QUANTUM FIELD THEORY IN
CONDENSED-MATTER PHYSICS
R. E. S. Otadoy and F. A. Buot
Contents
2 INTRODUCTION .............................................................................................................................. 2
3 TIME EVOLUTION OPERATOR (S-MATRIX) FOR A SINGLE PARTICLE................................................. 3
3.1 BASIC DEFINITION AND PROPERTIES .......................................................................................................... 3
3.2 SOLUTION OF THE SCHROEDINGER EQUATION ............................................................................................. 5
4 PICTURES ........................................................................................................................................ 8
4.1 THE SCHROEDINGER PICTURE .................................................................................................................. 9
4.2 THE HEISENBERG PICTURE ...................................................................................................................... 9
4.3 THE INTERACTION PICTURE ................................................................................................................... 12
5 PROPAGATOR FOR A SINGLE PARTICLE ......................................................................................... 14
5.1 PROPERTIES OF THE PROPAGATOR .......................................................................................................... 16
5.2 PROPAGATOR AND THE GREEN’S FUNCTION ............................................................................................. 18
5.3 PROPAGATOR AS A TRANSITION AMPLITUDE ............................................................................................. 18
6 IDENTICAL PARTICLES ................................................................................................................... 19
6.1 EXCHANGE DEGENERACY ...................................................................................................................... 19
6.2 PERMUTATION OPERATORS................................................................................................................... 20
6.3 PROPERTIES OF P21 .............................................................................................................................. 21
6.4 THE N-PARTICLE SYSTEM...................................................................................................................... 21
7 SECOND QUANTIZATION ............................................................................................................... 21
7.1 CLASSICAL FIELD THEORY ...................................................................................................................... 21
7.1.1
The Lagrangian as a Functional ............................................................................................ 21
7.1.2
Variation of a Functional: The Functional Derivative............................................................ 22
7.1.3
Hamilton’s Principle (Principle of Least Action) for Fields ..................................................... 22
7.2 THE HAMILTON FORMALISM ................................................................................................................. 25
7.2.1
Poisson Bracket of Functionals ............................................................................................. 26
7.2.2
An Important Special Case .................................................................................................... 27
7.2.3
Time Evolution and Poisson Bracket of Fields ....................................................................... 27
7.3 CANONICAL QUANTIZATION .................................................................................................................. 28
7.3.1
Introduction .......................................................................................................................... 28
7.3.2
Quantization Rules for Bosons .............................................................................................. 30
7.3.3
Quantization Rules for Fermions .......................................................................................... 34
8 ZERO-TEMPERATURE GREEN’S FUNCTION .................................................................................... 36
8.1.1
Definition of Green’s Function of Many-Body System .......................................................... 36
8.1.2
Analytic Properties of the Green’s Function.......................................................................... 36
8.1.3
Retarded and Advanced Green’s Function ............................................................................ 36
8.2 QUANTUM CORRELATION FUNCTIONS IN MANY-BODY THEORY ................................................................... 36
8.3 PERTURBATION THEORY: FEYNMAN DIAGRAMS ........................................................................................ 36
9 QUANTUM SUPERFIELD THEORY .................................................................................................. 36
9.1 BASIC CONCEPTS IN QUANTUM SUPERFIELD THEORY ................................................................................. 37
9.1.1
Single-Particle ....................................................................................................................... 37
9.1.2
Many-Body System: The Thermal Liouville Space ................................................................. 40
1
9.2 QUANTUM DYNAMICS IN LIOUVILLE SPACE .............................................................................................. 47
9.2.1
Time Evolution Equations ..................................................................................................... 47
9.2.2
The Unitary Superoperator ................................................................................................... 49
9.2.3
The Time Evolution Superoperator ....................................................................................... 50
9.2.4
The Super-Heisenberg and Super-Interaction Pictures ......................................................... 51
9.2.5
Super S-Matrix Theory and Variational Principle in Liouville Space ...................................... 53
9.3 SUPER-GREEN’S FUNCTIONS ................................................................................................................. 55
9.4 QUANTUM TRANSPORT EQUATIONS ....................................................................................................... 59
9.5 ENERGY BAND DYNAMICS OF BLOCH ELECTRONS ...................................................................................... 60
9.5.1
Bloch and Wannier Functions ............................................................................................... 60
9.5.2
Lattice Weyl-Wigner Formulation of Electron Band Dynamics ............................................. 63
9.5.3
Weyl Transformation in Continuous Phase Space ................................................................ 73
9.5.4
Integral Form of the Weyl Transform of the Commutator and Anticommutator ................. 85
BIBLIOGRAPHY .................................................................................................................................... 86
10
REFERENCES ............................................................................................................................. 89
1 Introduction
Equation Chapter 1 Section 0
See Chapter 1 of McMahon (McMahon, 2008) (introductory
comments on quantum field theory as a theoretical framework
combining quantum mechanics and the special theory of relativity):
Quantum field theory is a theoretical framework that combines quantum
mechanics and Einstein’s special theory of relatativity (McMahon, 2008). The
key ideas in quantum theory are the operators representing measurable
quantities known also as observables, the uncertainty principle, and the
commutation relations of operators. Measurable quantities in classical
mechanics become operators in quantum mechanics. The transition from
measurable quantities to operators is called quantization. For instance, the
energy equation
H  T U
(1.0.1)
becomes the Schroedinger’s equation in
i
2
Y

 2Y U Y
t
2m
in quantum mechanics where, the following transformations are used:

H i
t
2 2

T 
2m
U U
2
(1.0.2)
In the special theory of relativity, the mass-energy relation
E  mc 2
(1.0.3)
and its variant
(1.0.4)
E 2  p2c2  mc2
implies conversion of mass to energy and vice versa, that is, matter can now be
created or destroyed. The number of particles therefore is no longer fixed.
The first attempt to construct a relativistic quantum equation based on Eq.
(1.0.4), which results to
m02c 2
1 
2
(1.0.5)




2
c 2 t 2
Continue reading page 3 of McMahon (McMahon, 2008).
2 Time Evolution Operator (S-Matrix) for a Single Particle
Equation Chapter 2 Section 1
2.1 Basic Definition and Properties
The nature of time is one of the greatest ironies in physics. In classical
mechanics, time is considered as a parameter whereas in the special theory of
relativity, time has been elevated as a dynamical variable. In quantum
mechanics, time is still considered as a parameter, so it makes no sense to talk
about its eigenvalues and eigenstates. With its elevation as a dynamical
variable in the special theory of relativity, it is expected that in quantum field
theory time would emerge as an observable. However, the concept of time
operator is still untenable even in quantum field theory.
To take into account the time evolution of a state, we write a time-dependent
state ket as
  t , t0 
(2.1.1)
This physically means that the state y evolves in time from t0 to t . At t 0 we
would have
y t t0 ,t0   y t0 ,t0   y t0 
(2.1.2)
As a matter of notation, when t0  0 we write
and
  t  t0  0,t0  0     t0  0,t0  0     t0  0   
(2.1.3)
  t , t0  0     t 
(2.1.4)
3
Another way of describing the time evolution of states is to introduce atime
evolution operator otherwise known as the S-matrix U  t ,t0  and define its action
on state kets as
  t , t0   U  t , t0    t0 
(2.1.5)
The properties of this operator can be obtained from physical considerations:
1. Unitarity
Recall that in order for the probabilistic interpretation of quantum mechanics to
hold ground, we require that states kets are normalized, that is,
  1
(2.1.6)
For quantum theory to be consistent we should require this to hold true
regardless of time,
  t , t0    t , t0     t0    t0      1
(2.1.7)
Since the corresponding bra of Eq. (2.1.5) is
  t , t0     t0  U †  t , t0 
(2.1.8)
Eq. (2.1.7) implies that,
  t , t0    t , t0     t0  U †  t , t0  U  t , t 0    t 0   1
U†  t ,t0  U  t ,t0   I
(2.1.9)
where I is the identity operator. Eq. (2.1.9) can only be true if
U†  t ,t0   U1  t ,t0 
(2.1.10)
that is, the adjoint of the time evolution operator is equal to its inverse. An
operator whose adjoint is equal to its inverse is called unitary. It also follows
that
U  t0 ,t0   U  t ,t = I
(2.1.11)
which means that when the arguments are equal, the state does not evolve.
2. Composition
The composition property stems from the evolution of the state from t 0 to t1 then
from t1 to t . This can be accomplished by successive application of the time
evolution operator as follows,
U  t ,t1  U  t1 ,t0    t0 
(2.1.12)
The evolution of the state from t 0 to t1 then from t1 to t is equivalent to its
evolution from t 0 to t represented by Eq. (2.1.5). Thus one can have
and therefore,
U  t ,t1  U  t1 ,t0    t0   U  t ,t0    t0 
(2.1.13)
U  t ,t1  U  t1 ,t0   U  t ,t0 
(2.1.14)
4
It follows that when we slice time N times
U  t ,tN  U  tN ,t0  U tN 1 ,t0  U tN 1 ,t0  U t1 ,t0  U t1 ,t0   U t ,t0 
(2.1.15)
Eqs. (2.1.14) and (2.1.15) are called composition property.
It is to be noted also that from Eq. (2.1.11) and from the composition property,
U  t, t   U  t, t0  U t0 , t   I
which shows that,
U†  t, t0 = U1  t, t0   U t0 , t 
(2.1.16)
3. Time Evolution
The state ket evolves in time according to the Schroedinger equation which in a
more familiar form is given by,
  t
(2.1.17)
i
 H  t
t
According to our notation this is written as
   t , t0 
(2.1.18)
i
 H   t , t0 
t
Substituting Eq. (2.1.5),
U  t ,t0    t0 
i
 HU  t ,t0    t0 
t
 U  t ,t0  
i
   t0   HU  t ,t0    t0 

t


Thus, the time evolution operator also obeys the Schroedinger equation,
U  t ,t0 
(2.1.19)
i
 HU  t ,t0 
t
2.2 Solution of the Schroedinger Equation
Case 1. The Hamiltonian is independent of time
The solution to Eq. (2.1.19)can be obtained by treating U  t ,t0  as ordinary
variable1, after which its operator nature would be restored and the resulting
1
In treating a function of operator, the operator is first considered as an ordinary variable. The
function of operator is then treated as an ordinary function of the variable in which a power
series expansion is well defined. The operator character of the dynamical variable or observable
is then returned and the action of the function of the operator to a state ket is determined by the
series expansion. For reference see any book in quantum mechanics, e.g. (Cohen-Tannoudji et
al. 1970, Sakurai 1994).
5
solution would be interpreted based on the series expansion of the operator.
For a time-independent Hamiltonian the solution is
iH t t
(2.1.20)
U  t , t0   e  0 
on the understanding that

e
iH tt0 
2
3
 1 i

i
1 i
 I  H t t0     H t t0      H t t0   
2! 
 3! 

(2.1.21)
Case 2. The Hamiltonian is time-dependent but the Hamiltonian at different
times commute
Taking U  t ,t0  in Eq. (2.1.19) as an ordinary function,
i
U  t ,t0 
d U  t ,t0 
i
 HU  t , t 0  
  Hdt
t
U  t ,t0 
d U  t , t0 
i

U t0 ,t0  U t , t
 0
U  t ,t0 

 H  t  dt 
ln U  t ,t0   ln U  t0 ,t0   
From Eq. (2.1.11),
t
t0
i
 H t  dt 
t
(2.1.22)
t0
ln U  t0 ,t0   ln1  0
Thus, Eq. (2.1.22) becomes
ln U  t ,t0   
i
 H  t  dt   U  t ,t   e
t

i
0
t0
t
t0 Ht dt 
(2.1.23)
Case 3. The Hamiltonian is time-dependent and the Hamiltonian at different
times do not commute.
In this case, Eq. (2.1.19) cannot be solved as in Case 2 since the differential
equation cannot be separated. We can, however, solve this equation as follows:
U  t ,t0 
U  t ,t0 
i t
i
 HU  t ,t0   
d U  t ,t0     H  t1  U  t1 ,t0  dt1
U  t0 ,t0 
t0
t
i t
U  t ,t0   U  t0 ,t0     H  t1  U  t1 ,t0  dt1
t0
I
U  t , t0   I 
i
 H  t  U  t ,t  dt
t
1
t0
1
0
1
(2.1.24)
As can be seen in the above equation, the time evolution operator cannot be
completely solved in closed form. However, we can arrive at an approximate
solution by iteration. Following Eq. (2.1.24), U  t1 ,t0  can be written as
U  t1 ,t0   I 
i

t1
t0
H  t2  U  t2 ,t0  dt2
Substituting Eq. (2.1.25) to Eq. (2.1.24) one obtains,
6
(2.1.25)
U  t , t0   I 
i
i

t0 H  t1   I 
t
 H  t  U  t ,t  dt
t1
2
t0
2
0
2

 dt1
2
t1
 i t
 i t
U  t ,t0   I      H  t1  dt1      dt1  H  t1  H  t2  U  t2 ,t0  dt2 (2.1.26)
t0
t0
t0




One may note that U  t2 ,t0  appears in the second integral of Eq. (2.1.26). An
expression for this can be derived by the replacement t1  t2 and t2  t3 in
Eq.(2.1.25). When the resulting expression is substituted to Eq. (2.1.26), one
obtains
 i t
U  t ,t0   I      H  t1  dt1

 t0
 i
  


2
t1
i

dt1  H  t1  H  t2   I 
t0
t0


t

t2
t0

H  t3  U  t3 ,t0  dt3  dt2

Simplifying
 i t
U  t ,t0   I      H  t1  dt1

 t0
 i
  


2

t
t0
dt1  H  t1  H  t2  dt2
t1
t0
3
t1
t2
 i t
     dt1  dt2  H  t1  H  t2  H  t3  U  t3 ,t0  dt3
t0
t0

 t0
If the process is iterated, we arrive at the so-called Dyson series,
(2.1.27)
n

t1
tn1
 i t
U  t ,t0   I       dt1  dt2  dtnH  t1  H  t2  H  t3  H  tn  (2.1.28)
t0
t0
 t0
n 1 
At this juncture it is convenient to introduce the time-ordering operator T useful
in dealing with product of operators that do not commute at different times like
the integrand in Eq.(2.1.28). It is just an instruction to arrange the product so
that the right-most operator corresponds to the earliest time. For instance, if
t2  t1 ,
T H  t1  H  t2    H  t2  H  t1 
To include different possibilities, we introduce the Heaviside unit-step
defined by
1 t1  t2
  t1  t2   
0 t2  t1
and that
0 t1  t2
   t1  t2     t2  t1   
1 t2  t1
The action of T is then expressed as
T H t1  H t2     t1  t2  H  t1  H  t2     t2  t1  H  t2  H  t1 
for t1 t2
for t1 t2
7
function
(2.1.29)
(2.1.30)
(2.1.31)
The step functions assure that the two possibilities do not occur at the same
time. Now consider the integral  dt1  dt2 T H  t1  H  t2   :

t
t0
dt1  dt2 T H  t1  H  t2   
t0

t


t


t

t
t

t0
t0
t0
t0
t
t
t0
t0
dt1  dt2   t1  t2  H  t1  H  t2     t2  t1  H  t2  H  t1  
t0
t
dt1  dt2  t1  t2  H  t1  H  t2    dt1  dt2  t2  t1  H  t2  H  t1 
t
t
t
t0
t0
t0
dt1  dt2  t1  t2  H  t1  H  t2    dt2  dt1  t2  t1  H  t2  H  t1 
t
t0
t
t
t0
t0
dt1  dt2H  t1  H  t2    dt2  dt1H  t2  H  t1 
t1
t
t2
t0
t0
t0
Interchanging the indices 1 and 2 in the second integral shows that the two
integrals are actually the same. Thus,
t0 dt1 t0 dt2T H t1 Ht2   2t0 dt1 t0 dt2H t1 H t2 
t
t
t
t1
(2.1.32)
If we follow the same steps, one can write Eq.(2.1.28) as
n

t
t
1 i t
U  t ,t0   I       dt1  dt2  dtn T H  t1  H  t2  H  t3  H  tn   (2.1.33)
t0
t0
 t0
n 1 n ! 
This can further be written symbolically as
 i t

U  t ,t0   T exp    dt H  t   
(2.1.34)
t0


3
Pictures
Equation Chapter 2 Section 2
The expectation value of an operator (average value of a dynamical variable) is
given by
A   A
(2.2.1)
The time-dependent expectation value can also be written as
A  t     t ,t0  A   t ,t0     t0  U †  t ,t0  AU  t ,t0    t0 
(2.2.2)
As can be seen in Eq. (2.2.2), one can take note two schemes of describing the
dynamics of the expectation value:
Scheme 1

 
A  t     t0  U †  t , t0  A U  t , t 0    t0 
Scheme 2
A  t     t0  U †  t ,t0  A U  t ,t0    t0 
8

(2.2.3)
(2.2.4)
3.1 The Schroedinger Picture
Scheme 1 is the more famaliar picture of quantum mechanics in which state
kets are time-dependentwhile operators associated with dynamical variables
are time-independent,
  t , t0   U  t , t0    t0 
(2.2.5)
It is actually the picture we are actually following so far. It is called the
Schroedinger picture. It is also well known that even though state kets are timedependent, eigenkets (which constitute a basis) remain time-independent,
A un  an un
(2.2.6)
If the state ket is initially given by
  t0    un un   t0    cn un
n
it evolves as
n
cn
  t ,t0   U  t ,t0    t0   U  t ,t0   cn un
(2.2.7)
n
For time-independent Hamiltonian and if un is its eigenket, we have
  t ,t0   eiHt t0 
c
n
n
un   cn e
 iH t t0 
un
n
  t ,t0    cn eiE t t  un
n
n
(2.2.8)
0
cn  t 
Thus, the time evolution operator introduces a phase change of the coefficients.
3.2 The Heisenberg Picture
The Heisenberg picture adopts Scheme 2 of describing quantum dynamics in
which state kets are time-independent and operators are time-dependent,
according to Eq.(2.2.4),
A H  t   U†  t ,t0  AS U  t ,t0 
(2.2.9)
where the subscript H indicates that the operator is in the Heisenberg picture
while the subscript S indicates that the operator is in the Schroedinger picture.
To simplify things, we take t0  0 and assume that the Hamiltonian is timeindependent. Thus one can have,
 iHt 
U  t ,t0  0   exp 
(2.2.10)



Let us give a special designation to Eq. (2.2.10),
 iHt 
U  t   exp 
(2.2.11)



In this notation, Eqs.(2.2.5) and(2.2.9) become, respectively,
  t  S  U  t    0  S  eiHt   0 
S
which describes the time evolution of Schroedinger picture state ket and
9
(2.2.12)
A H  t   U †  t  AS U t   eiHt AS eiHt
which describes the time evolution of Heisenberg picture operator.
(2.2.13)
At t  0 the Heisenberg and the Schroedinger pictures coincide,

H
   0
S

S
A H  0  AS
such that Eq. (2.2.13) can also be written as
A H  t   U †  t  A H  0 U  t   eiHt A H  0 eiHt
(2.2.14)
(2.2.15)
(2.2.16)
In general, for t  t0 , the Schroedinger and the Heisenberg pictures coincide,

H
   t0 
S
A H  t0   U†  t0 ,t0  A S U  t0 ,t0   A S
and that Eq.(2.2.9) can also be written as
A H  t   U†  t ,t0  A H  t0  U  t ,t0 
(2.2.17)
(2.2.18)
(2.2.19)
The Heisenberg Equation of Motion
The Heisenberg-picture operators evolve in time according to the equation,
d AH 1
1
  A H , U †HU    A H ,H
(2.2.20)
dt
i
i
This is called the Heisenberg equation of motion.
PROOF OF HEISENBERG EQUATION OF MOTION
Let us use Eq. (2.2.9) in evaluating the derivative of Heisenberg-picture
operators:
d AH d
  U †  t , t0  A S U  t ,t0  
(2.2.21)
dt
dt
†
d U  t , t0 
d A H d U  t , t0 

