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

Подождите немного. Документ загружается.

where h is Planck’s constant, c the speed of light,

and m the particle mass. Using from now on units so

that h = c = 1, this can be abbreviated as

¼0





 m

ðxÞ¼0; x ¼ðx

; x



¼ @=@x



;¼ 0; 1; 2; 3 ½5

The relativistic invariance of this equation can be

understood as follows. First, since the equation

does not explicitly involve the spacetime coordi-

nates, it is invariant under spacetime translations.

(If (t, x) solves [5], then also (t  a

, x  a)isa

solution for all (a

, a) 2 R

.) Second, it is invariant

under Lorentz trans formations (rotations and

boosts). Indeed, if (x) is a solution and L 2

SO(1, 3), then S(L)(L

1

x) solves [5] too, where

S(L) denotes a (suitably normalized) matrix

satisfying

SðLÞ

1





SðLÞ¼

¼0









½6

(The matrices 

0

on the right-hand side of [6] satisfy

the -algebra [1]. From this, the existence of a

representation S(L) of SO(1, 3) satisfying [6] is

readily deduced.)

As a consequence, the Poincare´ group (inhomo-

geneous Lorentz group) acts in a natural way on the

space of solutions to the time-dependent Dirac

equation, expressing its independence of the choice

of inertial frame. For quantum mechanical purposes,

however, one needs to choose a frame and use the

associated time variable to rewrite the equation as a

Hilbert space evolution equation.

The relevant Hilbert space

H is the space of four-

component functions that are square integrable over

space,

H¼L

ðR

; dxÞC

½7

To obtain a self-adjoint Hamiltonian on

H, one multi-

plies [5] by 

and introduces the Hermitian matrices

 ¼ 

;

¼ 



; k ¼ 1; 2; 3 ½8

Then, one obtains the Schro¨ dinger type equation

H ½9

where

H is the Dirac operator,

H ¼i rþm ½10

Under Fourier transformation,

F :

H!L

ðR

; dpÞC

ðxÞ7!ðpÞ¼ð2Þ

3=2

dx expðix  pÞ ðxÞ

½11

eqn [9] turns into

 ¼ DðpÞ; D ðpÞ¼a  p þ m ½12

The matrix D(p) is Hermitian and has square E

where E

is the relativistic energy,

¼ðp  p þ m

1=2

½13

corresponding to a momentum p. Now, we have

DðpÞ¼DðpÞU

½14

where U

is the charge conjugation matrix,

¼ i

½15

Hence, the four eigenvalues of D(p) are given by

, E

, E

, and E

. Therefore, the matrices



ðpÞ¼



DðpÞ



½16

are projections on the positive and negative spectral

subspaces of D(p).

As orthonormal base for the positive-energy sub-

space, we can now choose

þ;j

ðpÞ¼

þ m



1=2

ðpÞb

; j ¼ 1; 2 ½17

where

ﬃﬃﬃ

; b

ﬃﬃﬃ

½18

Next, setting

;j

ðpÞ¼U

þ;j

ðpÞ; j ¼ 1; 2 ½19

an orthonormal base w

,1

(p), w

,2

(p) for the

negative-energy subspace of D(p) is obtained; cf. [14].

The upshot is that the time-independent Dirac

equation

H ¼ E ½20

76 Dirac Operator and Dirac Field

gives rise to bounded eigenfunctions

þ;j

ðx; pÞ

¼ð2Þ

3=2

expðix  pÞw

þ;j

ðpÞ; j ¼ 1; 2

;j

ðx; pÞ

¼ð2Þ

3=2

expðix  pÞw

;j

ðpÞ; j ¼ 1; 2

½21

with eigenvalues E = E

and E = E

, resp. Clearly,

they are not square-integrable, but they can be used

as the kernel of a unitary transformation between

H (7) and the Hilbert space

H¼H

H



¼ P

HP



; H



¼ L

ðR

; dpÞC

½22

Specifically, we have

W : H!

f ðpÞ¼ðf

ðpÞ; f



ðpÞÞ

7! ðxÞ¼

¼þ;

j¼1;2

dp e

;j

ðx; pÞf

;j

ðpÞ½23

which entails

ðW

1

;j

ðpÞ¼

dx e

;j

ðx; pÞ ðxÞ½24

(Here and throughout this article, a bar denotes

complex conjugation.)

From the above, it is clear that the Dirac

Hamiltonian

H acting on the Hilbert space

H is

unitarily equivalent to the multiplication operator on

H [22] given by

ðHf Þ



ðpÞ¼E



ðpÞ;¼þ; ½25

Indeed, W is a diagonalizing transformation for

the relation

H ¼ W

1

HW ½26

yielding an explicit realization of the spectral

theorem.

