<< . .

. 2
( : 78)



. . >>

general quantum ¬eld theory applicable to non-equilibrium states.
The scope of the book is not so much to dwell on a detailed application of the non-
equilibrium theory to a single topic, but rather to show the versatility and universality
of the method by applying it to a broad range of core topics of physics. One purpose
of the book is to demonstrate the utility of Feynman diagrams in non-equilibrium
quantum statistical mechanics using an approach appealing to physical intuition. The
real-time description of non-equilibrium quantum statistical mechanics is therefore
adopted, and the diagrammatic technique for systems out of equilibrium is developed
systematically, and a representation most appealing to physical intuition applied.
Though most examples are taken from condensed matter physics, the book is intended
to contribute to the cross-fertilization between all the ¬elds of physics studying the
in¬‚uence of ¬‚uctuations, be they quantum or thermal or purely statistical, and to
establish that the convenient technique to use is in fact that of non-equilibrium
quantum ¬eld theory. The book should therefore be of interest to a wide audience of
physicists; in particular the book is intended to be self-contained so that students of
physics and physicists in general can bene¬t from its detailed expositions. It is even
contended that the method is of importance for other ¬elds such as chemistry, and
of course useful for electrical engineers.
A complete allocation of the credit for the progress in developing and applying
the real-time description of non-equilibrium states has not been attempted. However,
the references, in particular the cited review articles, should make it possible for the
interested reader to trace this information.
The book is intended to be su¬ciently broad to serve as a text for a one- or
two-semester graduate course on non-equilibrium statistical mechanics or condensed
matter theory. It is also hoped that the book can serve as a useful reference book
for courses on quantum ¬eld theory, physics of disordered systems, and quantum
transport in general. It is hoped that this attempt to make the exposition as lucid as
possible will be successful to the point that the book can be read by students with
only elementary knowledge of quantum and statistical mechanics, and read with
bene¬t on its own. Exercises have been provided in order to aid self-instruction.
I am grateful to Dr. Joachim Wabnig for providing ¬gures.



Jørgen Rammer
1

Quantum ¬elds

Quantum ¬eld theory is a necessary tool for the quantum mechanical description
of processes that allow for transitions between states which di¬er in their particle
content. Quantum ¬eld theory is thus quantum mechanics of an arbitrary number of
particles. It is therefore mandatory for relativistic quantum theory since relativistic
kinematics allows for creation and annihilation of particles in accordance with the
formula for equivalence of energy and mass. Relativistic quantum theory is thus in-
herently dealing with many-body systems. One may, however, wonder why quantum
¬eld theoretic methods are so prevalent in condensed matter theory, which consid-
ers non-relativistic many-body systems. The reason is that, though not mandatory,
it provides an e¬cient way of respecting the quantum statistics of the particles,
i.e. the states of identical fermions or bosons must be antisymmetric and symmet-
ric, respectively, under the interchange of pairs of identical particles. Furthermore,
the treatment of spontaneously symmetry broken states, such as super¬‚uids, is fa-
cilitated; not to mention critical phenomena in connection with phase transitions.
Furthermore, the powerful functional methods of ¬eld theory, and methods such as
the renormalization group, can by use of the non-equilibrium ¬eld theory technique
be extended to treat non-equilibrium states and thereby transport phenomena.
It is useful to delve once into the underlying mathematical structure of quantum
¬eld theory, but the upshot of this chapter will be very simple: just as in quan-
tum mechanics, where the transition operators, |φ ψ|, contain the whole content of
quantum kinematics, and the bra and ket annihilate and create states in accordance
with
(|φ ψ|) |χ = ψ|χ |φ (1.1)
we shall ¬nd that in quantum ¬eld theory two types of operators do the same job.
One of these operators, the creation operator, a† , is similar in nature to the ket in
the transition operator, and the other, the annihilation operator, a, is similar to
the action of the bra in Eq. (1.1), annihilating the state it operates on. Then the
otherwise messy obedience of the quantum statistics of particles becomes a trivial
matter expressed through the anti-commutation or commutation relations of the
creation and annihilation operators.