A S U  t , t0   U †  t , t0  A S
dt
dt
dt
From Eq. (2.1.19)
d U  t , t0  1
(2.2.22)
 HU  t , t0 
dt
i
d U †  t , t0 
1
  U †  t , t0  H
(2.2.23)
dt
i
Substituting to Eq.(2.2.21)
d AH
1
1

  U †  t , t0  HA S U  t ,t0   U †  t ,t0  A S  HU  t ,t0  
dt
i
i

d AH 1
   U †  t ,t0  HA S U  t ,t0   U †  t ,t0  A S HU  t ,t0  
dt
i
10
d AH 1
  U †  t ,t0  A S HU  t ,t0   U †  t ,t0  HA S U  t ,t0  
dt
i
d AH 1
  U †  t ,t0  A S U  t ,t0  U †  t ,t0  HU  t ,t0   U †  t ,t0  HU  t ,t0  U †  t ,t0  A S U  t ,t0  
dt
i
d AH 1
  A H  t  U †  t ,t0  HU  t ,t0   U †  t ,t0  HU  t ,t0  A H  t  
dt
i
d AH 1
  A H  t  , U †  t , t0  HU  t , t0  
(2.2.24)
dt
i
Base Kets in the Heisenberg Picture
The eigenvalue equation for the Schroedinger-picture observable is given by Eq.
(2.2.6) and it can be considered as a Heisenberg-picture observable at t  0 ,
A H  0 un  an un
(2.2.25)
For finite time t ,
U †  t  A H  0 un  an U †  t  un
U † A H  0 UU † un  an U † un
U † A H  0  U U † un  an U † un
A H t 
un  t 
A H  t  un  t 
H
H
un  t 
 an un  t 
H
(2.2.26)
H
Whereas the state ket is time-independent in the Heisenberg picture, the basis
kets change with time as
(2.2.27)
un  t  H  U †  t  un S
In general, one can have the result for the basis kets
un  t , t0 
H
 U †  t ,t0  un
(2.2.28)
S
Transition Amplitudes
Another quantity which is important in our future studies of many-particle
system is the transition amplitude, which is the probability amplitude 2 for a
particle initially prepared in the state   t ,t0  to be found in the eigenstate un
of observable A . It is given in the Schroedinger picture by
T  S un   t , t0  S  S u n U  t , t 0    t 0  S
(2.2.29)
For t0  0
T
S
un   t 
In the Heisenberg picture,
T  H un  t  
H
S
 S un U  t    0 
S
un  0  U  t  
H

H
 S un U  t  

H
un U  t  
(2.2.30)
S
H
(2.2.31)
Thus, the expressions for the transitition amplitude are the same in both
Schroedinger and Heisenberg pictures.If  is an eigenstate v j of B ,
2
The probability is the square of the probability amplitude.
11
Tnj 
H
un  t  v j
H

H
un  0 U  t  v j
H

H
un U  t  v j
H
(2.2.32)
3.3 The Interaction Picture
The interaction picture is most useful when the Hamiltonian can be split into,
H  H0  W
(2.2.33)
where H0 is the Hamiltonian whose eigenkets and eigenvalues are already
known while W is the remaining part of the total Hamiltonian whose eigenkets
and eigenvalues are unknown. The state ket in the interaction picture is defined
by
(2.2.34)
  t ,t0  I  eiH t   t ,t0  S
0
The subscript I indicates that the quantity is in the interaction picture. Obviously,
at t  0 the two pictures coincide. For operators representing observables,
transformation from the Schroedinger picture to interaction picture is defined by
A I  eiH0t A S e iH0t
(2.2.35)
The unknown part of the total Hamiltonian can therefore be written in the
interaction picture as
WI  eiH0t WS eiH0t
(2.2.36)
The evolution of the state ket is given by
   t , t0  I
i
 WI   t ,t0  I
(2.2.37)
t
which shows that the state ket evolves as,
 i t

  t ,t0  I  Teexp    WI  t   dt     t0  I
(2.2.38)
t0


PROOF OF THE TIME EVOLUTION EQUATION FOR THE STATE KET IN
THE INTERACTION PICTURE
   t ,t0  I  iH t
 e 0   t ,t0  S 
t
t
   t , t0 
I

iH0
eiH0 t   t ,t0  S  eiH0t
   t , t0 
S
t
t
In the above equation, we substitute the Schroedinger equation for the state ket
in the Schroedinger picture,
   t ,t0  I iH0 iH t
1

e 0   t ,t0  S  eiH0t
H0  W    t ,t0  S
t
i
   t , t0  I
i
 H0eiH0 t   t ,t0  S  eiH0t  H0  W    t ,t0  S
t
12
i
i
   t , t0 
t
   t , t0 
t
i
 eiH0 t W   t ,t0 
I
I
S
 eiH0 t WeiH0 t eiH0 t   t ,t0 
   t , t0 
 WI   t ,t0 
I
t
S
I
Likewise the time evolution of the operator is given by
d AI
i
  A I , H0 
dt
(2.2.39)
PROOF OF THE TIME EVOLUTION EQUATION OF AN OPERATOR
d A I d iH0 t

e
A S e iH0 t
dt
dt

  iH
0
 iH 
eiH0t A S e iH0t  eiH0t A S   0  e iH0t


d A I iH0 iH0 t
iH0

e
A S e iH0 t  eiH0 t A S e iH0 t
dt
d A I iH0
iH

AI  AI 0
dt
d AI i
i
  H0 A I  A I H0   H0 , A I 
dt
d AI
i
  A I , H0 
dt
which is Eq.(2.2.39).
One can see that in this picture, both the state ket and the operator are timedependent contrary to the Schroedinger and Heisenberg pictures.
The real value of the interaction picture lies in its use of its corresponding timeevolution operator. We define it as
(2.2.40)
  t ,t0  I  UI  t ,t0    t0  I
and from Eq.(2.2.37) it satisfies the equation
d UI  t ,t0 
(2.2.41)
i
 WI UI  t ,t0 
dt
Following the same steps as in the derivation of Eq.(2.1.28), the solution of
Eq.(2.2.41) can be written as

 i
UI  t ,t0   I     

n 1 
n

t
t0
t1
dt1  dt2
t0
13

tn1
t0
dtn W  t1  W  t2  W  t3 
W  tn  (2.2.42)
n

t
1 i t
UI  t ,t0   I       dt1  dt2
t0
 t0
n 1 n ! 
 i t

 T exp    dt1W  t1  
t0



t
t0
dtn T  W  t1  W  t2  W  t3 
W  tn  
(2.2.43)
Let us explore the connection between the time evolution operator in the
Schroedinger picture to that of the interaction picture. To this end we can make
use of Eq.(2.2.34)
  t ,t0  I  eiH t   t ,t0  S
0
  t ,t0  I  eiH t U  t ,t0    t0 
(2.2.44)
0
S
On the right-hand side we can again invoke Eq.(2.2.34) to write   t0 
  t0  I  eiH t
  t0 
0 0
S
   t0 
S
Substituting Eq.(2.2.45) to Eq.(2.2.44)
  t ,t0  I  eiH t U  t ,t0  e iH t
0
0 0
 e iH0t0   t0 
  t0 
I
I
which upon comparison with Eq.(2.2.40), one can have
UI  t ,t0   eiH0t U  t ,t0  eiH0t0
S
as
(2.2.45)
(2.2.46)
(2.2.47)
Lastly, let us find the connection between the interaction and the Heisenberg
representations. From Eq. (2.2.9) one can solve for the Schroedinger-picture
operator:
A S  U  t, t0  A H  t  U†  t, t0 
(2.2.48)
Substituting this to Eq. (2.2.35), one obtains
A I  eiH0t U  t, t0  A H  t  U†  t, t0  eiH0t
(2.2.49)
4 Propagator for a Single Particle
Equation Chapter 4 Section 1
Let us consider the case where the operator A commutes 3 with a timeindependent Hamiltonian,
(4.1.1)
 A,H  0
which implies that un are also energy eigenkets, that is,
With
 u  as
n
H un  En un
basis, ( I   un un is the completeness relation), the time
n
evolution operator can then be written as
3
(4.1.2)
When an operator commutes with another operator, they are said to be compatible.
14
U  t , t0   e
 iH t t0 
e
 iH t t0 
u
n
n
un   un e
 iEn  t t0 
un
(4.1.3)
n
The Schroedinger picture state ket can also be written as,


  t ,t0   U  t ,t0    t0     un eiEn t t0  un    t0 
 n

  t ,t0    un un   t0  e iEn t t0 
(4.1.4)
n
In the position representation,
 iE t t
r    t ,t0    r  un un   t0  e n  0 
n
  r ,t    un  r   un   t0  e iEn t t0 
(4.1.5)
n
Inserting the completeness relation for position basis,
above equation
  r,t   un  r  un
n
 d r r r   t  e
3
d r r
3
r  I , in the
 iEn  t t0 
0
  r ,t    un  r    d 3r  un r  r    t0  e iEn t t0 
n
  r ,t    d 3r  un  r   un  r   e iEn t t0    r ,t0 
(4.1.6)
n
Eq. (4.1.6) shows that a wavefunction   r ,t  can be obtained from an initial
wavefunction   r ,t0  by the application of an integral operator the kernel of
which is called propagator and is denoted by K  r,t ; r,t0  . Thus,
K  r ,t ; r ,t0    un  r   un  r   e
 iEn  t t0 
n
  r  un un r  e
 iEn  t t0 
(4.1.7)
n
  r ,t    d 3r K  r ,t ; r ,t0   r ,t0 
Eq.(4.1.7) can further be written as,



 iE t t
 iH t t
K  r ,t ; r ,t0   r    un un e n  0   r   r    un e  0 
 n

 n
K  r,t ; r,t0   r U t ,t0  r
(4.1.8)

un  r  (4.1.9)

(4.1.10)
Eq.(4.1.10) can also be arrived at in general by noting that
  t ,t0   U  t ,t0    t0   r    t ,t0   r  U  t ,t0    t0 
r    t ,t0   r  U  t ,t0   d 3r  r  r    t0 
r    t ,t0    d 3r  r  U  t ,t0  r  r    t0 
  r ,t    d 3r  r  U  t ,t0  r    r ,t0 
from which we can obtain Eq.(4.1.10).
15
(4.1.11)
We can now generalize the equations not only to an initial time t 0 and final time
t but also to any times t and t . In the above equations we implement the
replacement t  t and t0  t  .
  r ,t     d 3r K  r ,t ; r ,t    r ,t  
(4.1.12)
K  r,t ; r,t   r U t ,t  r
(4.1.13)
Eq. (4.1.12) is the analogue of Huygen’s principle in optics. It shows that a
wavefunction at position-time coordinates  r ,t   can be completely determined
from position-time coordinates  r,t  provided the propagator is known. It also
shows that the wavefunction at  r ,t   is a superposition of the wavefunctions
for all position-time coordinates with the propagator taking the role of the
coefficients.
4.1 Properties of the Propagator
1. Causality
To take into account causlity (the cause precedes the effect) we require that
0
t   t 

(4.1.14)
K  r ,t ; r ,t    
 K  r ,t ; r ,t   t   t 
2. Composition
The composition property for the propagator is expressed as
K  r ,t ; r ,t     d 3rK  r ,t ; r ,t  K  r ,t ; r ,t  
(4.1.15)
PROOF OF COMPOSITION PROPERTY
This follows from the composition property of the time-evolution operator
(2.1.14)
K  r,t ; r,t  r U  t,t  r  r U t,t  U t ,t  r
Using the completeness relation for the position basis,
K  r ,t ; r ,t     d 3r r  U  t ,t  r r U  t ,t   r 
K  r ,t ;r ,t 
K  r ,t ;r ,t  
K  r ,t ; r ,t     d 3rK  r ,t ; r ,t  K  r ,t ; r ,t  
which is Eq.(4.1.15). The composition property can also be derived by iterating
Eq.(4.1.12) by writing
(4.1.16)
  r ,t     d 3rK  r ,t ; r ,t   r ,t 
and substituting back to Eq. (4.1.12),
  r ,t     d 3r K  r ,t ; r ,t    d 3rK  r ,t ; r ,t   r ,t 
16
  r ,t    d 3r  d 3r K  r ,t ; r ,t  K  r ,t ; r ,t   r ,t 
K  r ,t ;r ,t 
K  r ,t ; r ,t    d r K  r ,t ; r ,t   K  r ,t ; r ,t 
3
(4.1.17)
which is the same as Eq.(4.1.15).
3. Another property of the propagator can be obtained by noting that when
t   t  in Eq.(4.1.13),
(4.1.18)
lim K  r ,t ; r ,t    r  U  t ,t   r   r  r     r   r  
t t 
I
4. For t  t , the propagator obeys the Schroedinger equation,
2
K  r ,t ; r ,t  
i

 2r  K  r ,t ; r ,t    V  r   K  r ,t ; r ,t  
t 
2m
(4.1.19)
PROOF OF THE SCHROEDINGER EQUATION FOR THE PROPAGATOR
The proof follows from the Schroedinger equation for the time-evolution
operator, Eq.(2.1.19),
 r  U  t ,t   r 
 U  t ,t0 

r  i
 HU  t ,t0   r   i
 r  HU  t ,t   r 
t
t 


 r  U  t ,t   r 
i
 r  H d 3r r r U  t ,t   r 
t 
 r  U  t ,t   r 
i
  d 3r r  H r r U  t ,t   r 
t 
 r  U  t ,t   r 
 p2

i
  d 3r r  
 V  r   r r U  t ,t   r 
t 
 2m

2
K  r ,t ; r ,t  
p
i
  d 3 r r 
r r U  t ,t   r    d 3r r  V  r  r r U  t ,t   r 
t 
2m
K  r ,t  ;r  ,t  


2
K  r ,t ; r ,t  
3 
2

i
 d r 
 r  r r  r U  t ,t   r    d 3rV  r  r  r K  r ,t ; r ,t  
 2m

t 
  r  r  
  r  r 

i
K  r ,t ; r ,t  

t 
2

3 
2
3

d
r



r

r




 r U  t ,t   r    d rV  r   r   r  K  r ,t ; r ,t  
r
  2m
 K  r ,t ;r ,t 
i
2
K  r ,t ; r ,t  

 2r  K  r ,t ; r ,t    V  r   K  r ,t ; r ,t  
t 
2m
17
which is Eq.(4.1.19).
4.2 Propagator and the Green’s Function
The requirement for causality expressed by Eq.(4.1.14) can be collectively
written as
K  r,t ; r,t     t   t  K  r,t ; r,t 
(4.1.20)
where   t   t   is the Heaviside unit-step function defined by
1 t   t 
  t   t    
0 t   t 
(4.1.21)
It is to be noted that
  t   t  
(4.1.22)
   t   t  
t 
The causality requirement, together with Property 3, enables us to write the
Schroedinger’s equation for the propagator as
   2 2

 
 r   V  r    K  r ,t ; r ,t    i   t   t    r   r   (4.1.23)
i

 t   2m
Thus, apart from the factor i , the propagator is the Green’s function of the
Schroedinger equation for the wavefunction and provides another mathematical
justification for Eq.(4.1.12)4.
4.3 Propagator as a Transition Amplitude
Let us go back to Eq.(4.1.7)
K  r ,t ; r ,t     r  un un r  e iEn t
 t  
n
K  r ,t ; r ,t     r  eiHt  un un eiHt  r 
n
If we consider the position eigenkets as the basis, we have
eiHt r  r,t 
(4.1.24)
H
and
r eiHt 
H
and thus,
K  r ,t ; r ,t     H r ,t  un un r ,t 
n
r,t 
H

H
(4.1.25)
r ,t   un un r ,t 
K  r,t ; r,t   H r,t  r,t 
4
I
H
Recall that the Green’s function of a differential equation
d 
f   G  x, x0     x  x0  and its solution is y  x    dx0G  x, x0  y  x0  .
 dx 
18
H
n
(4.1.26)
d 
f   y  x   0 is given by
 dx 
In this form, the propagator can be interpreted as the probability amplitude in
going from one position-time coordinates to another position-time coordinates.
The result (4.1.26) can actually be directly obtained from Eq.(4.1.13),
K  r,t ; r,t   r U t ,t  r   K  r ,t ;r ,t   r  U t ,t0  U t0 ,t   r 
K  r ,t ; r ,t    r  U  t ,t0  U†  t ,t0  r 
H
K  r,t ; r,t  
r ,t 
H
r ,t 
r,t  r,t 
H
H
(4.1.27)
5 Identical Particles
In the previous sections we were dealing with a single particle which is not the
domain of many-body physics and is a clear overkill with more difficult concepts
of S-matrix and propagators. We could have used the quantum mechanics we
know in an elementary quantum mechanics course and not bother about these
concepts. The previous sections however lay the foundation as the same
concepts will be used in developing a many-body theory, which we start in this
section by studying a system composed of the same species 5 of particles
known as identical particles.
In classical mechanics, identical particles are distinguishable in the sense that
we can assign a trajectory to each particle. Since the concept of trajectory
breaks down in quantum mechanics, there is no way that we can label the
individual particles. In this sense, identical quantum particles are
indistinguishable.
5.1 Exchange Degeneracy
Equation Chapter 5 Section 1
Let us first consider a system of three nonidentical particles. With each of the
three particles taken separately we can associate a state space and
observables acting in that space. If we number the particles 1,2, and 3 the state
spaces are respectively, E 1 , E  2 , & E  3 . The state spaces of the threeparticle system is given by the tensor product,
E  E 1  E  2  E  3
(5.1.1)
Now consider an observable B 1 defined in state space E 1 . If we consider
the three particles as identical, then correspondingly there are observables B  2 
and B  3 in state spaces E  2 and E  3 respectively (identical particles have the
same intrinsic properties). Furthermore, these particles have the same set of
5
The particles have the same properties.
19
eigenvalues. If the basis in the state spaces E 1 , E  2 , & E  3 are respectively,
 b 1  ,  b  2  , and  b 3  , the basis in the state space E
i
j
k
 b 1  b  2  b 3    b 1 ;b  2 ;b 3 
i
j
k
i
j
k
is
(5.1.2)
where i, j , k  0,1, 2,3,
are the associated individual or collective quantum
numbers. The vectors bi 1 ;b j  2  ;bk  3 are the eigenvectors of the extension of
B 1 , B  2  and B  3 in the state space E with the respective eigenvalues bi , b j ,
and bk . The fact that the particles are identical, the individual observables B 1 ,
B  2  and B  3 cannot be measured since the numbering has no physical
significance. However, we can measure the observable B of the system, that is,
we cannot distinguish that the result of the measurement belongs to particle 1,
or 2, or 3. Now suppose that a measurement on the system results on three
different values bm ,bn , and b p . Since the particles are identical the three
eigenvalues belong to any of the following eigenkets:
 bm 1 ;bn  2  ;bp  3
bp 1 ;bm  2  ;bn  3
bn 1 ;bp  2  ;bm  3 


(5.1.3)
 bm 1 ;bp  2  ;bn  3

b
1
;
b
2
;
b
3
b
1
;
b
2
;
b
3












p
n
m
n
m
p


Thus, one set of eigenvalues correspond to more than one set of eigenkets,
that is the states are degenerate. This sort of thing is called exchange
degeneracy.
5.2 Permutation Operators
Before we continue, let us first study the tool indispensable in the discussion of
identical particles. To facilitate discussion consider first a system of two
particles. Using the same convention as in the last section, the state space of
the system can be considered as the tensor product of E 1 and E  2 , that is,