Using the same notational convention, the

momentum, charge conjugation, parity, and time-

reversal operators on

H, given by

ÞðxÞ¼i@

ðxÞ½27

C ÞðxÞ¼U



ðxÞ½28

P ÞðxÞ¼U

ðxÞ; U

¼ 

½29

T ÞðxÞ¼U



ðxÞ; U

¼ 



½30

transform into the operators

ðP

f Þ



ðpÞ¼p



ðpÞ;¼þ; ½31

ðCf Þ



ðpÞ¼





ðpÞ;¼þ; ½32

ðPf Þ



ðpÞ¼f



ðpÞ;¼þ; ½33

ðTf Þ



ðpÞ¼i





ðpÞ;¼þ; ½34

Note that P

, P,andT leave the positive- and

negative-energy subspaces H

and H



invariant,

whereas C interchanges them.

To conclude this section, we describe some salient

features of the unitary representation of the (identity

component of the) Poincare´ group on H, which

follows from the representation on solutions to [5]

already sketched. The spacetime translations over

a 2 R

are represented by the unitary operator

exp (ia

H þ ia P); explicitly,

ðexpðia

H þ ia PÞf Þ



ðpÞ

¼ expðiða

 a  pÞÞf



ðpÞ;¼þ; ½35

The representation of the Lorentz group involves

unitary 2  2matricesU(k, A), where k is an

arbitrary 4-vector satisfying k



= 1andA the

matrix in SL(2, C) representing L 2 SO(1, 3). (Recall

that SL(2, C) can be viewed as a 2-fold cover of

SO(1, 3).) In particular, U(k, A) does not depend on

k for rotations,

Uðk; AÞ¼A



; 8A 2 SUð2Þ½36

(Here and henceforth, we use  to denote the

Hermitian adjoint of matrices and ope rators.) For

boosts, however, there is dependence on the vector

k, which is the image of the vector (1, 0) under the

boost. We refrain from a more detailed description

of U(k, A), as this would carry us too far afield.

The unitary SO(1, 3) representation leaves the

decomposition H= P

HP



H invariant. On the

positive-energy subspace H

, it is given by

ðUð LÞf Þ

ðpÞ¼



1=2

; A





ðp

Þ½37

where

p ¼ðE

; pÞ ; p

¼ L

1

p ½38

On H



, it is given by the complex-conjugate

representation,

ðUð LÞf Þ



ðpÞ¼



1=2

; A





ðp

Þ½39

Dirac Operator and Dirac Field 77

just as for the spacetime translations, cf. [35]. (The

superscript t is used to denote the transpose matrix.)

This feature is crucial for the second-quantized

Dirac theory, which is discussed next.

The Free Dirac Field in R

The free Dirac field is an operator-valued distribution

on a Fock space that describes an arbitrary number of

spin-1/2 particles and antiparticles in terms of momen-

tum space wave functions. Since spin-1/2 particles are

fermions (which encodes the Pauli exclusion principle),

an M-particle wave function F

þ,...,þ

,...,j

, ..., p

)(where

2 {1, 2} is the spin index) is antisymmetric under any

interchange of a pair (j

, p

)and(j

, p

). Likewise,

N-antiparticle wave functions F

,...,

,...,k

, ..., q

)

are antisymmetric. But a wave function F

þ

j,k

(p, q)

describing a particle–antiparticle pair need not have

any symmetry property, since a particle and an

antiparticle can be distinguished by their charge.

The rel evant Fock space is therefore the tensor

product of two antisymmetric Fock spaces built over

the one-particle a nd one-antiparticle spaces

,dp)  C

. For later purposes, it is important

to view these spaces as the summands H

and H



the space H from the previous section. Thus, the

arena for the free Dirac field is the Hilbert space

ðHÞ ’ F

ðH

ÞF

ðH



Þ½40

where, for example,

ðHÞ ¼ ðC HðHHÞ

Þ



½41

where the bar denotes the completion of the infinite

direct sum in the obvious inner product. The tensor

(1, 0, 0, ...) is viewed as the vacuum (the ‘‘filled

Dirac sea’’) and denoted by .

To get around in Fock space, one employs the

creation and annihilation operators c

()

(f ), f 2H.

The creation operator c



(f ), f 2H, is defined by

linear and continuous extension of its action on the

vacuum  and on elementary antisymmetric tensors,

recursively given by



ðf Þ ¼ f ; c



ðf Þf

¼ f ^ f

; ...



ðf Þf

^^f

¼ f ^ f

^^f

; ...

½42

Its adjoint, the annihilation operat or c(f ), satisfies

cðf Þ ¼ 0; cðf Þf

¼ðf ; f

Þ; ...

cðf Þf

^^f

j¼1

ðÞ

j1

ðf ; f

 f

^^

^^f

; ... ½43

Accordingly, the operators c

()

(f ) satisfy the canoni-

cal anticommutation relatio ns (CARs) over H,

fcðf Þ; cðgÞg ¼ 0;

fcðf Þ; c



ðgÞg ¼ ðf ; gÞ; 8f ; g 2H ½44

where {A, B} denotes the anticommutator AB þ BA.

(From this, one readily deduces that c

()

(f )is

bounded with norm kf k.)

