Pagina principale Concepts of theoretical solid state physics

# Concepts of theoretical solid state physics

,
Anno: 2001
Edizione: free web version, lecture notes
Lingua: english
Pagine: 412
File: PDF, 5.46 MB

## Most frequently terms

You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
1

### Band theory of solids: symmetry

Anno: 1991
Lingua: english
File: DJVU, 2.65 MB
2

### Concepts of theoretical solid state physics

Anno: 2001
Lingua: english
File: DJVU, 3.82 MB
```Con epts of Theoreti al Solid State
Physi s1

ψ0
gψ0
g(x)ψ0
λ

Alexander Altland and Ben Simons

Copyright (C) 2001: Permission is granted to anyone to make verbatim opies of this do ument provided that the opyright noti e and this permission noti e are preserved.
1

ii

Chapter 1
Colle tive Ex itations: From
Parti les to Fields
The goal of this se tion is to introdu e some fundamental on epts of lassi al eld theory
in the framework of a simple model of latti e dynami s. In doing so, we will be ome a quainted with the notion of a ontinuum a tion, elementary ex itations, olle tive modes,
symmetries and universality | on epts whi h will pervade the rest of the ourse.

One of the more remarkable fa ts about ondensed matter physi s is that phenomenology of phantasti omplexity is born out of a Hamiltonian that does not look parti ularly
impressive. Indeed, it is not in the least diÆ ult to write down mi ros opi \ ondensed
matter Hamiltonians" of reasonable generality. E.g. a prototypi al metal might be des ribed by

H = He + Hi + Hei ;
X p2
X
i +
He =
Vee (ri rj );
2
m
i
i6=j
X P2
X
I +
Hi =
Vii (RI RJ );
2
M
I
I 6=J
Hei =

X
iI

Vei (RI

ri );

(1.1)

where, ri (RI ) are the oordinates of i; I = 1; : : : ; N valen e ele trons (ion ores) and
He ; Hi , and Hei des ribe the dynami s of ele trons, ions and the intera tion of ele trons
and ions, respe tively. (For details of the notation, see Fig. 1.1.) Of ourse, the Hamiltonian (9.30) an be made \more realisti ", e.g. by remembering that ele trons and
ions arry spin, adding disorder or introdu ing host latti es with multi-atomi unit- ells,
however for developing our present line of thought the prototype H will do ne.
The fa t that an inno uously looking Hamiltonian like (9.30) is apable of generating
a vast panopti um of metalli phenomenology an be read in reverse order: one will
normally not be able to make theoreti al progress by approa hing the problem in an \ab
1

2

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

ri
RI

Figure 1.1: One-dimensional artoon of a (metalli ) solid. Positively harged ions lo ated
at positions RI are sourrounded by a ondu tion ele tron loud (ele tron oordinates
denoted by riP
). Both ele trons
P and ions are free to move, as des ribed by the kineti
energy terms p2i =(2m) and P2i =(2M ) of Eq. (9.30), respe tively. While, the motion
of the ions is massively onstrained by the latti e potential Vii (indi ated by solid lines) the
dynami s of the ele trons is a e ted by their mutual intera tion (Vee ) and the intera tion
with the ore ions (Vei ).
initio" manner, i.e. an approa h that treats all mi ros opi onstituents as equally relevant
degrees of freedom. But how then, an su essful analyti al approa hes be developed?
Mu h of the answer to this question lies in a number of basi prin iples inherent to generi
ondensed matter systems
1. Stru tural redu ibility of the problem. Whi h simply means that not all ompounds of the Hamiltonian (9.30) need to be treated simultaneously. E.g. when the
interest is foremostly in the vibrational motion of the ion latti e, the dynami s of
the ele tron system an often be negle ted or, at least, be treated in a simplisti
manner. Similarly, mu h of the dynami s of the ele trons is independent of the ion
latti e, et .
2. In the majority of ondensed matter appli ations one is not so mu h interested in the
full pro le of a given system but rather in its energeti ally low lying dynami s. This
is partly motivated by pra ti al aspe ts (In daily life, iron is normaly en outered
at room temperature and not at its melting point.), partly by the tenden y of
large systems to behave \universal" at low temperatures. Universality means that
systems di ering in mi ros opi detail (e.g. di erent types of intera tion potentials,
ion spe ies et .) exhibit identi al olle tive behaviour. As a physi ist, one will
normally seek for unifying prin iples in olle tive phenomena rather than to des ribe
the spe ialties of individual spe ies. Hen e the fundamental importan e of the
universality prini iple. However, universality is equally important in the pra ti e of
ondensed matter theory. It implies, e.g., that at low temperatures, details of the
fun tional form of mi ros opi intera tion potentials are of se ondary importan e,
i.e. that one may employ simple model Hamiltonians.

1.1.

CLASSICAL HARMONIC CHAIN: PHONONS

3

3. For most systems of interest, the number of degrees of freedom, N , is formidably
large, e.g. N = O(1023 ). However, ontrary to the rst impression, the magnitude
of this gure is rather an advantage. The reason is that in addressing ondensed
matter problems we may make use of the on epts of statisti s and that (pre isely
due to the largeness of N ) statisti al errors tend to be negligibly small1.
4. Finally, ondensed matter systems typi ally possess a number of intrinsi symmetries. E.g. our prototype Hamiltonian above is invariant under simultaneous translation and rotation of all oordinates whi h expresses the global Galilei invarian e of
the system (a ontinuous set of symmetries). Spin rotation invarian e ( ontinuous)
and time reversal invarian e (dis rete) are other examples of frequently en ountered
symmetries. The general importan e of symmetries needs no stressing: symmetries
entail onservation laws and onservation laws simplify any problem. Yet in ondensed matter physi s, symmetries are \even more" important. The point is that a
onserved observable is generally tied to an energeti ally low-lying ex itation. In the
universal low temperature regimes we will typi ally be interested in, it is pre isely
the dynami s of these low level ex itations that governs the gross behaviour of the
system. In subsequent se tion, the sequen e \symmetry ! onservation law ! lowlying ex itations" will be en ountered time and again. At any rate, identi ation
of the fundamental symmetries will typi ally be step no.1 in the analysis of a solid
state system.
Employing a hain of harmoni ally bound atoms as an example, we next attempt to
illustrate how su h prin iples an be applied to onstru t \e e tive low energy" models
of solid state systems. I.e. models that are universal, en apsulate the essential low energy
dynami s, and an be related to experimentally observable data. We will also observe that
the low energy dynami s2 of large systems naturally relates to on epts of eld theory;
In a way, this and the next few hapters represent a rst introdu tion to the use of eld
theoreti al methods in solid state physi s.

1.1 Classi al Harmoni Chain: Phonons
Coming ba k to our prototype Hamiltonian (9.30), let us fo us on dynami al behaviour
of the positively harged ore ions onstituting the host latti e. For the moment, let us
negle t the fa t that atoms are quantum obje ts, i.e. treat the ions as lassi al. To further
simplify the problem, we onsider an atomi hain rather than a generi d-dimensional
solid. I.e. the positions of the ions are given by a sequen e of points with average spa ing
a. Relying on the redu tion prin iple (1.) we next argue that to understand the behaviour
1 The importan e of this point is illustrated by the empiri al observation that the most resistive system

lasses in physi al s ien es are of medium (and not large) s ale. E.g. metalli lusters, medium size nu lei
or large atoms onsist of O(101 2 ) fundamental onstituents. Su h problems are well beyond the rea h
of few body quantum me hani s while not yet a essible to reliable statisti al modelling. Often the only
viable path to approa hing systems of this type is massive use of phenomenology.
2 In this ourse we will fo us on the dynami al behaviour of large systems, as opposed to stati stru tural
properties. E.g., we will not address questions related to the formation of de nite rystallographi
stru tures in solid state systems.

4

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

of the ions the dynami s of the ondu tion ele troni se tor of is of se ondary importan e,
i.e. we set He = Hei = 0.
At striktly zero temperature, the system of ions will be frozen out, i.e. the onedimensional ion oordinates RI  R I = Ia de ne a regularly spa ed array. Any deviation
from a perfe tly regular on guration has to be payed for by a prize in potential energy.
For low enough temperatures (prini iple 3.), this energy will be approximately quadrati
in the small deviation from the equilibrium position (the dashed line in Fig. 1.1.) The
redu ed low energy Hamiltonian3 of our system then reads

H=

N 
X
I =1

PI2 ks
+ (R
2M 2 I

RI +1

a)2



;

(1.2)

where the oeÆ ient ks determines the steepness of the latti e potential. Noti e that H
an be interpreted as the Hamiltonian of N parti les of mass M elasti ally onne ted by
springs with spring onstant ks (see Fig. 1.2).
φn

xn-1

(n-1)a

na

m

ks

(n+1)a

Figure 1.2: Toy model of a one-dimensional solid: A hain of elasti ally bound massive
point parti les. hange x \$ R

1.1.1 Lagrangian Formulation and Equations of Motion
What are the elementary low energy ex itations of th is system? To answer this question
we might, in prini ple, solve Hamiltons equations of motion; this is possible be ause H is
quadrati in all its oordinates. However, we must keep in mind that few of the problems
en ountered in general solid state physi s enjoy this property. Further, it seems unlikely
that the low energy dynami s of a ma ros opi ally large hain { whi h we know from our
experien e will be governed by large s ale wave type ex itations { is adequately des ribed
in terms of an \atomisti " language; the relevant degrees of freedom will be of di erent
type. Rather what we should do is make mu h more ex essive use from our basi prin iples
1.-4. Notably, we have so far neither payed attention to the intrinsi symmetry of the
problem nor to the fa t that N is large.
3
Sir William Rowan Hamilton 18051865; a mathemati ian redited with
the dis overy of quaternions, the rst
non- ommutative algebra to be studied. He also invented important new
methods in Me hani s.

1.1.

5

CLASSICAL HARMONIC CHAIN: PHONONS

Now omes a very imporant point: To redu e a mi ros opi ally formulated model
down to an e e tive low energy model, the Hamiltonian is often not a very onvenient
starting point. It is usually more eÆ ient to start out from an a tion. As usual, the
Lagrangian a tion4 of our system is de ned as

S=

Z T

L(R; R_ )dt;

0

where (R; R_ )  fRI ; R_ I g symboli ally represents the set of all oordinates and their time
derivatives. The Lagrangian L related to the Hamiltonian (1.2) is given by

L=T

U=

N 
X
I =1

PI2
2M

ks
(R
2 I

RI +1

a)2



;

(1.3)

where T and U stand for kineti and potential energy, respe tively.

Re apitulate the onne tion between Hamiltonian and Lagrangian in N parti le lassi al me hani s.
.

Exer ise.

For onvenien e we assume that our atomi hain has the topology of a ring, i.e. adopt
periodi boundary onditions RN +1 = R1 . Further, anti ipating that the e e t of latti e
vibrations on the solid is weak (i.e. long-range atomi order is maintained) we assume that
the deviation from the equilibrium position is small (jRI (t) R I j  a), i.e. the integrity
of the solid is maintained. With RI (t) = R I + I (t) (I +1 = 1 ) the Lagrangian (1.3)
simpli es to

L=

N 
X
M
I =1

2

_ 2I

ks
(
2 I +1

I )

2



:

To make further progress, we now use that we are not on erned with the behaviour
of our system on `atomi ' s ales. (In any ase, for su h purposes a modelling like the
one above would be mu h too primitive!) Rather, we are interested in experimentally
observable behaviour that manifests itself on ma ros opi length s ales (prin iple 2.).
For example, one might wish to study the spe i heat of the solid in the limit of innitely many atoms (or at least a ma ros opi ally large number, O(1023 )). Under these
onditions, mi ros opi models an usually be substantially simpli ed (prin iple 3.). In
parti ular it is often permissible to subje t a dis rete latti e model to a so- alled ontinuum limit, i.e. to negle t the dis reteness of the mi ros opi entities of the system and
to des ribe it in terms of e e tive ontinuum degrees of freedom.
4

Joseph-Louis Lagrange 1736-1813; Lagrange was a mathemati ian
who ex elled in all elds of analysis, number theory, analyti al, and
elestial me hani s. In 1788 he published Me anique analytique,
whi h summarised all the work done in the eld of me hani s sin e
the time of Newton and is notable for its use of the theory of
di erential equations. In it he transformed me hani s into a bran h
of mathemati al analysis.

6

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

φn

Continuum Limit φ(x)

(n-1)a na (n+1)a

Figure 1.3: Continuum limit of the harmoni hain. For larity, the (horizontal) distortion
of the point parti les has been plotted against the verti al.
In the present ase, taking a ontinuum limit amounts to des ribing the latti e u tuations I in terms of smooth fun tions of a ontinuous variable x (Fig. 1.3). Clearly su h a
des ription makes sense only if relative u tuations on atomi s ales are weak. (Otherwise
the smoothness ondition would be violated.) However, if this ondition is met { as it will
be for suÆ iently large values of the sti ness onstant ks { the ontinuum des ription is
mu h more powerful than the dis rete en oding in terms of the 've tor' fI g. All steps
we need to take to go from the Lagrangian to on rete physi al predi tions will be mu h
easier to formulate.