E  E 1  E  2
If the basis in E 1 is um 1 and that in E  2 is
(5.1.4)
 u  2  , the basis in E is
n
um 1  un  2   um 1 ;un  2   un  2  ;um 1
It has to be noted that
um 1 ;un  2   un 1 ;um  2 
mn
(5.1.5)
(5.1.6)
The exchange of labels can be effected by the permutation operator P21 defined
as a linear operator whose action on the basis ket is:
(5.1.7)
P21 um 1 ;un  2   um  2  ;un 1  un 1 ;um  2 
Its action on an arbitrary ket in the state space E can be obtained by expanding
the ket in the basis (5.1.5).
20
5.3 Properties of P21
1. P12 is the inverse of itself
PROOF
This section is taken from my lecture notes on Identical Particles. To continue,
read my lecture notes.
5.4 The N-Particle System
See Section 1.1 of Balisot and Ripka (Balizot & Ripka). Also see
Coleman (Coleman, 2013).
6 Second Quantization
6.1 Classical Field Theory
Equation Chapter 6 Section 1
6.1.1 The Lagrangian as a Functional
See Page 32 of Greiner et al. (Greiner & Reinhardt, 1996).
In particle mechanics the Lagrangian is a function of the generalized
coordinates q and generalized velocity q . In field theory, the Lagrangian is a
function of the field   r , t  and its time derivative   r , t  . A field is a function of
the coordinates. So, the Lagrangian is a function of a function. In short, it is a
mapping or correspondence between a function and real numbers (in general a
complex number), a quantity called functional. We can think of a functional as a
machine whose input is a function and whose output is a real number.
More formally, a functional is a mapping of a normed linear space of functions
(this linear space of functions is a Banach space 6 ) to the field 7 of real or
complex numbers. This statement can written symbolically as
(6.1.1)
F : M  or
6
A Banach space is a linear vector space endowed with a norm [see Section 1.2 of Tarasov
(Tarasov, 2008)].
7 Field in this context is different from the field  You can think of field here as a set. Thus a
field of complex numbers simply means a set of complex numbers.
21
where M    x, y, z  : x, y, z 
 , that is the linear space of functions.
The Lagrangian in field theory belongs to this class of quantities, that is, the
Lagrangian in field theory is a functional,
L t  L  r, t ,  r, t 
(6.1.2)
The functional depends on the values of the field  and its time derivative at
each point in space with the coordinates  x, y, z acting as continuous index. It
does not depend explicitly on the coordinates.
6.1.2 Variation of a Functional: The Functional Derivative
See Section 2.3 of Greiner et al. (Greiner & Reinhardt, 1996).
To facilitate discussion let us consider a function of a single variable x :   x  .
There must no problem extending this to three dimensions except that we have
to extend our imagination a bit. Suppose that a variation of the field  is
performed as shown in Figure 6.1.
The
corresponding
functional
Figure 6.1 Variation of the field
Reinhardt, 1996)].
is
 F    F      F    

[taken from (Greiner &
F
  xdx

variation of the
defined by
(6.1.3)
Eq. (6.1.3) implies that the variation of F   due to the variation of  is a
summation of the local changes of F   over the whole range of x . The
quantity  F  is called the functional or Frechet derivative. The functional
derivatives obey the same rules as ordinary derivatives.
6.1.3 Hamilton’s Principle (Principle of Least Action) for Fields
The variation of the Lagrangian can be evaluated using Eq. (6.1.3),
L
L 
 L  t    d 3 r     
 
 
The variation of the action
S   Ldt
22
(6.1.4)
(6.1.5)
can be written as
 S    Ldt   S    Ldt
Using Eq. (6.1.4)
L
 S    d 3r 
 
Using   d   dt
L
 S    d 3r 
 
 
L 
 dt
 
 L d   
dt
 dt 
 
Integrating the second term by parts with:
L
 L 
u
 du  
 dt

t   
d  
dv 
dt  v  
dt


t2

L 
L
 L 
 S    d 3r
 dt   d 3 r         dt 

t    
 
t1


0
L
 L 
 S    d 3r
 dt   d 3 r     dt

t   
L
 S    d 3r 
 

   L 

   dt
t    
(6.1.6)
The principle of least action  S  0  results to
L  L 

0
 t   
(6.1.7)
Following McMahon (McMahon, 2008), one can write the Lagrangian in terms
of the Lagrangian density
L  t    d 3 r L  ,  ,     d 3 r L  ,    
(6.1.8)
This is true for local field theories. The variation of the Lagrangian is then given
by
 L

L
L
L
L
 L  t    d 3 r     
   x  
   y  
   z   (6.1.9)
 


   x 
   z 
   y 


In Minkowski notation,
 L

L
 L  t    d 3 r   
     
(6.1.10)
 

    


Noting that i   i
23
 L
 L t    d 3r 
 

L
L
L
L
 
 x   
 y   
 y    (6.1.11)


   x 
   z 
   y 

 

L
Consider 
 x   dx . This can be integrated by parts by letting
   x 
L
 L
L
u
 du 
dx   x
dx
   x 
x    x 
   x 
dv   x   dx  v  
Thus,
x2
 L

 

    x   x1
L
    x   x   dx 
   x
L
dx
   x 
0 since the variation of  vanishes at the endpoints
L
L
         dx          dx
(6.1.12)
This is similarly true for the other integrals:
L
L
    y   y   dy     y    y  dy
(6.1.13)
L
L
         dz          dz
(6.1.14)
x
x
x
x
z
z
z
z
Eq. (6.1.11) becomes
 L

L
L
L
L
 L  t    d 3 r       x
   y
   z
 
 

   x 
   z  
   y 


 L
L
L
L 
L 
  
 L  t    d 3 r 
 x
 y
 z
  (6.1.15)
   x 
   z  
 
   y 
 


Comparing this with Eq. (6.1.4) one can identify
 L L
L
L
L
L
L

 x
 y
 z

 i
(6.1.16)
 
   x 
   z  
   i 
   y 
 L L

 
(6.1.17)
Putting these equations into (6.1.7)
L
L
L
L
L
L
L
 x
 y
 z
 t
0
 
 0 (6.1.18)

   x 
   z 


    
   y 
which is the Euler-Lagrange equations for fields. If the Lagrangian density
depends on more than one independent fields  s for s  1, 2, 3, , an EulerLagrange equation can be written for each field  s
24
L
L
 
0
 s
    s 
(6.1.19)
The Euler-Lagrange equation can also be derived directly using (6.1.8) to write
the action as8
S   Ldt     d 3 r L  ,    dt   dtd 3 r L  ,      d 4 x L  ,    (6.1.20)
Variation of the action yields
 L

L
(6.1.21)
 S   d 4 x L  ,      d 4 x   
     
    
 

Using the same steps as in Eqs. (6.1.11)-(6.1.14),
 L
L 
(6.1.22)
 S   d 4 x   

     
 
The principle of least action then results to Eq. (6.1.18).
6.2 The Hamilton Formalism
Equation Chapter 6 Section 2
We define the canonical momentum in analogy with discrete Hamiltonian
mechanics as
L
(6.2.1)


Using Eq. (6.1.17) in Eq. (6.2.1),

 L L

 
(6.2.2)
From Eq. (6.1.7)
L   L 
L
 
  0  
 t  

Furtheremore, using (6.1.16)

 L L
L

 i
 
  i 
(6.2.3)
(6.2.4)
The Hamiltonian is obtained by Legendre transformation for fields (summation
is replaced by integration)
H t   d3r  L t
(6.2.5)
This can also be written as
H t   d3rH r, t with H r, t    L
(6.2.6)
If we take the variation of H,
 H t   d3r      L t
(6.2.7)
Using Eq. (6.1.4) for the variation of the Lagrangian,
8
See Chapter 2 of McMahon (McMahon, 2008).
25
L
 H  t    d 3 r       d 3 r 
 
 
L 

 
(6.2.8)
and using Eqs. (6.2.3) and (6.2.1)
 H  t    d 3 r       d 3 r    
 H  t    d 3 r    
Since the H  H  ,  
H
 H   d 3r 
 
(6.2.9)
H 
 


(6.2.10)
 
Comparing the above two equations, we obtain the Hamilton’s equations of
motion,
H

(6.2.11)

H
(6.2.12)
 

On the other hand, the variation of the Hamiltonian can be executed by using
Eq. (6.2.6) and noting that H  H  ,  ,  ,    H  ,  i ,  ,  i 
 H
 H t     d 3r H   d 3r 
 

H
H
H
   i  
 
   i  
   i 

   i 

 
Noting that   i   i   and integrating by parts following Eqs. (6.1.11)-
(6.1.14)
 H
 H  t    d 3 r 
 i
 H
H 
H  
 i
   
  
   i  







i

 
 
Comparing Eqs. (6.2.10) and (6.2.13)
 H H
H

 i
 
  i 
 H H
H

 i
 
   i 
The Hamilton’s equations of motion then become,
 H H
H


 i
 
   i 
 H
H

   
 i

   i  
 
6.2.1 Poisson Bracket of Functionals
In field theory, the Poisson bracket is defined by
  F G  F G 

F , GPB   d 3 r 

     
26
(6.2.13)
(6.2.14)
(6.2.15)
(6.2.16)
(6.2.17)
(6.2.18)
for functionals F  ,   and G  ,   . The time evolution of a functional is
F
F
1
F 
 F t    d 3r 
 
 
t
t

 

  F   F  
F 
3 F
F t    d 3r 



  d r
 
   t   t 
 
F H F H 
F t    d 3r 

  F , H PB
     
6.2.2 An Important Special Case
Using the definition of the Dirac-delta function,
  r , t    d 3 r  r , t    r  r  
(6.2.19)
(6.2.20)
We can then consider   r , t  as a functional of   r , t  , that is   r , t    r , t   ,
a functional depending on itself. Taking the variation of  [see Eq. (6.1.3)]
  r , t 
  r , t    d 3 r
  r , t 
(6.2.21)
  r , t 
The variation of Eq. (6.2.20) yields
(6.2.22)
  r , t    d 3 r  r , t    r  r    d 3r  r , t    r  r  
0
Comparing Eqs. (6.2.21) and (6.2.22)
  r , t 
   r  r
  r , t 
Using the same steps,
  r , t 
   r  r
  r , t 
  r , t    r , t 

0
  r , t    r , t 
(6.2.23)
(6.2.24)
(6.2.25)
6.2.3 Time Evolution and Poisson Bracket of Fields
Using Eq. (6.2.19) we can obtain the Hamilton’s equations of motion for fields
already given by Eqs. (6.2.11) and (6.2.12):
   r , t   H
  r , t   H 
  r , t     r , t  , H PB   d 3 r  


   r , t    r , t    r , t    r , t  
(6.2.26)
  r , t    d 3 r   r  r  
H
H

  r , t    r , t 
Similarly,
27
(6.2.27)
 r ,t   
H
  r , t 
(6.2.28)
The mutual Poisson bracket of fields is given by,
   r , t    r , t    r , t    r , t  
  r , t  ,   r , t PB   d 3r    r , t  r , t   r , t  r , t  (6.2.29)
  
   
 
  r , t  ,   r , t    d r   r  r    r   r  
  r , t  ,   r, t     r  r
3
PB
PB
and
  r , t  ,  r, t 
PB
   r , t  ,   r , t PB  0
(6.2.30)
(6.2.31)
6.3 Canonical Quantization
Equation Chapter 6 Section 3
6.3.1 Introduction
The following is taken from Chapter 3 of (Greiner & Reinhardt,
1996). See also Chapter 2 of (McMahon, 2008).
The process of transforming classical physical quantities into quantum
mechanical operators and Poisson bracket into commutator is called first
quantization. This process also results to the Schroedinger equation
2
y r,t
i

2 y r,t U r  y r,t
(6.3.1)
t
2m
In the context of quantum field theory, the wavefunction y r,t can be
considered as a complex classical field. We can then use the mathematical
machinery introduced in Section 6.1 but in the inverse mode by searching for a
Lagrangian density that will result to the Schroedinger equation (6.3.1). It turns
out that the appropriate Lagrangian density that will result to the Schroedinger
equation (Greiner & Reinhardt, 1996) is
L  ,     L  ,  ,   i   
with
2
2m
   U r 
(6.3.2)

 t
t
This can be verified by substituting (6.3.2) into Eq. (6.1.19) with  and  
considered as independent fields:
 L 
L
  
0

      

28
 L   L 
 L 
 L 
L
(6.3.3)
 t 
 z 
  x 
  y 
0

    y       z  
    
    x  
 L 
L
0


 

 
      
 L 
 L 
 L 
 L 
L






  0 (6.3.4)









t
x
y
z 




 
    
    x  
    y  
    z  
Let us write (6.3.2) in the form,

L  ,     i   
Working on Eq. (6.3.3)
L


 
2
 
2m
x

 x  y y  z  z  U r 
2



i



    U  r   

2m


L
 U  r  

L
i 

(6.3.5)
(6.3.6)
(6.3.7)
2

L
 


i



 x  x   y  y   z  z   U  r   


   x     x  
2m

2
L

 x 
   x 
2m
(6.3.8)
2
L

 y 
2
m
   y 
(6.3.9)
2
L

 z 
   z 
2m
(6.3.10)
Similarly,
If we put together Eqs. (6.3.6)-(6.3.10) into Eq. (6.3.3) we obtain
2
 
i

2   U  r  
t
2m
In the same manner, Eq. (6.3.4) results to the Schroedinger equation:
L

i
 U


t
L
0
 
2
L

 x
2m
   x  
29
(6.3.11)
(6.3.12)
(6.3.13)
(6.3.14)
2
L


 y
2m
   y  
(6.3.15)
2
L

 z
2m
   z  
(6.3.16)
2


2  U  r 
t
2m
The canonical momentum   r , t  is
i
 r ,t  
L
i 

(6.3.17)
(6.3.18)
The Hamiltonian density is
2


H    L  i    i   
    U  r   
2m


H
2
    U  r  
2m
The Hamiltonian can be determined in the following:
2

3
3 
H  d rH  d r
    U  r   
 2m

H
2
d r
2m 
3

   d 3 r U  r 
(6.3.19)
(6.3.20)
The first integral can be evaluated by integration by parts using the identity
           2             2 (6.3.21)
H
2
2m 
2
d 3 r       2    d 3 r U  r 
3

 d r     
2

 dSnˆ     
2
d 3 r  2   d 3 r U  r 
2m
2m 
Using divergence theorem to the first integral
H
H 
2
 d r
3
2   d3r U r 

2m
2m
The surface integral vanishes since the wavefunction decreases rapidly at the
surface
2
2

3
 2
3

3

(6.3.22)
H 
d
r




d
r

U
r


d
r


2 U 






2m
 2m

6.3.2 Quantization Rules for Bosons
We quantize the fields   r , t  and   r , t  by promoting them into operators
ˆ  r , t  and ˆ  r , t  , respectively. In this context the fields   r , t  and   r , t 
are considered classical fields. However, we already know that   r , t  itself is a
30
wavefunction, which is the by-product of quantizing classical physical quantities
into operators. This transformation of physical quantities into operators is called
first quantization while the quantization of   r , t  and   r , t  into field operators
ˆ  r , t  and ˆ  r , t  , respectively, is called second quantization. Moreover, the
Poisson brackets becomes commutators divided by i .
 ,  PB

1
ˆ , ˆ 
i
(6.3.23)
From Eqs. (6.2.30) and (6.2.31)
1
ˆ  r , t  , ˆ  r , t      r  r  
(6.3.24)
i 
1
1
ˆ  r , t  ,ˆ  r , t    ˆ  r , t  , ˆ  r , t    0
(6.3.25)
i
i
We also quantize the complex conjugate of the field  the canonical
conjugate/adjoint of ˆ
(6.3.26)
  ˆ †
From Eq. (6.3.18)
(6.3.27)
ˆ  i ˆ †
Substituting Eq. (6.3.27) to the commutation relations (6.3.24) and (6.3.25)
1
ˆ  r , t  , i ˆ †  r , t      r  r  
i
 yˆ r,t , yˆ † r,t  d r  r 
(6.3.28)


1
1
ˆ  r , t  ,ˆ  r , t    i ˆ †  r , t  , i ˆ †  r , t    0
i
i
 yˆ r,t , yˆ r,t    yˆ † r,t , yˆ † r,t   0
(6.3.29)

 

Particles that obey the above commutation relations are called bosons. They
have integral spins.  † and ˆ † are known as field operators.
The time evolution of the field operators can be obtained from Eq. (6.2.26) with
the Poisson bracket replacement following Eq. (6.3.23):
1
ˆ  ˆ , Hˆ 
(6.3.30)
i
1
ˆ  ˆ , Hˆ 
(6.3.31)
i
The latter is the Hermitian conjugate of the former in the following sense:
ˆ  i ˆ †
(6.3.32)
   
   
†
1

 1   ˆ ˆ †
ˆ  i  ˆ , Hˆ    i 
 ,H
i

 i  
†
ˆ   ˆ , Hˆ     Hˆ † ,ˆ †     Hˆ ,ˆ † 
1
ˆ  ˆ † , Hˆ   ˆ , Hˆ 
i
31
(6.3.33)
This expression also begets the time evolution equation for ˆ †
1
i ˆ †  i ˆ † , Hˆ 
i
1
ˆ †  ˆ † , Hˆ 
i
Using Eq. (6.3.22) in quantized form
2

3
† 
ˆ
H   d rˆ 
2  U ˆ
 2m

to Eq. (6.3.30)
2



1
ˆ  r , t   ˆ  r , t  ,  d 3 r ˆ †  r , t    2  U  r   ˆ  r , t  
i 
 2m


(6.3.34)
(6.3.35)
2




i ˆ  r , t    d 3 r  ˆ  r , t  ,ˆ †  r , t   
2  U  r   ˆ  r , t  
 2m



Using the commutator identity  A,BC =  A,BC+B A,C
2



i ˆ  r , t    d 3 r   ˆ  r , t  ,ˆ †  r , t    
2  U  r   ˆ  r , t 
 2m


2




ˆ †  r , t  ˆ  r , t  ,  
2  U  r   ˆ  r , t   

 2m



2



i ˆ  r , t    d 3 r     r  r    
2  U  r   ˆ  r , t 
 2m



2


ˆ †  r , t   
2  U  r    ˆ  r , t  ,ˆ  r , t   

 2m

0

2


(6.3.36)
i ˆ  r , t    
2  U  r  ˆ  r , t 
 2m

The time evolution of ŷ is given by the Schroedinger equation. However, this
equation is now a derived equation not a postulate unlike the Schroedinger
equation for the wavefunction in first quantization.
In the same way,
2
 †



