Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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can write n

(1)

(r, r

) =



(r)(r

) þ

(1)

(r, r

), where 

is the order parameter of the condensate (



(r)(r

)

being of order N), while

(1)

(r, r

) vanishes for large

jr  r

j. This is equivalent to say that the bosonic field

operator splits in two parts,

ðrÞ¼ðrÞþ

ðrÞ½18

where the first term is a complex function and the

second one is the field operat or associated with

the noncondensed particles. This decomposition is

particularly useful when the depletion of the

condensate, that is, the fraction of noncondensed

particles, is small. This happens when the interac-

tion is weak, but also for particles with arbitrary

interaction, provided the gas is dilute. In this case,

one can expand the many- body Hamiltonian by

treating the operator

 as a small quantity.

A suitable strategy consists in writing the Heisen-

berg equation for the evolution of the field opera-

tors, ih@

=[

,

H], using the many-body

Hamiltonian [17]:

ih@

ðr; tÞ



ðr

; tÞVðr  r

ðr

; tÞ





ðr; tÞ½19

The zeroth-order is thus obtained by replacing the

operator

 with the classical field .Intheintegral

containing the interaction V(r  r

), this replacement is,

in general, a poor approximation when short distances

(r  r

) are involved. In a dilute and cold gas, one can

nevertheless obtain a proper expression for the inter-

action term by observing that, in this case, only binary

collisions at low energy are relevant and these collisions

are characterized by a single parameter, the s-wave

scattering length, a, independently of the details of the

two-body potential. This allows one to replace V(r  r

)

H with an effective interaction V(r  r

) = g(r  r

where the coupling constant g is given by g = 4h

a=m.

The scattering length can be measured with several

experimental techniques or calculated from the exact

two-body potential. Using this pseudopotential and

replacing the operator

 with the complex function  in

the Heisenberg equation of motion, one gets

ih@

ðr; tÞ

¼

h

þ V

ext

ðrÞþgjðr; tÞj

ðr; tÞ½20

This is known as Gross–Pitaevskii (GP) equation and

it was first introduced in 1961. It has the form of a

nonlinear Schro¨ dinger equation, the nonlinearity

coming from the mean-field term, proportional to

jj

. It has been derived assuming that N is large

while the fraction of noncondensed atoms is negli-

gible. On the one hand, this means that quantum

fluctuations of the field operator have to be small,

which is true when njaj

 1, where n is the particle

density. In fact, one can show that, at T = 0the

quantum depletion of the condensate is proportional

to (njaj

)

1=2

. On the other hand, thermal fluctuations

have also to be negligible and this means that the

theory is limited to temperatures much lower than

. Within these limits, one can identify the total

density with the condensate density.

The stationary solution of eqn [20] corresponds to

the condensate wave function in the ground state. One

can write (r, t) =

(r)exp(it=h), where  is the

chemical potential. Then the GP equation [20] becomes



h

þV

ext

ðrÞþgj

ðrÞj



ðrÞ¼

ðrÞ½21

where n(r)=j

(r)j

is the particle density. The same

equation can be obtained by minimizing the energy of

the system written as a functional of the density:

E½n¼

h

ﬃﬃﬃ

þ nV

ext

ðrÞþ

½22

The first term on the right corresponds to the

quantum kinetic energy coming from the uncertainty

principle; it is usually named ‘‘quantum pressure’’

and vanishes for uniform systems.

The next order in

 gives the excited states of the

condensate. In a uniform gas the ground-state order

parameter, 

, is a constant and the first-order

expansion of

H was introduced by N Bogoliubov in

1947. In particular, he found an elegant way to

diagonalize the Hamiltonian by using simple linear

combinations of particle creation and annihilation

operators. These are known as Bogoliubov’s trans-

formations and stay at the basis of the concept of

quasiparticle, one of the most important concepts in

quantum many-body theory.

A generalization of Bogoliubov’s approach to the

case of nonuniform condensates is obtained by

considering small deviations around the ground

state in the form

ðr; tÞ¼e

it=h



ðrÞþuðrÞe

i!t

þ v



ðrÞe

i!t



½23

Inserting this expression into eqn [20] and keeping

terms linear in the complex functions u and v, one gets

h!uðrÞ¼½

  þ 2g

ðrÞuðrÞþg

ðrÞvðrÞ½24

h!vðrÞ¼½

  þ 2g

ðrÞvðrÞþg

ðrÞuðrÞ½25

Bose–Einstein Condensates 315

These coupled equations allow one to calculate the

energies " =h! of the excitations. They also give the

so-called quasiparticle amplitudes u and v,whichobey

the normalization condition

dr½u



ðrÞu

ðrÞv



ðrÞv

ðrÞ= 

In a uniform gas, u and v are plane waves and one

recovers the famous Bogoliubov’s spectrum

h! ¼

h

þ 2gn

!"#

1=2

½26

where q is the wave vector of the excitations.

For large momenta the spectrum coincides with the

free-particle energy h

=2m. At low momenta, it

instead gives the phonon dispersion ! = cq, where

c = [gn=m]

1=2

is the Bogoliubov sound velocity. The

transition between the two regimes occurs when the

excitation wavelength is of the order of the healing

length,

 ¼½8na

1=2

¼ h=ðmc

ﬃﬃﬃ

Þ½27

which is an important length scale for superfluidity.

