given by
aσ (q) =
a†σ (q) =
Z
Z
d3 xe−iqx ψσ (x),
d3 xeiqx ψσ† (x).
(35)
Since a† (q) and a(q) are the creation and destruction operators for a given momentum one expects their linear
combination ψ † (x) and ψ(x) playing the role of creation and destruction operators at a given space location. This
guess is confirmed by the canonical commutation relations
Z Z 3 3
d qd r iqx−iry
e
[a(q), a† (r)]ξ = δ(x − y)
ψ(x), ψ † (y) ξ =
(2π)6
Z Z 3 3
d qd r iqx+iry
[ψ(x), ψ(y)]ξ =
e
[a(q), a(r)]ξ = 0
(2π)6
Z Z 3 3
†
d qd r −iqx−iry †
ψ (x), ψ † (y) ξ =
e
[a (q), a† (r)]ξ = 0
(36)
(2π)6
As a little exercise, one can construct simple one particle states, such as
|xi = ψ † (x)|0i
(37)
6
which is localized at x and
|Ψ1 i =
Z
d3 xΨ1 (x)ψ † (x)|0i
(38)
whose Schrödinger wave function is Ψ1 (x). The removal of a particle from the state |xi at the location y yields
ψ(y)|xi = ψ(y)ψ † (x)|0i = [ξψ † (x)ψ(y) + δ(x − y)]|0i = δ(x − y)|0i,
(39)
the null vector when y 6= x and the no-particle ground state for y = x.
Another simple exercise is to check the exchange statistics. Let us define the localized two particle state as
|y, xi = ψ † (y)|xi.
(40)
ψ(z)|y, xi = ψ(z)ψ † (y)ψ † (x)|0i
= [ξψ † (y)ψ(z) + δ(y − z)]ψ † (x)|0i = ξδ(z − x)|yi + δ(z − y)|xi
(41)
The removal of a particle at z leads to the state
The exchange statistical factor ξ appears in the first term because the first placed particle is removed first which
necessitate the exchange of the two particles in the state.
It is easy to verify that
n(x) = ψ † (x)ψ(x)
(42)
plays the role of particle density operator and the states |xi and |x, yi are eigenstates of the total particle number
operator
Z
Ntot = d3 xn(x)
(43)
with eigenvalues 1 and 2, respectively.
C.
1.
Observables
One-body operators
Let us now consider the simplest observables
which are constructed from one particle observables in an additive
manner. The operator FS (p) = FS ~i ∇x , the function of the momentum operator of the usual Schrödinger picture
defines a one-particle operator. Its additive extension for many body system acts on the Fock-space,
Z
d3 k †
FS (p) → FF ≡
a (k)a(k)FS (~k)
(2π)3
Z
d3 k 3 3 †
=
d xd yψ (x)ψ(y)eik(x−y) FS (~k)
(2π)3
Z
d3 k 3 3 †
~
ik(x−y)
=
−
d
xd
yψ
(x)ψ(y)F
∇
S
y e
(2π)3
i
Z
~
=
d3 xd3 yψ † (x)ψ(y)FS − ∇y δ(x − y)
i
Z
~
=
d3 xd3 yψ † (x)δ(x − y) FS
∇y ψ(y)
i
Z
~
=
d3 xψ † (x)FS
(44)
∇x ψ(x)
i
where a partial integration was performed in arriving at the fifth equation, looks formally as an expectation value in
first quantized Quantum Mechanics for the state described by the wave function ψ(x). But this ’wave function’ is
now operator valued because we need the quantum version of the harmonic oscillators due to the discreteness of the
7
k
q
k
r
F (x)
S
FS (p)
FIG. 1: Graphical representation of one-particle operators. fS (p) is diagonal in momentum space but fS (x) changes the
momentum.
particle number. It is as if the ’wave function’ ψ(x) appearing in the naive interpretation of Eq. (44) would have to
be quantized again, the second time. In another way arriving at similar conclusion is to interpret the Fourier mode
in the Fourier integral of Eq. (33) as the wave function of a particle with momentum p = ~q. This is the historical
source of the name ’second quantization’. But it is misleading for it holds for non-interactive particles only. When
interactions are present one can not isolate one-particle states anymore due to the entanglement and this analogy is
completely lost.
Similar construction for first quantized operator FS (x) which is diagonal in the coordinate representation is
Z
FS (x) → FF ≡
d3 xψ † (x)ψ(x)fS (x)
Z 3 3
d qd r 3 †
=
d xa (q)a(r)eix(r−q) FS (x)
(2π)6
Z 3 3
d qd r †
a (q)a(r)F̃S (q − r),
(45)
=
(2π)6
where
F̃S (q) =
Z
d3 xe−iqx FS (x)
(46)
Examples:
1. Kinetic energy:
~2 k 2
2m
Z
Z
d3 k †
~
3
†
=
a
(k)a(k)H
(~k)
=
d
xψ
(x)H
∂ ψ(x)
S
S
(2π)3
i
HS (~k) =
HF
(47)
2. Potential energy:
US → UF =
Z
3
†
d xψ (x)ψ(x)US (x) =
Z
d3 qd3 r †
a (q)a(r)ŨS (q − r)
(2π)6
(48)
3. Particle density:
nF (x) =
Z
Z
d3 x′ ψ † (x′ ) δ(x′ − x) ψ(x′ ) = ψ † (x)ψ(x)
| {z }
nS (x′ )
3
3
d qd r 3 †
d xa (q)a(r)e−i(q−r+s)x
(2π)6
Z 3 3
d qd r †
a (q)a(r)δ(q − r + s)
=
(2π)6
Z
d3 q †
=
a (q)a(q + s)
(2π)3
ñF (s) =
(49)
8
4. Current density:
~
[ψ † (x)∇ψ(x) − ∇ψ † (x)ψ(x)]
2mi
Z 3 3
~
d qd r 3 †
J˜F (s) =
d xa (q)a(r)e−i(q−r+s)x i(q + r)
2mi
(2π)6
Z 3 3
~
d qd r †
=
a (q)a(r)δ(q − r + s)(q + r)
2m
(2π)6
Z
~
d3 q †
s
=
a
(q)a(q
+
s)
q
+
m
(2π)3
2
JF (x) =
2.
(50)
Two-body operators
The most important two-body operator represents the interactions among pairs of particles,
US =
1X
1X
N
U (xi − xj ) =
U (xi − xj ) + U (0).
2 i,j
2
2
(51)
i6=j
A characteristic difference between classical and quantum physics is that the last term is usually not considered in
the classical theory but is necessarily present in the quantum case where the particles are indistinguishable and is not
possible to separate i = j from i 6= j. It clearly plays entirely different role in the dynamics than ’real’ interactions,
taking place between different particles. To separate them in an automatic manner one introduces the normal ordered
product, A · B →: A · B :, for operators composed of creation and destruction operators. The rule is to order all
destruction operators right to the creation operators and inserting the statical phase factor ξ for each exchange, eg.
: ψ(x)ψ(y) : = ψ(x)ψ(y)
: ψ † (x)ψ(y) : = ψ † (x)ψ(y)
: ψ(x)ψ † (y) : = ξψ † (y)ψ(x),
(52)
according to the canonical commutation relations (36) and neglecting their right hand side of the first equation. The
additive potential energy acting on the Fock-space can then be written as
Z
1
UF =
d3 xd3 yψ † (x)ψ(x)U (x − y)ψ † (y)ψ(y)
2
Z
Z
1
1
=
d3 xd3 yψ † (x)ψ † (y)U (x − y)ψ(y)ψ(x) +
d3 xψ † (x)ψ(x)U (0)
2
2
N
(53)
= : U : + U (0),
2
the normal order prescription identifying the ’real’ interaction.
As an application of this formalism let us now calculate the matrix element of the potential energy, hΨ| : U : |Φi,
between two-particle states
Z
Z
1
1
d3 xd3 yΨ(x, y)ψ † (x)ψ † (y)|0i,
|Φi = √
|Ψi = √
d3 xd3 yΦ(x, y)ψ † (x)ψ † (y)|0i.
(54)
2
2
By means of the relation
ψ(u)ψ(v)ψ † (x)ψ † (y)|0i = ψ(u) δ(v − x) + ξψ † (x)ψ(v) ψ † (y)|0i
= [δ(v − x)δ(u − y) + ξδ(v − y)δ(u − x)] |0i
(55)
1
1
1
2
2
2
Ψ
Φ
Ψ
x
y
(a)
x
1
1
2
2
Φ
Ψ
x
1
1
2
2
Φ
Ψ
x
y
y
(b)
1
9
y
(c)
2
Φ
(d)
FIG. 2: Symbolic representation of the four contributions of the second equation of Eq. (56), (a): B · Z, (b): A · Z, (c): B · W ,
(d): A · W .
we have
Z
1
d3 z1 d3 z2 d3 x1 d3 x2 d3 y1 d3 y2 Ψ∗ (x1 , x2 )U (z1 − z2 )Φ(y1 , y2 )
hΨ| : U : |Φi =
2
×h0|ψ(x2 )ψ(x1 )ψ † (z1 )ψ † (z2 )ψ(z2 )ψ(z1 )ψ † (y1 )ψ † (y2 )|0i
Z
1
d3 z1 d3 z2 d3 x1 d3 x2 d3 y1 d3 y2 Ψ∗ (x1 , x2 )U (z1 − z2 )Φ(y1 , y2 )
=
2
A
B
z
}|
{
}|
{
z
×[δ(x1 − z1 )δ(x2 − z2 ) +ξ δ(x1 − z2 )δ(x2 − z1 )]
×[δ(z1 − y1 )δ(z2 − y2 ) +ξ δ(z1 − y2 )δ(z2 − y1 )]
{z
}
|
{z
}
|
W
Z
Z
=
d3 xd3 yU (x − y) [Ψ∗ (x, y)Φ(x, y) + ξΨ∗ (x, y)Φ(y, x)]
(56)
for U (x − y) = U (y − x).
This result can simply be represented by graphs, shown in Fig. 2. Note that each particle created at a y coordinate
is removed by one of the vertices, at a v coordinate. In a similar manner, the particles created at the vertices
are removed at an x coordinate point. Let introduce the pairing among field operators appearing in the vacuum
expectation values in such a manner that an operator creating a particle is paired with the operator which removes
the same particle. Clearly ψ are paired with ψ † and never ψ with ψ or ψ † with ψ † .
It important to realize that this structure, namely that the action of a product of field operators can be represented
graphically by distributing the operators in pairs, remains valid as long as the operator product is sandwiched between
the vacuum state. The states |Ψi and |Φi are obtained by acting with the same number of ψ or ψ † operators on
the vacuum state because the operator (53) preserves the particle number. It is an advantage of the graphical
representation that the contribution of particles whose state is not changed during the action of the operators in
Eq. (53) is factorizable and can be left out as a trivial facor. This becomes important when macroscopic objects are
considered where the typical particle number is in the order of the Avogadro number.
To construct a graph representing a contribution to the matrix element (56) we place all operator into the drawing
and connect the pairs. Each pair is replaced by the Dirac-delta for the difference of the coordinates of the pairs and
the value of the graph is the product of the pairs. When placed appropriately, like in Fig. 2 then each crossing stands
for a particle exchange and brings a multiplicative factor ξ. At the end we insert the wavefunctions for the initial and
final states and the potential. Finally, integrations over the coordinates has to be carried out.
The analogous calculation in the first quantized formalism gives identical result,
1
hx, y|Ψi = √ [Ψ(x, y) + ξΨ(y, x)]
2
Z
1
d3 x1 d3 y1 d3 x2 d3 y2 hΨ|x1 , y1 ihx1 , y1 |U |x2 , y2 ihx2 , y2 |Φi
hΨ| : U : |Φi =
2
Z
1
=
d3 x1 d3 y1 d3 x2 d3 y2 [Ψ∗ (y1 , x1 ) + ξΨ∗ (x1 , y1 )]
2
×δ(x1 − y2 )δ(y1 − x2 )U (x1 − y1 ) [Φ(x2 , y2 ) + ξΦ(y2 , x2 )]
Z
=
d3 xd3 yU (x − y)[Ψ∗ (y, x)Φ(y, x) + ξΨ∗ (y, x)Φ(x, y)]
(57)
10
q
s
s−q
r
t
FIG. 3: Graphical representation of the first, direct term in the two-particle interaction energy, Eq. (61), in momentum space.
In particular, for the Slater-determinant
1
hx, y|Ψi = √ [ψ1 (x)ψ2 (y) − ψ2 (x)ψ1 (y)]
2
(58)
we have
Z
d3 xd3 yU (x − y)[ψ1∗ (y)ψ1 (y)ψ2∗ (x)ψ2 (x) − ψ1∗ (y)ψ2 (y)ψ2∗ (x)ψ1 (x)]
∗
Z
1 ψ1 (x) ψ1 (y)
1 ψ1 (x) ψ1 (y)
3
3
=
d xd y √ .
U (x − y) √ 2 ψ2 (x) ψ2 (y)
2 ψ2 (x) ψ2 (y)
hΨ| : U : |Ψi =
(59)
The matrix element in momentum space reads
Z
i
h
d3 qd3 rd3 sd3 t
∗
i(s−q)x+i(t−r)y
i(t−q)x+i(s−r)y
U
(x
−
y)
Ψ̃
(q,
r)
Φ̃(s,
t)
e
+
ξe
hΨ| : U : |Φi = d3 xd3 y
(2π)12
where
Φ̃(q, r) =
Z
d3 xd3 ye−iqx−iry Φ(x, y)
(60)
The introduction of the center of mass and relative coordinates, rr = x − y and R = (x + y)/2, respectively, factorizes
the Fourier transform of the potential,
Z
d3 qd3 rd3 sd3 t
hΨ| : U : |Φi =
d3 Rd3 rr
U (rr )Ψ∗ (q, r)Φ(s, t)
(2π)12
i
h
rr
rr
rr
rr
× ei(s−q)(R+ 2 )+i(t−r)(R− 2 ) + ξei(t−q)(R+ 2 )+i(s−r)(R− 2 )
Z
h
i
d3 qd3 rd3 sd3 t
∗
3
i(s−q−t+r) r2r
i(t−q−s+r) r2r
U
(r
)Ψ
(q,
r)Φ(s,
t)(2π)
δ(t
+
s
−
q
−
r)
e
+
ξe
=
d3 rr
r
(2π)12
Z 3 3 3 3
d qd rd sd t
q−t−r+s
q−s−r+t
3
∗
+ ξ Ũ
=
(2π) δ(t + s − q − r)Ψ (q, r)Φ(s, t) Ũ
(2π)12
2
2
Z 3 3 3 3
d qd rd sd t
(2π)3 δ(t + s − q − r)Ψ∗ (q, r)Φ(s, t)[Ũ (q − s) + ξ Ũ (q − t)].
(61)
=
(2π)12
3.
Electron gas
For point like charges interacting vie the Coulomb potential,
U (x) =
e2
|x|
(62)
11
the Hamiltonian
H = H0 + H1 =
XZ
σ
Z
~2
1 X
d3 xψσ† (x) −
d3 xd3 yψσ† 1 (x)ψσ† 2 (y)U (x − y)ψσ2 (y)ψσ1 (x) (63)
∆ ψσ (x) +
2m
2 σ ,σ
1
2
gives in momentum space the kinetic energy
H0 =
XZ
σ
~2 k 2
d3 k †
a
(k)
aσ (k)
σ
(2π)3
2m
(64)
and interaction term
H1 =
1X
2 ′
σ,σ
=
=
Z
Z
1X
2
d3 xd3 y
d3 qd3 rd3 sd3 t ix(s−q)+iy(t−r)
e
U (x − y)a†σ (q)a†σ′ (r)aσ′ (t)aσ (s)
(2π)12
d3 Rd3 rr
σ,σ ′
1X
2 ′
σ,σ
Z
d3 qd3 rd3 sd3 t i(R+ 1 rr )(s−q)+i(R− 1 rr )(t−r)
2
2
e
U (rr )a†σ (q)a†σ′ (r)aσ′ (t)aσ (s)
(2π)12
d3 qd3 rd3 sd3 t
(2π)3 δ(q + r − s − t)Ũ (t − r)a†σ (q)a†σ′ (r)aσ′ (t)aσ (s)
(2π)12
(65)
with
Ũ (p) =
4πe2
.
p2
(66)
One has perturbation expansion as the only systematic, analytical method to deal with interactions and it is
essential to establish the limit of its applicability. Though the perturbation series are not convergent they are at least
asymptotic meaning that their truncation at a finite order leads to an error which is bounded by the first omitted
order. Therefore one needs an estimate of the order of magnitudes of different orders in the series. Even this is usually
too much to ask for and what is left is to find a better estimate of the small parameter which organizes the series.
The formal parameter, like the charge for the Coulomb gas is not very useful because what matters is the relative
importance of the interactions compared to the free Hamiltonian, the kinetic energy. Let us for instance suppose that
the average separation of electrons is r0 . Then the kinetic energy of an electron is proportional to 1/r02 and a typical
value of the Coulomb potential contains the factor 1/r0 . Hence one expects perturbation expansion working at most
for high density only. A more explicit way to arrive at such a conclusion is to remove the length dimensions from the
Hamiltonian. For this end we express the interparticle distance in units of the Bohr radius by using rs = r0 /a0 with
a0 = ~2 /me2 , introduce the dimensionless wave numbers q̃ = r0 q, etc., volume Ṽ = V /r03 with 4πr03 /3 = V /N and
write the Hamiltonian in a finite volume, constructed by operators satisfying the commutation relations of Eqs. (12),
H=
X ~2 k 2
k,σ
2m
a†kσ akσ +
1
2
X
δq+r,s−t
q,r,s,t,σ,σ ′
4πe2 † †
a a ′ atσ′ asσ
(t − r)2 qσ rσ
(67)
as