1
†
† ˆ
†
3
†


ˆ
ˆ
ˆ
ˆ
ˆ


   , H   i   r , t     r , t  ,  d r   r , t    2  U  r   ˆ  r , t  
i
 2m



2
 †



†
3
†
ˆ
ˆ
ˆ


i   r , t    d r   r , t  ,  r , t   
2  U  r   ˆ  r , t  
 2m




2



i ˆ  r , t    d r   ˆ †  r , t  ,ˆ †  r , t    
2  U  r   ˆ  r , t 
 2m


0

†
3
32
2
 †




†
ˆ
ˆ

  r , t    r , t  ,  
2  U  r   ˆ  r , t   
 2m




2


†
3
†
ˆ
ˆ


i  r ,t    d r  r ,t  
2  U  r    ˆ †  r , t  ,ˆ  r , t  
 2m

2


i ˆ †  r , t     d 3 r ˆ †  r , t   
2  U  r    ˆ  r , t  ,ˆ †  r , t  
 2m

  r  r 


i yˆ † r,t   
2 U r   yˆ † r,t
(6.3.37)
2m


after an integration by parts. It turns out that this equation is the Hermitian
conjugate of Eq. (6.3.36).
2
6.3.2.1 Representation of the field operators
In order to apply the field operators to different situations, we need to find its
representation. This can be done by choosing a particular basis. Suppose that
we use the energy eigenfunctions ui r as the complete orthonormal basis.
By the generalized Fourier decomposition,
yˆ r,t   aˆi tui r 
(6.3.38)
i
yˆ † r,t  aˆi† tui r 
(6.3.39)
i
where the basis ui r satisfies
2
 2ui r  V r ui r   Eiui r 
(6.3.40)
2m
The operator property of ŷ and ŷ † are now borne by the Fourier coefficients aˆi

and aˆi† respectively. Substituting Eqs. (6.3.38) and (6.3.39) into the
commutations relations (6.3.28) and (6.3.29)


†

aˆi tui r, aˆ j tuj r    d r  r 
j
 i


†
ui r uj r  aˆi t ,aˆ j t   d r  r 

ij
(6.3.41)
Using the completeness relation
u ru r  d r  r
i

i
(6.3.42)
i
the commutation relation in Eq. (6.3.41) is a Kronecker delta dij , that is,
aˆi t ,aˆ j† t  dij


(6.3.43)
aˆi t ,aˆ j t  aˆi† t ,aˆ j† t  0

 

(6.3.44)
Similary, it can be shown that
33
The Hamiltonian can also be written in terms of aˆi and aˆi† by substituting Eqs.
(6.3.38) and (6.3.39) into Eq. (6.3.35)
 2 2

H   d3raˆi† tui r   
 U  aˆ j tuj r 
i
 2m
 j
2


H  aˆi† taˆ j t  d3rui r   
2 U uj r 
i, j
 2m

†
3

H  aˆi t aˆ j t Ej d rui r uj r
i, j
 
   
The integral in the above expression is the orthonormality condition. Thus
H  aˆi† taˆ j t Ejdij
i, j
H   Eiaˆi† taˆi t
(6.3.45)
i
To continue see pages 60-65 of Greiner and Reinhardt (Greiner &
Reinhardt, 1996).
6.3.3 Quantization Rules for Fermions
Another class of particles obey the so-called anticommutation relations given
below:
(6.3.46)
yˆ r,t , yˆ † r,t  d r  r
yˆ r,t , yˆ r,t  yˆ r,t , yˆ r,t  0
†
†
(6.3.47)
These particles are known as fermions, which are found to have half-integral
spin. The equations of motion can be obtained also from Eqs. (6.3.30) and
(6.3.34). For ˆ
2




i yˆ r,t   d3r  yˆ r,t , yˆ † r,t  
2 U r   yˆ r,t 
(6.3.48)
 2m



Let us use the following identity connecting the commutator with the
anticommutator:
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
 A,
(6.3.49)
 BC  A, B C - B A, C
   
ˆˆ C
ˆ=A
ˆ B,
ˆ  - A,
ˆ C
ˆ B
ˆ C
ˆ
 AB,



2



†
ˆ
ˆ
ˆ


i   r , t    d r    r , t  ,  r , t   
2  U  r   ˆ  r , t 
 2m


  r  r 

3
34
(6.3.50)
2






†
ˆ
ˆ

  r , t    r , t  ,  
2  U  r   ˆ  r , t   



 2m



2



i ˆ  r , t    d 3 r     r  r    
2  U  r   ˆ  r , t 
 2m



2


2
ˆ  r , t   
  U  r    ˆ  r , t  ,ˆ  r , t  

2
m


0

†
2


(6.3.51)
i ˆ  r , t    
2  U  r  ˆ  r , t 
2
m


Thus, the field operator also obeys the Schroedinger equation just like the case
for bosons.
On the other hand, the time evolution of ˆ † can be obtained as follows:
1
ˆ †  ˆ † , Hˆ 
i
2




i ˆ †  r , t   ˆ †  r , t  ,  d 3 r ˆ †  r , t   
2  U  r   ˆ  r , t  
(6.3.52)
2
m




2




i ˆ †  r , t    d 3 r  ˆ †  r , t  ,ˆ †  r , t   
2  U  r   ˆ  r , t  
 2m



Using Eq. (6.3.49)

2


i ˆ †  r , t    d 3 r   ˆ †  r , t  ,ˆ †  r , t   
2  U  r   ˆ  r , t 

 2m

0

2




 †

†
ˆ
ˆ

  r , t    r , t  ,  
2  U  r   ˆ  r , t   



 2m



2


†
3
†
ˆ
ˆ


i  r, t    d r r , t  
2 U r   ˆ † r, t ,ˆ r, t
 2m

 rr 
After integration by parts,
2


i ˆ † r, t   
2 U r ˆ † r, t
 2m

Read Balizot and Ripka (Balizot & Ripka).
35
(6.3.53)
7 Zero-Temperature Green’s Function
7.1.1 Definition of Green’s Function of Many-Body System
Read pages 49-57 of Zagoskin (Zagoskin, 1998).
See also pages 153-177 of Gross, Runge, and Heinonen (Gross,
Runge, & Heinonen, 2005).
Eq. (4.1.13) is the propagator or the Green’s function of a single particle. How
can we construct the Green’s function for a many-particle system? To answer
this question, we take note that Eq. (4.1.13) is the defining expression for the
propagator of a single variable.
For the non-equilibrium Green’s function formalism, read Chapter 8 of Datta
(Datta, 1995). Read also Chapter 3 of (Datta, 1995).
7.1.1.1 The Unperturbed Green’s Function
(Zagoskin pages 57-58)
7.1.2 Analytic Properties of the Green’s Function
See pages 58-62 of (Zagoskin, 1998)
7.1.2.1 Poles of Green’s Function and Quasiparticle Excitations
See page 62 of (Zagoskin, 1998)
7.1.3 Retarded and Advanced Green’s Function
7.1.3.1 Quasiparticle Excitations and Advanced and Retarded Green’s Functions
7.2
Quantum Correlation Functions in Many-Body Theory
Click this link
7.3 Perturbation Theory: Feynman Diagrams
See page 66 of (Zagoskin, 1998)
8 Quantum Superfield Theory
The complete theoretical description of open, nonequilibrium systems is still an
open problem in many-particle physics. The practical importance of this subject
is more apparent in the studies of quantum transport in nanostructures. Most of
the current theoretical and computational researches in the study of
nanostructures and their device applications rely on conventional equilibrium
transport techniques or near equilibrium quantum transport physics at best. If
36
ever an attempt to analyze nonequilibrium quantum tranport is done,
nonequilibrium Green’s function technigue is usually adopted (Kadanoff & Baym,
1962; Mahan G. D., 1990), which leads to the quantum Boltzmann equation.
This technique however suffers from a serious drawback when applied to
nanoscale structures, where there is an abrupt variation of physical quantities.
The resulting transport equations are expressed in gradient expansion, which is
applicable only for slow spatial variation of physical quantities and slow
temporal variation of the fields. The formalism is also not applicable to describe
ultrafast processes in nanodevices. Therefore, a wide spectrum of novel highly
nonlinear, nonequilibrium quantum transport phenomena in nanostructures is
not captured and their full application potentials for new functionalities are not
realized.
In this article we use quantum superfield theory(QSFT) combined with lattice
Weyl transform techniques in solid-state physics. In QSFT, ordinary quantum
field operators become superoperators while the von Neumann density operator
evolution equation in Hilbert-Fock space H becomes super-statevector
quantum dynamical equation in Liouville space L .
Notes:
Read Arimitsu and Umezawa (Arimitsu & Umezawa, 1987; Arimitzu & Umezawa, 1987).
Read Section 29.1 of Buot (Buot F. A., 2009).
Read Chapter 8 of Datta (Datta, 1995) for a clear exposition of the nonequilibrium
Green’s function.
8.1 Basic Concepts in Quantum Superfield Theory
Equation Chapter 3 Section 1
8.1.1 Single-Particle
9In quantum mechanics, the state of a particle is represented by a state vector
 in Hilbert space H d while physical quantities are represented by operators
acting on  . These operators also constitute a linear vector space, called the
Liouville space L attached to H d . An element in L , which is an operator10 A in
H d , is denoted by A
and known as superstate vector. Recall that we can
also construct an operator in H d by the outer product or dyad formed by the
state vectors u and v as follows,
9This
10
discussion is based on Schmutz (Schmutz, 1978) and Section 1.2 of Audretsch (Audretsch, 2007).
An operator in Hilbert space is written in bold.
37
u v
In Liouville space this should become a superstate vector denoted by
u v
(3.1.1)
(3.1.2)
Similarly, we can write the dyad ui uj formed by the orthonormal basis
u 
i
as a superstate vector in L as
ui uj
(3.1.3)
Now recall that an operator A in H d can be expressed in dyad decomposition
as
(3.1.4)
A   ui ui A uj uj   ui A uj ui uj
i, j
i, j
Aij
The corresponding superstate vector is written as
A   A ij ui uj
The set
u
i
(3.1.5)
i, j
uj
 therefore span the Liouville space. We also anticipate that
the superstate vectors in the set are linearly independent11, which implies that
the set constitutes a basis in L . If the number of state vectors in the basis12
 u  is d , then the number of
i
superstate vectors in the set
u
i
uj
 is d .
2
The dimension of L is therefore d2 .
By definition, L is itself a Hilbert space and is therefore endowed with an inner
product. We define the inner as
A B  Tr A †B
(3.1.6)
with the Hilbert-Schmidt norm given by
A  A A
12
(3.1.7)
The following requirements:
AB  BA
A c1B1  c2B2
AA

(3.1.8)
 c1 A B1  c2 A B2
 0 im plies A
(3.1.9)
0
(3.1.10)
are satisfied by (3.1.6).
Let us use Eq. (3.1.6) to determine the inner product of the basis set
u
i
uj
:
ui uj ui uj
11
12


 Tr  ui uj

They are, in fact, orthonormal which will be shown later.
The dimension of H d is d .
38
 u
†
i


uj 




 Tr  uj ui ui uj 


 Tr  u j ui ui u j   ii Tr u j u j
 Tr  u j  ui ui u j    ii  uk u j  u j uk
k
u
Thus, the basis set
uj

  ii j j
ui u j  ui u j
i

(3.1.11)
 is an orthonormal set.
For consistency of terminology, we call the basis set in Liouville space,
superbasis set. In general, a superbasis may not be in dyadic form
Q
Suppose that
, s  1, 2,
s
,d 2

u
i
uj
.
is an orthonormal superbasis set. Any
superstate vector A can be written as a linear combination of the superstate
vectors in the superbasis,
d2
A   cs Qs
(3.1.12)
s 1
Since
Qs Qt
  st , the coefficient can be written as,
cs  Qs A
(3.1.13)
Substitution of Eq. (3.1.13) back to Eq. (3.1.12) results to the closure relation
d2
Q
s
s 1
Qs  1
(3.1.14)
where 1 is the identity operator in Liouville space. Operators in Liouville space
are called superoperators denoted by A and 1 is an example, which is
consequently called identity superoperator. The quantity Qs
Qs should be a
superoperator, the analogue of the dyad ui ui . We call this superoperator
superdyad.
Following Eq. (3.1.13), the coeficient in A ij in Eq. (3.1.5) can be written as
Aij 
ui u j A
(3.1.15)
Using Eq. (3.1.6) for the definition of the inner product in L ,
†
Aij  Tr  ui u j A   Tr  u j ui A    uk u j ui A uk

 k








A ij   uk u j ui A uk    kj ui A uk  ui A u j
k
k
A ij  ui A u j
(3.1.16)
which is indeed the formula for A ij as can be seen from Eq. (3.1.4). In the
superbasis
u
i
uj
 , the closure relation is written as
39
d

ui u j  1
ui u j
(3.1.17)
i, j
8.1.2 Many-Body System: The Thermal Liouville Space
If we deal with a many-particle (or many-body system) the Hilbert space H d
becomes the Hilbert-Fock space H . We choose the orthonormal basis
 n1, n2 , n3 ,  in which
 n   a1† 
n1 , n2 , n3 ,
n1
a 
a 
† n2
2
† ni
i
0
(3.1.18)
for fermions and
 n 
n1, n2 , n3 ,
1
n1 ! n2 ! ni !
for bosons. The vectors  n1, n2 , n3 ,

a  a 
† n1
1
a 
† n2
2
† ni
i
0
(3.1.19)
are the eigenvectors of the number
†
operator Ni  ai ai ,
 ni n1, n2 , n3 ,
Ni n1, n2 , n3 ,
(3.1.20)
From now on, we simply use n to mean n1, n2 , n3 ,
. The annihilation and
†
creation operators ai and a i satisfy the (anti)commutation relations
 ai , a†j    ij

ai , a j   ai† , a†j   0

(3.1.21)
(3.1.22)

where
1 for bosons
1 for fermions
In this case, the Liouville space L consists of operators in H . The superbasis is
written as
m n
(3.1.23)
 
in analogy with
i

j

 in the single-particle case. To simplify the notation we
simply write expression (3.1.23) as,
m n
 m, n
(3.1.24)
and
m, n  m, n
†

m
n

†

n m
(3.1.25)
The orthonormality and completeness relations can be written, respectively, as,
(3.1.26)
mn mn   mm nn
 m, n
m, n  1
m, n
40
(3.1.27)
In the same way as Eq. (3.1.5), we can also expand an arbitrary superstate
vector A in the basis m, n


A   m, n
m, n A
(3.1.28)
m, n
in which following Eqs. (3.1.15) and (3.1.16)
m, n A  m A n
The bra
(3.1.29)
A is the adjoint of the corresponding ket,
A  A†
†
(3.1.30)
A   A m, n
(3.1.31)
m, n
m, n
A m, n
 n A m
(3.1.32)
When A  1 , special cases of Eqs. (3.1.28) and (3.1.31) are, respectively,
obtained
1   m, n
m, n 1
m,n
1   m, n  mn   n, n
m, n
(3.1.33)
n
1   1 m, n
m, n
m,n
1    mn m, n   n, n
m, n
(3.1.34)
n
We define annihilation superoperators aˆi and ai (Schmutz, 1978; Arimitsu &
Umezawa, 1987) such that
(3.1.35)
aˆi m, n  aˆi m n  ai m n
ai m,n  ai m n
where
   1 m n ai†
(3.1.36)
  m j  n j 
(3.1.37)
j
From the above equations we can write down the corresponding expressions
aˆi† m, n and ai† m, n (Schmutz, 1978; Arimitsu & Umezawa, 1987). The
equivalent expression for the former can be derived as follows:
m, n aˆi† m, n  m, n aˆi m, n


 m, n ai m n
Using (3.1.29) with A  ai m n
m, n aˆi† m, n   m ai m n n
 m ai m
41



n n

 m ai† m n n
Another application of Eq. (3.1.29) now with A  ai† m n
results to
m, n aˆi† m, n  m, n ai† m n
aˆi† m, n  ai† m n
On the other hand, one can evaluate ai† m, n
(3.1.38)
as follows:
m, n ai† m, n  m, n ai m, n


   1 m, n m n ai†
where      mj  nj  . Using (3.1.29) with A  m n ai†
j

m, n ai† m, n    1 m m m n ai† n


   1 m m

n ai† n

  1 m m n ai n
Using (3.1.29) again but now A  m n ai
m, n ai† m, n    1 m, n m n ai
Taking note that
   1    mj  nj   1   mj   nj  1
j
j
Letting mj  m j and nj  1  n j , we get    1   . Thus,
ai† m, n    m n ai
(3.1.39)
The (anti)commutation, also called   commutation (Arimitsu & Umezawa,
1987), relations satisfied by the annihilation and creation superoperators can be
determined from Eqs. (3.1.35), (3.1.36), (3.1.38), and (3.1.39). First note that
  ij m, n
 aˆi , aˆ †j  m, n   ai , a †j  m, n


 ai , a †j  m, n     1 

2
 m n  ij
  ij m, n
m n  ai , a †j 

  ij m n
  ij m, n
Thus,
aˆi , aˆ †j   ai , a†j   ij


Similarly, one can show that
aˆi , aˆ j   aˆi† , aˆ †j   ai , a j   ai† , a †j   0




We also have, from Eqs. (3.1.35) and (3.1.38), the following relations:
aˆi† aˆi m, n  ai†ai m n  mi m n  mi m n
42
(3.1.40)
(3.1.41)
aˆi†aˆi m, n  mi m, n

ai† ai m, n  ai† ai m, n
a 
(3.1.42)
 1
†
i
m n ai†
ai† ai m, n    1ai† m n ai†
To evaluate ai† m n ai†
(3.1.43)
, we have to take note that,
n ai†  ai n  ai n1, n2 , n3 , ni
 ni n1, n2 , n3 , ni  1
 in Eq. (3.1.39) can then be replaced by


   mj  n j   1   1
 j

(3.1.45)
 m   n  1 
   m j  n j   m1  n1    m2  n2  
i
(3.1.44)
i
j
   1
We then have,
   1 m n ai†ai
ai† m n ai†
and thus Eq. (3.1.43) becomes
(3.1.46)


ai† ai m, n    1  1 m n ai†ai
    1
 ni m n
 ni m, n
ai† ai m, n
2
m n ni
(3.1.47)
We can also obviously have,
aˆi 0, 0  ai 0, 0  0
(3.1.48)
where
0, 0  0 0
(3.1.49)
is the supervacuum.The supervector m, n
can be constructed from the
supervacuum using Eqs. (3.1.18), (3.1.19), and (3.1.35)-(3.1.41). For bosons,
we have
1
n
n
n
n  0  ai  i  a2  2  a1  1
(3.1.50)
n1 ! n2 ! ni !
 0
m n 
a  a 
1
m1 ! m2 ! mi !
m n 
 ai 
1
m1 ! m2 ! mi !
  a1† 
m1
a 
† m2
2
ni
† m1
1
 a2   a1 
n2
† m2
2
n1
a 
† mi
i
0
1
n1 ! n2 ! ni !
1
n1 ! n2 ! ni !
a 
† mi
i
43
0 0
 ai 
ni
 a2   a1 
n2
n1

m n
1
m1 !m2 ! mi !
  a1† 
m n

a 
m1
  aˆ1† 
m1
a 
† m2
2
1
m1 !m2 ! mi !
1
n1 !n2 ! ni !
† mi
i
 a2   a1 
ni
n2
n1
1
n1 !n2 ! ni !
 aˆ 
 aˆ 
† m2
2
a  a 
† mi
i
† n1
1
and for fermions, we also have
m1
m2
n
  1i i  aˆ1†   aˆ2† 
m n
 ai 
0 0
a 
† n2
1
† ni
i
a  a 
† n1
1
0 0
a 
† n2
1
† ni
i
0 0
(3.1.51)
(3.1.52)
From the above discussion, one can realize that the number of annihilation and
creation operators in Hilbert-Fock space has been doubled in Liouville space,
 aˆi , ai 

ai
 aˆ , a 
ai† 
†
i
†
i
Since the degrees of freedom is determined by the annihilation and creation
operators, there is doubling of the number of degrees of freedom in Liouville
space.
Using Eqs. (3.1.51) and (3.1.52), we get
 aˆ a   aˆ a 

† † m1
1 1
m, m  m m
m1 !
and
 m, m

m
 m, m
m
m1
 m, m
† †
2 2
m1 !
m

m1
 aˆ a 
† †
1 1
m1 !