Next, recalling the direct sum decomposition [22],

a notation change

ðÞ

ðP

f Þ!a

ðÞ

ðP

f Þ; c

ðÞ

ðP



f Þ!b

ðÞ

ðP



f Þ½45

is made, thus indicating that a

()

and b

()

should be

viewed as the creation/annihilation operators of

particles and antiparticl es, resp. Since H

and H



are copies of L

,dp)  C

, a given function

(p), f

(p)) in the latter space can occur both as an

argument of a

()

() and of b

()

(); it can also be

viewed as a smearing funct ion for unsmeared

quantities a

()

(p) and b

()

(p), j = 1, 2, that are often

referred to as operators as well (even though they

are only quadratic forms). Thus, one has, for

example,



ðf Þ¼

j¼1

dpb



ðpÞf

ðpÞ

bðf Þ¼

j¼1

dpb

ðpÞ



ðpÞ

½46

As explained shortly, the smeared time-zero Dirac

field takes the form

ðf Þ¼aðP

f Þþb



ðKP



f Þ; f 2H ½47

Here and below, K denotes complex conjugation on

H, H

,andH



. Just as the operators c

()

(f ), the

operators 

()

(f ) satisfy the CARs over H,

fðf Þ; ðgÞg ¼ 0

fðf Þ; 



ðgÞg ¼ ðf ; gÞ; 8f ; g 2H

½48

as is readily verified using [44]–[45]. But this

-representation is not unitarily equivalent to the

c-representation [44]. This becomes clear in parti-

cular from the consideration of a crucial type of

CAR automorphism that is considered next.

To this end, we fix a unitary operator U on H.

Then it is plain that the operators

ðÞ

ðf Þ¼c

ðÞ

ðUf Þ½49



ðÞ

ðf Þ¼

ðÞ

ðUf Þ½50

78 Dirac Operator and Dirac Field

also satisfy the CARs. The CAR-algebra automorph-

ism c

()

(f ) 7!

()

(f ) can be unitarily implemented in

(H), since one has

ðÞ

ðf Þ¼ðUÞc

ðÞ

ðf ÞðU



Þ½51

where (U) denotes the Fock-space product opera-

tor corresponding to U. Thus, for example,

ðUÞ ¼ ; ðUÞf ¼ Uf ; ...

ðUÞf

^^f

¼ Uf

^^Uf

; ...

½52

For the CAR automorphism 

()

(f ) 7!



()

(f )thisis

not true, however. Rewriting it in terms of the

annihilation and creation operators a

()

and b

()

via

[47], it amounts to a linear transformation (Bogoliubov

transformation), whose unitary implementability has

been clarified several decades ago. To be specific, the

necessary and sufficient condition for unitary imple-

mentability is that the off-diagonal parts

þ

¼ P



; U

þ

¼ P



½53

in the 2  2 matrix decomposition of operators on

H be Hilbert–Schmidt operators. Therefore, no

problem arises when U is diagonal with respect to

this decomposition. Indeed, in that case one can

choose as unitary implementer the product operator

ðUÞ¼ðU

þþ

ÞðKU



KÞ½54

(cf. the tensor product structure [40] of F

(H)).

In particular, the automorphism

ðf Þ7!ðe

itH

f Þ½55

where H is the free diagonalized Dirac Hamiltonian

[25], is implemented by the operator

ðe

itH

Þ¼ðe

itE

Þðe

itE

Þ½56

where E denotes multiplication by E

on H

and H



The change of CAR representation, therefore, entails

that the unphysical negative energies of the one-

particle theory are replaced by positive energies of

antiparticles. Hence, we obtain a mathematically

precise version of Dirac’s hole theory substitut ion

(p) ! b



(p), b



(p) ! b

(p).

More genera lly, if one chooses for U the Poincare´

group representation (given by [35] and [37]–[39]),

then the Fock-space implementer [54] is the tensor

product of two product operators with the same

action on F

,dp)  C

). Observe that this is

also true for the Fock-space version

(T) =(T)of

the time-reversal operator [34] . By contrast, the

Fock-space parity operator

(P) =(P) gives rise to

two product operators with slightly different

actions, cf. [33]. Accordingly, particles and anti-

particles have opposite parity.

The map

ðf Þ7!ðCf Þ



½57

also yields a CAR automorphism. It is unitarily

implemented by the Fock-space charge-conjugation

operator

C¼



½58

which interchanges particles and antiparticles.

Notice that C is unitary, whereas C is antiunitary.

It remains to establish the precise relation of the

above to the customary free Dirac field (t, x). This

is a quadratic form on F

(H) given by

ðt ; xÞ¼ð2Þ

3=2

j¼1;2



ðpÞw

þ;j

ðpÞe

iE

tþipx

þb



ðpÞw

;j

ðpÞe

tipx



½59

(Its expectation hF

, (t, x)F

i is, for example, well

defined for F

, F

in the dense subspace of F

(H)

that consists of vectors with finitely many particles

and antiparticles and wave functions in Schwartz

space.) It satisfies the time-dependent Dirac equation

 ¼ði rþmÞ ½ 60

in the sense of quadratic forms. Furthermore,

smearing it with a function (x) in the Hilbert

space

H (7), we obtain

dx ðxÞðt; xÞ

¼ ðe

itH

1

ðe

itH

ÞðW

1

ðe

itH

Þ 2

H½61

As announced, the time evolution of the free Dirac

field is, therefore, given by the unitary one-

parameter group [56], whose generator (the sec-

ond-quantized Dirac Hamiltonian) has spectrum

{0} [ [m, 1).