rs
e2 X k̃2 †
ak̃σ ak̃σ +
H=
2
a0 rs
2
2Ṽ
k̃,σ
X
q̃,r̃,s̃,t̃,σ,σ

4π
a† a† ′ a ′ a  ,
δq̃+r̃,s̃−t̃
2 q̃σ r̃σ t̃σ s̃σ
(
t̃
−
r̃)
′
(68)
indicating that the electron gas is definitely non-perturbative at low densities. It is expected that electrons form
an inhomogeneous ground state, a Wigner lattice with periodic rearrangement in space. At high density one would
expect a power series in rs in perturbation expansion. But the usual short distance divergences of quantum field
theory plague the limit rs → 0 rendering the higher order contributions divergent. Careful calculations actually reveal
the presence the terms O (rsn ln rs ) in the perturbation series.
It is known from classical electrodynamics that an infinite, homogeneous charge distribution electrostatic energy
density is infinite in the vacuum due to the long range nature of the unscreened Coulomb potential. This infrared
divergence appears in our expressions as well and it is removed by evoking that the solids are electrically neutral, the
charge of the electrons is always compensated for by that of the ion core. This ion charge distribution is inhomogeneous
and polarizes the electron states. In order to separate the two problems, the infrared problem due to the non-vanishing
electron charge, considered now, and the polarization effect of the ion core, addressed in the band structure theory,
12
we assume here the presence of classical positive charges distributed by a homogeneous, static density n = N/V (N
is the number of electron and V is the volume) which neutralizes the electron gas. The homogeneity assures that no
polarization effects arises due to the presence of this artificial device introduce to eliminate the infrared divergence
only. The Hamiltonian is now
H = Hnn + Hne + : Hee :
where the self interaction of the classical background charge is given by
Z
Z
n2
n2
n2
Hnn =
V Ũ (0)
d3 xd3 yU (x − y) =
d3 Rd3 rU (r) =
2
2
2
(69)
(70)
and the interaction energy between the electrons and the background charges is
Hne = −n
N Z
X
d3 yU (y − xi ) = −nN Ũ (0) = −n2 V Ũ (0)
i=1
(71)
The Hamiltonian in the normal ordered expression of the right hand side of Eq. (69), Hee , stands for the self interaction
of the electrons, given by Eq. (65) and its infrared contribution is
Z
1
n2
h0| : Hee :I.R. |0i =
d3 xd3 yψ † (x)ψ † (y)U (x − y)ψ(y)ψ(x) ≈
V Ũ (0)
(72)
2 |x−y|→∞
2
and the vacuum expectation value of the Hamiltonian (69) is infrared finite, O V 0 .
D.
Representations of the time evolution
Once the basic observables have been identified and the Hamiltonian is known Schrödinger equation, describes the
time dependence of the state vector |Ψ(t)i. By analogy with classical field theory one expects that our quantum fields
become time dependent as well in order to reproduce the usual time dependent averages known from classical field
theory. How can we place the time dependence, generated in the state vector by the Schrödinger equation, into the
field operator? The answer is given by recognizing that the time dependence can freely be distributed between the
state vector and the observables and our expectation corresponds to one particular choice in this question.
1.
Schrödinger representation
The time is a classical, external parameter in non-relativistic Quantum Mechanics. In the usual representation of
the time dependence the observables are considered time independent functions of the canonical pairs of coordinates
and momenta,
∂t AS = 0
(73)
and the time evolution is entirely placed on the state vector, by the Schrödinger equation
i~∂t |Ψ(t)iS = H|Ψ(t)iS
(74)
|Ψ(t)iS = U (t, t0 )|Ψ(t0 )iS
(75)
whose solution is
where U (t, t0 ) satisfies the equation of motion
i~∂t U (t, t0 ) = HU (t, t0 )
(76)
together with the initial condition U (t0 , t0 ) = 11. The solution of these equations is
i
U (t, t0 ) = e− ~ (t−t0 )H .
(77)
13
2.
Heisenberg representation
Another extremity is to place all time dependence in the observables and leave the state vector time independent.
This can be achieved by performing a time dependent unitary transformation,
i
|Ψ(t)iH = e ~ (t−t0 )H |Ψ(t)iS = |Ψ(t0 )iS
(78)
where the two representations agree for t = t0 . This transformation does not affect the expectation values,
i
i
AH = e ~ (t−t0 )H AS e− ~ (t−t0 )H ,
(79)
i~∂t |Ψ(t)iH = 0.
(80)
but stops the state vector,
The equation of motion for the time dependent observables is the Heisenberg equation,
i~∂t AH = [AH , H].
Let us consider as a simples exercise non-interacting, spinless particles whose Hamiltonian density is
Z
d3 k †
a (t, k)E(~k)a(t, k).
H=
(2π)3
(81)
(82)
It is the conservation of the particle number what allows the replacement of the operators by their time dependent
version in this expression and the value of the time is left arbitrary. Note that the canonical commutation relations
(13) are valid for arbitrary time,
[a(t, q), a† (t, r)]ξ = (2π)3 δ(q − r),
[a(t, q), a(t, r)]ξ = 0.
(83)
i~∂t ψ(t, x) = [ψ(t, x), H]
Z 3 3
d kd q iqx
e E(~k)[a(t, q), a† (t, k)a(t, k)].
=
(2π)6
(84)
The Heisenberg equation becomes
The use of the identity [A, BC] = B[A, C]+[A, B]C and the canonical commutation relation (83) leads to the equations
of motion
Z
d3 q iqx
i~∂t ψ(t, x) =
e i~∂t a(t, q)
(2π)3
Z
d3 q iqx
e E(~q)a(t, q)
(85)
=
(2π)3
or
i~∂t a(t, q) = E(~q)a(t, q)
(86)
whose solution is satisfying the initial condition
a(t0 , q) = a(q)
(87)
is
i
a(t, q) = e− ~ E(~q)(t−t0 ) a(q).
Therefore, the time dependence of the quantum field is
Z
d3 q − i E(~q)t+iqx
e ~
a(q)
ψ(t, x) =
(2π)3
for t0 = 0.
(88)
(89)
14
The time dependence obtained above make spossible to achieve a formally similar treatment for the trivial vacuum and the Fermi-sphere. Fermions with finite density are described in the grand canonical ensemble where the
Hamiltonian is modified by the chemical potential,
XZ
H →H −µ
d3 xψσ† (t, x)ψσ (t, x).
(90)
σ
The Fermi-sphere
|µi = ⊗
Y
a† (k)|0i
(91)
~|k|≤pF
of the electron gas with Fermi energy µ is defined by the properties
aσ (k)|µi = 0 for E(~k) > µ,
a†σ (k)|µi = 0
and
for E(~k) < µ.
(92)
We redefine the zero level of the one particle energy by the replacement E(~k) → E(~k) − µ because this latter which
appears in the exponent of the Gibbs factor in the grand canonical enseble. This convention assignes the lowest energy
for the Fermi-sphere and a hole, the lack of a particle within the Fermi-sphere with negative energy will appear as a
state with positive energy, an excitation.
It is advantageous to perform a canonical transformation aσ (k) → bσ (k) of the creation and destruction operators
(canonical transformations preserves the canonical commutation relations) where
(
a†σ (−k) |k| ≤ kF
bσ (k) =
.
(93)
aσ (k)
|k| > kF
The negative sign in a†σ (−k) will proven to be useful later because a hole, the lack of a particle with energy momentum
~k appears as an excitation of momentum −~k. We have a more homogeneous notation in this manner because b†σ (k)
and bσ (k) are the creation and destruction operators, respectively, as can be seen from the canonical commutation
relation,
[bσ (k), b†σ (k′ )]− = (2π)3 δ(k − k′ )
(94)
bσ (k)|µi = 0.
(95)
and from
The fermion field operator is then written as
Z
ψσ (t, x) =
e−iωk t+kx aσ (k) = ψ (−) (t, x) + ψ (+) (t, x)
(96)
k
where
ψ (+) (t, x) =
ψ
(−)
(t, x) =
Z
Z
e−iωk t+kx bσ (k)
k>kF
ei|ωk |t−kx b†σ (k)
(97)
k≤kF
denote the positive and negative energy part of the field, defined by the condition
ψ (+) |µi = ψ (−)† |µi = 0
(98)
It may happen, for instance in the case of phonons that the dispersion relation, the energy-momentum relation for
a free particle is quadratic in the energy, E 2 = (~ωk )2 . This makes the energy spectrum of such particles formally
unbounded from below, E = ±~ωk . By anticipating that the negative frequency/energy components of the field
operator, the general solution of the equation of motion for a free particle, will play a role different from that of the
positive frequency/energy component it is advantegous to separate them as it was done for a Fermi-sphere. Therefore
the field operator for such particles is written in the form
ψ(t, x) = ψ (−) (t, x) + ψ (+) (t, x)
(99)
15
where
Z
ψ (+) (t, x) =
ψ
(−)
Z
(t, x) =
e−iωk t+kx c(k)
k
eiωk t−kx d† (k)
(100)
k
When the canonical quantization procedure is performed then the operators c(k) and d(k) belong to two different
kind of particles, called particle and anti-particle in the relativistic treatment. If the field is real, ψ † (t, x) = ψ(t, x)
then we have to impose d(k) = c(k) and the anti-particle becomes equivalent with the original particle.
3.
Interaction representation
We considered non-interacting particles so far. When interactions are introduced, H = H0 + H1 , the Hamiltonian
(82) which is identified with the non-perturbed part, H0 , is completed by some higher than quadratic order terms in
the quantum field, H1 . In this case the algebraic manipulation leading to Eq. (85) give nonlinear differential equation
whose solution is not easy to find. One may turn to perturbation expansion by assuming that the non-quadratic
part of the Hamiltonian is small compared to the quadratic part. Though the usual perturbation expansion of the
Schrödinger for the state vector allows us to make some progress in this direction the perturbative solution of the
Heisenberg operator equation generates a perturbation series which is too involved to work with. Thus one looks for
a compromise to save the perturbation expansion by leaving the time dependence coming from the non-interacting
part of the Hamiltonian in the operators and putting the more complicated but weak time dependence, generated by
the interaction Hamiltonian into the state vector for which the perturbation expansion is manageable.
The removal of the time dependence due to the on-interacting dynamics is achieved by the unitary transformation
i
|Ψ(t)ii = e ~ (t−t0 )H0 |Ψ(t)iS
(101)
which transforms the operators transform as
i
i
Ai = e ~ (t−t0 )H0 AS e− ~ (t−t0 )H0 .
(102)
The equation of motion for the state vector and the operators read
i
i~∂t |Ψ(t)ii = e ~ (t−t0 )H0 (−H0 + H0 + H1 )|Ψ(t)iS
= H1i (t)|Ψ(t)ii ,
(103)
i~∂t Ai = [Ai , H0 ]
(104)
and
respectively. We have thus proven that the unitary transformation (101) indeed leads to a Schrödinger equation
involving the interaction Hamiltonian (taken in the interaction representation) and the free Heisenberg equation.
Notice that the free and the interaction part of the Hamiltonian do not commute and H1i (t) is time dependent.
The formal solution of the Schrödinger equation with time-dependent Hamiltonian can be given easier in terms of
chronological products. We have already introduced a modified multiplication for operators build up from the creation
and destruction operators, the normal ordering. Now we introduce a third multiplication for operators which depend
on the time,
T [A(t1 )B(t2 )] = Θ(t1 − t2 )A(t1 )B(t2 ) + ξΘ(t2 − t1 )B(t2 )A(t1 )
(105)
Such a multiplication law allows us to write the general solution of time evolution operator Ui (t, t0 ) which satisfies
the equation of motion
i~∂t Ui (t, t0 ) = H1i (t)Ui (t, t0 )
(106)
and expresses the solution of the Schrödinger equation (103) as
|Ψ(t)ii = Ui (t, t0 )|Ψ(t0 )i
(107)
16
in a simple form,
i
h i Rt ′
dt H1i (t′ )
−
.
Ui (t, t0 ) = T e ~ t0
(108)
The definition of the chronological product of a function of the operators is given by means of the Taylor expanded
form,
"Z
n #
∞
t
h i Rt ′
i
X
(− ~i )n
− ~ t dt H1i (t′ )
′
′
0
T e
dt H1i (t )
T
=
n!
t0
n=0
Z t
Z
∞
X
(− ~i )n t
dtn T [H1i (t1 ) · · · H1i (tn )]
(109)
dt1 · · ·
=
n!
t0
t0
n=0
which leads to Eq. (106),
Z t
Z t
∞
X
(− ~i )n
dtn−1 T [H1i (t1 ) · · · H1i (tn−1 )H1i (t)]
dt1 · · ·
n
n!
t0
t0
n=0
Z t
Z
∞
X
(− ~i )n t
i
= − H1i (t)
dtn T [H1i (t1 ) · · · H1i (tn )]
dt1 · · ·
~
n!
t0
t0
n=0
∂t Ui (t, t0 ) =
i
= − H1i (t)Ui (t, t0 ).
~
(110)
The chronological product is a simple device to place always the Hamiltonian taken at the time t at the very left of
the expressions and thereby avoiding the ambiguities arising from the non-commutativity of H1i (t) taken at different
time.
4.
Finite temperature
Finally, let us mention here another issue of time dependence, the relation between the averages of a canonical
ensemble and the analytically continued Quantum Mechanics for imaginary time. The average of an observable A in
the canonical ensemble described by the density matrix
ρ=
e−βH
Z
(111)
is
≪ A ≫= Tr[ρA] =
where β =
1
kB T
1
Tr[e−βH A]
Z
(112)
and
Z = Tre−βH .
(113)
These expressions can be obtained by performing the analytic continuation for imaginary time, Wick rotation,
ER = iEI ,
tR = −itI ~
(114)
of the time evolution operator U (t − t0 ) = U (t, t0 ) for time independent Hamiltonian as
i
UR (t) = e− ~ tH → UI (τ ) = UR (−i~τ ) = e−τ H
Z = TrUI (β)
1
≪A≫ =
Tr[UI (β)A]
Z
(115)
The lesson is that static or equal time quantum statistical averages can be obtained by means of the Wick-rotated
time evolution operator. Perturbation expansion and the interaction representation can be worked out for imaginary
time with no difficulty.
17
E.
Phonons
The part of the dynamics of solid state crystal where the particle number is not conserved is related to the ion
core where the fluctuations around the equilibrium position of the ions are represented as ’elementary particles’. The
elementary excitations of the ions are either internal excitations or spatial motion of the ions. Internal excitations
tend to be more energetic than the spatial motion therefore, we shall consider the latter.
Let us consider a perfect cubic single atom crystalline structure with lattice spacing a where the coordinate of the
atom at the lattice site n is written as
rn = na + χn .
(116)
We shall assume that the fluctuations do not lead to the melting of the crystall, |χn | ≪ a. The classical Hamiltonian
of the ion positions is taken to be
X p2
X
n
H=
V (rn − rn′ )
(117)
+
2m
′
n
n,n
where V (r) is a pair potential. By assuming that the potential changes slowly within the range of the typical
deplacement of the atom we have
X p2
X
n
H =
+
V (na − n′ a) + (χn − χn′ )∇V (na − n′ a)
2m
n
j,n
1
i
j i j
′
3
(118)
+ (χn − χn′ ) (χn − χn′ ) ∇ ∇ V (na − n a) + O χ .
2
The second and the third terms on the right hand side can be dropped, the former being an irrelevant constant and
the latter is vanishing in the equilibrium position around which the expansion is made. The two quadratic forms,
the kinetic and the potential energies can be diagonalized simultaneously because the kinetic energy is h and nindependent. The basis where both are diagonal gives the normal modes, the phonons, labeled by a continuous wave
number k, restricted into the first Brillouin zone, |k j | < 2π
a , a discrete quantum number, the band index λ takes care
of the rest of the wave number space and an index h = 1, 2, 3 handles the different phonon polarisations,
s
X Z d3 k
h
h
~
iωλ
(k)t+iank †
χn (t) =
[ehλ (k)e−iωλ (k)t+iank chλ (k) + eh∗
cλ (−k)],
(119)
λ (−k)e
h
3
(2π)
2mω
(k)
λ
λ,h
where ωλh (k) denotes the frequency of the normal mode and the polarisation vectors satisfy the completeness relation
X
hj∗
i,j
ehi
(120)
λ (k)eλ (−k) = δ .
h
The momentum, defined by
d
χn (t)
dt
r
X Z d3 k
h
h
~mωλh (k) h
iωλ
(k)t+iank †
= −i
[eλ (k)e−iωλ (k)t+iank chλ (k) − eh∗
cλ (−k)],
λ (−k)e
3
(2π)
2
πn (t) = m
(121)
λ,h
is supposed to satisfy the Heisenberg commutation relation
′
′
[χℓλ (t, n), πλℓ ′ (t, n′ )] = i~δλ,λ′ δ ℓ,ℓ δn,n′ .
(122)
The canonical commutation relation impose the commutator
′ ′
′
ℓ,ℓ
hℓ
′
(2π)3 δ(k + k′ )
[χhℓ
λ (t, k), πλ′ (t, k )] = i~δλ,λ′ δ
(123)
for
s
h
h
~
iωλ
(k)t †
[ehλ (k)e−iωλ (k)t chλ (k) + eh∗
cλ (−k)],
λ (−k)e
h
2mω
(k)
λ
h
r
X ~mω h (k)