When the order parameter is forced to vanish at some

point (by an impurity, a wall, etc.), the healing length

provides the typical distance over which it recovers its

bulk value. In a nonuniform condensate the excitations

are no longer plane waves but, at low energy, they have

still a phonon-like character, in the sense that they

involve a collective motion of the condensate.

The GP equation [20] is the starting poin t for an

accurate mean-field description of BEC in dilute

cold gases, which is rigorous at T = 0 and for

njaj

1. Static and dynamics properties of con-

densates in different geometries can be calculated by

solving the GP equatio n numerically or using

suitable approximated methods. The inclusion of

effects beyond mean field is a highly nontrivial and

interesting problem. A rather extreme case is

represented by liquid

He, which is a dense system

where the interaction between atoms causes a large

depletion of the condensate even at T = 0(N

being less than 10%) and thus a full many-body

treatment is required for its rigorous description.

Nevertheless, even in this case, the general defini-

tions of the section ‘‘What is BEC?’’ are still useful .

Superfluidity and Coherence

With the word superfluidity, one summarizes a

complex of macroscopic phenomena occurring in

quantum fluids under particular conditions: persis-

tent currents, equilibrium states at rest in rotating

vessels, viscousless motion, quantized vorticity, and

others. These features can also be observed in BEC.

The link between BEC and superfluidity is given by

the phase of the order parameter [11]. To under-

stand this point, let us consider a uniform system. If

(r, t) is a solution of the Heisenberg equation [19]

with V

ext

= 0, then



ðr; tÞ¼

ðr  vt; tÞexp

h

mv r 



½28

where v is a constant vector, is also a solution. This

equation gives the Galilean transformation of

the field operator and also applies to its condensate

component . At equilibrium, the ground-state

order parameter is give n by 

ﬃﬃﬃ

exp (it=h),

where n is a constant independent of r. In a frame

where the condensate moves with velocity v, the

order parameter instead takes the form 

ﬃﬃﬃ

exp (iS), with S(r, t) =h

1

[mv r (mv

=2 þ)t].

The velocity of the condensate can thus be identified

with the gradient of the phase S:

vðr; tÞ¼

h

=Sðr; tÞ½29

This definition is also valid for v varying slowly in

space and time. The modulus of the order para-

meter plays a minor role in this definition and it is

not necessary to assume the gas to be dilute and

close to T = 0. Indeed, the relation [29] between the

velocity field and the phase of the order parameter

also applies in the presence of large quantum

depletion, as in superfluid

He, and at T 6¼0. In

this case, n should not be identified with the

condensate density. Conversely, in dilute gases at

T = 0, n is the condensate density and the velocity

[29] can be simply obtained by applying the usual

definition of current density operator,

j, to the order

parameter [11].

The velocit y [29] describes a potential flow and

corresponds to a collective motion of many particles

occupying a single quantum state. Being equal to the

gradient of a scalar function, it is irrotational

(=  v

= 0) and satisfies the Onsager–Feynman

quantization condition

 dl = h=m, with 

non-negative integer. These conditions are not

satisfied by a classical fluid, where the hydro-

dynamic velocity field, v(r, t) = j(r, t)=n(r, t), is the

average ov er many different states and does not

correspond to a potential flow.

By using the definition of the phase S and velocity

v, together with particle conservation, one can show

that the dynamics of a condensate, as far as

macroscopic motions are concerned, is governed by

the hydrodynamic equations of an irrotational

316 Bose–Einstein Condensates

nonviscous fluid. Within the mean-field theory, this

can be easily seen by rewriting the GP equation [20]

in terms of the density n = jj

and the velocity

[29]. Neglecting the quantum pressure term r

ﬃﬃﬃ

(hence limiting the description to length scales

larger than the healing length ), one gets

n þ= ðvnÞ¼0 ½30

and

v þ = V

ext

þ ðnÞþ



¼ 0 ½31

with the local chemical potential (n) = gn. These

equations have the typical structure of the dynamic

equations of superflui ds at zero temperature and can

be viewed as the T = 0 case of the more general

Landau’s two-fluid theory.

One of the most striking evidences of superflui dity

is the observation of quantized vortices, that is,

vortices obeying the Onsager–Feynman quantization

condition. A vast literature is devoted to vortices in

superfluid helium and, more recently, vortices have

also been produced and studied in condensates of

ultracold gases, including nice configuration s of

many vortices in regular triangular lattices, similar

to the Abrikosov lattices in superconductors. Other

phenomena, such as the reduction of the moment of

inertia, the occurrence of Josephson tunneling

through barriers, the existence of thresholds for

dissipative processes (Landau criterion), and others,

are typical subjects of intense investigation.