1
2 1. Quantum ¬elds


1.1 Quantum mechanics
A short discussion of quantum mechanics is ¬rst given, setting the scene for the
notation. In quantum mechanics, the state of a physical system is described by a
vector, |ψ , providing a complete description of the system. The description is unique
modulo a phase factor, i.e. the state of a physical system is properly represented by
a ray, the equivalence class of vectors ei• |ψ , di¬ering only by an overall phase factor
of modulo one.
We consider ¬rst a single particle. Of particular intuitive importance are the
states where the particle is de¬nitely at a given spatial position, say x, the corre-
sponding state vector being denoted by |x . The projection of an arbitrary state onto
such a position state, the scalar product between the states,

ψ(x) = x|ψ , (1.2)

speci¬es the probability amplitude, the so-called wave function, whose absolute square
is the probability for the event that the particle is located at the position in question.1
The states of de¬nite spatial positions are delta normalized

= δ(x ’ x ) . (1.3)
x|x

Of equal importance is the complementary representation in terms of the states
of de¬nite momentum, the corresponding state vectors denoted by |p . Analogous to
the position states they form a complete set or, equivalently, they provide a resolution
ˆ
of the identity operator, I, in terms of the momentum state projection operators

ˆ
dp |p p| = I . (1.4)

The appearance of an integral in Eq. (1.4) assumes space to be in¬nite, and the
(conditional) probability amplitude for the event of the particle to be at position x
given it has momentum p is speci¬ed by the plane wave function
1
e p·x ,
i
= (1.5)
x|p 3/2
(2π )
the transformation between the complementary representations being Fourier trans-
formation. The states of de¬nite momentum are therefore also delta normalized2

= δ(p ’ p ) . (1.6)
p|p

The possible physical momentum values are represented as eigenvalues, p|p =
ˆ
p|p , of the operator
dp p |p p|
ˆ
p= (1.7)

1 Treating space as a continuum, the relevant quantity is of course the probability for the particle
being in a small volume around the position in question, P (x)”x = |ψ(x)|2 ”x, the absolute square
of the wave function denoting a probability density.
2 If the particle is con¬ned in space, say con¬ned in a box as often assumed, the momentum states

are Kronecker normalized, p|p = δp,p .
1.1. Quantum mechanics 3


representing the physical quantity momentum. Similarly for the position of a particle.
The average value of a physical quantity is thus speci¬ed by the matrix element of
its corresponding operator, say the average position in state |ψ is given by the three
real numbers composing the vector ψ|ˆ |ψ . In physics it is customary to interpret a
x
scalar product as the value of the bra, a linear functional on the state vector space,
on the vector, ket, in question.3
The complementarity of the position and momentum descriptions is also expressed
by the commutator, [ˆ , p] ≡ x p ’ p x, of the operators representing the two physical
xˆ ˆˆ ˆˆ
quantities, being the c-number speci¬ed by the quantum of action


[ˆ , p] = i . (1.8)

The fundamental position and momentum representations refer only to the kine-
matical structure of quantum mechanics. The dynamics of a system is determined
ˆ pˆ
by the Hamiltonian H = H(ˆ , x), the operator speci¬ed according to the correspon-

dence principle by Hamilton™s function H(ˆ , x), i.e. for a non-relativistic particle of
mass m in a potential V (x) the Hamiltonian, the energy operator, is
ˆ2
ˆ = p + V (ˆ ) .
H (1.9)
x
2m
It can often be convenient to employ the eigenstates of the Hamiltonian
ˆ
H| |
= . (1.10)
» » »

The completeness of the states of de¬nite energy, | , is speci¬ed by their resolution
»
of the identity
ˆ
| » »| = I (1.11)
»
here using a notation corresponding to the case of a discrete spectrum.
At each instant of time a complete description is provided by a state vector, |ψ(t) ,
thereby de¬ning an operator, the time-evolution operator connecting state vectors at
di¬erent times
ˆ
|ψ(t) = U (t, t ) |ψ(t ) . (1.12)
Conservation of probability, conservation of the length of a state vector, or its nor-
malized scalar product ψ(t)|ψ(t) = 1, under time evolution, determines the evo-
lution operator to be unitary, U ’1 (t, t ) = U † (t, t ). The dynamics is given by the
Schr¨dinger equation
o
d|ψ(t) ˆ
= H |ψ(t)
i (1.13)
dt
and for an isolated system the evolution operator is thus the unitary operator

U (t, t ) = e’ H(t’t ) .