The Dirac field (t, x) can also be smeared with a

test function F(t, x) in the Schwartz space S(R

)

yielding a bounded operator

ðFÞ¼

dxFðxÞðxÞ½62

Then one obtains the relativistic covariance relation

ðUða; LÞÞðFÞ

ðUða; LÞÞ



¼ ðF

a;L

Þ½63

where

a;L

ðxÞ¼SðL

1

FðL

1

ðx  aÞÞ ½64

Dirac Operator and Dirac Field 79

and U(a, L) denotes the Poincare´ group representa-

tion on H, cf. [35] and [37]–[39]. Likewise, one gets

the inversion formulas

ðIÞðFÞ

ðIÞ



¼ ðF

Þ; I ¼ P; T ½65

with

ðt; xÞ¼U

Fðt; xÞ; F

ðt; xÞ¼U



Fðt; xÞ½66

while the Fock-space charge-conjugation operator

[58] transforms the Dirac field as

CðFÞC ¼ ð F



½67

with

ðxÞ¼U



FðxÞ½68

Finally, let us consider the global U(1) gauge

transformations f 7!e

i

f , where  2 R and f 2H.

They can be implemented by

ðe

i

Þ¼ðe

i

Þðe

i

Þ½69

and one has

ðe

i

ÞðFÞ

ðe

i



¼ ðF



Þ½70

with



ðxÞ¼e

i

FðxÞ½71

The generator Q of the one-parameter group

 7!

(e

i

) is the charge operator: on wave functions

describing N

particles and N



antiparticles, it has

eigenvalue N

 N



More on the One-Particle Dirac Theory

Even for the free one-particle setting, the account

given earlier is far from complete. To begin with,

the free D irac equation admits a specialization to

massless particles. In the Weyl representa tion of

the -algebra adopted above, the choice m = 0

entails that the p-space equation [12] decouples

into tw o 2  2 equations for spinors that can be

labeled by their chirality (‘‘handedness’’). This

refers to their eigenvalue with respect to the

chirality m atrix



¼ i



0 1



½72

and this notion derives from the noninvariance of

the separate 2  2 equations under parity. (A

positive-chirality spinor is mapped to a negative-

chirality spinor under the parity operator

P (33) and

vice versa.) Since the weak interaction breaks parity

symmetry, the two 2  2 equations (often called

Weyl equations) do have physical relevance. Indeed,

the associated quantum fields are a crucial ingredi-

ent of the standard model.

Next, we point out that it is possible to switch to

a representa tion in which the gamma matrices are

real. This so-called Majorana representation is

convenient (but not indispensable) in the description

of neutral spin-1/2 particles. By definition, such

particles are equal to their antiparticles, so that the

second-quantized formalism of the previous section

must be adapted: one needs the neutral CAR algebra

over H (also known as self-dual CAR).

For vari ous purposes, it is important to formulate

the free Dirac equation for a spacetime whose

spatial dimension is arbitrary. Then one needs, first

of all, gamma matrices satisfying the (Minkowski)

Clifford algebra relations









þ 







¼ 2g





; g ¼ diagð1; 1

Þ½73

where n is the space dimension and the minimal size

   of the gamma matrices is to be determined.

Clearly, for n = 1andn = 2, one can take =2,

choosing, for example,





;

10





i 0

0 i



½74

to fulfill [73]. For n = 4, one can take =4, just

as for n = 3, supplementing [1] with the matrix i

cf. [72].

More generally, for n = 2N  1andn = 2N, one

can take =2

in [73]. Indeed, a representation on

the 2

-dimensional fermion Foc k space F

)

(cf. [41]) is readily constructed using the creation

and annihilation operators described in the previous

section. Once this has been taken care of, most

of the discussion on the free one-particle Dirac

equation in R

can be easily generalized. Of special

importance in this regard is the straightforward

adaptation of the formulas [7]–[26], which form

the foundation for the second-quantized version.

Indeed, the discussion of the last section applies

nearly verbatim for arbitrary spacetime dimension.

In several applications, the so-called Euclidean

version of the free Dirac theory in spacetime

dimension n þ 1 is important. Basically, this version

is obtained upon replacing i@

by @

nþ1

in the Dirac

equation, a substitution that changes the character

of the equation from hyperbolic to elliptic. Pro-

vided that the mass vanishes, the Euclidean Dirac

equation admits a reinterpretation as a time-

independent zero-eigenvalue Weyl equation in a

Minkowski spacetime of dimension n þ 2. (This

equation is often called the zero-mode equation.)

80 Dirac Operator and Dirac Field

Let us now turn to the description of the

interaction with an external electromagnetic poten-

tial A



(t, x). This can be taken into account via the

minimal substitution,



! @



þ ieA



½75

also known as the covariant derivative, in the time-

dependent Dirac equation [5].