h
h
iωλ
(k)t †
λ
πλ (t, k) = −i
[ehλ (k)e−iωλ (k)t chλ (k) − eh∗
cλ (−k)].
λ (−k)e
2
χλ (t, k) =
X
h
(124)
18
The detailed form,
h′ ℓ ′
′
[χhℓ
λ (t, k), πλ′ (t, k )]
= −i
X
h,h′
s
~2
′
′
2mλ ωλh (k)2mλ′ ωλh′ (k′ )
′ ′
h′
h
hℓ∗
−ehℓ
(−k′ )e−i[ωλ (k)−ωλ
λ (k)eλ′
′ ′
h′
h
hℓ
′ i[ωλ (k)−ωλ
+ehℓ∗
λ (−k)eλ′ (k )e
′ ′
(k′ )]t
(k′ )]t
h′
h
′ ′
h
h′
hℓ
′ −i[ωλ (k)+ωλ
mλ′ ωλh′ (k′ )(ehℓ
λ (k)eλ′ (k )e
hℓ∗
−ehℓ∗
(−k′ )ei[ωλ (k)+ωλ
λ (−k)eλ′
(k′ )]t
′
[chλ (k), chλ′ (k′ )]
′
[chλ (k), chλ′ † (−k′ )]
′
h
′
[ch†
λ (−k), cλ′ (k )]
(k′ )]t
′
h†
′
[ch†
λ (−k), cλ′ (−k )])
(125)
shows that the Heisenberg commutation relation are equivalent with
′
′
[chλ (k), chλ′ † (k′ )]+ = δ h,h δλ,λ′ (2π)3 δ(k − k′ ).
The quadratic part of the Hamiltonian defines the noninteracting phonon dynamics,
X Z d3 k π h2 (k) mω h2
h2
λ
λ
+
χλ (k)
Hph =
(2π)3
2m
2
λ,h
X Z d3 k
h
~ω h (k)ch†
=
λ (k)cλ (k),
(2π)3 λ
(126)
(127)
λ,h
after dropping an unimportant constant, coming from the zero point fluctuations.
The quantum field for electrons propagating in the presence of the periodic core potential is
Z
d3 q
ψσ (x) =
aσ (q)φσq (x)
(2π)3
(128)
where φσq (x) is the Bloch wave function and the Hamiltonian containing their kinetic and the potential energy reads
Z
XZ
1 X
~2
3
†
d3 xd3 yψσ† 1 (x)ψσ† 2 (y)Uee (x − y)ψσ2 (y)ψσ1 (x).
(129)
∆ ψσ (x) +
H=
d xψσ (x) −
2m
2
σ
σ ,σ
1
2
The interaction energy between electrons and the ion core,
XZ
Hi =
d3 xn(x)Uen (x − rn ),
(130)
n
with n(x) = ψσ† (x)ψσ (x) being the density, is rewritten by expanding in the fluctuations of the ions around the
equilibrium positions,
XZ
XZ
3
Hi =
d xn(x)Uen (x − na) −
d3 xn(x)χn ∇Uen (x − na) + O χ2 .
(131)
n
n
The first term on the right hand side is responsible for the formation of the band structure and can be ignored when
the Bloch wave functions are used for the one-particle states.
In fact the Bloch wavefunctions already take into
account the preiodic ion core potential. The ignored O χ2 terms renormalize the phonon dispersion relation ωλ (k)
and introduce further phonon-electron interactions. They will be ignored. Therefore, we are lead to
XZ
Hi = −
d3 xn(x)χn ∇Uen (x − na)
(132)
n
what is rewritten by inserting the electron field in terms of the creation and destruction operators,
XZ
d 3 p d 3 p′ ∗
Hi = −
d3 x
φ (x)φσp′ (x)a†σ (p)aσ (p′ )χn ∇Uen (x − na).
3 (2π)3 σp
(2π)
n,σ
(133)
In the next step we perform the shift x → x + na of the integral variable,
XZ
X Z d 3 p d 3 p′
†
′
a (p)aσ (p )
d3 xφ∗σp (x + na)φσp′ (x + na)χn ∇Uen (x).
Hi = −
3 (2π)3 σ
(2π)
n
σ
(134)
19
By means of the periodicity of the Bloch functions,
φσp (x + na) = φσp (x)eipna ,
(135)
we have
Hi =
XZ
σ
X
′
d 3 p d 3 p′ †
aσ (p)aσ (p′ )
ei(p −p)na χn W (p, p′ )
3
3
(2π) (2π)
n
(136)
with
′
W (p, p ) = −
Z
d3 xφ∗σp (x)φσp′ (x)∇Uen (x).
(137)
The insertion of the phonon coordinate in terms of the creation and destruction operators at t = 0, the simplest value
of the arbitrary time argument in the conserved Hamiltonian,
s
X
~
†
χλ (k) =
[eh (k)chλ (k) + eh∗
(138)
λ (−k)cλ (−k)],
h (k) λ
2mω
λ
h
and the relation
X
eiqja = N
X
δq,K =
(139)
K
K
j
(2π)3 X
δ(q − K)
a3
where N denotes the number of lattice sites and the summation on the right hand side is over the reciprocal lattice
produces
X Z d 3 p d 3 p′
X
X Z d3 k
†
′
i(p′ −p)na
′
Hi =
a (p)aσ (p )
χλ (k)eikja
e
W (p, p )
(2π)3 (2π)3 σ
(2π)3
σ
n
λ
X
X Z d 3 p d 3 p′
†
′ 1
′
a (p)aσ (p ) 3
W (p, p )χλ (p − p′ + K),
(140)
=
3 (2π)3 σ
(2π)
a
σ
λ
K being such a vector in the reciprocal lattice for which p − p′ + K falls into the first Brillouin zone. Finally, one
introduces the form factor
s
1
~
h
′
′ h
′
(141)
gλ (p, p ) = W (p, p )eλ (p − p + K) 3
a
2mωλh (p − p′ + K)
what allows us to write
XZ
Hi =
σ,λ,h
d 3 p d 3 p′ †
′
a (p)aσ (p′ )[gλh (p, p′ )chλ (p − p′ + K) + gλ (p′ , p)ch†
λ (−p + p − K)].
(2π)3 (2π)3 σ
(142)
This interaction Hamiltonian, shown graphically in Fig. 4 preserves the momentum. Furthermore, the same vertex
function, gλ (p, p′ ), parameterizes other processes, too, such as phonon-hole interactions, electron-hole pair annihilations or creations.
III.
GREEN FUNCTIONS
The construction of observables in terms of canonical conjugate coordinates and momenta or the creation and
destruction operators, together with the canonical commutation relations define the algebraic structure in which any
usual quantum mechanical calculation can in principle be carried out. But even a modest goal leads to enormous complications as soon as interactions are introduced. Therefore, one needs some new elements to render the perturbation
expansion easier. Such a help will come from the introduction of Green functions.
The need of considering transition amplitudes rather than quantum states and observables in a separated manner actually comes from Relativistic Quantum Mechanics. An unexpected obstacle is encountered when the causal
20
p−p’+K
p
p’−p−K
p’
p
g
p’
g
p
p
p−p’+K
p’−p−K
g
g
p’
p’
FIG. 4: Graphical representation of phonon-electron interaction vertex, gλ (p, p′ ) in momentum space where the phonon is
absorbed (upper left), created (upper right) or an electron-hole pair is created (lower left) or annihilated (lower right).
structure of Special Relativity is imposed on Quantum Mechanics: the state vector of the system is not relativistic
covariant and even worst, not always well defined during the time the system evolves. Despite this complication the
transition amplitudes between states observed at certain times preserve the expected relativistical structure. Therefore, one expresses the expectation values of observables in terms of well defined transition amplitudes, called Green
functions. Apart of such a conceptual need, this change in the formalism responds to the need mentioned in the
preceding paragraph and makes the calculation of expectation values simpler.
A related advantage arising from the use of Green functions is the decoupling of the spectator particles from the
active ones. Let us suppose that we are interested in the expectation value of an n-body operator. In a many
body system where the elementary interactions are supposed to be simple, pair interactions one expects that at low
order of the perturbation expansion few particles participate only in the interactions with our n particles. These
few particles are called active ones and the particles in the rest of the system are the spectators. The perturbation
expansion should be organized in such a manner that the increased accuracy, the increased order of the expansion
should increase gradually the number of active particles, the amount of work to obtain the result. We shall see that
the introduction of Green functions which correspond to transition amplitudes for a given number of particles seems
to be just the right object to achieve this goal.
We shall introduce first the Green functions for non-interacting particles in this chapter. The spectral representation,
discussed in Appendix B, offers an insight into the structure of the interacting propagators. The explicit form of the
interactive Green functions will be given by means of perturbation expansion in the Chapter IV.
A.
Electrons
1.
Definition
Let us consider two states of an electron,
(e)
|Ψf i = ψα† f (tf , pf )|µi,
(h)
|Ψf i = ψαf (tf , pf )|µi,
|Ψi i = ψα† i (ti , pi )|µi,
(e)
(143)
(h)
(144)
and for a hole,
|Ψi i = ψαi (ti , pi )|µi,
for tf > ti . Their overlaps,
A(e) = hµ|ψαf (tf , pf )ψα† i (ti , pi )|µi,
A(h) = hµ|ψα† f (tf , pf )ψαi (ti , pi )|µi
(145)
21
are the transition amplitudes of the initial state into the final one. In fact, these amplitudes, written in the Schrödinger
representation,
i
i
i
i
A(e) = hµ|e ~ (tf −t0 )H ψαf (pf )e− ~ (tf −t0 )H e ~ (ti −t0 )H ψα† i (pi )e− ~ (ti −t0 )H |µi
i
= hµ|ψαf (pf )e− ~ (tf −ti )H ψα† i (pi )|µi,
i
A(h) = hµ|ψα† f (pf )e− ~ (tf −ti )H ψαi (pi )|µi,
(146)
give the amplitude of propagation of an electron or hole. Naturally these amplitudes are non-vanishing for pi 6= pf in
the presence of an external potential only. Both amplitudes are defined for ti < tf which allows to pack then together
into a single function, called the causal or Feynman propagator,
i~Gαβ ((t, p), (t′ p′ )) = hµ|T [ψα (t, p)ψβ† (t′ , p′ )]|µi = Θ(t − t′ )hµ|ψα (t, p)ψβ† (t′ , p′ )|µi + ξΘ(t′ − t)hµ|ψβ† (t′ , p′ )ψα (t, p)|µi
(147)
whose real-space version is
i~Gαβ (x, x′ ) = h0|T [ψα (x)ψβ† (x′ )]|0i = h0|Θ(t − t′ )ψα (x)ψβ† (x′ ) + ξΘ(t′ − t)ψβ† (x′ )ψα (x)|0i
(148)
will be used in the perturbation expansion therefore, it is defined in terms of the chronological product, cf. Eq. (109),
the starting point for this expansion. The vacuum state |0i should naturally be replaced by the Fermi sphere |µi at
finite density, µ 6= 0. The chronological product allows us to keep both the particle and the hole propagation amplitude
in the same, common function. But there is a price for this convenience. The causal propagator i~Gαβ (x, x′ ) gives the
transition amplitude of a particle created at time t′ and captured at time t. In case of holes the process of creation
is at time t and the hole is captured at time t′ . This remark, without regarding into Eq. (148) gives the impression
that holes propagate in an opposite direction in time than particles. Such a formal point of view is further supported
i
by noting that the hole excitations must have positive energies. Thus the phase factor e− ~ Et corresponding to the
time evolution of an energy eigenstate requires the change t → −t in order to produce the expected time evolution
for a state with E = E(~k) − µ < 0. But there is no well-defined meaning in the word ’direction of time’ as used
in the preceding sentences. This is simply another problem which is settled by the second quantized formalism in a
consistent manner.
The numerical value of the causal propagator is a transition amplitude. But most many body system problem aims
at expectation values of observables. To obtain an expectation value the retarded and advanced propagators,
†
′
′
′
i~GR
αβ (x, x ) = h0|[ψα (x), ψβ (x )]− 0iΘ(t − t )
†
′
′
′
i~GA
αβ (x, x ) = −h0|[ψα (x), ψβ (x )]− 0iΘ(t − t)
(149)
prove to be useful. Naturally, the knowledge of the causal propagator is sufficient in principle but the resulting
expressions for the expectation values are simpler when written in terms of retarded or advanced Green functions.
Finally, thermal averages in equilibrium can easily be expressed in terms of the imaginary time propagator,
†
′
′
~GE
αβ (x, x ) = −h0|T [ψα (x)ψβ (x )]|0i,
(150)
where the time argument parameterizes the imaginary time axis and the chronological product is defined according
to this parameter.
Green functions can be introduced for more particles, in particular the two-particle causal Green function is
−~2 Gα1 α2 β1 β2 (x1 , x2 , x′1 , x′2 ) = h0|T [ψα1 (x1 )ψα2 (x2 )ψβ† 1 (x′1 )ψβ† 2 (x′2 )]|0i,
(151)
etc.
2.
Relation to expectation values
Vacuum expectation values of one-body operators can be expressed in terms of one-particle causal functions, eg.
the expectation value of the density
X
n(x) =
ψα† (x)ψα (x)
(152)
α
22
is
h0|n(x)|0i = −i~trG((t, x), (t + η, x))
with η = 0+ . The expectation value of the kinetic energy
Z
~
∇ ψα (x)
T = d3 xψα† (x)E
i
(153)
(154)
can be written as
h0|T |0i = −i~
Z
3
3
d xd yδ(x − y)trE
~
∇x G((x, t), (y, t + η)
i
(155)
The vacuum expectation values of two-body operators are expressed by means of two-particle Green function, eg.
the interaction energy
Z
1
: U (t) :=
d3 xd3 yψα† (x, t)ψβ† (y, t)U (x − y)ψβ (y, t)ψα (x, t)
(156)
2
gives the leading order energy shift
h0| : U (t) : |0i =
~2
2
Z
d3 xd3 yGβααβ ((y, t), (x, t), (x, t + 2η), (y, t + η))U (x − y).
(157)
in the vacuum.
The retarded and advanced propagators are useful to express expectation values of observables in excited states.
The imaginary time Green functions facilitate the calculation of expectation values in thermal equilibrium.
3.
Equation of motion
An explicit expression for the causal propagator can be obtained by means of the equation of motion
~
i~∂t − E
∇ ψ(x) = 0
i
(158)
for the field ψ(x), the equation which governs the time evolution of the transition amplitudes. Let us start with the
relation
~
~
† ′
i~∂t − E
∇x T [ψ(x)ψ (x )] = i~∂t − E
∇x [Θ(t − t′ )ψ(x)ψ † (x′ ) + ξΘ(t′ − t)ψ † (x′ )ψ(x)]
i
i
~
† ′
= T i~∂t − E
∇x ψ(x)ψ (x ) + i~δ(t − t′ ) [ψ(x), ψ † (x′ )]ξ
i
|
{z
}
δ(x−x′ )11
= i~δ(x − x′ )11
(159)
where the canonical commutation relation was used in the second equation. The vacuum expectation value of this
equation,
~
i~∂t − E
∇ Gαβ (x, x′ ) = δ(x − x′ )
(160)
i
follows for the causal propagator, justifying the name Green function, borrowed from the theory of linear differential
equations. One can actually
extract more information from Eq. (159). The null-space of the equation of motion
operator, i~∂t − E ~i ∇x , consists of the time dependent one-particle wave functions which solve the Schrödinger
equation and this equation shows that the Fock-space operator T [ψ(x)ψ † (x′ )] is a c-number times the identity operator
of the Fock-space for wave function outside of the null-space.
The formal solution of Eq. (160), given in terms of the Fourier integral
Z 3
′
′
d kdω e−iω(t−t )+ik(x−x )
′
i~G(x, x ) = i
(161)
(2π)4
ω − ω(k)
23
with ~ω(k) = E(~k), is not well defined due to the zero modes of the differential operator i~∂t − E( ~i ∇). In fact, one
can add the expression,
Z
′
′
d3 k
′
′
i~G(x, x ) → i~G(x, x ) +
f (k)e−iω(k)(t−t )+ik(x−x )
(162)
3
(2π)
to the propagator with an arbitrary f (k) without affecting the defining relation (160). The zero modes where the
propagator is not yet well defined are just the free one-particle wave functions, the most important type of functions
as far as the physical relevance is concerned.
The way out from this dilemma and the actual form of f (k) comes from the boundary conditions in time on the
propagator, just as Green functions are uniquely defined in the theory of linear differential equations. Our goal is to
develop a formalism to deal with, among other problems, the interactions of atoms and photons. We know in this
case that the atoms should be found in their ground state after being exposed to the interaction with electromagnetic
field for long enough time. In fact, let us suppose that an atom is in its excited state at the initial time. With some
probability it will be de-excited and one or several photons will be emitted. The probability of capturing these photons
by the same or another atom is vanishing in the thermodynamical limit, where the volume available for the photons
tends to infinity. Thus the final boundary condition in time is that all excited state must decay and we recover the
ground state. Holes or negative energy states appear as propagating backward in time therefore, the should be no
holes, or anti-particles in the language of high energy physics, in the initial state. The form of the function f (k)
should be chosen in such a manner that these boundary condition are satisfied.
From the point of view of mathematics the extra additive contribution, parameterized by the function f (k), can
be build into the Fourier integral (161) by adding Dirac-delta contributions. This latter can be obtained by certain
prescription about avoiding the poles of the integrand of the Fourier integral appearing at the zero modes. The
simplest, heuristic way of finding this prescription is to note that the slow, adiabatic decay of excited states can be
achieved by assigning an infinitesimally small, negative imaginary part of the one-particle energies, ω(k) → ω(k) − iǫ
for ω(k) > 0 with ǫ = 0+ . In fact, the time dependence of the state vector of the excited state, e−it(ω(k)−iǫ) |Ψi