Another important consequence of the fact that

BEC is described by an order parameter with a well-

defined phase is the occurrence of cohere nce effects

which, in different words, mean that condensates

behave like matter waves. For instance, one can

measure the phase difference between two conden-

sates by means of interference. This can be done in

coordinate space by confining two condensates in

two potential minima, a and b, at a distance d. Let

us take d along z and assum e that, at t = 0, the order

parameter is given by the linear combination

(r) =

(r) þ exp (i)

(r)with

and 

real

and without overlap. Then let us switch off the

confining potentials so that the condensates expand

and overlap. If the overlap occurs when the density

is small enough to neglect interactions, the motion

is ballistic and the phase of each condensate evolves

as S(r, t) ’ mr

=(2ht), so that v = r=t. This implies

a relative phase  þ S(x, y, z þ d = 2)  S(x, y , z 

d=2) =  þ mdz=ht. The total density n = jj

thus

exhibits periodic modulations along z with wave-

length ht=md. This interference pattern has indeed

been observed in condensates of ultracold atoms. In

these systems it was also possible to measure the

coherence length, that is, the distance jr  r

j at which

the one-body density vanishes and the phase of the

order parameter is no more well defined. In most

situations, the coherence length turns out to be of the

order of, or larger than the size of the condensates.

However, interesting situations exist when the coher-

ence length is shorter but the system still preserves some

features of BEC (quasicondensates).

Final Remarks

Bose–Einstein condensates of ultracold atoms are

easily manipulated by changing and tuning the

external potentials. This means, for instance, that one

can prepare condensates in different geometries,

including very elongated (quasi-1D) or disk-shaped

(quasi-2D) condensates. This is conceptually impor-

tant, since BEC in lower dimensions is not as simple as

in three dimensions: thermal and quantum fluctua-

tions play a crucial role, superfluidity must be properly

re-defined, and very interesting limiting cases can be

explored (Tonks–Girardeau regime, Luttinger liquid,

etc.). Another possibility is to use laser beams to

produce standing waves acting as an external periodic

potential (optical lattice). Condensates in optical

lattices behave as a sort of perfect crystal, whose

properties are the analog of the dynamic and transport

properties in solid-state physics, but with controllable

spacing between sites, no defects and tunable lattice

geometry. One can investigate the role of phase

coherence in the lattice, looking, for instance, at

Josephson effects as in a chain of junctions. By tuning

the lattice depth one can explore the transition from a

superfluid phase and a Mott-insulator phase, which is

a nice example of quantum phase transition. Control-

ling cold atoms in optical lattice can be a good starting

point for application in quantum engineering, inter-

ferometry, and quantum information.

Another interesting aspect of BECs is that the key

equation for their description in mean-field theory,

namely the GP equation [20], is a nonlinear Schro¨-

dinger equation very similar to the ones commonly

used, for instance, in nonlinear quantum optics. This

opens interesting perspectives in exploiting the analo-

gies between the two fields, such as the occurrence of

dynamical and parametric instabilities, the possibility

to create different types of solitons, the occurrence of

nonlinear processes like, for example, higher harmonic

generation and mode mixing.

A relevant part of the current research also involves

systems made of mixtures of different gases, Bose–Bose

or Fermi–Bose, and many activities with ultracold

atoms now involve fermionic gases, where BEC can

Bose–Einstein Condensates 317

also be realized by condensing molecules of fermionic

pairs. An extremely active research now concerns the

BCS–BEC crossover, which can be obtained in Fermi

gases by tuning the scattering length (and hence the

interaction) by means of Feshbach resonances.

Ten years after the first observation of BEC in

ultracold gases, it is almost impossible to summarize

all the researches done in this field. A larg e amount

of work has already been devoted to characterize the

condensates and several new lines have been opened.

Rather detailed review articles and books are

already available for the interested readers.

See also: Interacting Particle Systems and Hydrodynamic

Equations; Quantum Phase Transitions; Quantum

Statistical Mechanics: Overview; Renormalization:

Statistical Mechanics and Condensed Matter; Superfluids;

Variational Techniques for Ginzburg–Landau Energies.

Further Reading

Cornell EA and Wieman CE (2002) Nobel lecture: Bose–Einstein

condensation in a dilute gas, the first 70 years and some recent

experiments. Reviews of Modern Physics 74: 875.

Dalfovo F, Giorgini S, Pitaevskii LP, and Stringari S (1999)

Theory of Bose–Einstein condensation in trapped gases.

Reviews of Modern Physics 71: 463.

Griffin A, Snoke DW, and Stringari S (1995) Bose–Einstein

Condensation. Cambridge: Cambridge University Press.

Huang K (1987) Statistical Mechanics, 2nd edn. New York:

Wiley.

Inguscio M, Stringari S, and Wieman CE (1999) Bose–Einstein

Condensation in Atomic Gases, Proceedings of the Inter-

national School of Physics ‘‘Enrico Fermi,’’ Course CXL.

Amsterdam: IOS Press.

Ketterle W (2002) Nobel lecture: when atoms behave as waves:

Bose–Einstein condensation and the atom laser. Reviews of

Modern Physics 74: 1131.

Landau LD and Lifshitz EM (1980) Statistical Physics, Part 1.

Oxford: Pergamon Press.

Leggett AJ (2001) Bose–Einstein condensation in the alkali gases:

some fundamental concepts. Reviews of Modern Physics

73: 307.

Lifshitz EM and Pitaevskii LP (1980) Statistical Physics, Part 2.

Oxford: Pergamon Press.

Pethick CJ and Smith H (2002) Bose–Einstein Condensation in

Dilute Gases. Cambridge: Cambridge University Press.

Pitaevskii LP and Stringari S (2003) Bose–Einstein Condensation.

Oxford: Clarendon Press.

