Download Field Formulation of Many-Body Quantum Physics {ffmbqp

A common mistake that people make
whenFIELDS
trying to design something completely foolproof
H. Kleinert, PARTICLES AND QUANTUM
November 19, 2016 ( /home/kleinert/kleinert/books/qft/nachspa1.tex)
is to underestimate the ingenuity of complete fools.
Douglas Adams (1952-2001)
2
Field Formulation of Many-Body
Quantum Physics
{
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A piece of matter composed of a large number of microscopic particles is called a
many-body system. The microscopic particles may either be all identical or of different species. Examples are crystal lattices, liquids, and gases, all of these being
aggregates of molecules and atoms. Molecules are composed of atoms which, in turn,
consist of an atomic nucleus and electrons, held together by electromagnetic forces,
or more precisely their quanta, the photons. The mass of an atom is mostly due to
the nucleus, only a small fraction being due to the electrons and an even smaller
fraction due to the electromagnetic binding energy. Atomic nuclei are themselves
bound states of nucleons, held together by mesonic forces, or more precisely their
quanta, the mesons. The nucleons and mesons, finally, consist of the presently most
fundamental objects of nuclear material, called quarks, held together by gluonic
forces. Quarks and gluons are apparently as fundamental as electrons and photons.
It is a wonderful miracle of nature that this deep hierarchy of increasingly fundamental particles allows a common description with the help of a single theoretical
structure called quantum field theory.
As a first step towards developing this powerful theory we shall start from the
well-founded Schrödinger theory of nonrelativistic spinless particles. We show that
there exists a completely equivalent formulation of this theory in terms of quantum
fields. This formulation will serve as a basis for setting up various quantum field
theoretical models which can eventually explain the physics of the entire particle
hierarchy described above.
2.1
Mechanics and Quantum Mechanics
for n Distinguishable Nonrelativistic Particles
For a many-body system with only one type of nonrelativistic spinless particles of
mass M, which may be spherical atoms or molecules, the classical Lagrangian has
the form
L(xν , ẋν ; t) =
n
X
M 2
ẋν − V (x1 , . . . , xn ; t),
ν=1 2
82
(2.1) {3.1LA}
2.1 Mechanics and Quantum Mechanics for n Nonrelativistic Particles
83
where the arguments xν , ẋν in L(xν , ẋν ; t) stand, pars pro toto, for all positions xν
and velocities ẋν , ν = 1, . . . , n. The general n-body potential V (x1 , . . . , xn ; t) can
usually be assumed to consist of a sum of an external potential V1 (xν ; t) and a pair
potential V2 (xν − xµ ; t), also called one- and two-body-potentials, respectively:
V (x1 , . . . , xn ; t) =
X
V1 (xν ; t) +
ν
1X
V2 (xν − xµ ; t).
2 ν,µ
(2.2) {3.1b}
The second sum is symmetric in µ and ν, so that V2 (xν − xµ ; t) may be taken as
a symmetric function of the two spatial arguments — any asymmetric part would
not contribute. The symmetry ensures the validity of Newton’s third law “actio est
reactio”. The two-body potential is initially defined only for µ 6= ν, and the sum is
restricted accordingly, but for the development to come it will be useful to include
also the µ = ν -terms into the second sum (2.2), and compensate this by an appropriate modification of the one-body potential V1 (xν ; t), so that the total potential remains the same. Such a rearrangement excludes pair potentials V2 (xν − xµ ; t) which
are singular at the origin, such as the Coulomb potential between point charges
V2 (xν − xµ ; t) = e2 /4π|xν − xµ |. Physically, this is not a serious obstacle since
all charges in nature really have a finite charge radius. Even the light fundamental particles electrons and muons possess a finite charge radius, as will be seen in
Chapter 12.
At first we shall consider all particles to be distinguishable. This assumption
is often unphysical and will be removed later. For the particle at xν , the EulerLagrange equation of motion that extremizes the above Lagrangian (2.1) reads [recall
Eq. (1.8)]
∂V1 (xν ; t) X ∂
M ẍν = −
−
V2 (xν − xµ ; t).
(2.3) {3.2}
∂xν
µ ∂xν
The transition to the Hamiltonian formalism proceeds by introducing the canonical
momenta [see (1.10)]
∂L
= M ẋν ,
(2.4) {3.3}
pν =
∂ ẋν
and forming the Legendre transform [see (1.9)]
H(pν , xν ; t) =
"
X
ν
=
X
ν
pν ẋν − L(xν , ẋν ; t)
p2ν
2M
+
X
ν
#
V1 (xν ; t) +
ẋν =pν /M
1X
V2 (xν − xµ ; t).
2 ν,µ
(2.5) {3.4S}
From this, the Hamilton equations of motions are derived as [see (1.17)]
∂V1 (xν ; t) X ∂
−
V2 (xν − xµ ; t),
∂xν
µ ∂xν
pν
∂H
=
,
ẋν = {H, xν } =
∂pν
M
ṗν = {H, pν } = −
(2.6) {3.5}
(2.7) {nolabel}
84
2 Field Formulation of Many-Body Quantum Physics
with {A, B} denoting the Poisson brackets defined in Eq. (1.20):
{A, B} =
n
X
ν=1
∂A ∂B
∂B ∂A
−
∂pν ∂xν
∂pν ∂xν
!
.
(2.8) {3.6}
An arbitrary observable F (pν , xν ; t) changes as a function of time according to the
equation of motion (1.19):
∂F
dF
= {H, F } +
.
(2.9) {3.7}
dt
∂t
It is now straightforward to write down the laws of quantum mechanics for the
system. We follow the rules in Eqs. (1.236)–(1.238), and take the local basis
|x1 , . . . , xn i
(2.10) {3.8}
as eigenstates of the position operators x̂ν :
x̂ν |x1 , . . . , xn i = xν |x1 , . . . , xn i,
ν = 1, . . . , n.
(2.11) {3.9}
They are orthonormal to each other:
hx1 , . . . , xn |x′1 , . . . , x′n i = δ (3) (x1 − x′1 ) · · · δ (3) (xn − x′n ),
(2.12) {3.10}
and form a complete basis in the space of localized n-particle states:
Z
d3 x1 · · · d3 xn |x1 , . . . , xn ihx1 , . . . , xn | = 1.
(2.13) {3.11}
An arbitrary state is denoted by a ket vector and can be expanded in this basis by
multiplying the state formally with the unit operator (14.245), yielding the expansion
|ψ(t)i ≡ 1 × |ψ(t)i =
Z
d3 x1 · · · d3 xn |x1 , . . . , xn ihx1 , . . . , xn |ψ(t)i. (2.14) {3.12}
The scalar products
hx1 , . . . , xn |ψ(t)i ≡ ψ(x1 , . . . , xn ; t)
(2.15) {@}
are the wave functions of the n-body system. They are the probability amplitudes
for the particle 1 to be found at x1 , particle 2 at x2 , etc. .
The Schrödinger equation reads, in operator form,
Ĥ|ψ(t)i = H(p̂ν , x̂ν ; t)|ψ(t)i = ih̄∂t |ψ(t)i,
(2.16) {3.13}
where p̂ν are Schrödinger’s momentum operators, whose action upon the wave functions is specified by the rule
hx1 , . . . , xn |p̂ν = −ih̄∂xν hx1 , . . . , xn |.
(2.17) {3.14}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
85
2.2 Identical Particles: Bosons and Fermions
Multiplication of (14.247) from the left with the bra vectors hx1 , . . . , xn | yields via
(14.248) the Schrödinger differential equation for the wave functions:
H(−ih̄∂xν , xν ; t)ψ(x1 , . . . , xn ; t)
"
#
X h̄2
X
X
1
= −
∂x2ν +
V1 (xν ; t) +
V2 (xν − xµ ; t) ψ(x1 , . . . , xn ; t)
2M
2 ν,µ
ν
ν
(2.18) {3.15S}
= ih̄∂t ψ(x1 , . . . , xn ; t).
In many physical systems, the potentials V1 (xν ; t) and V2 (xν −xµ ; t) are independent
of time. Then we can factor out the time dependence of the wave functions with
fixed energy as
ψ(x1 , . . . , xn ; t) = e−iEt/h̄ ψE (x1 , . . . , xn ),
(2.19) {3.16}
and find for ψE (x1 , . . . , xn ) the time-independent Schrödinger equation
H(−ih̄∂xν , xν ; t)ψE (x1 , . . . , xn ) = EψE (x1 , . . . , xn ).
2.2
(2.20) {3.17S}
Identical Particles: Bosons and Fermions
The quantum mechanical rules in the last section were written down in the previous section under the assumption that all particles are distinguishable. For most
realistic n-body systems, however, this is an unphysical assumption. For example,
there is no way of distinguishing the electrons in an atom. Thus not all of the
solutions ψ(x1 , . . . , xn ; t) of the Schrödinger equation (2.18) can be physically permissible. Consider the case where the system contains only one species of identical
particles, for example electrons. The Hamilton operator in Eq. (2.18) is invariant
under all permutations of the particle labels ν. In addition, all probability amplitudes hψ1 |ψ2 i must reflect this invariance. They must form a representation space
of all permutations. Let Tij be an operator which interchanges the variables xi and
xj .
Tij ψ(x1 , . . . , xi , . . . , xj , . . . , xn ; t) ≡ ψ(x1 , . . . , xj , . . . , xi , . . . , xn ; t).
(2.21) {3.18}
It is called a transposition. The invariance may then be expressed as hTij ψ1 |Tij ψ2 i =
hψ1 |ψ2 i, implying that the wave functions |ψ1 i and |ψ2 i can change at most by a
phase:
Tij ψ(x1 , . . . , xn ; t) = eiαij ψ(x1 , . . . , xn ; t).
(2.22) {3.19}
But from the definition (2.21) we see that Tij satisfies
Tij2 = 1.
(2.23) {3.20}
Thus eiαij can only have the values +1 or −1. Moreover, due to the identity of all
particles, the sign must be the same for any pair ij.
The set of all multiparticle wave functions can be decomposed into wave functions
transforming in specific ways under arbitrary permutations P of the coordinates.
86
2 Field Formulation of Many-Body Quantum Physics
It will be shown in Appendix 2A that each permutation P can be decomposed
into products of transpositions. Of special importance are wave functions which
are completely symmetric, i.e., which are obtained by applying, to an arbitrary
n-particle wave function, the operation
S=
1 X
P.
n! P
(2.24) {@SYO}
The sum runs over all n! permutations of the particle indices. Another important
type of wave function is obtained by applying the antisymmetrizing operator
A=
1 X
ǫP P
n! P
(2.25) {@ASY}
to the arbitrary n-particle wave function. The symbol ǫP is unity for even permutations and −1 for odd permutations. It is called the parity of the permutation.
{ppppf}
By applying S or A to an arbitrary n-particle wave function one obtains completely symmetric or completely antisymmetric n-particle wave functions. In nature,
both signs can occur. Particles with symmetric wave functions are called bosons,
the others fermions. Examples for bosons are photons, mesons, α-particles, or any
nuclei with an even atomic number. Examples for fermions are electrons, neutrinos,
muons, protons, neutrons, or any nuclei with an odd atomic number.
In two-dimensional multi-electron systems in very strong magnetic fields, an
interesting new situation has been discovered. These systems have wave functions
which look like those of free quasiparticles on which transpositions Tij yield a phase
factor eiαij which is not equal to ±1: These quasiparticles are neither bosons nor
fermions. They have therefore been called anyons. Their existence in two dimensions
is made possible by imagining each particle to introduce a singularity in the plane,
which makes the plane multisheeted. A second particle moving by 3600 around this
singularity does not arrive at the initial point but at a point lying in a second sheet
below the initial point. For this reason, the equation (2.23) needs no longer to be
fulfilled.
Distinguishing the symmetry properties of the wave functions provides us with
an important tool to classify the various solutions of the Schrödinger equation (2.18).
Let us illustrate this by looking at the simplest nontrivial case in which the particles
have only a common time-independent external potential V1 (xν ) but are otherwise
noninteracting, i.e., their time-independent Schrödinger equation (2.20) reads
X
ν
h̄2 2
−
∂ + V1 (xν ) ψE (x1 , . . . , xn ) = EψE (x1 , . . . , xn ).
2M xν
"
#
(2.26) {3.21}
It can be solved by the factorizing ansatz in terms of single-particle states ψEαν of
energy Eαν ,
ψE (x1 , . . . , xn ) =
n
Y
ν=1
ψEαν (xν ),
(2.27) {3.22}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
87
2.2 Identical Particles: Bosons and Fermions
with the total energy being the sum of the individual energies:
E=
n
X
(2.28) {3.23}
Eαν .
ν=1
The single-particle states ψEαν (xν ) are the solutions of the one-particle Schrödinger
equation
#
"
h̄2 2
(2.29) {3.24}
∂ + V1 (x) ψEα (x) = Eα ψEα (x).
−
2M x
If the wave functions ψEα (x) form a complete set of states, they satisfy the oneparticle completeness relation
X
α
ψEα (x)ψE∗ α (x′ ) = δ (3) (x − x′ ).
(2.30) {3.25}
The sum over the labels α may, of course, involve an integral over a continuous
part of the spectrum. It is trivial to verify that the set of all products (14.266) is
complete in the space of n-particle wave functions:
X
α1 ,...,αn
ψEα1 (x1 ) · · · ψEαn (xn ) ψE∗ αn (x′n ) · · · ψE∗ α1 (x′1 )
= δ (3) (x1 − x′1 ) · · · δ (3) (xn − x′n ).
(2.31) {3.nbwf}
For identical particles, this Hilbert space is greatly reduced. In the case of bosons,
only the fully symmetrized products correspond to physical energy eigenstates. We
apply the symmetrizing operation S of Eq. (2.24) to the product of single-particle
Q
wave functions nν=1 ψEαp(ν) (xν ) and normalize the result to find
S
ψ{E
(x1 , . . . , xn )
α}
=
S
N{E
S
α}
n
Y
ν=1
ψEαp(ν) (xν ) =
S
N{E
α}
n
1 XY
(xν ).
ψE
n! P ν=1 αp(ν)
(2.32) {3.27}
The sum runs over n! permutations of the indices ν = 1, . . . , n, the permuted indices
being denoted by p(ν).
S
Note that ψ{E
(x1 , . . . , xn ) no longer depends on the order of labels of the enα}
ergies Eα1 , . . . , Eαn . This is a manifestation of the indistinguishability of the particles in the corresponding one-particle states indicated by the curly-bracket notation
S
ψ{E
. Also the normalization factor is independent of the order and has been
α}
S
denoted by N{E
.
α}
For fermions, the wave functions are
A
A
ψ{E
(x1 , . . . , xn ) = N{E
A
α}
α}
n
Y
ν=1
A
ψEαp(ν) (xν ) = N{E
α}
n
1 X Y
ψEαp(ν) (xν ), (2.33) {3.28}
ǫP
n! P
ν=1
where ǫP has the values ǫP = ±1 for even or odd permutations, respectively.
88
2 Field Formulation of Many-Body Quantum Physics
Instead of the indices on the labels αν of the energies, we may just as well
symmetrize or antisymmetrize the labels of the position variables:
S,A
ψ{E
(x1 , . . . , xn )
α}
=
S,A
N{E
α}
1 X
n! P
(
1
ǫP
)
n
Y
(2.34) {3.27x}
ψEαν (xp(ν) ).
ν=1
See Appendix 2A for more details.
The completely antisymmetrized products (2.33) can also be written in the form
of a determinant first introduced by Slater:
A
A
ψ{E
(x1 , . . . , xn ) = N{E
α}
α}
1
n!
ψEα1 (x1 ) ψEα1 (x2 ) . . . ψEα1 (xn )
..
..
..
.
.
.
ψEαn (x1 ) ψEαn (x2 ) . . . ψEαn (xn )
.
(2.35) {3.29}
To determine the normalization factors in (2.32)–(2.35), we calculate the integral
Z
S ∗
S
d3 x1 · · · d3 xn ψ{E
(x1 , . . . , xn )ψ{E
(x1 , . . . , xn )
α}
α}
2
S
= N{E
α}
n Z
1 XY
d3 xν ψE∗ α (xν )ψEαq(ν) (xν ).
2
p(ν)
n! P,Q ν=1
(2.36) {3.nori0}
Due to the group property of permutations, the double-sum contains n! identical
terms with P = Q and can be rewritten as
Z
S ∗
S
d3 x1 · · · d3 xn ψ{E
(x1 , . . . , xn )ψ{E
(x1 , . . . , xn )
α}
α}
2
S
= N{E
α}
n Z
1 XY
d3 xν ψE∗ αν (xν )ψEαp(ν) (xν ).
n! P ν=1
(2.37) {3.nori}
If all single-particle states ψEαν (xν ) are different from each other, then only the
identity permutation P = 1 with p(ν) = ν survives, and the normalization integral
(2.37) fixes
S 2
N{E
= n!.
(2.38) {3.31}
α}
Suppose now that two of the single-particle wave functions ψEαν , say ψEα1 and ψEα2 ,
coincide, while all others are different from these and each other. Then only two
permutations survive in (2.37): those with p(ν) = ν, and those in which P is a
transposition Tij of two elements (see Appendix 2A). The right-hand side of (2.38)
is then reduced by a factor of 2:
2
S
N{E
=
α}
n!
.
2
(2.39) {3.32}
Extending this consideration to n1 identical states ψEα1 , . . . , ψEαn1 , all n1 ! permutations among these give equal contributions to the normalization integral (2.37) and
lead to
n!
S 2
= 1.
(2.40) {3.33}
N{E
=
α}
n1 !
H. Kleinert, PARTICLES AND QUANTUM FIELDS
89
2.2 Identical Particles: Bosons and Fermions
Finally it is easy to see that, if there are groups of n1 , n2 , . . . , nk identical states, the
normalization factor is
n!
S 2
.
(2.41) {3.34}
N{E
=
α}
n1 !n2 ! · · · nk !
For the antisymmetric states of fermion systems, the situation is much simpler.
Here, no two states can be identical as is obvious from the Slater determinant (2.35).
Similar considerations as in (2.36), (2.37) for the wave functions (2.33) lead to
2
A
N{E
= n!.
α}
(2.42) {3.35}
The projection into the symmetric and antisymmetric subspaces has the following
effect upon the completeness relation (2.31): Multiplying it by the symmetrization
or antisymmetrization operators
P̂ S,A
1 X
=
n! P
(
1
ǫP
)
(2.43) {3.36}
,
which may be applied upon the particle positions x1 , . . . , xn as in (2.34), we calculate
X h
α1 ,...,αn
S,A
N{E
α}
i2
∗
S,A
S,A
ψ{E
(x1 , . . . , xn )ψ{E
(x′1 , . . . , x′n ) = δ (3)S,A (x1 , . . . , xn ; x′1 , . . . , x′n ),
α}
α}
(2.44) {3.37}
where the symmetrized or antisymmetrized δ-function is defined by
δ
(3)S,A
(x1 , . . . , xn ; x′1 , . . . , x′n )
1 X
≡
n! P
(
1
ǫP
)
δ (3) (x1 − x′p(1) ) · · · δ (3) (xn − x′p(n) ).
(2.45) {3.38}
Since the left-hand side of (2.44) is independent of the order of Eα1 , . . . , Eαn , we
may sum only over a certain order among the labels
X
α1 ,...,αn
−
−−→ n!
X
α1 >...>αn
.
(2.46) {3.39}
If there are degeneracy labels in addition to the energy, these have to be ordered in
the same way.
For antisymmetric states, the labels α1 , . . . , αn are necessarily different from each
other. Inserting (2.42) and (2.46) into (2.45), this gives directly
X
α1 >...>αn
∗
A
A
ψ{E
(x1 , . . . , xn )ψ{E
(x′1 , . . . , x′n ) = δ (3)A (x1 , . . . , xn ; x′1 , . . . , x′n ). (2.47) {3.40}
α}
α}
For the symmetric case we can order the different groups of identical states and denote their common labels by αn1 , αn2 , . . . , αnk , with the numbers nν indicating how
often the corresponding label α is present. Obviously, there are n!/n1 !n2 ! · · · nk !
permutations in the completeness sum (2.44) for each set of labels αn1 > αn2 >
90
2 Field Formulation of Many-Body Quantum Physics
. . . > αnk , the divisions by nν ! coming from the permutations within each group
of n1 , . . . , nk identical states. This combinatorial factor cancels precisely the normalization factor (2.41), so that the completeness relation for symmetric n-particle
states reads
X
X
n1 ,...,nk αn1 >...>αnk
∗
S
S
ψ{E
(x1 , . . . , xn )ψ{E
(x′1 , . . . , x′n )
α}
α}
= δ (3)S (x1 , . . . , xn ; x′1 , . . . , x′n ).
(2.48) {3.41}
The first sum runs over the different breakups of the total number n of states into
identical groups so that n = n1 + . . . + nk .
It is useful to describe the symmetrization or antisymmetrization procedure directly in terms of Dirac’s bra and ket formalism. The n-particle states are direct
products of single-particle states multiplied by the operator P̂ S,A , and can be written
as
|ψ
S,A
i = P̂
S,A
|Eα1 i · · · |Eαn i = N
S,A
1 X
n! P
(
1
ǫP
)
|Eαp(1) i · · · |Eαp(n) i.
(2.49) {3.42}
The wave functions (2.32) and (2.33) consist of scalar products of these states with
the localized boson states |x1 , . . . , xn i, which may be written as direct products
|x1 , x2 , . . . , xn i = |x1 i ×
⊙ |x2 i ×
⊙ ... ×
⊙ |xn i.
(2.50) {3.43}
In this state, the particle with number 1 sits at x1 , the particle with number 2 at x2 ,
. . . , etc. The symmetrization process wipes out the distinction between the particles
1, 2, . . . , n.
Let us adapt the symbolic completeness relation to the symmetry of the wave
functions. The general relation
Z
d3 x1 · · · d3 xn |x1 ihx1 | ×
⊙ ... ×
⊙ |xn ihxn | = 1
(2.51) {3.44}
covers all square integrable wave functions in the product space. As far as the physical Hilbert space is concerned, it can be restricted as follows:
Z
d3 x1 · · · d3 xn |x1 , . . . , xn iS,A
S,A
hx1 , . . . , xn | = P̂ S,A ,
(2.52) {3.45}
where
|x1 , . . . , xn i
S,A
1 X
=
n! P
(
1
ǫP
)
|xp(1) , . . . , xp(n) i.
(2.53) {3.46}
The states are orthonormal in the sense
S,A
hx1 , . . . , xn |x′1 , . . . , x′n iS,A = δ (3)S,A (x1 , . . . , xn ; x′1 , . . . , x′n ).
(2.54) {3.47}
This basis will play an essential role for the introduction of quantum fields.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.3 Creation and Annihilation Operators for Bosons
91
While the formalism presented so far is applicable to any number of particles,
practical calculations usually present a tremendous task. The number of particles is
often so large, of the order 1023 , that no existing computer could even list the wave
functions. On the other hand, macroscopic many-body systems containing such a
large number of microscopic particles make up our normal environment, and our
experience teaches us that many global phenomena can be predicted quite reliably.
They should therefore also be calculable in simple terms. For example, for most
purposes a crystal follows the laws of a rigid body, and nothing in theses laws records
the immense number of degrees of freedom inherent in a microscopic description.
If the solid is excited, there are sound waves in which all the many atoms in the
lattice vibrate around their equilibrium positions. Their description requires only a
few bulk parameters such as elastic constants and mass density. Phenomena of this
type are called collective phenomena.
To describe such phenomena, an economic way had to be found which does
not require the solution of the Schrödinger differential equation with 3n ∼ 1023
coordinates. We shall later see that field theory provides us with an elegant and
efficient access to such phenomena. After a suitable choice of field variables, simple
mean-field approximations will often give a rough explanation of many collective
phenomena. In the subsequent sections we shall demonstrate how the Schrödinger
theory of any number of particles can be transformed into the quantum field theory
of a single field.
There is further motivation at a more fundamental level for introducing fields.
They offer a natural way of accounting for the symmetry properties of the wave
functions, as we shall now see.
2.3
Creation and Annihilation Operators for Bosons
When dealing with n-particle Schrödinger equations, the imposition of symmetry
upon the Schrödinger wave functions ψ(x1 , . . . , xn ; t) seems to be a rather artificial
procedure. There exists an alternative formulation of the quantum mechanics of
n particles in which the Hilbert space automatically carries the correct symmetry.
This formulation may therefore be viewed as a more “natural” description of such
quantum systems. The basic mathematical structure which will serve this purpose
was first encountered in a particular quantum mechanical description of harmonic
oscillators which we now recall. It is well-known that the Hamilton operator of an
oscillator of unit mass
1 2 ω2 2
Ĥ = p̂ + q̂
(2.55) {@harmopham
2
2
can be rewritten in the form
1
Ĥ = h̄ω ↠â +
,
(2.56) {3.48}
2
where
√
√
√
√
ω q̂ − ip̂/ ω
ω q̂ + ip̂/ ω
†
√
√
â =
,
â =
(2.57) {3.49}
2h̄
2h̄
92
2 Field Formulation of Many-Body Quantum Physics
are the so-called raising and lowering operators. The canonical quantization rules
[p̂, p̂] = [x̂, x̂] = 0,
[p̂, x̂] = −ih̄
(2.58) {3.49x@}
imply that â, ↠satisfy
[â, â] = [↠, ↠] = 0,
[â, ↠] = 1.
(2.59) {3.ccr}
The energy spectrum of the oscillator follows directly from these commutation rules.
We introduce the number operator
N̂ = ↠â,
(2.60) {3.51}
[N̂, ↠] = ↠,
(2.61) {3.52}
which satisfies the equations
[N̂, â] = −â.
(2.62) {3.53}
These imply that ↠and â raise and lower the eigenvalues of the number operator
N̂ by one unit, respectively. Indeed, if |νi is an eigenstate with eigenvalue ν,
N̂ |νi = ν|νi,
(2.63) {@}
we see that
N̂ ↠|νi = (↠N̂ + ↠)|νi = (ν + 1)↠|νi,
N̂ â|νi = (âN̂ − â)|νi = (ν − 1)â|νi.
(2.64) {nolabel}
(2.65) {@}
Moreover, the eigenvalues ν must all be integer numbers n which are larger or equal
to zero. To see this, we observe that ↠â is a positive operator. It satisfies for every
state |ψi in Hilbert space the inequality
hψ|↠â|ψi = ||↠|ψi||2 ≥ 0.
(2.66) {@}
Hence there exists a state, usually denoted by |0i, whose energy cannot be lowered
by one more application of â. This state will satisfy
(2.67) {3.54}
â|0i = 0.
As a consequence, the operator N̂ applied to |0i must be zero. Applying the raising
operator ↠any number of times, the eigenvalues ν of N̂ will cover all integer numbers
ν = n with n = 0, 1, 2, 3, . . . . The corresponding states are denoted by |ni:
N̂|ni = n|ni,
n = 0, 1, 2, 3, . . . .
(2.68) {3.55}
Explicitly, these states are given by
|ni = Nn (↠)n |0i,
(2.69) {3.56}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.3 Creation and Annihilation Operators for Bosons
93
with some normalization factor Nn , that can be calculated using the commutation
rules (2.59) to be
1
(2.70) {3.57}
Nn = √ .
n!
By considering the commutation rules (2.59) between different states |ni and inserting intermediate states, we derive the matrix elements of the operators â, ↠:
√
hn′ |â|ni = n δn′ ,n−1 ,
(2.71) {3.58}
√
(2.72) {3.59}
hn′ |↠|ni = n + 1 δn′ ,n+1.
In this way, all properties of the harmonic oscillator are recovered by purely algebraic
manipulations, using (2.59) with the condition (2.67) to define the ground state.
This mathematical structure can be used to describe the complete set of symmetric localized states (2.32). All we need to do is reinterpret the eigenvalue n of
the operator N̂. In the case of the oscillator, n is the principal quantum number
of the single-particle state that counts the number of zeros in the Schrödinger wave
function. In quantum field theory, the operator changes its role and its eigenvalues
n count the number of particles contained in the many-body wave function. The
operators ↠and â which raise and lower n are renamed creation and annihilation
operators, which add or take away a single particle in the state |ni. The ground
state |0i contains no particle. It constitutes the vacuum state of the n-body system.
In the states (2.32), there are n particles at places x1 , . . . , xn . We therefore introduce the spatial degree of freedom by giving ↠, â a spatial label and defining the
operators
â†x , âx ,
which permit the creation and annihilation of a particle localized at the position x.1
The operators at different locations are taken to be independent, i.e., they commute as
[âx , âx′ ] = [â†x , â†x′ ] = 0,
(2.73) {3.lcr00}
[âx , â†x′ ] = 0,
x 6= x′ .
(2.74) {3.lcr0}
[âx , â†x′ ] = δ (3) (x − x′ ).
(2.75) {3.lcr1}
The commutation rule between
and âx′ for coinciding space variables x and x′ is
specified with the help of a Dirac δ-function as follows:
â†x
We shall refer to these x-dependent commutation rules as the local oscillator algebra.