Introdu ing ontinuum degrees of freedom (x), and applying a rst order Taylor
expansion,5 we de ne

I !

a1=2 (x)

x=Ia

;

I +1

I !

a3=2 x (x)

x=Ia

;

N
X
I =1

!

1
a

Z L
0

dx;

where L = Na. Note that, as de ned, the fun tions (x; t) have dimensionality [Length℄1=2 .
Expressed in terms of the new degrees of freedom, the ontinuum limit of the Lagrangian
Z L
2
L[℄ =
dx L(; x; _ );
(1.4)
L(; x; _ ) = m2 _ 2 ks2a (x)2 ;
0
where the Lagrangian density L has dimensionality [energy℄=[length℄ and we have designated the parti le mass by the more ommon symbol m  M . Similarly, the lassi al
a tion assumes the ontinuum form

S [℄ =

Z

dt L[℄ =

Z

dt

Z L
0

dx L(; x; _ ):

(1.5)

We have thus su eeded in abandoning the N -point parti le des ription in favour of one
involving ontinuous degrees of freedom, a ( lassi al) eld. The dynami s of the latter
is spe i ed by the fun tionals L and S whi h represent the ontinuum generalisations
of the dis rete lassi al Lagrangian and a tion, respe tively.

. Info. The ontinuum variable  is our rst en ounter with a eld. Before pro eeding
with our example, let us pause to make some preliminary remarks on the general de nition of

5 Indeed, for reasons that will be ome lear, higher order ontributions to the Taylor expansion are

immaterial in the long-range ontinuum limit.

1.1.

7

CLASSICAL HARMONIC CHAIN: PHONONS

these obje t s. This will help to pla e the subsequent dis ussion of the atomi hain into a larger
ontext.
Mathemati ally speaking, a eld is a mapping
: M ! T;
z 7! (t);
from a ertain manifold M , often alled the 'base manifold', into a target or eld manifold T , see
Fig. 1.4. In our present example, M = [0; L℄  [0; T ℄  R2 is the produ t of intervals in spa e and
time, and T = R is the real numbers. In general appli ations, the base manifold will be a (subset
of) some d-dimensional spa e-like manifold R multiplied by a time-like interval: M  R  R.
(E.g. in our present example, R ' S1 is isomorphi to the unit ir le S1 .) Sometimes, espe ially
in problems relating to statisti al me hani s, M  R is just spa elike. However, we are always
free to assume that lo ally M is isomorphi to some subsset of d +1- or d-dimensional real ve tor
spa e. In ontrast, the target manifold an be just any (di erentiable) manifold. From real or
omplex numbers, over ve torspa es and groups to the fan iest obje ts of mathemati al physi s.
In applied eld theory, elds do not appear as nal obje ts but rather as input to fun tionals
(see Fig. 1.4) Mathemati ally, a fun tional S :  7! S [℄ 2 R is a mapping that that takes a
eld as its argument and maps it into the real numbers. The fun tional pro le S [℄ essentially
determines the hara ter of a eld theory. Noti e that the argument of a fun tional is ommonly
indi ated in angular bra kets [ : ℄.

T
φ
S

S[φ]

M
Figure 1.4: S hemati vizualization of a eld: a mapping  from a base manifold M into
a target spa e T (here the real numbers, but T an be more ompli ated). A fun tional
assigns to ea h  a real number S [℄. The grid embedded into M indi ates that elds in
ondensed matter physi s arise as ontinuum limits of dis rete mappings.

While these formulations may appear unne essarily abstra t, remembering the dry mathemati al ba kbone of the theory often helps to avoid onfusion. At any rate, it takes some
time and pra ti e to get used to the on ept of elds and fun tionals. Con eptual diÆ ulties

8

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

in handling these obje ts an be over ome by remembering that any eld in ondensed matter
physi s arises as the limit of a dis rete mapping. E.g. in our example, the eld (x) obtained as
ontinuum approximation of the dis rete ve tor fI g 2 RN ; the fun tional L[℄ is the ontinuum
limit of the fun tion L : RN ! R, et . While in pra ti al al ulations elds are usually easier to
handle than their dis rete analogs, it is sometimes easier to think about problems of eld theory
in a dis rete language. Within the disr rete pi ture, the mathemati al aparatus of eld theory
redu es to nite dimensional al ulus.
||||||||||||||{

Although Eq. (1.4) ontains the full information about the model, we have not yet
learned mu h about its a tual behaviour. To extra t on rete physi al information from
Eq. (1.4) we need to derive equations of motion. At rst sight, it may not be entirely
lear what is meant by the term `equations of motion' in the ontext of an in nite dimensional model: The equations of motion relevant for the present problem obtain as
generalization of the onventional Lagrange equations of N -parti le lassi al me hani s
to a model with in nitely many degrees of freedom. To derive these equations we need to
generalize Hamilton's extremal prin iple, i.e. the route from an a tion to the asso iated
equations of motion, to in nite dimensions. As a warmup, let us brie y re apitulate how
the extremal prin iple worked for a system with one degree of freedom:
Suppose the dynami s of a lassi al point parti leRwith oordinate x(t) is des ribed by
the lassi al Lagrangian L(x; x_ ), and a tion S [x℄ = dtL(x; x_ ). Hamilton's extremal
prin iple states that the on gurations x(t) that are a tually realised are those that
extremise the a tion ÆS [x℄ = 0. This means that for any smooth urve t 7! y (t),
1
lim
(S [x + y ℄ S [x℄) = 0:
(1.6)
!0 
I.e. to rst order in  the a tion has to remain invariant. Applying this ondition, one
nds that it is ful lled if and only if x solves Lagrange's equation of motion

d
( L) x L = 0:
dt x_
. Exer

ise.

(1.7)

Re apitulate the derivation of (1.7) from the lassi al a tion.

In Eq. (1.5) we are dealing with a system of in nitely many degrees of freedom (x; t).
Yet Hamilton's prin iple is general and we may see what happens if (1.5) is subje ted to
an extremal prin iple analogous to Eq. (1.6). To do so, we substitute

(x; t) ! (x; t) +  (x; t)
into Eq. (1.5) and demand vanishing of the rst order ontribution to an expansion in 
(see Fig. 1.5). When applied to the spe i Lagrangian (1.4), substituting the `varied'

S [ +  ℄ = S [℄ + 

Z

dt

Z L
0



dx m_ _



ks a2 x x  + O(2 ):

1.1.

9

CLASSICAL HARMONIC CHAIN: PHONONS

φ

φ(x,t)

x

εη(x,t)
T
t

L

Figure 1.5: S hemati diagram showing the variation of the eld asso iated with the
a tion fun tional. Noti e that the variation  is supposed to vanish on the boundaries
of the base M = [0; L℄  [0; T ℄.
Integrating by parts and demanding that the ontribution linear in  vanishes, one obtains
Z

dt

Z





dx m ks a2 x2   = 0:

(Noti e that the boundary terms vanish identi ally.) Now, sin e  was de ned to be an
arbitrary smooth fun tion, the integral above an only vanish if the term in parentheses
is globally vanishing. Thus the equation of motion takes the form of a wave equation

mt2



ksa2 x2  = 0:

(1.8)
p

The solutions of Eq. (1.8) have the general form + (x vt)+ (x+vt) where v = a ks =m,
and  are arbitrary smooth fun tions of the argument. From this we an dedu e that the
basi low energy elementary ex itations of our model are latti e vibrations propagating
as sound waves to the left or right at a onstant velo ity v (see Fig. 1.6)6 . The trivial
behaviour of our model is of ourse a dire t onsequen e of its simplisti de nition { no
to the general lassi al theory of latti e vibrations (see, e.g., Ref. [?℄). Finally, noti e
that the elementary ex itations of the hain have little in ommon with its \mi ros opi "
onstitutents (i.e. the atomi os illators.) They rather are olle tive ex itations, i.e.
ex itations omprising a ma ros opi ally large number of mi ros opi degrees of freedom.

The 'relevant' ex itations of a ondensed matter system an but need not be of
olle tive type. E.g. the intera ting ele tron gas, a system to be dis ussed in detail below,
supports mi ros opi ex itations { viz. harged quasi-parti les standing in 1-1 orresponden e
.

Info.

6 Striktly speaking the modeling of our system enfor es a periodi ity onstraint  (x + L) =  (x).

However, in the limit of a large system, this aspe t be omes inessential.

10

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

with the ele trons of the original mi ros opi system { and olle tive ex itations { plasmon modes
of large wavelength. The nature of the fundamental ex itations is often far from obvious from
the mi ros opi de nition of a model. In fa t, the mere identi ation of the relevant ex itations
often represents the most important step in the solution of a ondensed matter problem.
||||||||||||||{

x=-vt

Φ-

Φ+

x=vt

Figure 1.6: Visualisation of the fundamental left and right moving ex itations of the
lassi al harmoni hain.

1.1.2 Hamiltonian Formulation
An important hara teristi of any ex itation is its energy. How mu h energy is stored in
the sound waves of the harmoni hain? To address this question, we need to swit h ba k
to a Hamiltonian formulation. This is, again, done by generalizing standard manipulations
from point me hani s to the ontinuum. Remembering that for a Lagrangian, L(x; x_ ) of
a point parti le, p  x_ L is the momentum onjugate to the oordinate x, we onsider
the Lagrangian density and de ne



 L(; x ; _ )
:
 _

(1.9)

as the anoni al momentum asso iated to . (In the eld theory literature it is popular
to denote the momentum by Greek letters.) In ommon with  the momentum  is a ontinuum degree of freedom. At ea h spa e point it may take an independent value. Noti e
that  (x) is nothing other than the ontinuum generalization of the latti e momentum
PI of Eq. (1.2). (I.e. applied to PI , a ontinuum approximation like I ! (x) would
produ e  (x).)
The Hamiltonian density is then de ned as usual through Legendre transformation,


H(; x; ) = _ L(; x; _ )
from where the full Hamiltonian obtains as H =

RL
0


_ =_ (;)

;

(1.10)

Hdx.

Verify that the transition L ! H is a straightforward ontinuum generalization of the Legendre transformation of the N -parti le Lagrangian L(fI g; f_ I g).
. Exer

ise.

1.1.

11

CLASSICAL HARMONIC CHAIN: PHONONS

Having introdu ed a Hamiltonian, we are in a position to determine the energy of
sound waves. Appli ation of (1.9) and (1.10) to the Lagrangian of the atomi hain yields
 (x; t) = m_ (x; t) and

H [; ℄ =

Z





1 2 ksa2
dx
 +
   :
2m
2 x x

(1.11)

We next evaluate this fun tion on a sound wave, i.e. on a spe i solution of the equations
of motion. Considering for de niteness a right-moving ex itation, (x; t) = + (x vt),
we nd  (x; t) = mvx + (x vt) and

H [; ℄ = ks

a2

Z

dx[x + (x

vt)℄2

= ks

a2

Z

dx[x +(x)℄2 ;

i.e. a positive de nite time independent expression as one would expe t.
Noti e an interesting feature of the energy fun tional: in the limit of an in ntelty
shallow ex itation, x + ! 0, the energy vanishes. This brings the symmetry prin iple,
4.), not onsidered so far, onto the stage: The Hamiltonian of an atomi hain is invariant
under simultaneous translation of all atom oordinates by a xed in rement: I ! I + Æ ,
where Æ is onstant. This expresses the fa t that a global translation of the solid as a
whole does not a e t the internal energy. Now, the ground state of any spe i realization of the solid will be de ned through a stati array of atoms, ea h lo ated at a xed
oordinate RI = Ia ) I = 0. We say that the translational symmetry is \spontaneously
broken", i.e. the solid has to de ide where exa tly it wants to rest. However, spontaneous breakdown of a symmetry does not imply that the symmetry disappeared, on the
ontrary: In nite wavelength deviations from the pre-assigned ground state ome lose
to global translations of (ma ros opi ally large portions of) the solid and, therefore, ost
a vanishingly small amount of energy. This is the reason for the vanishing of the sound
wave energy in the limit x  ! 0. It is also our rst en ounter with the aforementioned
phenomenon that symmetries lead to the formation of soft, i.e. low-energy ex itations. A
mu h more systemati exposition of these onne tions will be given in hapter ** below.
To on lude our dis ussion of the lassi al harmoni hain, let us onsider the spe i
heat, i.e. a quantity dire tly a essible to experimental measurement. An rought estimate
of this quantity an readily be obtained from our initial harmoni Hamiltonian (1.2).
A ording to the prin iples of statisti al me hani s, the thermodynami energy density is
given by
R

Z
1 Rd e H H
1
H
u=
;
 ln
d e
=
L d e H
L
where

7

= 1=T , Z

R

 de

H

is the Boltzmann partition fun tion7 and the phase

Ludwig Boltzmann 1844-1906: physi ist whose greatest
a hievement was in the development of statisti al me hani s, whi h explains and predi ts how the properties of
atoms (su h as mass, harge, and stru ture) determine the
visible properties of matter (su h as vis osity, thermal ondu tivity, and di usion).