introduced by this imaginary part makes the state slowly disappearing, e−it(ω(k)−iǫ) |Ψi → 0, as t → ∞. For negative
energy states the modification of the energy level is ω(k) → ω(k) + iǫ, leading to e−it(ω(k)+iǫ) |Ψi → 0 for t → −∞.
A more systematical way to arrive to the same result is to calculate directly (148) by means of the Fourier representation
Z
dω e−iωt
Θ(t) = i
(163)
2π ω + iǫ
of the Heavyside-function. We use the Fourier integral
Z
d3 k −iω(k)t+ikx
e
a(k)
ψ(x) =
(2π)3
(164)
for the quantum field and
~ω(k) =
~2 k 2
−µ
2m
(165)
which yield spin-independent propagators,
Gαβ (x, x′ ) = δαβ G(x, x′ )
′
R
′
GR
αβ (x, x ) = δαβ G (x, x )
′
A
′
GA
αβ (x, x ) = δαβ G (x, x )
′
E
′
GE
αβ (x, x ) = δαβ G (x, x ).
(166)
24
The causal propagators reads
i~G(x, x′ ) = h0|Θ(t − t′ )ψ(x)ψ † (x′ ) − Θ(t′ − t)ψ † (x′ )ψ(x)|0i
Z
Z
′
′
d3 kdω e−iω(t−t ) −iω(k)(t−t′ )+ik(x−x′ )
d3 kdω e−iω(t −t) −iω(k)(t−t′ )+ik(x−x′ )
= i
e
−
i
e
4
4
ω + iǫ
ω + iǫ
|k|>kF (2π)
|k|<kF (2π)
Z
Z
′
′
′
′
d3 kdω e−i(ω(k)−ω)(t−t )+ik(x−x )
d3 kdω e−i(ω(k)+ω)(t−t )+ik(x−x )
−i
= i
4
4
ω + iǫ
ω + iǫ
|k|<kF (2π)
|k|>kF (2π)
Z
Z
′
′
′
′
d3 kdω e−iω(t−t )+ik(x−x )
d3 kdω eiω(t−t )+ik(x−x )
= i
−i
4
4
ω − ω(k) + iǫ
ω + ω(k) + iǫ
|k|>kF (2π)
|k|<kF (2π)
Z
Z
′
′
′
′
d3 kdω e−iω(t−t )−ik(x−x )
d3 kdω e−iω(t−t )+ik(x−x )
+
i
= i
4
4
ω − ω(k) + iǫ
ω − ω(k) − iǫ
|k|<kF (2π)
|k|>kF (2π)
Z
Z
3
−iω(t−t′ )+ik(x−x′ )
d k
dω e
= i
.
(167)
3
(2π)
2π ω − ω(k) + isign(ω(k))ǫ
Few remarks are in order at this point.
1. The propagator ~G(x, x′ ), a measure of quantum fluctuations, is O (~) because ~G(x, x′ ) was obtained by
applying the canonical commutation relations whose non-vanishing part is O (~).
2. The order of the integration is written explicitely in the last equation because Fubini theorem does not hold for
this multiple integral without uniform convergence. One has always to perform the frequency integration first
in dealing with propagators, followed by the integration of the spatial quantum numbers.
3. The relation
1
1
= P ∓ iπδ(x)
x ± iǫ
x
(168)
where P denotes the principal value integral allows as to write
i~G(x, x′ ) = i
Z
d3 k
P
(2π)3
Z
′
′
dω e−iω(t−t )+ik(x−x )
+ sign(ω(k))π
2π
ω − ω(k)
Z
d3 k −iω(k)(t−t′ )+ik(x−x′ )
e
,
(2π)3
(169)
cf. Eq. (162).
4. The systematic derivation, based on the Fourier-integral representation of the Heavyside-function does not
contain any consideration on the boundary conditions in time. But gives no clue about the direction of the
flow of the physical time, neither. What the iǫ term does is an infinitesimally weak breakdown of the time
inversion symmetry. Such a symmetry breaking renders the differential operator i~∂t − E( ~i ∇) invertible and
the propagator well defined. Had we broken the time inversion symmetry in the opposite way (suppression
of excited and negative energy states for t → ∞ and t → ∞, respectively) the experimentally observed time
evolution would have been reproduced after a time inversion transformation, t → −t.
5. The infinitesimally weak explicit symmetry breaking leaving finite trace in the dynamics is the hallmark of spontaneous symmetry breaking. This suggests that the breakdown of the time inversion symmetry is spontaneous
in Quantum Field Theory. This idea seems to be further supported by the need of the thermodynamical limit
mentioned in the heuristic argument, to break the symmetry. (No spontaneous symmetry breaking in finite
systems!) This scenario is confirmed by more detailed calculations.
The calculation of the retarded and advanced propagators,
R
′
i~G (x, x ) = i
= i
= i
Z
Z
Z
′
d3 kdω e−iω(t−t ) −iω(k)(t−t′ )+ik(x−x′ )
e
(2π)4 ω + iǫ
′
′
d3 kdω e−i(ω(k)+ω)(t−t )+ik(x−x )
(2π)4
ω + iǫ
′
′
d3 kdω e−iω(t−t )+ik(x−x )
,
(2π)4
ω − ω(k) + iǫ
(170)
25
Im
Re
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
FIG. 5: Wick rotation.
and
i~GA (x, x′ ) = −i
= −i
= −i
= i
= i
Z
Z
Z
Z
Z
′
d3 kdω e−iω(t −t) −iω(k)(t−t′ )+ik(x−x′ )
e
(2π)4 ω + iǫ
′
′
′
′
d3 kdω e−i(ω(k)−ω)(t−t )+ik(x−x )
(2π)4
ω + iǫ
d3 kdω e−i(ω(k)+ω)(t−t )+ik(x−x )
(2π)4
−ω + iǫ
′
′
d3 kdω e−i(ω(k)+ω)(t−t )+ik(x−x )
(2π)4
ω − iǫ
′
′
d3 kdω e−iω(t−t )+ik(x−x )
,
(2π)4
ω − ω(k) − iǫ
(171)
proceeds in similar manner,
The imaginary time propagator and be obtained with no difficulty,
Z
Z
′
′
d3 kdω e−iω(t−t ) −ω(k)(t−t′ )+ik(x−x′ )
d3 kdω e−iω(t −t) −ω(k)(t−t′ )+ik(x−x′ )
~G(x, x′ ) = −i
e
+
i
e
4
4
ω + iǫ
ω + iǫ
|k|>kF (2π)
|k|<kF (2π)
Z
Z
′
′
′
′
d3 kdω e−i(−iω(k)+ω)(t−t )+ik(x−x )
d3 kdω e−i(−iω(k)−ω)(t−t )+ik(x−x )
= −i
+i
4
4
ω + iǫ
ω + iǫ
|k|>kF (2π)
|k|<kF (2π)
Z
Z
′
′
′
′
d3 kdω eiω(t−t )+ik(x−x )
d3 kdω e−iω(t−t )+ik(x−x )
+i
= −i
4
4
ω + iω(k) + iǫ
ω − iω(k) + iǫ
|k|<kF (2π)
|k|>kF (2π)
|
|
{z
}
{z
}
ω→ω+iω(k)
= −i
Z
3
ω→ω−iω(k)
−iω(t−t′ )+ik(x−x′ )
|k|>kF
d kdω e
(2π)4 ω + iω(k) + iǫ
3
−iω(t−t′ )+ik(x−x′ )
−i
′
Z
′
d3 kdω e−iω(t−t )−ik(x−x )
4
ω + iω(k) − iǫ
|k|<kF (2π)
|
{z
}
ω→−ω
=
Z
d kdω e
(2π)4
iω − ω(k)
(172)
The poles of the causal propagators is shown on Fig. 5 together with the analytical continuation, called Wick
rotation tR = −itI and ωR = iωI , yielding the imaginary time Green functions. All poles of the retarded and
advanced Green functions are on the lower and upper half of the complex frequency plane, respectively.
It is interesting to observe that the propagator, considered on the complex frequency plane, displays (anti)periodicity
in the imaginary direction with (anti)period length β = 1/kB T . Let us consider the purely imaginary time difference
for simplicity,
~GE ((t, x), (0, x′ )) = −T re−βH T [ψ(t, x)ψ † (0, x′ )]
(173)
~GE ((t, x), (0, x′ )) = −T re−βH ψ(t, x)ψ † (0, x′ ).
(174)
for 0 ≤ t ≤ β what we write as
26
We can shift the imaginary time argument,
~GE ((t − β, x), (0, x′ )) = −ξT re−βH ψ † (0, x′ )ψ(t − β, x)
= −ξT re−βH ψ † (0, x′ )e−βH ψ(t, x)eβH
= −ξT re−βH ψ(t, x)ψ † (0, x′ ),
(175)
and the result yields the the Kubo-Martin-Schwinger condition,
GE ((t − β, x), (0, x′ )) = ξGE ((t, x), (0, x′ )).
(176)
The Fourier series over the frequency for imaginary time is carried out with the Matsubara spectrum, ωnB = 2πkB T n
or ωnF = 2πkB T (n + 12 ) for bosons or fermions, respectively. Therefore, the imaginary time fermion propagator can
be written
Z −iωF (τ −τ ′ )+ik(x−x′ )
1X
e n
′
~GE
(x,
x
)
=
.
(177)
αβ
β n k
iωnF − ω(k)
One can check that the Euclidean Green function, made periodic with period length β is
′
−βH0
T [ψα (x)ψβ† (x′ )]],
~GE
αβ (x, x ) = −T re
(178)
where H0 is the free Hamiltonian.
IV.
PERTURBATION EXPANSION
The expectation values of observables can be expressed in terms of Green functions. Therefore, the perturbation
expansion will be presented for Green functions only.
A.
Interactive Green functions
Let us consider an n-point function
i~G(t1 , . . . , tn ) = h0|T [A1 (t1 ) · · · An (tn )]|0i
(179)
where |0i is the ground state. Let us assume for simplicity that tn ≥ tn−1 ≥ · · · ≥ t2 ≥ t1 ≥ t0 , when one finds in the
Heisenberg representation
i~G(t1 , . . . , tn ) = h0|T [An (tn ) · · · A1 (t1 )]|0i
(180)
what can be written as
i
i
i
i
i
i
i~G(t1 , . . . , tn ) = h0|e ~ (tn −t0 )H AnS e− ~ (tn −t0 )H · · · e ~ (t1 −t0 )H A1S e− ~ (t1 −t0 )H |0i
i
i
= h0|e ~ (tn −t0 )H AnS e− ~ (tn −tn−1 )H · · · e− ~ (t2 −t1 )H A1S e− ~ (t1 −t0 )H |0i
(181)
by means of the operators of the Schrödinger representation. The interaction representation gives
− ~i
R tn
dt′ H1 (t′ )
− ~i
R tn
dt′ H1 (t′ )
− ~i
R tn
dt′ H1 (t′ )
])† An (tn )T [e
i
] · · · T [e− ~
R t2
dt′ H1 (t′ )
R t2
dt′ H1 (t′ )
i
]A1 (t1 )T [e− ~
R t1
dt′ H1 (t′ )
R t1
dt′ H1 (t′ )
]|0i.
(182)
In order to have a simple perturbation series one would need a single time ordered product between the vacuum state,
i
ie. the first operator in the second line of (181) should be replaced by e− ~ (tf −tn )H where tf > tn .
Luckily this can be made easily since the vacuum is an eigenstate of the Hamiltonian with vanishing eigenvalue,
i
e− ~ (tf −tn )H |0i = |0i and we find
i~G(t1 , . . . , tn ) = h0|(T [e
t0
i
i~G(t1 , . . . , tn ) = h0|T [e− ~
= h0|T [e
= h0|T [e
R tf
tn
− ~i
R tf
− ~i
R tf
tn
t0
dt′ H1 (t′ )
′
′
′
′
dt H1 (t )
dt H1 (t )
]An (tn )T [e
An (tn )e
− ~i
R tn
tn−1
tn−1
tn−1
′
′
dt H1 (t )
A1 (t1 ) · · · An (tn )]|0i
i
] · · · T [e− ~
···e
− ~i
R t2
t1
′
t1
t1
′
dt H1 (t )
i
]A1 (t1 )T [e− ~
A1 (t1 )e
− ~i
R t1
t0
′
t0
t0
′
dt H1 (t )
]|0i
]|0i
(183)
27
in the interaction representation. In the more realistic cases our initial state contains a large number of excitations
and the reduction of the expectation value of the chronological product (181) does not agree with the transition
amplitude (183). These cases requires the use of the Heisenberg representation and the resulting Schwinger-Keldysh
formalism leads in a natural manner to the linear response formulae derived later. We continue here with the simpler
porblems where the reduction (181) → (183) applies.
The expansion in the interaction Hamiltonian yields for t0 = −∞, tf = ∞
i
R∞
′
′
i~G(t1 , . . . , tn ) = h0|T [e− ~ −∞ dt H1 (t ) A1 (t1 ) · · · An (tn )]|0i
Z
∞
X
(− ~i )m ∞ ′
=
dt1 · · · dt′m h0|T [H1 (t′1 ) · · · H1 (t′m )A1 (t1 ) · · · An (tn )]|0i.
m!
−∞
m=0
(184)
The interaction Hamiltonian and the operators A1 ,...,An can always be written as sum or integral of product of local
field operators for a local theory and the vacuum expectation value in the last line is a Green function. The crucial
observation is that this Green function corresponds to a free model because the time dependence of the operators
is generated by the free part of the Hamiltonian, in other words the field operators are constructed by means of
non-interacting fields. The perturbation series obtains the interactive Green function as an infinite sum of free Green
functions.
Let us consider as simple example the electron propagator in a system of electrons interacting with a potential
U (x),
i
h0|T [ψα (x)ψβ† (y)]|0i = h0|T [e− 2~
=
P
σ,σ ′
R
†
†
dtd3 vd3 zψσ
(t,v)ψσ
′ (t,z)U (v−z)ψσ ′ (t,z)ψσ (t,v)
ψ(x)ψ † (y)]|0i
∞
i m
X
)
(− 2~
dtd3 vm d3 zm U (v1 − z1 ) · · · U (vm − zm )
dtd3 v1 d3 z1 · · ·
m!
′
′
m=0
σm ,σm
σ1 ,σ1
†
†
×h0|T ψσ1 (t1 , v1 )ψσ′ (t1 , z1 )ψσ1′ (t1 , z1 )ψσ1 (t1 , v1 ) · · ·
1
†
†
†
×ψσm (tm , vm )ψσm
(185)
′ (tm , zm )ψσ ′ (tm , zm )ψσm (tm , vm )ψ(x)ψ (y) |0i.
m
X Z
X Z
B.
Wick theorem
Let us ignore first any other degrees of freedom than electrons for the sake of simplicity. Then the interacting Green
functions are given in (184) in terms of the noninteracting m + n-point functions
Gα1 ,...,αn ,α′1 ,...,α′n (x1 , · · · , xm , x′1 , · · · , x′n ) = h0|T [ψα1 (x1 ) · · · ψαm (xm )ψα† ′ (x′1 ) · · · ψα† ′n (x′n )]|0i.
1
(186)
It is easy to see that the Green function is vanishing for m 6= n. In fact, each field operator creates or removes
an excitation. One starts at the right and ends at the left with the vacuum, the state of no excitation. Therefore,
each excitation created by a field operator must be removed by another operator. Excitations created by ψ (ψ † ) are
removed by ψ † (ψ). Thus we have non-vanishing number of excitation in the state
T [ψα1 (x1 ) · · · ψαm (xm )ψα† ′ (x′1 ) · · · ψα† ′n (x′n )]|0i
1
(187)
for m 6= n and its overlap with |0i will be vanishing. The 2n-point function can be interpreted as the propagator,
the transition amplitude, for k electrons and n − k holes with k = 0, . . . , n, where k depends on the order of the time
argument of the space-time vectors, the variables of the Green function.
The Wick theorem expresses the general noninteracting Green functions by means of free causal propagators. The
essence of the theorem is the observation that the free Green functions express the transition amplitude for a system
of electrons and holes and this factorizes for individual particles or holes in the absence of interactions. Let us consider
the field operator with the earliest time argument in the state (187). It acts on the vacuum and the excitation, created
by it must be removed by one of the other field operators. The operators which create or removes the same excitation
are called pairs. Thus the earliest operator will be paired up with another operator of the chronological product of
(187). Once this pair is identified, the excitation corresponding to them can be factorized off from the rest of the
Green functions because this is a non-interactive system. The factorization leads to the product of a propagator
for an excitation and a 2n − 2-point function. The iterative application of this argument yields the product of n
propagators at the end. Since there are several operators which should remove of a given excitation we have to sum
over all possibility. This is due to the linear superposition principle of Quantum Mechanics which gives the transition
amplitudes as sums over the transition amplitudes of possible processes.
28
Note that the concept of pairing has already been introduced in representing the matrix elements of two-body
operators, c.f. the discussion of Fig. 2. The strategy followed there was the iterative application of the canonical
commutation relation (13) to move all destruction and creation operator to the right and to the left of the product,
respectively. When the order of a creation and destruction operator was changed then one has to include the right
hand side of the commutator. What is important is that it is a c-number times the identity operator in the Fock-space.
Therefore, this c-number can be factorized outside of the vacuum expectation value which contains two operator less.
The repeated application of this step leads to sum of contributions, each of them represented by pairing, a graph.
The implementation of this strategy for Green functions consists of the following three steps.
1. A pairing distribution is defined by regrouping all creation and annihilation operators in pairs.
2. The order of the field operators in the Green functions is changed in such a manner that operators within each
pair are moved beside each others.
3. Such a modified order of the operators allows us to write the Green function as a product of two point functions,
one factor for each pair because the vacuum state is always regenerated in between the pairs.
z}|{
This plan can easiest be realized by a generalization of the time ordered product, T [AB] → AB , called contraction
which satisfies the following two criterias:
1. The vacuum expectation value of the contraction should agrees with that of the time ordered product,
z}|{
h0|T [AB]|0i = h0| AB |0i to recover the expected propagator for each contracted pairs of operator.
2. The contraction should be the identity operator acting in the Fock-space up to space-time dependent c-number
z}|{
coefficient, AB = f (x, y)11, in order to factorize it out from the vacuum expectation value.
3. The contraction should be vanishing for commuting operators.
The coefficient function appearing in the latter condition must be the propagator, f (x, y) = h0|T [AB]|0i according
to the first property. We already know that T [AB] = h0|T [AB]|0i11 beyond the null-space of the equation of motion,
hence all we need to do is to modify the time-ordered product within the null-space to preserve this property there.
This can be achieved by adding a c-number times the normal ordered product of the two operators in question,
z}|{
AB = T [AB]− : AB :
(188)
because the normal order product always lies within the null-space. (The only reason Green-functions can extend
beyond the null-space is the time ordering opearation, not used in the normal orded product.) The coefficient −1 of
the normal ordered product comes from the third property.
Let us check the second property for A = ψ(x) and B = ψ † (y) where we have
z
}|
{
z }| {