The δ-function singularity in (2.75) is dictated by the fact that we want to
preserve the raising and lowering commutation rules (2.61) and (2.62) for the particle
number at each point x, i.e., we want that
[N̂ , â†x ] = â†x ,
1
(2.76) {3.64}
The label x in configuration space of the particles bears no relation to the operator q̂ in the
Hamiltonian (2.55), which here denotes an operator in field space, as we shall better understand
in Section 2.8.
94
2 Field Formulation of Many-Body Quantum Physics
[N̂, âx ] = −âx .
The total particle number operator is then given by the integral
N̂ =
Z
d3 x â†x âx .
(2.77) {3.65}
(2.78) {3.63}
Due to (2.74), all parts in the integral (2.78) with x′ different from the x in (2.76)
and (2.77) do not contribute. If the integral is supposed to give the right-hand
sides in (2.76) and (2.77), the commutator between âx and â†x has to be equal to a
δ-function.
The use of the δ-function is of course completely analogous to that in Subsection 1.4. [recall the limiting process in Eq. (1.160)]. In fact, we could have introduced
local creation and annihilation operators with ordinary unit commutation rules at
each point by discretizing the space into a fine-grained point-lattice of a tiny lattice
spacing ǫ, with discrete lattice points at
nν = 0, ±1, ±2, . . . .
xn = (n1 , n2 , n3 )ǫ,
(2.79) {3.67}
And for the creation or annihilation of a particle in the small cubic box around xn we
could have introduced the operators â†n or ân , which satisfy the discrete commutation
rules
[ân , ân′ ] = [â†n , â†n′ ] = 0,
(2.80) {nolabel}
(3)
[ân , â†n′ ] = δnn′ .
(2.81) {3.68}
For these the total particle number operator is
N̂ =
X
â†n ân .
(2.82) {3.69}
n
This would amount to identifying ân with a discrete subset of the continuous set of
operators âx as follows:
√
ân = ǫ3 âx .
(2.83) {3.70}
x≡xn
Then the discrete and continuous formulations of the particle number operator would
be related by
Z
X †
X †
N̂ =
ân ân ≡ ǫ3
âxn âxn −
−−→ d3 x â†x âx .
(2.84) {3.71}
n
xn
ǫ→0
In the same limit, the commutator
1
1 (3)
[ân , â†n′ ] ≡ [âxn , â†xn’ ] = 3 δnn′
3
ǫ
ǫ
(2.85) {3.72}
would tend to δ (3) (x − x′ ), which can be seen in the same way as in Eq. (1.160).
We are now ready to define the vacuum state of the many-particle system. It is
given by the unique state |0i of the local oscillator algebra (2.73) and (2.74), which
contains no particle at all places x, thus satisfying
âx |0i ≡ 0,
h0|â†x ≡ 0.
(2.86) {3.73}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.4 Schrödinger Equation for Noninteracting Bosons in Terms of Field Operators
95
It will always be normalized to unity:
h0|0i = 1.
(2.87) {3.74}
We can now convince ourselves that the fully symmetrized Hilbert space of all localized states of n particles may be identified with the states created by repeated
application of the local creation operators â†x :
1
|x1 , . . . , xn iS ≡ √ â†x1 · · · â†xn |0i.
n!
(2.88) {3.75}
The such-generated Hilbert space will be referred to as the second-quantized Hilbert
space for reasons to be seen below. It decomposes into a direct sum of n-particle
sectors. The symmetry of these states in the position variables is obvious, due to
the commutativity (2.73) of all â†xν , â†xµ among each other.
Let us verify that the generalized orthonormality relation is indeed fulfilled by
the single-particle states. Using the local commutation rules (2.73), (2.74), and the
definition of the vacuum state (2.86), we calculate for a single particle
S
hx|x′ iS = h0|âx â†x′ |0i
= h0|δ (3) (x − x′ ) + â†x′ âx |0i
= δ (3) (x − x′ ).
(2.89) {3.76}
For two particles we find
S
1
h0|âx2 âx1 â†x′ â†x′ |0i
1
2
2!
h
i
1 (3)
=
δ (x1 − x′1 )h0|âx2 â†x′ |0i + h0|âx2 â†x′ âx1 â†x′ |0i
2
1
2
2!
h
i
1 (3)
δ (x1 − x′1 )δ (3) (x2 − x′2 ) + δ (3) (x2 − x′1 )δ (3) (x1 − x′2 )
=
2!
(2.90) {3.77a}
= δ (3)S (x1 , x2 ; x′1 , x′2 ).
hx1 , x2 |x′1 , x′2 iS =
The generalization to n particles is straightforward, although somewhat tedious. It
is left to the reader as an exercise. Later in Section 7.17.1, rules will be derived in a
different context by a procedure due to Wick, which greatly simplifies calculations
of this type.
2.4
Schrödinger Equation for Noninteracting Bosons
in Terms of Creation and Annihilation Operators
Expressing the localized states in terms of the local creation and annihilation operators â†x , âx does not only lead to an automatic symmetrization of the states. It also
brings about an extremely simple unified form of the Schrödinger equation, which
does not require the initial specification of the particle number n, as in Eq. (2.18).
96
2 Field Formulation of Many-Body Quantum Physics
This will now be shown for the case of identical particles with no two-body interactions V2 (xν − xµ ; t).
In order to exhibit the unified Schrödinger equation for any number of particles,
let us first neglect interactions and consider only the motion of the particles in an
external potential with the Schrödinger equation:
(
X
ν
h̄2 2
−
∂ + V1 (xν ; t)
2M xν
"
#)
ψ(x1 , . . . , xn ; t) = ih̄∂t ψ(x1 , . . . , xn ; t).
(2.91) {3.77}
We shall now demonstrate that the ↠, â -form of this equation, which is valid for
any particle number n, reads
Ĥ(t)|ψ(t)i = ih̄∂t |ψ(t)i,
(2.92) {3.78}
where Ĥ(t) is simply the one-particle Hamiltonian sandwiched between creation and
annihilation operators â†x and âx and integrated over x, i.e.,
Ĥ(t) =
Z
d
3
xâ†x
"
h̄2 2
∂x + V1 (x; t) âx .
−
2M
#
(2.93) {3.79}
The operator (2.93) is called the second-quantized Hamiltonian, equation (2.92) the
second-quantized Schrödinger equation, and the state |ψ(t)i is an arbitrary n-particle
state in the second-quantized Hilbert space, generated by multiple application of â†x
upon the vacuum state |0i, as described in the last section. The operator nature of
âx , â†x accounts automatically for the many-body content of Eq. (2.92).
This statement is proved by multiplying Eq. (2.92) from the left with
S
1
hx1 , . . . , xn | = √ h0|âxn · · · âx1 ,
n!
which leads to
1
1
√ h0|âxn · · · âx1 Ĥ(t)|ψ(t)i = ih̄∂t √ h0|âxn · · · âx1 |ψ(t)i.
n!
n!
(2.94) {3.80}
Here we make use of the property (2.86) of the vacuum state to satisfy h0|â†x = 0.
As a consequence, we may rewrite the left-hand side of (2.94) with the help of a
commutator as
1
√ h0|[âxn · · · âx1 , Ĥ]|ψ(t)i.
n!
This commutator is easily calculated using the operator chain rules
[Â, B̂ Ĉ] = B̂[Â, Ĉ] + [Â, B̂]Ĉ,
[ÂB̂, Ĉ] = Â[B̂, Ĉ] + [Â, Ĉ]B̂.
(2.95) {3.abc}
These rules can easily be memorized, noting that their structure is exactly the
same as in the Leibnitz rule for derivatives. In the first rule we may imagine A to
be a differential operator applied to the product BC, which is evaluated by first
H. Kleinert, PARTICLES AND QUANTUM FIELDS
97
2.5 Second Quantization and Symmetrized Product Representation
differentiating B, leaving C untouched, and then C, leaving B untouched. In the
second rule we imagine C to be a differential operator acting similarly to the left
upon the product AB. Generalizing this rule to products of more than two operators
we derive
[âxn · · · âx1 , â†y âz ] = [âxn · · · âx1 , â†y ]âz + â†y [âxn · · · âx1 , âz ]
= âxn · · · âx3 âx2 [âx1 , â†y ]âz + [âxn · · · âx3 âx2 , â†y ]âx1 âz + . . .
X
=
ν
δ (3) (xν − y)âxn · · · âxν+1 âz âxν−1 · · · âx1 .
Multiplying both sides by
h
i
δ (3) (y − z) −h̄2 ∂z 2 /2M + V1 (z; t) ,
and integrating over d3 y d3 z using (2.93) and (2.75) we find
[âxn · · · âx1 , Ĥ(t)] =
X
ν
h̄2 2
∂ + V1 (xν ; t) âxn · · · âxν+1 âxν âxν−1 · · · âx1 , (2.96) {3.81}
−
2M xν
#
"
so that (2.94) becomes
1
1
h̄2 2
∂xν + V1 (xν ; t) √ h0|âxn · · · âx1|ψ(t)i =ih̄∂t √ h0|âxn · · · âx1 |ψ(t)i,
−
2M
n!
n!
ν
(2.97) {3.82}
which is precisely the n-body Schrödinger equation (2.91) for the wave function
X
"
#
1
ψ(x1 , . . . , xn ; t) ≡ √ h0|âxn · · · âx1 |ψ (t)i.
n!
2.5
(2.98) {3.83}
Second Quantization and Symmetrized Product
Representation
{sqspr}
It is worth pointing out that the mathematical structure exploited in the process of
second quantization is of a very general nature.
Consider a set of matrices Mi with indices α′ , α
(Mi )α′ α
which satisfy some matrix commutation rules, say
[Mi , Mj ] = ifijk Mk .
(2.99) {2B.1}
Let us sandwich these matrices between creation and annihilation operators which
satisfy
[âα , âα′ ] = [â†α , â†α′ ] = 0,
[âα , â†α′ ] = δαα′ ,
(2.100) {3.68g}
98
2 Field Formulation of Many-Body Quantum Physics
and form the analogs of “second-quantized operators” (2.93) by defining
M̂i ≡ â†α′ (Mi )α′ α âα = ↠Mi â.
(2.101) {2B.2}
In expressions of this type, repeated indices α, α′ imply a summation over all α,
α′ . This is commonly referred to as Einstein’s summation convention. On the righthand side of (2.101) we have suppressed the indices α, α′ completely, for brevity. It
is now easy to verify, using the operator chain rules (2.95), that the operators M̂i
satisfy the same commutation rules as the matrices Mi :
h
M̂i , M̂j
i
h
i
↠Mi â, ↠Mj â
=
i
h
h
i
↠Mi â, ↠Mj â + ↠Mj ↠Mi â, â
=
= ↠Mi Mj â − ↠Mj Mi â
= ↠[Mi , Mj ] â = ifijk ↠Mk â = ifijk M̂k .
(2.102) {2B.3}
Thus the “second-quantized” operators M̂i generate an operator representation of
the matrices Mi . They can be sandwiched between states in the “second-quantized”
Hilbert space generated by applying products of creation operators â†n upon the
vacuum state |0i. Thereby they are mapped into an infinite-dimensional matrix
representation. On each subspace spanned by the products of a fixed number of
creation operators, they generate the symmetrized part of the direct product representation.
The action of the “second-quantized” operators M̂i upon the large Hilbert space
is very simple to calculate. The only commutation rules required are
h
M̂i , â†α
h
âα′ , M̂i
i
= â†α′ (Mi )α′ α ,
i
(2.103) {2B.4}
= (Mi )α′ α âα .
From this property we calculate directly the action upon “single-particle states”:
h
M̂i â†α |0i =
i
M̂i , â†α |0i = â†α′ |0i(Mi )α′ α ,
h
i
h0|âα′ M̂i = h0| âα′ , M̂i = (Mi )α′ α h0|âα .
(2.104) {2B.6}
Thus the states â†α |0i span an invariant subspace and are transformed into each
other via the matrix (Mi )α′ α .
Consider now a state with two particles:
â†α1 â†α2 |0i.
(2.105) {2B.7}
Applying M̂i to this state yields
h
M̂i , â†α1 â†α2
i
=
h
i
h
M̂i , â†α1 â†α2 + â†α1 M̂i , â†α2
h
i
i
= â†α′ â†α′ (Mi )α′1 α1 δα′2 α2 + δα′1 α1 (Mi )α′2 α2 .
1
2
(2.106) {2B.9}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.5 Second Quantization and Symmetrized Product Representation
99
Multiplying Eq. (2.106) by |0i from the right, we find the transformation law for the
two-particle states â†α1 â†α2 |0i. They are transformed via the representation matrices
(Mi )α′1 α′2 ,α1 α2 = (Mi )α′1 α1 δα′2 α2 + δα′1 α1 (Mi )α′2 α2 .
(2.107) {2B.9b}
Omitting the indices, we may also write the matrices as
(2)
Mi
= Mi × 1 + 1 × Mi ,
(2.108) {2B.10}
which is the well-known way of forming representations of a matrix algebra in a
direct product space.
Since a†α1 and a†α2 commute with each other, the invariant space constructed in
this way contains only symmetric tensors, and only the symmetrized part of the
matrices (Mi )α′1 α′2 ,α1 α2 contribute.
The alert reader will have realized that the same operator structure can be obtained for the antisymmetrized parts of the matrices (Mi )α′1 α′2 ,α1 α2 by using creation
and annihilation operators a†α1 and aα2 which satisfy the fermionic version of the
algebraic rules (2.100):
{âα , âα′ } = {â†α , â†α′ } = 0,
{âα , â†α′ } = δαα′ .
(2.109) {3.68ga}
In these, curly brackets are used to abbreviate anticommutators
{A, B} ≡ AB + BA.
(2.110) {3.141a}
The first two lines in (2.102) are unchanged since the operator chain rules (2.95)
hold for both Bose and Fermi operators. To derive the third line we must use the
additional rules
[Â, B̂ Ĉ] = {Â, B̂}Ĉ − B̂{Â, Ĉ},
[ÂB̂, Ĉ] = Â{B̂, Ĉ} − {Â, Ĉ}B̂.
(2.111) {3.abcanti}
This fermionic version of the commutation relation (2.102) will form, in Section 2.10,
the basis for constructing a second-quantized representation for the n-particle wave
functions and Schrödinger operators of fermions.
If Mi are chosen to be representation matrices Li of the generators of the rotation group (to be discussed in detail in Section 4.1), the law (2.108) represents the
quantum mechanical law of addition of two angular momenta:
(2)
Li = Li × 1 + 1 × Li .
The generalization to any number of angular momenta is obvious.
Incidentally, any operator which satisfies the same commutation rules with M̂i ,
as âα in (2.103), i.e., satisfies the commutation rules
h
M̂i , Ôα†
h
Ôα′ , M̂i
i
i
= Ôα† ′ (Mi )α′ α ,
= (Mi )α′ α Ôα ,
(2.112) {2B.5s}
100
2 Field Formulation of Many-Body Quantum Physics
will be referred to as a spinor operator. Generalizing this definition, an operator Ôα1 α2 ...αn which commutes with M̂i like a product Ôα1 Ôα2 · · · Ôαn , generalizing
Eq. (2.106), is called a multispinor operator of rank n.
Another type of operators which frequently occurs in quantum mechanics and
quantum field theory is a vector operator. This is any operator Ôj which commutes
with M̂i in the same way as M̂j does in (2.102), i.e.,
h
i
(2.113) {2B.5}
M̂i , Ôj = ifijk Ôk .
Its generalization Ôj1 j2 ...jn , which commutes with M̂i as the product of operators
M̂j1 M̂j2 . . . M̂jn , is called a tensor operator of rank n.
The many-particle version of the Schrödinger theory is obtained if we view the
one-particle Schrödinger equation
H(−ih̄∂x , x)ψ (x, t) = ih̄∂t ψ (x, t)
(2.114) {2B.11}
as a matrix equation in the discretized x-space with the lattice positions x =
(n1 , n2 , n3 )ǫ, so that the wave functions ψ (x, t) correspond to vectors ψn (t). Then
the differential operator ∂i ψ(x) becomes simply ∇i ψ(x) = [ψ (x + iǫ) − ψ(x)] /ǫ =
[ψn+i − ψn ] /ǫ where i is the unit vector in the ith direction, and ǫ the lattice spacing.
The Laplacian may be viewed as the continuum limit of the matrix
¯ i ∇i ψ(x) =
∇
3
1 X
[ψ (x + iǫ) − 2ψ(x) + ψ (x − iǫ)]
ǫ2 i=1
3
1 X
[ψn+i − 2ψn + ψn−i ] .
= 2
ǫ i=1
(2.115) {2B.12}
The Schrödinger equation (2.114) is then the ǫ → 0 -limit of the matrix equation
Hnn′ ψn′ (t) = ih̄∂t ψn (t).
(2.116) {2B.13}
The many-particle Schrödinger equation in second quantization form reads
Ĥ|ψ(t)i = ih̄∂t |ψ(t)i,
(2.117) {2B.13a}
with the Hamiltonian operator
Ĥ = â†n′ Hn′ n ân .
(2.118) {2B.13b}
It can be used to find the eigenstates in the symmetrized multispinor representation
space spanned by
â†n1 . . . â†nN |0i.
(2.119) {2B.14}
Applying Ĥ to it we see that this state is multiplied from the right by a directproduct matrix
H × 1 × . . . × 1 + 1 × H × . . . × 1 + . . . 1 × 1 × . . . × H.
(2.120) {2B.15}
Due to this very general relation, the Schrödinger energy of a many-body system
without two- or higher-body interactions is the sum of the one-particle energies.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
101
2.6 Bosons with Two-Body Interactions
2.6
Bosons with Two-Body Interactions
We now include two-body interactions. For simplicity, we neglect the one-body
potential V1 (x; t) which can be added at the end and search for the second-quantized
form of the Schrödinger equation
"
−
X
ν
h̄2 2
1X
∂xν +
V2 (xν − xµ ; t) ψ(x1 , . . . , xn ; t) = ih̄∂t ψ(x1 , . . . , xn ; t). (2.121) {3.84}
2M
2 µ,ν
#
It is easy to see that such a two-body potential can be introduced into the secondquantized Schrödinger equation (2.92) by adding to the Hamilton operator in (2.93)
the interaction term
Z
1
d3 xd3 x′ â†x â†x′ V2 (x − x′ ; t)âx′ âx .
(2.122) {3.85}
Ĥint (t) =
2
To prove this we work out the expectation value
1
1
√ h0|âxn . . . âx1 Ĥint (t)|ψ(t)i = √ h0|[âxn . . . âx1 , Ĥint (t)]|ψ(t)i.
n!
n!
(2.123) {3.86}
We do this by using the local commutation rules (2.73), (2.75), and the vacuum
property (2.86). First we generalize Eq. (2.96) to
[âxn · · · âx1 , â†y2 â†y1 âz1 âz2 ]
= [âxn · · · âx1 , â†y2 â†y1 ]âz1 âz2 + â†y2 â†y1 [âxn · · · âx1 , âz1 âz2 ]
= [âxn · · · âx1 , â†y2 ]â†y1 âz1 âz2 + â†y2 [âxn · · · âx1 , â†y1 ]âz1 âz2
X
=
ν
δ (3) (xν − y2 )âxn · · · âxν+1 âxν−1 · · · âx1 â†y1 âz1 âz2
+ â†y2
X
ν
δ (3) (xν − y1 )âxn · · · âxν+1 âxν−1 · · · âx1 âz1 âz2 .
(2.124) {3.87}
The second piece does not contribute to Eq. (2.123) since â†y2 annihilates the vacuum
on the left. For the same reason, the first piece can be written as
X
ν
δ (3) (xν − y2 )[âxn · · · âxν+1 âxν−1 · · · âx1 , â†y1 ]âz1 âz2 ,
(2.125) {@}
as long as it stands to the right of the vacuum. Using the commutation rule (2.96),
this leads to
h0|âxn · · · âx1 â†y2 â†y1 âz1 âz2 |ψ(t)i =
X
µ,ν
δ (3) (xµ − y1 )δ (3) (xν − y2 )
×h0|âxn · · · âxν+1 âxν−1 · · · âxµ+1 âxµ−1 . . . âx1 âz1 âz2 |ψ(t)i.
(2.126) {nolabel}
After multiplying this relation by V2 (y2 − y1 ; t)δ (3) (y1 − z1 )δ (3) (y2 − z2 )/2, and
integrating over d3 y1 d3 y2 d3 z1 d3 z2 , we find
h0|âxn · · · âx1 Ĥint |ψ(t)i =
1X
V2 (xν − xµ ; t)h0|âxn · · · âx1 |ψ (t) i,
2 µ,ν
(2.127) {3.88}
102
2 Field Formulation of Many-Body Quantum Physics
which is precisely the two-body interaction in the Schrödinger equation (2.121).
Adding now the one-body interactions, we see that an n-body Schrödinger equation with arbitrary one- and two-body potentials can be written in the form of a
single operator Schrödinger equation
Ĥ(t)|ψ(t)i = ih̄∂t |ψ(t)i,
(2.128) {3.90}
with the second-quantized Hamilton operator
Ĥ(t) =
Z
d
3
xâ†x
"
1
h̄2 2
∂x + V1 (x; t) âx +
−
2M
2
#
Z
d3 x d3 x′ â†x â†x′ V2 (x − x′ ; t)âx′ âx .
(2.129) {3.89}
The second-quantized Hilbert space of the states |ψ(t)i is constructed by repeated
multiplication of the vacuum vector |0i with particle creation operators â†x . The
order of the creation and annihilation operators in this Hamiltonian is such that the
vacuum, as a zero-particle state, has zero energy:
Ĥ(t)|0i = 0,
h0|Ĥ(t) = 0,
(2.130) {@zeropen}
as in the original Schrödinger equation.
A Hamiltonian which is a spatial integral over a Hamiltonian density H(x) as
H=
Z
d3 x H(x),
(2.131) {@}
is called a local Hamiltonian. In (2.129), the free part is local, but the interacting
part is not. It consists of an integral over two spatial variables, thus forming a bilocal
operator.
2.7
Quantum Field Formulation of Many-Body
Schrödinger Equations for Bosons
The annihilation operator âx can now be used to define a time-dependent quantum
field ψ̂(x, t) as being the Heisenberg picture of the operator âx (which itself is also
referred to as the Schrödinger picture of the annihilation operator). According to
Eq. (1.285), the Heisenberg operator associated with âx is
axH (t) ≡ [Û (t, ta )]−1 âx Û (t, ta ).
(2.132) {1.defhbo3}
ψ̂(x, t) ≡ axH (t).
(2.133) {3.defhbo}
Thus we define
Choosing the time variable ta = 0, the quantum field ψ̂(x, t) coincides with âx at
t = 0:
ψ̂(x, 0) ≡ âx .
(2.134) {3.defhbop}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.7 Quantum Field Formulation of Many-Body Schrödinger Equations for Bosons
103
The time dependence of ψ̂(x, t) is ruled by Heisenberg’s equation of motion (1.280):
∂t ψ̂(x, t) =
i
[ĤH (t), ψ̂(x, t)].
h̄
(2.135) {3.91}
For simplicity, we shall at first assume the potentials to have no explicit time dependence, an assumption to be removed later. Then Eq. (2.135) is solved by
ψ̂(x, t) = eiĤt/h̄ ψ̂(x, 0)e−iĤt/h̄ = eiĤt/h̄ âx e−iĤt/h̄ .
(2.136) {3.92}
The Hermitian conjugate of this determines the time dependence of the Heisenberg
picture of the creation operator:
ψ̂ † (x, t) = eiĤt/h̄ ψ̂ † (x, 0)e−iĤt/h̄ = eiĤt/h̄ â†x e−iĤt/h̄ .
(2.137) {3.92cr}
At each given time t, the field ψ̂(x, t) fulfills the same commutation rules (2.73) and
(2.74) as âx :
[ψ̂(x, t), ψ̂(x′ , t)] = 0,
[ψ̂ † (x, t), ψ̂ † (x′ , t)] = 0,
[ψ̂(x, t), ψ̂ † (x′ , t)] = δ (3) (x − x′ ).
(2.138) {3.93}
Consider now the Hamiltonian operator (2.129) in the Heisenberg representation.
Under the assumption of no explicit time dependence in the potentials we may simply
multiply it by eiĤt/h̄ and e−iĤt/h̄ from the left and right, respectively, and see that
h̄2 2
∂x + V1 (x) ψ̂(x, t)
ĤH (t) =
d x ψ̂ (x, t) −
2M
Z
1
d3 xd3 x′ ψ̂ † (x, t)ψ̂ † (x′ , t)V2 (x − x′ )ψ̂(x′ , t)ψ̂(x, t).
+
2
Z
3
†
#
"
(2.139) {3.96}
Since Ĥ commutes with itself, the operator ĤH (t) is time independent, so that
ĤH (t) ≡ Ĥ.
(2.140) {@}
The important point about the expression (2.139) for Ĥ is now that by containing the time-dependent fields ψ̂(x, t), it can be viewed as the Hamilton operator of a
canonically quantized Heisenberg field. This is completely analogous to the Hamiltonian operator Ĥ ≡ H(p̂H (t), x̂H (t), t) in (1.278). Instead of pH (t) and qH (t), we
are dealing here with generalized coordinates and their canonically conjugate momenta of the field system. They consist of the Hermitian and anti-Hermitian parts
of the field, ψ̂R (x, t) and ψ̂I (x, t), defined by
ψ̂R ≡
(ψ̂ + ψ̂ † )
√
,
2
ψ̂I ≡
(ψ̂ − ψ̂ † )
√
.
2i
(2.141) {3.94}
104
2 Field Formulation of Many-Body Quantum Physics
They commute like
[ψ̂I (x, t), ψ̂R (x′ , t)] = −iδ (3) (x − x′ ),
[ψ̂I (x, t), ψ̂I (x′ , t)] = 0,
[ψ̂R (x, t), ψ̂R (x′ , t)] = 0.
(2.142) {3.95}
These commutation rules are structurally identical to those between the quasiCartesian generalized canonical coordinates q̂iH (t) and p̂iH (t) in Eq. (1.97).
In fact, the formalism developed there can be generalized to an infinite set of
canonical variables labeled by the space points x rather than i, i.e., to canonical
variables px (t) and qx (t). Then the quantization rules (1.97) take the form
[p̂x (t), q̂x′ (t)] = −ih̄δ (3) (x − x′ ),
[p̂x (t), p̂x′ (t)] = 0,
[q̂x (t), q̂x′ (t)] = 0,
(2.143) {3.101}
which is a local version of the algebra (2.58). The replacement i → x can of course
be done on a lattice with a subsequent continuum limit as in Eqs. (2.79)–(2.85).
When going from the index i to the continuous spatial variable x, the Kronecker δij
turns into Dirac’s δ (3) (x − x′ ), and sums become integrals.
By identifying
p̂x (t) ≡ h̄ψ̂I (x, t),
q̂x (t) ≡ ψ̂R (x, t),
(2.144) {3.142@}
we now obtain the commutation relations (2.142). In quantum field theory it is
customary to denote the canonical momentum variable px (t) by the symbol πx (t),
and write
p̂x (t) = h̄ψ̂I (x, t) ≡ π̂(x, t).
(2.145) {3.103}
Thus the many-body nature of the system may be considered as a consequence
of quantizing the fields qx (t) = ψ̂R (x, t) and p̂x (t) = h̄ψ̂I (x, t) canonically via
Eq. (2.143).
2.8
Canonical Formalism in Quantum Field Theory
{cfqft}
So far, the commutation rules have been imposed upon the fields ψ̂(x, t) and ψ̂ † (x′ , t)
by the particle nature of the n-body Schrödinger theory. It is, however, possible to
derive these rules by applying the standard canonical formalism to the fields ψR (x, t)
and ψI (x, t), treating them as generalized Lagrange coordinates. To see this, let us
recall once more the general procedure for finding the quantization rules and the
Schrödinger equation for a general Lagrangian system with an action
A=
Z
dt L(q(t), q̇(t)),
(2.146) {3.104}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
105
2.8 Canonical Formalism in Quantum Field Theory
where the Lagrangian L is some function of the independent variables q(t) =
(q1 (t), . . . , qN (t)) and their velocities q̇(t) = (q̇1 (t), . . . , q̇N (t)). The conjugate momenta are defined, as usual, by the derivatives
pi (t) =
∂L
.
∂ q̇i (t)
(2.147) {3.105}
The Hamiltonian is given by the Legendre transformation
H(p(t), q(t)) =
X
i
pi (t)qi (t) − L(q(t), q̇(t)).
(2.148) {3.106}
If q(t) are Cartesian or quasi-Cartesian coordinates, quantum physics is imposed in
the Heisenberg picture by letting pi (t), qi (t) become operators p̂iH (t), q̂iH (t) which
satisfy the canonical equal time commutation rules
[p̂iH (t), q̂jH (t)] = −iHh̄δij ,
[p̂iH (t), p̂jH (t)] = [q̂iH (t), q̂jH (t)] = 0,
(2.149) {3.107a}
and postulating the Heisenberg equation of motion
i
∂
d
ÔH = [ĤH , ÔH ] + ÔH
dt
h̄
∂t
(2.150) {3.heqm}
ÔH (t) ≡ O(p̂H (t), q̂H (t), t).