12

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

spa e volume element

d =

N
Y
I =1

dRI dPI :

The spe i heat then obtains as = T u. To determine the temperature dependen e of
this quantity, we use that upon res aling integration variables,

RI !

1=2 X

I

PI !

1=2 Y ;
I

the exponent H (R; P ) ! H (X; Y ) be omes independent of temperature. (Noti e that
this relies on the quadrati Q
dependen e of H on both R and P .) The integration measure
N
N
1 d 0 . Expressed in terms of the res aled
transforms as d !
I =1 dXI dYI 
variables, the energy density reads as

1
u=
 ln N K = T;
L
R

where  = NL is the density of atoms and we have used that the onstant K  d 0 e H (X;Y )
is independent of temperature. We thus nd a temperature independent spe i heat
= . Noti e that is fully universal, i.e. independent of the material onstants M and
ks determining H . (IN fa t, we ould have anti ipated this result from the equipartition
theorem of lassi al me hani s, i.e. the law that in a system with N degrees of freedom
the energy s ales like U = NT .)
How do these ndings ompare with experiment? Fig. 1.7 shows the spe i heat
of various insulatoring, semi ondu ing and metali solids8. For large temperatures, the
spe i heat approa hes a onstant value, in a ord with our so far analysis. However,
for lower temperatures, drasti deviations from = onst: o ur. The strong temperature
dependen e invalidates attempts to explain the deviations by ina urate modeling. It is
rather indi ative of a quantum phenomenon. Indeed, we have so far totally negle ted the
quantum nature of the atomi os illators. In the next hapter we will ure this deÆ ien y
and dis uss how the the e e tive low energy theory of the harmoni hain an be promoted
to a quantum eld theory. However, before pro eeding with the development of the theory
let us pause to introdu e a number of mathemati al on epts that surfa ed above in a
way that survives generalization to more interesting problems.

1.2 Fun tional Analysis and Variational Prin iples
Let us revisit the derivation of the equations of motion (1.8). Although straightforward,
the al ulation was neither eÆ ient, nor did it reveal general stru tures. In fa t, what
we did | expanding expli itly to rst order in the variational parameter  | had the
same status as evaluating derivatives by expli itly taking limits: f 0 (x) = lim!0 (f (x +
) f (x))=. Moreover, the derivation made expli it use of the parti ular form of the
Lagrangian, thereby being of limited use with regard to a general understanding of the
onstru tion s heme. Given the importan e atta hed to extremal prin iples in all of eld
8 In metals, the spe i

heat due to latti e vibrations ex eeds the spe
ele tron for temperatures larger than a few Kelvin.

heat of the free ondu tion

1.2.

FUNCTIONAL ANALYSIS AND VARIATIONAL PRINCIPLES

13

Figure 1.7: Spe i heat of various di erent solids. At large temperatures, the spe i
heat approa hes a onstant value, as predi ted by the analysis of the lassi al harmoni
model system. However, for small temperatures, deviations from = onst: are grave:
quantum e e ts.
theory, it is worthwhile investing some e ort in onstru ting a more eÆ ient s heme for
general variational analysis of ontinuum theories. In order to arry out this program
we rst need to introdu e a mathemati al tool of fun tional analysis, viz. the on ept of
fun tional di erentiation.
In working with fun tionals, one is often on erned with how a given fun tional behaves
under (small) variations of its argument fun tion. where h is an 'in nitely small' in rement
fun tion. In ordinary analysis, questions of this type are ommonly addressed by exploring
derivatives, i.e. what we need to do is generalize the on ept of a derivative to fun tionals.
This is a hieved by the following de nition: A fun tional F is alled di erentiable if

F [f + g ℄ F [f ℄ = DFf [g ℄ + O(2 );
where the di erential DFf is a linear fun tional (i.e. one with DFf [g1 + g2 ℄ = DFf [g1 ℄ +
DFf [g2 ℄),  a small parameter and g an arbitrary fun tion. The subs ript indi ates that
the di erential generally depends on the 'base argument' f . A fun tional F is said to be
stationary on f , i DFf = 0.
In prin iple, the de nition above answers our question for a stationarity ondition.
However, to make a tual use of the de nition, we still need to know how to ompute the
di erential DF and how to relate the di erentiability riterion to the on epts of ordinary
al ulus. In order to understand how answers to these questions an be systemati ally
found, it is helpful to temporarily return to a dis rete way of thinking, i.e. to interpret
the argument f of a fun tional F [f ℄ as the limit N ! 1 of a dis rete ve tor f =

14

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

ffn  f (xn); n = 1; : : : N g, where fxng denotes a dis retisation of the support of f ( .f.
Fig. 1.3  \$ f ). Prior to taking the ontinuum limit, N ! 1, f has the status of a

N -dimensional ve tor and F (f ) is a fun tion de ned over N -dimensional spa e. After the
ontinuum limit, f ! f be omes a fun tion itself and F (f ) ! F [f ℄ be omes a fun tional.
Now, within the dis rete pi ture it is lear how the variational behaviour of fun tions
is to be analysed. E.g. the ondition that for all  and all ve tors g, the linear expansion
of F (f + g) ought to vanish, is simply to say that the ordinary di erential, dFf , de ned
through
F (f + g) = F (f ) + dFf g + O(2 )
must be zero. In pra ti e, one often expresses onditions of this type in terms of a ertain
basis. E.g. in a Cartesian basis of N unit ve tors, e^n ; n = 1; : : : ; N ,
dFf g  hrFf ; gi;

P

where hg; f i  Nn=1 fn gn is the standard s alar produ t,
and fn F the partial derivative de ned through

rF jf = ffn F g the gradient

1
fn F (f )  lim
[F (f + e^n ) F (f )℄ :
!0 
From these identities, the di erential is identi ed as

dFf g =

X
n

fn F (f )gn :

(1.12)

(1.13)

Vanishing of the di erential amounts to vanishing of all partial derivatives fn F = 0.
Eqs. (1.12) and (1.13) an now be straightforwardly generalized by taking ontinuum
limits. In the ontinuum limit, the summation de ning the nite dimensional s alar
produ t be omes an integral,
N
X
n=1

fn g n !

Z

dxf (x)g (x):

The analog of the n-th unit ve tor is a Æ -distribution,

en ! Æy ;
where Æy (x) = Æ (x y ), as an be seen from the following orresponden e,

fn =!
f (y ) =!

X
Zm

fm (en )m !

dxf (x)Æy (x):

Here (en )m = Ænm stands for the m-th omponent of the n-th unit ve tor. The orresponden e (unit ve tor \$ Æ -distribution) is easy to memorize: While the omponents of
en vanish, save for the n-th omponent that equals unity, Æy is a fun tion that vanishes

1.2.

FUNCTIONAL ANALYSIS AND VARIATIONAL PRINCIPLES

15

everywhere, save for y where it equals in nity. That a unit omponent is repla ed by 'innity' has to do with the fa t that the support
of the Æ -distribution is in nitely narrow.
R
I.e. to obtain a unit-normalized integral Æy , the fun tion must be singular.
As a onsequen e of these identities, the ontinuum version of (1.13), i.e. the ontinuum di erential, is given by

dFf [g ℄ =

Z

dx

ÆF [f ℄
g (x);
Æf (x)

(1.14)

where the generalization of the partial derivative,

ÆF [f ℄
Æf (x)

1
lim
(F [f + Æx ℄ F [f ℄) :
!0 



(1.15)

is ommonly denoted by a urly Æ instead of  .
Eqs. (1.14) and (1.15) establish the on eptual onne tion between ordinary and
fun tional di erentiation. Noti e, that we have not yet learned how to pra ti ally al ulate
the di erential, i.e. evaluate expressions like (1.15) for on rete fun tionals. Nevertheless,
the identities above are very useful. They enable us to generalize more omplex derivative
operations of ordinary al ulus by straightforward trans ription. E.g. the generalisation
of the standard hain rule,

fn F (g(f )) =

ÆF [g [f ℄℄
=
Æf (x)

X

Z

m

gm F (g)jg=g(f ) fn gm (f )

dy

ÆF [g ℄
Æg (y )

Æg (y )[f ℄
:
g=g[f ℄ Æf (x)

(1.16)

Here g [f ℄ is the ontinuum generalization of an R m -valued fun tion, g : R n ! R m , i.e.
a fun tion whose omponents g (y )[f ℄ are fun tionals by themselves. Furthermore, given
some fun tional F [f ℄, we an Taylor expand it as

F [f ℄ = F [0℄ +
where

Z

ÆF [f ℄
dx1
f (x ) +
Æf (x1 ) 1

Z

dx1

Z

dx2

1 Æ 2 F [f ℄
f (x )f (x2 ) +    ;(1.17)
2 Æf (x2 )Æf (x1 ) 1

1
Æ 2 F [f ℄
(F [f + 1 Æx1 + 2 Æx2 ℄ F [f ℄)
=  lim
Æf (x2 )Æf (x1 ) 1;2 !0 1 2
generalizes a two-fold partial derivative. The validity of these identities an be made
plausible by applying the trans ription table 1.1 to the orresponding relations of standard
al ulus. To a tually verify the formulae, one has to take the ontinuum limit of ea h
step taken in the proof of the dis rete ounterparts. At any rate, experien e shows that
it takes some time to get used to the on ept of fun tional di erentiation. However, after
some pra ti e it will be ome lear that this operation is not only extremely useful but
also as easy to handle as onventional partial di erentiation.

16

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

entity
dis rete
ontinuum
argument
ve tor f
fun tion f
fun tion(al)
multi-dimensional fun tion F (f ) fun tional F [f ℄
di erential
dFf g
DFf [g ℄
^n
Cartesian basis
Æx
R
Pe
s alar produ t, h ; i
dxf
(x)g (x)
f
g
n n n
ÆF [f ℄
`partial derivative'
fn F (f )
Æf (x)
Table 1.1: Summary of basi de nitions of dis rete and ontinuum al ulus.
We nally address the question how to ompute the fun tional di erential for the
fun tionals ommonly en ountered in eld theory. What we will use is that in all but a
few ases, these fun tionals are of the stru ture,

S [℄ =

Z

dtL((t); _ (t));

where L is an ordinary fun tion. To establish onta t with the dis ussion of the previous
se tions, we have hanged the notation, F ! S , f ! , x ! t. The essential point is
that the information arried by the fun tional is en oded in an lo al fun tion. In general
appli ations both, the eld manifold and the base manifold will be higher dimensional.
However, the fun tional S will still be des ribed by a ertain fun tion L.
Owing to the spe i form of S , the fun tional derivative an be related to an ordinary
derivative of the fun tion L. To a hieve this, all what we have to do is evaluate the general
de nition (1.14) on the fun tional S :

S [ + ℄ S [℄ =
=

Z



Z

h

i

dt L( + ; _ + _) L(; _ ) =


L
L
 + _ _  + O(2 ) =
dt



Z



L
dt




d L
 + O(2 ):
_
dt 

(1.18)

Comparison with (1.14) identi es the fun tional derivative as

L
ÆS [℄
=
Æ(t) (t)

d L
:
dt  _ (t)

(1.19)

This equation answers our initial question of the stationarity of fun tionals: The fun tional
is stationary, if its di erential vanishes. A ording to (3.26), this is guaranteed if the rhs
of Eq. (1.19) vanishes for all t. Conversly, if the di erential vanishes, Eq. (3.26) must
vanish for all smooth fun tions g . This requires vanishing of the rhs of Eq. (1.19) for
all t. We thus on lude that vanishing of the rhs of (1.19) is a suÆ ient and ne essary
ondition for the stationarity of the fun tional S . Needless to say that

8t :

L
(t)

d L
=0
dt  _ (t)

(1.20)

1.2.

FUNCTIONAL ANALYSIS AND VARIATIONAL PRINCIPLES

17

is the familiar Euler-Lagrange equation.

. Exer ise. Consider a one-dimensional lassi al ontinuum system. The a tion fun tional is then given by S [℄ = R dxdtL(; x ; _ ), i.e. the Lagrangian density L assumes the role
of the fun tion L above. Show that the generalization of Eq. (1.20) produ es the Euler-Lagrange
equations
L
ÆS [℄
=
Æ(x; t) 
. Exer

ise.

L = L(; x2 ; _ ).

d L
dt  _

d L
= 0:
dx  (x )

(1.21)

Write down the equation of motion orresponding to the Lagrangian density

Eq. (1.21) represents the generalisation of Lagrange's equation of motion of point
me hani s to lassi al one-dimensional eld theory. Noti e that the equation is invariant
under ex hange of spa e and time oordinates. This is be ause, mathemati ally, we are
varying a fun tional that depends on elds  and rst derivatives  ;  = x; t, i.e.
mathemati s does not are about the fa t that in lassi al me hani s spa e and time play
fundamentally di erent roles. However, the stru tural symmetry of the equations is more
than mathemati al oin iden e! E.g. in relativisti theories, spa e and time appear in a
uni ed manner and a sensible variational equation must re e t this feature. To make this
stru ture more expli it, let us introdu e a two omponent ve tor x ;  = 0; 1 with x0 = t
and x1 = x. Eq. (1.21) then assumes the form
L
ÆS [℄  L
=

= 0:
(1.22)
Æ(x) 
 ( )
whi h not only expresses spa e-time symmetry but also is easy to remember.