For the electron in the Coulomb field of a nucleus

of charge Ze, one has

¼ 0; k ¼ 1; 2; 3; A

¼

4jxj

½76

and the time-independent equation

ia rþm 

4jxj



¼ E ½77

can be solved explicitly. This leads to a bound-state

spectrum that is more accurate than its nonrelativis-

tic counterpart. In particular, one finds that energy

levels that are degenerate in the nonr elativistic

theory split up into slightly different levels. The

resulting fine structure of the Di rac levels can be

understood as a consequence of the coupling

between the spin of the electron and its orbital

motion.

In spite of this better agreement with the

experimental levels, the phy sical interpretation of

the Dirac electron in a Coulomb field is enigmatic.

This is not only because of the persistence of the

negative-energy states of the free theory (which

turn into scattering states), but also because of

unphysical properties of the position operator.

More general time-independent external fields

(such as step potentials A

(x) with a step height

larger than 2m) can cause transitions between

positive- and negative-energy states (Klein para-

dox). This phenomenon is enhanced when time

dependence is allowed. In particular, any external

field that is given by functions in C

)leadsto

a scattering operator S on the one-particle space H

[22] that has nonzero off-diagonal parts S



Hence, a positive-energy wave packet scattering

at such a tim e- and space-l ocalize d field has a

nonzero probability to show up as a negative-

energy wave packet.

When one tensors the one-particle space

H with

an internal symmetry space C

, one can also

couple external Yang–Mills fields A



taking values

in the k  k matrices via the substitution [75].

(From a geometric viewpoint, this can be

rephrased as tensoring the spinor bundle with a

vector bundle equipped with a connection A .) The

generalization of this external gauge field coupling

to a Minkowski spacetime or Euclidean space of

arbitrary dimension is stra ightforward. An adapta-

tion of the resulting interacting one-particle Dirac

theory in arbitrary dimension to quite general

geometric settings also yields a crucial starting

point for index t heory.

Before turning to the latter area, we conclude this

section with another striking application of the one-

particle framework, namely the massless Dirac

equation in two spacetime dimensions with special

external fields. Specifically, the relevant Dirac

operator is of the form

iqðxÞ

irðxÞi

½78

where r(x)andq(x) are not necessarily real valued.

(Note that this operator is in general not self-

adjoint.) With suitable restrictions on r and q,

the direct and inve rse scatt ering theor y associa-

ted with the Dirac operator [78] can be applied

to various nonlinear PDEs in two spacetime

dimensions to solve their Cauchy problems in

considerable detail. As a crucial special case,

initial conditions yielding vanishing reflection

give rise to soliton solutions for the pertinent

equation.

The first example in this framework was found by

Zakharov and Shabat (the nonlinear Schro¨ dinger

equation); with other choices of r and q several other

soliton PDEs (including the sine-Gordon and mod-

ified Korteweg–de Vries equations) were handled by

Ablowitz, Kaup, Newell, and Segur, who studied a

quite general class of external fields r and q.

The Dirac Operator and Index Theory

Thus far, we have considered various versions

of the Dirac operator associated with the spaces

for some l  1. For applications in the area

of index theory, however, one needs to generalize

this base manifold. Indeed, one can d efine a Dirac

operator for any l-dimensional oriented Rieman-

nian manifold M that adm its a spin struc ture.

This is a lifting of the transition functions of the

tangent bundle TM (which may be assumed to

take values in SO(l)) to the simply connected

twofold cover Spin(l) (ta king l  3).

Choosing first l = 2N þ 1, the spin group has a

faithful irreducible representation on C

. Hence,

one obtains a C

-bundle over M, the spinor

bundle. The Levi-Civita connection on M derived

from the m etric can now be lifted to a connection

on the spinor bundle. From the covariant deriva-

tive corresponding t o the spin connection and the

Dirac Operator and Dirac Field 81

Clifford algebra generators 

, ..., 

,onecan

then construct a first-order elliptic differential

operator that acts on sections of the spinor

bundle. (For the case M= R

2Nþ1

with its Eucli-

dean metric, this construction yields the massless

positive-chirality Dirac operator acting on wave

functions with 2

components, as considered

above.)

The massless Dirac operator thus obtained is

self-adjoint as an operator on the L

-space H

associated with the spinor bundle, and it has

infinite-dimensional positive and negative spectral

subspaces H

and H



. (In this section the check

accent on position-space quantities is omitted.)

Specializing to the case of compact M, a contin-

uous map from M to C



gives rise to a Fredholm

operator on H

, and more generally a continuous

map from M to GL(k, C) yields a Fredholm

operator on H

 C

For a smooth map, the Fredholm index of this

operator can be written in terms of an integral over

M involving certain closed differential forms. The

value of this integral does not change when exact

forms are added, since M has no boundary. Hence,

one is dealing with de Rham cohomology classes. In

this context, the class invol ved (‘‘characteristic

class’’) is determined by the Riemann curvature

tensor of M and the topological (‘‘winding’’)

characteristics of the map.