(2.151) {3.hbobs}
for any observable
This formalism holds for any number of Cartesian or quasi-Cartesian variables.
It can therefore be generalized to functions of space variables xn lying on a lattice
with a tiny width ǫ [see (2.79)]. Suppressing the subscripts of xn , the canonical
momenta (2.147) read
∂L
px (t) =
,
(2.152) {3.110}
∂ q̇x (t)
and the Hamiltonian becomes
H=
X
x
px (t)q̇x (t) − L(qx , q̇x ).
(2.153) {3.hamni}
The canonical commutation rules (2.149) become the commutation rules (2.143) of
second quantization.
In quantum field theory, the formalism must be generalized to continuous space
variables x. For a Hamiltonian (2.153), the action (2.146) is
A=
Z
dt L(t) =
Z
dt
Z
3
d x ψI (x, t)h̄∂t ψR (x, t) −
Z
dt H[ψI , ψR ],
(2.154) {3.117}
where H[ψI , ψR ] denotes the classical Hamiltonian associated with the operator
HH (t) in Eq. (2.139). The derivative term can be written as an integral over a
kinetic Lagrangian Lkin (t) as
Akin =
Z
dtLkin (t) =
Z
dt
Z
d3 x Lkin (x, t) ≡
Z
dt
Z
d3 x ψI (x, t)h̄∂t ψR (x, t). (2.155) {3.117S}
106
2 Field Formulation of Many-Body Quantum Physics
Then the lattice rule (2.152) for finding the canonical momentum has the following
functional generalization to find the canonical field momentum:
px (t) =
∂L
∂ q̇x (t)
→ π(x, t) ≡
∂Lkin
= h̄ψI (x, t),
∂∂t ψR (x, t)
(2.156) {3.118}
in agreement with the identification (2.152) and the action (2.155). The canonical
quantization rules
[π(x, t), ψ̂R (x′ , t)] = −iδ (3) (x − x′ ),
[π(x, t), π(x′ , t)] = 0,
[ψ̂R (x, t), ψ̂R (x′ , t)] = 0
(2.157) {3.95x}
coincide with the commutation rules (2.142) of second quantization. Obviously, the
Legendre transformation (2.153) turns L into the correct Hamiltonian H.
More conveniently, one expresses the classical action in terms of complex fields
A=
Z
dt L(t) =
Z
dt
Z
d3 x ψ ∗ (x, t) ih̄∂t ψ(x, t) −
Z
dt H[ψ, ψ ∗ ],
(2.158) {3.120}
and defines the canonical field momentum as
π(x, t) ≡
∂Lkin
= h̄ψ ∗ (x, t).
∂∂t ψ(x, t)
(2.159) {3.118c}
Then the canonical quantization rules become
[ψ(x, t), ψ † (x′ , t)] = −iδ (3) (x − x′ ),
[ψ(x, t), ψ(x′ , t)] = 0,
[ψ † (x, t), ψ † (x′ , t)] = 0.
(2.160) {3.95xC}
We have emphasized before that the canonical quantization rules are applicable
only if the field space is quasi-Cartesian (see the remark on page 15). For this, the
dynamical metric (1.94) has to be q-independent. This condition is violated by the
interaction in the Hamiltonian (2.139). There are ambiguities in ordering the field
operators in this interaction. These are, however, removed by the requirement that,
after quantizing the field system, one wants to reproduce the n-body Schrödinger
equation, which requires that the zero-body state has zero energy and thus satisfies
Eq. (2.130).
The equivalence of the n-body Schrödinger theory with the above-derived canonically quantized field theory requires specification of the ordering of the field operators
after having imposed the canonical commutation rules upon the fields.
By analogy with the definition of a local Hamiltonian we call an action A local
if it can be written as a spacetime integral over a Lagrangian density L(x, t):
A=
Z
dt
Z
d3 x L(x, t),
(2.161) {3.localact}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.8 Canonical Formalism in Quantum Field Theory
107
where L(x, t) depends only on the fields ψ(x, t) and their first derivatives. The
kinetic part in (2.158) is obviously local, the interacting part is bilocal [recall (2.139)].
For a local theory, the canonical field momentum (2.162) becomes
π(x, t) ≡
∂L
= h̄ψ ∗ (x, t).
∂∂t ψ(x, t)
(2.162) {3.118c}
The formal application of the rules (2.143) leads again directly to the commutation
rules (2.138) without prior splitting into kinetic part and remainder.
In the complex-field formulation, only ψ(x, t) has a canonical momentum, not
ψ ∗ (x, t). This, however, is an artifact of the use of complex field variables. Later,
in Section 7.5.1 we shall encounter a more severe problem, where the canonical
momentum of a component of the real electromagnetic vector field vanishes as a
consequence of gauge invariance, requiring an essential modification of the quantization procedure.
Let us calculate the classical equations of motion for the continuous field theory.
They are obtained by extremizing the action with respect to ψ(x) and ψ ∗ (x). To
do this we need the rules of functional differentiation. These rules are derived
as follows: we take the obvious differentiation rules stating the independence of
generalized Lagrange variables qi (t), which read
∂qi (t)
= δij ,
∂qj (t)
(2.163) {3.111}
and generalize them to lattice variables
∂qx (t)
= δxx′ .
∂qx′ (t)
(2.164) {3.113}
For continuous field variables, these become
∂ψ(x, t)
= δ(x − x′ ).
′
∂ψ(x , t)
(2.165) {3.113f}
The entire formalism can be generalized, thus considering the action as a local functional of fields living in continuous four-dimensional spacetime. Then the
derivative rules must be generalized further to functional derivatives whose variations satisfy the basic rules
δψ(x, t)
= δ (3) (x − x′ )δ(t − t′ ) = δ (4) (x − x′ ).
δψ(x′ , t′ )
(2.166) {3.113fR}
The functional derivatives of actions which depend on spacetime-dependent fields
ψ(x, t) are obtained by using the chain rule of differentiation together with (2.166).
The formalism of functional differentiation and integration will be treated in detail
in Chapter 14.
108
2 Field Formulation of Many-Body Quantum Physics
For a local theory, where the action has the form (2.161), and the fields and
their canonical momenta (2.162) are time-dependent Lagrange coordinates with differentiation rules (2.165), the extremality conditions lead to the Euler-Lagrange
equations
∂L
δL
∂A
=
− ∂t
= 0,
(2.167) {3.125}
∂ψ(x, t)
∂ψ(x, t)
∂ ∂t ψ(x, t)
∂A
∂L
δL
=
− ∂t ∗
= 0.
∗
∗
∂ψ (x, t)
∂ψ (x, t)
∂ ∂t ψ(x, t)
(2.168) {3.126}
The second equation is simply the complex-conjugate of the first.
Note that these equations are insensitive to surface terms. This is why, in spite
of the asymmetric appearance of ψ and ψ ∗ in the action (2.158), the two equations
(2.167) and (2.168) are complex conjugate to each other. Indeed, the latter reads
explicitly
"
h̄2 2
ih̄∂t +
∂x − V1 (x) ψ(x, t) =
2M
#
Z
dx′ ψ ∗ (x′ , t)V2 (x − x′ ; t)ψ(x′ , t) ψ(x, t), (2.169) {3.127}
and it is easy to verify that (2.167) produces the complex conjugate of this.
After field quantization, the above Euler-Lagrange equation becomes an equation
for the field operator ψ(x′ , t) and its conjugate ψ ∗ (x′ , t) must be replaced by the
Hermitian conjugate field operator ψ †∗ (x′ , t).
Let us also remark that the equation of motion (2.168) can be used directly to
derive the n-body Schrödinger equation (2.91) once more in another way, by working
with time-dependent field operators. As a function of time, an arbitrary state vector
evolves as follows:
|ψ(t)i = e−iĤt/h̄ |ψ(0)i.
(2.170) {@}
Multiplying this by the basis bra-vectors
1
√ h0|âxn · · · âx1 ,
n!
(2.171) {3.1280a}
we obtain the time-dependent Schrödinger wave function
(2.172) {3.1280}
ψ(x1 , . . . , xn ; t).
Inserting between each pair of âxν -operators in (2.171) the trivial unit factors 1 =
e−iĤt/h̄ eiĤt/h̄ , each of these operators is transformed into the time-dependent field
operators ψ̂(xν , t), and one has
1
ψ(x1 , . . . , xn ; t) = √ h0|e−iĤt/h̄ ψ̂(xn , t) · · · ψ̂(x1 , t)|ψ(0)i.
n!
(2.173) {3.1281}
Using the zero-energy property (2.130) of the vacuum state, this becomes
ψ(x1 , . . . , xn ; t) = hx1 , . . . , xn ; t|ψ(0)i.
(2.174) {3.128}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
109
2.9 More General Creation and Annihilation Operators
The bra-states arising from the application of the time-dependent field operators
ψ̂(xi , t) to the vacuum state on the left
1
√ h0|ψ̂(xn , t) · · · ψ̂(x1 , t),
n!
(2.175) {3.statest}
define a new time-dependent basis
hx1 , . . . , xn ; t|,
(2.176) {@}
with the property
hx1 , . . . , xn ; t|ψ(0)i ≡ hx1 , . . . , xn |ψ(t)i.
(2.177) {3.128b}
If we apply to the states (2.175) the differential operator (2.169) and use the
canonical equal-time commutation rules (2.138), we may derive once more that
ψ(x1 , . . . , xn ; t) obeys the Schrödinger equation (2.18).
The difference between the earlier way (2.98) of defining the wave function and
the formula (2.174) is, of course, the second-quantized version of the difference
between the Schrödinger and the Heisenberg picture for the ordinary quantum mechanical wave functions. In Eq. (2.98), the states |ψ(t)i are time-dependent but
the basis ket vectors hx1 , .., xn | are not, and with them also the field operators
ψ̂(x, 0) = âx generating them. In Eq. (2.174), on the contrary, the states hψ(0)|
are time-independent (and may be called Heisenberg states), but the local basis bra
states hx1 , . . . , xn ; t| are not, and with them the field operators ψ̂(x, t) generating
them. Whatever representation we use, the n-body wave function ψ(x1 , . . . , xn , t)
remains the same and obeys the Schrödinger equation (2.18). The change of picture
is relevant only for the operator properties of the many-particle description.
Certainly, there is also the possibility of changing the picture in the Schrödinger
wave function ψ(x1 , . . . , xn ; t). But the associated unitary transformation would
take place in another Hilbert space, namely in the space of square integrable functions of n arguments, where p̂ and x̂ are the differential operators −ih̄∂x and x.
When going through the proof that (2.174) satisfies the n-body Schrödinger
equation (2.18), we realize that at no place do we need the assumption of timeindependent potentials. Thus we can conclude that the canonical quantization
scheme for the action (2.158) is valid for an arbitrary explicit time dependence
of the potentials in the Hamiltonian operator Ĥ [see (2.5)]. It is always equivalent
to the Schrödinger description for an arbitrary number of particles.
2.9
More General Creation and Annihilation Operators
In many applications it is possible to solve exactly the Schrödinger equation with
only the one-body potential V1 (x; t). In these cases it is useful to employ, instead
of the creation and annihilation operators of particles at a point, another equivalent
set of such operators which refers, right away, to the corresponding eigenstates. We
110
2 Field Formulation of Many-Body Quantum Physics
do this by expanding the field operator into the complete set of solutions of the
one-particle Schrödinger equation
ψ̂(x, t) =
X
ψEα (x, t)âα .
α
(2.178) {3.129}
If the one-body potential is time-independent and there is no two-body potential,
the states have the time dependence
ψEα (x, t) = ψEα (x) e−iEα t/h̄ .
(2.179) {3.130}
The expansion (2.178) is inverted to give
âα =
Z
d3 x ψE∗ α (x, t)ψ̂(x, t),
(2.180) {@}
which we shall write shorter in a scalar-product notation as
âα = (ψEα (t), ψ̂(t)).
(2.181) {inversiona}
As opposed to the Dirac bracket notation to denote basis-independent scalar products, the parentheses indicate more specifically a scalar product between spatial
wave functions.
From the commutation rules (2.138) we find that the new operators âα , â†α satisfy
the commutation rules
[âα , âα′ ] = [â†α , â†α′ ] = 0,
[âα , â†α′ ] = δα,α′ .
(2.182) {3.131}
Inserting (2.178) into (2.139) with V2 = 0, we may use the orthonormality relation
among the one-particle states ψEα (x) to find the field operator representation for
the Hamilton operator
X
Ĥ =
Eα â†α âα .
(2.183) {3.132}
α
The eigenstates of the time-independent Schrödinger equation
Ĥ|ψ(t)i = E|ψ(t)i
(2.184) {3.133}
are now
1
(â†α1 )n1 · · · (â†αk )nk |0i,
n1 ! · · · nn !
where the prefactor ensures the proper normalization. The energy is
|n1 , . . . , nn i = √
E=
k
X
(2.185) {3.134}
(2.186) {3.135}
Eαi ni .
i=1
Finally, by forming the scalar products
1
1
√ h0|âxn · · · âx1 (â†α1 )n1 · · · (â†αk )nk |0i √
,
n1 ! · · · nk !
n!
(2.187) {3.136}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.10 Quantum Field Formulation of Many-Fermion Schrödinger Equations
111
we recover precisely the symmetrized wave functions (2.32) with the normalization
factors (2.41).
Similar considerations are, of course, possible in the Heisenberg picture of the
operators â†α , âα which can be obtained from
âα (t) =
Z
d3 xψE∗ α (x)ψ̂(x, t).
(2.188) {@}
In the field operator description of many-body systems, the Schrödinger wave
function ψ(x, t) has become a canonically quantized field object. Observe that the
field ψ(x, t) by itself contains all relevant quantum mechanical information of the
system via the derivative terms of the action (2.158),
ψ̂ † (x, t)ih̄∂t ψ̂(x, t) +
h̄2 †
ψ̂ (x, t)∂x2 ψ̂(x, t).
2M
(2.189) {3.137}
This fixes the relation between wavelength and momentum, and between frequency
and energy. The field quantization which introduces the additional processes of
particle creation and annihilation is distinguished from this and often referred to as
second quantization.
It should be kept in mind that, for a given n-body system, second quantization
is completely equivalent and does not go beyond the usual n-body Schrödinger
theory. It merely introduces the technical advantage of collecting the wave equations
for any particle number n in a single operator representation. This advantage is,
nevertheless, of great use in treating systems with many identical particles. In
the limit of large particle densities, it gives rise to approximations which would be
very difficult to formulate in the Schrödinger formulation. In particular, collective
excitations of many-particle systems find their easiest explanation in terms of a
quantum field formulation.
The full power of quantum fields, however, will unfold itself when trying to
explain the physics of relativistic particles, where the number of particles is no
longer conserved. Since the second-quantized Hilbert space contains any number of
particles, the second-quantized formulation allows naturally for the description of the
emission and absorption of fundamental particles, processes which the Schrödinger
equation is unable to deal with.
2.10
Quantum Field Formulation of Many-Fermion
Schrödinger Equations
{manyferm
The question arises whether an equally simple formalism can be found which automatically leads to the correct antisymmetric many-particle states
1 X
|x1 , . . . , xn iA = √
ǫP |xp(1) i ×
⊙ ... ×
⊙ |xp(n) i.
n! P
(2.190) {3.138}
This is indeed possible. Let us remember that the symmetry of the wave functions
was a consequence of the commutativity of the operators ψ̂(x, t) for different position
112
2 Field Formulation of Many-Body Quantum Physics
values x. Obviously, we can achieve an antisymmetry in the coordinates by forming
product states
1
(2.191) {3.139}
|x1 , . . . , xn iA = √ â†x1 · · · â†xn |0i
n!
and requiring anticommutativity of the particle creation and annihilation operators:
{â†x , â†x′ } = 0,
{âx , âx′ } = 0.
(2.192) {3.140}
The curly brackets denote the anticommutator defined in Eq. (2.110). To define a
closed algebra, we require in addition, by analogy with the third commutation rule
(2.75) for bosons, the anticommutation rule
{âx , â†x′ } = δ (3) (x − x′ ).
(2.193) {3.142}
As in the bosonic case we introduce a vacuum state |0i which is normalized as in
(2.87) and contains no particle [cf. (2.86)]:
âx |0i = 0 , h0|â†x = 0.
(2.194) {3.143}
The anticommutation rules (2.192) have the consequence that each point can at
most be occupied by a single particle. Indeed, applying the creation operator twice
to the vacuum state yields zero:
â†x â†x |0i = {â†x , â†x }|0i − â†x â†x |0i = 0.
(2.195) {@}
This guarantees the validity of the Pauli exclusion principle.
The properties (2.192), (2.193), and (2.194) are sufficient to derive the manybody Schrödinger equations with two-body interactions for an arbitrary number of
fermionic particles. It is easy to verify that the second-quantized Hamiltonian has
the same form as in Eq. (2.129). The proof proceeds along the same line as in
the symmetric case, Eqs. (2.94)–(2.129). A crucial tool is the operator chain rule
(2.111) derived for anticommutators. The minus sign, by which anticommutators
differ from commutators, cancels out in all relevant equations.
As for bosons we define a time-dependent quantum field for fermions in the
Heisenberg picture as
ψ̂(x, t) = eiĤt/h̄ ψ̂(x, 0)e−iĤt/h̄
= eiĤt/h̄ âx e−iĤt/h̄ ,
(2.196) {3.144}
and find equal-time anticommutation rules of the same type as the commutation
relations (2.138):
{ψ̂(x, t), ψ̂(x′ , t)} = 0,
{ψ̂ † (x, t), ψ̂ † (x′ , t)} = 0,
{ψ̂(x, t), ψ̂ † (x′ , t)} = δ (3) (x − x′ ).
(2.197) {3.93f}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
113
2.11 Free Nonrelativistic Particles and Fields
The Hamiltonian has again the form Eq. (2.139).
There is only one place where the fermionic case is not completely analogous
to the bosonic one: The second-quantized formulation cannot be derived from a
standard canonical formalism of an infinite number of generalized coordinates. The
standard formalism of quantum mechanics applies only to true physical canonical
coordinates p(t) and q(t), and these can never account for anticommuting properties
of field variables.2 Thus an identification analogous to (2.144),
p̂x (t) ≡ ψ̂I (x, t),
q̂x (t) ≡ ψ̂R (x, t),
(2.198) {3.145}
is at first impossible.
The canonical formalism may nevertheless be generalized appropriately. We may
start out with exactly the same classical Lagrangian as in the boson case, Eq. (2.158),
but treat the fields formally as anticommuting objects, i.e.,
ψ(x, t)ψ(x′ , t′ ) = −ψ(x′ , t′ )ψ(x, t).
(2.199) {@}
In mathematics, such objects are called Grassmann variables. Using these, we define
again classical canonical momenta
px (t) ≡
δL
= −ih̄ψ † (x, t) ≡ π(x, t).
δ ψ̇(x, t)
(2.200) {3.146}
Together with the field variable qx (t) = ψ † (x, t), this is postulated to satisfy the
canonical anticommutation rule
{px (t), qx′ (t)} = −ih̄δ (3) (x − x′ ).
2.11
(2.201) {3.147}
Free Nonrelativistic Particles and Fields
An important way to approach interacting theories is based on perturbative methods. Usually, these begin with the free theory and prescribe how to calculate successive corrections due to the interaction energies. A detailed discussion of how
and when this works will be given later. It seems intuitively obvious, however, that
at least for weak interactions, the free theory may be a good starting point for an
approximation scheme. It is therefore worthwhile to study a few properties of the
free theory in detail.
The free-field action is, according to Eqs. (2.158) and (2.139) for V1 (x) = 0 and
V2 (x − x′ ) = 0:
A=
2
Z
h̄2 2
∂x ψ(x, t).
dtd x ψ (x, t) ih̄∂t +
2M
3
∗
"
#
(2.202) {3.120ff}
For a detailed discussion of classical mechanics with supersymmetric Lagrange coordinates see
A. Kapka, Supersymmetrie, Teubner, 1997.
114
2 Field Formulation of Many-Body Quantum Physics
The quantum field ψ̂(x, t) satisfies the field operator equation
h̄2 2
∂x ψ̂(x, t) = 0,
ih̄∂t +
2M
(2.203) {3.148}
with the conjugate field satisfying
←
ψ̂ † (x, t) − ih̄ ∂ t +
h̄2 ←2
∂x
2M
(2.204) {3.149}
= 0.
The equal-time commutation rules for bosons and fermions are
[ψ̂(x, t), ψ̂(x′ , t)]∓ = 0,
[ψ̂ † (x, t), ψ̂ † (x′ , t)]∓ = 0,
†
′
(3)
(2.205) {@cancomrel
′
[ψ̂(x, t), ψ̂ (x , t)]∓ = δ (x − x ),
where we have denoted commutator and anticommutator collectively by [ . . . , . . . ]∓ ,
respectively.
In a finite volume V , the solutions of the free one-particle Schrödinger equation
are given by the time-dependent version of the plane wave functions (1.185) [compare
(2.179)]:
hx, t|â†pm i
m
ψpm (x, t) = hx, t|p i =
pm 2
i
1
pm x −
t
= √ exp
h̄
2M
V
(
!)
.
(2.206) {3.150}
These are orthonormal in the sense
Z
d3 x ψp∗ m (x, t)ψpm′ (x, t) = δpm ,pm′ ,
(2.207) {nolabel}
and complete, implying that
X
pm
ψpm (x, t)ψp∗ m (x′ , t) = δ (3) (x − x′ ).
(2.208) {3.152}
As in Eq. (2.178), we now expand the field operator in terms of these solutions as
ψ̂(x, t) =
X
ψpm (x, t)âpm .
pm
(2.209) {3.153}
This expansion is inverted with the help of the scalar product (2.181) as
âpm = (ψpm (t), ψ̂(t)) =
Z
d3 x ψp∗ m (x, t)ψ̂(x, t).
(2.210) {inversionab}
In the sequel we shall usually omit the superscript of the momenta pm if their
discrete nature is evident from the context. The operators âp and â†p , obey the
canonical commutation rules corresponding to Eq. (2.182):
[âp , âp′ ]∓ = 0,
[â†p , â†p′ ]∓ = 0,
[âp , â†p′ ]∓ = δpp′ ,
(2.211) {3.154d}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
115
2.11 Free Nonrelativistic Particles and Fields
where we have used the modified δ-functions introduced in Eq. (1.196).
In an infinite volume, we use the time-dependent version of the continuous wave
functions (1.195)
ψp (x, t) = hx, t|pi =
hx, t|â†p i
pm 2
i
1
pm x −
t
= √ exp
h̄
2M
V
(
!)
,
(2.212) {3.150c}
which are orthonormal in the sense
Z
d3 x ψp∗ (x, t)ψp′ (x, t) = (2πh̄)3 δ (3) (p − p′ )
δ- (3) (p − p′ ),
(2.213) {3.151}
d-3 p ψp (x, t)ψp∗ (x′ , t) = δ (3) (x − x′ ).
(2.214) {3.152}
≡
and complete, as expressed by
Z
d3 p
ψp (x, t)ψp∗ (x′ , t) ≡
3
(2πh̄)
Z
In terms of these continuum wave functions, we expand the field operator as
ψ̂(x, t) =
Z
d-3 p ψp (x, t)â(p),
(2.215) {3.153c}
and have the inverse
â(p) = (ψp (t), ψ̂(t)) =
Z
d3 x ψp∗ (x, t)ψ̂(x, t).
(2.216) {inversionabc
The discrete-momentum operators âp , and â†p and the continuous ones â(p) and
↠(p), are related by [recall Eq. (1.190)]
√
√
↠(p) = V â†p .
(2.217) {identification
â(p) = V âp ,
For the continuous-momentum operators â(p) and â(p)† , the canonical commutation
rules in Eq. (2.182) take the form
[â(p), â(p′ )]∓ = [↠(p), ↠(p′ )]∓ = 0,
(3)
[â(p), ↠(p′ )] = δ- (p − p′ ).
∓
(2.218) {3.154}
The time-independent many-particle states are obtained, as in (2.185), by repeatedly applying any number of creation operators â†p [or ↠(p)] to the vacuum
state |0i, thus creating states
|np1 , np2 , . . . , npk i = N S,A (â†p1 )np1 · · · (â†pk )npk |0i,
(2.219) {3.155a}
where the normalization factor is determined as in Eq. (2.185). For bosons with np1
identical states of momentum p1 , with np2 identical states of momentum p2 , etc.,
the normalization factor is
1
NS = q
.
(2.220) {3.156}
np1! · · · npk!
116
2 Field Formulation of Many-Body Quantum Physics
The same formula can be used for fermions, only that then the values of npi are
restricted to 0 or 1, and the normalization constant N A is equal to 1. The timeindependent wave functions are obtained as
N S,A
hx1 , . . . , xn |np1 , np2 , . . . , npk iS,A = √
(2.221) {3.155}
n!
× h0|ψ̂(xn , 0) · · · ψ̂(x1 , 0)(â†p1 )np1 · · · (â†pk )npk |0i,
and the time-dependent ones as
hx1 , . . . , xn |np1 , np2 , . . . , npk ; tiS,A = hx1 , . . . , xn |np1 , np2 , . . . , npk iS,A e−iEt/h̄
N S,A
= √ h0|ψ̂(xn , 0) · · · ψ̂(x1 , 0)e−iĤt/h̄ (â†p1 )np1 . . .(â†pk )npk|0i
n!
S,A
N
= √ h0|ψ̂(xn , 0) · · · ψ̂(x1 , 0)[â†p1 (t)]np1 · · · [â†pk (t)]npk |0i,
n!
with the time-dependent creation operators being defined by
â†p (t) ≡ eiĤt/h̄ â†p e−iĤt/h̄ .
(2.222) {3.164}
The energy of these states is
E=
k
X
(2.223) {3.157}
ni εpi ,
i=1
where εp ≡ p2 /2M are the energies of the single-particle wave functions (2.212). The
many-body states (2.219) form the so-called occupation number basis of the Hilbert
space. For fermions, ni can only be 0 or 1, due to the anticommutativity of the
operators âp and â†p among themselves. The basis states are properly normalized:
hnp1 np2 np3 . . . npk |n′p1 n′p2 n′p3 . . . n′pk i = δnp1 n′p1 δnp2 n′p2 δnp3 n′p3 . . . δnpk n′pk .
(2.224) {3.160}
They satisfy the completeness relation:
X
p1 p2 p3 ...
X
np1 np2 np3 ...npk
|np1 np2 np3 . . . npk ihnp1 np2 np3 . . . npk | = 1S,A ,
(2.225) {3.161}
where the unit operator on the right-hand side covers only the physical Hilbert space
of symmetric or antisymmetric n-body wave functions.
2.12
Second-Quantized Current Conservation Law
{SQCCL}
In Subsection 1.3.4 of Chapter 1 we have observed an essential property for the
probability interpretation of the Schrödinger wave functions: The probability current density (1.107) and the probability density (1.108) are related by the local
conservation law (1.109):
∂t ρ(x, t) = −∇ · j(x, t).
(2.226) {probclaw3}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
117
2.13 Free-Particle Propagator
This followed directly from the Schrödinger equation (2.169). Since the same equation holds for the field operators, i.e., with ψ ∗ (x′ , t) replaced by ψ̂ † (x′ , t), the field
operators of charge and current density,
ρ̂(x, t) = ψ̂ † (x, t)ψ̂(x, t),
↔
h̄ †
ψ̂ (x, t) ∇ ψ̂(x, t)
ĵ(x, t) = −i
2M
(2.227) {quantizcurr}
satisfy the same relation. When integrating (2.226) over all x, and using Green’s
theorem as done in Eq. (1.110), we obtain a global conservation law that ensures the
time independence of the particle number operator
N̂ =
Z
3
d x ρ̂(x, t) =
Z
d3 x ψ̂ † (x, t)ψ̂(x, t).
(2.228) {partnbschr}
Since this is time-independent, we can use (2.134) to rewrite
N̂ =
Z
3
d x ρ̂(x, 0) =
Z
3
†
d x ψ̂ (x, 0)ψ̂(x, 0) =
Z
d3 x â†x âx ,
(2.229) {2.222a}
so that N̂ coincides with the particle number operator (2.78). The original form
(2.228) is the Heisenberg picture of the particle number operator, which coincides
with (2.229), since the particle number is conserved.
2.13
Free-Particle Propagator
The perturbation theory of interacting fields to be developed later in Chapter 10
requires knowledge of an important free-field quantity called the free-particle propagator. It is the vacuum expectation of the time-ordered product of two free field
operators. As we shall see, the calculation of any observable quantities can be reduced to the calculation of some linear combination of products of free propagators
[see Section 7.17.1]. Let us first extend the definition (1.249) of the time-ordered
product of n time-dependent operators to allow for fermion field operators. Suppose
that the times in an operator product Ân (tn ) · · · Â1 (t1 ) have an order
tin > tin−1 > . . . > ti1 .