Assume we were dealing with a eld  : Rd  R ! R; (x; t) 7! (x; t) de ned
over a d +1-dimensional base manifold and an asso iated fun tional L[; xi ; t ℄. De ning x
by x0 = t and x=1;2;:::d = x1;2;:::d show that Eq. (1.23) is the equation of motion, i.e that the
equation survives generalization to higher dimensional problems.
. Exer

ise.

Finally, we will frequently, en ounter problems where the eld manifold T is more
omplex than the real numbers. (For example, T  R N might be a subset of an N dimensional ve tor spa e.) Suppose, we had parameterized a eld  2 T through some
oordinates i , i = 1; : : : N . The Lagrangian L(i;  i ) would then be a fun tion of the
oeÆ ients of the elds and their derivatives. The variational prin iple demands that the
a tion be stationary under variation of all eld omponents individually. This produ es
N independent variational equations,

ÆS [℄  L
=
Æi (x) i



L
= 0;
 ( i )

i = 1; : : : ; N:

(1.23)

18

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

Eq. (1.23) expresses the variational prin iple in its most general form.

. Exer

ise.

omponents i(x).

Verify this equation by varying the a tion S [℄ = R L(i ;  i) wrt the eld

In the next se tion we will illustrate how the general variational prin iple (1.23) works
on a higher dimensional problem:

1.3 Maxwell Theory from Variational Prin iples
As a se ond example, let us onsider the ar hetype of lassi al eld theory, lassi al ele trodynami s. The idea not only is to exemplify the appli ation of ontinuum variational
prin iples on a problem we are all well a quainted with but also to illustrate the unifying
potential of the approa h: That problems as di erent as the low-lying vibrational modes
of a solid and ele trodynami s an be des ribed by almost identi al language indi ates
that we are dealing with a useful formalism.
Spe i ally, what we wish to explore is how the equations of motion of ele trodynami s,
the inhomogeneous Maxwell9 equtions,

rE

= ;
r  B t E = j;

(1.24)

an be obtained from variational prin iples. (For simpli ity, we restri t ourselves to a
va uum theory, i.e. E = D and B = H. Further, we have set the velo ity of light
to unity, = 1. Within the framework of the variational prin iple the homogeneous
equations,

r  E + t B
rB

= 0;
= 0;

(1.25)

are regarded as ab initio onstraints imposed on the 'degrees of freedom' E and B.)
What we need to des ribe Maxwell theory by variational prin iple is 1.) a set of suitable
'generalized oordinates' and 2.) an a tion. As for oordinates, the natural hoi e will be
the oeÆ ients of the ele tromagneti (EM) 4-potential, A = fA g, where A0 =  is the
s alar and A=1;2;3 = A1;2;3 (the negative of) the ve tor potential. The potential A is
un onstrained and uniquely determines the elds E and B through the standard equations
E = r t A and B = r A. (In fa t, the set of oordinates A is 'overly free', in the
9
James C. Maxwell 1831-1879;
Amongst many other a hievements in the elds of physi s and
mathemati s, he is redited with
the formulation of the theory of
ele tromagnetism.

1.3.

MAXWELL THEORY FROM VARIATIONAL PRINCIPLES

19

sense that gauge transformations A ! A +  , where is a fun tion, leave the physi al
elds invariant. We will omment on this point later on.) The onne tion between A and
the physi al elds an be expressed in a more symmetri way by introdu ing the EM eld
tensor,
2

6

F = fF g = 6
4

0
E1
E2
E3

E1
0
B3
B2

E2
B3
0
B1

3

E3
B2 7
7
B1 5 :
0

(1.26)

The relation between elds and potential now reads F =  A  A , where, as above,
0 = t and =1;2;3 = r1;2;3 .

Che k that this onne tion follows from the de nition of the ve tor potential.
To verify that the onstraint (1.25) is automati ally in luded in the de nition (1.26), ompute
the onstru t
 F +  F +  F ;
where () are arbitrary but di erent integers drawn from the set (0; 1; 2; 3). This produ es four di erent terms, identi ed as the lhs of (1.25). Evaluation of the same onstru t
on F   A  A produ es zero by the symmetry of the right hand side.
. Exer

ise.

As for the stru ture of the a tion S [A℄ we an pro eed in di erent ways. One option
would be to regard Maxwell's equations as fundamental, i.e. to postulate an a tion that
produ es these equations upon variation (similarly to the situation in lassi al me hani s,
were the a tion fun tional was designed so as to reprodu e Newton's equations.) However,
we an also be a little bit more ambitious and ask whether the stru ture of the a tion
an be motivated independently of Maxwells equations. In fa t, there is just one prin iple
in ele trodynami s equally 'fundamental' as Maxwells equations, symmetry: A theory of
ele tromagnetism must be Lorentz-invariant, i.e. invariant under relativisti oordinate
transformations.

. Info. Referring for a thorough dis ussion of relativisti theories to hapter * below, let
us brie y re apitulate the notion of Lorentz invarian e10 : Suppose we are given a 4-ve tor
X . A linear oordinate transformation X ! X0  T X is a Lorentz transformation if it
leaves the 4-metri
2
3
1
6
7
1
7
(1.27)
g = fg g = 6
4
1 5
1
10

Hendrik Antoon Lorentz 18531928; 1902 Nobel Laureate in
Physi s (with Pieter Zeeman) in
re ognition of the extraordinary
servi e they rendered by their
resear hes into the in uen e of

20

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

invariant: T T gT = g. To on isely formulate the invarian e properties of relativisti theories, is
is ommon to introdu e the notion of raised and lowered indi es. De ning X   g X , Lorentz
invarian e is expressed as X  X = X 0 X0 .
||||||||||||||{

Aided by the symmetry riterion, we an try to onje ture the stru ture of the a tion
from three basi presumptions, all independent from Maxwell's equations: The a tion
should be invariant under (a) Lorentz transformations, (b) gauge transformations, and
( ) simple. The most elementary hoi e ompatible with these onditions is

S [A℄ =

Z

d4 x ( 1 F F  + 2 A j  ) ;

(1.28)

Q

where the measure d4 x =  dx = dtdx1 dx2 dx3 , the 4- urrent j is de ned through
j0 =  and j=1;2;3 = j1;2;3 , and 1;2 are undetermined onstants. Up to quadrati order
in A, Eq. (1.28) de nes, in fa t, the only possible stru ture onsisetent with gauge and
Lorentz invarian e.

Using the ontinuity equation  j  = 0, verify that
the Aj - oupling is gauge
invariant. Hint: integrate by parts. Verify that a ontribution like R A A would not be gauge
invariant.
. Exer

ise.

Having de ned a trial a tion, we an apply the variational prin iple, i.e. Eq. (1.23), to
ompute equations of motion. In the present ontext, the role of the eld  is taken over
by the four- omponents of A. Variation of the a tion w.r.t. A obtains four equations of
motion,
L
L

= 0;
 = 0; : : : ; 3;
(1.29)
A
 ( A )
R
where the Lagrangian density is de ned through S = d4 xL.

Following the logi s of se tion 1.2, verify that, irrespe tive of the form of the
Lagrangian L(A ;  A ), the generalization of Eq. (1.23) to a ve torial eld,  ! A , is given
by (1.29).
. Exer

ise.

With our spe i form of L, it is straightforward to verify that A L = 2 j  and
 A L = 4 1 F  . Substituion of these building blo ks into the equations of motion
nally yields
4 1   F = 2 j :
Comparing this with the de nition of the eld tensor (1.26) and setting 12 = 41 we arrive
the Maxwell equations (1.24). and

S [A℄ =

Z

d4 x





1
F F  + A j  ;
4 

(1.30)

1.4.

SUMMARY AND OUTLOOK

21

as the nal result for the Lagrangian a tion of the ele tromagneti eld. Here we
have xed the overall multipli ative onstant 1 = 2 =4, not determined by the variational
prin iple, by requiring that the Hamiltonian density asso iated to L oin ides with the
known energy of the EM eld. (See the problem se tion.)

. Exer

ise.

Verify the statements made in the previous paragraph.

At rst sight, this result does not at all look spe ta ular; After all, Maxwell's equations
an be found on p1 of most text books on ele trodynami s. However, a se ond thought
shows that what we have a hieved is a tually quite remarkable. The only pie e of input
that went into the onstru tion was symmetry. This was enough to x the algebrai stru ture of Maxwell's equations unambigously. We have thus proven that Maxwell's equations
are relatvisti ally invariant, a fa t not obvious from the equations itself. Further we have
shown that (1.24) are the only equations of motion linear in the urrent/density distribution and onsistent with the invarian e prin iple. One might obje t that in addition to
symmetry we also imposed an ad ho 'simpli ity' riterion on the a tion S [A℄. However,
later on we will see that that was motivated by more than mere aestheti prin iples.
Finally, noti e that the symmetry oriented modelling that led to (1.28) stands exemplari for a popular onstru tion s heme in modern eld theory. The symmetry oriented
approa h stands omplementary to the \mi ros opi " formulation exempli ed in se tion
1.1. Roughly speaking, these are the two prin ipal approa hes to onstru ting e e tive
low energy eld theories:

. The mi ros opi route: Starting from a mi ros opi ally de ned system, one proje ts
out the degrees of freedom one believes relevant for the low energy dyanmi s. Ideally,
this 'belief' is ba ked up by a small expansion parameter stabilizing the mathemati al parts of the analysis.
pro: The method is rigorous and xes the resulting eld theory ompletely.
ontra: The mi ros opi route is slow and, for suÆ iently omplex systems, not even
viable.
. The phenomenologi al route: Given a physi al system one has already de ided
what its relevant degrees of freedom are. (Sometimes this has to be done on the
basis of mere phenomenologi al reasoning.) The stru ture of the e e ti low energy
a tion is then usually xed by symmetries.
pro: The method is fast and elegant.
ontra: It is less expli it than the mi ros opi approa h. Most importantly, it does
x the oeÆ ients of the di erent ontributions to the a tion.

1.4 Summary and Outlook
We have introdu ed some basi on epts of eld theoreti al modelling in ondensed matter
physi s. Starting from a mi ros opi model Hamiltonian, we have exempli ed how priniples of universality and symmetry an be applied to destill e e tive ontinuum theories
apturing the low energy ontent of the system. We have formulated su h theories in the

22

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

language of Langrangian and Hamiltonian ontinuum me hani s, respe tively, and shown
how variational prin iples an be applied to extra t on rete physi al information. Also,
we have seen that eld theory provides a unifying framework whereby analogies between
seemingly di erent physi al systems an be un overed.
In the next hapter we will dis uss how the formalism of lassi al eld theory is
promoted to the quantum level.

1.5.

23

PROBLEM SET

1.5 Problem Set
1.5.1 Questions on Colle tive Modes and Field Theories
Q1 In obtaining the spe trum of olle tive phonon ex itations for the latti e Lagrangian
(??), a ontinuum approximation was employed. However, sin e the degrees of
freedom are oupled linearly, the equations of motion an be solved expli itly, even
for the dis rete model. By onstru ting the equations of motion, identify the normal
modes of the system and obtain the exa t spe trum of ex itations. Identify the
limit in whi h the spe trum of the dis rete latti e model oin ides with that obtained
for the ontinuum approximation of the model. In what limit does the ontinuum
approximation fail and why?
mA

(n-1)a

mB

na

(n+1)a

Figure 1.8: Latti e with two atoms of mass mA and mB per unit ell.

Q2 In latti es with two atoms (of di erent mass mA and mB ) per unit ell (see g. 1.8)
the spe trum of elementary ex itations splits into an a ousti and opti bran h.
For this model, show that the latti e Lagrangian an be written as

L=

N 
X
mA
n=1

2

(_ (nA) )2

m
+ B (_ (nB) )2
2

ks  (A)

2 n+1

(nB)

2

ks (B)

2 n

2
(nA)



:

Applying the Euler-Lagrange equation, obtain the equations of motion. Taking the
boundary onditions to be periodi , and swit hing to Fourier representation, show
that the exa t (i.e. dis rete) spe trum an be obtained from the 2  2 se ular
equation for ea h k value
2
ika
A !k 2ks ks (1 + e )
det km(1
ika ) m ! 2 2k = 0:
s +e
B k
s
By nding an expression for the spe trum, obtain the asymptoti dependen e as
k ! 0. In this limit, des ribe qualitatively the symmetry of the normal modes.