† † mi
i i
m2 !
 aˆ a   aˆ a 
† †
1 1
 aˆ a 
† † m2
2 2
m2
mi !
 aˆ a 
† †
i i
m2 !
m1

m2

 aˆ a 
† †
2 2

m2 !
 exp aˆ1† a1† exp aˆ2† a2†


mi
0, 0
 aˆ a 
† †
i i

(3.1.53)
mi
mi !
m2
0, 0
mi !
exp aˆi† ai†
mi

0, 0
0, 0
m
 m, m
m


 exp   aˆi† ai†  0, 0
 i

(3.1.54)
From Eq. (3.1.33)


1  exp   aˆi† ai†  0, 0
 i

(3.1.55)
The corresponding bra is


1  0, 0 exp   ai aˆi 
 i

44
(3.1.56)
ai and creation
Let us now construct the annihilation
ai† superstate vectors
†
associated with the annihilation ai and a i creation operators in Hilbert-Fock
space, respectively. For the annihilation superstate vector, we note that from Eq.
(3.1.35),
aˆi 1  aˆi  m, m  aˆi  m m   ai m m
m
m
m
aˆi 1  ai  m m  ai
(3.1.57)
m
Thus, we can construct the annihilation supervector through Eq.(3.1.57)
ai  aˆi 1
(3.1.58)
We can also derive its alternative expression as,
ai† 1    
0
m
m ai  ai
m
ai  ai† 1
(3.1.59)
Thus, the annihilation superstate vector can be written in two forms:
ai  aˆi 1  ai† 1
(3.1.60)
The creation superstate vector can be constructed using either Eq. (3.1.58) or
(3.1.59) as follows:
aˆi† 1 
a
†
i
m m  ai†  m m
m
m
†
i
a
 aˆ 1
†
i
(3.1.61)
ai 1    01 m m ai†
m
ai 1  
m
m ai†   ai†
m
ai†   ai 1
The creation superstate vector can be written in these forms:
ai†  aˆi† 1   ai 1
(3.1.62)
(3.1.63)
Continue here…..from page 36 of (Arimitsu & Umezawa, 1987).
Suppose that A is an operator in H . Upon second quantization, this operator
takes a general form
A  a † , a    c  p1, pm ; q1, , qn  a †p1 , , a †pm aq1 , , aqn
(3.1.64)
mn
pq
We define the superoperators
ˆ  A  aˆ † , aˆ 
A
A  A  a † , a 
45
(3.1.65)
(3.1.66)
where the complex conjugation sign is understood to be applied to the constant
c. Let us now consider two arbitrary operators, A and B, in H . They become
super-statevectors, A and B , in L .The super-operators associated with A are
given by Eqs. (3.1.65) and (3.1.66) while the super-statevectors can be written
as a linear combination of the superbasis  m, n  .
Given operators A and B , there exists an operator A  A, B  such that
AB  Aˆ B
(3.1.67)
BA  Aˆ † B
(3.1.68)
A 1  A
(3.1.69)
where
To prove the first experssion, we have
Aˆ B  Aˆ  m, n m, n B
m,n
  Aˆ m, n m, n B   Aˆ m n
m ,n
m, n B
m,n



m, n B    m, n m, n  AB
 m ,n


  A aˆi† , aˆi m n
m ,n
1
This implies,
ˆ B  AB
A
(3.1.70)
The second one can be proven as follows,
A†  A  a † , a 
(3.1.71)
A† B  A†  m, n m, n B
m,n

  A† m, n m, n B   A a † , a
m,n

  m n A a† , a
m,n

m n
m, n B
m ,n



m, n B    m, n m, n  AB
 m ,n

1
BA  A† B or A B  BA†
(3.1.72)
When B  1 , we have from Eqs.(3.1.70) and (3.1.72),
A  Aˆ 1
(3.1.73)
A  A† 1
(3.1.74)
For fermions,
46
A 1   1
 1
2
A
(3.1.75)
where   m  n . If the operator in Eq. (3.1.64) conserves the fermion number,
m  n    0 , and therefore
A 1  A
(3.1.76)
From Eqs. (3.1.61) and (3.1.62) one can also conclude that given a


†
†
superoperator A aˆ , aˆ , a , a in Liouville space, there exists an unique operator
A  a † , a  such that,
A  aˆ † , aˆ , a † , a  1  A  a † , a 
(3.1.77)
In the subsequent discussions, we assume that   0 for fermions.
In the following, equations in Liouville space will be developed in terms of the
field creation and annihilation superoperators ˆ † , ˆ ,  † , and  .
8.2 Quantum Dynamics in Liouville Space
Equation Chapter 3 Section 2
Read Arimitzu and Umezawa (Arimitsu & Umezawa, 1987; Arimitzu &
Umezawa, 1987). These two references introduced the quantum superfield
theory.
8.2.1 Time Evolution Equations
In conventional quantum mechanics, the state of a system is denoted by a state
ket, say  or in the position representation by a wavefunction   q  . The
density operator is then expressed in terms of the state ket or of the
wavefunction. However, it is more convenient to consider the density or
statistical operator as the fundamental quantity representing the state of the
system. The state ket or the wavefunction can then be considered as derived
quantity (Fain 2002). The (first) postulate in quantum mechanics, which states
that (Tannoudji et al 1989):
“The state of a physical system at time t is represented by a state ket
  t  .”
is replaced by the statement:
“The state of a physical system at time t is represented by a density
operator ρ  t  .”
47
The latter form of the first postulate is more appropriate in the QSFT formalism.
Thus, the state of a physical system is represented by the density superstatevector
ρ t 
The expectation value of an operatorAin Hilbert-Fock space is written as (Blum,
1981)
A  Tr  Aρ 
(3.2.1)
which can also be written as
A  Tr 1Aρ 
(3.2.2)
Using Eq. (3.1.6), this can be written as
A  1 A
(3.2.3)
Furthermore, we can invoke Eq. (3.1.67) to write it in the form,
A  1 Aˆ 
(3.2.4)
where the superoperator  is related to the operator A according to Eq. (3.1.65).
The time evolution of the density operator, and therefore of the state, is given
by the von Neumann equation (Blum, 1981)
ρ
i
  H, ρ  i ρ  Hρ  ρH
(3.2.5)
t
where H is the Hamiltonian. In converting this to the density super-statevector
equation, we write
ρ
i
 Hρ  ρH
t
With the help of Eqs. (3.1.67) and (3.1.68),
ρ
(3.2.6)
i
 Hˆ ρ  H ρ  Hˆ  H ρ  L ρ
t
where L is called the Liouvillian in analogy with the Hamiltonian H :
ˆ H
(3.2.7)
L H
Eq. (3.2.6) is called thesuper-Schroedinger equation in analogy with the
Schroedinger equation in Hilbert-Fock space. We then say that the system of
bosons or fermions13 is characterized by the Liouvillian L in the same way that
a system of bosons or fermions is characterized by the Hamiltonian H .


In arriving at Eq. (3.2.6) we make use of the hermiticity of the Hamiltonian, that
is,
(3.2.8)
H  H†
Eq. (3.2.7) implies that
Hˆ , H  0
(3.2.9)
 
13
We can call them superbosons or superfermions.
48
The corresponding density super-operator is constructed as follows:
ρ ρ 1
(3.2.10)
Note that
ρ 1  ρ 11  ρ
which satisfies Eq. (3.1.77). In general, in constructing a superoperator A in L
from an operator A in H we simply have14,
A A 1
(3.2.11)
In analogy with Eqs. (3.2.1) and (3.2.4) we also define the expectation value of
a super-operator A as15
A  Tr Aρ  1 A ρ
(3.2.12)
It can also be shown that16
A  Tr Aρ  A  Tr Aρ
(3.2.13)
Let us now determine the time evolution equation of the density superoperator
ρ . From Eqs. (3.2.6) and (3.2.10),
ρ

i
i
ρ 1  L ρ 1  Lρ
(3.2.14)
t
t
This equation can be written in a more suggestive form by noting that,
1 L  1 Hˆ  H  H  H  0


We can use this expression to show that
ρL  ρ 1 L  0
Combining Eqs. (3.2.14) and (3.2.15),
ρ
i
 Lρ  0  Lρ  ρL   L , ρ 
t
(3.2.15)
(3.2.16)
8.2.2 The Unitary Superoperator
A unitary operator U is defined by the relation
UU†  1  U†  U1
In general, this operator is of the form
U  exp icA 
(3.2.17)
(3.2.18)
U†  U1  exp  icA 
(3.2.19)
where c is a real number and A is a Hermitian operator. The time evolution
operator discussed in Section 2 is an example of a unitary operator. The
14
I am not sure of this. Justify this later.
This is Eq. 3.8 of (Schmutz, 1978). Justify this expression later.
16 I am not sure of this. Justify this later. This is Eq. (3.9) and (3.10) of (Schmutz, 1978).
15
49
construction of the corresponding unitary superoperator follows from Eqs.
(3.1.65) and (3.1.66). Usually, U can be written as
ˆ
(3.2.20)
U  UU
Thus, using Eq. (3.2.18)
Uˆ  exp icAˆ
(3.2.21)
 

U  exp icA

  
(3.2.22)

(3.2.23)
Using Eq. (3.1.73) and (3.1.74)
Uˆ 1  exp icAˆ 1  exp  icA 
(3.2.24)
ˆ  exp icAˆ exp icA
U  UU
 


U† 1  exp icA† 1  exp  icA 
We then have,
(3.2.25)
   
U 1  exp  icAˆ  exp  icA   exp  icAˆ  exp  icA 
ˆ 1  exp icAˆ exp icA 1
U 1  UU
†
U 1  exp  icA  exp  icA   1

 
(3.2.26)

U† 1  U†Uˆ † 1  exp icA† exp icAˆ † 1



U† 1  exp icA† exp icAˆ †

 exp  icA  exp icA 
U† 1  1
(3.2.27)
8.2.3 The Time Evolution Superoperator
We define the time evolution superoperator in analogy with Section 2. All the
properties of the time evolution operator in Hilbert space should be satisfied by
the time evolution superoperation in Liouville space. Being a unitary
superoperator we expect that it can be written in the form (3.2.20)
ˆ
U  UU
(3.2.28)
It is defined such that the superstatevector evolves as,
ρ  t   U  t , t0  ρ  t0 
(3.2.29)
In the same way that the wave function satisfies the Schroedinger’s equation
(2.1.18), the superstatevector satisfies the Liouville equation (3.2.6). This gives
us the differential equation
U  t , t0 
i
 LU  t , t0 
(3.2.30)
t
for the time evoluton superoperator. This is of the same form as Eq. (2.1.19)
satisfied by the time evolution operator in Hilbert space.
50
In analogy with Eq. (2.1.34), the general solution of Eq. (3.2.30)can be written
as,
 i t

 i t

U  t, t0   T exp    dt1L  t1    T exp    dt1 Hˆ  t1   H  t1  
(3.2.31)
t0
t0






which shows that,
ˆ
U  UU
(3.2.32)
t
 i

Uˆ  t, t0   T exp    dt1Hˆ  t1  
(3.2.33)
t0


i t

U  t, t0   T exp   dt1H  t1  
(3.2.34)
t0


The fact that Eq. (3.2.31) can be written in the form Eq. (3.2.32)is justified from
the Campbell-Baker-Hausdorff identity (Buot F. A., 2009; Sakurai, 1994),
 1

exp  A  B   exp  A  exp B  exp    A, B 
(3.2.35)
 2

wherein
Hˆ , H  0
(3.2.36)
 
It also follows that
 Uˆ  t, t0  , U  t , t0   0
(3.2.37)


In addition, Eqs. (3.2.31) - (3.2.34) obey Eqs. (3.1.65) and (3.1.66).
8.2.4 The Super-Heisenberg and Super-Interaction Pictures
In the Heisenberg representation, the state is not a function of time. We impose
the same for the super-statevector. Thus,
ρ
H
 ρ  t0 
S
H
 ρ  0
S
(3.2.38)
For t0  0
ρ
(3.2.39)
In analogy with Eq. (2.2.9), we define the super-Heisenberg representation of a
superoperator A as
A H  t   U†  t, t0  A S U  t, t0 
(3.2.40)
By direct differentiation with respect to time, it can be shown that this obeys the
super-Heisenberg equation of motion,
i A H  t    A H  t  , LH  t 
analogous to Eq. (2.2.20). From Eq. (3.1.73)
Aˆ S 1  A S  Aˆ S U  t, t0  U †  t , t0  1  A S
With Eq. (3.2.27),
51
(3.2.41)
Aˆ S U  t , t0  1  A S  U †  t , t0  Aˆ S U  t , t0  1  U †  t , t0  A S
Aˆ H  t 
Aˆ H  t  1  U†  t , t0  U †  t , t0  A S
Aˆ H  t  1  U †  t , t0  U †  t , t0  A S
Aˆ H  t  1  U †  t , t0  A S U  t , t0 
Aˆ H  t  1  A H  t 
(3.2.42)
Using similar steps, it can also be shown that,
A† H  t  1  A H  t 
In particular,
(3.2.43)
†
aiH  t   aˆiH  t  1  aiH
t  1
(3.2.44)
†
aiH
 t   aˆiH†  t  1   aiH  t  1
(3.2.45)
To define the super-interaction picture, we let
L  L0  L1
(3.2.46)
following Eq. (2.2.33). The relation between the super-statevector in the superinteraction picture to that in the Schroedinger picture is written as (see Eq.
(2.2.34)),
i t

ρ  t, t0  I  T exp   L0  t  dt  ρ  t, t0  S  U0†  t, t0  ρ  t, t0  S
(3.2.47)
t0


For the super-operator, we have the corresponding relation
A I  t   U0†  t, t0  A S  t0  U0 t, t0 
(3.2.48)
The time evolution of the supers-statevector can be obtained by time
differentiation of Eq. (3.2.47) making use of Eqs. (3.2.30) and (3.2.6) on the
understanding that in the latter the super-statevector is in the Schroedinger
picture and in the former L should be replaced by L0 . The result is
where L1I
(3.2.48)
 ρ  t, t0 
 L1I ρ  t, t0  I
(3.2.49)
t
follows from the construction (3.2.48). In the same way, from Eq.
i
I
d AI
(3.2.50)
  A I , L0 
dt
The time evolution super-operator in the super-interaction picture follows from
Eq. (3.2.49):
UI  t, t0 
i
 L1I UI  t, t0  I
(3.2.51)
t
The solution of Eq. (3.2.51) is obviously,
i
52
 i t

UI  t, t0   T exp    L1I  t  dt 
t0


In the same manner as in Section 3.3, it can be shown that,
UI  t, t0   U0†  t, t0  US  t, t0  U0  t, t0 
(3.2.52)
(3.2.53)
One can also write in analogy with Eq. (2.2.49) the equation,
A I  U0†  t, t0  U t, t0  A H t  U † t, t0  U0 t, t0 
(3.2.54)
which gives the relation between super-operators in the super-Heisenberg and
super-interaction pictures.
8.2.5 Super S-Matrix Theory and Variational Principle in Liouville Space
8.2.5.1 Super S-Matrix Theory
Consider now the Liouvillian (3.2.46). Let us interpret the second term as a
perturbation that is turned on infinitely slowly at t0 and that for times before t0
the system is acted on by the unperturbed Liouvillian L0 only. This implies that
the perturbation can be written as
 t t
L1  We  0 
t  t0 and   0
(3.2.55)
  t   0 
L1  We
t0  t  0
where 0 is an arbitrary initial time before t0 .This type of thought experiment is
called adiabatic turning on of state. Then in the Schroedinger picture we can
write,
 i t0

ρ  t0 ,0  S  T exp    L0  t   dt  ρ 0  S  U0  t0 ,0  ρ 0  S
(3.2.56)
0


For t  t0 , the full Liouvillian acts and we have,
 i t

 T exp    L  t   dt  ρ  t0 ,0  S  U  t ,t0  ρ  t0 ,0  S
(3.2.57)
t0


We would like to express the above equations in the super-interaction picture.
Following the prescription (3.2.47)
i t

ρ  t , t0  I  T exp   L0  t   dt  ρ  t , t0  S  U0†  t , t0  ρ  t , t0  S
(3.2.58)
t0


Now substituting Eq. (3.2.57), Eq. (3.2.58) becomes,
ρ  t ,t0  I  U0†  t ,t0  U  t ,t0  ρ  t0  S
(3.2.59)
ρ  t , t0 
S
and with Eq. (3.2.56) we finally obtain,
ρ  t ,t0  I  U0†  t ,t0  U  t ,t0  U0  t0 ,0  ρ 0 
S
In the time interval  t0 ,0  , the interaction and the Schroedinger pictures
coincide since the system is acted only by the unperturbed Liouvillian and so
we have
53
ρ  t ,t0  I  U0†  t ,t0  U  t ,t0  U0  t0 ,0  ρ 0 
I
By defining the super S-matrix (in the super-interaction picture) as,
S  t ,0   U0†  t ,t0  U  t ,t0  U0  t0 ,0 
(3.2.60)
(3.2.61)
This definition is reminiscent Eq. (3.2.53). Eq. (3.2.60)then becomes
ρ  t ,t0  I  S  t ,0  ρ 0 
I
(3.2.62)
If we let 0   and interpret ρ    I  ρeq I , the latter being the equilibrium
super-statevector, then Eq. (3.2.62) becomes,
ρ  t ,t0  I  S  t ,   ρeq
I
(3.2.63)
The left-hand side is the super-statevector at any time t after the interaction
has been switched on.
8.2.5.2 Time-Dependent Variational Principle in Liouville Space
In quantum field theory, correlation functions are derived using variational
principles. In this section let us explore the variational principle in Liouville
space. Our guide is the generalized variational principle as reviewed by Gerjouy
et al. (Gerjouy, Rau, & Spruch, 1983) and by Balian and Veneroni (Balian &
Veneroni, 1988). We construct the estimate of the expectation value of an
operator O
t1
 

 ρ  t  ,   t   ρ  t1      t   i
 L  ρ t 
t0
 t

t1
 

 ρ  t  ,   t   Tr  Oρ  t1       t   i
 L  ρ t 
t0
 t

t1
 

 ρ  t  ,   t   1 O ρ  t1      t   i
 L  ρ t 
t0
 t

t1
 

 ρ  t  ,   t   O  t  ρ  t1      t   i
 L  ρ t 
t0
 t









(3.2.64)
acting as constraint and Λ  t  as the Lagrange multiplier. Optimization of 
results to the following:
i   t1   O  t 
(3.2.65)
i ρ t   L ρ t 
(3.2.66)
i Λ t   L Λ t 
(3.2.67)
The variational principle, therefore, defines a dual super-statevector   t  .
The solutions of Eqs. (3.2.66) and (3.2.67), respectively are
 i t

ρ  t   T exp    L0  t   dt  S  t ,   ρeq
0


0
 i

Λ  t   T ac exp    L0  t   dt  S  t ,   O
t


54
(3.2.68)
(3.2.69)
where we let t1   and t2   . From the above equations, a transition
probability can be defined,
Λ t  ρ t  
1
1
O S  ,   ρeq 
i
i
H
Oρ
H
(3.2.70)
Whereit is assumed that the interaction is turned on at t0  0 and therefore,
(3.2.71)
S  0,   ρeq  ρ  0,   I  ρ H
H
O I O    O S  , 0
(3.2.72)
Using Eq. (3.1.77),
O O 1
which implies,
Λ t  ρ t  
1
1 OS  ,   ρeq
i
(3.2.73)
Of partcular interest is the situation in which,
Oi 1
This introduces a canonically conjugate pair Λ  t  , ρ  t  defined by the
transition probability
Λ  t  ρ  t   1 1S  ,   ρeq  1 S  ,   ρeq  1 S  , 0 S  0,   ρeq (3.2.74)