Bosons and Fermions in External Fields

E Langmann, KTH Physics, Stockholm, Sweden

Introduction

In this artic le we d iscuss quantum theories which

describe systems of nondistinguishable particles

interacting with external fields. Such models are

of interest also in the nonrelativistic case (in

quantum statistical mechanics, nuclear physics,

etc.), but the relativistic case has a dditional,

interesting complications: relativistic models are

genuine quantum field theories, that is, quantum

theories with an infinite number of degrees of

freedom, with nontrivial features like diverge nces

and anomalies. Since interparticle interactions are

ignored, such models can be re garded as a first

approximation to more com plicated theories, and

they can be studied by mathematically precise

methods.

Models of relativistic particles in external electro-

magnetic fields have received considerable attention

in the physics literature, an d interesting phenomena

like the Klein paradox or particle–an tiparticle pair

creation in overcritical fields have been studied; see

Rafelski et al. (1978) for an extensive review. We

will not discuss these physics questions but only

describe some prototype examples and a general

Hamiltonian framewo rk which has been used in

mathematically precise work on such models. The

general framework for this latter work is the

mathematical theory of Hilbert space operators

(see, e.g., Reed and Simon (1975)), but in our

discussion we try to avoid presupposing knowledge

of that theory. As mentioned briefly in the end, this

work has had close relations to various topics of

recent interest in mathematical physics, including

anomalies, infinite-dimensional geome try and group

theory, conformal field theory, and noncommutative

geometry.

We restrict our discussion to spin-0 bosons and

spin-1/2 fermions, and we will not discuss models

of particles in external gravitational fields but

only re fer the interested reader to DeWitt (2003).

We also only mention in passing that external

field problems have also been studied using

functional integral approaches, and mathemati-

cally precise work on this can be found in the

extensive lit erature on determinants of differ ential

operators.

Examples

Consider the Schro¨ dinger equation describing a

nonrelativistic particle of mass m and charge e

318 Bosons and Fermions in External Fields

moving in three-dimensional space and interacting

with an external vector and scalar potentials A and

, respectively,

¼ H ; H ¼

ðirþeAÞ

 e ½1

(we set h = c = 1, @

= @=@t,and ,, and A can

depend on the space and time variables x 2 R

and

t 2 R). This is a standard quantum-mechanical

model, with the one-particle wave function

allowing for the usual probabilistic interpretation.

One interesting generalization to the relativistic

regime is the Klein–Gordon equation

þ eðÞ

ðirþeAÞ

 m

¼ 0 ½2

with a C-valued function . There is another

important relativistic generalization, the Dirac

equation

þ eðÞðirþeAÞa þ m½ ¼ 0 ½3

with a = (

, 

)and Hermitian 4  4

matrices satisfying the relations



þ 



¼ 

;

 ¼

;

¼ 1 ½4

and a C

-valued function (we also write 1 for the

identity). These two relativistic equations differ by

the transformation properties of under Lorentz

transformations: in [2] it transforms like a scalar

and thus describes spin-0 particles, and it transforms

like a spinor describing spin-1/2 particles in [3]. While

these equations are natural relativistic generaliza-

tions of the Schro¨ dinger equation, they no longer

allow to consistently interpret as one-particle

wave functions. The physical reason is that, in a

relativistic theory, high-energy processes can create

particle–antiparticle pairs, and this makes the

restrictiontoafixedparticlenumberinconsistent.

This problem can be remedied by constructing a

many-body model allowing for an arbitrary number

of particles and antiparticles. The requirement that

this many-body model should have a ground state is

an important ingredient in this construction.

It is obviously of interest to formulate and study

many-body models of nondistinguishable particles

already in the nonrelativistic case. An important

empirical fact is that such particles come in two

kinds, bosons and fermions, distinguished by their

exchange statistics (we ignore the interesting possi-

bility of exotic statistics). For example, the fermion

many-particle version of [1] for suitable  and A is a

useful model for electrons in a metal. An elegant

method to go from the one- to the many-particle

description is the formalism of second quantization:

one promotes to a quantum field operator with

certain (anti-) commutator relations, and this is a

convenient way to construct the appropriate many-

particle Hilbert space, Hamiltonian, etc. In the

nonrelativistic case, this formalism can be regarded

as an elegant reformulation of a pedestrian con-

struction of a many-body quantum-mechanical

model, which is useful since it provides convenient

computational tools. However, this formalism nat-

urally generalizes to the relativistic case where the

one-particle model no longer has an acceptable

physical interpretation, and one finds that one can

nevertheless give a consistent physical interpretation

to [2] and [3] provided that are interpreted as

quantum field operators describing bosons and

fermions. This particular exchange statistics of the

relativistic particles is a special case of the spin-

statistics theorem: integer-spin particles are bosons

and half-integer spin particles are fermions. While

many structural features of this formalism are

present already in the simpler nonrelativistic models,

the relativistic models add some nontrivial features

typical for quantum field theories.

In the following, we discuss a precise mathema-

tical formulation of the quantum field theory models

described above. We emphasize the functorial nature

of this construction, which makes manifest that it

also applies to other situations, for example, where

the bosons and fermions are also coupled to a

gravitational background, are considered in other

spacetime dimensions than 3 þ 1, etc.