ψ(x)ψ † (x′ ) = [ψ (+) (x) + ψ (−) (x)][ψ (+)† (x′ ) + ψ (−)† (x′ )]
z
}|
{ z
}|
{
= ψ (+) (x)ψ (+)† (x′ ) + ψ (−) (x)ψ (−)† (x′ )
after ignoring the commuting operators in the product. Hence we find for t > t′
}|
{
z
ψ (+) (x)ψ (+)† (x′ ) = ψ (+) (x)ψ (+)† (x′ ) − ξψ (+)† (x′ )ψ (+) (x) = [ψ (+) (x), ψ (+)† (x′ )]ξ
}|
{
z
ψ (−) (x)ψ (−)† (x′ ) = ψ (−) (x)ψ (−)† (x′ ) − ψ (−) (x)ψ (−)† (x) = 0,
(189)
(190)
′
and for t > t
z
}|
{
ψ (+) (x)ψ (+)† (x′ ) = ξψ (+)† (x′ )ψ (+) (x) − ξψ (+)† (x′ )ψ (+) (x) = 0,
z
}|
{
(−)
(−)† ′
ψ (x)ψ
(x ) = ξψ (−)† (x′ )ψ (−) (x) − ψ (−) (x)ψ (−)† (x′ ) = −[ψ (−) (x), ψ (−)† (x′ )]ξ .
(191)
The right hand side consists of space integrals of the canonical commutation relation (13) and is indeed a c-number
times the identity operator,
z }| {
ψ(x)ψ † (x′ ) = f (x, x′ )11.
(192)
29
The Wick theorem states that the free 2n-point function can be written as sum of the product of contractions,
h0|T [φ1 · · · φ2n ]|0i =
X′
π∈S2n
z }| { z
}|
{
ξ σ(π) φπ(1) φπ(2) · · · φπ(2n−1) φπ(2n)
(193)
where the summation is over those permutations of 2n objects which yield different pairing structure, eg. the exchange
z }| {
z }| { z }| {
z }| {
of the two points in a contraction, φ1 φ2 → φ2 φ1 , or the permutation of the contractions themselves, φ1 φ2 φ3 φ4 →
z }| { z }| {
φ3 φ4 φ2 φ1 , do not lead to a new pairing. The order of the permutation σ(π) is 0 or 1, and is determined in the
π(1),...,π(k)
following manner. Any permutation π = (1,...,k
) can be written as the product of exchanges of neighbouring
231
213
elements, eg. (123 ) = (12)(13) and (123 ) = (12), where (ℓ, m) denotes the permutation changing ℓ and m, leaving other
object unchanged. Such a decomposition of permutations is not unique but the number of pair exchanges needed to
obtain a given permutation has a unique value modulo 2. This unique value is its order, σ(π), eg. σ[(231
123 )] = 0 and
σ[(213
123 )] = 1. Eq. (193) is the realization of the heuristic argument mentioned above that the contributions of the
noninteracting elementary excitations to the propagator factorizes.
Let us check the theorem in a simple case of a four-point function. We assume t1 > t2 > t3 > t4 for the sake of
simplicity and consider the Green function
i~G(1, 2, 3, 4) = h0|T [ψ1 ψ2 ψ3† ψ4† ]|0i = h0|ψ1 ψ2 ψ3† ψ4† |0i
(194)
z}|{
describing the propagation of two electrons. The repeated application of the rule AB = T [AB]− : AB : gives
(+)
h0|ψ1 ψ2 ψ3† ψ4† |0i = h0|(ψ1
(−)
(+)
+ ψ1 )(ψ2
(−)
=
(−) (−) (+)† (+)†
h0|ψ1 ψ2 ψ3 ψ4 |0i
=
(−)
(−) (+)†
h0|ψ1 (ψ2 ψ3
z
}|
{
(+)†
+ ψ2 )(ψ3
(−)
(+)†
+ : ψ2 ψ3
(−)†
+ ψ3
(+)†
)(ψ4
(−)†
+ ψ4
)|0i
(+)†
:)ψ4
|0i
z }| {
(−) (+)†
(−) (+)†
(−) (+)† (−) (+)†
= ψ2 ψ3 h0|ψ1 ψ4 |0i + ξh0|ψ1 ψ3 ψ2 ψ4 |0i
z }| { z }| {
z }| {
(−) (+)†
(−) (+)†
(−) (+)†
(−) (+)†
(−) (+)†
(−) (+)†
= ψ2 ψ3 h0| ψ1 ψ4 + : ψ1 ψ4
: |0i + ξh0|ψ1 ψ3 (ψ2 ψ4 + : ψ2 ψ4
:)|0i
z }| { z }| {
z }| { z }| {
(−) (+)† (−) (+)†
(−) (+)† (−) (+)†
(195)
= ψ2 ψ3 ψ1 ψ4 +ξ ψ1 ψ3 ψ2 ψ4 .
This exercises shows the strategy followed in the more complicated cases, too. For each creation and destruction
operator pair appearing beside each other in the wrong order in the time ordered product ie. creation operator right
of the destruction operator, Eq. (188) is used. The normal ordered product appearing on the right hand side of Eq.
(188) puts the operators in the right order and the price of this exchange of order is the contraction. This latter can
be factorized as a c-number and then the procedure is repeated until all destruction (creation) operator are at right
(left) between the vacuum.
The Feynman graphs corresponding to this expression, shown in Fig. 6, represent the direct and the exchange
contributions, the latter involving the exchange statistical factor ξ. The lines stand for the contractions. The single
particle propagator is not a symmetric function, h0|T [ψ(x)ψ † (y)]|0i =
6 h0|T [ψ(y)ψ † (x)]|0i, thus the line is oriented
and the arrow point from the space-time position where an electron was created to the point where it is removed.
As an application of the Wick theorem we consider the two-electron propagator,
A = h0|T [ψ(x1 )ψ(x2 )ψ † (y2 )ψ † (y1 )]|0i
(196)
with spin ignored. The perturbation series is
i
R
3
3
†
†
A = h0|T [e− 2~ dtd vd zψ (t,v)ψ (t,z)U (v−z)ψ(t,z)ψ(t,v) ψ(x1 )ψ(x2 )ψ † (y2 )ψ † (y1 )]|0i
Z
∞
i m Z
X
(− 2~
)
=
dtd3 v1 d3 z1 · · · dtd3 vm d3 zm U (v1 − z1 ) · · · U (vm − zm )
m!
m=0
†
×h0|T ψ (t1 , v1 )ψ † (t1 , z1 )ψ(t1 , z1 )ψ(t1 , v1 ) · · ·
×ψ † (tm , vm )ψ † (tm , zm )ψ(tm , zm )ψ(tm , vm )ψ(x1 )ψ(x2 )ψ † (y2 )ψ † (y1 ) |0i
= h0|T [ψ(x1 )ψ(x2 )ψ † (y2 )ψ † (y1 )]|0i
Z
i
dtd3 vd3 zU (v − z)h0|T [ψ † (t, v)ψ † (t, z)ψ(t, z)ψ(t, v)ψ(x1 )ψ(x2 )ψ † (y2 )ψ † (y1 )]|0i + · · · .
−
2~
(197)
30
2
3
2
3
1
4
1
4
FIG. 6: Graphical representation of the free two-electron propagator (195).
x1
v
y
1
y1
x1
y1
x1
●
●
v
v
●
x2
z
y2
x2
x1
y2
y1
v●
●
z
●
y2
x2
●
●
z
x2
z
y2
FIG. 7: First order contributions to the two-electron propagator of Eq. (199).
Zeroth order contribution, depicted in Fig. 6, is just the non-interactive Green function,
A(0) = h0|T [ψ(x1 )ψ(x2 )ψ † (y2 )ψ † (y1 )]|0i.
The first order contributions are
Z
i
dtd3 vd3 zU (v − z)h0|T [ψ † (t, v)ψ † (t, z)ψ(t, z)ψ(t, v)ψ(x1 )ψ(x2 )ψ † (y2 )ψ † (y1 )]|0i
A(1) = −
2~
z
Z
}|
{z
}|
{z
}|
{z
}|
{
i
= −
dtd3 vd3 zU (v − z) ψ † (t, v)ψ(x1 ) ψ † (t, z)ψ(x2 ) ψ(t, z)ψ † (y2 ) ψ(t, v)ψ † (y1 )
~
z
}|
{z
}|
{z
}|
{z
}|
{
+ ψ(x2 )ψ † (t, z) ψ † (t, v)ψ(t, v) ψ(t, z)ψ † (y2 ) ψ(x1 )ψ † (y1 )
}|
{z
}|
{z
}|
{z
}|
{
z
+ ψ(x2 )ψ † (t, z) ψ(t, z)ψ † (t, v) ψ(t, v)ψ † (y2 ) ψ(x1 )ψ † (y1 )
z
}|
{z
}|
{z
}|
{z
}|
{
+ξ ψ † (t, v)ψ(t, z) ψ † (t, z)ψ(t, v) ψ(x2 )ψ † (y2 ) ψ(x1 )ψ † (y1 ) + ξ(x1 ←→ x2 ) + ξ(y1 ←→ y2 ),
(198)
(199)
the corresponding Feynman graphs are shown in Fig. 7.
C.
Feynman rules
The calculation presented above can be generalized. The resulting procedure can be summarized in the following manner. Suppose that we are calculating the k-th order contribution to the fermion Green function
h0|T [ψ(x1 ) · · · ψ(xn )ψ † (y1 ) · · · ψ † (yn )]|0i.
1. Place 2n points, one for each space-time variable in the Green function, external legs and k pairs of points,
vertices, on the plane. Add a short line running out,←−•, to n external legs which will stand for ψ † (yi ) and draw
short lines running in to the remaining n external legs,−→•, to represent ψ(xj ). For each vertex point draw a
short line in and line out in addition to a dashed line, − − − − − − −, representing the potential connecting
the pair of points.
2. Distribute the contractions between ψ and ψ † ie. the connections of the external leg and vertex points in
such a manner that every point is connected to another one with the opposite type of short line. There is
31
p4
●
u
p3
x
z
.
y
●
p2
p1
FIG. 8: Hartree contribution to the electron propagator.
a unique correspondence between the such contraction distributions and contributions to the Green function.
Each oriented line between the points represents a free electron propagator i~G and the line − − − − − − −
stands for −iU/~.
3. Insert a −1 factor for each fermionic closed loop, eg.
T [ψ1† ψ1 ψ2† ψ2 ψ3† ψ3 ] = ξT [ψ1 ψ2† ψ2 ψ3† ψ3 ψ1† ]
(200)
4. The interaction Hamiltonian is given in a normal ordered form hence one has to make the replacement G(x, y) →
G(x, y + η t̂), η = 0+ in each fermion line which is closed or connects the same potential U (x − y) = δ(x0 −
y 0 )U (x − y) taken at identical time.
5. Divide the contribution with the order of the symmetry group of the graph, the number of permutations of the
points which leave the graph invariant.
Let us consider, as examples, the leading order contributions to the electron propagator, shown in Fig. 8 and 9,
respectively. The Hamiltonian is
H=
XZ
σ
vertex
vertex
z
}|
{
}|
{
z
2
X Z
~
1
ψσ† (x) −
∆ − µ ψσ (x) +
: ψσ† 1 (x)ψσ2 (x) U (x − y) ψσ† 2 (y)ψσ1 (y) :
2m
2 σ ,σ x,y |
{z
}
{z
}
|
x
1
2
←−•←−
(201)
←−•←−
The O (U ) Hartree and Fock contributions are
Z
i
λλ′
dzdu(−i~)Gλ′ λ (u, u + η t̂)Uγγ
i~GH
(x,
y)
=
−
′ (z, u)i~Gαγ (x, z)i~Gγ ′ β (z, y)
αβ
~
Z
i
λλ′
dzdui~Gαλ (x, z)Uγγ
i~GF
(x,
y)
=
−
′ (z, u)i~Gλ′ γ (z, u + η t̂)i~Gγ ′ β (u, y).
αβ
~
In the next step we go over energy-momentum space by the the Fourier transformation
Z
dk −ik(x−y)
G(x, y) =
e
G̃(k)
(2π)4
(202)
(203)
and we find
i~GH
αβ (x, y) = −
= −
i
~
Z
i
~
Z
′
λλ
dzdu(−i~) Gλ′ λ (u, u + η t̂) Uγγ
′ (z, u) i~ Gαγ (x, z) i~ Gγ ′ β (z, y)
|
{z
} | {z } | {z } | {z }
p4
dzdu
p3
p2
p1
dp1 dp2 dp3 dp4 iηp04
λλ′
e
(−i~)G̃λ′ λ (p4 )e−ip3 (z−u) Ũγγ
′ (p3 )
(2π)16
×e−ip2 (x−z) i~G̃αγ (p2 )e−ip1 (z−y) i~G̃γ ′ β (p1 ).
(204)
32
p1
u●
p
x
z
p
y
p2
3
●
4
FIG. 9: Fock contribution to the electron propagator.
The integration over the vertex positions z and u generates the energy-momentum conservation,
Z
0
i
dp1 dp2 dp3 dp4
(2π)4 δ(p3 − p2 + p1 )(2π)4 δ(p3 )eip1 y−ip2 x eiηp4
i~GH
(x,
y)
=
−
αβ
16
~
(2π)
′
λλ
×(−i~)G̃λ′ λ (p4 )Ũγγ
′ (p3 )i~G̃αγ (p2 )i~G̃γ ′ β (p1 )
Z
0
i
dp1 dp4
λλ′
= −
(−i~)e−ip1 (x−y)+iηp4 G̃λ′ λ (p4 )Ũγγ
′ (0)i~G̃αγ (p1 )i~G̃γ ′ β (p1 )
8
~
(2π)
(205)
and we find
i
λλ′
i~G̃H
αβ (p) = − i~G̃βγ (p)i~G̃γ ′ α (p)Ũγγ ′ (0)
~
.
Z
dq
(−i~)G̃λ′ λ (q)
(2π)4
(206)
We find in a similar manner for the Fock contribution
Z
′
i
F
i~Gαβ (x, y) = −
i~ Gλ′ λ (x, z) U λλ′ (z, u) i~ Gαγ (z, u + η t̂) i~ Gγ ′ β (u, y)
~ z,u | {z } | γγ {z } |
{z
} | {z }
p4
i
= −
~
Z
p3
p2
p1
dp1 dp2 dp3 dp4 −ip4 (x−z)
λλ′
−ip2 (z−u−η t̂)
e
i~G̃αλ (p4 )e−ip3 (z−u) Ũγγ
i~G̃λ′ γ (p2 )
dzdu
′ (p3 )e
(2π)16
×e−ip1 (u−y) i~G̃γ ′ β (p1 )
Z
0
i
dp1 dp2 dp3 dp4
= −
(2π)4 δ(p2 + p3 − p4 )(2π)4 δ(p1 − p2 − p3 )eip1 y−ip4 x+ip2 η
~
(2π)16
′
λλ
×i~G̃αλ (p4 )Ũγγ
′ (p3 )i~G̃λ′ γ (p2 )i~G̃γ ′ β (p1 )
Z
dp1 dp2 −ip1 (x−y) ip02 η
i
λλ′
e
e
i~G̃αλ (p1 )Ũγγ
= −
′ (p1 − p2 )i~G̃λ′ γ (p2 )i~G̃γ ′ β (p1 )
~
(2π)8
(207)
and
i~GF
αβ (p)
i
=−
~
Z
dq iq0 η
λλ′
e
i~G̃αλ (p)Ũγγ
′ (p − q)i~G̃λ′ γ (q)i~G̃γ ′ β (p).
(2π)4
(208)
It is instructive to check explicitely the emergence of the energy-momentum conserving Dirac-deltas which arise
from the integration of a vertex position. The three-point function generated by the vertex in the space-time,
Z
V (x, y, v) =
G(x − z)G(z − y)U (z − v)
Zz
=
Gp eip(x−z) Gq eiq(z−y) Ur eir(z−v)
z,p,q,r
Z
=
G(p)eipx G(q)e−iqy U (r)e−irv (2π)4 δ(q + r − p)
(209)
p,q,r
33
gives
V (p, q, r) =
Z
x,y,v
V (x, y, v)e−ipx+iqy+irv = G(p)G(q)U (r)(2π)4 δ(q + r − p)
(210)
after Fourier transformation.
This exercise explains the following modification of the Feynman rules in momentum space compared to the rules
given above:
1. The external legs are given the energy-momentum carried by the particle in question
2. Insert an energy-momentum conserving Dirac-delta at each vertex.
3. Each propagator or potential is replaced by its Fourier transform.
4. Integrate over the energy-momentum variables corresponding to the internal lines.
D.
Schwinger-Dyson equation
The structure of the Feynman graph allows a partial resummation of the perturbation series for the propagator.
Let us imagine the infinitely many graphs contributing to the propagator G(p) in momentum space. We shall regroup
the series in the following manner. The zeroth order group contains free propagator. The first group contains all
graphs which do not fall into two disconnected parts after cutting an internal electron or potential line. After having
separated of these contributions the remaining graphs fall into two parts by the cutting of internal lines. The second
group contains graphs which fall into two disconnected parts by cutting an internal line. In general, the n-th group
contains graphs which fall into n disconnected parts by cutting n − 1 internal lines.
Consider now the sum of graphs of the first class and divide by G2 (p), remove its two external legs. What is left
behind are called one-particle irreducible graphs whose sum is the self energy Σ(p)/i~, eg. the sum of graph if Figs.
8 and 9 after the removal of the external legs. A little exercising with examples shows that the full propagator is a
geometrical series,
G(p) = G0 (p) + G0 (p)Σ(p)G0 (p) + G0 (p)Σ(p)G0 (p)Σ(p)G0 (p) + · · ·
G0 (p)
=
1 − Σ(p)G0 (p)
1
=
−1
G0 (p) − Σ(p)
(211)
and we have the Schwinger-Dyson equation
G−1 = G−1
0 − Σ.
(212)
The advantage of this equation is that any approximation to the self-energy together with the solution of this equation
for the propagator generates a partial resummation of the perturbation series.
V.
EXAMPLES
We present finally few simple model calculations.
A.
Collective modes in Coulomb gas
Let us consider now the dressing, the modification of the pair potential U (x) in an electron gas by the interaction.
We shall use the Hamiltonian of Eq. (201). The potential entering into a Feynman graph between the space-time
points x and x′ is represented by the factor −iU0 (x − x′ )/~. More complicated graphs give higher order corrections to
this factor. The first two corrections of the Schwinger-Dyson resummed potential using the leading order self energy
term is shown in Fig. 10. The full geometric series,
i
i
i
i
i
i
i
− U = − U0 + − U0 L̃ − U0 + − U0 L̃ − U0 L̃ − U0 + · · ·
(213)
~
~
~
~
~
~
~
34
+
+
+ ...
FIG. 10: The dressing of the two-body potential. The dashed line represents −iU0 /~ and the first two graphs of the SchwingerDyson partial resummation of the perturbation series, based on the one-loop self energy are shown.
with
L̃(x, y) = tr[G(x, y)G(y, x)]
(214)
where the trace is over the spin indices represents the particle-hole loop can be written as
U = U0 + U0 G̃U0 + U0 G̃U0 G̃U0 + · · ·
(215)
i
i
G̃(x, y) = − L̃(x, y) = − tr[G(x, y)G(y, x)]
~
~
(216)
where
This geometric series gives after resummation
U = U0
1
1
.
= −1
1 + G̃U0
U0 − G̃
It is advantegous to use the Fourier transform, defined by
Z
d4 k −ik(x−x′ )
′
e
G̃(k)
G̃(x, x ) =
(2π)4
where k = (ω, k), x = (t, x), and k(x − x′ ) = ω(t − t′ ) − k(x − x′ ). Its form is
Z
ins
d4 q
G̃(k) = −
G(q)G(q − k)
~
(2π)4
(217)
(218)
(219)
where ns = 2 stands for the spin degeneracy factor and G(q) is the free propagator. We consider the electron gas at
vanishing temperature and finite density where the propagator, given by Eq. (167) yields
Z 3
1
1
ns
d qdω ′
(220)
G̃(k) = −i
~
(2π)4 ω ′ − ω(q) + iǫsign(ω(q)) ω ′ − ω − ω(q − k) + iǫsign(ω(q − k))
where ~ω(q) = (~q)2 /2m − µ. It is a basic rule in dealing with Feynmang raphs that the frequency itegrals have to be
carried out before the momentum integration. By closing the integration contour upward on the complex frequency
plane we find
Z
ns
n(q) − n(q − k)
d3 q
G̃(k) =
(221)
~
(2π)3 ω(q) − ω − ω(q − k) + iǫ[sign(ω(q − k)) − sign(ω(q))]
in terms of the occupation number density n(q) = Θ(−ω(q)). The identity
1
1
= P − iπδ(ω)
ω + iǫ
ω
(222)
35
where P stands for the principal value gives the real and imaginary parts
Z
ns
n(q) − n(q + k)
d3 q
ℜG̃(k) =
P
= L(k)
3
~
(2π) ω + ω(q) − ω(q + k)
Z
d3 q
ns
π
sign([sign(ω(q)) − sign(ω(q + k))])[n(q) − n(q + k)]δ(ω + ω(q) − ω(q + k))
ℑG̃(k) =
~
(2π)3
(223)
after making the transformation q → q + k in the integrals. The relation
sign([sign(ω(q)) − sign(ω(q + k))])[n(q) − n(q + k)] = n(q)[n(q + k) − 1] + n(q + k)[n(q) − 1]
(224)
and the function
R(k) =
ns
π
~
Z
d3 q
n(q)[1 − n(q + k)]δ(ω + ω(q) − ω(q + k))
(2π)3
(225)
G̃(k) = L(k) − i[R(k) + R(−k)]
(226)
allows us to write
where
Z
ns
d3 q
π
n(q)[1 − n(q − k)]δ(−ω + ω(q) − ω(q − k))
~
(2π)3
Z
ns
d3 q
=
n(q + k)[1 − n(q)]δ(−ω + ω(q + k) − ω(q)).
π
~
(2π)3
R(−k) =
The real part, called Lindhard function is
#
"
Z
ns
1
d3 q
1
L(k) =
+
P
n(q)
2
2
~kq
~kq
~
(2π)3
ω − ~k
−ω − ~k
2m − m
2m − m
Z 1
Z ∞
n(q)
ns
2
dC
dqq
P
=
+ (ω → −ω)
~k2
4π 2 ~
ω − 2m − ~|k|qC
−1
0
m
Z kF
2mω − ~k2 − 2~|k|q ns m
= − 2
dqq
ln
P
2mω − ~k2 + 2~|k|q + (ω → −ω)
4π |k|~2
0