{@freepprop}
(2.230) {3.165}
Then the time-ordered product of the operators is defined by
T̂ Ân (tn ) · · · Â1 (t1 ) ≡ ǫP Âin (tin ) · · · Âi1 (ti1 ).
(2.231) {3.166}
With respect to the definition (1.250), the right hand side carries a sign factor
ǫP = ±1 depending on whether an even or an odd permutation P of the fermion
field operators is necessary to reach the time-ordered form. For bosons, εP ≡ 1.
The definition of the time-ordered products can be given more concisely using
the Heaviside function Θ(t) of Eq. (1.313).
118
2 Field Formulation of Many-Body Quantum Physics
For two operators, we have
T̂ Â(t1 )B̂(t2 ) = Θ(t1 − t2 )Â(t1 )B̂(t2 )±Θ(t2 − t1 )B̂(t2 )Â(t1 ),
(2.232) {3.167}
with the upper and lower sign applying to bosons and fermions, respectively. The
free-particle propagator can now be constructed from the field operators as the
vacuum expectation value
G(x, t; x′ , t′ ) = h0|T̂ ψ̂(x, t)ψ̂ † (x′ , t′ )|0i.
(2.233) {3.168}
Applying the free-field operator equations (2.203) and (2.204), we notice a remarkable property: The free-particle propagator G(x, t; x′ , t′ ) coincides with the Green
function of the Schrödinger differential operator. Recall that a Green function of a
homogeneous differential equation is defined by being the solution of the inhomogeneous equation with a δ-function source (see Section 1.6). This property may easily
be verified for the free-particle propagator, which satisfies the differential equations
h̄2 2
ih̄∂t +
∂x G(x, t; x′ , t′ ) = ih̄δ(t − t′ )δ (3) (x − x′ ),
2M
←
h̄2 ←2
′ ′
′
G(x, t; x , t ) − ih̄ ∂t +
= ih̄δ(t − t′ )δ (3) (x − x′ ),
∂x
2M
(2.234) {3.emo}
(2.235) {3.emo1}
thus being a Green function of the free-particle Schrödinger equation: The righthand side follows directly from the fact that the field ψ̂(x, t) satisfies the Schrödinger
equation and the obvious formula
∂t Θ(t − t′ ) = δ(t − t′ ).
(2.236) {3.170}
With the help of the chain rule of differentiation and Eq. (2.232), we see that
h̄2 2
ih̄∂t +
∂x h0|T̂ ψ̂(x, t)ψ̂ † (x′ , t′ )|0i
2M
h
i
= ih̄ ∂t Θ(t − t′ )h0|ψ̂(x, t)ψ̂ † (x′ , t′ )|0i ± ∂t Θ(t′ − t)h0|ψ̂ † (x′ , t′ )ψ̂(x, t)|0i
= ih̄δ(t − t′ )h0|[ψ̂(x, t), ψ̂ † (x′ , t)]∓ |0i = ih̄δ(t − t′ )δ (3) (x − x′ ),
(2.237) {3.emop}
where the commutation and anticommutation rules (2.138) and (2.197) have been
used, together with the unit normalization (2.87) of the vacuum state.
In the theory of differential equations, Green functions are introduced to find
solutions for arbitrary inhomogeneous terms. These solutions may be derived from
superpositions of δ-function sources. In quantum field theory, the same Green functions serve as propagators to solve inhomogeneous differential equations that involve
field operators.
Explicitly, the free field propagator is calculated as follows: Since ψ̂(x, t) annihilates the vacuum, only the first term in the defining Eq. (2.232) contributes, so
that we can write
G(x, t; x′ , t′ ) = Θ(t − t′ )h0|ψ̂(x, t)ψ̂ † (x′ , t′ )|0i.
(2.238) {3.171}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
119
2.13 Free-Particle Propagator
Inserting the expansion Eq. (2.215) with the wave functions (2.212), and using
(2.218), the right-hand side becomes
Θ(t − t′ )
Z
′ ′
2
′2 ′
d-3 p d-3 p′ ei[(px−p x )−(p t/2M −p t /2M )]/h̄ h0|â(p)↠(p′ )|0i
= Θ(t − t′ )
Z
′
2
′
d-3 p ei[p(x−x )−p (t−t )/2M ]/h̄ .
(2.239) {3.172}
By completing the square and using the Gaussian integral
1
2
d-3 p e−ap /2h̄ = √
3,
2πh̄a
Z
(2.240) {3.174}
we find
1
iM (x−x′ )2 /2h̄(t−t′ )
G(x, t; x′ , t′ ) = Θ(t − t′ ) q
3e
2πih̄(t − t′ )/M
= G(x − x′ , t − t′ ).
(2.241) {3.175}
The right-hand side is recognized as the usual quantum-mechanical Green function
of the free-particle Schrödinger equation of Eq. (1.350). Indeed, the factor after
Θ(t − t′ ) is simply the one-particle matrix element of the time evolution operator
′
h0|ψ̂(x, t)ψ̂ † (x′ , t′ )|0i = h0|ψ̂(x)e−iĤ(t−t )/h̄ ψ̂ † (x′ )|0i
= hx|Û(t, t′ )|x′ i.
(2.242) {3.173}
This is precisely the expression discussed in Eqs. (1.310)–(1.312). It describes the
probability amplitude that a single free particle has propagated from x to x′ in the
time t − t′ > 0. For t − t′ < 0, G vanishes.
There exists a more useful way of writing the Fourier representation of the propagator than that in Eq. (2.239). It is based on the integral representation (1.319)
of the Heaviside function:
′
Θ(t − t ) =
Z
∞
−∞
- e−iE(t−t′ )/h̄
dE
ih̄
.
E + iη
(2.243) {3.176}
As discussed in general in Eqs. (1.317)–(1.319), the iη in the denominator ensures
the causality. For t > t′ , the contour of integration can be closed by an infinite
semicircle below the energy axis, thereby picking up the pole at E = −iη, so that
we obtain by the residue theorem
Θ(t − t′ ) = 1,
t > t′ .
(2.244) {3.177}
For t < t′ , on the other hand, the contour may be closed above the energy axis and,
since there is no pole in the upper half-plane, we have
Θ(t − t′ ) = 0,
t < t′ .
(2.245) {3.177b}
120
2 Field Formulation of Many-Body Quantum Physics
Relation (2.243) can be generalized to
′
Θ(t − t′ )e−iE0 (t−t )/h̄ =
Z
∞
−∞
- e−iE(t−t′ )/h̄
dE
ih̄
.
E − E0 + iη
(2.246) {3.178}
Using this with E0 = p2 /2M we find from (2.239) the integral representation
G(x − x′ , t − t′ ) =
Z
d-3 p
Z
∞
−∞
- eip(x−x′ )/h̄−iE(t−t′ )/h̄
dE
ih̄
E−
p2 /2M
+ iη
.
(2.247) {3.179}
In this form we can trivially verify the equations of motion (2.234) and (2.235). This
expression agrees, of course, with the quantum mechanical time evolution amplitude
(1.344).
The Fourier-transformed propagator
G(p, E) =
=
Z
d3 x
E−
Z
∞
−∞
dt e−i(px−Et)/h̄ G(x, t)
ih̄
p2 /2M
+ iη
(2.248) {3.180}
has the property of being singular when the variable E is equal to a physical particle
energy E = p2 /2M . This condition is often called the energy shell condition. It is a
general property of Green functions that their singularities in the energy-momentum
variables display the spectra of the particles of the system.
2.14
Collapse of Wave Function
A related Green function can be used to illustrate the much discussed phenomenon of
the collapse of the wave function in quantum mechanics [10]. If we create a particle
at some time t′ , we create a Schrödinger wave function that fills immediately the
entire space. If we annihilate the particle at some later time t, the wave function
disappears instantaneously from the entire space. This phenomenon which is hard
to comprehend physically is obviously an artifact of the nonrelativistic Schrödinger
theory. Let us see how it comes about in the formalism. If we measure the particle
density at a time t′′ that lies only slightly later than the later time t in the above
Green function, we find
G(z, t′′ ; x, t; x′ , t′ ) = h0|T̂ ρ̂(z, t′′ )ψ̂(x, t)ψ̂ † (x′ , t′ )|0i
= h0|T̂ ψ̂ † (z, t′′ )ψ̂(z, t′′ )ψ̂(x, t)ψ̂ † (x′ , t′ )|0i.
{COLLWF}
(2.249) {3.168}
The second and the fourth field operators yield a time-ordered Green function
hT̂ ψ̂(z, t′′ )ψ̂ † (x′ , t′ )i= Θ(t′′ − t′ )G(z, t′′ ; x′ t′ ), which is zero for t′′ > t′ . This is multiplied by the time-ordered Green function of the fist and the third field operators
which is hT̂ ψ̂ † (z, t′′ )ψ̂(x, t)i= Θ(t − t′′ )G(z, t′′ ; x t), and thus vanishes for t′′ > t.
Applying Wick’s theorem to (2.249) shows that by the time t′′ that is later than t
and t′ , the wave function created at the initial time t′ has completely collapsed.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
121
2.15 Quantum Statistics of Free Nonrelativistic Fields
2.15
Quantum Statistics of Free Nonrelativistic Fields
2.15.1
{qstfp}
Thermodynamic Quantities
Consider the grand-canonical partition function introduced in Eq. (1.493):
ZG (T, µ) = Tr(e−(Ĥ−µN̂ )/kB T ).
(2.250) {3.181}
The trace has to be taken over the complete set of basis states (2.219):
X
p1 p2 p3 ...
X
np1 np2 np3 ...npk
hnp1 np2 np3 . . . |e−(Enp1np2np3... −µ
P
i
npi )/kB T
|np1 np2 np3 . . . i.
Using the additivity of the energies of all single-particle states found in Eq. (2.223),
this can be written as an infinite product
ZG (T, µ) =
X
p2
p2
p2
2
3
1 −µ)n
−( 2M
p1 /kB T −( 2M −µ)np2 /kB T −( 2M −µ)np3 /kB T
X
e
e
e
p1p2p3 ... np1np2np3 ...npk
··· .
(2.251) {3.183}
Each Boltzmann factor leads to the partition function associated with the available
single-particle momentum p1 , p2 , . . . . The product is turned into a sum by taking
the logarithm of this and considering the grand-canonical free energy
FG (T, µ) ≡ −kB T log ZG (T, µ) = −kB T
X
log
X
p2
e−( 2M −µ)n/kB T .
n
p
(2.252) {3.184}
We now distinguish between Bose and Fermi particles. In the first case, the
occupation numbers ni run over all integers 0, 1, 2, . . . up to infinity:
∞
X
1
p2
e−( 2M −µ)n/kB T =
n=0
p2
−( 2M
1−e
−µ)/kB T
.
(2.253) {3.186}
In the second case, ni can be only zero or one, so that
1
X
p2
p2
e−( 2M −µ)n/kB T = 1 + e−( 2M −µ)/kB T .
n=0
(2.254) {nolabel}
Thus we obtain
FG (T, µ) = ±kB T
X
p
p2
log[1 ∓ e−( 2M −µ)/kB T ].
(2.255) {3.188}
Because of the frequent appearance of the energy combination p2 /2M − µ, it will
often be useful to define the quantity
ξp =
p2
− µ = εp − µ.
2M
(2.256) {@XIDEf}
122
2 Field Formulation of Many-Body Quantum Physics
This will abbreviate calculations in grand-canonical ensembles.
In a large volume, momentum states lie so close to each other that the sum may
be approximately evaluated as an integral with the help of the limiting formula
X
−
−−→ gV
V →∞
p
d3 p
= gV
(2πh̄)3
Z
Z
d-3 p.
(2.257) {3.189}
In writing this we have allowed for a degeneracy number g for each momentum state
p. It accounts for extra degrees of freedom of the particles in each momentum state.
In the absence of internal quantum numbers, g counts the different spin polarization
states. If s denotes the spin, then its third component can run from −s to s so that
(2.258) {3.190}
g = 2s + 1.
Moreover, the limit (2.257) is certainly valid only for sums over sufficiently smooth
functions. We shall see in Section 2.15.3 that the limit fails for a Bose gas near
T = 0, where the limit requires the more careful treatment in Eq. (2.337).
As an alternative to the momentum integral (2.257), we may integrate over the
single-particle energies. With the energy εp = p2 /2M, the relation between the
integration measures is
Z
4π
(2π)3h̄3
1
d-3 p =
Z
∞
0
dp p2
2
= q
3√
π
2πh̄2 /M
Z
0
∞
Z
√
dε ε ≡ gε dε.
(2.259) {3.191}
In the last expression we have introduced the quantity
2 √
1
ε,
gε ≡ q
3√
π
2
2πh̄ /M
(2.260) {3.ninteg}
which is the density of states per unit energy interval and volume. With the help
of this quantity we may write (2.255) as an energy integral
FG (T, µ, V ) = gV
Z
∞
0
(2.261) {3.192}
dεgε Fε (T, µ),
where
Fε (T, µ) ≡ ±kB T log[1 ∓ e−(ε−µ)/kB T ]
(2.262) {@}
is the grand-canonical free energy of an individual single-particle energy state.
According to the thermodynamic rule (1.502), the average particle number is
found from the derivative of FG (T, µ, V ) with respect to the chemical potential.
Using (2.255) or the integral representation (2.261), we find
N = gV
X
p
≡ gV
Z
0
1
p2
( 2M
e
∞
−µ)/kB T
dεgε fε ,
∓1
= −gV
Z
0
∞
dεgε
∂
Fε (T, µ)
∂µ
(2.263) {3.pn}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
123
2.15 Quantum Statistics of Free Nonrelativistic Fields
where
fε ≡ −
∂
1
Fε (T, µ) = (ε−µ)/k T
B
∂µ
e
∓1
(2.264) {3.198}
are the average Bose and Fermi occupation numbers of a level of energy ε, respectively. They are plotted in Figs. 2.1 and 2.2.
Figure 2.1 Average Bose occupation number nB (ε − µ). Note that free bosons have a
negative chemical potential µ.
nF (ε)
{3.nbb}
e−(ε−µ)/kB T
ε
Figure 2.2 Average Fermi occupation number nF (ε). Fermions have a positive chemical
potential µ.
{3.nbf}
The internal energy of the system can be calculated from the integral
E = g
Z
0
∞
dεgε fε ε
√
2 Z∞
dε ε ε
.
= gq
3√
(ε−µ)/kB T ∓ 1
π
e
0
2
2πh̄ /M
V
(2.265) {3.199}
124
2 Field Formulation of Many-Body Quantum Physics
On the other hand, we find by a partial integration of the integral in (2.261):
Z
0
∞
dεε
1/2
(ε−µ)/kB T
log[e
2
∓ 1] = −
3
Z
dε ε3/2
∞
0
e(ε−µ)/kB T ∓ 1
,
(2.266) {@}
so that the grand-canonical partition function can be rewritten as
FG (T, µ, V ) = g
Z
∞
0
2
dεgε Fε = − g
3
Z
0
∞
dεgεfε ε.
(2.267) {3.200}
This implies the general thermodynamic relation for a free Bose or Fermi gas:
2
FG = − E.
3
(2.268) {3.201}
Recalling the definition of the pressure (1.527), we have thus found the equation of
state for a free Bose or Fermi gas:
2
pV = E.
3
(2.269) {3.203}
To evaluate the energy integral, we introduce the variable z ≡ ε/kB T and write
(2.263) as
µ
1
V
∓
,
(2.270) {3.206}
I3/2
N = N(T, µ) ≡ g 3
λ (T ) Γ(3/2)
kB T
where
1
λ(T ) ≡ q
2πh̄2 /MkB T
(2.271) {@c2.lambda
is the thermal length associated with mass M and temperature T , and In∓ (µ/kB T )
denotes the function
Z ∞
z n−1
∓
In (α) ≡
dz z−α
.
(2.272) {3.207}
e
∓1
0
After expanding the denominator in a power series, each term can be integrated
and leads for bosons to a series representation
In− (α) = Γ(n)
∞
X
ekα
.
n
k
k=1
(2.273) {7.@207ex}
The sum can be expressed in terms of the polylogarithmic function [8]3
ζn (z) ≡
∞
X
zk
n
k=1 k
(2.274) {@polylog}
as
In− (α) = Γ(n)ζn (eα ).
3
(2.275) {7.@207ex1}
A frequently used notation for ζn (z) is Lin (z).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
125
2.15 Quantum Statistics of Free Nonrelativistic Fields
The sum in (2.273) converges only for α < 0. In the limit α → −0, it has the
limit
In− (0) = Γ(n)ζ(n),
n > 1,
(2.276) {@zeroptc}
where ζ(n) is Riemann’s zeta function
ζ(ν) ≡
∞
X
1
.
ν
k=1 k
(2.277) {@Rzeta}
For n ≤ 1, the function In− (α) diverges like (−α)n−1 in the limit α → −0. In the
opposite limit α → −∞, it goes to a constant:
In− (α) → eα Γ(n).
(2.278) {@largeneg}
This limit is needed to find the high-temperature behavior of the free Bose gas at a
fixed average particle number N, as we see from Eq. (2.270). For large T , the ratio
µ/kB T diverges to −∞.
In the case of fermions, the expansions (2.273), (2.275) read
In+ (α)
= Γ(n)
∞
X
(−1)k−1
k=1
ekα
= −Γ(n)ζn (−eα ),
n
k
(2.279) {7.@207exf}
the only difference with respect to bosons being the alternating signs in the sum.
For α = 0, this becomes
In+ (0) =
∞
∞
X
X
1
1
1
1
1
1
1−n
1−n
ζ(n). (2.280) {7.@Inferm}
− n + n − n +... =
−
2
=
1
−
2
n
n
1 2
3
4
k=0 k
k=0 k
In the opposite limit α → −∞, the sum yields the same constant as in (2.278):
In+ (α) → eα Γ(n).
(2.281) {@largene}
At a fixed particle number N, the chemical potential changes with temperature
in a way determined by the vanishing of the derivative of (2.270), which yields the
equation
∓
I3/2
( µ )
µ
T ∂T
(2.282) {@dmudt}
= −3 ∓ kBµT .
kB T
I1/2 ( kB T )
Here we have used the property
∓
∂z In∓ (z) = (n − 1)In−1
(z),
(2.283) {@invsinm1}
which follows directly from the series expansion (2.273). For large T , we obtain from
Eqs. (2.276) and (2.278) the limit
T ∂T
3
µ
−
−−→ − .
kB T T →∞ 2
(2.284) {@}
126
2 Field Formulation of Many-Body Quantum Physics
Relation (2.269) can also be obtained from the general thermodynamic calculation:
∂
∂ZG
=
βFG ,
(2.285) {@}
E − µN = −ZG−1
∂β
∂β
which follows directly from (2.250) by differentiating with respect to β = 1/kB T .
The grand-canonical free energy is from (2.267), using (2.260), (2.264), (2.271),
and (2.272):
FG (T, µ, V ) = −pV
V
2 1
µ
∓
= −g 3
kB T
I5/2
λ (T )
3 Γ(3/2)
kB T
µ
∓
2 I5/2 ( kB T )
.
= −N(T, µ) kB T
∓
3 I3/2
( kBµT )
(2.286) {3.204}
For large T where α → −∞, the limiting formula (2.278) shows that FG has the
correct Dulong-Petit-like behavior −NkB T , implying the ideal-gas law
(2.287) {3.206i}
pV = NkB T.
Due to the relation (2.268), the energy is
E = N(T, µ) kB T
∓
I5/2
( kBµT )
∓
I3/2
( kBµT )
(2.288) {@3.204E}
.
For large temperatures, this has the correct Dulong-Petit limit of free particles
3NkB T /2.
We may check the first line in Eq. (2.286) by differentiating it with respect
to µ and using the relation (2.283) to reobtain the thermodynamic relation N =
−∂FG /∂µ.
According to Eq. (1.520), the entropy S is obtained from the negative derivative
of (2.286) with respect to T . This yields
µ
∓
2 IS k B T
= gkB N(T, µ) ∓ µ ,
3 I3/2
(2.289) {3.208}
5 ∓
5 ∓
3 ∓
∓ ′
IS∓ (α) ≡ I5/2
(α) − αI5/2
(α) = I5/2
(α) − αI3/2
(α).
2
2
2
This agrees with the general thermodynamic relation
(2.290) {3.209}
V
2
µ
√ IS∓
S = kB g 3
λ (T ) π
kB T
where
kB T
FG (T, µ) = E(T, µ) − µN(T, µ) − T S(T, µ).
(2.291) {@}
Adiabatic processes are defined by the condition S/N = const. which implies by
Eq. (2.290) that the ratio µ/kB T is also a constant. Inserting this into (2.270) with
(2.271) we find that, at a constant particle number, an adiabatic process satisfies
V T 3/2 |adiab = const.
(2.292) {3.210}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
127
2.15 Quantum Statistics of Free Nonrelativistic Fields
Combining this with (2.287) leads to
pV 5/3 |adiab = const.
(2.293) {3.211}
The specific heat at a constant volume and particle number is found from the
entropy by forming the derivative CV = T ∂T S of Eq. (2.289). Using (2.282), this
leads to


∓
∓
I3/2
(α)
1  I5/2 (α)
,
5 ∓
−9 ∓
CV = gkB N
2 I3/2 (α)
I1/2 (α)
α=
µ
.
kB T
(2.294) {@spheatI}
For large T , we use (2.278) again to show that this becomes a constant
3
CV −
−−→ CVDP = g NkB ,
T →∞
2
(2.295) {@DULP}
which complies with the classical rule of Dulong and Petit (kB /2 per degree of
freedom).
2.15.2
Degenerate Fermi Gas Near T = 0
Consider the Fermi gas close to zero temperature which is called the degenerate
limit. Then the occupation number (2.264) reduces to
fε =
(
)
1
0
(
for
ε<µ
ε>µ
)
= Θ(ε − µ).
(2.296) {3.212}
All states with energy lower than µ are filled, all higher states are empty. The
chemical potential µ at zero temperature is called Fermi energy εF :
µ
T =0
≡ εF .
(2.297) {@e312}
The Fermi energy for a given particle number N is found from (2.260), (2.263), and
(2.296):
N = gV
Z
0
∞
dεgεfε = gV
Z
0
εF
dεgε = gV
where
pF =
√
3/2
2M 3/2 εF
3π 2h̄3
q
2MεF ≡ h̄kF
= gV
p3F
,
6π 2 h̄3
(2.298) {3.213}
(2.299) {3.213e}
is the Fermi momentum associated with the Fermi energy. Equation (2.298) is solved
for εF by
εF =
6π 2
g
!2/3 N
V
2/3
h̄2
,
2M
(2.300) {Eq.VAREP}
128
2 Field Formulation of Many-Body Quantum Physics
and for the Fermi momentum by
pF ≡ kFh̄ =
6π 2
g
!1/3 N
V
1/3
h̄.
(2.301) {Eq.Fm}
In two dimensions, we find
εF =
2π ρ
.
g M
(2.302) {Eq.VAREP}
Note that in terms of the particle number N, the density of states per unit energy
interval and volume (2.260) can be written as
3N 1
ggε ≡
2 V εF
s
ε
.
εF
(2.303) {3.nintegn}
As the gas is heated slightly, there is a softening of the degeneracy in the particle
distribution (2.296). In order to study this quantitatively it is useful to define a
characteristic temperature associated with the Fermi energy εF , the so-called Fermi
temperature
εF
1 pF 2
h̄2 kF 2
TF ≡
=
=
.
(2.304) {3.215}
kB
kB 2M
kB 2M
For electrons in a metal, kF is of the order of 1/Å. Inserting M = me = 9.109558 ×
10−28 g, kB = 1.380622 × 10−16 erg/K, and h̄ = 6.0545919 × 10−27 erg sec, we see
that the order of magnitude of TF is
TF ≈ 44 000 K.
(2.305) {3.216}
T /TF ≪ 1
(2.306) {3.217}
Hence the relation
is quite well fulfilled even far above room temperature, and T /TF can be used as
an expansion parameter in evaluating the thermodynamic properties of the electron
gas at nonzero temperature.
Let us do this to calculate the corrections to the above equations at small T .
Eliminating the particle number in (2.270) in favor of the Fermi temperature with
the help of Eqs. (2.298) and (2.304), we find the temperature dependence of the
chemical potential from the equation
T
1=
TF
3/2
3 +
µ
I3/2
2
kB T
.
(2.307) {3.206c}
For T →0, the chemical potential µ approaches the Fermi energy εF , so that small
T corresponds to a large reduced chemical potential µ̄ = µ/kB T . Let us derive an
+
expansion for I3/2
(µ̄) in powers of 1/µ̄ in this regime. For this we set
z − µ̄ ≡ x,
(2.308) {3.219}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
129
2.15 Quantum Statistics of Free Nonrelativistic Fields
and write In+ (µ̄) as
In+ (µ̄)
=
Z
∞
−µ̄
(µ̄ + x)n−1
dx x
=
e +1
µ̄
Z
0
(µ̄ − x)n−1
+
dx −x
e +1
Z
∞
0
dx
(µ̄ + x)n−1
. (2.309) {Eq.In}
ex + 1
In the first integral on the right-hand side we substitute
1
1
=
1
−
e−x + 1
ex + 1
(2.310) {3.221}
and obtain
In+ (µ̄) =
Z
µ̄
0
n−1
dx x
+
Z
∞
0
(µ̄+x)n−1 − (µ̄−x)n−1
dx
+
ex + 1
Z
∞
µ̄
(µ̄−x)n−1
dx x
. (2.311) {3.222}
e +1
In the limit µ̄→∞, only the first term survives, whereas the last term
L(µ̄) ≡
Z
∞
µ̄
(µ̄−x)n−1
= −(−1)n
dx x
e +1
Z
0
∞
xn−1
= (−1)n Γ(n)ζn (−e−µ̄ ) (2.312) {@2.312}
dx x+µ̄
e
+1
is exponentially small, so that it can be ignored in an expansion in powers of 1/µ̄.
The second term is expanded as
∞ X
n−1
X
xk
(n−1)! n−1−k
=
2
µ̄
(1 − 2−k )ζ(k + 1).
x
k
e
+
1
(n−1−k)!
0
k=odd
k=odd
(2.313) {3.223}
At even positive and odd negative integer arguments, the zeta function is related to
the Bernoulli numbers by4
2
µ̄n−1−k
Z
∞
dx
B2n
,
ζ(1 − 2n) = −
2n
(2π)2n
ζ(2n) =
|B2n |.
2(2n)!
(2.314) {197bn}
The two equations go over into each other via the identity
ζ(x) = 2x π x−1 sin(πx/2)Γ(1 − x)ζ(1 − x),
(2.315) {197bnp}
which can also be written as
ζ(x) = 2x−1 π x ζ(1 − x)/Γ(x) cos
xπ
.
2
(2.316) {197ex}
The lowest values of ζ(k + 1) occurring in the expansions (2.313) are5
ζ(2) =
4
π2
π4
π6
, ζ(4) = , ζ(6) =
,
6
90
945
(2.317) {3.zetafval}
These and the subsequent formulas are found in I.S. Gradshteyn and I.M. Ryzhik, op. cit.,
Formulas 9.542 and 9.535.
5
Other often-needed values are ζ(0) = −1/2, ζ ′ (0) = − log(2π)/2, ζ(−2n) = 0, ζ(3) ≈
1.202057, ζ(5) ≈ 1.036928, . . . .
130
2 Field Formulation of Many-Body Quantum Physics
so that In+ (µ̄) starts out for large µ̄ like
In+ (µ̄) =
1
7
1 n
µ̄ + 2(n−1) ζ(2)µ̄n−2 + 2(n−1)(n−2)(n−3) ζ(4)µ̄n−4 + . . . . (2.318) {@}
n
2
8
Inserting this with n = 3/2 into Eq. (2.307), we find
T
1=
TF
3/2
"
3 2
µ
2 3 kB T
3/2
µ
π2
+
12 kB T
−1/2
7π 4
µ
+
+
3 · 320
kB T
−5/2
#
... .
(2.319) {3.206d}
From this we derive, by inversion, a small-T /TF expansion of µ/kB T . Rewriting the
latter ratio as a product µ/εF × TF /T , we obtain for µ the series
µ = εF
π2 T
1−
12 TF
"
2
7π 4 T
+
720 TF
4
#
(2.320) {3.206df}
+ ... .
Only exponentially small terms in e−TF /T coming from the integral L(µ̄) of (2.312)
are ignored. Inserting the chemical potential (2.320) into the grand-canonical free
energy FG , we obtain by reexpanding the first line in (2.286):
5π 2
FG (T, µ, V ) = FG (0, µ, V ) 1 +
8
"
T
TF
2
7π 4 T
−
384 TF
4
#
+ ... ,
(2.321) {3.228}
where
FG (0, µ, V )
√
kµ3
2
2M 3/2 5/2
2
µ = − gV 2 µ.