The simplest example of this state of affairs arises

for l = 1andM= S

with its obvious spin structure

(periodic boundary conditions). Writing 2H= L

)as

ðzÞ¼

n2Z

; z 2 S

½79

the Dirac operator H on H reads

H ¼ z

½80

It has eigenfunctions z

, n 2 Z. Thus, we may

choose

ðP

ÞðzÞ¼

n0

; ðP



ÞðzÞ¼

n<0

½81

As a consequence, the functions in H



) are

-boundary values of holomorphic functions in

jzj < 1(jzj >1). Operators of the form

¼ P

½82

where is a continuous function on S

and M

denotes multiplication by , are called Toeplitz

operators. It is not hard to see that they are Fredholm

(viewed as operators on H

), provided that does not

vanish on S

. (Recall a bounded operator B is

Fredholm if it has finite-dimensional kernel K and

cokernel C. Its Fredholm index is given by

indexðBÞ¼dim K  dim C ½83

and is norm continuous and invariant under addi-

tion of a compact operator.) Assuming (S

)  C



from now on, the curve (S

) has a well-defined

winding number w( ) with respect to the origin. The

equality

indexðT

Þ¼wð Þ½84

between objects from the area of analysis on the

left-hand side and from the areas of topology and

geometry on the right-hand s ide is the simplest

example of an Atiyah–Singer type index formula.

When is not only continuous but also smooth,

the index formula can be rewritten as

indexðT

Þ¼

2i

½85

yielding a characteristic class version.

It should be noted that the operator M

on H

has a bounded inverse M

when 0 =2 (S

), hence a

trivially vanishing index. Therefore, the compres-

sion [82] involving the spectral projection of the

Dirac operator is needed to get a nonzero index.

Observe also that the equality [84] is quite easily

verified for the case (z) = z

, since T

yields a

power of the right (n > 0) or left (n < 0) shift on

H’l

(N).

We proceed to the case of even-dimensional

manifolds, l = 2N. Then the fibe r C

of the spinor

bundle splits into a direct sum of even and odd

spinors, corresponding to two distinct representa-

tions of Spin(2N)onC

N1

. (Here it is assumed

that N > 3; recall the Lie algebra isomorphisms

so(4) ’ so(3)  so(3) and so(6) ’ su(4).) With respect

to this decomposition, the Dirac operator can be

written as

H ¼

0 D



D 0



½86

where D and D



are again first-order elliptic

differential operators expressed in terms of Clifford

algebra generators and the spin connection. Tensor-

ing the spinor bundle with a vector bundle equipped

with a connection A, one can define a Dirac

operator on the tensor product which involves A

and takes the form

0 D





½87

82 Dirac Operator and Dirac Field

with respect to the even/odd spinor decomposition.

Once more, the index of D

(viewed as a Fredholm

operator between tw o different Hilbert spaces) can

be expressed as an integral over M involving

characteristic classes that depend on the curvatures

of the two connections.

Probably the simplest example of the construc-

tions just sketched is given by the torus M= S

 S

with its flat metric. Employing the above coordinate

and spin structure on S

, one can take

H¼L

ðS

S

ÞC

; D¼z

þiz

½88

Since the curvature vanishes, the index theorem for

this situation implies index(D)= 0. (Note that this is

also plain from [88]: both kernel and cokernel of D

are spanned by the constant sections.) On the other

hand, when one tensors the spinor bundle with a line

bundle with connection A, the index formula reads

indexðD

Þ¼

2

S

F ½89

where F is the curvature 2-form corresponding to A.

The Atiyah–Singer index theorem for Dirac

operators has far-reaching applications. It can be

used to derive other results in this area, such as the

Gauss–Bonnet–Chern theorem, the Hirzebruch sig-

nature theorem, and (when M is a Ka¨ hler manifold)

Riemann–Roch type theorems. From this, one can

obtain information on various questions, such as the

existence of positive scalar curvature metrics or

zeros of vector fields on M. Other applications

include insights on topological invariants of mani-

folds obtained from ‘‘simple’’ manifolds (such as

spheres and tori) by glueing or covering operations.

This hinges on the additive properties of the index

that are clear from its being given by an integral

over the manifold. Conversely, the integrality of

Fredholm indices can be used to deduce that certain

rational cohomology classes are actually integral on

manifolds that admit the structure that is required

for the pertinent index theorem to apply, that

certain manifolds do not admit such structures,

since one knows that the relevant class is not

integral, etc.

More on the Dirac Field

As mentioned earl ier, the free-field formalism can be

easily generalized to an arbitrary spacetime dimen-

sion d. For d > 4, however, no renormalizable

interacting quantum field models involving the

Dirac field are known. For the physical case d = 4

the standard model involves various Dirac fields

interacting with quantized gauge fields and Klein–

Gordon fields. Although its perturbation theory is

renormalizable, its mathematical existence is to date

wide open.

It is far beyond the scope of this article to

elaborate on the analytical difficulties of relativistic

quantum field theories, let alone those associated

with the standard model. Even for d = 2 and 3, a

nonperturbative construction of interacting quan-

tum field models involving the Dirac field is an

extremely difficult enterprise. Apart from some

rigorous results on certain self-interacting Dirac

field models, the only interacting model that is

reasonably well understood from the constructive

field theory viewpoint is the Yukawa model for

d = 2 and 3. This describes the interaction between

the Dirac field  and a Klein–Gordon field , the

interaction term being formally given by g(





).