Q3 Ele trodynami s an be des ribed by Maxwell's equations or, equivalently, by wave
type equations for the ve tor potential. In the Lorentz gauge, t  = r  A, these

(t2 ) = ;
(t2 )A = j;
where  = r  r is the three-dimensional Lapla e operator. Using relativistially
ovariant notation, the form of the equations an be ompressed further to
   A = j;
   A = 0:

24

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

Starting from the Lagrangian a tion,

S [A℄ =

Z

d4 x





1
F F  + A j  ;
4

obtain these equations by applying the variational prini ple. Compare the Lorentz
gauge representation of the a tion of the eld with the a tion of the elasti hain.
What are the di eren es/parallels?

Q4 Consider the ele tormagneti eld in the absen Re of matter, j = 0. Verify that the
total energy stored in the eld is given by H  d3 xH(x) where

H(x) = E2(x) + B2(x);
is the familiar expression for the EM energy density. Hint: Use the va uum form of
Maxwell's equations and that for an in nite system the energy is de ned only up to
surfa e terms.

1.5.

25

PROBLEM SET

A1 Applying the Euler-Lagrange equation dt (_ n L) n L = 0 to the dis rete Lagrangian of the latti e model we nd the N equations of motion whi h take the
form of a three-term di eren e equation,
mn = ks (n+1 2n + n 1 ) :

As in the ontinuum theory, the latter an be brought to diagonal form by turning
to the Fourier representation. Applying the Ansatz
1 X i(!k t kna)
n (t) = p
k
e
N k
where the dis rete quasi-momentum k = 2m=Na take values from the range m =
[ N=2; N=2℄ (i.e. the rst Brillouin Zone), we nd


m!k2



2ks (1

os(ka)) k = 0:

From this equation we obtain the dispersion relation
p

!k = 2 ks=mj sin(ka=2)j:
In the limit k ! 0, this result ollapses to the linear dispersion relation !k = v jkj
obtained from the ontinuum theory. This an be understood simply by omparing
the wavelength of the latti e vibration  = 2=k with the latti e spa ing a. When
  a, the relative displa ement of the atomi sites is small and the ontinuum
approximation is justi ed. When   a, the relative displa ement is large and the
ontinuum theory be omes inappli able.
In the two-dimensional generalisation the displa ement takes the form of a twoomponent ve tor n . In this ase, the dis rete Lagrangian assumes the form

L=

X

"

n

#

1 _2 X 1
m +
k (
2 n i=x;y 2 s n+^ei

n )2 :

In this ase, the Euler-Lagrange equations lead to the di eren e equation

mn =

X

i=x;y

ks (n+^ei

2n + n

e^i ) ;

p

with the soloution !k = 2 ks =m(sin2 (kxa) + sin2 (ky a))1=2 . In the low-energy limit,
the spe trum redu es to the relativisti form !k = v jkj.

A2 Applying the Euler-Lagrange equations we obtain the 2  N oupled equations of
motion





mA (nA) = ks (nB)

2(nA)

mB n(B) = ks (nA+1)

2(nB) + n(A) :



+ (nB)1



;

26

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

p

P i(! t kna) (A=B )
k
k , we
ke



(kA) = 0:
ks(1 + eika )
mB !k2 2ks
(kB)

Applying the Ansatz (nA=B) = (1= N )


obtain

mA !k2 2ks
ks(1 + e ika )
Diagonalizing the 2  2 matrix we obtain the se ular equation shown in the question
and from whi h we obtain the dispersion relation (see Fig. 1.9)
!k()

"



4mA mB
sin2 (ka=2)
2
(mA + mB )

= !0 1  1

1=2 #1=2

:

p

where !0 = ks = where  = 1=(mA1 + mB1 ) denotes the redu ed mass. It is
instru tive to note how the standard phonon dispersion relation is re overed when
the masses are set equal.

ωk
mB =mA

B.Z.
Optic

Acoustic
−π/a

π/a

k

Figure 1.9: Spe trum of the two-atom dis rete linear hain. Note that when mA = mB
we re over the spe trum of the single atom hain with period a=2. For mA 6= mB , a gap
opens at the Brillounin zone boundary. The lower energy band is known as the a ousti
bran h where atoms in ea h unit ell move in phase. The higher energy opti bran h
involves atoms in ea h ell moving in antiphase.
An expansion in the limit k ! 0 yields

8
p 
mA mB
>
2
>
+ O(k4 )
< 2 1
2 (ka)
8(
m
+
m
)
()
A
B
!k ! !0 r m m
1
>
A B
>
:
j
kaj + O(k3 )
2 (mA + mB )
from whi h we dedu e that the lower bran h des ribes a ousti phonons with a linear
dispersion relation, while the opti phonons are massive with a quadrati spe trum.

A3 Using that F =  A  A and integrating by parts the a tion assumes the form
S [A℄ =

Z

d4 x



1
A [   A
2  



  A ℄ + j



A



:

1.5.

27

PROBLEM SET

Due to the Lorentz gauge ondition, the se ond ontribution in the angular bra kets
vanishes and we obtain

S [A℄ =

Z

d4 x





1
 A   A + j A ;
2

where we have again integrated by parts. Appli ation of the general variational
equation (1.23) nally obtains the wave equation.

A4 Following the anoni al pres ription, we onsider the Lagrangian density,

L=

1
1
F F  = ( A  A )(  A
4
4
3
1X
( A  A )( A  A )
=
2 i=1 0 i i 0 0 i i 0

  A ) =
3
1X
( A
4 i;j =1 i j

j Ai )(i Aj

j Ai );

where we have lowered all indi es on a ount of introdu ing minus signs in the rst
group of terms. We next determine the omponents of the anon ial momentum
through  = 0 A L:

0 = 0;
i = 0 Ai

i A0 = Ei :

Using that i Aj j Ai is a omponent of the magneti eld, the Hamiltonian density
an now be written as
H~ =  0 A L = 21 (2E  0 A E2 + B2) =1)
2) 1
1) 1
= (2E  r + E2 + B2 ) = (2[E  r + r  E℄ + E2 + B2 )
2
2
3) 1
= (2r  (E) + E2 + B2 );
2
where 1) is based on addition and subtra tion of a a term 2E r, 2) on r E = 0 and
3) on the identity r (af ) = r af + a rf (valid for general ve tor (s alar) fun tions
a (f)). Substitution of this expression into the de nition of the Hamiltonian yields
1
H=
2

Z

d3 x

2r 


(E) + E2 + B2

1
=
2

Z



d3 x E2 + B2 ;

where we have used that the ontribution r  (E) is a surfa e term that vanishes
upon integration by parts.

28

CHAPTER 1.

COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS

Chapter 2
From Classi al to Quantum Fields
The on ept of eld quantization is introdu ed. Again employing the atomi hain and
the ele tromagneti eld as examples, we will explore how ex itations in the ontinuum
Hilbert spa e a quire the meaning of quantum 'parti les'.

In the previous hapter we had seen that at low temperatures the ex itation pro le of
the lassi al atomi hain di ers drasti ally from what is observed in experiment. Generally, in ondensed matter physi s, low energy phenonmena with pronoun ed temperature
sensitivity are indi ative of a quantum me hanism at work. To introdu e and exempify
a general pro edure whereby quantum me hani s an be in orportated into ontinuum
models, we next onsider the low energy physi s of the

2.1 Quantum Chain
The rst question to ask is a on eptual one: in whi h way an a model like Eq. (1.4)
be quantised in general? As a matter of fa t there exists a standard pro edure of quantising ontinuum theories whi h losely resembles the quantisation of Hamiltonian point
me hani s: Consider the de ning equation (1.9) and (1.10) for the anoni al momentum
and the Hamiltonian, respe tively. Classi ally, the momentum  (x) and the oordinate
(x) are anoni ally onjugate variables: f (x); (x0 )g = Æ (x x0 ) where f ; g is the Poisson bra ket and the Æ -fun tion arises through ontinuum generalization of the dis rete
identity fPI ; RI 0 g = ÆII 0 , I; I 0 = 1; : : : N . The theory is quantized by generalization of the
anoni al quantization prodedure for the dis rete pair of onjugate oordinates (RI ; PI )
to the ontinuum: (i) promote (x) and  (x) to operators:  7! ^,  7! ^ , and (ii)
generalize the anoni al ommutation relation [PI ; RI 0 ℄ = i~ÆII 0 to1
[^ (x); ^(x0 )℄ = i~Æ (x x0 ) :

(2.1)

1 Note that the dimensionality of both the quantum and lassi al ontinuum elds is ompatible with
the dimensionality of the Dira Æ-fun tion, [Æ(x x0 )℄ = [Length℄ 1 , i.e. [(x)℄ = [I ℄  [Length℄ 1=2 and
similarly for .

29

30

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

Operator-valued fun tions like ^ and ^ are generally referred to as quantum elds. For
larity, the relevant relations between anoni ally onjugate lassi al and quantum elds
are summarized in Tab. 2.1.
lassi al
fPI ; RI 0 g = ÆII 0

quantum
dis rete
[PI ; RI 0 ℄ = i~ÆII 0
ontinuum f (x); (x0 )g = Æ (x x0 ) [^ (x); ^(x0 )℄ = i~Æ (x x0 )
Table 2.1: Relations between dis rete and ontinuum anoni ally onjugate variables/operators

By introdu ing quantum elds, we have left the on eptual framework laid on
page 6: Being operator valued, the quantized eld no longer represents a mapping into an
ordinary di erentiable manifold2. It is thus legitimate to ask why we bothered to give a lengthy
exposition of elds as 'ordinary' fun tions. The reason is that in the not too distant future,
after the framework of eld fun tional integration has been introdu ed, we will be ba k on the
omfortable ground of the de nition of page 6.
.

Info.

||||||||||||||{

Employing these de nitions, the lassi al Hamilonian density (1.10) be omes the quantum operator
2

H^ (;^ ^ ) = 21m ^ 2 + ks2a (x^)2:

(2.2)

The Hamiltonian above represents a quantum eld theoreti al formulation of the problem
but not yet a solution. In fa t, the development of a spe trum of methods for the analysis
of quantum eld theoreti al models will represent a major part of this text. At this point
the obje tive is merely to exemplify how physi al information an be extra ted from models like (2.2). As a word of pre aution, let us mention that the following manipulations,
while mathemati ally not diÆ ult, are on eptually deep. To disentangle di erent aspe ts
of the problem, we will rst on entrate on plain operational aspe ts. 'What has really
happened' will then be dis ussed in se tion * below.
As with any fun tion, operator valued fun tions an be represented in a variety of
di erent ways. In parti ular they an be subje ted to a Fourier transform,


^k
^k



1
L1=2

Z L
0

P

dx efikx



^(x) ;
^ (x)





^(x) = 1 X efikx ^k ;
^ (x) L1=2 k
^k

(2.3)

where k represents the sum over all Fourier oeÆ ients indexed by quantised momenta
k = 2m=L; m 2 Z . (Do not onfuse the momenta k with the `operator momentum'
^ !) Note that the real lassi al eld (x) quantises to an Hermitian quantum eld ^(x)
2 At least if we ignore the mathemati al subtlety that a linear operator an be interpreted as element

of a ertain manifold, too.

2.1.

31

QUANTUM CHAIN

implying that ^k = ^y k (and similarly for ^k ). The Fourier representation of the anoni al
[^k ; ^k0 ℄ = i~Ækk0
(2.4)

. Exer

ise.

Verify this identity.

Figure 2.1: S hemati visualisation of the os illator mode k = 8=L of the harmoni
hain. Arrows indi ate the distortion of individual atoms (for larity plotted in the verti al
dire tion.)
When expressed in the Fourier representation, making use of the identity
Æk+k0;0
z Z
}|
{
Z
X
X
X
1
0
dx ( ^)2 = ( ik^k )( ik0 ^k0 )
dxe i(k+k )x =
k2 ^k ^ k =
k2 j^k j2
L
k
k
k;k0
R

together with a similar relation for dx ^ 2 , the Hamiltonian assumes the near diagonal
form


X 1
m!k2 ^ ^
^
(2.5)
;
^ ^ +
H=
2m k k
2 k k
k
q

where !k = v jkj and v = kms a denotes the lassi al sound velo ity. In this form, the
Hamiltonian an be identi ed as nothing but a superposition of independent harmoni
os illators.3 This result is a tually not diÆ ult to understand (see Fig. 2.1): Classi ally,
3 The only di eren e to the anon ial form of an os illator Hamiltonian H^ = p^2 + m!2 x^2 is the presen e
2m 2
of the sub-indi es k and k. (Whi h is a onsequen e of ^yk = ^ k .) As we will show momentarily, this

di eren e is ompletely inessential.

32

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

the system supports a dis rete set of wave ex itations, ea h indexed by a wave number
k = 2m=L. Within the quantum pi ture, ea h of these ex itations is des ribed by an
os illator Hamilton operator with k-dependent frequen y. (However, it is important not to
onfuse the atomi onstituents, also os illators albeit oupled ones, with the independent
olle tive os illator modes des ribed by H^ .)