≡ − gV
5
5 6π
3π 2h̄3
(2.322) {3.229}
Here kµ is the analog of kF, to which it reduces for T = 0 [compare (2.299)]:
kµ ≡
1q
2Mµ.
h̄
(2.323) {@}
At T = 0 where εF , we see from (2.299) that
2
FG (0, εF, V ) = − NεF .
5
(2.324) {3.228c}
This can also be obtained from (2.286) using the limiting behavior (2.281).
By differentiating FG with respect to the temperature at fixed µ, we obtain the
low-temperature behavior of the entropy
π2 T N
+ ... ,
2 TF V
(2.325) {3.230}
π2 T
∂S =
k
N .
B
∂T V,N
2 TF
(2.326) {3.231}
S = kB
and from this the specific heat
CV
= T
H. Kleinert, PARTICLES AND QUANTUM FIELDS
131
2.15 Quantum Statistics of Free Nonrelativistic Fields
This is a linear behavior with a slope CVDP of Eq. (2.295) associated with the hightemperature Dulong-Petit law:
CV = CVDP ×
π2 T
.
3g TF
(2.327) {@DULFE}
The linear growth is a characteristic feature of the electronic specific heat at low
temperature sketched for all temperatures in Fig. 2.3. It is due to the progressive softening of the Fermi distribution with temperature, and this makes a linearly
increasing number of electrons thermally excitable. This is directly observable experimentally in metals at low temperature. There the contribution of lattice vibrations
are frozen out since they behave like (T /TD )3 . The temperature TD is the Debye
temperature which characterizes the elastic stiffness of a crystal. It ranges typically from TD ≈ 90 K in soft metals like lead, over TD ≈ 389 K for aluminum, to
TD ≈ 1890 K for diamonds. The measured experimental slope is usually larger than
that for a free electron gas in (2.326). This can be explained mainly by the effect of
the lattice which leads to an increased effective mass Meff > M of the electrons.
Note that the quantity FG (0, µ, V ) is temperature-dependent via the chemical
potential µ. Inserting (2.320) into (2.321), we find the complete T -dependence

5π 2
FG (T, µ, V ) = FG (0, εF, V ) 1 +
12
T
TF
2
π4
−
16
kB T
εF
!4

+ . . .
(2.328) {3.228b}
with FG (0, εF , V ) given by Eq. (2.324) at µ = εF .
Recalling the relation (2.268), the above equation supplies us also with the lowtemperature behavior of the internal energy:
3
5π 2
E = NεF 1 +
5
12
"
T
TF
2
π4 T
−
16 TF
4
#
+ ... .
Figure 2.3 Temperature behavior of the specific heat of a free Fermi gas.
(2.329) {@}
{3.tbfg}
132
2 Field Formulation of Many-Body Quantum Physics
The first term is the energy of the zero-temperature Fermi sphere. Using the relation
CV = ∂E/∂T , the second term yields once more the leading T → 0 behavior (2.326)
of the specific heat.
This behavior of the specific heat can be observed in metals where the conduction
electrons behave like a free electron gas. Due to Bloch’s theorem, a single electron
in a perfect lattice behaves just like a free particle. For many electrons, this is still
approximately true, if the mass of the electrons is replaced by an effective mass.
Another important macroscopic system, where the behavior (2.326) can be observed, is a liquid consisting of the fermionic isotope 3 He. There are two electron
spins and an odd number of nucleon spins which make this atom a fermion. The
atoms interact strongly in the liquid, but it turns out that these interactions produce a screening effect after which the system may be considered approximately as
an almost-free gas of quasiparticles which behave like free fermions whose mass is
about 8 times that of the strongly interacting atoms [10].
2.15.3
Degenerate Bose Gas Near T = 0
For bosons, the low-temperature discussion is quite different. As we can see from
Eq. (2.263), the particle density remains positive for all ε ∈ (0, ∞) only if the
chemical potential µ is negative. A positive µ would also cause a divergence in the
integrals (2.270), (2.272). At high temperatures, the chemical potential has a large
negative value, which moves closer to zero as the temperature decreases (see Fig.
2.4, compare also with Fig. 2.1).
Figure 2.4 Temperature behavior of the chemical potential of a free Bose gas.
The chemical potential vanishes at a critical temperature Tc . From Eqs. (2.263),
(2.270), and (2.276), this is determined by the equation
N
V
= g
Z
d-3 p
1
p2
e( 2M −µ)/kB T − 1
2 −
V
µ
√ I3/2
,
= g 3
λ (T ) π
kB T
=g
Z
0
∞
{degbgas}
{3.tbbg}
dεgε fε
(2.330) {3.232}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
133
2.15 Quantum Statistics of Free Nonrelativistic Fields
at µ = 0, where it yields, via Eq. (2.276), the particle density
N
V
= g
V
λ3(T )
(2.331) {3.2320}
ζ(3/2).
Thus the critical temperature Tc satisfies the equation
−2/3
Tc = [gζ(3/2)]
N
V
2/3
2πh̄2
,
kB M
(2.332) {@tcrit2}
with ζ(3/2) = 2.61238 . . . .
It is interesting to rewrite this equation in natural variables. We may introduce
an average distance a between the bosons by the relation N/V ≡ 1/a3 . There is an
energy associated with it,
εa ≡
h̄2
,
2Ma2
(2.333) {@}
and a temperature Ta ≡ εa /kB . In these natural units, the critical temperature
(2.332) of the free Bose gas is simply
Tc = [gζ(3/2)]−2/3 4πTa .
(2.334) {CRITTc@}
We may rewrite Eq. (2.330) in the form
1=
T
Tc
3/2
I3/2 (µ̄)
,
I3/2 (0)
µ̄ =
µ
.
kB T
(2.335) {@crit2’}
For µ̄ between 0 and 1, this equation yields T /Tc > 1, while Eq. (2.294) gives us
the associated specific heat. The result is displayed in Fig. 2.5. To understand the
figure we must realize what happens in the regime of low temperatures T < Tc where
(2.270) has no solution.
Figure 2.5 Temperature behavior of the fraction of zero-momentum bosons in a free
Bose gas.
{3.tbc}
134
2 Field Formulation of Many-Body Quantum Physics
A glance at Eq. (2.331) shows that a phase transition takes place when the
average distance a between atoms becomes smaller than the De Broglie wavelength
of thermal motion (2.271). In natural units, this may be expressed as
1
λ(T ) = √
4π
s
Ta
.
T
(2.336) {@DEBOTW
For helium, this length scale has roughly the value 5.64 Å.6
For T < Tc , Eq. (2.330) has no solution even though the physical system can be
cooled further. The apparent contradiction has its origin in a failure of the integral
approximation (2.257) to replace the sum over momenta for T < Tc . The reason is
that the state with p = 0 is not properly included in the energy integral over all
states in Eq. (2.270). To avoid this we have to write more properly:
N = Np=0 + Np6=0 ≡ Ns + gV
Z
∞
0
dεgε fε |µ=0 ,
(2.337) {3.234}
where Ns ≡ Np=0 is the number of Bose particles at zero momentum and energy.
Below Tc , a finite fraction of all particles Ns /N accumulates in this single degenerate
state. It can be calculated from the modified Eq. (2.330):
N − Ns = V g
Z
∞
0
dεgε fε |µ=0
V
2 −
T
√ I3/2
= g 3
(0) = N
λ (T ) π
Tc
3/2
.
(2.338) {3.235}
Thus we find that the number of degenerate bosons has the temperature behavior
"
T
Ns = N 1 −
Tc
3/2 #
,
(2.339) {3.236}
which is plotted in Fig. 2.5.
The phenomenon of a macroscopic accumulation of particles in a single state
is called Bose condensation and plays a central role in the understanding of the
phenomenon of superfluidity in liquid helium consisting of the bosonic atoms of
4
He. In fact, the temperature Tc calculated from Eq. (2.332) is Tc ∼ 3.1 K, which is
roughly of the same order as the experimental value7
Tcexp ≡ 2.18K.
(2.340) {3.237}
The discrepancy is due to the strong interactions between the 4 He atoms in the
liquid state, which have all been neglected in deriving Eq. (2.340).
There exists a phenomenological two-fluid description of superfluidity in which
the condensate of the p = 0 -bosons is identified with the superfluid component of
6
See p. 256 in the textbook Ref. [11].
The mass density is ρ ≈ 0.145 g/cm3 . With the mass of the helium atoms being M4 He ≈
4mp ≈ 4 × 1.6762 × 10−24 g, this implies a volume per atom of V /N ≈ 46.2Å3 .
7
H. Kleinert, PARTICLES AND QUANTUM FIELDS
{GFCM1-2}
135
2.15 Quantum Statistics of Free Nonrelativistic Fields
Figure 2.6 Temperature behavior of the specific heat of a free Bose gas. For comparison
we show the specific heat of the strongly interacting Bose liquid 4 He, scaled down by a
factor of 2 to match the Dulong-Petit limit of the free Bose gas.
{3.tbcd}
the liquid. This is the reason why we have used the subscript s in (2.338). The
complementary piece
Nn = N − Ns
(2.341) {3.238}
is usually referred to as the normal component of the superfluid.
For T < Tc , the energy of the normal liquid is equal to the total energy. Using
Eq. (2.288) with µ = 0 we obtain
E = En
−
3/2
I5/2
(0)
2
T
V
−
√ kB T I5/2 (0) = g −
NkB T
= g 3
λ (T ) π
I3/2 (0)
Tc
T
≈ g 0.7703 NkB T
Tc
3/2
≡ Ec
T
Tc
3/2
,
(2.342) {3.240}
where we have expressed In− (0) via Eq. (2.276). From this energy we derive the
specific heat below Tc :
∂S ∂E =
∂T V,N
∂T V,N
5E
=
∝ T 3/2 .
2T
Integrating this with respect to the temperature gives the entropy
CV
= T
5E
,
3T
and the free energy F = E − T S takes the simple form
S=
2
F = − E.
3
(2.343) {3.241x}
(2.344) {3.242}
(2.345) {3.243}
136
2 Field Formulation of Many-Body Quantum Physics
This is consistent with the general relation (2.268), since for µ = 0 the grand canonical free energy FG coincides with the free energy F [recall the Euler relation (1.529)].
The special role of the Bose condensate lies in the fact that it provides the system
with a particle reservoir, with the relation (2.270) being replaced by Eq. (2.330) and
Eq. (2.338).
Inserting (2.342) into (2.343), the low-temperature behavior of the specific heat
becomes explicitly
∂E
5 Γ(5/2)ζ(5/2) T
CV =
= gkB N
∂T
2 Γ(3/2)ζ(3/2) Tc
3/2
T
≈ gkB N 1.92567
Tc
3/2
.
(2.346) {@338x}
Comparing (2.346) with (2.294) and using the fact that I1/2 (0) = ∞, we see that at
Tc the peak in (2.346) has the same maximal value as in the T > Tc -solution (since
I1/2 (0) = ∞).
As T passes Tc , the chemical potential becomes negative. To calculate the be−
−
havior of −µ in this regime, we use Eq. (2.335), and replace I3/2
(µ̄) by I3/2
(0) +
−
∆I3/2 (µ̄) with
−
∆I3/2
(µ̄)
=
Z
∞
0
dz z
1/2
1
1
.
− z
z−µ̄
e
−1 e −1
(2.347) {3.245}
This function receives its main contribution from z ≈ 0, where it can be approximated by8
Z ∞
√
1
−
= −π −µ̄.
(2.348) {3.248}
∆I3/2 (µ̄) ≈ µ̄
dz 1/2
z (z − µ̄)
0
Hence we obtain from (2.335) the relation
T
1=
Tc
3/2

1 +
−
∆I3/2
µ
kB T
−
I3/2
(0)

(2.349) {3.244x}
.
Inserting here the small-µ̄ behavior (2.348), we see that, for T slightly above Tc , the
negative chemical potential −µ becomes nonzero behaving like
1
−
−µ ≈ 2 kB Tc [I3/2
(0)]2
π
"
T
Tc
3/2
#2
−1 ,
(2.350) {3.249}
−
where I3/2
(0) is given by Eq. (2.276).
Let us use this result to find the internal energy slightly above the critical temperature Tc . With the help of relation (2.268) we calculate the derivative of the
energy with respect to the chemical potential as
3 ∂FG 3
∂E =−
= gN.
∂µ T,V
2 ∂µ T,V
2
8
(2.351) {3.250}
In general, the small-µ̄ expansion of Iν−P
(µ̄) = Γ(n)ζν (eµ̄ ) follows from the so-called Robinson
∞
µ̄
ν−1
expansion: ζν (e ) = Γ(1 − ν)µ̄
+ ζ(ν) + k=1 (−µ̄)k ζ(ν − k)/k!, derived in Subsec. 2.15.6 of the
textbook in Ref. [1].
H. Kleinert, PARTICLES AND QUANTUM FIELDS
{@ROBINs}
137
2.15 Quantum Statistics of Free Nonrelativistic Fields
This allows us to find the internal energy slightly above the critical temperature Tc ,
where −µ is small, as
3
E ≈ Ec + gNµ
2
" 3/2
#2
3
T
−
2
= Ec − 2 gNkB Tc [I3/2 (0)]
−1 .
2π
Tc
(2.352) {3.251}
Here Ec is the internal energy at the critical point defined in Eq. (2.342). Forming
the derivative with respect to the temperature as in (2.343) we see that at Tc the
specific heat has a kink. Its slope jumps by
∂CV
∆
∂T
!
≈
27 −
kB
kB
[I3/2 (0)]2 gN
≡ 3.6658 gN ,
2
4π
Tc
Tc
(2.353) {3.252}
the slopes being below Tc from (2.342):
∂CV
∂T
!
=
5 3 I5/2 (0) gNkB
53
gNkB
gNkB
≈
0.7703
≈ 2.8885
,
2 2 I3/2 (0) Tc
22
Tc
Tc
T ≤ Tc ,
(2.354) {@}
and above Tc from (2.352):
∂CV
∂T
!
(
5 3 I5/2 (0)
27 −
=
− 2 [I3/2
(0)]2
2 2 I3/2 (0) 4π
)
gNkB
gNkB
≈ −0.7715
,
Tc
Tc
T > Tc . (2.355) {@}
Let us compare the behavior of the specific heat of the free Bose gas with the
experimental results for the Bose liquid 4 He (see Fig. 2.6). The latter also rises like
T 3 for small T , but it has a sharp singularity at T = Tc . Considering the crudeness of
the free-gas approximation, the similarity of the curves is quite surprising, indicating
the physical relevance of the above idealized quantum-statistical description.
2.15.4
High Temperatures
At high temperatures, the particles are distributed over a large volume in phase
space, so that the occupation numbers of each energy level are very small. As
a consequence, the difference between bosons and fermions disappears, and the
distribution functions (2.264) become
fε ≡ −
∂
Fε (T, µ) ≈ e(µ−ε)/kB T ,
∂µ
(2.356) {3.198bf}
for either statistics. The high-temperature limit of the thermodynamic quantities
can therefore all be calculated from the fermion expressions. The corresponding
limit in the functions In∓ (µ̄) in Eq. (2.272) is µ̄ → −∞, for which we obtain
Z
In∓ (µ̄) −
−−→
µ̄→−∞ 0
∞
z n eµ̄−z = n!eµ̄ .
(2.357) {@}
138
2 Field Formulation of Many-Body Quantum Physics
Inserting this into (2.307) for n = 1/2, we find that at a fixed particle number N,
the chemical potential has the large-T behavior
√ !2/3 
3 π
.
4
(2.358) {@}
FG (T, µ, V ) ≈ −gN(T, µ) kB T.
(2.359) {@}

3
T
µ
≈ log 
kB T
2
TF
In the same limit, the grand-canonical free energy (2.286) behaves like
With the definition of the pressure (1.527), this is the equation of state for the ideal
gas. Using the relation (2.268), we obtain from this the internal energy at a fixed
particle number
3
E ≈ gN kB T.
(2.360) {@}
2
This equation is a manifestation of the Dulong-Petit law: Each of the 3N degrees of
freedom of the system carries an internal energy kB T /2.
The corresponding specific heat per constant unit volume is
3
CVDP = gNkB
2
(2.361) {@}
[compare with the fermion formula (2.295)]. The low-temperature behavior (2.346)
is related to this by the factor [compare with (2.327)]:
CV
2.16
≈
small T
CVDP
ζ(5/2) T
×5
ζ(3/2) Tc
3/2
≈
CVDP
T
× 1.2838
Tc
3/2
.
(2.362) {@DULPB}
Noninteracting Bose Gas in a Trap
In 1995, Bose-Einstein condensation was observed in a dilute gas in a way that fits
the above simple theoretical description [9]. When 87 Rb atoms were cooled down
in a magnetic trap to temperatures less than 170 nK, about 50 000 atoms were
observed to form a condensate, a kind of “superatom”. Such condensates have been
set into rotation and shown to become perforated by vortex lines [12,13,14] just as
in rotating superfluid helium II.
2.16.1
{sec@BECon
Bose Gas in a Finite Box
Consider first the condensation process in a finite number N of bosons enclosed in
a large cubic box of size L. Then the momentum sum in Eq. (2.263) has to be
carried out over the discrete momentum vectors pn = h̄π(n1 , n2 , . . . , nD )/L with
ni = 1, 2, 3, . . . :
N=
X
VD box
1
ζ
(z)
≡
,
2 /2M −βµ
D/2
D
βp
n
le (h̄β)
−1
pn e
(2.363) {@zetabox}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
139
2.16 Noninteracting Bose Gas in a Trap
where z ≡ eµβ is the so-called fugacity. This can be expressed in terms of the onedimensional auxiliary partition function of a particle in a one-dimensional “box”:
Z1 (b) ≡
∞
X
e−bn
2 /2
b ≡ βh̄2 π 2 /ML2 = πle2 (h̄β)/2L2 .
,
n=1
(2.364) {@Z1Bset}
Using this, we can rewrite (2.363) in the form of a so-called fugacity expansion
N=
X
VD
box
ζ
(z)
≡
Z1D (wb)z w ,
leD (h̄β) D/2
w
(2.365) {@}
where the function Z1 (b) is related to the elliptic theta function
ϑ3 (u, z) ≡ 1 + 2
∞
X
2
z n cos 2nu
(2.366) {@ELLTH}
n=1
by Z1 (b) = [ϑ3 (0, e−b/2 ) − 1]/2. The small-b behavior of this function is easily
calculated as follows. We rewrite the sum as a sum over integrals
−b/2
ϑ3 (0, e
∞
X
) =
−k 2 b/2
e
k=−∞
=
s
=
Z
∞
X
∞
m=−∞ −∞
dk e−k
2 b/2+2πikm
∞
X
2π
2 2
e−2π m /b .
1+2
b
m=1
!
Thus, up to exponentially small corrections, we may replace ϑ3 (0, e−b/2 ) by
√
so that for small b (i.e., large L/ β):
Z1 (b) =
r
π
1
2
− + O(e−2π /b ).
2b 2
(2.367) {@smBEx1}
q
2π/b,
(2.368) {@smBEx}
For large b, Z1 (b) falls exponentially fast to zero.
In the sum (2.363), the lowest energy level with p1,...,1 = h̄π(1, . . . , 1)/L plays a
special role. Its contribution to the total particle number is the number of particles
in the condensate:
Ncond (T ) =
1
eDb/2−βµ
−1
=
zD
,
1 − zD
zD ≡ eβµ−Db/2 .
(2.369) {@7numco}
box
This number diverges for zD → 1, where the box function ζD/2
(z) has a pole
1/(Db/2 − βµ). This pole prevents βµ from becoming exactly equal to Db/2 when
solving the equation (2.363) for the particle number in the box.
For a large but finite system near T = 0, almost all particles will go into the
condensate, so that Db/2−βµ will be very small, of the order 1/N, but not zero. The
thermodynamic limit can be performed smoothly by defining a regularized function
box
ζ̄D/2
(z) in which the lowest (singular) term in the sum (2.363) is omitted. Let us
define the number of normal particles which have not condensed into the state of
140
2 Field Formulation of Many-Body Quantum Physics
zero momentum as Nn (T ) = N − Ncond (T ). Then we can rewrite Eq. (2.363) as an
equation for the number of normal particles:
Nn (T ) =
VD
box
ζ̄D/2
(z(β)),
D
le (h̄β)
(2.370) {@thiseq1}
which reads more explicitly
Nn (T ) = SD (zD ) ≡
∞
X
w
[Z1D (wb)ewDb/2 − 1]zD
.
w=1
(2.371) {@thiseq12}
A would-be critical point may now be determined by setting here zD = 1 and
equating the resulting Nn with the total particle number N. If N is sufficiently
large, we need only the small-b limit of SD (1) which is calculated in Appendix 2B
[see Eq. (2B.14)], so that the associated temperature Tc(1) is determined from the
equation
s
3
π
3π
N=
(2.372) {@thieq12AB
ζ(3/2) + (1) log C3 bc + . . . ,
2bc
4bc
where C3 ≈ 0.0186. In the thermodynamic limit, the critical temperature Tc(0) is
obtained by ignoring the second term, yielding
N=
s
π
(0)
2bc
3
(2.373) {@TherML}
ζ(3/2),
in agreement with Eq. (2.331) for Tc , if we recall b from (2.364). Using this we
rewrite (2.372) as
!3/2
3 π
Tc(1)
+
log C3 b(0)
(2.374) {@thieq12AB
1≡
c .
(0)
(0)
2N
Tc
2bc
Expressing b(0)
c in terms of N from (2.373), this implies
δTc(1)
(0)
Tc
≈
2
N 2/3
1
log
.
ζ 2/3 (3/2)N 1/3
πC3 ζ 2/3 (3/2)
(2.375) {@thieq12AB
Experimentally, the temperature Tc(1) is not immediately accessible. What is
easy to find is the place where the condensate density has the largest curvature, i.e.,
where d3 Ncond /dT 3 = 0. The associated temperature Tcexp is larger than Tc(1) by a
factor 1 + O(1/N), so that it does not modify the leading finite-size correction which
is of the order 1/N 1/3 .
Alternatively we may use the phase space formula
∞
X
dD p
1
Nn = d x
=
(2πh̄)D eβ[p2 /2M +V (x)] − 1 n=1
Z
∞
X
1
=
dD x e−nβV (x) ,
q
D
2
n=1
2πh̄ nβ/M
Z
D
where the spatial integration produces a factor
side becomes again (T /Tc(0) )D N.
Z
dD p −nβ[p2 /2M +V (x)]
d x
e
(2πh̄)D
D
(2.376) {@insinto}
q
2π/Mω 2 nβ so that the right-hand
H. Kleinert, PARTICLES AND QUANTUM FIELDS
141
2.16 Noninteracting Bose Gas in a Trap
2.16.2
Harmonic and General Power Trap
For a D-dimensional harmonic trap V = Mω 2 x2 /2, the critical temperature is
reached if Nn is equal to the total particle number N where
kB Tc(0)
"
N
= h̄ω
ζ(D)
#1/D
(2.377) {@lowestor}
.
This formula has a solution only for D > 1.
The equation (2.376) for the particle number can be easily calculated for a more
general trap where the potential has the anisotropic power behavior
D
|xi |
M 2 2X
ω̃ ã
V (x) =
2
ai
i=1
!pi
(2.378) {@anharmp}
.
The parameter ω̃ denotes some frequency parameter, and ã the geometric average
h
ã ≡ ΠD
i=1 ai
D Z
Y
i1/D
∞
i=1 −∞
. Inserting (2.378) into (2.376) we encounter a product of integrals:
dx e−nβM ω̃
2 ã2 (|x
p
i |/ai ) i /2
=
D
Y
ai
Γ(1 + 1/pi ),
2 2
1/pi
i=1 (βM ω̃ ã /2)
(2.379) {@}
so that the right-hand side of (2.376) becomes (T /Tc(0) )D̃ N, with the critical temperature
kB Tc(0)
M ã2 ω̃ 2
=
2
h̄ω̃
M ã2 ω̃ 2
!D/D̃ "
Here D̃ is the dimensionless parameter
Nπ D/2
Q
ζ(D̃) D
i=1 Γ(1 + 1/pi )
#1/D̃
D
D X
1
D̃ ≡
+
2
i=1 pi
.
(2.380) {@gentrt2c}
(2.381) {@tildeD}
that takes over the role of D in the harmonic formula (2.377). A harmonic trap,
that has different oscillator frequencies ω1 , . . . , ωD along the D Cartesian axes, is a
special case of (2.378) with pi ≡ 2, ωi2 = ω̃ 2 ã2 /a2i , and D̃ = D, and formula (2.380)
reduces to (2.377), with ω replaced by the geometric average ω̃ ≡ (ω1 · · · ωD )1/D of
the frequencies. The parameter ã disappears from the formula. A free Bose gas in
Q
D D
a box of size VD = D
i=1 (2ai ) = 2 ã is described by (2.378) in the limit pi → ∞,
where D̃ = D/2. Then Eq. (2.380) reduces to
kB Tc(0)
πh̄2
N
=
2
2M ã ζ(D/2)
"
#2/D
N
2πh̄2
=
M VD ζ(D/2)
"
#2/D
,
(2.382) {@}
in agreement with Eq. (2.331).
Another interesting limiting case is that of a box of length L = 2a1 in the xdirection with p1 = ∞, and two different oscillators of frequency ω2 and ω3 in the
142
2 Field Formulation of Many-Body Quantum Physics
2
other two directions. To find Tc(0) for such a Bose gas we identify ω̃ 2 ã2 /a22,3 = ω2,3
in the potential (2.378), so that ω̃ 4 /ã2 = ω22 ω32 /a21 , and we obtain
kB Tc(0)
2.16.3
πh̄
= h̄ω̃
2M ω̃
!1/5 "
N
a1 ζ(5/2)
#2/5
2πλω1 λω2
= h̄ω̃
L2
!1/5 "
N
ζ(5/2)
#2/5
. (2.383) {@7FORMU}
Anharmonic Trap in Rotating Bose-Einstein Gas
Another interesting potential can be prepared in the laboratory by rotating a Bose
condensate [13] with an angular velocity Ω around the z-axis. The vertical trapping
frequency is ωz ≈ 2π × 11.0 Hz ≈ 0.58× nK, while the horizontal one is ω⊥ ≈ 6 × ωz .
The centrifugal forces create an additional repulsive harmonic potential, bringing
the rotating potential to the form
4
κ r⊥
Mωz2 2
2
z + 36ηr⊥
+
V (x) =
2
2 λ2ωz
!
4
2
z2
r⊥
κ r⊥
+
36η
+
λ2ωz
λ2ωz
2 λ4ωz
h̄ωz
=
2
!
,
(2.384) {@7POt}
2
2
where r⊥
= x2 + y 2 , η ≡ 1 − Ω2 /ω⊥
, κ ≈ 0.4, and λωz ≡ 3.245 µm ≈ 1.42 × 10−3 K.
For Ω > ω⊥ , η turns negative and the potential takes the form of a Mexican hat as
2
shown in Fig. 2.7, with a circular minimum at rm
= −36ηλ2ωz /κ. For a large rotation
speed Ω, the potential may be approximated by a circular harmonic well, so that
we may apply formula (2.383) with a1 = 2πrm , to obtain the η-independent critical
temperature
#2/5
1/5 "
N
κ
(0)
.
(2.385) {@}
kB Tc ≈ h̄ωz
π
ζ(5/2)
Figure 2.7 Rotating trap potential for ω 2 > 0 and ω 2 < 0, pictured for the case of two
components x1 , x2 . The right-hand figure looks like a Mexican hat or the bottom of a
champagne bottle.
ω2 > 0
ω2 < 0
V (x)
V (x)
x2
x2
x1
x1
{Fig. 7.2E}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
143
2.17 Temperature Green Functions of Free Particles
For κ = 0.4 and N = 300 000, this yields Tc ≈ 53nK.
q At the critical rotation speed Ω = ω⊥ , the potential is purely quartic r⊥ =
(x2 +y 2). To estimate Tc(0) we approximate it for a moment by the slightly different
potential (2.378) with the powers p1 = 2, p2 = 4, p3 = 4, a1 = λωz , a2 = a3 =
λωz (κ/2)1/4 , so that formula (2.380) becomes
kB Tc(0)
= h̄ωz
"
π2κ
16Γ4 (5/4)
#1/5 "
N
ζ(5/2)
#2/5
.
(2.386) {@isresu}
It is easy to change this result so that it holds for the potential ∝ r 4 = (x + y)4
rather than x4 + y 4. We use the semiclassical formula for the number of normal
particles in the form
Z ∞
ρcl (E)
Nn =
dE E/k T
,
(2.387) {@7.NEQ}
e B −1
Emin
where Emin is the classical ground state energy, and
ρcl (E) =
M
2πh̄2
D/2
1
Γ(D/2)
Z
dD x [E − V (x)]D/2−1
(2.388) {7.r0}
is the semiclassical density of states. For a harmonic trap, the spatial integral can
be done, after which the energy integral on the right-hand side of (2.387) yields
[kB T /h̄ω]D ζ(D) = (T /Tc(0) )D N, thus leading back to (2.377). Hence the critical
temperature for the potential ∝ r 4 = (x + y)4 rather than x4 + y 4 is obtained by
multiplying the right-hand side of Eq. (2.387) for N by a factor
4
2π rdrdxdy e−r
π 3/2
R
=
.
dxdy e−x4 −y4
Γ[5/4]2
R
(2.389) {@}
This factor arrives inversely in front of N in Eq. (2.390), so that we obtain the
critical temperature of the critically rotating Bose gas:
kB Tc(0)
= h̄ωz
4κ
π
1/5 "
N
ζ(5/2)
#2/5
.