Second Quantization:

Nonrelativistic Case

Consider a quantum system of nondistinguishable

particles where the quantum-mechanical descrip-

tion of one such particle is known. In general, this

one-particle description is given by a Hilbert space

h and one-particle observables and transforma-

tions which are self-adjoint and unitary operators

on h, respectively. The most important observable

is the Hamiltonian H.Wewilldescribeageneral

construction of the corresponding many-body

system.

Example As a motivating example we take the

Hilbert space h = L

) of square-integrable func-

tions f (x), x 2 R

, and the Hamiltonian H in [1].A

specific example for a unitary operator on h is the

gauge transformation (Uf )(x) = exp(i(x))f (x) with

 a smooth, real-valued function on R

In this example, the corresponding wave functions

for N identical such particles are the L

-functions

, ..., x

), x

2 R

. It is obvious how to extend

Bosons and Fermions in External Fields 319

one-particle observables and transformations to such

N-particle states: for example, the N-particle Hamil-

tonian corresponding to H in [1] is

j¼1

ðir

þ eAðt; x

ÞÞ

 eðt; x

Þ½5

and the N-particle gauge transformation U

is defined

through multiplication with

j = 1

exp(i(x

)).

For systems of indistinguishable particles it is

enough to restrict to wave functions which are even

or odd under particle exchanges,

ðx

; ...; x

¼f

ðx

; ...; x

Þ½6

for all 1  j < k  N, with the upper and lower

signs corresponding to bosons and fermions, respec-

tively (this empirical fact is usually taken as a

postulate in nonrelativistic many-body quantum

physics). It is convenient to define the zero-particle

Hilbert space as C (complex numbers) and to

introduce a Hilbert space containing states with all

possible particle numbers: this so-called Fock space

contains all states

ðx

; x

ðx

; x

½7

with f

2 C. The definition of H

and U

then

naturally extends to this Fock space; see below.

General Construction

The construction of Fock spaces and many-particle

observables and transformations just outlined in a

specific example is conceptually simple. An alter-

native, more efficient construction method is to use

‘‘quantum fields,’’ which we denote as (x) and

(x), x 2 R

. They can be fully characterized by the

following (anti-) commutator relations:

½ ðxÞ;

ðyÞ



¼ 

ðx  yÞ; ½ ðxÞ; ðyÞ



¼ 0 ½8

where [a, b]



 ab  ba, with the commutator and

anticommutators (upper and lower signs, respec-

tively) corresponding to the boson and fermion case,

respectively. It is convenient to ‘‘smear’’ these fields

with one-particle wave functions and define

ðf Þ¼

xf ðxÞ ðxÞ

ðf Þ¼

ðxÞf ðxÞ

½9

for all f 2 h. Then the relations characterizing the

field operators can be written as

½ ðf Þ ;

ðgÞ



¼ðf; g Þ

½ ðf Þ ; ðgÞ



¼ 0

8f ; g 2 h

½10

where

ðf ; gÞ¼

xf ðxÞg ðxÞ

is the inner product in h. The Fock space F



(h) can

then be defined by postulating that it contains a

normalized vector  called ‘‘vacuum’’ such that

ðf Þ ¼ 0 8f 2 h ½11

and that all

(y)

(f ) are operators on F



(h) such that

(f ) = (f )



, where  is the Hilbert space adjoint.

Indeed, from this we conclude that F



(h), as vector

space, is generated by

^ f

^^f



ðf

Þ

ðf

Þ ½12

with f

2 h and N = 0, 1, 2, ..., and that the Hilbert

space inner product of such vectors is

^ f

^^f

; g

^ g

^^g

¼ 

N;M

P2S

ð1Þ

jPj

j¼1

ðf

; g

Þ½13

with S

the permutation group, with (þ1)

jPj

= 1

always, and (1)

jPj

= þ1and1 for even and odd

permutations, respectively. The many-body Hamil-

tonian q(H) corresponding to the one-particle Hamil-

tonian H can now be defined by the following relations:

qðHÞ ¼ 0; ½qðHÞ;

ðf Þ ¼

ðHf Þ½14

for all f 2 h such that Hf is defined. Indeed, this

implies that

qðHÞf

^ f

^^f

j¼1

^ f

^^ðHf

Þ^^f

½15

which defines a self-adjoint operator on F



(h), and

it is easy to check that this coincides with our down-

to-earth definition of H

above. Similarly, the

many-body transformation Q(U) corresponding to

a one-particle transformation U can be defined as

QðUÞ ¼ ; QðUÞ

ðf Þ¼

ðUf ÞQðUÞ½16

for all f 2 h, which implies that

QðUÞf

^ f

^^f

¼ðUf

Þ^ðUf

Þ^^ðUf

½17

320 Bosons and Fermions in External Fields

and thus coincides with our previous definition of

While we presented the construction above for a

particular example, it is important to note that it

actually does not make reference to what the one-

particle formalism actually is. For example, if we

had a model of particles on a space M given by

some ‘‘nice’’ manifold of any dimension and with M

internal degrees of freedom, we would take

h = L

(M)  C

and replace [9] by

ðf Þ¼

dðxÞ

j¼1

ðxÞ

ðxÞ½18

and its Hermitian conjugate, with the measure  on

M defining the inner product in h,

ðf ; gÞ¼

dðxÞ

ðxÞg

ðxÞ

With that, all formulas after [9] hold true as they stand.