H
1ρ
(3.2.75)
H
An important result in the variational principle is the equation,
i
 1 S  ,   ρeq  exp 





i 
w  t   dt   exp  W 



(3.2.76)
where w is a Lagrange multiplier introduced during the variation and W is called
the effective action, the generating super-functional of the super-Green’s
function to be introduced later. Following Eqs. (3.2.52), (3.2.53), and (3.2.61),
 i
1 S  ,   ρeq  1 T exp 



L
 1I
 t dt ρeq

i 
 exp  W 


(3.2.77)
The above equation relates the effective action and the interaction term of the
Liouvillian.
8.3 Super-Green’s Functions
Equation Chapter 3 Section 4
Let us introduce the 4-component second quantized quantum field superoperator
 ˆ 1 
 † 
 1
 1   † 
(3.4.1)
ˆ 1 


  1 
55
where the argument is a short-hand notation for  r1, t1, 1  ;  1 is a quantum
number (or a set of quantum numbers).To identify the components of the 4component field super-operator, we adopt the notation that  1 represents its
components, where   1, 2, 3, 4 . Thus we can have the following identification
1 1  ˆ 1
 2 1   † 1
(3.4.2)
 3 1  ˆ † 1
 4 1   1
The commutation (anticommutation) relations of the 4-component field superoperator are the following:


(3.4.3)
 1 ,   2     r  ,   r '     r  r '   4

 

  1 ,   2     r  ,   r '     r  r '   4  x 



(3.4.4)
 † 1 , †  2   †  r  , †  r '     r  r '   4 



(3.4.5)
†
†

1
where
0
0

4  
 

0
0
0 0  
(3.4.6)
0 0 0

1 0 0
0  
 1 0
x  
;  
(3.4.7)


 0 
 0 1
The 4-component quantum field superoperator can be used to construct a
super Green’s function G1, 2 as
G1, 2  
where
0 1
I H T  H 1  H  2  ρ H
I H H
I H  1 S  , 0 
(3.4.8)
(3.4.9)
the subscript H indicates that the operators are in the super-Heisenberg
representation, and S  t , t0  is the S-matrix. Eq. (3.4.8)is the analogue of the
one-particle Green’s function in many-particle theory. The Green’s function is
the time-ordered ensemble average of cartesian or tensor product  H 1  H  2 ,
that is T  H 1  H  2  :
56
 ˆ 1  ˆ  2  
 †  †

 1   2  

 H 1  H  2   †
ˆ 1 ˆ †  2  




  1    2  
T  H 1  H  2 








ˆ H 1ˆ H  2 
ˆ H 1 †  2 
ˆ H 1ˆ †  2 
ˆ H 1 H  2  

 H 1 H  2 
 H 1 H  2  
 (3.4.10)
†
†
†
†
†
†
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
 H 1 H  2   H 1 H  2   H 1 H  2   H 1 H  2  

†
†
 H 1ˆ H  2   H 1 H  2   H 1ˆ H  2   H 1 H  2  
c
 g hh
1, 2  ghh 1, 2  G c 1, 2   G  1, 2  
 

ac
g hh 1, 2 
g hh
1, 2  G  1, 2   G ac 1, 2  

i
(3.4.11)

  G cT 1, 2   G T 1, 2  g eec 1, 2   g hh
1, 2  
 T

acT

ac
 G 1, 2  G 1, 2   g ee 1, 2  g ee 1, 2  
The identifications in Eq. (3.4.11) is based on Section 7.2. For a systematic
accounting of the different correlation functions, we define the following
matrices:
c
 g hh
1, 2  ghh 1, 2  
F 1, 2    
(3.4.12)

ac
 g hh 1, 2  g hh 1, 2  

 g eec 1, 2   g hh
1, 2  
†
F 1, 2    
(3.4.13)

ac

g
1
,
2
g
1
,
2




ee
ee


 H 1ˆ H  2 
†
H
†
H
 H 1ˆ H†  2 
†
†
 G c 1, 2   G  1, 2  
G 1, 2    

ac
 G 1, 2   G 1, 2  
  G cT 1, 2   G T 1, 2  
T
 G 1, 2    T

acT
 G 1, 2  G 1, 2  
†
(3.4.14)
(3.4.15)
and Eq. (3.4.8)can be written as,
 F 1, 2 
G 1, 2  
G 1, 2   T  L 1  L  2   i  T
(3.4.16)

†
  G 1, 2  F 1, 2  
The Green’s function is one of the class of functions called the moment
quantum distribution functions (MQDF), which is defined by
G1, 2, 3, , n   T  H 1  H  2   H  2   H  n 
(3.4.17)
The super Green’s function(3.4.8)is called the two-point MQDF, which is a
special case of Eq.(3.4.17).We also introduce the so-calledcorrelation quantum
distribution functions (CQDF),
57
K 1  G1
(3.4.18)
1
 PG1 G 2 

2! P
1
n     PG1 G 2 
n! P
K 1, 2   G1, 2  
K 1, 2, 3,
n   G1, 2, 3,
(3.4.19)
G n 
(3.4.20)
whose existence can be proven by graph theory. The quantity G i  , called the
one-point MQDF, is also a special case of (3.4.17) and is given by
G i    H  i 
(3.4.21)
Both classes of functions are called the generalized quantum distribution
functions (GQDF).
The MQDF can be generated from the effective action (the generating superfunctional) through the equation



, N   i
i
  u  N   u  N  1
i 
exp  W  G 1, 2, 3,


i

i 
 exp  W 
 u 1 



(3.4.22)
A recursion relation of the MQDF can be written down as,
G 1, 2, 3,


 W 

, n   i
i
 G 1, 2, 3,
 u  n  

  u n
where

W
 u  n
i
 ln 1 S  ,   ρeq
 u n
, n  1
 G n   K  n 
(3.4.23)
(3.4.24)
and  u  n  is the Schwinger external source.
The CQDF similarly can also be generated from
K 1, 2,
n  i

 n ln 1 S  ,   ρeq
n
 u  n   u  n  1  u  n  2   u 1
with the recursion relation
K 1, 2,
n  i
 K 1, 2, n  1
 u  n
(3.4.25)
(3.4.26)
The equations of motion of the MQDF can be obtained from the equation of
motion of the 4-component field super-operator  H  t  , which is given by
i
Th
  4 

 1

 H       H    , L 
t 
  r   r1 
M
 N 1
   N Dn 1 v 1, t  ; 2t  ;
N 1  n0

N , t   

58
(3.4.27)
 H 1; t    H  2; t  
 H  N ; t  
(3.4.28)
where Dn 1 is an operator whose action is to transpose the index 1, n places to
the right in v. Taking the average of the 4-component field super-operator gives
the equation of motion of the MQDF.
i

G     4    r   r1 
 1
t 

v 1, 2, N  
 N 1

    t   ti 
 G  2, 3, 4,
 N  1 ! 
N 1 
 n 0
M
where
G 2, 3, 4,
N    H  2  H  3 
v 1, 2,
N
(3.4.29)
H  N 
(3.4.30)
N
(3.4.31)
  1, 2 
(3.4.32)
N     P v 1, 2,
P
Eq. (3.4.29) can be brought into the form,
G01 1, 2    1, 2  
K 1, 2 
i
where
G01 1, 2   4  i  1, 2
1

 V 1, 2
t2
(3.4.33)
and  1, 2  , the super self-energy. Evaluation of the super self-energy is
outlined by Buot (Buot F. A., 2009). It is noted from Eq. (3.4.19) that
K 1, 2  G1, 2 for fermions.
8.4 Quantum Transport Equations
Equation Chapter 3 Section 4
Eq. (3.4.32) is the starting point of the quantum transport equations. It can be
recast into the form,
K 1 1, 2 K 1, 2   1, 2
(3.4.1)
with
K 1 1, 2 
1 1
G0 1, 2   1, 2
i 
Let us define the following:
  1, 2 
K 1, 2   i  T
  G 1, 2 

(3.4.2)
G 1, 2  

 1, 2  
(3.4.3)

 1, 2   F 1, 2  1, 2 

†

 1, 2   F 1, 2   1, 2  

G 1, 2   G 1, 2   1, 2 

 G T 1, 2    G T 1, 2  T 1, 2  
(3.4.4)
†
†
59
 1 1 
1
 P g h 1 g h  2  


i 2! P
 1 1
(3.4.5)

 1 1 
1
P 

1, 2  
  gh 1 gh  2 1 1 
i 2! P


†
In the above expressions gn i  ˆ i   i is the condensate wavefunction.
1, 2  
i 1,2  and i 1, 2 are the one-particle reduced density matrix and the
anomalous reduced density matrix of the boson condensate, respectively. They
are zero for fermions.
8.5 Energy Band Dynamics of Bloch Electrons
Equation Chapter 3 Section 5
8.5.1 Bloch and Wannier Functions
To reconcile the discussion of electron dynamics in crystalline solid in common
books on solid-state physics (e.g. Ashcroft and Mermin 1975) with that of Buot
(Buot F. A., 2009), we first follow the notation adopted in Ashcroft and Mermin
(Ashcroft and Mermin 1975). Then we will give the correspondence between
the notation adopted in the latter to that of the former (Buot 2009).
The dynamics of an electron in a periodic potential is given by Bloch’s theorem
which states that the electron’s wavefunction, the Bloch function, can be written
as,
 nk r   eikrunk r 
(3.5.1)
where
unk r  R   unk r 
(3.5.2)
and R is the lattice vector in the Bravais lattice. k is called the crystal
momentum and is a vector in the reciprocal lattice. Its values, in accord with the
Born-von Karman boundary conditions, are given by
3
m
(3.5.3)
k   i bi
i1 N i
with bi as the basis in recirprocal space. The other form of Bloch’s theorem is
 nk r  R   eikR nk r 
(3.5.4)
To show consistency with Buot’s notation, we make the following substitution:
R  q
k  p
and replace the band index n by  . We also note that k is the crystal
momentum which justifies the above replacement and r is the position variable.
We also write,
 nk r  b r, p
60
If one keeps r fixed, Bloch’s theorem implies that  nk r is a periodic function
of k . This means that  nk r can be expanded in Fourier series. Thus,

b r, p  N
3
1
2
i
qp
 e

w  r,q 
q
(3.5.5)
The Fourier coefficient w  r,q is called the Wannier function while b  r , p  is
obviously the Bloch function. Making use of the identity (actually the
completeness relation of plane wave-basis in the lattice),
e
i
 p q  q
 N 3 q ,q
(3.5.6)
p
we can derive the inverse relation,
w  r , q    N
 e
3 1 2
i
 q p
b  r , p .
(3.5.7)
q
The steps leading to Eq. (3.5.7) are the following:
e
i
 p q 
b  r , p    N
 e
3 1 2
i
i
q  p  p q 
e
w  r , q 
q
e
i
 p q 
b  r , p    N

3 1 2
p

i
 q q  p
w  r , q 

i
 q q  p
w  r , q 
 e
q
p
With the use of Eq. (3.5.6)
e
i
 p q 
b  r , p    N
  e
3 1 2
p
q
p
N 3 q ,q
e
p
i
 p q 
b  r , p    N
 N
3 1 2
 q,q w  r , q    N
3

3 12
w  r , q 
q
Changing q  q we obtain Eq. (3.5.7).
We also take note of the orthogonality condition for the plane waves in the
lattice,
e
i
 q  p  p
 N 3 p , p
(3.5.8)
q
It is clear that the phase space variables are discrete.
In Dirac notation, the Bloch and Wannier functions can be written as
b  r , p   r , p
w  r , q   r , q
where the basis kets (eigenkets) are given by:
, p  p
, q  q
61
(3.5.9)
(3.5.10)
(3.5.11)
(3.5.12)
In the above equations we define the collective variable p   , p  and q    , q  .
The orthonormality relations are
, p , p    pp
(3.5.13)
, q , q    qq
(3.5.14)
by17
The completeness (closure) relationsare also given
 q q   , q , q  1

(3.5.15)
 ,q
q
p   , p , p  1
p
(3.5.16)
, p
p
The transformation equation between the two bases is defined as,
 , p  , q   N

3 1 2
e
i
 p q
  
(3.5.17)
Using the closure relation (3.5.15) and transformation equation (3.5.17), we
could have written Eq. (3.5.5) as,




 , p    q q   , p     , q  , q   , p
 q

  q

 , p    , q  , q  , p
 q
, p  
 q
 , q  N

3 1 2
i
e
p q
     N
 e
3 1 2
i
p q
, q
(3.5.18)
q
The position representation of Eq. (3.5.18) is Eq. (3.5.5). The former equation
can be interpreted in a more general point of view. It suggests two bases
  , p  labeled by the band index  and p , and   , q  labeled by the band
index  and q , connected by a Fourier transformation. The wavefunctions
b  r , p  and w  r , q  can then be considered as position representations of the
general basis states  , p and  , q , respectively, in which the Bloch and
Wannier functions are special cases of these complete sets.
Likewise one can also have,


q   p p  q   p p q
p
 p

 , q    , p  , p  , q
p 
 , q    , p  N

3 1 2
p 
, q   N
 e
3 1 2
e
i
 p q
i
 p q
  
, p
(3.5.19)
p
17For
continuous eigenvalues of the position and momentum, the summations are replaced by
integrals.
62
8.5.2 Lattice Weyl-Wigner Formulation of Electron Band Dynamics
Using the closure relations (3.5.15) and (3.5.16), any operator A can be written
as
(3.5.20)
A   p p q q A q q p p
qqpp

A
  p   p  q  q A  q  q   p   p
(3.5.21)
q q pp    
   
A
qqpp    
  p  N
A  N


3 1 2
3 1
  e
 
i
 q p
  p e
 q A  q  N
i
 q p
 q A  q e

3 1 2
i
q p
i
q p
   e
i
q p
  p
  p
qqpp  
Changing   to   one obtains,
A  N

3 1
 
  p e
i
 q p
p  p  u
p  p  u
the above equation becomes

3 1
(3.5.22)
 , p  u e

i
q  q  v
q  q  v
 q v  p u 
(3.5.23)
 , q  v A  , q  v e
i
 q v  p u 
 , p  u
qqpp  
A  N
A  N
 
  p
qqpp  
Introducing the notation,
A  N
 q A  q e
 
2i
3 1
e
qu
2i
e
pv
 , q  v A  , q  v  , p  u  , p  u
qpuv   
  e
3 1
2i
pv
 , q  v A  , q  v
qp  v
e
2i
qu
 , p  u  , p  u (3.5.24)
u
with   change to  . Letting
A    p, q    e
2i
pv
 , q  v A  , q  v
(3.5.25)
 , p  u  , p  u
(3.5.26)
v
Δ    p, q    e
2i
q u
u
Eq. (3.5.24) can be written as
A  N
  A  p, q  Δ  p, q 
3 1
qp 
 
 
(3.5.27)
Using Eq. (3.5.19), one can write the lattice-Weyl transform A   p, q  in
alternative form. We first write down , q  v followingEq. (3.5.19) in the form,
 , q  v   N

3 1 2
e
p
63
i
 p q  v 
 , p
(3.5.28)
A    p, q    e
2i
 , q  v A  N
pv
p
v
A    p, q    N

3 1 2
  e
3 1 2
i
2i
 p q  v 
pv
e
e
i
 p q  v 
 , p
 , q  v A  , p
(3.5.29)
 , p
(3.5.30)
p
v
Similarly, , q  v can be written as,
, q  v   N

3 1 2
e
i
p q  v 
p
Substituting Eq. (3.5.30) into Eq. (3.5.29), one obtains
A    p, q    N
  e
3 1 2
3 1 2

3 1

3 1
 e
p
p q v 
 , p A  , p
i
i
2i
p q v   p q  v 
pv
e
 , p A  , p
e
p
 e
p
i
p
v
A    p, q    N
 N  e
e
p
v
A    p, q    N
i
2i
 p q  v 
pv

i
 p pq
e
p
 p2 p  pv

i
3

 , p A  , p
(3.5.31)
v
From Eq. (3.5.8),
e
i

 p2 p  pv
 N
(3.5.32)
p,2 p  p
v
Using the result (3.5.32) into Eq. (3.5.31),
A    p, q    N
  e


i
3 1
p
 p pq
p
A    p, q    e
p
i
 N 
3
 2 p  p pq
p,2 p  p
 , p A  , p
 , p A  , 2 p  p
p
A    p, q    e

2i
 p  pq
 , p A  , 2 p  p
(3.5.33)
p
Replacing p  with p  u ,
A   p, q    e

2i
 p  p u q
 , p  u A  , 2 p   p  u 
u
A    p, q    e
2i
qu
 , p  u A  , p  u
(3.5.34)
u
Similarly, Eq. (3.5.26) can be written in the form,
Δ    p, q    e
2i
pv
 , q  v  , q  v
v
using Eq. (3.5.18) to write  , p  u as,
, p  u   N

3 1 2
e
q
and  , p  u as,
64
i
 p u q
 , q
(3.5.35)
 , p  u   N
 e
3 1 2
i
 p u q
 , q
(3.5.36)
q
A  p, q  is the lattice Weyl transform of the operator A . One can also note
that Δ  p, q  is an operator,in fact it is a projection operator. It is for this
reason that we call it the phase-space point projector. Thus, Eq. (3.5.27)
means that an operator A is expanded in terms of the phase-space point
projector and that the expansion coefficients are the lattice-Weyl transform of
A . It has to be noted that the phase-space points  p, q  are discrete. Thus,
the operator A is cast into discrete phase space by virtue of Eq. (3.5.27). In
summary, quantum mechanics can be cast into lattice (discrete) phase space
by the bases   , p  and   ,q  and the lattice-Weyl transformation of
operator A .
8.5.2.1 The Trace of an Operator
The representation of an operator A in the form of Eq. (3.5.27) is convenient
in evaluating its trace.
Tr A   p A p     p A   p
p 
Tr A   N
 
3 1
 A   p, q  Δ   p, q    p
  p
  p

qp 

where we substitute Eq. (3.5.27) for A . Simplifying
Tr A   N
   A  p, q    p Δ  p, q    p
3 1
  p qp 
 
 
Substituting Eq. (3.5.26) for Δ  p, q 
TrA   N
TrA   N
   A  p, q    p  e
3 1
  p qp 
 
   e
2i
qu
  p qp  u
TrA   N
   e
2i
qu
  p qp  u

3 1
Tr A   N
qu
 , p  u  , p  u   p
A    p, q    p  , p  u  , p  u   p
3 1
TrA   N
2i
u
3 1
  e
2i
A    p, q     p, p u    p u , p
qu
p qp  u

3 1
Tr A   N
(3.5.37)
  p
 e
2i
A    p, q   p, p u   p u , p
qu
qp  u
  e
3 1
2i
A    p, q     p u , p u
qu
qp u
65
A   p, q   p u , p u
(3.5.38)
Tr A   N
  A  p, q 
3 1
(3.5.39)

qp
In some applications, the trace of a product of operators is required. For each
operator one can have, using Eq. (3.5.27),
A  N
B  N
The trace of AB is,
  A  p ,q Δ  p ,q 
(3.5.40)
 
(3.5.41)
3 1
 
p1q1 
3 1
p2 q2  
1
 
1
1
1
B   p2 , q2  Δ    p2 , q2 
Tr  AB     p AB  p
(3.5.42)
p
Tr  AB   N
3 2
   p  A  p ,q Δ  p ,q  
Tr  AB   N
3 2
p
  
 p p1q1  p2 q2  
Note that
Tr  AB    N
 
p1q1 
  
3 2
p1q1  p2 q2  
1
 
1
1
1
p2 q2  
B   p2 , q2  Δ    p2 , q2   p
A   p1, q1  B   p2 , q2   p Δ   p1, q1  Δ    p2 , q2   p
A   p1, q1  B   p2 , q2    p Δ   p1, q1  Δ    p2 , q2   p
p