(2.390) {@isresu}
The critical temperature at Ω = ω⊥ is therefore by a factor 41/5 ≈ 1.32 larger
than at infinite Ω. Actually, this limit is somewhat academic in a semiclassical
approximation since it ignores the quantum nature of the oscillator.
For more details see Chapter 7 in the textbook [1].
2.17
Temperature Green Functions of Free Particles
As argued in Section 1.7, all properties of a system in thermodynamic equilibrium
are calculable by continuing the quantum theory to imaginary times t = −iτ , with
real τ . We shall see later that the calculation of small interaction effects to the
144
2 Field Formulation of Many-Body Quantum Physics
free-particle results, presented in the last section, can be done perturbatively. It
involves the analytically continued analog of the free-particle propagator (2.249).
In a grand-canonical ensemble, the relevant quantity is the so called finitetemperature Green function or finite-temperature propagator of the free particles:
h
Tr e−ĤG /kB T T̂τ ψ̂(x, τ )ψ̂ † (x′ , τ ′ )
G(x, τ ; x′ τ ′ ) =
Tr(e−ĤG /kB T )
h
i
= eFG /kB T Tr e−ĤG /kB T T̂τ ψ̂(x, τ )ψ̂ † (x′ , τ ′ )
i
.
(2.391) {3.253}
Here T̂τ is the τ -ordering operator defined in complete analogy to the time-ordering
operator T̂ in (2.231). The fields ψ̂(x, τ ), ψ̂ † (x, τ ) are defined in analogy to (2.133)
and (2.134) via an analytically continued time evolution operator as follows:
ψ̂(x, τ ) = eĤG τ /h̄ ax e−ĤG τ /h̄ ,
†
ψ̂ (x, τ ) =
e−ĤG τ /h̄ â†x eĤG τ h̄ .
(2.392) {3.254}
(2.393) {3.254b}
At equal imaginary time, these satisfy canonical commutation relations analogous
to (2.205):
[ψ̂(x, τ ), ψ̂(x′ , τ )]∓ = 0,
[ψ̂ † (x, τ ), ψ̂ † (x′ , τ )]∓ = 0,
[ψ̂(x, τ ), ψ̂ † (x′ , τ )]∓ = δ (3) (x − x′ ).
(2.394) {@cancomrel
The time evolution of these field operators is governed by the grand-canonical
Hamiltonian
ĤG = Ĥ − µN̂ .
(2.395) {@}
Note that while ψp† in (2.393) at τ = 0 is the Hermitian conjugate of ψp in (2.392),
this is no longer true for τ 6= 0. The advantages of using these non-Hermitian fields
will become apparent later when we discuss perturbation theory.
When it comes to calculating physical phenomena in thermal equilibrium, all
Green functions will be needed whose imaginary time τ lies in the interval
τ ∈ (0, h̄/kB T ) .
(2.396) {@}
Differentiating (2.392) and (2.393) with respect to τ , we obtain the Heisenberg
equations
∂ˆτ ψ̂(x, τ ) = [ĤG , ψ̂(x, τ )],
(2.397) {nolabel}
†
∂ˆτ ψ̂ (x, τ ) = −[ĤG , ψ̂(x, τ )].
(2.398) {@}
Using the canonical field commutation relations (2.394), the fields ψ̂(x, τ ) and
ψ̂ † (x, τ ) are seen to satisfy the analytically continued Schrödinger equations
h̄2 2
−h̄∂τ +
∂ + µ ψ̂(x, τ ) = 0,
2M x
!
†
ψ̂ (x, τ )
←
h̄ ∂ τ
h̄2 ← 2
+
∂x + µ
2M
!
= 0.
(2.399) {3.255}
(2.400) {3.256}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
145
2.17 Temperature Green Functions of Free Particles
If we apply these differential operators to the Green function (2.391), we obtain the
equation
h̄2 2
∂ + µ G(x, τ ; x′ τ ′ )
−h̄∂τ +
2M x
!
n
h
= eFG /kB T Tr e−ĤG /kB T δ(τ − τ ′ ) ψ̂(x, τ ), ψ̂ † (x′ , τ ′ )
Using the commutation rule (2.395), this becomes
io
.
h̄2 2
−h̄∂τ +
∂ + µ G(x, τ ; x′ τ ′ ) = δ(τ − τ ′ )δ (3) (x − x′ ),
2M x
!
(2.401) {@}
(2.402) {@TEMPGF}
which is the general defining equation of a Green function [recall (1.315)].
The plane-wave solutions of Eqs. (2.399) and (2.400) are the analyticallycontinued versions of the plane-wave solutions (2.206),
ψ̂p (x, τ ) = âp e(ipx−ξ(p)τ )/h̄ ,
ψ̂p† (x, τ ) = â†p e(−ipx+ξ(p)τ )/h̄ ,
(2.403) {3.257}
where
p2
ξ(p) ≡
−µ
(2.404) {3.259}
2M
with particle energies counted from the chemical potential µ rather than from zero.
The canonical field operators solving (2.399) and (2.400) have momentum-space
expansions of the type (2.215), namely
ψ̂(x, τ ) =
ψ̂ † (x, τ ) =
Z
Z
d-3 p eipx/h̄−ξ(p)τ /h̄ âp ,
d-3 p e−ipx/h̄+ξ(p)τ /h̄ â†p .
(2.405) {3.260}
Inserting these into (2.391), we now calculate the Green function for τ > τ ′ :
G(x, τ ; x′ , τ ′ ) = G(x − x′ , τ − τ ′ )
h
= eFG /kB T Tr e−ĤG /kB T ψ̂(x, τ )ψ̂ † (x′ , τ ′ )
= eFG /kB T
Z
Z
i
(2.406) {3.262}
′ ′
′ ′
d-3 p d-3 p′ Tr(e−ĤG /kB T âp â†p′ )ei[px−p x ]/h̄−i[ξ(p)τ −ξ(p )τ ]/h̄.
Here we observe that the expression
eFG /kT Tr e−ĤG /kB T â†p âp′ =
Tr e−ĤG /kB T â†p âp′
Tr(e−ĤG /kB T )
(2.407) {nolabel}
is simply the average particle number of Eq. (2.264) for the energy ξ(p):
fξ(p) =
1
eξ(p)/kB T
∓1
,
(2.408) {3.pnu}
146
2 Field Formulation of Many-Body Quantum Physics
apart from a Dirac δ-function in the momenta δ (3) (p − p′ ). Hence we find for τ > 0:
G(x, τ ) ≡
Z
d-3 peipx/h̄−ξ(p)τ /h̄ (1 ± fξ(p) ).
(2.409) {3.262b}
For τ < 0, the operator order is reversed, and we obtain directly
G(x, τ ) ≡ ±
Z
d-3 peipx/h̄−ξ(p)τ /h̄ fξ(p) .
(2.410) {3.263}
From these expressions we can derive an important property of the temperature
Green function. Using the identity
eξ/kB T fξ = 1 ± fξ ,
(2.411) {3.264}
we see that G(x, τ ) is periodic or antiperiodic under the replacement
τ →τ+
h̄
,
kB T
(2.412) {perantiper}
depending on whether the particles are bosons or fermions, respectively. Explicitly.
the Green function satisfies the relation
G(x, τ ) = ±G(x, τ + h̄/kB T ),
τ ∈ (−h̄/kB T , h̄/kB T ].
(2.413) {3.265}
At zero imaginary time this reduces to G(x, 0) = ±G(x, h̄/kB T ), implying that
G(x, τ ) has a Fourier transform
G(x, τ ) =
kB T X −iωm τ
e
G(x, ωm )
h̄ ωm
(2.414) {3.266}
with the frequencies
(
2π
ωm ≡ ωm =
h̄β
m
m+
1
2
for bosons,
for fermions,
(2.415) {3.267}
where m runs through all integer values m = 0, ±1, ±2, . . .. These are known as
Matsubara frequencies.
The Fourier components are given by the integrals
G(x, ωm ) =
Z
h̄/kB T
0
dτ eiωm τ G(x, τ ).
(2.416) {3.268}
The full Fourier representation in space and imaginary time reads
G(x, τ ) =
kB T X Z -3 −iωm τ +ipx/h̄
d pe
G(p, ωm ),
h̄ ωm
(2.417) {3.268b}
with the components
G(p, ωm ) =
=
Z
d3 x
Z
0
h̄/kB T
dτ eiωm τ −ipx/h̄ G(x, τ )
n
o
h̄
(1 ± fξ(p) ) e[ih̄ωm −ξ(p)]/kB T − 1 .
ih̄ωm − ξ(p)
(2.418) {3.269}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
147
2.17 Temperature Green Functions of Free Particles
Inserting (2.408) and the explicit form of the Matsubara frequencies (2.415), we
obtain
h̄
.
(2.419) {3.270}
G(p, ωm ) = −
ih̄ωm − ξ(p)
Due to a marvelous cancellation, this has become very simple. In fact, the result
can be obtained from the Fourier transform (2.248) of the quantum field theoretic
Green function G(p, E) in two steps: First, we continue (2.248) analytically in the
energy E to the imaginary off-shell values E = ih̄ωm , and second, we shift the
single-particle energy from ε(p) = p2 /2M to ξ(p) = ε(p) − µ, this being a trivial
consequence of the use of ĤG = Ĥ − µN̂ instead of Ĥ.
As a cross check, let us calculate G(x, τ ) for a very small negative τ = −η from
the Fourier representation (2.417) with the components (2.419) for bosons:
G(x, −η) = −
Z
kB T X iωm η
h̄
d-3 p eipx/h̄
.
e
h̄ ωm
−iωm + ξ(p)
(2.420) {3.266}
The phase factor eiωm η is necessary to ensure convergence of the otherwise
logarithmically-divergent sums. The sum can be performed by rephrasing it as
the contour integral
−kB T
X
eiωm η
ωm
kB T
1
=∓
iωm − ξ(p)
2πi
Z
C
dz
eηz
ez/kB T
1
.
∓1z−ξ
(2.421) {te-3.18a}
The contour of integration C encircles the imaginary z-axis in the positive sense,
thereby enclosing all integer or half-integer valued poles at z = iωm . The upper
signs on the right-hand side of (14.196) holds for bosons, the lower for fermions.
The closed contour C may be viewed as the result of two straight contours, which
lie next to each other: one runs upwards to the right of the imaginary axis and the
other that runs downwards to the left of it. The two parts may be closed by infinite
semicircles Γl and Γr at infinity (see Fig. 2.8). These contribute nothing since the
right-semicircle is suppressed by an exponential factor e−z/kB T , and the left-hand by
a factor eηz . The two resulting closed contours may now be distorted and shrunk
to zero. There is a pole only on the right-hand side, at z = ξ, which contributes by
Cauchy’s residue theorem:
∓kB T
X
eiωm η
ωm
1
1
= ± ξ(p)/k T
= ±fξ(p) .
B
iωm − ξ(p)
e
∓1
(2.422) {te-3.18X}
Via Eq. (2.420) these lead to the Bose and Fermi distribution functions represented
by Eq. (2.410) for small negative τ .
In the opposite limit τ = +η, the phase factor in the sum would be e−iωn η . In
this case, the sum is converted into the contour integral
−kB T
X
ωn
iωn η
e
kB T Z
1
e−ηz
1
=±
,
dz −z/k T
B
iωn − ξ(p)
2πi C e
∓1z −ξ
(2.423) {te-3.18an}
148
2 Field Formulation of Many-Body Quantum Physics
Figure 2.8 Contour C in the complex z-plane for evaluating the Matsubara sum (2.422).
The semicircles at infinity Γl and Γr do not contribute. After shrinking the contours, only
the pole on the right-hand side contributes via Cauchy’s residue theorem.
{te-matsub}
from which we would find 1±fξ(p) , corresponding to Eq. (2.409) for small positive τ .
While the phase factors e±iωm η are needed to make the logarithmically-divergent
sums converge, they become superfluous if the two sums are combined. Indeed,
adding the two sums yields
−kB T
X
ωn
"
"
#
X
1
1
1
1
eiωn η
= −kB T
+ e−iωn η
+
e
iωn − ξ
iωn − ξ
iωn − ξ −iωn − ξ
ωn
kB T X 2ξ
=
= 1 ± 2 fξ
2 + ξ2
h̄ ωm ωm
iωn η
=
(
coth(h̄ξ/kB T )
tanh(h̄ξ/kB T )
)
for
(
bosons
fermions
)
.
#
(2.424) {te-3.18an2}
The right-hand side is the thermal expectation value of âp â†p + â†p âp .
2.18
Calculating the Matsubara Sum via Poisson Formula
There exists another way of calculating the Matsubara sum in the finite-temperature
propagator (2.417). At very low temperatures, the Matsubara frequencies ωm =
2mπkB T /h̄ or ωm = (2m + 1)πkB T /h̄ move infinitely close to each other, so that
the sum over ωm becomes an integral
Z ∞
∞
kB T X
dωm
→
h̄ n=−∞
−∞ 2π
(2.425) {@sumeqin}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
149
2.18 Calculating the Matsubara Sum via Poisson Formula
for Bose and Fermi fields. The propagator (2.417) becomes therefore, with (2.419),
kB T
h̄
G(x, τ ) =
Z
→
∞
−∞
Z
∞
X
i
ωm + iξ(p)/h̄
m=−∞
Z
dωm
i
d-3 pe−iωm τ +ipx/h̄
.
2π
ωm + iξ(p)/h̄
d-3 pe−iωm τ +ipx/h̄
(2.426) {3.268bs}
The integral over ωm can be performed trivially with the help of the residue theorem,
as in Eq. (2.243), yielding
Z
G(p, τ ) =
∞
−∞
dωm −iωm τ
i
e
= Θ(τ )e−ξ(p)τ /h̄ .
2π
ωm + iξ(p)/h̄
(2.427) {@}
For finite temperatures we make use of Poisson’s summation formula (1.213) to write
∞
X
f (m) =
m=−∞
Z
∞
−∞
∞
X
dµ
e2πiµn f (µ),
(2.428) {3.47}
(−)n e2πiµn f (µ).
(2.429) {3.47b}
n=−∞
from which we derive
∞
X
f (m + 1/2) =
m=−∞
Z
∞
−∞
dµ
∞
X
n=−∞
A direct application of this formula shows that
∞
∞
X
i
kB T X
=
e−iωm τ
h̄ m=−∞
ωm + iξ(p)/h̄ n=−∞
(
1
(−1)n
)
Θ(τ + nh̄β)e−ξ(p)(τ +nh̄β)/h̄ . (2.430) {@2.426}
Thus the finite-temperature Green function is obtained from the zero-temperature
function by making it periodic or antiperiodic by forming a simple sum over all
periods, with equal or alternating signs. This guarantees the property (2.413). The
sum over n on the right-hand side of (2.430) is a geometric series in powers of
e−ξ(p)nh̄β , which can be performed trivially. For τ ∈ (0, h̄β), the Heaviside function
forces the sum to run only over positive n, so that we find
G(p, τ ) =
∞
X
n=0
(
1
(−1)n
)
−ξ(p)τ /h̄ −nξ(p)β
Θ(τ + nh̄β)e
e
e−ξ(p)τ /h̄
=
.
1 ∓ e−ξ(p)β
(2.431) {@PERGF}
This can also be rewritten in terms of the Bose and Fermi distribution functions
(2.408) as
G(p, τ ) = e−ξ(p)τ /h̄ 1 ± fξ(p) .
(2.432) {@}
For free particles with zero chemical potential where ξ(p) = p2 /2M, the momentum integral can be done at zero temperature as in Eq. (2.239), and we obtain
the imaginary-time version of the Schrödinger propagator (2.241):
1
−M (x−x′ )2 /2h̄2 (τ −τ ′ )
G(x, τ ; x′ , τ ′ ) = Θ(τ − τ ′ ) q
3e
2πh̄2 (τ − τ ′ )/M
= G(x − x′ , τ − τ ′ ).
(2.433) {3.175e}
150
2 Field Formulation of Many-Body Quantum Physics
This Gaussian function coincides with the end-to-end distribution of random walk
of length proportional to h̄(τ − τ ′ ). Thus the quantum-mechanical propagator is
a complex version of a particle performing a random walk. The random walk is
caused by quantum fluctuations. This fluctuation picture of the Schrödinger theory
is exhibited best in the path-integral formulation of quantum mechanics.9 In the
imaginary-time formulation of quantum field theory, we describe ensembles of particles. They correspond therefore to ensembles of random walks of fixed length. For
this reason, nonrelativistic quantum field theories can be used efficiently to formulate theories of fluctuating polymers. In this context, they are called disorder field
theories.10 .
2.19
Nonequilibrium Quantum Statistics
{NEQS}
The physical systems which can be described by the above imaginary-time Green
functions are quite limited. They must be in thermodynamic equilibrium, with a
constant temperature enforced by a thermal reservoir. Only then can a partition
function and a particle distribution be calculated from an analytic continuation
of quantum-mechanical time evolution amplitudes to an imaginary time tb − ta =
−ih̄/kB T . In this section we want to go beyond such equilibrium physics and extend
the path-integral formalism to nonequilibrium time-dependent phenomena.
2.19.1
Linear Response and Time-Dependent Green
Functions for T =
/0
If the deviations of a quantum system from thermal equilibrium are small, the
easiest description of nonequilibrium phenomena proceeds via the theory of linear
response. In operator quantum mechanics, this theory is introduced as follows.
First, the system is assumed to have a time-independent Hamiltonian operator Ĥ.
The ground state is determined by the Schrödinger equation, evolving as a function
of time according to the equation
|ΨS (t)i = e−iĤt |ΨS (0)i
{sec18.1}
(2.434) {18.1}
(in natural units with h̄ = 1, kB = 1). The subscript S denotes the Schrödinger
picture.
Next, the system is slightly disturbed by adding to Ĥ a time-dependent external
interaction,
Ĥ → Ĥ + Ĥ ext (t),
(2.435) {18.2}
where Ĥ ext (t) is assumed to set in at some time t0 , i.e., Ĥ ext (t) vanishes identically
for t < t0 . The disturbed Schrödinger ground state has the time dependence
−iĤt
|Ψdist
ÛH (t)|ΨS (0)i,
S (t)i = e
9
10
(2.436) {18.3}
See the textbook Ref. [1].
See the textbooks Refs. [1] and [11].
H. Kleinert, PARTICLES AND QUANTUM FIELDS
151
2.19 Nonequilibrium Quantum Statistics
where ÛH (t) is the time translation operator in the Heisenberg picture. It satisfies
the equation of motion
˙
ext
iÛH (t) = ĤH
(t)ÛH (t),
(2.437) {18.4}
with11
ext
ĤH
(t) ≡ eiĤt Ĥ ext (t)e−iĤt .
(2.438) {18.5}
To lowest-order perturbation theory, the operator ÛH (t) is given by
ÛH (t) = 1 − i
Z
t
t0
ext ′
dt′ ĤH
(t ) + · · · .
(2.439) {18.6}
In the sequel, we shall assume the onset of the disturbance to lie at t0 = −∞.
Consider an arbitrary time-independent Schrödinger observable Ô whose Heisenberg
representation has the time dependence
ÔH (t) = eiĤt Ôe−iĤt .
(2.440) {18.7}
Its time-dependent expectation value in the disturbed state |Ψdist
S (t)i is given by
†
dist
iĤt
hΨdist
Ôe−iĤt ÛH (t)|ΨS (0)i
S (t)|Ô|ΨS (t)i = hΨS (0)|ÛH (t)e
≈ hΨS (0)| 1 + i
× 1−i
Z
t
−∞
t
Z
−∞
ext ′
dt′ ĤH
(t ) + . . . ÔH (t)
ext ′
dt′ ĤH
(t ) + . . . |ΨS (0)i
= hΨH |ÔH (t)|ΨH i − ihΨH |
Z
t
−∞
h
i
ext ′
dt′ ÔH (t), ĤH
(t ) |ΨH i + . . . .
(2.441) {18.8}
We have identified the time-independent Heisenberg state with the time-dependent
Schrödinger state at zero time in the usual manner, i.e., |ΨH i ≡ |ΨS (0)i. Thus the
expectation value of Ô deviates from equilibrium by
dist
δhΨS (t)|Ô|ΨS (t)i ≡ hΨdist
S (t)|Ô(t)|ΨS (t)i − hΨS (t)|Ô(t)|ΨS (t)i
= −i
Z
t
−∞
h
i
ext ′
dt′ hΨH | ÔH (t), ĤH
(t ) |ΨH i.
(2.442) {18.9}
If the left-hand side is transformed into the Heisenberg picture, it becomes
δhΨS (t)|Ô|ΨS (t)i = δhΨH |ÔH (t)|ΨH i = hΨH |δ ÔH (t)|ΨH i,
11
ext
Note that after the replacements H → H0 , HH
→ HIint , Eq. (2.437) coincides with the
equation for the time evolution operator in the interaction picture to appear in Section 9.1.3. In
contrast to that section, however, the present interaction is a nonpermanent artifact to be set equal
to zero at the end, and H is the complicated total Hamiltonian, not a simple free one. This is why
we do not speak here of an interaction picture.
152
2 Field Formulation of Many-Body Quantum Physics
so that Eq. (2.442) takes the form
hΨH |δ ÔH (t)|ΨH i = −i
Z
t
−∞
h
i
ext ′
dt′ hΨH | ÔH (t), ĤH
(t ) |ΨH i.
(2.443) {18.10}
It is useful to use the retarded Green function of the operators ÔH (t) and ĤH (t′ ) in
the state |ΨH i
h
i
′
′
′
GR
OH (t, t ) ≡ Θ(t − t )hΨH | ÔH (t), ĤH (t ) |ΨH i.
(2.444) {18.11}
Then the deviation from equilibrium is given by the integral
hΨH |δ ÔH (t)|ΨH i = −i
Z
∞
−∞
′
dt′ GR
OH (t, t ).
(2.445) {18.12}
Suppose now that the observable ÔH (t) is capable of undergoing oscillations.
Then an external disturbance coupled to ÔH (t) will in general excite these oscillations. The simplest coupling is a linear one, with an interaction energy
Ĥ ext (t) = −ÔH (t)δj(t),
(2.446) {18.13}
where j(t) is some external source. Inserting (2.446) into (2.445) yields the linearresponse formula
hΨH |δ ÔH (t)|ΨH i = i
Z
∞
−∞
′
′
dt′ GR
OO (t, t )δj(t ),
(2.447) {18.14}
where GR
OO is the retarded Green function of two operators Ô:
h
i
′
′
′
GR
OO (t, t ) = Θ(t − t )hΨH | ÔH (t), ÔH (t ) |ΨH i.
(2.448) {18.15}
At frequencies where the Fourier transform of GOO (t, t′ ) is singular, the slightest
disturbance causes a large response. This is the well-known resonance phenomenon
found in any oscillating system. Whenever the external frequency ω hits an eigenfrequency, the Fourier transform of the Green function diverges. Usually, the eigenfrequencies of a complicated N-body system are determined by calculating (2.448)
and by finding the singularities in ω.
It is easy to generalize this description to a thermal ensemble at a nonzero
temperature. The principal modification consists in the replacement of the ground
state expectation by the thermal average
hÔiT ≡
Tr(e−Ĥ/T Ô)
.
Tr(e−Ĥ/T )
Using the free energy
F = −T log Tr(e−Ĥ/T ),
H. Kleinert, PARTICLES AND QUANTUM FIELDS
153
2.19 Nonequilibrium Quantum Statistics
this can also be written as
hÔiT = eF/T Tr(e−Ĥ/T Ô).
(2.449) {@}
In a grand-canonical ensemble, Ĥ must be replaced by Ĥ − µN̂ and F by its grandcanonical version FG (see Section 1.16). At finite temperatures, the linear-response
formula (2.447) becomes
δhÔ(t)iT = i
Z
∞
−∞
′
′
dt′ GR
OO (t, t )δj(t ),
(2.450) {18.16}
′
where GR
OO (t, t ) is the retarded Green function at nonzero temperature defined by
[recall (1.313)]
h
n
′
R
′
′
F/T
GR
Tr e−Ĥ/T ÔH (t), ÔH (t′ )
OO (t, t ) ≡ GOO (t − t ) ≡ Θ(t − t ) e
io
. (2.451) {18.17}
i
In a realistic physical system, there are usually many observables, say ÔH
(t) for
i = 1, 2, . . . , l, which perform coupled oscillations. Then the relevant retarded Green
function is some l × l matrix
n
h
j
′
R
′
′
F/T
i
(t′ )
GR
Tr e−Ĥ/T ÔH
(t), ÔH
ij (t, t ) ≡ Gij (t − t ) ≡ Θ(t − t ) e
io
.
(2.452) {18.ret}
After a Fourier transformation and diagonalization, the singularities of this matrix
render the important physical information on the resonance properties of the system.
The retarded Green function at T 6= 0 occupies an intermediate place between
the real-time Green function of field theories at T = 0, and the imaginary-time Green
function used before to describe thermal equilibria at T 6= 0. The Green function
(2.452) depends both on the real time and on the temperature via an imaginary
time.
2.19.2
Spectral Representations of Green Functions for T =
/0
The retarded Green functions are related to the imaginary-time Green functions of
1
equilibrium physics by an analytic continuation. For two arbitrary operators ÔH
,
2
ÔH , the latter is defined by the thermal average
h
i
1
2
G12 (τ, 0) ≡ G12 (τ ) ≡ eF/T Tr e−Ĥ/T T̂τ ÔH
(τ )ÔH
(0) ,
{sec18.2}
(2.453) {18.18}
where ÔH (τ ) is the imaginary-time Heisenberg operator
ÔH (τ ) ≡ eĤτ Ôe−Ĥτ .
(2.454) {18.19}
To see the relation between G12 (τ ) and the retarded Green function GR
12 (t), we take
1
2
a complete set of states |ni, insert them between the operators Ô , Ô , and expand
G12 (τ ) for τ ≥ 0 into the spectral representation
G12 (τ ) = eF/T
X
n,n′
e−En /T e(En −En′ )τ hn|Ô 1|n′ ihn′ |Ô 2|ni.
(2.455) {18.20}
154
2 Field Formulation of Many-Body Quantum Physics
Since G12 (τ ) is periodic under τ → τ + 1/T , its Fourier representation contains only
the discrete Matsubara frequencies ωm = 2πmT :
Z
G12 (ωm ) =
1/T
0
F/T
= e
dτ eiωm τ G12 (τ )
X
n,n′
e−En /T 1 − e(En −En′ )/T hn|Ô 1 |n′ ihn′ |Ô 2|ni
×
−1
.
iωm − En′ + En
(2.456) {18.21}
The retarded Green function satisfies no periodic (or antiperiodic) boundary condition. It possesses Fourier components with all real frequencies ω:
GR
12 (ω)
=
Z
∞
−∞
= eF/T
iωt
dt e
Z
0
∞
F/T
Θ(t)e
dt eiωt
Xh
n,n′
−Ĥ/T
Tr e
i h
1
2
ÔH
(t), ÔH
(0)
∓
e−En /T ei(En −En′ )t hn|Ô 1|n′ ihn′ |Ô 2|ni
i
∓e−En /T e−i(En −En′ )t hn|Ô 2|n′ ihn′ |Ô 1 |ni . (2.457) {18.22}
In the second sum we exchange n and n′ and perform the integral, after having
attached to ω an infinitesimal positive-imaginary part iη to ensure convergence.
The result is
F/T
GR
12 (ω) = e
X
n,n′
i
h
e−En /T 1 − e(En −En′ )/T hn|Ô 1 |n′ ihn′ |Ô 2|ni
×
i
.
ω − En′ + En + iη
(2.458) {18.23}
By comparing this with (2.456), we see that the thermal Green functions are obtained from the retarded ones by the replacement (for a discussion see [15])
−1
i
→
.
ω − En′ + En + iη
iωm − En′ + En
(2.459) {18.24}
A similar procedure holds for fermion operators Ô i (which are not observable). There
are only two changes with respect to the boson case. First, in the Fourier expansion
of the imaginary-time Green functions, the bosonic Matsubara frequencies ωm in
(2.456) become fermionic. Second, in the definition of the retarded Green functions
(2.452), the commutator is replaced by an anticommutator, i.e., the retarded Green
i
function of fermion operators ÔH
is defined by
h
j
′
R
′
′ F/T
i
GR
Tr e−Ĥ/T ÔH
(t), ÔH
(t′ )
ij (t, t ) ≡ Gij (t − t ) ≡ Θ(t − t )e
i +
. (2.460) {18.retb}
These changes produce an opposite sign in front of the e(En −En′ )/T -term in both of
the formulas (2.456) and (2.458). Apart from that, the relation between the two
Green functions is again given by the replacement rule (2.459).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
155
2.19 Nonequilibrium Quantum Statistics
At this point it is customary to introduce the spectral function
ρ12 (ω ′ ) =
′
1 ∓ e−ω /T eF/T
×
X
n,n′
e−En /T 2πδ(ω − En′ + En )hn|Ô 1|n′ ihn′ |Ô 2|ni,
(2.461) {18.26}
where the upper and the lower sign hold for bosons and fermions, respectively. Under
an interchange of the two operators it behaves like
ρ12 (ω ′) = ∓ρ12 (−ω ′ ).