The des ription above, albeit perfe tly valid still su ers from a deÆ ien y: What
we are doing, see Fig. 2.1, amounts to expli itly des ribing the e e tive low energy
ex itations of the system (the waves) in terms of their mi ros opi onstituents (the
atoms). Indeed the di erent ontributions to H^ keeps tra k of details of the mi ros opi
os illator dynami s of individual k-modes. However, it would be mu h more desirable
to develop a pi ture where the relevant ex itations of the system, the waves, appear as
fundamental units, without expli it a ount of underlying mi ros opi details. (Like, e.g.
in hydrodynami s information is en oded in terms of olle tive density variables rather
than through individual mole ules.) As a preparation to the onstru tion of this improved
formulation of the system let us temporarily fo us on a single os illator mode.

2.1.1 Harmoni Os illator Revisited
Consider a standard harmoni os illator (HO) Hamiltonian de ned through
1
m! 2 2
H^ = p^2 +
x^ :
2m
2


The few rst energy levels n = ! n + 21 and the asso iated Hermite-polynomial eigenfun tions are displayed s hemati ally in Fig. 2.2. (To simplify the notation we hen eforth
set ~ = 1.)

ω

Figure 2.2: S hemati diagram showing the low lying energy levels/states of the harmoni
os illator.
In quantum me hani s, the HO has, of ourse, the status of a single-parti le problem.
However, the fa t that the energy levels are equi-distant suggests an alternative interpretation: Think of a given energy state n as an a umulation of n elementary entities, or
quasi-parti les, ea h having energy ! . What an be said about the features of these
new obje ts? First, they are stru tureless, i.e. the only 'quantum number' identifying the
quasiparti les is their energy ! . (Otherwise n-parti le states formed of the quasi-parti les
would not be equi-distant.) This implies that the quasi-parti les must be bosons. (The
same state ! an be o upied by more than one parti le, see Fig. 2.2.)

2.1.

33

QUANTUM CHAIN

This idea an be formulated in quantitative terms by employing the formalism of
so- alled ladder-operators: De ne a pair of Hermitian adjoint operators through

a^ 

r





m!
i
x^ +
p^ ;
2
m!

a^y 

r



m!
x^
2



i
p^ :
m!

(Up to a fa tor of i,) the transformation (^x; p^) ! (^a; a^y ), is anoni al, i.e. the new
operators obey the anoni al ommutation relation
[^a; a^y ℄ = 1:

(2.6)

More importantly, the a-representation of the Hamiltonian is very simple, viz.




1
H^ = ! a^y a^ + ;
2

(2.7)

as an be he ked by dire t substitution. Suppose now, we had been given a zero eigenvalue state j0i of the operator a^: a^j0i = 0. As a dire t onsequen e, H^ j0i = !2 j0i, i.e.
j0i is identi ed as the ground state of the os illator4. The omplete hierar hy of higher
energy states an now generated by setting
jni  (n!)11=2 a^ynj0i:

. Exer

ise.

and hnjni = 1.

Using the anoni al ommutation relation, verify that H^ jni = !(n + 1=2)jni

So far, what we have a hieved is onstru ting yet another way of solving the HOproblem. However, the 'real' advantage of the a-representation is that it naturally a ords
a many parti le interpretation: Temporarily forgetting about the original de nition of
the os illator, let us de lare j0i to represent a 'va uum' state, i.e. a state with zero
parti les present. Next, imagine that a^y j0i was a state with a single featureless parti le
(the operator a^y does not arry any quantum number labels) of energy ! . Similarly, a^yn j0i
is onsidered as a many body state with n of these parti les, i.e. within the new pi ture,
a^y is an operator that reates parti les. The total energy of these states is given by ! 
(o upation number). Indeed, it is straightforward to verify that a^y a^jni = njni, i.e. the
Hamiltonian basi ally ounts the number of parti les. The new interpretation, while at
rst sight unfamiliarly looking, is internally onsistent. In parti ular it does what we
had asked for above, i.e. interpreting the ex ited states of the HO as an superposition of
independent stru tureless entities.
The representation above illustrates the possibility to think about individual quantum
problems in omplementary pi tures. This prin iple nds innumerable appli ations in
modern ondensed matter physi s. To get used to it one has to realize that interpreting

4 ... as an be veri ed by expli it onstru tion: Going into a real spa e representation one solves
q
2 e x2 m!=2 whi h is the well known ground state wave fun tion of the
[x + x =(m!)℄hxj0i to hxj0i = m!
os illator.

34

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

a give system in di erent dire tions is by no means hereti but, rather, stands in the best
spirit of quantum me hani s. Indeed, it is one of the prime prin iples of quantum theories
that there is no su h thing like 'the real system' underneath the surfa e of phenomenology.
The only thing that matters is observable phenomena. For example, we will see later on
that the ' ti ious' quasi-parti le states of os illator systems behave like 'real' parti les, i.e.
they have dynami s, they an intera t, be dete ted experimentally et . From a quantum
point of view there is a tually no fundamental di eren e between these obje ts and 'real'
parti les.

2.1.2 Quasi-Parti le Interpretation of the Quantum Chain
Turning ba k to the os illator hain, we transform the Hamiltonian (2.5) to a form analogous to (2.7). This is a hieved by de ning the ladder operators5

a^k 

r





1
m!k ^
k + i
^
2
m!k

k

a^y

;

k



r



m!k ^

2



k

1
i
^ :
m!k k

With this de nition, applying the ommutation relations (2.4), one nds that the ladder
ε

ω3
ω2

ω1
k1

k2

k3

k

Figure 2.3: S hemati diagram visualizing an ex ited state of the hain. The number of
reated quasi-parti les de reases with in reasing energy !k .
operators obey ommutation relations generalizing (2.6):
[ak ; ayk0 ℄ = Ækk0 ;

[ak ; ak0 ℄ = [ayk ; ayk0 ℄ = 0:

(2.8)

Expressing the operators (^k ; ^k ) in terms of (^ak ; a^yk ) it is now straightforward to bring
the Hamiltonian into the quasi-parti le os illator form

H^ =

X
k





1
y
:
~!k ak ak +
2

(2.9)

Equations (2.9) and (2.8) represent the nal result of our analysis. The Hamiltonian H^
takes the form of a sum of harmoni os illators with hara teristi frequen ies !k . Noti e
5 As for the onsisten y of these de nitions, re all that ^y = ^
k

and ^ky = ^ k . Under these onditions
the se ond of the de nitions below indeed follows from the rst upon taking the Hermitian adjoint.
k

2.2.

QUANTUM ELECTRODYNAMICS

35

that !k =! 0 in the limit of u tuations with long wavelength, k ! 0. Ex itations with
this property are generally alled massless ex itations.
An ex ited state of the system is represented through a set (n1 ; n2 ; : : :) of nk quasiparti les with energy !k . Physi ally, the quasi-parti les of the harmoni hain are to
be identi ed with the phonon modes of the solid. A plot ot 'real' phonon ex itation
energies is shown in Fig. 2.4. Noti e that at low momenta, !k  jkj in agreement with
our simplisti model and in spite of the fa t that the spe trum was re orded for a threedimensional solid with non-trivial unit ell (universality!). While linear dispersion was a
feature already of the lassi al sound waves, the low temperature spe i heat behaves
altogether non- lassi al: It is left as an (answered) ex er ise to verify that the quantum
nature of the phonons resolves the problem with the low temperature pro le of the spe i
heat dis ussed in se tion 1.1.2. For further dis ussion of phonon modes in atomi latti es
we refer to Kittel, Chapter 2.

Figure 2.4: Typi al phonon spe tra of a rystal with a BCC latti e. The data is sensitive
to both longitudinal (LA) and transverse (TA) a ousti phonons. Noti e that for small
momenta, the dispersion is linear.

2.2 Quantum Ele trodynami s
The generality of the pro edure outlined above suggests that the quantization of the EM
eld a tion (1.30) pro eeds in analogy to the dis ussion of the phonon system. However,
there are are a number pra ti al di eren es whi h make quantization of the EM eld a
mu h harder (but also more interesting!) experien e: First, the ve torial hara ter of the
ve tor potential, in ombination with the imperative ondition of relativisti ovarian e,
gives the problem a non-trivial internal geometry. Closely related, the gauge freedom
of the ve tor potential introdu es redundant degrees of freedom whose removal on the
quantum level is not easily a hieved. E.g. quantization in a setting where only physi al

36

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

degrees of freedom { i.e. the two polarization dire tions of the transverse photon eld {
are kept is te hni ally umbersome, the reason being that the relevant gauge ondition is
not relativisti ally ovariant. In ontrast, a manifestly ovariant s heme, while te hni ally
more onvenient, introdu es spurious 'ghost degrees of freedom' whi h are diÆ ult to get
rid of. In order to not get aught up in a dis ussion of these problems we will not dis uss
the problem of EM eld quantization in full detail6.
On the other hand, the quantum aspe ts of the photon eld play a mu h too important
role in various areas of ondensed matter physi s to drop the problem altogether. We will
therefore aim at an intermediate exposition, largely insensitive to the problems outlined
above but suÆ iently general to illustrate the main prini iples.

2.2.1 Wave Guide Quantization

R

Consider the Lagrangian of the matter-free EM eld, L = 14 d3 xF F  . As a rst
step towards quantization of this system we x a gauge. E.g., in the absen e of harge
a parti ularly onvenient hoi e is the Coulomb gauge, r  A = 0; plus vanishing of the
s alar omponent,  = 0. Keep in mind that on e a gauge has been set we annot expe t
further results to display 'gauge invarian e'.
Using the gauge onditions and integrating by parts one veri es that the Lagrangian
assumes the form
Z
1
L=
d3 x [t A  t A + A  A℄ :
(2.10)
2
In analogy to our dis ussion of the atomi hain, one would next pro eed to 'de ouple' the
theory by expanding in terms of eigenfun tions of the Lapla e operator. The di eren e to
our previous dis ussion is that we are dealing (i) with the full three-dimensional Lapla ian
(instead of a simple se ond derivative) a ting on (ii) the ve torial quantity A whi h is (iii)
subje t to the onstraint r  A = 0. It is these aspe ts whi h lead to the ompli ations
outlined above.
We an navigate around these diÆ ulties by onsidering problems where the system
geometry redu es the omplexity of the eigenvlue problem. This restri tion is less arti ial than it might appear. E.g. in anisotropi ele tromagneti wave guides, systems
of undisputable te hnologi al relevan e, the solutions of the eigenvalue equation an be
formulated as7
Rk (x) = k Rk (x);

(2.11)

where k 2 R is a one-dimensional
index parameter and the ve tor-valued fun tions Rk
R
are real and ortho-normalized, Rk  R0k = Ækk0 . The dependen e of the eigenvalues k on
k depends on details of the geometry (see Eq. (2.14) below) and needs not be spe i ed
for the moment.

ex ellent introdu tions, e.g. the book by Ryder.
7 More pre isely, one should say that (2.11) de nes the set of eigenfun tions relevant for the low energy
dynami s of the wave guide. More omplex eigenfun tions of the Lapla e operator exist but they arry
mu h higher energy.

2.2.

37

QUANTUM ELECTRODYNAMICS

. Info. An ele trodynami wave guide is a quasi one-dimensional avity with metalli
boundaries ( f. Fig. 2.5.) The great pra ti al advantage of wave guides is that they are very
good at on ning EM waves. At large frequen ies where the wavelengths are of order meters
or less radiation loss in onventional ondu tors are high. In these frequen y domains, hollow
ondu tors provide the only pra ti al way of transmitting radiation.
EM eld propagation inside a wave guide is onstrained by boundary onditions. E.g.,
assuming the walls of the system to be perfe tly ondu ting,
Ek (xb ) = 0;
(2.12)
(2.13)
B? (xb ) = 0;
where xb is a point at the system boundary and Ek (B?) is the parallel (perpendi ular) omponent of the ele tri (magneti ) eld.
Ly
Ly

z
y
x

Figure 2.5: Re tangular EM wave guide. The stru ture of the eigenmodes of the EM eld
is determined by boundary onditions at the walls of the avity.

For on reteness, and turning ba k to our problem of eld quantization, let us onsider a
avity with uniform re tangular ross se tion Ly  Lz . To onveniently represent the Lagrangian
of the system, we wish to express the ve tor potential in terms of eigenfun tions Rk that
are onsistent with the boundary onditions (2.12). A omplete set of fun tions ful lling this
ondition is given by
0
1
1 os(kx x)sin(ky y )sin(kz z )
Rk = Nk  2 sin(kx x) os(ky y)sin(kz z ) A :
3 sin(kx x)sin(ky y ) os(kz z )
Here, ki = ni=Li, ni 2 N ; i = x; y; z, Nk is a fa tor normalizing Rk to unity, and the oeÆ ients
i are subje t to the ondition 1 kx + 2 ky + 3 kz = 0. Indeed, it is straightforward to verify
that a general superposition of the type A(x; t)  Pk k(t)Rk (x), k(t) 2 R, is divergen eless,
and generates an EM eld ompatible with (2.12). Substitution of Rk into (2.11) identi es the
eigenvalues to
k = (kx2 + ky2 + kz2 ):
In the physi s and ele tro-engineering literature, eigenfun tions of the Lapla e operator in a
quasi one-dimensional geometry are ommonly denoted as modes. As we will see shortly, the
energy of a mode (i.e. the Hamiltonian evaluated on a spe i mode on guration) grows
with jk j; In ases where one is interested in the low energy dynami s of the EM eld, only
on gurations with small jkj are relevant. E.g. let us onsider a massively anisotropi wave
guide with Lz < Ly  Lx. In this ase the modes with smallest jk j are those with kz = 0,
ky = =Ly and kx  k  Lz;y1 . (Why is it not possible to set both ky and kz to zero?) With this
hoi e,
k =

k2 +




Ly

2 !