ˆ (1.14)

Here we have presented the operator calculus approach to quantum dynamics, the
equivalent path integral approach is presented in Appendix A.
3 For a detailed introduction to quantum mechanics we direct the reader to chapter 1 in reference
[1].
4 1. Quantum ¬elds


In order to describe a physical problem we need to specify particulars, typically in
the form of an initial condition. Such general initial condition problems can be solved
through the introduction of the Green™s function. The Green™s function G(x, t; x , t )
represents the solution to the Schr¨dinger equation for the particular initial condition
o
where the particle is de¬nitely at position x at time t

lim ψ(x, t) = δ(x ’ x ) = x, t |x , t . (1.15)
t t

The solution of the Schr¨dinger equation corresponding to this initial condition there-
o
fore depends parametrically on x (and t ), and is by de¬nition the conditional prob-
ability density amplitude for the dynamics in question4
ˆ ≡ G(x, t; x , t ) .
ψx ,t (x, t) = x, t|x , t = x|U (t, t )|x (1.16)

The Green™s function, de¬ned to be a solution of the Schr¨dinger equation, satis-
o
¬es

’ H(’i ∇x , x) G(x, t; x , t ) = 0
i (1.17)
‚t
where, according to Eq. (1.3), the Hamiltonian in the position representation, H, is
speci¬ed by the position matrix elements of the Hamiltonian
ˆ = H(’i ∇x , x) δ(x ’ x ) .
x|H|x (1.18)

The Green™s function, G, is the kernel of the Schr¨dinger equation on integral
o
form (being a ¬rst order di¬erential equation in time)

ψ(x, t) = dx G(x, t; x , t ) ψ(x , t ) (1.19)

as identi¬ed in terms of the matrix elements of the evolution operator by using the
resolution of the identity in terms of the position basis states

ˆ x |ψ(t ) .
x|ψ(t) = dx x|U (t, t )|x (1.20)

The Green™s function propagates the wave function, and we shall therefore also refer
to the Green™s function as the propagator. It completely speci¬es the quantum
dynamics of the particle.
We note that the partition function of thermodynamics and the trace of the
evolution operator are related by analytical continuation:

= Tr e’H/kT = dx x|e’H/kT |x
ˆ ˆ ˆ
Z = Tr U(’i /kT, 0)

dx G(x, ’i /kT ; x, 0)
= (1.21)

4 In
the continuum limit the Green™s function is not a normalizable solution of the Schr¨dinger
o
equation, as is clear from Eq. (1.15).
1.2. N-particle system 5


showing that the partition function is obtained from the propagator at the imaginary
time „ = ’i /kT . The formalisms of thermodynamics, i.e. equilibrium statistical
mechanics, and quantum mechanics are thus equivalent, a fact we shall take advan-
tage of throughout. The physical signi¬cance is the formal equivalence of quantum
and thermal ¬‚uctuations.
Quantum mechanics can be formulated without the use of operators, viz. using
Feynman™s path integral formulation. In Appendix A, the path integral expressions
for the propagator and partition function for a single particle are obtained. Various
types of Green™s functions and their properties for the case of a single particle are
discussed in Appendix C, and their analytical properties are considered in Appendix
D.


N -particle system
1.2
Next we consider a physical system consisting of N particles. If the particles in an
assembly are distinguishable, i.e. di¬erent species of particles, an orthonormal basis
in the N -particle state space H (N ) = H1 — H2 — · · · — HN is the (tensor) product
states, for example speci¬ed in terms of the momentum quantum numbers of the
particles
|p1 , p2 , . . . , pN ≡ |p1 — |p2 — · · · — |pN ≡ |p1 |p2 · · · |pN . (1.22)
We follow the custom of suppressing the tensorial notation.
Formally everything in the following, where an N -particle system is considered,
is equivalent no matter which complete set of single-particle states are used. In prac-
tice the choice follows from the context, and to be speci¬c we shall mainly explicitly
employ the momentum states, the choice convenient in practice for a spatially trans-
lational invariant system.5 These states are eigenstates of the momentum operators
pi |p1 , p2 , . . . , pN = pi |p1 , p2 , . . . , pN ,
ˆ (1.23)
where tensorial notation for operators are suppressed, i.e.
ˆ ˆ ˆ ˆ
pi = I1 — · · · Ii’1 — pi — Ii+1 — · · · IN ,
ˆ ˆ (1.24)
each operating in the one-particle subspace dictated by its index. In particular the
N -particle momentum states are eigenstates of the total momentum operator
N
ˆ ˆ
PN = (1.25)
pi
i=1
5 In the next sections we shall mainly use the momentum basis, and refer in the following to
the quantum numbers labeling the one-particle states as momentum, although any complete set
of quantum numbers could equally well be used. The N -tuple (p1 , p2 , . . . , pN ) is a complete
description of the N -particle system if the particles do not posses internal degrees of freedom. In
the following, where we for example have electrons in mind, we suppress for simplicity of notation
the spin labeling and simply assume it is absorbed in the momentum labeling. If the particles
have additional internal degrees of freedom, such as color and ¬‚avor, these are included in a similar
fashion. If more than one type of species is to be considered simultaneously the species type, say
quark and gluon, must also be indicated.
6 1. Quantum ¬elds