(2.462) {@}
Using this spectral function, we may rewrite the Fourier-transformed retarded and
thermal Green functions as the following spectral integrals:
GR
12 (ω) =
Z
∞
G12 (ωm ) =
Z
∞
−∞
−∞
i
dω ′
ρ12 (ω ′)
,
2π
ω − ω ′ + iη
(2.463) {18.27}
−1
dω ′
ρ12 (ω ′)
.
2π
iωm − ω ′
(2.464) {18.28}
dω ′
′
ρ12 (ω ′ )e−iω t ,
2π
(2.465) {18.29}
These equations show how the imaginary-time Green functions arise from the retarded Green functions by a simple analytic continuation in the complex frequency
plane to the discrete Matsubara frequencies, ω → iωm . The inverse problem is not
solvable. It is impossible to reconstruct the retarded Green functions in the entire
upper half-plane of ω from the imaginary-time Green functions defined only at the
Matsubara frequencies ωm . The problem is solvable only approximately if other
information is available [16]. For instance, the sum rules for canonical fields, which
will be derived later in Eq. (2.499) with the ensuing asymptotic condition (2.500), ref(2.499)
lab(18.41)
are sufficient to make the continuation unique [17].
est(18.59)
Going back to the time variables t and τ , the Green functions are
GR
12 (t) = Θ(t)
G12 (τ ) =
Z
∞
−∞
Z
∞
−∞
X −iω τ
dω ′
−1
e m
ρ12 (ω ′)T
.
2π
iωm − ω ′
ωm
(2.466) {18.30}
The sum over even or odd Matsubara frequencies on the right-hand side of G12 (τ )
was evaluated before as [recall (2.422)]
T
X
e−iωm τ
X
e−iωm τ
n
1
−1
= Gpω,e (τ ) = e−ω(τ −1/2T )
iωm − ω
2 sin(ω/2T )
−ωτ
= e (1 + fω ),
(2.467) {@}
and
T
n
1
−1
= Gaω,e (τ ) = e−ω(τ −1/2T )
iωm − ω
2 cos(ω/2T )
−ωτ
= e (1 − fω ),
(2.468) {@}
156
2 Field Formulation of Many-Body Quantum Physics
with the Bose and Fermi distribution functions (2.422)
fω =
1
eω/T
∓1
(2.469) {18.bfdisf}
,
respectively.
2.20
Other Important Green Functions
In studying the dynamics of systems at finite temperature, several other Green
functions are useful. Let us derive their spectral functions and general properties.
By complete analogy with the retarded Green functions for bosonic and fermionic
operators, we may introduce their counterparts, the so-called advanced Green functions (compare page 39):
h
′
A
′
′
F/T
1
2
GA
Tr e−Ĥ/T ÔH
(t), ÔH
(t′ )
12 (t, t ) ≡ G12 (t − t ) = −Θ(t − t)e
i ∓
. (2.470) {18.34}
Their Fourier transforms have the spectral representation
GA
12 (ω)
=
Z
∞
−∞
i
dω ′
ρ12 (ω ′ )
,
2π
ω − ω ′ − iη
(2.471) {18.35}
differing from the retarded case (2.463) only by the sign of the iη-term. This makes
the Fourier transforms vanish for t > 0, so that the time-dependent Green function
has the spectral representation [compare (2.465)]
GA
12 (t) = −Θ(−t)
Z
∞
−∞
dω
ρ12 (ω)e−iωt .
2π
(2.472) {18.36}
By subtracting retarded and advanced Green functions, we obtain the thermal
expectation value of commutator or anticommutator:
′
F/T
C12 (t, t ) = e
−Ĥ/T
Tr e
h
i 1
2
ÔH
(t), ÔH
(t′ )
∓
′
A
′
= GR
12 (t, t ) − G12 (t, t ). (2.473) {18.38}
Note the simple relations:
′
′
′
GR
12 (t, t ) = Θ(t − t )C12 (t, t ),
′
′
′
GA
12 (t, t ) = −Θ(t − t)C12 (t, t ).
(2.474) {18.arc}
(2.475) {18.arcb}
When inserting into (2.473) the spectral representations (2.463) and (2.472) of GR
12 (t)
A
and G12 (t) and using the identity (1.337),
i
i
η
−
=
2
= 2πδ(ω − ω ′ ),
ω − ω ′ + iη ω − ω ′ − iη
(ω − ω ′ )2 + η 2
(2.476) {18.37}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
157
2.20 Other Important Green Functions
we obtain the spectral integral representation for the commutator function:12
C12 (t) =
∞
Z
−∞
dω
ρ12 (ω)e−iωt .
2π
(2.477) {18.40}
Thus a knowledge of the commutator function C12 (t) determines directly the spectral
function ρ12 (ω) by its Fourier components
(2.478) {18.39}
C12 (ω) = ρ12 (ω).
An important role in studying the dynamics of a system in a thermal environment
is played by the time-ordered Green functions. They are defined by
h
i
1
2
G12 (t, t′ ) ≡ G12 (t − t′ ) = eF/T Tr e−Ĥ/T T̂ ÔH
(t)ÔH
(t′ ) .
(2.479) {18.45}
Inserting intermediate states as in (2.456) we find the spectral representation
G12 (ω) =
+
Z
∞
−∞
∞
Z
−∞
F/T
= e
n
1
2
dt eiωt Θ(t) eF/T Tr e−Ĥ/T ÔH
(t)ÔH
(0)
2
1
dt eiωt Θ(−t)eF/T Tr e−Ĥ/T ÔH
(t)ÔH
(0)
Z
∞
Z
0
0
± eF/T
n
o
dt eiωt
X
e−En /T ei(En −En′ )t hn|Ô 1|n′ ihn′ |Ô 2|ni
X
e−En /T e−i(En −En′ )t hn|Ô 2 |n′ ihn′ |Ô 1|ni .
n,n′
−∞
dt eiωt
o
n,n′
(2.480) {18.22T}
Interchanging again n and n′ , this can be written in terms of the spectral function
(2.461) as
G12 (ω) =
Z
∞
−∞
dω ′
1
1
i
i
. (2.481) {18.27T}
ρ12 (ω ′ )
+
′ /T
′ /T
−ω
′
ω
2π
1∓e
ω − ω + iη 1 ∓ e
ω − ω ′ − iη
#
"
Let us also write down the spectral decomposition of a further operator expression complementary to C12 (t) of (2.473), in which boson or fermion fields appear
with the “wrong” commutator:
h
1
2
A12 (t − t′ ) ≡ eF/T Tr e−Ĥ/T ÔH
(t), ÔH
(t′ )
i ±
(2.482) {18.42}
.
1
2
This function characterizes the size of fluctuations of the operators OH
and OH
.
Inserting intermediate states, we find
A12 (ω) =
Z
∞
−∞
= eF/T
iωt F/T
dt e e
Z
∞
−∞
−Ĥ/T
Tr e
dt eiωt
Xh
n,n′
h
i 1
2
ÔH
(t), ÔH
(0)
±
e−En /T ei(En −En′ )t hn|Ô 1|n′ ihn′ |Ô 2|ni
i
±e−En /T e−i(En −En′ )t hn|Ô 2 |n′ ihn′ |Ô 1|ni . (2.483) {18.22}
12
Due to the relation (2.474), the same representation is found by dropping the factor Θ(t) in
(2.465).
158
2 Field Formulation of Many-Body Quantum Physics
In the second sum we exchange n and n′ and perform the integral, which now runs
over the entire time interval and therefore gives a δ-function:
A12 (ω) = eF/T
X
n,n′
i
h
e−En /T 1 ± e(En −En′ )/T hn|Ô 1 |n′ ihn′ |Ô 2|ni
(2.484) {18.23A}
× 2πδ(ω − En′ + En ).
In terms of the spectral function (2.461), this has the simple form
A12 (ω) =
∞
Z
−∞
ω′
ω
dω ′
tanh∓1
ρ12 (ω ′ ) 2πδ(ω − ω ′) = tanh∓1
ρ12 (ω). (2.485) {18.43}
2π
2T
2T
Thus the expectation value (2.482) of the “wrong” commutator has the time dependence
′
′
A12 (t, t ) ≡ A12 (t − t ) =
Z
∞
−∞
ω −iω(t−t′ )
dω
ρ12 (ω) tanh∓1
e
.
2π
2T
(2.486) {18.44}
There exists another way of writing the spectral representation of the various
A
Green functions. For retarded and advanced Green functions GR
12 , G12 , we decompose in the spectral representations (2.463) and (2.471) according to the rule (1.338):
i
P
=i
∓ iπδ(ω − ω ′ ) ,
′
ω − ω ± iη
ω − ω′
(2.487) {18.47}
where P indicates principal value integration across the singularity, leading to
GR,A
12 (ω)
=i
Z
∞
−∞
dω ′
P
ρ12 (ω ′ )
∓ iπδ(ω − ω ′) .
2π
ω − ω′
(2.488) {18.48}
Inserting (2.487) into (2.481) we find the alternative representation of the timeordered Green function
G12 (ω) = i
Z
∞
−∞
P
dω ′
∓1 ω
ρ12 (ω ′ )
−
iπ
tanh
δ(ω − ω ′) . (2.489) {18.46}
2π
ω − ω′
2T
The term proportional to δ(ω − ω ′ ) in the spectral representation is commonly
referred to as the absorptive or dissipative part of the Green function. The first
term proportional to the principal value is called the dispersive or fluctuation part.
The relevance of the spectral function ρ12 (ω ′ ) in determining both the fluctuation part as well as the dissipative part of the time-ordered Green function is the
content of the important fluctuation-dissipation theorem. In more detail, this may
be restated as follows: The common spectral function ρ12 (ω ′) which appears in the
commutator function in (2.477), in the retarded Green function in (2.463), and in
the fluctuation part of the time-ordered Green function in (2.489) determines, after
multiplication by a factor tanh∓1 (ω ′ /2T ), the dissipative part of the time-ordered
Green function in Eq. (2.489).
A
The three Green functions −iG12 (ω), −iGR
12 (ω), and −iG12 (ω) have the same real
parts. By comparing Eqs. (2.463) and (2.464), we see that retarded and advanced
H. Kleinert, PARTICLES AND QUANTUM FIELDS
159
2.21 Hermitian Adjoint Operators
Green functions are simply related to the imaginary-time Green function via an
analytic continuation. The spectral decomposition (2.489) shows that this is not
true for the time-ordered Green function, due to the extra factor tanh∓1 (ω/2T ) in
the absorptive term.
Another representation of the time-ordered Green function is useful. It is obtained by expressing tan∓1 in terms of the Bose and Fermi distribution functions
(2.469) as tan∓1 = 1 ± 2fω . Then we can decompose
G12 (ω) =
2.21
Z
∞
−∞
i
dω ′
ρ12 (ω ′ )
± 2πfω δ(ω − ω ′ ) .
2π
ω − ω ′ + iη
"
#
(2.490) {18.46n}
Hermitian Adjoint Operators
1
2
If the two operators ÔH
(t), ÔH
(t) are Hermitian adjoint to each other,
2
1
ÔH
(t) = [ÔH
(t)]† ,
(2.491) {@hermco}
the spectral function (2.461) can be rewritten as
′
ρ12 (ω ′) = (1 ∓ e−ω /T )eF/T
X
1
(t)|n′ ik2 .
e−En /T 2πδ(ω ′ − En′ + En )|hn|ÔH
×
n,n′
(2.492) {18.51}
This shows that
ρ12 (ω ′)ω ′ ≥ 0
′
for bosons,
≥0
ρ12 (ω )
for fermions.
(2.493) {@}
This property makes it possible to derive several useful inequalities between various
diagonal Green functions.
Under the condition (2.491), the expectation values of anticommutators and
commutators satisfy the time-reversal relations
′
GA
12 (t, t )
A12 (t, t′ )
C12 (t, t′ )
G12 (t, t′ )
=
=
=
=
′
∗
∓GR
21 (t , t) ,
±A21 (t′ , t)∗ ,
∓C21 (t′ , t)∗ ,
±G21 (t′ , t)∗ .
(2.494)
(2.495)
(2.496)
(2.497)
{18.timerevR
{18.timerevA
{18.timerevC}
{18.timerevG
Examples are the corresponding functions for creation and annihilation operators
which will be treated in detail below. More generally, these properties hold for any
1
2
interacting nonrelativistic particle fields ÔH
(t) = ψ̂p (t), ÔH
(t) = ψ̂p† (t) of a specific
momentum p.
Such operators satisfy, in addition, the canonical equal-time commutation rules
at each momentum:
h
i
ψ̂p (t), ψ̂p† (t) = 1.
(2.498) {18.cc}
160
2 Field Formulation of Many-Body Quantum Physics
Using (2.473) and (2.477), we derive from this the spectral function sum rule:
Z
∞
−∞
dω ′
ρ12 (ω ′ ) = 1.
2π
(2.499) {18.41}
For a canonical free field with ρ12 (ω ′) = 2πδ(ω ′−ω), this sum rule is of course trivially
fulfilled. In general, the sum rule ensures the large-ω behavior of imaginary-time
Green functions of canonically conjugate field operators, the retarded expressions
depending on real time, and the advanced expressions to be the same as for a free
particle, i.e.,
G12 (ωm ) −
−−→
ωm →∞
2.22
i
,
ωm
GR,A
−−→
12 (ω) −
ω→∞
1
.
ω
(2.500) {18.41b}
Harmonic Oscillator Green Functions for T =
/0
As an example, consider a single harmonic oscillator of frequency Ω or, equivalently,
a free particle at a point in the second-quantized field formalism. We shall start
with the second representation.
2.22.1
Creation Annihilation Operators
1
2
The operators ÔH
(t) and ÔH
(t) are the creation and annihilation operators in the
Heisenberg picture
â†H (t) = ↠eiΩt ,
âH (t) = âe−iΩt .
(2.501) {18.33beg}
The eigenstates of the Hamilton operator
1 2
ω †
1
Ĥ =
p̂ + Ω2 x̂2 =
â â + â↠= ω ↠⠱
2
2
2
(2.502) {@}
are
1
|ni = √ (↠)n |0i,
n!
(2.503) {@}
with the eigenvalues En = (n ± 1/2)Ω for n = 0, 1, 2, 3, . . . or n = 0, 1, if ↠and â
commute or anticommute, respectively. In the second-quantized field interpretation
the energies are En = nΩ and the final Green functions are the same. The spectral
function ρ12 (ω ′) is trivial to calculate. The Schrödinger√operator Ô 2 = ↠can connect
the state |ni only to hn + 1|, with the matrix element n + 1. The operator Ô 1 = â
does the opposite. Hence we have
ρ12 (ω ′) = 2πδ(ω ′ − Ω)(1 ∓ e−Ω/T )eF/T
∞,0
X
n=0
e−(n±1/2)Ω/T (n + 1).
(2.504) {18.ro}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.22 Harmonic Oscillator Green Functions for T 6= 0
161
Now we make use of the explicit partition functions of the oscillator whose paths
satisfy periodic and antiperiodic boundary conditions:
−F/T
ZΩ ≡ e
=
∞,1
X
−(n±1/2)Ω/T
e
n=0
=
(
[2 sinh(Ω/2T )]−1
2 cosh(Ω/2T )
bosons
fermions
for
)
. (2.505) {18.31}
These allow us to calculate the sums in (2.504) as follows
∞
X
n=0
0
X
−(n+1/2)Ω/T
e
!
−1
∂
1 −F/T (n + 1) = −T
e
= 1 ∓ e−Ω/T
+
e−F/T ,
∂Ω 2
e−(n−1/2)Ω/T (n + 1) = eΩ/2T = 1 + e−Ω/T
n=0
−1
e−F/T .
(2.506) {18.32}
The spectral function ρ12 (ω ′) of the single oscillator quantum of frequency Ω is
therefore given by
ρ12 (ω ′ ) = 2πδ(ω ′ − Ω).
(2.507) {singlefre}
With it, the retarded and imaginary-time Green functions become
′
′
′ −Ω(t−t )
GR
,
Ω (t, t ) = Θ(t − t )e
GΩ (τ, τ ′ ) = −T
∞
X
e−iωm (τ −τ
(2.508) {@}
′)
m=−∞
= e−Ω(τ −τ
′)



1
iΩm − Ω
1 ± nΩ
for τ
±nΩ
≥ ′
τ,
<
(2.509) {nolabel}
(2.510) {18.33}
with the average particle number fΩ of (2.469). The commutation function, for
instance, is by (2.477) and (2.507):
′
C12 (t, t′ ) = e−iΩ(t−t ) ,
(2.511) {18.44cf0}
and the correlation function of the “wrong commutator” is from (2.486) and (2.507):
AΩ (t, t′ ) = tanh∓1
Ω −iΩ(t−t′ )
e
.
2T
(2.512) {18.42sgp}
Of course, these harmonic-oscillator expressions could have been obtained directly by starting from the defining operator equations. For example, the commutator function
n
CΩ (t, t′ ) = eF/T Tr e−Ĥ/T [âH (t), â†H (t′ )]∓
o
(2.513) {@}
turns into (2.511) by using the commutation rule at different times:
[âH (t), â†H (t′ )] = e−iΩ(t−t ) ,
′
(2.514) {@cancomr}
162
2 Field Formulation of Many-Body Quantum Physics
which follows from (2.501). Since the right-hand side is a c-number, the thermodynamic average is trivial:
eF/T Tr(e−Ĥ/T ) = 1.
(2.515) {@}
After this, the relations (2.474) and (2.475) determine the retarded and advanced
Green functions
′
′
′
′ −iΩ(t−t )
GR
,
Ω (t − t ) = Θ(t − t )e
′
′
−iΩ(t−t )
GA
.
Ω (t − t ) = −Θ(t − t)e
For the Green function at imaginary times
h
i
GΩ (τ, τ ′ ) ≡ eF/T Tr e−Ĥ/T T̂τ âH (τ )â†H (τ ′ ) ,
the expression (2.510) is found using [see (2.518)]
(2.516) {@18.67}
(2.517) {@}
â†H (τ ) ≡ eĤτ ↠e−Ĥτ = ↠eΩτ ,
âH (τ ) ≡ eĤτ âe−Ĥτ = âe−Ωτ ,
(2.518) {18.19}
and the summation formula (2.506).
The “wrong” commutator function (2.512) can, of course, be immediately derived
from the definition
′
F/T
A12 (t − t ) ≡ e
−Ĥ/T
Tr e
i h
âH (t), â†H (t′ )
±
(2.519) {18.42sg}
using (2.501) and inserting intermediate states.
For the temporal behavior of the time-ordered Green function we find from
(2.481):
GΩ (ω) = 1 ∓ e−Ω/T
−1
Ω/T
GR
Ω (ω) + 1 ∓ e
and from this by a Fourier transformation:
GΩ (t, t′ ) = 1 ∓ e−Ω/T
h
−1
′
−1
Θ(t − t′ )e−iΩ(t−t ) − 1 ∓ eΩ/T
i
′
GA
Ω (ω),
−1
(2.520) {18.46a}
′
Θ(t′ − t)e−iΩ(t−t )
′
= Θ(t − t′ ) ± (eΩ/T ∓ 1)−1 e−iΩ(t−t ) = [Θ(t − t′ ) ± fΩ ] e−iΩ(t−t ) . (2.521) {18.50}
The same result is easily obtained by directly evaluating the defining equation using
(2.501) and inserting intermediate states:
h
GΩ (t, t′ ) ≡ GΩ (t − t′ ) = eF/T Tr e−Ĥ/T T̂ âH (t)â†H (t′ )
′
i
′
= Θ(t − t′ )hâ ↠ie−iΩ(t−t ) ± Θ(t′ − t)h↠âie−iΩ(t−t )
′
′
= Θ(t − t′ )(1 ± fΩ )e−iΩ(t−t ) ± Θ(t′ − t)fΩ e−iΩ(t−t ) ,
†
(2.522) {18.45Ome}
which is the same as (2.521). For the correlation function with a and a interchanged,
h
i
ḠΩ (t, t′ ) ≡ GΩ (t − t′ ) = eF/T Tr e−Ĥ/T T̂ â†H (t)âH (t′ ) ,
we find in this way
′
(2.523) {18.45Omei}
′
ḠΩ (t, t′ ) = Θ(t − t′ )h↠âie−iΩ(t−t ) ± Θ(t′ − t)hâ ↠ie−iΩ(t−t )
′
′
= Θ(t − t′ )fΩ e−iΩ(t−t ) ± Θ(t′ − t)(1 ± fΩ )e−iΩ(t−t )
in agreement with (2.497).
(2.524) {18.45Omei}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
2.22 Harmonic Oscillator Green Functions for T 6= 0
2.22.2
163
Real Field Operators
From the above expressions it is easy to construct the corresponding Green functions
for the position operators of the harmonic oscillator x̂(t). It will be useful to keep
the discussion more general by admitting oscillators which are not necessarily mass
points in space but can be field variables. Thus we shall use, instead of x̂(t), the
symbol ϕ(t), and call this a field variable. We decompose the field as
x̂(t) =
s
h̄ −iΩt
âe
+ ↠eiΩt ,
2MΩ
{@POSOPS}
(2.525) {18.decompph
using in this section physical units. The commutator function (2.473) is directly
C(t, t′ ) ≡ h[ϕ̂(t), ϕ̂(t′ )]∓ iρ = −
h̄
2i sin Ω(t − t′ ),
2MΩ
(2.526) {18.cons0}
implying a spectral function [recall (2.477)]
ρ(ω ′ ) =
1
2π [δ(ω ′ − Ω) − δ(Ω′ + Ω)].
2MΩ
(2.527) {18.spwe}
The real operator ϕ̂(t) behaves like the difference of a particle of frequency Ω and
−Ω, with an overall factor 1/2MΩ. It is then easy to find the retarded and advanced
Green functions of the operators ϕ̂(t) and ϕ̂(t′ ):
i
h̄
h̄ h R
′
GΩ (t, t′ ) − GR
Θ(t − t′ ) 2i sin Ω(t − t′ ), (2.528) {@retgf}
−Ω (t, t ) = −
2MΩ
2MΩ
i
h
h̄
h̄
′
′
A
GA (t, t′ ) =
GA
Θ(t − t′ ) 2i sin Ω(t′ − t). (2.529) {@}
Ω (t, t ) − G−Ω (t, t ) =
2MΩ
2MΩ
GR (t, t′ ) =
From the spectral representation (2.486), we obtain for the “wrong commutator”
A(t, t′ ) = h[ϕ̂(t), ϕ̂(t′ )]∓ i =
h̄
Ω
coth±1
2 cos Ω(t − t′ ).
2MΩ
2kB T
(2.530) {18.cons1}
The relation with (2.526) is again a manifestation of the fluctuation-dissipation
theorem (2.486).
The average of these two functions yields the time-dependent correlation function
at finite temperature, containing only the product of the operators:
GP (t, t′ ) ≡ hϕ̂(t)ϕ̂(t′ )i =
h̄
[(1 ± 2fΩ ) cos Ω(t − t′ ) − i sin Ω(t − t′ )] , (2.531) {18.cons1x}
2MΩ
with the average particle number fΩ of (2.469). In the limit of zero temperature
where fΩ ≡ 0, this reduces to
GP (t, t′ ) = hϕ̂(t)ϕ̂(t′ )i =
h̄ −iΩ(t−t′ )
e
.
2MΩ
(2.532) {18.cons1}
164
2 Field Formulation of Many-Body Quantum Physics
The time-ordered Green function is obtained from this by the obvious relation
1
[A(t, t′ ) + ǫ(t − t′ )C(t, t′ )] ,
2
(2.533) {@18.GCA}
′
where ǫ(t − t ) is the step function of Eq. (1.323). Explicitly, the time-ordered Green
function is
G(t, t′ ) = Θ(t − t′ )GP (t, t′ ) ± Θ(t′ − t)GP (t′ , t) =
G(t, t′ ) ≡ hT̂ ϕ̂(t)ϕ̂(t′ )i =
h̄
[(1 ± 2fΩ ) cos Ω|t − t′ | − i sin Ω|t − t′ |] ,
2MΩ
(2.534) {18.cons1x}
which reduces for T → 0 to
G(t, t′ ) = hT̂ ϕ̂(t)ϕ̂(t′ )i =
h̄ −iΩ|t−t′ |
e
.
2MΩ
(2.535) {18.cons1}
Thus, as a mnemonic rule, a finite temperature is introduced into a zerotemperature Green function by simply multiplying the real part of the exponential
function by a factor 1±2fΩ . This is another way of stating the fluctuation-dissipation
theorem.
There is another way of writing the time-ordered Green function (2.534) in the
bosonic case:
Ω
′
h̄ cosh 2 (h̄β − i|t − t |)
.
(2.536) {18.cons1xx}
G(t, t′ ) ≡ hT̂ ϕ̂(t)ϕ̂(t′ )i =
h̄Ωβ
2MΩ
sinh
2
For t − t′ > 0, this coincides precisely with the periodic Green function Gpe (τ, τ ′ ) =
Gpe (τ − τ ′ ) at imaginary-times τ > τ ′ [see (2.431)], if τ and τ ′ are continued analytically to it and it′ , respectively. Decomposing (2.534) into real and imaginary parts,
we see by comparison with (2.533) that anticommutator and commutator functions
are the doubled real and imaginary parts of the time-ordered Green function:
A(t, t′ ) = 2 Re G(t, t′ ),
C(t, t′ ) = 2i Im G(t, t′ ).
(2.537) {@ReGImG}
In the fermionic case, the hyperbolic functions cosh and sinh in numerator and
denominator are simply interchanged, and the result coincides with the analytically
continued antiperiodic imaginary-time Green function [see again (2.431)].
For real fields ϕ̂(t), the time-reversal properties (2.494)–(2.497) of the Green
functions become
GA (t, t′ )
A(t, t′ )
C(t, t′ )
G(t, t′ )
=
=
=
=
∓GR (t′ , t),
±A(t′ , t),
∓C(t′ , t),
±G(t′ , t).
(2.538)
(2.539)
(2.540)
(2.541)
H. Kleinert, PARTICLES AND QUANTUM FIELDS
{18.timerevR
{nolabel}
{nolabel}
{18.timerevG
165
Appendix 2A
Permutation Group and Representations
Appendix 2A
Permutation Group and Representations on
n-Particle Wave Functions
A permutation of n particles is given by
1
2
...
n
P =
,
p(1) p(2) . . . p(n)
{permgr}
(2A.1) {eq.2.A.1}
where p(i) are all possible one-to-one mappings of the integers 1, 2, 3, . . . , n onto themselves. In the
notation (2A.1) the order of the columns is irrelevant, i.e., the same permutation can be written
in any other form in which the columns are interchanged with each other, for example:
2
1
...
n
P =
.
(2A.2) {eq.2.A.2}
p(2) p(1) . . . p(n)
Given n particles at positions x1 , x2 , . . . , xn , the permutation P may be taken to change the
position x1 to xp(1) , x2 to xp(2) , etc., i.e., we define P to act directly on the indices:
P xi ≡ xp(i) .
(2A.3) {@}
Given an n-particle wave function ψ(x1 , x2 , . . . , xn ; t), it behaves under P as follows:
P ψ(x1 , x2 , . . ., xn ; t) =
=
ψ(P x1 , P x2 , . . ., P xn ; t)
ψ(xp(1) , xp(2) , . . ., xp(n) ; t).
(2A.4) {eq.2.A.3}
There exists a different but equivalent definition of permutations, to be denoted by P ′ , in which
the variables x1 , x2 , . . ., xn of the wave functions are taken from their places 1, 2, . . ., n in the list
of arguments of ψ(x1 , x2 , . . ., xn ; t) and moved to the positions p(1), p(2), . . ., p(n) in this list, i.e.,
P ′ ψ(x1 , x2 , . . ., xn ; t) ≡ ψ(. . ., x1 , . . ., x2 , . . .; t),
(2A.5) {eq.2.A.5}
where x1 is now at position p(1), x2 at position p(2), etc. The difference between the two definitions
is seen in the simple example:
1 2 3
ψ(x1 , x2 , x3 ; t) = ψ(x2 , x3 , x1 ; t),
(2A.6) {eq.2.A.6}
2 3 1
to be compared with
1
2
2 3
3 1
′
ψ(x1 , x2 , x3 ; t) = ψ(x3 , x1 , x2 ; t).