(2.14)

38

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

and a s alar index k suÆ es to label both eigenvalues and eigenfun tions Rk . A artoon of
the spatial stru ture the fun tions Rk is shown in Fig. 2.5. The dynami al properties of these
on gurations will be dis ussed in the text.
||||||||||||||{

Turning ba k to the problem posed by (2.10) and (2.11), we expand the ve tor potential
in terms of eigenfun tions Rk ,

A(x; t) =

X
k

k (t)Rk (x);

where the sum runs over all allowed values of the index parameter k. (E.g. in a wave
guide, k 2 L n, n 2 N where L is the length of the guide.) Substituting this expansion
into (2.10) and using the normalization properties of the Rk , we obtain

1X 2
L=
_ k + k k2 ;
2 k
i.e. a de oupled representation where the system is des ribed in terms of independent
dynami al systems with oordinates k . From this point on, quantization pro eeds along
the lines of the standard algorithm:
De ne momenta through k =  _ k L = _ k . This
P
produ es the Hamiltonian H = k (k k (k =2) k k ). We next quantize by promoting
^k to operators and de laring [^k ; ^ k0 ℄ = iÆkk0 . The quantum
k ! ^ k and k ! 
Hamilton operator, again of harmoni os illator type, then reads

2
X 1
m!
k
^k ^k +
^k ^k ;
H^ =
2
m
2
k
where the 'mass' parameter m = 1 and !k = jk j. Following the same logi s as in se tion
2.1.2, we de ne ladder operators
r





r



m!k
m!k
i
ak 
^k +
^k
^k ;
ayk 
2
m!k
2
whereupon the Hamiltonian assumes its nal form

X 
1
y
^
!k ak a^k + :
H=
2
k



i
^ ;
m!k k
(2.15)

For the spe i problem of the rst ex ited mode in a wave guide of witdth Ly , !k =
[k2 + (=Ly )2 ℄1=2 . Eq. (2.15) represents our nal result for the quantum Hamiltonian of
the EM wave guide. Before on luding this se tion let us make a few omments on the
stru ture of the result:
. First, noti e that the onstru tion above almost ompletely paralleld our previous
dis ussion of the harmoni hain8 . The lose stru tural simiarity between the two
8 Te hni ally, the only te hni al is that instead of index pairs (k; k) all indi es (k; k) are equal and

positive. This an be tra ed ba k to the fa t that we have expanded in terms of the real eigenfun tions of
the losed wave guide instead of the omplex eigenfun tions of the ir ular os illator hain. At any rate,
the di eren e is largely inessential.

2.2.

QUANTUM ELECTRODYNAMICS

39

systems roots in the fa t that the free eld Lagrangian (2.10) is quadrati and,
therefore, bound to map onto an os illator type Hamiltonian. That we obtained a
simple one-dimensional superposition of os illators is due to the boundary onditions
spe i to a narrow wave guide. For less restri tive geometries, e.g. free spa e, a
more omplex superposition of ve torial degrees of freedom in three-dimensional
spa e would have been obtained. However, the prin ipal mapping of the free EM
eld onto a superposition of os illators is independent of geometry.

. Physi ally, the quantum ex itations des ribed by (2.15) are, of ourse, the photons
of the EM eld. The unfamilar appearan e of the dispersion relation !k is again
a pe uliarity of the wave guide. However, in the limit of large longitudinal wave
numbers k  Ly 1 , the dispersion approa hes !k  jkj, i.e. the relativisti dispersion
of the photon eld. Also noti e that, due to the equality of the Hamiltonians (2.9)
and (2.15), all what had been said about the behaviour of the phonon modes of the
atomi hain arries over to the photon modes of the wave guide.
. As with their phononi analog, the os illators des ribed by (2.15) exhibit zero point
u tuations. It is a fas inating aspe t of quantum ele trodynami s that these osillations, aused by quantization of the most relativisti eld, surfa e at various
points of non-relativisti physi s. In the next se tion two prominent manifestations
of zero point u tuations in ondensed matter physi s will be brie y dis ussed.

2.2.2 The Quantum Va uum
One of the most prominent manifestations of va uum u tuations is the Casimir e e t9 :
Two parallel ondu ting plates embedded into the va uum ex ert an attra tive for e on
ea h other. This phenomenon is believed to be of some relevan e for, e.g., olloidal
hemistry. Namely the for e balan e of hydrophobi suspensions of parti les, sized 0:1
1m, in ele trolytes. At least it is problems of this type whi h prompted Casimir to his
famous analysis of the idealized va uum problem. Qualitatively, the origin of the Casimir
for e is readily understood (See our poor attempt at an illustration, Fig. 2.6.): Like
their lassi al analog, quantum photons exert a ertain radiation pressure on ma ros opi
media. The di eren e to the lassi al ase is that, due to zero point os illations, even the
quantum va uum is apable of reating radiation pressure. For a single ondu ting body
embedded into the in nite va uum, the net pressure vanishes by symmetry. However, for
two parallel plates, the situation is di erent. Mode quantization arguments similar to the
ones used in the previous se tion show that the density of quantum modes between the
plates is lower than in the half in nite outer spa es. Hen e, the for e (density) reated
by outer spa e ex eeds the ounter pressure from the inside; The plates 'attra t' ea h
9

Casimir.

40

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

other. For a quantiative formulation of this pi ture, see problem XX. A se ond important

Figure 2.6: Very s hemati attempt to illustrate the prin iple behind the asimir e e t.
Mode quantization implies that the density of EM eld modes between two parallel plates
in a va uum is lower than in outer spa e. Consequently, the radiation pressure from the
outside ex eeds the ounter pressure from the inside and the plates feel an attra tive for e.
phenonmenon where va uum u tuations play some role is Van der Waals for es10 :
Neutral atoms or mole ules attra t ea h other by a potential that, at small separation
r, s ales like r 6 . Early mi ros opi attempts to lassi ally explain the origin of the
phenomenon produ ed results for the attra tion for e that, in on i t with experimental
nding, were strongly temperature dependent. It was onsidered a major breakthrough
of the new quantum me hani s when London11 proposed a model whereby a temperature
independent r 6 -law was obtained. The essen e of Londons idea is easily explained:
Imagine ea h of the atoms/mole ules as an os illator. The zero point motion of these
os illators, measured in terms of some oordinate x, reates a dipole moment of strength
 x. At lose distan es, the systems intera t through a dipole-dipole intera tion. Writing
down oupled os illator equations (see problem XX), one readily obtains a lowering of the
ground state energy by a ontribution V (r)  r 6 .
However, there is a subtle point about the seemingly inno ous oupling of the quantum
os illator to the lassi al Coulomb potential. Close inspe tion (see Milonni's book for
details) shows that to maintain the quantum ommutation relations of the os illator
degrees of freedom, the EM eld must not be assumed entirely lassi al. In fa t, on e
the quantum nature of the EM eld has entered the stage, it be omes possible to explain
the for e without need for phenomenologi al introdu tion of atomi os illator degrees of
10

11

Johannes Diderik van der Waals 18371923, 1910 Nobel Laureate in physi s
in re ognition of his work on the state
equation of non-ideal gases.
Fritz Wolfgang London 1900-1954, famous for his work on van der Waals
for es and his phenomenologi al theory of super ondu tivity (in ollaboration with his brother Heinz, 19071970.)

2.2.

QUANTUM ELECTRODYNAMICS

41

freedom:

Figure 2.7: On the reation of van der Waals for es by EM va uum u tuations. An atom
or mole ule B is exposed to u tuations of the EM quantum va uum (visualized by wavy
lines.) This leads to the reation of a dipole element dB . A loseby partner atom A sees
both the va uum amplitude and the dipole eld reated by dB . The ooperation of the
two eld strengths leads to a net lowering of the energy of A.
Consider an atom A exposed to an ele tri eld E. The eld gives rise to the atomi
Stark12 e e t, i.e. a level shift of order W  hE(xA)  E(x A)i, where is the atomi
polarizability, E = t A is the operator representing the EM eld, and h: : :i a quantum
expe tation value.
For an atom in the empty va uum E  E0 is determined by the zero point amplitude
of the quantum os illator eld. However, in the neighbourhood of a se ond atom B ,
the eld amplitude is di erent. The point is that the zero point amplitude E0 indu es
a dipole element dB / E0(xB ). This dipole reates a eld EB whi h ontributes to
the total eld amplitude felt by atom A: E(xA ) = (E0 + EB )(xA ) (see Fig. 2.7.) The
dominant ontribution to the level shift due to the proximity of B is then given by

V (r) = W

hE0(xA)  E0(xA)i  hE0(xA)  EB (xB )  2hET0 (xA)A(r)E(xB )i;

(2.16)

where r  rB rA and A(r) is a matrix kernel des ribing the geometri details of the
dipole intera tion. (We have subtra ted the va uum level shift be ause it represents an undete table o set.) Noti e (2.16) predi ts the van der Waals intera tion to be proportional
to the square of the zero point eld amplitude.
Unfortunately, the quantiatitve evaluation of the matrix element, involving a geometri average over all polarization dire tions of the quantum eld amplitude is somewhat
involved. As a nal result of a al ulation (detailed, e.g., in Millnors book, se tion 3.11.)

12

Johannes Stark 1874-1957, Nobel laureate 1919 for his
dis overy of the Stark e e t in 1913. Otherwise, one of
the worst hara ters of Nazi German s ien e. In 1937
a used Heisenberg of ooperating with jews. It took
the personal intervention of Himmler, Rei hsfuhrer SS
(!), to get Heisenberg out of trouble.

42
one obtains

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

3!0 2
;
(2.17)
4r6
where !0 is the transition frequen y between the ground state and the rst ex ited state
of the atom. (This paramter enters the result through the dependen e of the mi ros opi
polarizability on the transition frequen ies.)

V (r) =

2.3 Summary and Outlook
In this hapter we have introdu ed the general algorithm whereby lassi al ontinuum
theories are quantized. Employing the elementary harmoni os illator as a prototypi al
example we have seen that the Hilbert spa es of these theories a ord di erent interpretations. Of parti ular use was a quasi-parti le pi ture whereby the olle tive ex itations
of the ontinuum theories a quired the status of elementary parti les.
Suspi iously, both examples dis ussed in this text, the harmoni quantum hain and
free quantum ele trodynami s, led to exa tly solvable, free eld theories. However,
it takes only little imagination to foresee that only few ontinuum theories will be as
simple. E.g. the exa t solvability of the atomi hain would have been lost had we
in luded higher order ontributions in the expansion in powers of the latti e distortion
amplitude. Su h terms would hinder the free wave-like propagation of the phonon modes.
Put di erently, phonons would begin to s atter o ea h other or intera t. Similarly, the
free status of ele trodynami s is lost on e the EM eld is allowed to intera t with a matter
eld. Needless to say, intera ting eld theories are in nitely more omplex, but also
more interesting, than the systems onsidered so far.
Te hni ally, we have seen that the phonon/photon interpretation of the eld theories
dis ussed in this hapter ould onveniently be formulated in terms of ladder operators.
However, the appli ations dis ussed so far provide only a glimpse on the full spe trum
of advantages of this language. In fa t, the formalism of ladder operators, ommonly
denoted as 'se ond quantization', represents a entral, and histori ally the oldest element
of quantum eld theory. The next hapter is devoted to a more omprehensive dis ussion
of both the formal aspe ts and appli ations of this language.

2.4.

43

PROBLEM SET

2.4 Problem Set
Q1 Applying the Euler-Lagrange equation, obtain the equation of motion of the Lagrangian densities:

_2

ksa2
m 2 2
(x )2
!
2
2
m_ 2  2 2
2
_
2:
L[; ; x ℄ = 2 2 x

_2
3:
L[; _ ℄ = m2 m2 !22 4 4

n 
X
m
1
2
2
2
4:
L[f_ ig; fxig℄ =
_ i
ksa (x i )
2
2
i=1
m
5:
L[;_ x℄ = 2 j_ j2 12 ksa2jxj2
[Note that in 5. the eld  is omplex.℄ Cast the Lagrangian density 4. in a more
ompa t form by making use of a relativisti 4-ve tor notation. Starting with the
Lagrangian 2., obtain the Hamiltonian density. Introdu ing anoni al ommutation
relations, quantise the Hamiltonian.