On the other hand, the interaction of the quantized

Dirac field with external classical fields is much more

easily understood and analytically controlled. As a

bonus, within this context, one can make contact

with various issues of physical and mathematical

relevance. We now proceed to sketch the external-

field framework and some of its applications.

Let us first consider the addition of an external

field term gV(t, x) to the free Dirac operator

H on

H¼L

ðR

; dxÞC



 C

½90

We assume from now on that the coupling g is real

and that V is a self-adjoint k  k matrix-valued

function on spacetime R

nþ1

with matrix elements

that are in C

nþ1

). Then the (interaction picture)

scattering operator S exists. It is unitary and has off-

diagonal Hilbert–Schmidt parts S



, so that a

unitary Fock-space S-operator

(S) implementing

the Bogoliubo v transformation generated by S

exists:

ðSÞðf Þ

ðSÞ



¼ ðSf Þ; 8f 2H ½91

The arbitrary phase in

(S) can be fixed by requiring

that the vacuum expectation value of

(S)be

positive. More precisely, this number is generically

nonzero and satisfies

jð;

ðSÞÞj ¼ detð1 þ T

1=2

½92

where T

is a positive trace class ope rator deter-

mined by S.

The vector

(S) is a superposition of wave

functions with an equal and arbitrary number of

particles and antiparticles. More generally, the

Fock-space S-operator

(S) leaves the subspaces of

(H)withafixedeigenvalueq 2 Z of the charge

operator Q i nva r ian t, and can create a nd

Dirac Operator and Dirac Field 83

annihilate an arbitrary number of particle–

antiparticle pairs.

The unitary propagator U(T

, T

) corresponding

to V(t , x) does not have Hilbert–Schmidt off-

diagonal parts (unless the spacetime dimension is

sufficiently small and special external fields are

chosen). Even so, the diagon al parts are Fredholm

with vanishing index, and the off-diagonal parts are

compact. Omitting the ill-defined determinantal

factor, these properties imply that one obtains a

renormalized quadratic form



rcn

(U(T

, T

)) satisfy-

ing the implementing relation



rcn

ðUðT

; T

ÞÞðf Þ

¼ ðUðT

; T

Þf Þ



rcn

ðUð T

; T

ÞÞ; 8f 2H ½93

in the quadratic form sense.

The above unitary operators on H yield Fredholm

diagonal parts whose indices vanish. (They are norm

continuous in g and reduce to the identity for g = 0.)

This is why their Fock-space implementers leave the

charge sectors invariant. Indeed, for a unitary

operator U on H with compact off-diagonal pa rts

the implementer maps the charge-q sector to the

charge-(q þ q(U)) sector, where

qðUÞ¼index ðU



Þ½94

Specializing to the case

n ¼ 2N  1;  ¼ 2;  ¼ 2

N1

½95

a unitary (k  k)-matrix multiplier

U on

H does

not have compact off-diagonal parts in general. But

when it is of the form

U ¼



 u

ðxÞ 0

0 1



 u



ðxÞ



½96

with respe ct to the chiral decomposition (the

generalization of the 

-decomposition [72] to even

spacetime dimension), then it suffices for compact-

ness of the off-diagonal parts that the matrices



(x) 2 U(k) are continuous and converge to 1

for

jxj!1.

Viewing R

2N1

as arising from S

2N1

via stereo-

graphic projection, the latter unitaries can be viewed

as continuous maps from S

2N1

to U(k), reducing to

at the north pole. As such, they yield elements of

the homotopy group 

2N1

(U(k)). By virtue of Bott’s

periodicity theorem, the latter group equals Z for

k  N. Thus, the maps u



have a well-defined

‘‘winding number’’ w(u



) 2 Z for k  N. From the

index formula

indexðU



Þ¼wðu

Þwðu



Þ½97

and [94] one now deduces that one can obtain

implementers



rcn

(U) effecting a nonzero charge

change from unitary maps with nonzero winding

number.

In particular, choosi ng k =  = 2

N 1

 N, there

exist quite special ‘‘kink maps’’

;a

ðxÞ2UðÞ;>0; a 2 R

2N1

½98

with winding number 1 and such that the quadratic

form implementers of the unitary multiplication

operators

þ;;a

¼



 u

;a

ðxÞ 0

0 1



 1



;;a

¼



 1



0 1



 u

;a

ðxÞ

½99

converge to (a linear combination of the chiral

components of) the free Dirac field (0, a) as the

kink size parameter  goes to 0.

For the special case N = 1, one can take

;a

ðxÞ¼

x  a  i

x  a þ i

½100

and the off-diagonal parts of U

,,a

are actually

Hilbert–Schmidt. Thus, the implementers can be

chosen to be unitary operators. But to get con-

vergence to the Dirac field components (0, a)



 ! 0, the unitary implementers

(U

,,a

) should be

renormalized by a multiplicative factor.