!2  ns mkF 
1 
k̃  1 + z − k̃2 1 
=
−1 +
ln 1− z−
1−
−
1 − z + k̃ 2k̃
4π 2 ~2 
2
2k̃
2
where
z=
vph
ωm
=
,
~kF |k|
vF
k̃
z+
2
!2 
1 + z +
 ln 1 − z −
(227)
k̃
2
k̃
2



(228)
(229)
is the ratio of the phase velocity vph = ω/|k| of the diagnostic external energy-momentum and the Fermi velocity
vF = ~kF /m and k̃ = |k|/kF . The imaginary part is constructed by the help of the function
Z
~qk ~q 2
ns
3
−
d qΘ(|k + q| − kF )δ ω −
R(k) =
8π 2 ~ q<kF
m
2m
Z
ns m
=
d2 kΘ(|k − q| − kF )|q3 =− mω + |k|
(230)
2
~|k|
8π 2 |k|~2 k<kF
mω
+ |k|
where the integral is the area of the region of the plan with qz = q3 = − ~|k|
2 which is inside of the Fermi-sphere
placed at the origin and is the outside of a similar the shifted in the z direction by k as shown in Fig. 11. Three
36
k
k
(a)
(c)
(b)
kF
FIG. 11: Integration domain in Eq. (230).
different cases are the following
(a) : |k| > 2kF
− kF < −
|k|
ωm
+
< kF
~|k|
2
k̃
<1
2
ωm
|k|
< −
+
< |k| − kF
~|k|
2
−1 < −z +
(b) : |k| < 2kF
− kF
k̃
< k̃ − 1
2
ωm
|k|
|k|
< −
+
<
~|k|
2
2
−1 < −z +
(c) : |k| < 2kF
|k| − kF
k̃ − 1 < −z +
k̃
k̃
< .
2
2
(231)
They give
R(k) =
Z
ns m
8π 2 |k|~2
q
where in cases (a,b): q2 =
kF2 − ( |k|
2 −
q
mω 2
kF2 − ( |k|
2 − ~|k| ) . We have finally
mω 2
~|k| ) ,
q2
2πdqq =
q1
ns m
(q 2 − q12 )
8π|k|~2 2
(232)
q1 = 0, and in case (c): q2 =

1 − ( k̃2 − z)2
ns mkF 
R(k) =
1 − ( k̃2 − z)2
8π k̃~2 

2z k̃
k̃ > 2,
k̃ < 2,
k̃ < 2,
kF2 − ( |k|
2 −
− 1 − k̃2 < −z < 1 − k̃2 ,
− 1 − k̃2 < −z < −1 + k̃2 ,
0 < z < 1 − k̃2 .
Notice the singularity of the self energy at k = 0,

k̃ 2
1−
1
k̃


−1 + k̃ 1 − 4 ln k̃2 

1+ 2
ns mkF
1+z G̃(k) =
2
2
−2
+
z
ln
−
iπΘ(1
−
z)z
+
O
k̃ 2

4π ~ 
1−z

 2
3z 2
q
mω 2
~|k| ) ,
q1 =
(233)
z = 0 and finite k
0 < z, k → 0
z k̃ =
mω
2
~kF
(234)
finite and k → 0
The dielectric constant κ, defined as
U0 (k)
U (k)
(235)
κ = 1 − G̃U0
(236)
κ(k) =
is of the form
37
FIG. 12: The function L(k)U0 (k)/rs shown for the Coulomb potential as the function of k̃ and Z.
according to Eq. (217). It is advantageous to introduce dimensionless parameters in the dielectric constant. For that
end we express the Fermi-wave vector kF by the average interparticle distance r0 , defined by 4πr03 /3 = V /N . The
particle number of an ideal gas
Z
ns V kF3
d3 k
Θ(k
−
k)
=
(237)
N = V ns
F
(2π)3
6π 2
gives
kF =
6π 2 N
V ns
1/3
=
9π
2ns
1/3
1
rs a0
(238)
The numerical factor in G̃(k)U0 (k) appearing for the Coulomb potential U0 (k) = 4πe2 /k2 can be expressed in terms
of rs = r0 /a0 where the Bohr radius a0 = ~2 /me2 as
ns mkF 4πe2
ns
=
= rs
2
2
2
4π ~ kF
πkF a0
2ns
9π
1/3
ns
π
(239)
yielding
κ(k) = 1 − rs
2ns
9π
1/3
k̃ 2
1−
1 − k̃4 ln k̃2 1+ 2
ns
1+z −
iπΘ(1
−
z)z
+
O
k̃ 2
−2
+
z
ln
2