Tr Δ   p1 ,q1  Δ    p2 ,q2 
Tr  AB   N
Continuing,
Tr  AB    N
 p
  
3 2
p1q1  p2 q2  
A   p1, q1  B   p2 , q2  Tr  Δ   p1, q1  Δ    p2 , q2  
  
3 2
 p p1q1  p2 q2  
e
2i
q1u1
A   p1, q1  B    p2 , q2 
 , p1  u1  , p1  u1  e
u1
Tr  AB    N
2i
q2 u2
 , p2  u2  , p2  u2  p
u1
     A  p , q B  p , q  e
3 2
 
 p p1q1  p2 q2   u1u2
1
1
 
2
2i
q1u1
2
2i
e
q2 u2
  p  , p1  u1  , p1  u1  , p2  u2  , p2  u2  p
Tr  AB    N
     A  p , q B  p , q  e
3 2
 
 p p1q1  p2 q2   u1u2
1
1
 
2
2i
q1u1
2
2i
e
q2 u2
  p , p1 u1   p1 u1 , p2 u2   p2 u2 , p
Tr  AB    N
     A  p , q B  p , q  e
3 2
 p p1q1  p2 q2   u1u2
 
1
1
 
2
2i
q1u1
2
2i
e
q2 u2
 p , p1 u1   p1 u1 , p2 u2   p2 u2 , p
Tr  AB    N
  
3 2

 A   p1, q1  B    p2 , q2  e
 p p1q1  p2 q2   u1u2
 p , p1 u1 p1 u1 , p2 u2   p2 u2 , p
66
2i
q1u1
2i
e
q2 u2
Tr  AB    N
    A  p , q  B  p , q  e
3 2
 
 p p1q1  p2 q2 u1u2
1
 
1
2
2i
q1u1
2
2i
e
q2 u2
 p , p1 u1 p1 u1 , p2 u2  p2 u2 , p
Tr  AB    N
    A  p , q B  p , q  e
3 2
 p p1q1  p2 q2
Tr  AB    N
1
 
1
2
2i
q1 p1  p 
2
    A  p , q B  p , q  e
3 2
 
 p p1q1  p2 q2
Tr  AB    N
Tr  AB    N
Tr  AB    N
 
1
 
1
2
2i
 
 p p1q1  q2
1
 
1
1
 p p1q1  q2
 
1
 
1
1
  p1q1 q2
 
1
 
1
1
1
q1 p1  p 

2i
2i
e

2i
1
2
2
q2  p  p2 
p p
1 2
q1 p  p1 
2i
e
q2  p  p1 
i
2i
 p 2 q1 2 q2   p1 q2  q1 
e
2
  A  p , q  B  p , q  e
3 2
 p  p  p , p  p  p 
2
    A  p , q B  p , q  e
3 2
q2  p  p2 
2
    A  p , q B  p ,q  e
3 2
2i
e
p1 q2  q1 
2
e
i
 p 2 q1  2 q2 
(3.5.43)
p
But from Eq. (3.5.6)
e
i
 p 2 q1 2 q2 
 N 3 2 q1 ,2 q2  N 3 q1 ,q2
(3.5.44)
p
Thus, Eq. (3.5.43) becomes
Tr  AB    N

3 1
 A    p1, q1  B   p1, q2  e

2i
p1 q2  q1 
q q
1 2
  p1q1 q2
Tr  AB   N
  A  p , q  B  p , q 
3 1
 
  p1q1
1
1
 
1
1
Implementing the change of variable,
q1  q
p1  p
  
gives the result
Tr  AB    N
  A  p, q  B  p, q 
3 1
 
  pq
 
(3.5.45)
8.5.2.2 The Lattice-Weyl Transform of Product of Operators
The lattice-Weyl transform of the product of operators AB can be written using
either Eq. (3.5.25)or Eq. (3.5.34). Thus we have, respectively,
 AB  p, q    e
2i
pv
 , q  v AB  , q  v
(3.5.46)
qu
 , p  u AB  , p  u
(3.5.47)
v
 AB  p, q    e
2i
u
We write the operators A and B as in Eq. (3.5.27):
67
A  N
B  N
  A  p ,q Δ  p ,q 
(3.5.48)
 
(3.5.49)
3 1
 
q1 p1 
3 1
q2 p2  
1
 
1
1
1
B   p2 , q2  Δ    p2 , q2 
If we use Eq. (3.5.46) to evaluate the lattice-Weyl transform, we should use Eq.
(3.5.35) for Δ . On the other hand, if we use Eq. (3.5.47) to evaluate the latticeWeyl transform, we should use Eq. (3.5.26) for Δ .Following the latter scheme,
 AB    p, q    e
2i
, p  u  N
qu
 
3 1
q1 p1 
u
A   p1, q1  Δ   p1, q1 
 N   B  p , q  Δ  p , q   , p  u
3 1
q2 p2  
 AB    p, q    N
 
2
 
2
2
2i
  
3 2
e
qu
u q1 p1  q2 p2  
2
A   p1, q1  B   p2 , q2 
 , p  u Δ   p1, q1  Δ    p2 , q2   , p  u
(3.5.50)
Using Eq. (3.5.26) for the Δ
Δ   p1, q1    e
2i
q1u1
 , p1  u1  , p1  u1
(3.5.51)
q2 u2
 , p2  u2  , p2  u2
(3.5.52)
u1
Δ    p2 , q2    e
2i
u2
Thus, one can have
 , p  u Δ   p1, q1  Δ    p2 , q2   , p  u 
, p  u
e
2i
q1u1
 , p1  u1  , p1  u1  e
u1
2i
q2 u2
 , p2  u2  , p2  u2  , p  u
u2
 , p  u Δ   p1, q1  Δ    p2 , q2   , p  u 
e
2i
 q1u1  q2 u2 
 , p  u  , p1  u1  , p1  u1  , p2  u2  , p2  u2  , p  u
u1u2
 , p  u Δ   p1, q1  Δ    p2 , q2   , p  u 
e
2i
 q1u1  q2 u2 
  p u , p u    p u , p u     p u , p u
1
1
1
1
2
2
2
2
u1u2
 , p  u Δ   p1, q1  Δ    p2 , q2   , p  u 
e
2i
 q1u1  q2  p2  p1  u1  
   p u , p u       p  p  p u , p u
1
1
2
2
1
1
u1
 , p  u Δ   p1, q1  Δ    p2 , q2   , p  u 
e
2i
 q1u1  q2  p2  q2  p1 q2 u1 
  p u , p u       2 p  p u , p u
1
1
u1
Substiuting the above equation to Eq. (3.5.50)
68
2
1
1
(3.5.53)
 AB    p, q    N
e
2i
2i
  
3 2
e
qu
uu1 q1 p1  q2 p2  
A   p1, q1  B   p2 , q2 
 q1u1  q2  p2 q2  p1 q2 u1
  pu , p u       2 p  p u , pu
1
 AB   p, q    N
   e
3 2
e
1
A   p1, q1  B   p2 , q2 
 2 p  p  p  p u , p u
2
   e
3 2
2i
qu
1
1
A   p1, q1  B   p2 , q2 
 q1 p1  p  u   q2  p2  q2  p1  q2  p1   p  u  
 2 p  2 p  p  u , p u
2
 AB    p, q    N
   e
3 2
2i
2i
q1 p1 q2 p2 
u
e
2i
q1 p1 q2 p2 
u
e
1
 q1 p1  p u   q2  p2  q2  p1  q2  p1   p u  
 AB    p, q    N
qu
1
A   p1, q1  B   p2 , q2 
 q1 p1  p  u   q2  p2  q2  p1  q2  p1   p  u  
u, p  p
1
 AB    p, q    N
e
2
qu
q1 p1 q2 p2 
u
2i
1
2i
 e
3 2
2i
q  p1  p2 
q1 p1 q2 p2 
2
A   p1, q1  B    p2 , q2 
2i 

 q1 p1  p   p1  p2    q2  p2  q2  p1  q2  p1   p   p1  p2   

 AB    p, q    N

 e
3 2

2i

 q  p1  p2   q1  p2  p   q2  p  p1  
q1 p1 q2 p2 
A   p1, q1  B  p2 , q2 
(3.5.54)
The bracketed quantity in the exponent can be written as
q   p1  p2   q1   p2  p   q2   p  p1   q1   p2  p   q  p1  q  p2  q2   p  p1 
 q1   p2  p   q  p1  q  p2  q2   p  p1   q  p  q  p
 q1   p2  p   q   p  p1   q   p2  p   q2   p  p1 
 q1   p2  p   q   p2  p    q2  q    p  p1 
  q1  q    p2  p    q2  q    p  p1 
Letting pˆ  p2  p and qˆ  q2  q , we obtain
q   p1  p2   q1   p2  p   q2   p  p1    q1  q   pˆ  qˆ   p  p1 
Substituting the above expression to Eq. (3.5.54), one gets
 AB    p, q    N

3 2
 e
2i
 pˆ  q1  q   qˆ  p1  p 
ˆˆ 
q1 p1 qp
A   p1 , q1  B   pˆ  p, qˆ  q  (3.5.55)
A simplification of the above equation can be carried out if we assume that
there are two continuous functions of p and q such that they are equal to A 
69
and B   at each lattice point. Thus, we are justified in performing a Taylor series
expansion of B  pˆ  p, qˆ  q  about  p, q  ,
1

 
 pˆ   qˆ   B   p, q 
p
q 
n 0 n ! 

B   pˆ  p, qˆ  q   


 
 exp  pˆ   qˆ   B    p, q 
q 
 p
where
(3.5.56)


  p and
  q . Eq. (3.5.55)then becomes,
p
q
 AB    p, q    N
  e
3 2
2i
 pˆ  q1  q   qˆ  p1  p  
ˆˆ 
q1 p1 qp
A   p1 , q1 


 
 exp  pˆ   qˆ   B   p, q 
(3.5.57)
q 
 p
To simplify things further, we note that
2i
 pˆ  q1  q  qˆ  p1  p 
 2i  pˆ  q1 q qˆ p1  p  2i 2i  pˆ  q1 q qˆ p1  p 
 2i  pˆ  q1 q qˆ p1  p 
ˆ
ˆ
e
 qe

e
 qe
p
2i p
 2i  pˆ  q1 q qˆ p1  p 
2i 2i  pˆ  q1 q qˆ p1  p 
ˆ
e
  pe
q
2i
 pˆ  q1  q  qˆ  p1  p 
 2i  pˆ  q1 q qˆ p1  p 
ˆ

e
  pe
2i q
Thus, we may write Eq. (3.5.57) as
 AB  p, q    N
  A  p , q 
3 2
ˆˆ 
q1 p1 qp

1
1
 2i  pˆ  q1 q qˆ p1  p 

   
  
 e
exp        B   p, q  
(3.5.58)
 2i  p q q p  


where the left arrows indicate that the derivative acts on the left, that is on the
exponential function, while the right arrows indicate derivatives acting on
B  p, q  .
2i
2i
pˆ  q1  q 
 qˆ  p1  p  

A
p
,
q
e
e


    1 1 

qˆ
q1 p1  
pˆ

   
  
exp        B   p, q 
 2i  p q q p  
Using Eqs. (3.5.6) and (3.5.8),


3 2
3
3

 AB    p, q    N    A   p1 , q1  N  2 q ,2 q1 N  2 p1 ,2 p 
q1 p1  

 q ,q1
p1 , p 

 AB  p, q    N
3 2
70
   
  
exp        B   p, q 
 2i  p q q p  
   
  
      B   p, q 
 2i  p q q p  
 AB    p, q    A   p, q  exp 

(3.5.59)
Using another notation,
    a   b    a   b   
 AB    p, q    exp       A   p, q  B   p, q  (3.5.60)

 2i  p q q p  
 
The superscript a  b  in the derivative operator means that it acts on A  B   .
If we consider the Weyl transforms as matrices, with the elements determined
by the band indices, that is, the band indices as row and column indices, one
can see that the summation over  together with the indices  and   implies
matrix multiplication. In matrix form, Eq. (3.5.60) can then be written as
    a   b    a   b   
(3.5.61)
 AB  p, q   exp       A  p, q  B  p, q 
 2i  p q q p  
8.5.2.3 The Lattice-Weyl Transform
Anticommutator of Operators
For the commutator:
of
the
Commutator
 A, B  p, q    AB  BA 
and
the
(3.5.62)
In matrix form:
   a   b   a   b  




  A  p, q  B  p, q 
 2i  p q q p  

 A, B  p, q   exp 
   b   a   b  a   
 exp  



  B  p, q  A  p, q 
 2i  p q q p  
Take note that the second exponential operator can be written as
   b    a   b    a   

   a   b    a   b   
exp  






   exp   
 
 2i  p q q p  
 2i  p q q p  
Thus,
    a   b   a   b  
A
,
B
p
,
q

exp



  
 
  A  p, q  B  p, q 
2
i

p

q

q

p
 
 

   a   b   a   b  
 exp   



  B  p, q  A  p, q 
 2i  p q q p  
 i    a   b   a   b   
 A, B  p, q   exp        A  p, q  B  p, q 
 2  p q q p  
71
(3.5.63)
i
 exp 
 2
   a   b   a   b   




  B  p, q  A  p, q 
 p q q p  
    a   b   a   b   




  A  p, q  B  p, q 
 2  p q q p  
    a   b   a   b 
i sin  



  A  p, q  B  p, q 
 2  p q q p  
 A, B  p, q   cos 
    a   b   a   b  
 cos  



  B  p, q  A  p, q 
 2  p q q p  
    a   b   a   b  
i sin  



  B  p, q  A  p, q 
 2  p q q p  
    a   b  a   b  




  A  p, q  B  p, q   B  p, q  A  p, q 
 2  p q q p  
 A, B  p, q   cos 
    a   b   a   b 
i sin  



  A  p, q  B  p, q   B  p, q  A  p, q  (3.5.64)
 2  p q q p  
For the anticommutator:
A, B  p, q    AB  BA 
(3.5.65)
In matrix form
    a   b   a   b   
A, B p, q   exp       A  p, q  B  p, q 
 2i  p q q p  
   b    a   b    a   
 exp  



  B  p, q  A  p, q 
 2i  p q q p  
    a   b   a   b   
A, B p, q   exp       A  p, q  B  p, q 
 2i  p q q p  

   a   b    a   b  
 exp   



  B  p, q  A  p, q 
 2i  p q q p  
 i    a   b   a   b 




  A  p, q  B  p, q 
 2  p q q p  
A, B p, q   exp  
i
 exp 
 2
   a   b    a   b   




  B  p, q  A  p, q 
 p q q p  
72
(3.5.66)
    a   b   a   b   
A, B p, q   cos       A  p, q  B  p, q 
 2  p q q p  
    a   b    a   b   
 i sin  



  A  p, q  B  p, q 
 2  p q q p  
    a   b   a   b 
 cos  



  B  p, q  A  p, q 
 2  p q q p  
 i    a   b   a   b 
i sin  



  B  p, q  A  p, q 
 2  p q q p  
    a   b   a   b  
A, B p, q   cos       A  p, q  B  p, q   B  p, q  A  p, q 
 2  p q q p  
 i    a   b  a   b  
i sin  



  B  p, q  A  p, q   A  p, q  B  p, q  (3.5.67)
 2  p q q p  
8.5.3 Weyl Transformation in Continuous Phase Space
For the sake of completeness we discuss in this section the Weyl
transformation in continuous phase space. Another reason for discussing it is
that in actual calculation, we will be adopting the continuous phase-space
approximation. The closure relation for continuous p and q are given by
  dp  p
  dq  q
where it is understood that
 p   dp p p  1
(3.5.68)
 q   dq  q  q  1
(3.5.69)
 dp    dp
(3.5.70)
 dq    dq

Recall also that
 q   p  q p 
1
 2 
32
eipq   
(3.5.71)
The analog of the completeness relation (3.5.6) can be obtained from Eq.
(3.5.68),


 , q    dp  p  p   q   , q  q
 

73

 dp  , q  p

  dp
1
 2 
 
eipq 
32
1
 p  q     q  q
1
 2 
dpe
 
 2
dpe
 
 2
e  ipq        q  q
e ipq     q  q
ipq
3
1
32
   q  q 
ip q  q
3
(3.5.72)
The orthogonality in the Dirac sense is also given by
p p   , p  , p     p  p


 , p    dq  , q  , q   , p     p  p




 dq  , p  , q

1
  dq  2 
 
32
1
 2
1
 2
 , q  , p     p  p
1
e ipq   
 2 
 i  p  pq
    p  p
 i  p  pq
   p  p
dqe
 
3
dqe
 
eipq     p  p
32
3
We also note that the position eigenkets can be written as,
q   , q    dp  , p  , p  , q
(3.5.73)
(3.5.74)

Using Eq. (3.5.71)
q   , q    dp  , p

q  ,q 
Similarly,
1
1
 2 
dpe
 
 2
32
 ipq
32
e  ipq   
, p
 , p    dq  , q  , q  , p
(3.5.75)
(3.5.76)

, p 
1
 2
, p 
 dq  , q
  
32
1
 2

dqe
 
ipq
32
Any operator A can then be written in the form
74
eipq   
, q
(3.5.77)
  dq dq dp dp   p
A
  p  q  q A  q  q   p   p
    
Using Eq. (3.5.71)

A
    
   
1
i
 2 
A
e
32
1
 2
 dq dq dp dp   p   
pq
1
 2 
i
 pq
 q A  q
  p
 dq dq dp dp   p e
     
3
e
32
i
 pq
 q A  q e
i
pq
  p (3.5.78)
 
We introduce the new variables
1
1
p  p  u
q  q  v
2
2
1
1
p  p  u
q  q  v
2
2
and   to  , so that the above equation can be written as,
A
1
 2 
3

 dq  dv  dp  due

i 1  1 
  p  u  q  v 
 2  2 

1
2
1
 2 
dq  dv  dp  due
3 
i
 pq
i
e2
1
1
 , q  v A  q  v e
2
2
1
2
u q  pv 
e

1
2
i
u v
4
1
2
 2 
dq  dp  dve
3 
i
u q
pq
i
e2
u q  pv 
1
2
i
pv
 
 due
i
 , p  u
, p  u
1
1
2
 , q  v A  , q  v e
 
A
i 1  1 
 p  u  q  v 
 2  2 
 , p  u
, p  u
A
(3.5.79)
1
2
, p  u
1
2
1
2
 , q  v A  , q  v
1
2
 , p  u
(3.5.80)
Defining,
A    p, q   A  p, q    dve
i
Δ    p, q   Δ  p, q    due
pv
i
qu
1
2
1
2
(3.5.81)
1
2
1
2
(3.5.82)
 , q  v A  , q  v
 , p  u  , p  u
Eq. (3.5.80) can be written as,
1
A
dq dpA    p, q  Δ    p, q 
3 
 2     
We can also write this in the form
75
(3.5.83)
i
e4
u v
A
using the notation
1
dqdpA    p, q  Δ    p, q 
3
 2  
(3.5.84)
 dqdp  
  dqdp .