For the N = 1 case, the unitary multipliers [96]

give rise to loop groups. Indeed, requiring

lim

x!1



ðxÞ¼1

;¼þ; ½101

we are dealing with continuous maps S

! U(k).

From the viewpoint of the Dirac theory, these

groups are local gauge groups. The convergence to

the Dirac field just sketched can be used to great

advantage to clarify the structure of the correspond-

ing Fock-spac e gauge groups. Their Lie algebras

yield representations of Kac–Moody algebras, a

topic which is considered shortly.

Before doing so, it should be pointed out that

under some mild smoothness assumptions all of

the above unitary matrix multipliers can also be

viewed as S-operators associated with very special

external fields. Indeed, the gauge-transformed Dirac

operator



U ½102

is of the form

H þ VðxÞ½103

84 Dirac Operator and Dirac Field

where V(x) is a self-adjoint k  k matrix on

2N1

(a ‘‘pure gauge’’ field). If one now defines a

time-dependent external field by

Vðt; xÞ¼

VðxÞ; t  0

0; t < 0



½104

then

U equals the S-operator for V(t, x). (Equiva-

lently,

U is the t !1wave operator for the time-

independent external field V(x).)

To conclude this section we sketch some applica-

tions of the second-quantized Dirac formalism for

the special case N = 1, m = 0, and positive chirality.

Even though we could stick to the massless positive-

chirality Dirac operator id=dx on the line, it is

simpler and more natural to start from its counter-

part on the circle already considered in the last

section, cf. [80]. (Under the Cayley transform, the

positive- and negative-energy subspaces of id=dx

on L

(R) correspond to those of zd=dz on L

given by [81].) Letting z = e

i

, we then obtain

H ¼id=d;

H¼L

ð½0; 2; dÞ

H¼l

ðZÞ; H

¼ l

ðNÞ; H



¼ l

ðZ



½105

and a corresponding Dirac field

ðt ;Þ¼ð2Þ

1=2

n¼0

intþin

n¼1



n

intin

ðt;Þ2R ½0; 2½106

where

¼ cðe

Þ; l  0; b

¼ cðe

Þ; l < 0 ½107

and {e

}

l2Z

is the canonical basis of l

(Z).

Consider now the group GL(H) of bounded

operators on H with bounded inverses. The

transformation





ðf Þ7!



ðGf Þ; ðfÞ7!ðG

1

f Þ

f 2H; G 2 GLðHÞ ½108

leaves the CAR [48] invariant. Provided that G

belongs to the subgroup

ðHÞ

¼fG 2 GLðHÞjG



Hilbert–Schmidtg½109

there exists an implementer

(G)onF

(H):

ðGÞ



ðf Þ¼



ðGf Þ

ðGÞ;

ðGÞðf Þ¼ðG

1

f Þ

ðGÞ; 8f 2H ½110

In particular, the multiplication operator

expðhðxÞÞ; hðxÞ¼

k¼1

k

; z ¼ e

i

½111

belongs to G

(H) provided the sequence x

vanishes

sufficiently fast as k !1. Thus, one obtains an

implementer

(e

h(x)

), the so-called KP evolution

operator. This designation is justified by the vacuum

expectation value

ð xÞ¼ð ;

ðe

hðxÞ

ðGÞÞ; G 2 G

ðHÞ ½112

being a tau-function solving the hierarchy of KP

evolution equations in Hirota bil inear form, as first

shown by Sato and his Kyoto school. For example,

the KP equation itself,

¼ @

 2uu



xxx



½113

has the bilinear form

 4

þ 3



ð x þ yÞðx  yÞ



y¼0

¼ 0 ½114

the relation being given by

¼ x; x

¼ y; x

¼ t; u ¼ 2@

ln  ½115

The class of solutions to [113] thus obtained

includes not only the rational and soliton solutions

(which correspond to choosing

G as multiplication by

a rational function of z = e

i

that does not vanish on

), but also the finite-gap solutions associated with

compact Riemann surfaces. Moreover, for suitable

subgroups of G

(H), one obtains tau-functions for

related soliton hierarchies, including the Korteweg–de

Vries, Boussinesq and Hirota–Satsuma hierarchies.

Even though the class of solutions associated with

(H) via the Dirac formalism is large, it should be

noted that from the perspective of the Cauchy problem

for the pertinent evolution equations the solutions are

nongeneric, inasmuch as the initial data are real-

analytic functions.

Finally, we consider Lie algebra representations

related to the above special starting point [105] for

the second-quantized Dirac framework. Assume that

exp(tA) is a one-parameter group of bounded

operators on H with generator A in the Lie algebra

of G

(H),

ðHÞ

¼fA bounded j A



Hilbert–Schmidtg½116

Then one can take

ðexpðtAÞÞ ¼ expðt d

ðAÞÞ ½117

where d

(A) is the Fock-space operator uniquely

determined up to an additive constant by its

commutation relation

½d

ðAÞ; 



ðf Þ ¼ 



ðAf Þ; 8f 2H ½118

Dirac Operator and Dirac Field 85