Given any one-particle Hilbert space h with inner

product ( , ), observable H, and transformation U, the

formulas above define the corresponding Fock spaces



(h) and many-body observable q(H) and transfor-

mation Q(U). It is also interesting to note that this

construction has various beautiful general (functorial)

properties: the set of one-particle observables has a

natural Lie algebra structure with the Lie bracket given

by the commutator (strictly speaking: i times the

commutator, but we drop the common factor i for

simplicity). The definitions above imply that

½qðAÞ; qðBÞ ¼ qð½A; BÞ ½19

for one-particle observables A, B, that is, the above-

mentioned Lie algebra structure is preserved under

this map q. In a similar manner, the set of one-

particle transformations has a natural group struc-

ture preserved by the map Q,

QðUÞQ ðVÞ¼QðUVÞ; QðUÞ

1

¼QðU

1

Þ½20

Moreover, if A is self-adjoint, then exp(iA)is

unitary, and one can show that

QðexpðiAÞÞ ¼ expðiqðAÞÞ ½21

For later use, we note that, if {f

}

n2Z

is some

complete, orthonormal basis in h, then operators A

on h can be represented by infinite matrices

)

m, n2Z

with A

= (f

, Af

), and

qðAÞ¼

m;n

½22

where

(y)

) obey

;





¼ 

m;n

;





¼ 0 ½23

for all m, n. We also note that, in our definition of

q(A), we made a convenient choice of normal-

ization, but there is no physical reason to not choose

a different normalization and define

ðAÞ¼qðAÞbðAÞ½24

where b is some linear function mapping self-adjoint

operators A to real numbers. For example, one may wish

to use another reference vector

 instead of  in the

Fock space, and then would choose b(A ) = h

, q(A)

i.

Then the relations in [19] are changed to

½q

ðAÞ; q

ðBÞ ¼ q

ð½A; BÞ þ S

ðA; BÞ½25

where S

(A, B ) = b([A, B]). However, the C-number

term S

(A, B) in the relations [25] is trivial, since it

can be removed by going back to q(A).

Physical Interpretation

The Fock space F



(h) is the direct sum of subspaces

of states with different particle numbers N,



ðhÞ¼

N¼0

ðNÞ



½26

where the zero-particle subspace h

(0)



= C is gener-

ated by the vacuum , and h

(N)



is the N-particle

subspace generated by the states f

^ f

^^

, f

2 h. We note that

Nqð1Þ½27

is the ‘‘particle-number operator,’’ NF

= NF

for

all F

2 h

(N)



. The field operators obviously change

the particle number:

(f ) increases the particle

number by one (maps h

(N)



to h

(Nþ1)



), and (f )

decreases it by one. Since every f 2 h can be interpreted

as one-particle state, it is natural to interpret

(f )and

(f ) as ‘‘creation’’ and ‘‘annihilation’’ operators,

respectively: they create and annihilate one particle in

the state f 2 h. It is important to note that, in the

fermion case, [10] implies that

(f )

= 0, which is a

mathematical formulation of the Pauli exclusion

principle: it is not possible to have two fermions in the

same one-particle state. In the boson case, there is no

such restriction. Thus, even though the formalisms

used to describe boson and fermion systems look very

similar, they describe dramatically different physics.

Applications

In our example, the many-body Hamiltonian

 q(H) can also be written in the following

suggestive form:

ðxÞðH ÞðxÞ½28

Bosons and Fermions in External Fields 321

and similar formulas hold true for other observables

and other Hilbert spaces h = L

(M)  C

.Itis

rather easy to solve the model defined by such

Hamiltonian: all necessary computations can be

reduced to one-particle computations. For example,

in the static case, where A and  are time

independent, a main quantity of interest in statistical

physics is the free energy

E

1

log tr exp ð½H

 NÞðÞðÞ½29

where >0 is the inverse temperature,  the

chemical potential, and the trace over the Fock

space F



(h). One can show that

E¼tr 

1

logð1  expð½H  ÞÞ



½30

where the trace is over the one-particle Hilbert space

h. Thus, to compute E, one only needs to find the

eigenvalues of H.

It is important to mention that the framework

discussed here is not only for external field

problems but can be equally well used to for-

mulate and study more complicated models with

interparticle interactions. For example, while the

model with the Hamiltonian H

above is often too

simple to describe systems in nature, it is easy to

write down more realistic models, for example, the

Hamiltonian

H¼H

þðe

=2Þ

ðxÞ

ðyÞ

jx  yj

1

ðyÞ ðxÞ½31

describes electrons in an external electromagnetic

field interacting through Coulomb interactions. This

illustrates an important point which we would like

to stress: the task in quantum theory is twofold,

namely to formulate and to solve (exact of other-

wise) models. Obviously, in the nonrelativistic case,

it is equally simple to formulate many-body models

with and without interparticle interactions, and only

the latter are simpler because they are easier to

solve: the two tasks of formulating and solving

models can be clearly separated. As we will see, in

the relativistic case, even the formulation of an

external field problem is nontrivial, and one finds

that one cannot formulate the model without at

least partially solving it. This is a common feature of

quantum field theories making them challenging and

interesting.

Relativistic Fermion and Boson Systems

We now generalize the formalism developed in the

previous section to the relativistic case.