(2A.7) {eq.2.A.7}
In the following we shall use only the first definition, but all statements to be derived would hold
as well if we use the second one throughout the remainder of this appendix.
For n elements there are n! different permutations P . Given any two permutations
1
...
n
P =
p(1) . . . p(n)
and
Q=
a product is defined by rewriting Q as
Q=
1
...
n
q(1) . . . q(n)
,
p(1)
...
p(n)
q(p(1)) . . . q(p(n))
166
2 Field Formulation of Many-Body Quantum Physics
and setting
QP
≡
≡
p(1)
q(p(1))
...
p(n)
. . . q(p(n))
1
q(p(1))
...
n
. . . q(p(n))
1
...
n
p(1) . . . p(n)
(2A.8) {eq.2.A.8}
.
Every element has an inverse. Indeed, if we apply first P and multiply it by
p(1) . . . p(n)
P −1 ≡
,
1
...
n
the operation P −1 P returns all elements to their original places:
p(1) . . . p(n)
1
...
n
1 ... n
−1
P P=
=
≡ I.
1
...
n
p(1) . . . p(n)
1 ... n
(2A.9) {eq.2.A.9}
(2A.10) {eq.2.A.10}
The right-hand side is defined as the identity permutation I.
It can easily be checked that for three permutations P QR, the product is associative:
(2A.11) {eq.2.A.11}
P (QR) = (P Q)R.
Thus the n! permutations of n elements form a group, also called the symmetric group Sn .
If P is such that only two elements p(i) are different from i, it can be written as
1 2 ...
i
...
j
... n
Tij =
.
(2A.12) {eq.2.A.12}
1 2 . . . p(i) . . . p(j) . . . n
It is called a transposition, also denoted in short by (i, j). Only the elements i and j are interchanged. Every permutation can be decomposed into a product of transpositions. There are many
ways of doing this. However, each permutation decomposes either into an even or an odd number
of transpositions. Therefore each permutation can be characterized by this property. As mentioned
on p. 86, it is called the parity of the permutation. It is useful to introduce the function
{parity}
1
P =even
ǫP =
for
,
(2A.13) {eq.2.A.13}
−1
P =odd
which indicates the parity. This function satisfies the identity
ǫP Q ≡ ǫP ǫQ .
(2A.14) {eq.2.A.14}
Indeed, if P and Q are decomposed into transpositions,
Y
T(ij) ,
n factors,
P =
(ij)
Q=
Y
T(i′ j ′ ) ,
(2A.15) {eq.2.A.15}
m factors,
(i′ j ′ )
then the product
PQ =
Y Y
T(ij) T(i′ j ′ )
(ij) (i′ j ′ )
(2A.16) {eq.2.A.16}
contains n × m transpositions. This number is even if n and m are both even or odd, and odd, if
one of them is odd and the other even. Since the identity is trivially even, the inverse P −1 of a
permutation has the same parity as P itself, i.e.,
ǫP −1 = ǫP .
(2A.17) {eq.2.A.17}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Appendix 2A
Permutation Group and Representations
167
Let us find the irreducible representations of the permutation group on the Hilbert space of
n-particle wave functions. The permutation on this space is defined by (2A.4). The irreducible
representations can be classified with the help of so-called Young tableaux. These are arrays of n
boxes of the form
1 2 3 4 5 6 7 8 9 · ·
· · ·
· · ·
(2A.18) {@}
· ·
·
n.
The boxes are filled successively with the numbers 1 to n. The numbers of boxes in the rows
m1 , m2 , m3 , . . . are ordered as
m1 ≥ m2 ≥ m3 ≥ . . .
n
X
mk = n.
k=1
Each tableau defines a symmetry type of a set of wave functions on which the group of permutations is represented irreducibly. The symmetry types are constructed as follows. Let pi be all
permutations of the numbers i in the row (including the identity), and let qj be the corresponding
operations in the column j. Then we define the operations P and Q as
Y X
P =
(
pi ),
Yi X
Q =
(
ǫqj qj ).
j
The product is taken over all rows i or all columns j, and ǫq is the parity function (2A.13). Now
apply the operation QP to the indices of the wave function ψ(x1 , x2 , . . ., xn ).
As an example, take the Hilbert space of three-particle wave functions. There are three different
Young tableaux corresponding to the following irreducible representations of S3 :
X
1 2 3
P =
p (sum over all 6 elements of S3 ), Q = 1
(2A.19) {eq.}
1 2
3
1
2
3
P = 1 + (1, 2),
P = 1, Q =
X
Q = 1 − (1, 3)
ǫq q (sum over all 6 elements of S3 ).
(2A.20) {nolabel}
(2A.21) {nolabel}
The wave functions associated with these have the form
1 2 3 ψ(x1 , x2 , x3 )
= ψ(x1 , x2 , x3 ) + ψ(x2 , x3 , x1 )
+ ψ(x3 , x1 , x2 ) + ψ(x1 , x3 , x2 )
+ ψ(x2 , x1 , x3 ) + ψ(x3 , x2 , x1 ),
1 2
ψ(x1 , x2 , x3 )
3
1
2 ψ(x1 , x2 , x3 )
3
(2A.22) {eq.1}
= ψ(x1 , x2 , x3 ) + ψ(x2 , x1 , x3 )
− ψ(x3 , x2 , x1 ) − ψ(x2 , x3 , x1 ),
(2A.23) {eq.2}
= ψ(x1 , x2 , x3 ) + ψ(x2 , x3 , x1 )
+ ψ(x3 , x1 , x2 ) − ψ(x1 , x3 , x2 )
− ψ(x2 , x1 , x3 ) − ψ(x3 , x2 , x1 ).
(2A.24) {nolabel}
168
2 Field Formulation of Many-Body Quantum Physics
These wave functions are easily normalized
√ by dividing them by the square root of the number
√
of terms in each expression, i.e., 6, 2, 6, in the three cases. The horizontal array leads to the
completely symmetrized wave function, the vertical array to the completely antisymmetrized one.
The second tableau results in a mixed symmetry.
Four-particle wave functions are classified with the following tableaux:
1 2 3 4 +
1 2 3
+
4
1 2
+
3 4
1 2
+
3
4
1
2
.
3
4
(2A.25) {@}
Consider, for instance, the third of these
1 2
.
3 4
This stands for the permutation operator QP with
P = (1 + (3, 4)) (1 + (1, 2)) ,
Q = (1 − (2, 4)) (1 − (1, 3)) ,
(2A.26) {nolabel}
so that
P ψ(x1 , x2 , x3 , x4 ) =
+
ψ(x1 , x2 , x3 , x4 ) + ψ(x2 , x1 , x3 , x4 )
ψ(x1 , x2 , x4 , x3 ) + ψ(x2 , x1 , x4 , x3 ),
QP ψ(x1 , x2 , x3 , x4 ) =
−
ψ(x1 , x2 , x3 , x4 ) − ψ(x3 , x2 , x1 , x4 )
ψ(x1 , x4 , x3 , x2 ) + ψ(x3 , x4 , x1 , x2 )
+
−
ψ(x1 , x2 , x4 , x3 ) − ψ(x3 , x2 , x4 , x1 )
ψ(x1 , x4 , x2 , x3 ) + ψ(x3 , x4 , x2 , x1 )
+
−
ψ(x2 , x1 , x3 , x4 ) − ψ(x2 , x3 , x1 , x4 )
ψ(x4 , x1 , x3 , x2 ) + ψ(x4 , x3 , x1 , x2 )
+
−
ψ(x2 , x1 , x4 , x3 ) − ψ(x2 , x3 , x4 , x1 )
ψ(x4 , x1 , x2 , x3 ) + ψ(x4 , x3 , x2 , x1 ).
(2A.27) {nolabel}
(2A.28) {nolabel}
There exists an altertive but mathematically equivalent prescription of forming the wave functions
of different symmetry types based on the permutations P ′ introduced in (2A.5). Instead of performing the permutations on the indices, one exchanges the positions of the arguments in the wave
functions to produce ψ(x1 , . . ., xi , . . ., xj , . . ., xn ). As an example, take the tableaux
1 2
3
,
whose associated wave function was written down in (2A.23). The alternative wave function would
be
1 2
3
′
ψ(x1 , x2 , x3 ) = ψ(x1 , x2 , x3 ) + ψ(x2 , x1 , x3 ) − ψ(x3 , x2 , x1 ) − ψ(x3 , x1 , x2 ), (2A.29) {eq.}
with the last two terms differing from those in (2A.23).
The following formula specifies the dimensionality with which the group elements of the permutation group are represented for these symmetry classes:
n!
,
i,j hij
d= Q
(2A.30) {nolabel}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Appendix 2B
169
Treatment of Singularities in Zeta-Function
where hij is the number of boxes to the right of the position ij, plus the number of boxes below
the position ij in the tableau, plus 1 for the box on the position ij itself.
As a useful check for the calculated dimensions we may use the dimensionality theorem, by
which the squares of the dimensions d of all inequivalent, irreducible, unitary representations add
up to the order of the group, here n!:
X
d2 = n! .
(2A.31) {qftc3.form}
This is a direct consequence of the great orthogonality theorem of such representations for any finite
group.
For the permutation group at hand, one has the additional property that the defining representation contains each irreducible representation with a multiplicity equal to its dimension.
For three particles, the symmetric and antisymmetric representations
1 2 3
and
1
2
3
are 3!/3 × 2 × 1 = 1-dimensional. The mixed representation
1 2
3
is 3!/3 × 1 × 1 = 2-dimensional. These dimensions fulfill the dimensionality formula (2A.31):
12 + 11 + 22 = 3! .
(2A.32) {@}
Similarly, one has for four particles the dimensionalities
4!
= 1,
4×3×2×1
1 2 3 4
d=
1 2 3
4
d=
4!
= 3,
4×2×1×1
1 2
3 4
d=
4!
= 2,
3×2×2×1
1 2
3
4
4!
d=
= 3,
4×1×2×1
1
2
3
4
d=
(2A.33) {@}
4!
= 1.
4×3×2×1
Again we check that these dimensions fulfill the dimensionality formula (2A.31):
12 + 32 + 22 + 32 + 12 = 4! .
Appendix 2B
(2A.34) {@}
Treatment of Singularities in Zeta-Function
Here we show how to evaluate the sums which determine the would-be critical temperatures of a
Bose gas in a box and in a harmonic trap.
{@APP71}
170
2 Field Formulation of Many-Body Quantum Physics
2B.1
Finite Box
According to Eqs. (2.364), (2.369), and (2.371), the relation between temperature T =
h̄2 π 2 /bM L2kB and the fugacity zD at a fixed particle number N in a finite D-dimensional box is
determined by the equation
zD
N = Nn (T ) + Ncond (T ) = SD (zD ) +
.
(2B.1) {@thiseq12A”’}
1 − zD
Here SD (zD ) is the subtracted infinite sum
SD (zD ) ≡
∞
X
w
[Z1D (wb)ewDb/2 − 1]zD
,
w=1
(2B.2) {@thiseq12A}
P∞
2
containing the Dth power of a one-particle partition function in the box Z1 (b) = k=1 e−bk /2 .
The would-be critical temperature is found by equating this sum at zD = 1 with the total particle
number N . We shall rewrite Z1 (b) as
h
i
(2B.3) {@Z1BAP}
Z1 (b) = e−b/2 1 + e−3b/2 σ1 (b) ,
where σ1 (b) is related to the elliptic theta function (2.366) by
σ1 (b) ≡
∞
X
e−(k
2
−4)b/2
=
k=2
i
e2b h
ϑ3 (0, e−b/2 ) − 1 − 2e−b/2 .
2
According to Eq. (2.368), this has the small-b behavior
r
π 2b
1
σ1 (b) =
e − e3b/2 − e2b + . . . .
2b
2
(2B.4) {@}
(2B.5) {@ExpANS}
The omitted terms are exponentially small as long as b < 1 [see the sum over m in Eq. (2.367)]. For
large b, these terms become important to ensure an exponentially fast falloff like e−3b/2 . Inserting
(2B.3) into (2B.2), we find
∞ X
D−1 2
(D−1)(D−2) 3
−3wb/2
−6wb/2
−9wb/2
.
σ1 (wb)e
+
σ1 (wb)e
+
σ1 (wb)e
SD (1) ≡ D
2
6
w=1
(2B.6) {@thiseq12Ab}
Inserting here the small-b expression (2B.5), we obtain
r
∞ X
π wb
π wb 1 wb
e −
e + e − 1 + ... ,
(2B.7) {@thiseq12AB}
S2 (1) ≡
2wb
2wb
4
w=1
!
r
r
3
∞
X
π
π 3wb/2 1 3wb/2
3 π 3wb/2 3
3wb/2
S3 (1) ≡
e
+
e
− e
− 1 + . . . , (2B.8) {@thiseq12AB2
e
−
2wb
2 2wb
4 2wb
8
w=1
the dots indicating again exponentially small terms. The sums are convergent only for negative b,
this being a consequence of the approximate nature of these expressions. If we evaluate them in
this regime, the sums produce polylogarithmic functions
ζν(z) ≡
∞
X
zw
,
wν
w=1
(2B.9) {@IZF}
and we find, using the property at the origin,13
ζ(0) = −1/2,
13
1
ζ ′ (0) = − log 2π,
2
(2B.10) {@zeta0p}
I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 9.541.4.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Appendix 2B
Treatment of Singularities in Zeta-Function
P∞
1 = ζ(0) = −1/2, − w=1 log w = ζ ′ (0) = − 12 log 2π, such that
r
π
1
b
S2 (1) = ζ1 (e ) −
ζ1/2 (eb ) + ζ0 (eb ) − ζ(0) + . . . .
2b
4
r
r 3
3 π
1
3 π
π
ζ1 (e3b/2 ) +
ζ1 (e3b/2 ) − ζ0 (e3b/2 ) − ζ(0) + . . . .
ζ3/2 (e3b/2 ) −
S3 (1) =
2b
2 2b
4 2b
8
which imply
P∞
171
w=1
(2B.11) {nolabel}
(2B.12) {@}
These expressions can now be expanded in powers of b with the help of the Robinson expansion
given in the footnote on p. 8. Afterwards, b is continued analytically to positive values and we
obtain
r
π
π
1
S2 (1) = − log(C2 b) −
ζ(1/2) + (3 − 2π) + O(b1/2 ),
(2B.13) {@thiseq12ABd2
2b
2b
8
r
r 3
3π
3 π
9
π
ζ(3/2) +
log(C3 b) +
ζ(1/2)(1 + π) + (1 + π) + O(b1/2 ). (2B.14) {@thiseq12ABd}
S3 (1) =
2b
4b
4 2b
16
′
The constants C2,3 , C2,3
inside the logarithms turn out to be complex, implying that the limiting
expressions (2B.7) and (2B.8) cannot be used reliably. A proper way to proceed goes as follows:
We subtract from SD (1) terms which remove the small-b singularities by means of modifications
of (2B.7) and (2B.8) which have the same small-b expansion up to b0 :
r
∞ X
π
4π − 3 −wb
π
S̃2 (1) ≡
e
+ ... ,
(2B.15) {@thiseq12ABx}
−
+
2wb
2wb
4
w=1
"r
#
r
3
∞
X
3
9
3 π
π
π
+ (1 + 2π)
− (1 + 2π) e−3wb/2 + . . . .
(2B.16) {@thiseq12AB2x
−
S̃3 (1) ≡
2wb
2
2wb
4
2wb
8
w=1
In these expressions, the sums over w can be performed for positive b, yielding
r
4π − 3
π
π
ζ1 (e−b ) −
ζ1/2 (e−b ) +
ζ0 (e−b ) + . . . ,
S̃2 (1) ≡
2b
2b
4
r 3
r
3
3 π
π
π
−3b/2
−3b/2
S̃3 (1) ≡
ζ1 (e
) + (1+2π)
ζ1/2 (e−3b/2 )
ζ3/2 (e
)−
2b
2 2b
4
2b
9
− (1+2π)ζ0 (e−3b/2 ) + . . . .
8
(2B.17) {@thiseq12ABy}
(2B.18) {@thiseq12AB2y
Inserting again the Robinson expansion in the footnote on p. 8, we obtain once more the above
expansions (2B.13) and (2B.14), but now with the well-determined real constants
C̃2 = e3/2π−2+
√
2
≈ 0.8973,
C̃3 =
3 −2+1/√3−1/π
≈ 0.2630.
e
2
(2B.19) {@thiseq12ABdx
The subtracted expressions SD (1) − S̃D (1) are smooth near the origin, so that the leading small-b
behavior of the sums over these can simply be obtained from a numeric integral over w:
Z ∞
Z ∞
1.1050938
,
dw[S3 (1) − S̃3 (1)] = 3.0441.
(2B.20) {@}
dw[S2 (1) − S̃2 (1)] = −
b
0
0
These modify the constants C̃2,3 to
C2 = 1.8134,
C3 = 0.9574.
(2B.21) {@}
The corrections to the sums over SD (1) − S̃D (1) are of order b0 and higher. They were already
included in the expansions (2B.13) and (2B.14), which only were unreliable as far as C2,3 is concerned.
172
2 Field Formulation of Many-Body Quantum Physics
Let us calculate from (2B.14) the finite-size correction to the critical temperature by equat(0)
ing S3 (1) with N . Expressing this in terms of bc via (2.373), and introducing the ratio
(0)
b̂c ≡ bc /bc which is close to unity, we obtain the expansion in powers of the small quantity
(0)
2/3
2bc /π = [ζ(3/2)/N ] :
s
p
(0)
(0)
2b
3
3
b̂c
2bc
c
log(C3 b(0)
ζ(1/2)(1 + π)b̂c + . . . .
(2B.22) {@}
b̂3/2
=1+
c b̂c ) +
c
π 2ζ(3/2)
π 4ζ(3/2)
To lowest order, the solution is simply
s
(0)
2bc
1
log(C3 b(0)
b̂c = 1 +
c ) + ... ,
π ζ(3/2)
(2B.23) {@7FstOr}
yielding the would-be critical temperature to first order in 1/N 1/3 as stated in (2.375). To next
order we insert, into the last term, the zero-order solution b̂c ≈ 1, and into the second term, the
first-order solution (2B.23). This leads to
s
(0)
1
3 2bc
log(C3 b(0)
b̂3/2
=
1
+
c )
c
2
π ζ(3/2)
(0)
i
1 h
3
2bc
2
(0)
(0)
ζ(1/2)(1 + π) +
2 log(C3 bc ) + log (C3 bc ) + . . . . (2B.24) {@7FstOr2}
+
π 4ζ(3/2)
ζ(3/2)
(0)
(0)
2/3
Replacing bc by (2/π) [ζ(3/2)/N ] , we obtain the ratio (Tc /Tc )3/2 between finite- and infinite(0)
size critical temperatures Tc and Tc . The first and second-order corrections are plotted in Fig. 2.9,
together with precise results from a numeric solution of the equation N = S3 (1).
1.4
(0)
Tc /Tc
1.3
1.2
1.1
0.05
0.1
0.15 1/N 1/3
Figure 2.9 Finite-size corrections to the critical temperature for N > 300, calculated once
from the formula N = S3 (1) (solid curve), and once from the expansion (2B.24) (short(0)
(0)
dashed up to the order [bc ]1/2 ∝ 1/N 1/3 , long-dashed up to the order bc ∝ 1/N 2/3 ).
The fat dots show the peaks in the second derivative d2 Ncond (T )/dT 2 . The small dots
show the corresponding values for canonical ensembles, for comparison.
{correfts}
2B.2
Harmonic Trap
The sum relevant for the would-be phase transition in a harmonic trap is
#
"
∞
X
1
w
− 1 zD
,
SD (b, zD ) =
−wb )D
(1−e
w=1
(2B.25) {1@7thfu1}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Appendix 2B
173
Treatment of Singularities in Zeta-Function
which determines the number of normal particles in the harmonic trap via
Nn (T ) = Zω (β)ζ̄D (βh̄ω; zD ) ≡ SD (βh̄ω, zD ).
(2B.26) {2@@AVNUnn}
We consider only the point zD = 1 which determines the critical temperature by the condition
Nn = N . Restricting ourselves to the physical cases D = 1, 2, 3, we rewrite the sum as
SD (b, 1) =
∞
X
(D − 1) −2wb (D − 1)(D − 2) −3wb
1
D e−wb −
.
e
+
e
2
6
(1 − e−wb )D
w=1
(2B.27) {@7thfu3c}
According to the method developed in the evaluation of the Robinson expansion in the footnote
on p. 8, we obtain such a sum in two steps. First we go to small b where the sum reduces to an
integral over w. After this we calculate the difference between sum and integral by a naive power
series expansion.
As it stands, the sum (2B.27) cannot be converted into an integral due to singularities at w = 0.
These must first be removed by subtractions. Thus we decompose SD (b, 1) into a subtracted sum
plus a remainder as
SD (b, 1) = S̄D (b, 1) + ∆D SD (b, 1) + b
D(3D − 1)
D
∆D−1 SD (b, 1) + b2
∆D−2 SD (b, 1). (2B.28) {@7thfu3c1}
2
24
Here
S̄D (b, 1) =
∞
X
w=1
×
D e
−wb
D − 1 −2wb (D − 1)(D − 2) −3wb
−
e
+
e
2
6
1
D
D(3D − 1)
1
− D D −
−
(1 − e−wb )D
w b
2wD−1 bD−1
24wD−2 bD−2
(2B.29) {@7thfu3c2}
is the subtracted sum and
∆D′ SD (b, 1) ≡
D
D−1
(D − 1)(D − 2)
−b
−2b
−3b
′ (e
′ (e
′ (e
)
−
ζ
)
+
)
ζ
ζ
D
D
D
bD
2
6
(2B.30) {@}
collects the remainders. The subtracted sum can now be done in the limit of small b as an integral
over w, using the well-known integral formula for the Beta function:
Z ∞
Γ(a)Γ(1 − b)
e−ax
= B(a, 1 − b) =
dx
.
(2B.31) {@}
−x
b
(1 − e )
Γ(1 + a − b)
0
This yields the small-b contributions to the subtracted sums
1
7
S̄1 (b, 1) →
γ−
≡ s1 ,
b→0
b
12
9
1
γ +log 2−
≡ s2 ,
S̄2 (b, 1) →
b→0
b
8
1
19
S̄3 (b, 1) →
γ +log 3−
≡ s3 ,
b→0
b
24
(2B.32) {@InTeG}
where γ = 0.5772 . . . is the Euler-Mascheroni number . The remaining sum-minus-integral is obtained by a series expansion of 1/(1 − e−wb )D in powers of b and performing the sums over w using
the formula:
Z ∞ ! nβh̄ω
∞
∞
X
X
1
e
=
(−βh̄ω)k ζ(ν − k) ≡ ζ̄ν (eβh̄ω ).
(2B.33) {@ZetaBar}
−
ν
n
k!
0
n=1
k=1
174
2 Field Formulation of Many-Body Quantum Physics
However, due to the subtractions, the corrections are all small of order (1/bD )O(b3 ). They will be
ignored here. Thus we obtain
SD (b, 1) =
sD
1
+ ∆SD (b, 1) + D O(b3 ).
bD
b
(2B.34) {@}
We now expand ∆D′ SD (b, 1) using Robinson’s formula stated in the footnote on p. 8 up to b2 /bD
and find
∆D′ S1 (b, 1) =
∆D′ S2 (b, 1) =
∆D′ S3 (b, 1) =
where
ζ1 (e−b ) =
ζ2 (e−b ) =
ζ3 (e−b ) =
1
ζD′ (e−b ),
b
1 2ζD′ (e−b ) − ζD′ (e−2b ) ,
2
b
1 3ζD′ (e−b ) − 3ζD′ (e−2b ) + ζD′ (e−3b ) ,
3
b
b2
b
+ ... ,
− log 1 − e−b = − log b + −
2 24
b2
+ ... ,
ζ(2) + b(log b − 1) −
4
2
b
3
b
ζ(3) − ζ(2) −
log b −
+ ... .
6
2
2
(2B.35) {nolabel}
(2B.36) {nolabel}
(2B.37) {@}
(2B.38) {nolabel}
(2B.39) {@}
The results are
S1 (b, 1) =
S2 (b, 1) =
S3 (b, 1) =
1
b
1
b2
1
b3
b
b2
(− log b + γ) + −
+ ... ,
4 144
7b2
1
+
ζ(2) − b log b − γ +
+ ... ,
2
24
3b
19
2
+ ... .
ζ(3) + ζ(2) − b log b − γ +
2
24
(2B.40) {nolabel}
Note that the calculation cannot be shortened by simply expanding the factor 1/(1 − e−wb )D
in the unsubtracted sum (2B.27) in powers of w, which would yield the result (2B.28) without the
first term S̄1 (b, 1), and thus without the integrals (2B.32).
Notes and References
For the second quantization in many-body physics and applications see
A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover, 1963;
A.L. Fetter and J.D. Walecka, Quantum Theory of Many Particle Systems, McGraw-Hill, New
York, 1971.
Good textbooks on statistical physics are
L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon, New York, 1958;
R. Kubo, Statistical Mechanics, North-Holland, Amsterdam, 1971;
K. Huang, Statistical Mechanics, Wiley, New York, 1987.
For representations of finite groups and, in particular, the permutation group see
H. Boerner, Darstellungen von Gruppen, mit Berücksichtigung der Bedürfnisse der modernen
Physik , Springer, Berlin, 1955;
A.O. Barut and R. Raczka, Theory of Group Representations and Applications, World Scientific,
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Notes and References
175
Singapore, 1986;
M. Hamermesh, Group Theory and its Application to Physical Problems, Dover, New York, 1989.
The individual citations refer to:
[1] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, Singapore, 2008 (http://klnrt.de/b5).
[2] F. Giacosa, Quanta 3, 156 (2014) (arXiv:1406.2344).
[3] H. Kleinert and V. Schulte-Frohlinde, Critical Phenomena in φ4 -Theory, World Scientific,
Singapore, 2001 (ibid.http/b8).
[4] T.D. Lee and K. Huang, and C.N. Yang, Phys. Rev. 106, 1135 (1957).
[5] U.C. Täuber and D.R. Nelson, Phys. Rep. 289, 157 (1997).
[6] O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).
[7] P. Nozières and D. Pines, The Theory of Quantum Liquids, Addison-Wesley, New York,
1990, Vol. II.
[8] In Wolfram’s program MATHEMATICA, this function is denoted by PolyLog[n, z] or Lin (z).
See http://mathworld.wolfram.com/Polylogarithm.html. We prefer the notation ζn (z)
to emphasize that ζn (1) = ζ(n) is Riemann’s zeta-function. The properties of ζn (z) are
discussed in detail in Section 7.2 of the textbook Ref. [1].
[9] The first observation was made at JILA with 87 Ru:
M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science
269, 198 (1995).
It was followed by a condensate of 7 Li at Rice University:
C.C. Bradley, C.A. Sackett, J.J. Tollet, and R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995),
and in 30 Na at MIT:
K.B. Davis, M.-O. Mewes, M.R. Andrews, and N.J. van Druten, D.S. Durfee, D.M. Kurn,
W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
[10] H. Kleinert, Collective Classical and Quantum Fields, World Scientific, Singapore, 2016
(http://klnrt.de/b7). Click on hel.pdf.
[11] H. Kleinert, Gauge Fields in Condensed Matter , Vol. I, World Scientific, 1989
(http://klnrt.de/b1).
[12] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, and W. Ketterle, Science 292, 476 (2001).
[13] V. Bretin, V.S. Stock, Y. Seurin, F. Chevy, and J. Dalibard, Phys. Rev. Lett. 92, 050403
(2004).
[14] V.S. Stock, B. Battelier, V. Bretin, Z. Hadzibabic and J. Dalibard, Laser Physics Letters 2,
275 (2005);
A. Aftalion, X. Blanc,and J. Dalibard, Physical Review A 71, 023611 (2005).
[15] Some authors define G12 (τ ) as having an extra minus sign and the retarded Green function
with a factor −i, so that the relation is more direct: GR
12 (ω) = G12 (ωm = −iω + η). See
A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Sov. Phys. JETP 9, 636 (1959); or
Methods of Quantum Field Theory in Statistical Physics, Dover, New York, 1975; also
A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill,
New York, 1971.
[16] E.S. Fradkin, The Green’s Function Method in Quantum Statistics, Sov. Phys. JETP 9, 912
(1959).
176
2 Field Formulation of Many-Body Quantum Physics
[17] G. Baym and N.D. Mermin, J. Math. Phys. 2, 232 (1961).
One extrapolation uses Padé approximations:
H.J. Vidberg and J.W. Serene, J. Low Temp. Phys. 29, 179 (1977);
W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in
Fortran, Cambridge Univ. Press (1992), Chapter 12.5.
Since the thermal Green functions are known only approximately, the continuation is not
unique. A maximal-entropy method that selects the most reliable result is described by
R.N. Silver, D.S. Sivia, and J.E. Gubernatis, Phys. Rev. B 41, 2380 (1990).
H. Kleinert, PARTICLES AND QUANTUM FIELDS