The Weyl transform can also be written in terms of the momentum by making
use of Eq. (3.5.75). We write
1 

 ip q  v 
1
1
 2 

 , q  v 
dp
e
32 
2
 2 
 , p
(3.5.85)
 , p
(3.5.86)
Likewise,
1 

ip q  v 
1
1
 2 

, q  v 
dp
e
32 
2
 2 
Thus, Eq. (3.5.81) can be written as,
A    p, q    dve
i
1
pv
 2
A    p, q  
1
 2 
A    p, q  
A   p, q  
A    p, q  

dpe
32 
 1 
ip q  v 
 2 
dvdpdpe
3 
1
 2 
3
1
 2 
1
 2 
3
i
pv
 2 
3
i
e
i1
1

  p p p v
2
2

1
2
 p pq
i
e
 1 
 ip q  v 
 2 
 , p A  , p
e
i 1
1 
 p  p p v
2 
 2
 dve
1
e

i  1 
 1 
p q  v   p q  v 
 2 
 2 
i  1
1 

 p  2 p 2 p v   p pq 



 dvdpdpe
dvdpdpe
3 
A    p, q    dpdp
i
 2
dpe
32 
i
 1 
 1 
 pv  p q  2 v   p q  2 v 





 dvdpdpe
1
1
 , p A
 , p A  , p
 , p A  , p
 , p A  , p
 p pq
 , p A  , p
  p p p 
2

1   i  p pq
1




A    p, q    dp dp   p   p  p   e
 , p A  , p
2
2




A    p, q   2 dpe
2i
 p  pq
 , 2 p  p A  , p
1
Letting p  p  u
2
 u
A    p, q   2 d    e
 2
2i  
1 
 p  p  u  q
  2 

76
1
2 
1
2
 , 2 p   p  u  A  , p  u
 , p
1
1
2
2
Since the integrand is evaluated over all the momentum space, the negative
sign has no effect on the integral. Thus,
i
u q
1
1
(3.5.87)
A   p, q    due
 , p  u A  , p  u
2
2
Similarly, using similar steps,
i
pv
1
1
(3.5.88)
Δ    p, q    dve
 , q  v  , q  v
2
2
A   p, q    d  u  e
i
u q
 , p  u A  , p  u
8.5.3.1 The Trace of an Operator in Continuous Phase Space
The trace of an operator can be written in terms of the basis
Using the latter,
 q  or  p  .
TrA   dp p A p    dp  , p A  , p
(3.5.89)

Inserting expression (3.5.83) or (3.5.84) for A
1
Tr A 
dp p  dqdpA    p, q  Δ    p, q  p
3
 2  
1
TrA 
dp  , p   dq  dpA    p, q  Δ    p, q   , p
3 
 2   
 
1
TrA 
dp dq dpA    p, q   , p Δ    p, q   , p
3 
 2     
Note that,
1
Tr A 
dq dpA    p, q    dp  , p Δ    p, q   , p
3 
 2     
 
Using Eq. (3.5.82) for Δ  p, q 
TrA 
1
 2 
3
 2
,


 
TrA 
Tr Δ    p ,q 

 dq  dpA  p, q 

  dp  , p
1
 due
i
qu
1
2
, p  u
 dq dpA  p, q  
 dp due
   

3
(3.5.90)



77
i
qu
1
2
 , p  u  , p
1
2
 , p  , p  u
1
2
 , p  u  , p
TrA 
TrA 
1
 2 
1
 2 
TrA 
1
 2
dq  dpA    p, q    dp due
3 
 
qu
i
qu
 dq dpA  p, q   dp due
   
i
3
 2
qu
 2 


 2 
3


1
 

1


i

1
qu
 

1
1


  p  u   p  u  
2
2


i

1
1
 
1
dq  dpA   p, q   due
3 
Tr A 

  p   p  u     p  u  p 
2
2
3
1


 dq dpA  p, q   due
   
TrA 
1
    p   p  u     p  u  p 
2
2

 
1

       p   p  u     p  u  p 
2
2

 
dq  dpA    p, q   dp due
3 
TrA 
i
qu

 u 
  dpdqA  p, q 
(3.5.91)
For single band, this becomes
Tr A 
1
dpdqA  p, q 
3
 2  
The trace of the product of operators AB is given by
Tr  AB   dp p AB p    dp  , p AB  , p
(3.5.92)
(3.5.93)

We can use Eq. (3.5.83) to write A and B as follows:
1
A
dq1 dp1A   p1, q1  Δ   p1, q1 
3 
 2     
1
B
dq2 dp2B   p2 , q2  Δ    p2 , q2 
3 
 2     
and the Tr  AB becomes
Tr  AB  
1
 2 
6
  dpdq dp dq dp
1
  
1
2
2
  , p A   p1, q1  Δ   p1, q1  B   p2 , q2  Δ    p2 , q2  , p
Take note that,
1
Tr  AB  
dq1dp1dq2dp2 A   p1, q1  B   p2 , q2 
6 
 2     
   dp  , p Δ   p1, q1  Δ    p2 , q2   , p


Tr Δ   p1 ,q1  Δ    p2 ,q2 

Now the Δ ’s can be written as follows, using Eq. (3.5.82):
78
(3.5.94)
(3.5.95)
Δ   p1, q1    du1e
Δ    p2 , q2    du2e
Tr  AB  
1
 2 
6
i
q1u1

i
q2 u2
Tr  AB  
1
1
  dpdu1du2e
i
q2 u2
 , p2  u2
1
2
1
2

2
1
1
2
1
2
6
1
2
 , p1  u1  , p1  u1
1
2
 , p2  u2
1

2
2
1
2
1
2
  dq dp dq dp A  p , q  B  p , q 
1
i
e
1
2

2
1
1
2
q2 u2
1

2
2
1
2
Simplifying the summations,
1
Tr  AB  
dq1dp1dq2dp2 A   p1, q1  B   p2 , q2 
6 
 2     
q1u1
i
q2 u2
e

     
1  
1
1  
1




  p   p1  u1     p1  u1   p2  u2     p2  u2  p 
2  
2
2  
2




1
Tr  AB  
dq1dp1dq2dp2 A   p1, q1  B  p2 , q2 
6 
 2    
 dpdu1du2e
i
q1u1
i
e
q2 u2
  
1  
1
1  
1




  p   p1  u1     p1  u1   p2  u2     p2  u2  p 
2  
2
2  
2




1
Tr  AB  
dq1dp1dq2dp2 A   p1, q1  B   p2 , q2 
6 
 2    
 dpdu1du2e
i
q1u1
i
e
q2 u2
1
2
 , p  , p1  u1  , p1  u1  , p2  u2

  dpdu1du2e
(3.5.97)
 , p2  u2  , p
  
q1u1
i
(3.5.96)
 , p1  u1  , p1  u1
 , p2  u2
 2 
i
1
2
q1u1
  dq dp dq dp A  p , q  B  p , q 
  
  dp  , p  du1e
 du2e
i
 
1  
1
1  
1




  p   p1  u1     p1  u1   p2  u2     p2  u2  p 
2  
2
2  
2




79
1
2
 , p2  u2  , p
Tr  AB  
1
 2 
6

 dq dp dq dp A  p , q  B   p , q 

1

1

2

2
1

1
1  
 
2
 
 
1


1
1  
1
1
 


 

 
To evaluate the integral with u1 as the variable, we let
1
u1  u1  du1  2du
2
1
Tr  AB  
dq1dp1dq2dp2 A   p1, q1  B   p2 , q2 
6 
 2    


 dpdu1du2e
i
q1u1
i
e
q2 u2


1
2
1


  p   p1  u1     p1  u1   p2  u2     p2  u2  p 
2
2
2
2

We can now simplify the integrations:
1
Tr  AB  
dq1dp1dq2dp2 A   p1, q1  B   p2 , q2 
6 
 2    
 dpdu1du2e
i
q1u1
i
e
2 dpdu1du2e
Tr  AB  
2i
1
 2 
6
q2 u2
q1u1
1
 2 
6
i
e
q2 u2

 
 
1


1


  u1   p1  p     p1  u1   p2  u2     p2  u2  p 
2
2


 dq dp dq dp A  p , q  B   p , q 

1

2 dpdu2e
Tr  AB  
  u1   p1  p     p1  u1   p2  u2     p2  u2  p 
2
2
2
2
2i
q1 p1  p 
1
i
e
2
q2 u2

2
1

1

2
2
1


 
 
1


  p1   p1  p    p2  u2     p2  u2  p 
2
2


 dq dp dq dp A  p , q  B   p , q 

1

2  dpdu2e
2i
1
q1 p1  p 
2
i
e

2
q2 u2
1

1

2
1
2


2
  1
  2


  2 p1  p   p2  u2     u2   p  p2  

1
We also let u2  u2 to evaluate the integral over u2 .
2
1
Tr  AB  
dq1dp1dq2dp2 A   p1, q1  B   p2 , q2 
6 
 2    
4 dpdu2e
2i
q1 p1  p 
Tr  AB  
4 dpe
2i
e
1
 2 
2i
6
q1 p1  p 
q2 u2
  2 p1  p   p2  u2     u2   p  p2  

 dq dp dq dp A  p , q  B   p , q 


2i
e
1
1
2
2

1
1

2
  2 p1  p   p2   p  p2   
q2  p  p2 
80
2
Tr  AB  
1
 2 
6

 dq dp dq dp A  p , q  B   p , q 

4 dpe
1

2i
q1 p1  p 
1
2i
e
2

2
1

1
2
2
q2  p  p2 
  2 p1  2 p2 
In the same manner we can also let p1  2 p1 to evaluate the integral over p1
Tr  AB  
4
 2
1
Tr  AB 
i
q1 p1 2 p 
1
2i
2

2
1

1

1
2
2
q2  p  p2 
  p1  2 p2 
e
2i
2i
q1 p2  p 
q2  p  p2 
1
dpdq
dq
dp
A
p
,
q
B
p
,
q
e
e







1
2
2

2
1


2
2
2   
4
 2 
1

e
Tr  AB 

 dpdq  2 dp  dq dp A  2 p, q  B   p , q 
  
6
6
2
 2 
dpdq1dq2dp2 A   p2 , q1  B   p2 , q2  e
6 
2i
 q1 p2  p  q2  p  p2 
 
Take note that the exponent can be written as:
q1   p2  p   q2   p  p2    q2  q1    p  p2 
q1   p2  p   q2   p  p2   p   q2  q1   p2   q2  q1 
Tr  AB 
Tr  AB 
Tr  AB 
2
 dpdq dq dp A  p , q  B   p , q  e
  
 2
6
2
 2 
2
 2 
1

2

2
2

1
2
2i
 p q2  q1  p2  q2  q1 
2
dpdq1dq2dp2 A   p2 , q1  B   p2 , q2  e
6 
2i
p q2  q1  
2i
e
p2  q2  q1 
 
2i
dq1dq2dp2 A   p2 , q1  B   p2 , q2   dpe
6 
p q2  q1  
e
2i
p2  q2  q1 
 
The integral over p can be evaluated by letting 2 p  p :
Tr  AB  
2
 2 
1 1
2  2
dq1dq2dp2 A   p2 , q1  B   p2 , q2 
3 
 

 dpe
3
i
p q2  q1 
e

  q2  q1 
Tr  AB 
1
 2
 dq dq dp A  p , q  B   p , q    q
  
Tr  AB 
3
1

1
 2 
Tr  AB  
2
2

2

1
2
1
2
dq1dp2 A   p2 , q1  B   p2 , q1  e
3 
 q1  e

2i
 2 
3

 dq dp A  p , q  B   p , q 


1
2
Changing the variables q1  q, p2  p ,
81

2
1

2
1
2i
p2  q1  q1 
 
1

p2  q2  q1 
2i
p2  q2  q1 
Tr  AB  
1
 2 
3

 dpdqA  p, q  B   p, q 



(3.5.98)

For a single band,
1
Tr  AB  
dpdq A  p, q  B  p, q 
3
 2  
(3.5.99)
8.5.3.2 Weyl Transform of Product of Operators in Continuous Phase Space
We follow the similar steps as in the previous section, using Eq. (3.5.81) to write
the Weyl transform as follows,
i
pv
1
1
(3.5.100)
 AB  p, q    dve  , q  v AB  , q  v
2
2
One can also write the Weyl transform using the Eq. (3.5.87)
i
qu
1
1
(3.5.101)
 AB  p, q    due  , p  u AB  , p  u
2
2
We also write the individual operators using Eq. (3.5.83) in the form,
1
A
dq1 dp1A   p1, q1  Δ   p1, q1 
(3.5.102)
3 
 2     
1
B
dq2 dp2B   p2 , q2  Δ    p2 , q2 
(3.5.103)
3 
 2     
In evaluating the Weyl transform of the product of operators, we can use either
use Eq. (3.5.100) together with Eq. (3.5.88) for the Δ ’s or use Eq. (3.5.101)
together with Eq. (3.5.82) for the Δ ’s. We follow the latter so that
i
qu
1
1
AB
p
,
q

due
, p  u
    
 dq1 dp1A  p1, q1  Δ   p1, q1 
2  2 3    

1
 2
1
dq  dp B  p , q  Δ   p , q   , p  u


2
 
3

 AB  p, q  
  due
i
q u
2
2
1
 2 
3

2

2
2
2
  dq dp dq dp A  p , q  B  p , q 
  
1
1
2
2
1
2

1
1

2
1
2
 , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u
2
(3.5.104)
Weyl transform of Δ   p1 , q1  Δ    p2 , q2 
1
1
Let us first consider the quantity,  , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u .
2
2
We use Eq. (3.5.82) for the Δ ’s:
i
q1 u1
1
1
(3.5.105)
Δ   p, q    du1e
 , p1  u1  , p1  u1
2
2
i
q2 u2
1
1
(3.5.106)
Δ    p, q    du2 e
 , p2  u2  , p2  u2
2
2
82
1
2
1
2
 , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u 
 du du e
1
i
q1 u1
2
i
e
1
2
q2 u2
1
2
1
2
1
2
 , p  u  , p1  u1  , p1  u1  , p2  u2
1
1
  , p2  u2  , p  u
2
2
1
2
1
2
 , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u 
 du1du2 e
i
q1 u1
i
e
q2 u2

1


1



1


1


    p  u   p1  u1      p1  u1   p2  u2  
2
2
2
2


1
1 


     p2  u2   p  u  
2
2 


1
2
1
2
 , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u 
         du1du2 e
i
q1 u1
i
e
  1
  2
1
1 


  p2  u2   p  u  
2
2 


q2 u2
1
2
1
2


1
2




  u1   p1  u  p    u2   p2  p1  u1  
1
2
1
2
 , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u 
2         du1e
i
q1 u1
2i
1 

q2  p2  p1  u1 
2 

1
2


1
2


  u1   p1  u  p  
e
1
1 


   p2  p2  p1  u1   p  u  
2
2 


1
1
  2 p2  p1  p  u1  u 
2
2


1
2
1
2
 , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u 
2i
4        e
1

 2 i q  p  p   p  1 u  p  
q1  p1  u  p 
2  2
1  1

2
2





e



1


1 
  2 p2  p1  p   p1  u  p   u 
2
2

1
2
1
2
 , p  u Δ   p1 , q1  Δ    p2 , q2   , p  u 
83

2i
4    e
1
1

 2i 

q1  p1  u  p 
q2  p2  2 p1  u  p 
2
2




e
  u   2 p1  2 p2  
(3.5.107)
Substituting Eq. (3.5.107) to Eq. (3.5.104)
1
dq1dp1dq2 dp2 A   p1 , q1  B   p2 , q2 
 AB    p, q  
3 
 2     
 due
i
q u
2i
4        e
 AB    p, q  
e
4
 2 
i
3
         dq dp dq dp A  p , q  B  p , q 
2i
e

 
1
1
2

2
1

1
2
2
1
1

 2i 

q1  p1   2 p1  2 p2   p 
q2  p2  2 p1   2 p1  2 p2   p 
2
2




e
4
 2 
e
 AB    p, q  
  u   2 p1  2 p2  
e
  
q  2 p1  2 p2 
 AB    p, q  
1
1

 2i 

q1  p1  u  p 
q2  p2  2 p1  u  p 
2
2




i
3

 dq dp dq dp A  p , q  B   p , q 

q  2 p1  2 p2 
4
 2 
3
2i
1

2i
e
1
2
2
q1  p2  p 
2i
e

1
1
 
2
2
q2  p  p1 

 dq dp dq dp A  p , q  B   p , q 


1
1
2
2

1
1
 
2
2
 q  p1  p2   q1  p2  p   q2  p  p1 
e
(3.5.108)
The exponent in Eq. (3.5.108) is exactly the same as in Eq. (3.5.54). Using
similar steps we arrive at an expression the same as Eqs. (3.5.60) and (3.5.61).
For the single-band case
    a   b   a   b  
 AB  p, q   exp       A  p, q  B  p, q  (3.5.109)
 2i  p q q p  
8.5.3.3 Weyl Transform of Commutator and Anticommutator of Operators
The formulas for the commutator and anticommutator turn out to be the same
as Eqs. (3.5.64) and (3.5.67), respectively. For a single band using Eq.
(3.5.109), we have
 i    a   b  a   b  
 A,B  p, q   AB  p, q   BA  p, q   exp        A  p, q  B  p, q 
 2  p q q p  
 i    a   b   a   b  
 exp  



  A  p, q  B  p , q 
 2  p q q p  
 i    a   b   a   b  




  A  p, q  B  p, q 
 2  q p p q  
 A,B  p, q   AB  p, q   BA  p, q   exp 
84
 i
 exp  
 2
   a   b  a   b 




  A  p, q  B  p, q 
 q p p q  
  a
A,B
p
,
q

2
i
sin
  
 
 2  q
Similarly, for the anticommutator,
  a
A,B
p
,
q

2
cos
  
 
 2  q

 b   a   b  


  A  p, q  B  p, q 
p p q  
(3.5.110)

 b   a   b  


  A  p, q  B  p, q 
p p q  
(3.5.111)
8.5.4 Integral Form of the Weyl Transform of the Commutator and
Anticommutator
Usually, it is more convenient computationally if the Weyl transform of the
commutator and the anticommutator are written in integral from. These forms
can be derived from the corresponding differential form by using the
displacement operator. The results are:
1
W  AB  
dpdqK A  p, q; p, q  B  p, q 
8 
 2 
(3.5.112)
1


dpdqA  p, q  KB  p, q; p, q 
8 
2

 
1
W  A, B 
dpdq  K A  p, q; p, q  B  p, q   B  p, q  K A  p, q ; p, q   (3.5.113)
8 
 2 
W A, B 
1
 2
dpdq  K  p, q; p, q  B  p, q  +B  p, q  K  p, q; p, q   (3.5.114)
 

A
8

A
where
KY  p, q; p, q 
1
 2
u
i
 
dudv exp   p  p   v   q  q   u  y  p  , q

2

 

8
v
 (3.5.115)
2
For scalar functions,
W  A, B 
W A, B 
1
 2
1
 2
dpdqK  p, q; p, q  B  p, q 
 
(3.5.116)
dpdqK  p, q; p, q  B  p, q 
 
(3.5.117)
s
A
8
c
A
8
where
i

K Ys , c  p, q; p, q    dudv exp   p  p   v   q  q   u 


85
 
u
v
 y p  ,q  
2
2
 
u
v 

y  p  , q  
2
2 

(3.5.118)
The other forms are:
W  A, B  
W A, B 
i
 2
dpdqK  p, q  q; p  p, q  B  p, q 
 
(3.5.119)
 2
dpdqK  p, q  q; p  p, q  B  p, q 
 
(3.5.120)
8
8
s
A
c
A
where
i

K Ys  p, q  q; p  p, q    dudv exp  u   q  q  


 
u
v
u
v     p  p   v 

  Y  p  , q    Y  p  , q    sin 

2
2
2
2  

 

i

K Yc  p, q  q; p  p, q    dudv exp  u   q  q  


  p  p   v 
 
u
v 
u
v 
  Y  p  , q   +Y  p  , q    cos 

2
2 
2
2 
 


(3.5.121)
(3.5.122)
See (Mahan G. D., 1987) for transport equation discussion.
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