L[; ;_ x℄ = m2

1:

Q2 Following the dis ussion in the le tures, a periodi one-dimensional quantum elasti
hain of length L is expressed by the Hamiltonian


Z
1 2 ks a2 ^ 2
^
^ +
( )
H = dx
2m
2 x
where the eld operators obey the anoni al ommutation relations
h

i

^ (x); ^(x0 ) = i~Æ (x x0 ):

(a) De ning the Fourier representation,


^k
^k



1
L1=2

Z L
0

dxefikx



^(x) ;
^ (x)

P





^(x) = 1 X efikx ^k ;
^ (x) L1=2 k
^k

where k represents the sum over all quantised quasi-momenta k = 2m=L; m 2 Z ,
show that the eld operators obey the ommutation relations [^k ; ^k0 ℄ = i~Ækk0 .
(b) In the Fourier representation, show that the Hamiltonian takes the form

H^ =

X
k

( ) De ning

ak 



1
k a2
^k ^ k + s k2 ^k ^
2m
2

r



1
m!k ^
^
k + i
2~
m!k


k



k

:

44

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

where !k = a(ks =m)1=2 jkj = v jkj show that the eld operators obey the anoni al
ommutation relations [ak ; ayk0 ℄ = Ækk0 , and [ak ; ak0 ℄ = 0.
(d) Finally, with this de nition, show that the Hamiltonian an be expressed in the
form

H^ =

X
k





1
y
~!k ak ak +
:
2

Q3 Londons phenomenologi al approa h to explaining the van der Waals for e starts
out from the Hamiltonian

p^2 m! 2
p^2
H^ = 1 + 1 + o (^x21 + x^22 ) + mK x^1 x^2 ;
2m 2m
2
where x^i is the oordinate of an os illator phenomenologi ally representing one of
two atoms i = 1; 2. The atoms are oupled through a dipole-dipole intera tion,
where the dipole operators are proportional to x^i and K = qe2 =(mr3 ) en apsulates
the details of the intera tion. Here r denotes the distan e between the atoms and q
is a dipole-dipole orientation fa tor.
Compute the spe trum of the two-parti le system des ribed by H^ and verify that
the dipole oupling leads to a lowering
V (r) =

K2
8!o3

of the ground state energy. Using that the 2lassi
al polarizability of an os illator is
2
!
q
2
2
given by = e =(m! ) this be omes V = 8r6 . Finally, using that the dire tional
average of the geometri fa tor q 2 equals 2 and multiplying by three to a ount for
the three dimensional hara ter of a 'real' atomi os illator, we re over the result
(2.17).

2.4.

45

PROBLEM SET

A1 To omplete this problem, one must generalise the Euler-Lagrange equation derived
in le tures. E.g., for ase 2., a variation of the a tion obtains

ÆS =

Z h

=

L(_ + ;_ x2  + x2 )

Z




_ _ L


+ x2 x2

L

L(;_ x2 )

=

Z





i

dt _ L + d2x x2  L



from whi h we obtain the Euler-Lagrange equation


dt _ L + d2x x2  L = 0:
Applied to the Lagrangian fun tionals at hand, one nds the equation of motion
k a2
1:
 s x2  + ! 2  = 0;
m

2:
 + x4  = 0;
m


3:
+ ! 2  + 3 = 0
m
2
ksa 2
  = 0:
4:
i
m x i
Finally, turning to ase 5., it is ne essary to generalise the Euler-Lagrange equation
to a ount for omplex elds. Sin e the real and imaginary parts u tuate independently, we an onsider a variation of ea h independently. Separating  = 0 + i00
into its real and imaginary parts, and applying a variation to ea h omponent, one
obtains
ks a2 2 0
ksa2 2 00
0
x  = 0;
00
  = 0:
m
m x
In fa t, this result shows that the omponents  and the omplex onjugate  an
be treated as independent. A variation of the a tion with respe t to  obtains
k a2
5:
 s x2  = 0:
m
For ase 2. the momentum onjugate to the eld  is given by
= _ L = m:_
The Hamiltonian orresponding to ase 2. is given by
2
2
H = _ L = 2m + ks2a (x2 )2 :
Applying the anoni al ommutation relations [^ (x); ^(x0 )℄ = i~Æ (x x0 ), and
promoting the lassi al elds to operators, one obtains the quantum Hamiltonian.
In Fourier spa e, the Hamiltonian takes the same form as that dis ussed in le tures
with q 2 7! q 4 .

46

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

A2 (a) Using the de nition provided, together with the anoni al ommutation relations
of the eld operators, one obtains
1
[^k ; ^k0 ℄ =
L

Z L
0

= i~

dx

Z L
0

z Z
1 L

L

0

dx0 e
Ækk0
}|

dx e

ikx+ik0 x0

{
0
i(k k )x0 =

i~Æ (x x0 )
}|
{
0
^
[^ (x); (x )℄
z

i~Ækk0 :

(b) Again, using the de nition, the kineti omponent of the Hamiltonian takes the
form
Z L
Z
X 1
^ 2 X 1 L
0 1
dx
dx ei(k+k )x
=
^k ^k0 =
^ ^ :
2m kk0 L 0
2m
2m k k
0
k
Similarly, applied to the potential omponent one obtains the Hamiltonian as advertised.
( ) Using the ommutation relation derived above, one nds

i ^
[ak ; ayk0 ℄ =
[ k ; ^ k0 ℄ [^k ^k0 ℄ = Ækk0 ;
[ayk ; ayk0 ℄ = 0:
2~
(d) Inverting the expression for the eld operators, one nds

^k =



~

2m!k
Using the identity

1=2 





m~!k
^k = i
2

ak + ay k ;

1=2 

ayk

a


k

:


1
k a2 
(^k ^ k + ^ k ^k ) + s k2 ^k ^ k + ^ k ^k
2m
2

1  y
= ~!k ak ak + ak ayk + ay k a k + a k ay k :
2
together with the ommutation relations, we obtain the harmoni os illator Hamiltonian.
A3 We begin by formulating the potential U^ of the os illator system in a matrix notation, U^ = x^ T Ax^ , where x^ T = (^x1 ; x^2 )T and




m !2 K
A=
:
2 K !2
Diagonalizing the matrix, A = UDU T and transforming oordinates x ! U T x = x0 ,
the system de ouples into two independent os illators,

H^ =

X  p^20
i

i=1;2



m! 2
+ i x^2i 0 ;
2m
2

2.4.

47

PROBLEM SET

where the new hara teristi frequen ies !1=2 = m2 (!o2  K )1=2 are determined by
the eigenvalues of the matrix D. The ground state of this system is given by
1
E0 = (!1 + !2 )  !o
2
whi h lies by an amount V =

K2
8!o3

K2
;
8!o3

lower than the energy of the isolated atoms.

48

CHAPTER 2.

FROM CLASSICAL TO QUANTUM FIELDS

Chapter 3
Se ond Quantisation
The aim of this se tion is to introdu e and apply the method of se ond quantisation. The
rst part of the se tion fo uses on methodology and notation, while the remainder is devoted to appli ations. We will investigate the physi s of harge density wave propagation in
one-dimensional quantum wires, spin waves in a quantum Heisenberg (anti)ferromagnet,
the weakly intera ting Bose gas, and the ele tron-phonon intera tion. In later hapters
most of these problem lasses will be revisited from di erent perspe tives.

In the previous hapter we have en ountered two eld theories that ould onveniently
be represented in the language of 'se ond quantization', i.e. a formulation based on the
formulation: First, se ond quantization provided a ompa t way of representing the many
body quasi-parti le spa e of ex itations in these systems. (There was no need to expli itly
symmetrize many parti le states as would be the ase in the formalism of 'ordinary' many
body quantum me hani s.) Se ond, the properties of the ladder operators were en oded
in a simple set of ommutation relations ( f. Eq. 2.8) rather than in some expli it Hilbert
spa e representation.
Apart from a ertain aestheti appeal, these observations would not be of mu h relevan e if it were not for the fa t that the formulation an be generalized to a omprehensive
and highly eÆ ient formulation of many body quantum me hani s in general. In fa t,
se ond quantization an be onsidered the rst major ornerstone on whi h the theoretial framework of quantum eld theory was built. As su h, extensive introdu tions to the
on ept an be found throughout the literature (see, e.g., Feynman [?℄). We will therefore
not develop the formalism in full mathemati al rigour but rather pro eed pragmati ally by
rst motivating and introdu ing its basi elements, followed by a dis ussion of the 'se ond
quantized' version of standard operations of quantum me hani s (taking matrix elements,
1 In the opinion of the authors the denotation 'se ond quantization' is hosen unfortunately. His-

tori ally, this denotation was motivated by the observation that the ladder operator algebra fosters an
interpretation of quantum ex itations as dis rete 'quantized' units. Fundamentally, however, there is no
su h thing like 'two' superimposed quantization steps in single- or many-parti le quantum me hani s.
What we are dealing with is basi ally a problem-adjusted representation of the ' rst and only quantized'
theory.

49

50

CHAPTER 3.

SECOND QUANTISATION

hanging bases, representing operators et .) The se ond part of the haper will be onerned with developing uen y in the method by addressing a number of appli ations.
Readers familiar with the formalism may dire tly pro eed to these se tions.

3.1 Introdu tion to Se ond Quantization
3.1.1 Motivation
We begin our dis ussion by re apitulating some fundamental notions of many body quantum me hani s, as formualted in the traditional language of symmetrized/anti-symmetrized
wave fun tions. Consider the (normalised) set of wavefun tions j n i of some single-parti le
Hamiltonian H^ ,
H^ j  i =  j  i:
where  are the eigenvalues. With this de nition, the normalized 2-parti le wavefun tion
of two fermions (bosons) populating the levels  and 0 6=  given by the anti-symmetrized
(symmetrized) produ t
1
F (x1 ; x2 ) = p (  (x1 ) 0 (x2 )
0 (x1 )  (x2 )) ;
2
1
B (x1 ; x2 ) = p (  (x1 ) 0 (x2 ) + 0 (x1 )  (x2 )) :
2
In the Dira Bra ket representation,
j; 0iF (B)  p1 (j  i j 0 i  j 0 i j i) ;
(3.1)
2
where  = 1 for fermions while  = 1 for bosons2 . Note that the expli it symmetrization
of the wave fun tions is ne essitated by quantum me hani al indistinguishability: For
(fermions) bosons the wave fun tion has to be anti-symmetri (symmetri ) under partile ex hange. More generally, an appropriately symmetrized N -parti le wavefun tion is
expressed in the form
X
1
(
Q1
N ! =0 n ! P

j1; 2; : : : N i  p

 )(1

par P )=2

j

P 1

i j

P 2

i::: j

P N

i

(3.2)

where n is the total number of parti les in state  (for fermions, Pauli ex lusion enfor es
the onstraint n = 1) { see g. 3.1. The summation runs over all N ! permutations of the
2

Enri o Fermi 1901-1954 (left) and Satyendranath
Bose 1894-1974 (right): Fermi was made 1938 Nobel Laureate in Physi s for his demonstrations
of the existen e of new radioa tive elements produ ed by neutron irradiation, and for his related
dis overy of nu lear rea tions brought about by
slow neutrons.

3.1.

51

INTRODUCTION TO SECOND QUANTIZATION

set of quantum numbers f1 ; : : : N g, and par P denotes the parity of the permutation
P . (par P = 1( 1) if the number of transpositions of two elements whi h brings the
permutationp(P1 ;Q
P2;   PN ) ba k to its original form (1; 2;    N ) is even (odd)). The
prefa tor 1= N !  n ! normalises the many-body wave fun tion. In the fermioni ase,
the many-body wavefun tion is known as a Slater determinant. Finally noti e that it
will be useful to assume the quantum numbers fi g de ning the state j1 ; 2 ; : : : ; N i to
be ordered a ording to some onvention. (E.g. for i = xi a one-dimensional oordinate
representation we might order a ording to x1  x2  : : :  xN .) Any initially nonordered state an be brought into an ordered form at the ost of, at most, a minus sign.
ελ
ε4
ε3
ε2
ε1
ε0

nλ

nλ

0

0

1

1

1

2

1

3

Fermions

Bosons

Figure 3.1: S hemati diagram showing o upation numbers for a generi fermion and
boson system.
While representations like (3.2) an be used to represent the full Hilbert spa e3 of
many body quantum me hani s, a moments thought shows that this formulation is not
in the least onvenient:

. It takes little imagination to anti ipate that a tual working in the language of (3.2)
will be diÆ ult. E.g to ompute the overlap of two wave fun tion one needs to form
(N !)2 di erent produ ts.
. The representation is tailor made to problems with xed parti le number N . However, we know from statisti al me hani s that for N = O(1023 ) it is mu h more
onvenient to work in a grand anoni al formulation whereby N is allowed to u tuate. Closely related,
. In appli ations one will often ask questions like, \What is the amplitude for inje tion
of a parti le into the ```
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.885021984577179, "perplexity": 19005.374764993074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664469.42/warc/CC-MAIN-20191112001515-20191112025515-00315.warc.gz"}