1−z
π k̃ 

 2



−1 +


1
k̃
3z 2
z = 0 and finite k
0 < z, k → 0
z k̃ =
mω
2
~kF
(240)
finite and k → 0
The Thomas-Fermi inverse screening length is defined by
kT2 F = 4πe2 lim G̃(0, k) =
k→0
2e2 ns mkF
.
π~2
(241)
The resummed potential (217) is time dependent due to the dynamical, propagating nature of the particle-hole
states and its pole structure on the complex frequency space is similar as that of a time ordered product, a Feynman
propagator. The retarded version, with poles on the lower half plane only is therefore
κr (k) = ℜκ(k) + isign(ℜω)ℑκ(k).
(242)
Collective exciations are defined in momentum space where the absolute magnitude of the Green function is large.
By assuming that ℜκ(k) ≫ ℑκ(k) we define quasiparticle dispersion relation by the condition ℜκ(k) = 0 and interpret
38
−ℑκ(k) on the dispersion relation curve as the inverse life-time. There are two kinds of collective modes for small
enough k̃, satisfying the condition
1/3
k̃ + Z 9π
π 3
(243)
k̃ + 2k̃ = Z ln k̃ − Z 2ns
rs ns
where Z = z k̃ = ωm/~kF2 . One dispersion relation approaches
2
Zpl
2
=
3
2ns
9π
1/3
rs ns
π
(244)
as k̃ → 0, yielding the plasma frequency
2
ωpl
~2 kF4 2
=
m2 3
2ns
9π
1/3
rs ns
π
(245)
which corresponds to oscillation where particles and holes make opposite displacement. Another dispersion relation
turns out to be at finite z, ω ≈ v0 k, where v0 = z0 ~kF /m and z0 is given by
1 + z0 .
(246)
2 = z0 ln 1 − z0 This belongs to the zero sound. The function L(k)U0 (k)/rs is plotted in Fig. 12. The contour on the plane (ω, |k|)
plane where this function assumes the values 1/rs is the quasi-particle line. Note that the plasmon and the zero-sound
lines always meet.
B.
Propagation in a disorded medium
Let us consider the propagation of a charge interacting with randomly placed static impurities with the potential
u(x − x′ ). We add a constant to this potential to ensure the relation
Z
d3 xu(x) = 0
(247)
which will suppresses the interactions with vanishing momentum transfer. The total potential acting on the charge is
X
U (x) =
u(x − xa )
(248)
a
where xa denotes the position of the impurities. We shall need the Fourier representation
X Z d3 k
U (x) =
eik(x−xa ) Ũ (k)
3
(2π)
a
for the potential. The second quantized one-body operator, acting in the Fock-space is
Z
U = d3 xψ † (x)U (x)ψ(x).
(249)
(250)
One can construct the interaction representation by considering the impurity potential as perturbation. The
propagator of a charge for a given impurity configuration {xa } is
i~G(x, y) = h0|T [ψ(x)ψ † (y)]|0i
∞ Z
X
i
i
dz1 · · · dzn i~G0 (x, z1 ) − U (z1 ) i~G0 (z1 , z2 ) · · · i~G0 (zn−1 , zn ) − U (zn ) i~G0 (zn , y),
=
(251)
~
~
n=0
where G0 (x, y) denotes the free propagator.This perturbation series is shown in Fig. 13.
39
=
+
●
+
●
●
+
● ● ●
FIG. 13: The propagator in the presence of a given impurity configuration.
In order to include the impurities we compare the characteristic time scales of the motion of the charges and the
impurities, tch and timp , respectively. We can safely assume that the impurities are slower degrees of freedom than
the charges, tch < timp . When tch and timp are not too far from each others then we have to include the impurity
dynamics in the description as usual, ie. by writing the total Hamiltonian as the sum of the kinetic energy for the
impurities, the charges and the interaction energy. Such an averaging over the impurities is called annealed average.
In the other extremity, tch ≪ timp , which is usually realized in real samples we assume that the quantum coherent scatterings among the charges take place much more frequently than the noticeable motion of the impurities.
Therefore, it makes sense to calculate the desired Green functions for the charges first for a given static impurity
configuration and then averaging this quantum expectation value over static impurity configurations. This is called
the quenched averaging over the impurities.
What is left to discuss is the choice of the impurity distribution. We have no knowledge about it thus we try to
reproduce the average over a large number of samples, each of which comes with its own static impurity configuration.
In order to relate the averaging over the impurity configuration with other, more physical averaging we rely on the
assumption of self averaging. This requires that the quenched average of our observable over the impurities is the same
as the spatial average carried out in one given large system. One is interested in general in bulk, extensive quantities
and the self averaging assumption assures that the measurement of a bulk quantity characterizing a large sample with
a given impurity configuration can be reproduced by the quenched averaging method. The Green functions are not
self averaging but their combinations making up a bulk quantity usually has this property.
Let us now return to the propagator (251) whose quenched average is
i~G(x, y) =
∞ Z
X
n=0
i
i
dz1 · · · dzn i~G0 (x, z1 ) · · · i~G0 (zn , y) − U (z1 ) · · · − U (zn ) ,
~
~
where the over-line denotes the average over the impurity configuration {xa },
X Z d3 k
U (x) =
eikx Ũ (k)e−ikxa .
3
(2π)
a
We use uniform distribution in space, p(xa ) = 1/V , for simplicity yielding
Z
Ni
d3 k ikx
U (x) =
e Ũ (k)(2π)3 δ(k) = nŨ (0) = 0
V
(2π)3
(252)
(253)
(254)
where Ni denotes the number of impurities and n = Ni /V . Thus the average of the second graph on the right hand
side of Fig. 13 is vanishing.
The second order term in the perturbation series (251) contains the average
X Z d3 k1 d3 k2
(255)
U (x1 )U (x2 ) =
eik1 x1 +ik2 x2 Ũ (k1 )Ũ (k2 )e−ik1 xa1 −ik2 xa2
3 (2π)3
(2π)
a ,a
1
2
and is shown in Fig. 14. The averaged Fourier modes
e−ik1 xa1 −ik2 xa2 =
(2π)3
δ(k1 + k2 )δa1 ,a2
V
(256)
40
●
●
FIG. 14: The O U 2 contribution to the propagator.
●
●
●
●
●
●
FIG. 15: The O n2 U 6 contribution to the propagator.
give
U (x1 )U (x2 ) = U (x1 )U (x2 )0
(257)
where
U (x1 ) · · · U (xm )0 = n
Z
d3 k1
d3 km
···
(2π)3 δ(k1 + · · · + km )eik1 x1 +···+ikm xm Ũ (k1 ) · · · Ũ (km )
3
(2π)
(2π)3
(258)
denotes the average over a single impurity center. The interaction vertices are connected by a dashed line with the
point representing the space-time location of the impurity on the graphs.
The systematics should be clear by now: The number of blobs of the graph, such as in Fig. 15, gives the order of
the contribution. The number of impurity points gives the power of the density because each independent averaging
over an impurity location generates the factor n = N/V ,
X Z d3 k1
d3 km ik1 x1 +···+ikm xm
U (x1 ) · · · U (xm ) =
·
·
·
e
Ũ (k1 ) · · · Ũ (km )e−ik1 xa1 −···−ikm xam
(2π)3
(2π)3
a1 ,...,,am
X
U (xa )U (xb )0 U (x1 ) · · · Û (xa ) · · · Û (xb ) · · · U (xm )0 +O n3 (259)
= U (x1 ) · · · U (xm )0 +
{z
} a6=b
|
O(n)
|
{z
}
O(n2 )
The hat above the potential indicates in this equation that that term is missing in the product.
It is instructive to carry out the partial resummation of the perturbation series by solving the Schwinger-Dyson
41
equation. The leading order contribution to the self energy whose first few graphs are shown in Fig. 16 is
2
Z
d3 k
i
Σ(ω, k) ≈ i~n
−
|Ũ (q)|2 i~G0 (ω, k − q) + O U 3
(2π)3
~
Z
d3 k
= n
|Ũ (q + k)|2 G0 (ω, q)
(2π)3
Z
d3 k
|Ũ (q + k)|2
= n
.
(2π)3 ω − ω(q) + iǫsign(ω(k))
(260)
The identity
1
1
= P ∓ iπδ(x)
x ± iǫ
x
gives
Z
d3 q |Ũ (q + k)|2
P
(2π)3
ω − ω(q)
Z
d3 q
ℑΣ(ω, k) = −sign(ω)nπ
|Ũ (q + k)|2 δ(ω − ω(q)).
(2π)3
ℜΣ(ω, k) = n
(261)
(262)
Note that the imaginary part of the self energy changes sign at the Fermi-level in agreement with Feynman iǫ
prescription. The partially resumed propagator
G(ω, k) =
1
ω − ω(q) − Σ(ω, k) + iǫsign(ω(k))
(263)
suggests to interpret the real part of the self energy at the Fermi level, −δµ = ℜΣ(0, kF ), as the renormalization of
the chemical potential up to a sign and the imaginary part evaluated at the Fermi level as the inverse life-time of the
charged particle,
τ −1 = −ImΣ(0+ , kF )
Z
d3 q
= nπ
|Ũ (q + kF )|2 δ(ω(q))
(2π)3
Z
Z
1
nπ
dEν(E)
dΩ|Ũ (kF Ω + kF )|2 δ(E − EF )
=
~
4π
nπ
ν(EF )|ŨF |2
=
~
(264)
with
1
|ŨF | =
4π
2
Z
dΩ|Ũ (kF Ω + kF )|2 .
(265)
How on the earth can it happen that a charged particle, propagating in a random static impurity potential acquires
finite life-time despite the charge conservation? The point is that the quenched averaging technique achieves something
highly non-trivial and subtle. The core of the random impurity problem is the non-homogeneity created by the
randomly distributed scattering centers. The loss of translation invariance changes the physics in a considerable
manner. The averaging over the impurity locations formally restores translation invariance. But the reminder of
formal nature of this restoration is betrayed by the finite life-time. In fact, the the lowest lying state of a charge,
propagating in a given random impurity configuration has no well-defined momentum. The quenched average of the
propagator at a given momentum and energy represent the mixture of the transition amplitudes of different, usually
excited states of the inhomogeneous problems which is unstable.
C.
Linear response formula for the electric conductivity
The calculation of the current generated by a weak external electric field will be carried out the interaction
representation where the external field effect represents the perturbation. We write the Hamiltonian as the sum
H(t) = H0 + H1 (t) and the expectation value of an observable
i
i
Oi (t) = e ~ H0 OS e− ~ H0 ,
(266)
42
●
●+●
●
●+ + ●
●●●
●
●
●+
●●●
FIG. 16: The first three graphs of the self energy.
defined in the interaction representation where H1 (t) is treated as perturbation is
hhO(t)ii = Trρi (t)Oi (t)
(267)
ρi (t) = Ui (t, ti )ρ(ti )U † (t, ti )
(268)
where
is the density matrix. The time evolution operator satisfies the equation of motion
i~∂t U (t, ti ) = H1i (t)U (t, ti )
(269)
according to the Schrödinger equation (103) and the corresponding equation of motion for the density matrix is
i~∂t ρi (t) = [H1i (t), ρi (t)].
The iterative integration of this equation generates the perturbation expansion whose first order piece,
Z
i t ′
2
ρi (t) = ρ0 −
,
dt [H1i (t′ ), ρ0 ]dt′ + O H1i
~ ti
(270)
(271)
gives
δhhO(t)ii = hhO(t)ii − Trρ0 O(t)
Z
i t ′
= −
dt Tr[H1i (t′ ), ρ0 ]Oi (t)dt′
~ 0
Z
i t ′
=
dt Trρ0 [H1i (t′ ), Oi (t)]dt′
~ 0
(272)
One can usually separate the time dependence of the perturbation, H1 (t) = h(t)Bi , and write
i
δhhAii =
~
= −
Z
t
dt′ h(t′ )Tr[Bi (t′ ), Oi (t)]ρ
t0
Z tf
t0
′
′
′
GR
O,B (t, t )h(t )dt
(273)
where the retarded Green-function is defined by
′
′
′
i~GR
A,B (t, t ) = Θ(t − t )hh[Ai (t), Bi (t )]ii.
(274)
Let us apply this formalism for the electric conductivity defined by the proportionality of a weak external electric
field and the current induced by it,
jk = σk,ℓ Eℓ .
(275)
43
The external electric field is written as
E = −∂t A,
(276)
which leads to the perturbation
H1 = −
Z
d3 xj(x)A(x).
(277)
The minimal coupling prescription,
p → p − eA
(278)
in taking into account the electromagnetic field gives the current operator
e~
(ψ † ∇ψ − ∇ψ † ψ)
2mi e~
ie
ie
†
†
→
ψ ∇− A ψ− ∇− A ψ ψ
2mi
~
~
2
e
= j − Aψ † ψ
m
j =
(279)
whose expectation value is given by the Kubo-formula,
hhjk (t, x)ii = −
Z
e2
Ahhψ † ψii +
m
′
′
dt′ d3 yGR
k,ℓ ((t, x), (t , y))Aℓ (t , y).
(280)
This expression identifies the conductivity,
σk,ℓ (ω, k) = −
ie2 n
i
+ GR (ω, k).
mω
ω
(281)
The Green function appearing in the Kubo formula contains the particle-particle interaction because only the
external perturbation is included in Hi of Eq. (270). The systematic treatment of the inter-particle interactions
requires to present all interactions in Hi and to ba able to go into higher order of the perturbation expansion. This
is a non-trivial task because the Wick theorem applies for the time ordered product of field operators and not for the
retarded or advanced Green functions. The generalization of the perturbation expansion for these Green function can
be achieved in the Closed Time Path (CTP) or Schwinger-Keldysh formalism.
The starting point to develop the perturbation expansion in the CTP formalism is the generalization of the expectation value of Eq. (181) for finite temperature,
i
i
i
i
i
i
i~G(t1 , . . . , tn ) = Tr[e ~ (tf −t0 )H e− ~ (tf −t0 )H e ~ (tn −t0 )H AnS e− ~ (tn −tn−1 )H · · · e− ~ (t2 −t1 )H A1S e− ~ (t1 −t0 )H ρi ]
(282)
where the initial density matrix is
ρi =
e−βH
Tre−βH
(283)
and tf > tn . The same quantity written in the interaction representation reads
i
i~G(t1 , . . . , tn ) = TrT [e− ~
R tf
t0
dt′ H1 (t′ )
A1 (t1 ) · · · An (tn )]ρi (T [e
− ~i
R tn
t0
dt′ H1 (t′ )
])†
(284)
One introduces at this step a formal redoubling of the degrees of freedom into CTP doublets, eg. ψ → (ψ + , ψ − ). The
two members of a doublet belong to the time evolution operator standing at the left and right of the density matrix.
We write this expectation value as
i
i~G(t1 , . . . , tn ) = TrT ∗ [e ~
R tf
t0
dtH1 (t)
i
]T [e− ~
R tf
t0
dt′ H1 (t′ )
A1 (t1 ) · · · An (tn )]ρi
(285)
where T ∗ denotes the anti-time ordered product which orders the factors in descending order of their time argument.
This form suggests the use of a generalized time ordering, T̄ which is T or T ∗ for the operatos with CTP index + or
-, respectively and places all + operator before the - one. This construction allows us to write
i
i~G(t1 , . . . , tn ) = TrT̄ [e− ~
R tf
t0
dt[H1+ (t)−H1− (t′ )]
+
A+
1 (t1 ) · · · An (tn )]ρi .
(286)
44
The reason of this chain of formal step is the introduction of this generalized time ordering which establishes a formal