Field Algebras and Quasifree Representations

In the previous section, we identified the field

operators

(y)

(f ) with particular Fock space opera-

tors. This is analogous to identifying the operators

= i@

and q

= x

on L

) with the generators

of the Heisenberg algebra, as usually done. (We

recall: the Heisenberg algebra is the star algebra

generated by P

and Q

, j = 1, 2, ..., M < 1, with

the well-known relations

½P

; P

¼i

; ½P

; P

¼½P

; Q

¼0

¼ P

; Q

¼ Q

½32

for all j, k.) Identifying the Heisenberg algebra with

a particular representation is legitimate since, as is

well known, all its irreducible representations are

(essentially) the same (this statement is made precise

by a celebrated theorem due to von Neumann).

However, in case of the algebra generated by the

field operators

(y)

(f ), there exist representations

which are truly different from the ones discussed in

the last section, and such representations are needed

to construct relativistic external field problems. It is

therefore important to distinguish the fields as

generators of an algebra from the operators repre-

senting them. We thus define the (boson or fermion)

field algebra A



(h) over a Hilbert space h as the star

algebra generated by 

(f ), f 2 h, such that the map

f ! (f ) is linear and the relations

½ðf Þ ; 

ðgÞ



¼ðf; g Þ

½ðf Þ; ðgÞ



¼ 0



ðf Þ

¼ ðf Þ

½33

are fulfilled for all f , g 2 h,withy the star

operation in A



(h). The particular representation

of this algebra discussed in the last section will be

denoted by 

, 

(

(y)

(f )) =

(y)

(f ). Other represen-

tations 



can be constructed from any projection

operators P



on h, that is, any operator P



on h

satisfying P





= P



= P



.Writing

(y)

(f )shortfor





(

(y)

(f )), this so-called quasifree representation

is defined by

ðf Þ¼

ðP

f Þþ ðP



f Þ

ðf Þ¼ ðP

f Þ

ðP



f Þ

½34

where the bar means complex conjugation. It is

important to note that, while the star operation is

identical with the Hilbert space adjoint  in the

fermion case, we have

ðf Þ

¼ ðFf Þ



with

F ¼ P

 P



for bosons

½35

322 Bosons and Fermions in External Fields

where F is a grading operator, that is, F



= F and F

= 1.

We stress that the ‘‘physical’’ star operation always is ,

that is, physical observables A obey A = A



The present framework suggests to regard quantiza-

tion as the procedure which amounts to going from a

one-particle Hilbert space h to the corresponding field

algebra A

(h). Indeed, the Heisenberg algebra is

identical with the boson field algebra A



)(since

the latter is obviously identical with the algebra of M

harmonic oscillators), and thus conventional quantum

mechanics can be regarded as boson quantization in the

special case where the one-particle Hilbert space is

finite dimensional. It is interesting to note that

‘‘fermion quantum mechanics’’ A



) is the natural

framework for formulating and studying lattice fer-

mion and spin systems which play an important role in

condensed matter physics.

In the following, we elaborate the naive inter-

pretations of the relativistic equations in [2] and [3]

as a quantum theory of one particle, and we discuss

why they are unphysical. For simplicity, we assume

that the electromagnetic fields , A are time inde-

pendent. We then show that quasifree representa-

tions as discussed above can provide physically

acceptable many-particle theories. We first consider

the Dirac case, which is somewhat simpler.

Fermions

One-particle formalism Recalling that i@

is the

energy operator, we define the Dirac Hamiltonian D

by rewriting [3] in the following form:

¼ D ; D ¼ðirþeAÞa þ m  e ½36

This Dirac Hamiltonian is obviously a self-adjoint

operator on the one-particle Hilbert space h = L

) 

, but, different from the Schro¨ dinger Hamiltonian in

[1], it is not bounded from below: for any E

> 1,

one can find a state f such that the energy expectation

value (f, Df )islessthanE

. This can be easily seen for

the simplest case where the external potential vanishes,

A =  = 0. Then the eigenvalues of D can be computed

by Fourier transformation, and one finds

E ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ m

; p 2 R

½37

Due to the negative energy eigenvalues we conclude

that there is no ground state, and the Dirac

Hamiltonian thus describes an unstable system,

which is physically meaningless.

To summarize: a (unphysical) one-particle

description of relativistic fermions is given by a

Hilbert space h together with a self-adjoint Hamil-

tonian D unbounded from below. Other observables

and transformations are given by self-adjoint and

unitary operators on h, respectively.

Many-body formalism We now explain how to

construct a physical many-body description from these

data. To simplify notation, we first assume that D has a

purely discrete spectrum (which can be achieved by

using a compact space). We can then label the eigen-

functions f

by integers n such that the corresponding

eigenvalues E

 0forn  0andE

< 0forn < 0.