corresponding to the total momentum eigenvalue
N
P= pi . (1.26)
i=1

The position representation of the momentum states is speci¬ed by the plane wave
functions, Eq. (1.5), the scalar product of the momentum states and the analogous
N -particle states of de¬nite positions being
N
x1 , x2 , . . . , xN |p1 , p2 , . . . , pN xi |pi
ψp1 ,...,pN (x1 , . . . , xN ) = =
i=1
N
1 p1 ·x1 p2 ·x2 pN ·xN
i i i
··· e
= e e . (1.27)
(2π )3/2

1.2.1 Identical particles
For an assembly of identical particles a profound change in the above description
is needed. In quantum mechanics true identity between objects are realized, viz.
elementary particle species, say electrons, are profoundly identical, i.e. there exists
nothing in Nature which can distinguish any two electrons. Identical particles are in-
distinguishable. States which di¬er only by two identical particles being interchanged
are thus described by the same ray.6 As a consequence of their indistinguishability,
assemblies of identical particles are described by states which with respect to inter-
change of pairs of identical particles are either antisymmetric or symmetric

|p1 , p2 , . . . , pN = ± |p2 , p1 , . . . , pN , (1.28)

this leaving the probability for a set of momenta of the particles, P (p1 , p2 , . . . , pN ), a
function symmetric with respect to interchange of any pair of the identical particles.
A word on notation: the particle we call the ¬rst particle is in the momentum
state speci¬ed by the ¬rst argument, and the particle we call the N th particle is
in the momentum state speci¬ed by the N th argument. Particles whose states are
symmetric with respect to interchange are called bosons , and for the antisymmetric
case called fermions.7
6 The quantum state with all of the electrons in the Universe interchanged will thus be the same
as the present one. A radical invariance property of systems of identical particles!
7 Quantum statistics and the spin degree of freedom of a particle are intimately connected as

relativistic quantum ¬eld theory demands that bosons have integer spin, whereas particles with
half-integer spin are fermions. This so-called spin-statistics connection seems in the present non-
relativistic quantum theory quite mysterious, i.e. unintelligible. It only gets its explanation in
the relativistic quantum theory, which we usually connect with high energy phenomena, where for
any particle relativity, through Lorentz invariance, requires the existence of an anti-particle of the
same mass and opposite charge (some neutral particles, such as the photon, are their own anti-
particle). Then, in fermion anti-fermion pair production the particles must be antisymmetric with
already existing particles as unitarity, i.e. conservation of probability, requires such a minus sign
[2]. Historically, the exclusion principle, which is a direct consequence of Fermi statistics, was
discovered by Pauli before the advent of relativistic quantum theory as a vehicle to explain the
periodic properties of the elements. Pauli was also the ¬rst to show that the spin-statistics relation
is a consequence of Lorentz invariance, causality and energy and norm positivity.
1.2. N-particle system 7


Any N -particle state |p1 , p2 , . . . , pN can be mapped into a state which is either
symmetric or antisymmetric with respect to interchange of any two particles. To
ˆ
obtain the symmetric state we simply apply the symmetrization operator S which
symmetrizes an N -particle state according to
1
ˆ
S |p1 , p2 , . . . , pN |pP 1 |pP 2 · · · |pP N
= (1.29)
N!
P

ˆ
and the antisymmetrization operator A antisymmetrizes according to
1
ˆ
A |p1 , p2 , . . . , pN (’1)ζP |pP 1 |pP 2 · · · |pP N .
= (1.30)
N!
P

The summations are over all permutations P of the particles. Permutations form
a group, and any permutation can be build by successive transpositions which only
permute a pair. In the case of antisymmetrization, each term appears with the sign
of the permutation in question
j’i
sign(P ) = . (1.31)
Pj ’ Pi
1¤i<j¤N

We have written this in terms of the number ζP which counts the number of trans-
positions needed to build the permutation P , since sign(P ) = (’1)ζP .
If the single-particle state labels in the N -particle state to be symmetrized on the
left in Eq. (1.29) are permuted, the same symmetrized state results, since if P can
be any of the N ! permutations, then P P for ¬xed permutation P will run through
ˆ ˆ
them all, S |pP 1 , pP 2 , . . . pP N = S |p1 , p2 , . . . , pN .

<< . .

. 2
( : 78)



. . >>