similarity with Eq. (184). This in turns allows the use of the Wick theorem again with the modification that the
original time ordering T in the definition of the free Green functions is replaced everywhere by the generalized time
ordering T̄ . The result is a perturbation expansion for quantum field theory where the fields have an additional
CTP index + or - which identify one of the two time axes where the operators belong and the propagators are block
matrices, eg.
′
′
i~Gσ,σ (x, x′ ) = Trψ σ (x)ψ †σ (x′ )ρ0 ,
(287)
for the electron field. The first order contributions in this perturbation expansion reproduces the Kubo formula and
the higher orders in the inter-particle interactions build up the interactive green functions which are automatically
retarded due to a destructive interference taking place between the two time axes for times later than the last operator
in the Green function.
Appendix A: Phonons
This Appendix contains the details of calculating the non-interactive phonon Green functions. We write the quantum
field controlling the phonon excitations in a given band λ as
s
Z
d3 k
~
χλ (x, t) =
[e−iωλ (k)t+ixk cλ (k) + eiωλ (k)t+ixk c†λ (−k)].
(A1)
(2π)3 2mλ ωλ (k)
It allows us to introduce the propagators
i~G(x, x′ )
i~GR (x, x′ )
i~GA (x, x′ )
~GE (x, x′ )
=
=
=
=
h0|T [χ(x)χ(x′ )]|0i
h0|[χ(x), χ(x′ )]+ |0iΘ(t − t′ )
−h0|[χ(x), χ(x′ )]+ |0iΘ(t′ − t)
−h0|T [χ(x)χ(x′ )]|0i.
The steps, analogous to the fermionic case yield the causal propagator
Z 3 3 ′
~
d kd k
p
i~G(x, x′ ) =
(2π)6 2m ω(k)ω(k′ )
′
′
(A2)
′
′
′
′
′
′
×h0|Θ(t − t′ )[e−iω(k)t+ixk c(k) + eiω(k)t+ixk c† (−k)][e−iω(k )t +ix k c(k′ ) + eiω(k )t +ix k c† (−k′ )]
′
′
′
′
′
′
′
′
+Θ(t′ − t)[e−iω(k )t +ix k c(k′ ) + eiω(k )t +ix k c† (−k′ )][e−iω(k)t+ixk c(k) + eiω(k)t+ixk c† (−k)]|0i
Z 3 3 ′
′ ′
′ ′
d kd k
~
p
=
h0|Θ(t − t′ )e−iω(k)t+ixk c(k)eiω(k )t +ix k c† (−k′ )
6
′
(2π) 2m ω(k)ω(k )
′
′
′
′
+Θ(t′ − t)e−iω(k )t +ix k c(k′ )eiω(k)t+ixk c† (−k)|0i
Z 3 3 ′
d kd k
~
p
(2π)3 δ(k + k′ )
=
(2π)6 2m ω(k)ω(k′ )
′
=
=
=
=
′
′
′
′
′
′
′
×[Θ(t − t′ )e−iω(k)t+ixk eiω(k )t +ix k + Θ(t′ − t)e−iω(k )t +ix k eiω(k)t+ixk ]
!
Z 3
′
′
′
′
d kdω ~i e−i(ω(k)+ω)(t−t )+ik(x−x )
e−i(ω(−k)+ω)(t −t)+ik(x−x )
+
(2π)4 2m
ω(k)(ω + iǫ)
ω(−k)(ω + iǫ)
!
Z 3
′
′
′
′
e−iω(t−t )+ik(x−x )
~i
e−iω(t −t)+ik(x−x )
d kdω
+
(2π)4 2mω(k)
ω − ω(k) + iǫ
ω − ω(k) + iǫ
Z 3
d kdω
~i
1
1
−iω(t′ −t)+ik(x−x′ )
e
−
(2π)4 2mω(k)
ω − ω(k) + iǫ ω + ω(k) − iǫ
Z 3
−iω(t−t′ )+ik(x−x′ )
~i
d kdω e
m
(2π)4 ω 2 − ω 2 (k) + iǫ
(A3)
45
The retarded propagators is
Z 3 3 ′
d kd k
~
p
i~GR (x, x′ ) =
6
(2π) 2m ω(k)ω(k′ )
′
′
′
′
′
′
′
′
×Θ(t − t′ )h0|[e−iω(k)t+ixk c(k) + eiω(k)t+ixk c† (−k)][e−iω(k )t +ix k c(k′ ) + eiω(k )t +ix k c† (−k′ )]
′
′
′
′
′
′
′
′
−[e−iω(k )t +ix k c(k′ ) + eiω(k )t +ix k c† (−k′ )][e−iω(k)t+ixk c(k) + eiω(k)t+ixk c† (−k)]|0i
Z 3 3 ′
′ ′
′ ′
d kd k
~
p
=
Θ(t − t′ )h0|e−iω(k)t+ixk c(k)eiω(k )t +ix k c† (−k′ )
(2π)6 2m ω(k)ω(k′ )
′
′
′
′
−e−iω(k )t +ix k c(k′ )eiω(k)t+ixk c† (−k)|0i
Z 3 3 ′
d kd k
~
p
=
(2π)3 δ(k + k′ )
(2π)6 2m ω(k)ω(k′ )
′
=
=
=
=
=
=
′
′
′
′
′
′
′
×Θ(t − t′ )[e−iω(k)t+ixk eiω(k )t +ix k − e−iω(k )t +ix k eiω(k)t+ixk ]
Z
′
′
′
′
d3 k
~
Θ(t − t′ )[e−iω(k)(t−t )+ik(x−x ) − e−iω(k)(t −t)+ik(x−x ) ]
3
(2π) k 2mω(k)
!
Z 3
′
′
′
′
e−i(ω(k)+ω)(t−t )+ik(x−x )
~i
d kdω
e−i(ω−ω(k))(t−t )+ik(x−x )
−
(2π)4 2mω(k)
ω + iǫ
ω + iǫ
!
Z 3
′
′
′
′
e−iω(t−t )+ik(x−x )
e−iω(t−t )+ik(x−x )
d kdω ~i
−
(2π)4 2m ω(k)(ω − ω(k) + iǫ) ω(−k)(ω + ω(−k) + iǫ)
Z 3
d kdω
~i
1
1
−iω(t−t′ )+ik(x−x′ )
e
−
(2π)4 2mω(k)
ω − ω(k) + iǫ ω + ω(k) + iǫ
Z 3
′
′
~i
d kdω e−iω(t−t )+ik(x−x )
m
(2π)4 ω 2 − ω 2 (k) + isign(ω)ǫ
Z 3
′
′
d kdω
e−iω(t−t )+ik(x−x )
~i
.
m
(2π)4 ω 2 − ω 2 (k) + isign(ω(k))ǫ
(A4)
46
The advanced version is obtained in a similar manner,
Z 3 3 ′
d kd k
~
A
′
p
i~G (x, x ) = −
6
(2π) 2m ω(k)ω(k′ )
×Θ(t′ − t)h0|[e−iω(k)t+ixk c(k) + eiω(k)t+ixk c† (−k)][e−iω(k
′
′
′
′
′
′
′
′
)t′ +ix′ k′
′
′
′
′
c(k′ ) + eiω(k )t +ix k c† (−k′ )]
′
−[e−iω(k )t +ix k c(k′ ) + eiω(k )t +ix k c† (−k′ )][e−iω(k)t+ixk c(k) + eiω(k)t+ixk c† (−k)]|0i
Z 3 3 ′
′ ′
′ ′
d kd k
~
p
= −
Θ(t′ − t)h0|e−iω(k)t+ixk c(k)eiω(k )t +ix k c† (−k′ )
(2π)6 2m ω(k)ω(k′ )
′
′
′
′
−e−iω(k )t +ix k c(k′ )eiω(k)t+ixk c† (−k)|0i
Z 3 3 ′
~
d kd k
p
(2π)3 δ(k + k′ )
= −
6
(2π) 2m ω(k)ω(k′ )
′
=
=
=
=
=
=
′
′
′
′
′
′
′
×Θ(t′ − t)[e−iω(k)t+ixk eiω(k )t +ix k − e−iω(k )t +ix k eiω(k)t+ixk ]
Z
′
′
′
′
d3 k
~
−
Θ(t′ − t)[e−iω(k)(t−t )+ik(x−x ) − e−iω(k)(t −t)+ik(x−x ) ]
3
(2π) 2mω(k)
!
Z 3
′
′
′
′
e−i(ω(k)−ω)(t−t )+ik(x−x )
~i
e−i(−ω−ω(k))(t−t )+ik(x−x )
d kdω
−
−
(2π)4 2mω(k)
ω + iǫ
ω + iǫ
!
Z 3
′
′
′
′
d kdω
e−i(ω(k)+ω)(t−t )+ik(x−x )
~i
e−i(ω−ω(k))(t−t )+ik(x−x )
−
(2π)4 2mω(k)
ω − iǫ
ω − iǫ
Z 3
d kdω
~i
1
1
−iω(t−t′ )+ik(x−x′ )
e
−
(2π)4 2mω(k)
ω − ω(k) − iǫ ω + ω(k) − iǫ
Z 3
′
′
d kdω e−iω(t−t )+ik(x−x )
~i
m
(2π)4 ω 2 − ω 2 (k) − isign(ω)ǫ
Z 3
′
′
e−iω(t−t )+ik(x−x )
~i
d kdω
.
m
(2π)4 ω 2 − ω 2 (k) − isign(ω(k))ǫ
(A5)
Finally, the imaginary time propagator is obtained by the Wick-rotation tR = −itI and ωR = iωI as
′
~GE
αβ (x, x ) =
Z −iωn (τ −τ ′ )+ik(x−x′ )
1X
e
β n k
ωnB2 + ωk2
(A6)
where ωnB = 2πkB T n is the bosonic Matsubara frequency.
Appendix B: Spectral representation
The calulation of the different propagators presented in Chapter III is valid for non-interacting particles only.
It is interesting to realize that there is a simple extension of the free propagators for interacting system which is
parameterized by a spectral weight function only with clear physical interpretation. Apart of being an important
tool, this representation offers another view of the propagators with or without interactions.
Let us denote the eigenstates of the Hamiltonian by |{n}i,
H|{n}i = ~ω{n} |{n}i
(B1)
and write the matrix elements of the quantum field as
ψ{m},{n} (x) = h{m}|ψ(0, x)|{n}i,
(B2)
where {n} a set of quantum numbers, such as the occupation number configuration n(q), identifying a basis element
of the Fock-space. The thermal and quantum average of an observable A is then
X
hhAii =
p{n} h{n}|A|{n}i
(B3)
{n}
47
where
p{n} = P
e−β~ω{n}
.
−β~ω{n}
{n} e
(B4)
We are first interested in the causal propagator at a given frequency,
Z
′
i~G(x, x , ω) =
dteiωt hhT [ψ(t, x)ψ † (0, x′ )]ii
Z 0
Z ∞
iωt
†
′
dteiωt hhψ † (0, x′ )ψ(t, x)ii.
dte hhψ(t, x)ψ (0, x )ii −
=
(B5)
−∞
0
The desired boundary conditions in time are implemented by the iǫ prescription,
Z 0
Z ∞
i(ω+iǫ)t
†
′
′
dtei(ω−iǫ)t hhψ † (0, x′ )ψ(t, x)ii
dte
hhψ(t, x)ψ (0, x )ii −
i~G(x, x , ω) =
−∞
0
Z ∞
X
†
dtei(ω+iǫ)t
p{m} eiω{m},{n} t ψ{m},{n} (x)ψ{n},{m}
(x′ )
=
0
−
= −
{m},{n}
Z
dtei(ω−iǫ)t
−∞
p{m}
X
p{m}
{m},{n}
Z ∞
dλ
−∞
+i
†
p{m} ψ{m},{n}
(x′ )ψ{n},{m} (x)eiω{n},{m} t
†
ψ{m},{n} (x)ψ{n},{m}
(x′ )
ω + iǫ + ω{m},{n}
{m},{n}
Z
X
{m},{n}
X
= i
= i
0
†
ψ{m},{n} (x)ψ{n},{m}
(x′ )
X
ω + iǫ + ω{m},{n}
+
†
ψ{m},{n}
(x′ )ψ{n},{m} (x)
+ p{n}
ω − iǫ + ω{n},{m}
†
ψ{n},{m}
(x′ )ψ{m},{n} (x)
ω − iǫ + ω{m},{n}
†
p{m} ψ{m},{n} (x)ψ{n},{m}
(x′ )δ(λ + ω{m},{n} )
{m},{n}
∞
X
dλ
−∞
!
1
ω + iǫ − λ
†
p{n} ψ{n},{m}
(x′ )ψ{m},{n} (x)δ(λ + ω{m},{n} )
{m},{n}
!
1
,
ω − iǫ − λ
(B6)
where ω{m},{n} = ω{m} − ω{n} . The relation (168) can be used to write
P
dλ
+ ω{m},{n} )
i~G(x, x , ω) = i
− iπδ(ω − λ)
ω−λ
−∞
{m},{n}
Z ∞
X
P
†
′
+ iπδ(ω − λ)
dλ
+i
p{n} ψ{n},{m} (x )ψ{m},{n} (x)δ(λ + ω{m},{n} )
ω−λ
−∞
{m},{n}
Z ∞
P
dλ A(x, x′ , λ)
= i
− iπB(x, x′ , λ)δ(ω − λ)
(B7)
ω−λ
−∞
′
Z
∞
X
†
p{m} ψ{m},{n} (x)ψ{n},{m}
(x′ )δ(λ
where
A(x, x′ , λ) =
X
†
(p{m} + p{n} )ψ{m},{n} (x)ψ{n},{m}
(x′ )δ(λ + ω{m},{n} )
X
†
(p{m} − p{n} )ψ{m},{n} (x)ψ{n},{m}
(x′ )δ(λ + ω{m},{n} ).
(B8)
{m},{n}
and
B(x, x′ , λ) =
{m},{n}
(B9)
48
The latter can be recasted in the form
X p{m} − p{n}
†
(p{m} + p{n} )ψ{m},{n} (x)ψ{n},{m}
(x)δ(λ + ω{m},{n} )
B(x, x′ , λ) =
p{m} + p{n}
{m},{n}
1 − e−βλ
1 + e−βλ
2
= A(x, x′ , λ) 1 −
,
1 + eβλ
= A(x, x′ , λ)
leaving the function A(x, x′ , λ) as the only unknown piece of the propagator.
The same function A(x, x′ , λ) determines other propagators, too. The retarded propagator is
Z ∞
R
′
dteiωt hh[ψ(t, x), ψ † (0, x′ )]+ iiΘ(t)
i~G (x, x , ω) =
−∞
Z ∞
X
dtei(ω+iǫ)t
→
p{m}
0
h
(B10)
{m},{n}
†
†
ψ{m},{n} (x)ψ{n},{m}
(x′ ) + ψ{m},{n}
(x′ )ψ{n},{m} (x)eiω{n},{m} t
!
†
†
X
ψ{m},{n} (x)ψ{n},{m}
(x′ ) ψ{m},{n}
(x′ )ψ{n},{m} (x)
= i
p{m}
+
ω + ω{m},{n} + iǫ
ω + ω{n},{m} + iǫ
{m},{n}
!
†
†
X
ψ{n},{m}
(x′ )ψ{m},{n} (x)
ψ{m},{n} (x)ψ{n},{m}
(x′ )
+ p{n}
= i
p{m}
ω + ω{m},{n} + iǫ
ω + ω{m},{n} + iǫ
{m},{n}
Z ∞
A(x, x′ , λ)
dλ
= i
.
ω + iǫ − λ
−∞
× e
iω{m},{n} t
i
(B11)
The advanced propagator is obtained in a similar manner,
Z ∞
dteiωt hh[ψ(t, x), ψ † (0, x′ )]+ iiΘ(−t)
i~GA (x, x′ , ω) = −
→ −
Z
−∞
0
dtei(ω−iǫ)t
−∞
X
p{m}
{m},{n}
i
h
†
†
× eiω{m},{n} t ψ{m},{n} (x)ψ{n},{m}
(x′ ) + ψ{m},{n}
(x′ )ψ{n},{m} (x)eiω{n},{m} t
!
†
†
X
ψ{m},{n} (x)ψ{n},{m}
(x′ ) ψ{m},{n}
(x′ )ψ{n},{m} (x)
= i
p{m}
+
ω + ω{m},{n} − iǫ
ω + ω{n},{m} − iǫ
{m},{n}
!
†
†
X
ψ{n},{m}
(x′ )ψ{m},{n} (x)
ψ{m},{n} (x)ψ{n},{m}
(x′ )
+ p{n}
= i
p{m}
ω + ω{m},{n} − iǫ
ω + ω{m},{n} − iǫ
{m},{n}
Z ∞
A(x, x′ , λ)
dλ
= i
ω − iǫ − λ
−∞
(B12)
49
Finally, the imaginary time propagator is
Z
X
1 0
†
dteiωt
p{n} e(ωm −ωn )t ψ{n},{m}
(x′ )ψ{m},{n} (x)
~GE (x, x′ , ω) =
2 −β
{m},{n}
Z β
X
1
†
−
dteiωt
p{m} e(ωm −ωn )t ψ{m},{n} (x)ψ{n},{m}
(x′ )
2 0
{m},{n}
= −
1
2
−
1
2
= −
−
=
1
2
+
=
1
2
+
=
Z
1
2
Z
Z
1
2
X
†
p{m} ψ{m},{n} (x)ψ{n},{m}
(x′ )
{m},{n}
1
2
1
2
Z
†
p{n} ψ{n},{m}
(x′ )ψ{m},{n} (x)
{m},{n}
Z
∞
dλ
−∞
dλ
−∞
−∞
X
†
p{m} δ(λ + ω{m},{n} )ψ{m},{n} (x)ψ{n},{m}
(x′ )
†
δ(λ + ω{m},{n} )ψ{n},{m}
(x′ )ψ{m},{n} (x)
{m},{n}
dλ
−∞
dλ
−∞
dλ
−∞
dλ
†
δ(λ + ω{m},{n} )ψ{m},{n} (x)ψ{n},{m}
(x′ )
X
†
δ(λ + ω{m},{n} )ψ{n},{m}
(x′ )ψ{m},{n} (x)
{m},{n}
∞
∞
X
X
e−βωm (eβω{m},{n} + 1)
iω − λ
e−βωm + e−βωn
iω − λ
†
δ(λ + ω{m},{n} )ψ{m},{n} (x)ψ{n},{m}
(x′ )
{m},{n}
eβ(iω+ω{m},{n} ) − 1
iω − λ
e−βωn (e−βω{m},{n} + 1)
iω − λ
{m},{n}
∞
−∞
†
p{n} δ(λ + ω{m},{n} )ψ{n},{m}
(x′ )ψ{m},{n} (x)
X
∞
e−β(iω+ω{m},{n} ) − 1
iω − λ
X
{m},{n}
dλ
Z
eβ(iω+ω{m},{n} ) − 1
iω + ω{m},{n}
{m},{n}
∞
∞
Z
e−β(iω+ω{m},{n} ) − 1
iω + ω{m},{n}
X
e−βωn + e−βωm
iω − λ
A(x, x′ , λ)
.
iω − λ
(B13)
The common function A(x, x′ , ω), characterizing the different propagators has a simple physical interpretation, it
gives the density of states. We have in coordinate space
X
†
A(x, x′ , ω) =
(p{m} + p{n} )ψ{m},{n} (x)ψ{n},{m}
(x′ )δ(ω + ω{m},{n} )
{m},{n}
=
X
†
†
[p{m} ψ{m},{n} (x)ψ{n},{m}
(x′ ) + p{n} ψ{n},{m}
(x′ )ψ{m},{n} (x)]δ(ω + ω{m},{n} )
(B14)
{m},{n}
where the first and the second terms in the second line represent the contributions of the positive energy, particle
excitations with ω = ωn − ωm and the negative energy, hole excitations because ω = −ωm − (−ωn ), respectively. The
density of state at point x and at energy ~ω is obviously
N (x, ω) = A(x, x, ω).
(B15)
In the case of a translation invariant system the density of state is constant in the coordinate space but its Fourier
transform yields the density of state,
Z
N (k, ω) = A(k, ω) = d3 xe−ikx A(x, 0, ω),
(B16)
in momentum space.
The expressions for the different propagators given in terms of the spectral density function A(x, x′ , λ) differ in the
50
infinitesimal imaginary part of the position of the poles on the complex frequency plan. In particular,
Z ∞
1
1
′
R
′
A
′
dλA(x, x , λ)
G (x, x , ω) − G (x, x , ω) =
−
ω + iǫ − λ ω − iǫ − λ
−∞
Z ∞
dλA(x, x′ , λ)δ(ω − λ)
= −2πi
−∞
= −2πiA(x, x′ , ω).
(B17)
Since GR (x, x′ , ω) = GA∗ (x, x′ , ω) the imaginary part of the retarded propagator,
1
A(x, x′ , ω) = A∗ (x, x′ , ω) = − ImGR (x, x′ , ω),
π
(B18)
determines the rest,
1
N (x, ω) = A(x, x, ω) = − ImGR (x, x, ω)
π
1
N (k, ω) = A(k, ω) = − ImGR (k, ω).
π
(B19)
The limit T → 0 gives
X
X
A(x, x′ , λ) →
hµ|ψ(x)|{n}ih{n}|ψ † (x′ )|µiδ(λ − ω{n} ) +
hµ|ψ † (x′ )|{m}ih{m}|ψ(x)|µiδ(λ + ω{m} )
{n}
A(k, λ) =
Z
{m}
d3 xe−ikx A(x, 0, λ) →
X
{n}
n|h{n}|a†k |µi|2 δ(λ − ωn ) +
X
{n}
n|h{n}|a−k |µi|2 δ(λ + ωn ),
(B20)
where translation invariance was assumed in deriving the second equation. The first equation gives
B(x, x′ , λ) = A(x, x′ , λ)
1 − e−βλ
→ sign(λ)A(x, x′ , λ)
1 + e−βλ
(B21)
and
G(x, x′ , ω) =
Z
∞
dλA(x, x′ , λ)
−∞
Z ∞
P
A(x, x′ , λ)
dλ
− iπsign(λ)δ(ω − λ) →
.
ω−λ
ω − λ + isign(ω)ǫ
−∞
(B22)
For non-interacting particles we have
X X
N (k, ω) =
(p{m} + p{n} )(ak ){m},{n} (a†q ){n},{m} δ(ω + ω{m},{n} )
k {m},{n}
=
XX
k m,n
=
X
m,n
=
X
m
(pm + pn )hm|a|nihn|a† |miδk,q δ(ω + ωm,n )
(pm + pn )hm|a|nihn|a† |miδ(ω + (m − n)ω(k))
(pm + pm+1 )(m + 1)δ(ω − ω(k))
(B23)
A(k, λ) = δ(λ − ω(k))
(B24)
which becomes
at vanishing temperature.
The decays due to the coupling of radiation fields or to reservoirs introduce a ’leaking’ of the excited states into the
environment which appears as a gradual decrease of the norm of the state, projected into the system’s Fock-space,
ignoring the environment. Such a ’leak’ generates a finite lint-time for the excited states. Exponential decays in time
with life-time τ = 1/γ can phenomenologically be reproduced by means of the spectral density
A(k, λ) = Zδγ (λ − ω(k)),
(B25)
51
where the Lorentzian spectral function
δγ (λ) =
γ
1
π λ2 + γ 2
(B26)
is introduced. In fact, the retarded and advanced propagators,
Z
R
R
~GA (k, ω) =
d3 xe−ikx ~GA (x, 0, ω)
Z ∞
A(k, λ)
dλ
=
ω
− λ ± iǫ
−∞
Z ∞
γ
dλ
= Z
π −∞ [(λ − ω(k))2 + γ 2 ](ω − λ ± iǫ)
Z
dλ
γ ∞
= −Z
π −∞ (λ − ω(k) + iγ)(λ − ω(k) − iγ)(λ − ω ∓ iǫ)
(
1
2iZγ −2iγ(ω(k)−iγ−ω)
=
1
−2iZγ 2iγ(ω(k)+iγ−ω)
=
Z
,
ω − ω(k) ± iγ
(B27)
give exponential decay in time,
R
i~GA (k, t) =
Z
R
dω −iωt
e
i~GA (k, ω)
2π
= Ze−i(ω(k)∓iγ)t ,
Note that the Lorentzian tends to the Dirac-delta for stable excitations, limγ→0 δγ (λ) = δ(λ).
(B28)