Using the naive representation of the fermion field

algebra discussed in the last section, we get (we use the

notation introduced in [22])

qðDÞ¼

n0



n<0

½38

which is obviously not bounded from below and thus

not physically meaningful. However,

= 1 

which suggests that we can remedy this problem by

interchanging the creation and annihilation operators

for n < 0. This is possible: it is easy to see that



8n  0and



8n < 0 ½39

provides a representation of the algebra in [23].We

thus define

qðDÞ

n2Z

: ½40

with the so-called normal ordering prescription

:

 ;





½41

where we made use of the freedom of normalization

explained after [23] to eliminate unwanted additive

constants. We get q(D) =

n2Z

, which is

manifestly a non-negative self-adjoint operator with

 as ground state. We thus found a physical many-

body description for our model. We can now define

for other one-particle observables,

qðAÞ

n2Z

: ½42

and, by straightforward computations, we obtain

qðAÞ;

qðBÞ ¼

qð½A; BÞ þ SðA; BÞ½43

where S(A, B) =

m<0

n0

 B

that is,

SðA; BÞ¼tr P



 P



ðÞ½44

with P



n<0

, ) the projection onto the

subspace spanned by the negative energy eigenvec-

tors of D and P

= 1  P



. One can show that

q(A)

is no longer defined for all operators but only if



and P



are

Hilbert–Schmidt operators ½45

(we recall that a is a Hilbert–Schmidt operator if

tr(a



a) < 1). The C-number term S(A,B)in[43] is

Bosons and Fermions in External Fields 323

often called Schwinger term and, different from the

similar term in [25], it is now nontrivial, that is, it is

no longer possible to remove it by a redefinition

(A) =

q(A)  b(A). This Schwinger term is an

example of an anomaly, and it has various interest-

ing implications.

In a similar manner, one can construct the many-

body transformations

Q(U) of unitary operators U

on h satisfying the very Hilbert–Schmidt condition

in [45], and one obtains

QðUÞ

QðVÞ¼ðU; VÞ

QðUVÞ½46

with interesting phase-valued functions .

More generally, for any one-particle Hilbert

space h and Dirac Hamiltonian D, the physical

representation is given by the quasifree representa-

tion 



in [34] with P



the projection onto the

negative energy subspace of D. The results about

and

Q mentioned hold true in any such

representation.

Thus the one-particle Hamiltonian D determines

which representation one has to use, and one

therefore cannot construct the ‘‘physical’’ represen-

tation without specific information about D. How-

ever, not all these representations are truly different:

if there is a unitary operator U on the Fock space

(h) such that





ð1Þ



ðyÞ

ðf ÞÞU ¼ 

ð2Þ



ðyÞ

ðf ÞÞ ½47

for all f 2 h, then the quasifree representations

associated with the different projections P

(1)



and

(2)



are physically equivalent: one could equally well

formulate the second model using the representation

of the first. Two such quasifree representations are

called unitarily equivalent, and a fundamental

theorem due to Shale and Stinespring states that

two quasifree represent ations 

(1, 2)



are unitarily

equivalent if and only if P

(1)



 P

(2)



is a Hilbert–

Schmidt operator (a similar result holds true in the

boson case).

Bosons

One-particle formalism Similarly as for the Dirac

case, the solutions of the Klein–Gordon equation in

[2] also do not define a physically acceptable one-

particle quantum theory with a ground state: the

energy eigenvalues in [37] for A =  = 0 are a

consequence the relativistic invariance and thus

equally true for the Klein–Gordon case. However,

in this case there is a further problem. To find the

one-particle Hamiltonian, one can rewrite the

second-order equation in [2] as a system of first-

order equations,

 ¼ K

 ¼





; K ¼

C i

iB



½48

with

ðirþeAÞ

þ m

; C e ½49

Thus, one sees that the natural one-particle Hilbert

space for the Klein–Gordon equation is

h = L

)  C

; here, and in the following, we

identify h with h

 h

, h

= L

), and use a

convenient 2  2 matrix notation naturally asso-

ciated with that splitting. However, the one-particle

Hamiltonian is not self-adjoint but rather obeys



¼ JKJ; J 

0 i



½50

with  the Hilbert space adjoint. It is important to

note that J is a grading operator. Thus, we can

define a sesquilinear form

ðf ; gÞ

ðf ; JgÞ8f ; g 2 h ½51

with ( , ) the standard inner product, and [50] is

equivalent to K being self-adjoint with respect to

this sesquilinear form; in this case, we say that K is

J-self-adjoint. Thus, in the Klein–Gordon case, this

sesquilinear form takes the role of the Hilbert space

inner product and, in particular, not (,) but (,)

preserved under time evolution. However, different

from 

, 

J is not positive definite, and it is

therefore not possible to interpret it as probability

density as in conventional quantum mechanics. For

consistency, one has to require that one-particle

transformations U are unitary with respect to (,)

that is, U

1

= JUJ. We call such operators J-unitary.

To summarize: a (unphysical) one-particle

description of relativistic bosons is given by a

Hilbert space of the form h = h

 h

, the grading

operator J in [50], and a J-self-adjoint Hamiltonian

K of the form as in eqn [48], where B  0 and C are

self-adjoint operators on h

. Other observables and

transformations are given by J-self-adjoint and

J-unitary operators on h, respectively.

Many-body form alism We first consider the quasi-

free representation 

(0)



of the boson field algebra



(h) so that the grading operator in [35] is

equal to J, that is, P

(0)



= (1  J)=2. Writing



(0)



(

(y)

(f )) =

(y)

(f ), one finds that

qðAÞ



¼ qðJAJÞ; QðU Þ



¼ Qð JU



JÞ½52

and thus J-self-adjoint operators and J-unitary

operators are mapped to proper observables and

transformations. In particular, q(K) is a self-adjoint

324 Bosons and Fermions in External Fields