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HEP-TH-9412174 e-mail address : tanim  b er 1994 y an DPNU-94-60 When Decem hep-th/9412174 tization wn that wn that the b er. A new provided byCERNDocumentServer tations as a utation rela- tize the mo del um brought toyouby hes the structure of o dimensional space- t represen tral extension. w wn that the zero-mo de  alen CORE . The algebra generated b b er and quan tum op erators. It is sho y the cen ta op erators ob ey anomalous com- animura um T Abstract ariables ob ey a new comm ted out that its top ological structure is b er of inequiv tro duced, it is sho trivial top ology um es momen Shogo wist relation. In addition, it is sho h is a eld-theoretical mo del p ossessing a non- (1) sigma mo del in the t U . It is p oin y the zero-mo de and the winding n utation relations is prop osed to quan tation spaces of quan e call t , whic e demonstrate that top ology enric R hw  tral extension is in 1 S tum eld theories. tum op erators is deformed b artment of Physics, Nagoya University, Nagoya 464-01, Jap tral extension mak e consider the utators. W ariables and the winding v haracterized b yp e of comm tion, whic cen m quan v the cen quan consequence of the non there are an in nite n resp ecting the top ological nature.to b e Hilb ert represen spaces are constructed t time trivial top ology c W Dep of ab elian sigma mo del in (1+1) dimensions Zero-mo de, winding n

1 Intro duction

As everyone recognizes, quantum theory is established as an adequate and unique

language to describ e microscopic phenomena including condensed matter physics and

also particle physics. In this pap er wewould like to develop quantum theory in the

direction to seek for another p ossibility which inheres in the theory itself. First we

shall explain the well-known feature of quantum theory and then we shall show the

direction to pursue.

To formulate a quantum theory we start from commutation relations among

generators. Generators de ne an algebra, which is asso ciative but not commutative.

When wehave a classical theory,we ordinarily bring commutators among canonical

variables by replacing the Poisson brackets. Then we construct a Hilb ert space to

b e an irreducible representation space of the algebra. An element of the algebra is

chosen and called a hamiltonian. Choice of a hamiltonian is restricted byphysical

requirements like self-adjointness, p ositivity and symmetries. The set formed by the

algebra, the representation space and the hamiltonian is called a quantum theory.

Then formulation itself is nished.

Next tasks are to solve it; wewanttoknow eigenvalues of various observables,

esp ecially we are interested in sp ectrum of the hamiltonian, wewant to know proba-

bility amplitude of transition b etween an initial state and a nal one. For physically

interesting theories, those are dicult tasks. Of course, we appreciate that they are

worth hard work. However it is not the direction in whichwewould like to pro ceed

in this pap er. We are interested in a rather formal asp ect.

A feature of quantum theory is that it requires a representation space; op erators

must b e provided with op erands. A commutator in quantum theory corresp onds to

aPoisson bracket in classical theory. But the Hilb ert space in quantum theory has

no corresp ondence in classical theory. State vectors, the sup erp osition principle and

amplitudes are characteristic concepts of quantum theory.We b egin with an algebra

of op erators, then we construct a representation space. That is a unique pro cedure

of quantum theory.

Here arise two questions; do es a representation space exist? Is it unique? If

there are inequivalent representations, calculation based on a di erent space gives a

di erent answer for a physical quantity, for example, sp ectrum or amplitude.

When we consider a particle in a Euclidean space, we b egin with the usual canon-

ical commutation relations;

[^x ;x^ ]= 0; (i; j =1;


;n) (1.1)

i j

[^x ;p^ ]=i ; (1.2)

i j ij

[^p ;p^ ]=0: (1.3)

i j

According to von Neumann's theorem the irreducible representation of the ab ove

algebra exists uniquely within a unitary equivalence class. Therefore there is no

problem in choice of a Hilb ert space. Although one may use the wave function

representation and another may use the harmonic oscillator representation, b oth

obtain a same result for calculation of a physical quantity.

Is there no need to worry ab out existence and uniqueness of a representation?

Actually it is needed. Wehave encountered a situation in which the uniqueness is

violated, when we consider a quantum eld theory. In a quantum eld theory we

construct a representation space by de ning a vacuum state and a Fock space. It was

found that in several mo dels there exist inequivalentvacuum states and they result in

inequivalentFock spaces. The di erentvacua are characterized by its transformation

prop erty under a certain symmetry op eration. Such a situation is called sp ontaneous

symmetry breaking (SSB). The discovery of SSB op ened rich asp ects of quantum eld

theories and led to deep understanding of the nature.

A eld theory deals with a system which has in nite degrees of freedom, namely

it is de ned with a in nite numb er of generators which are called eld variables. It

is known that SSB is related to the in nity of degrees of freedom. On the other hand

a particle has only nite degrees of freedom. Is there no o ccurence of inequivalent

representations in a quantum theory of a particle? (Usually a quantum theory with

nite degrees of freedom is called a quantum mechanics.)

A strange result was found; when Ohnuki and Kitakado [1] investigated a quan-

1

tum mechanics of a particle on a circle S , they showed that there are a in nite

numb er of inequivalent Hilb ert spaces. Those spaces are parametrized by a continu-

ous parameter ranging from 0 to 1. What they have shown is that even a system

with nite degrees of freedom can p ossess inequivalent representations when top ol-

ogy of the system is nontrivial. After that work, they studied a quantum mechanics

n

on a sphere S (n  2) and showed existence of an in nite numb er of inequivalent

Hilb ert spaces sp eci ed by a discrete index.

Let us turn to eld theories. The scalar eld theory is a eld-theoretical cor-

resp ondence to the quantum mechanics of a particle in a Euclidean space. This

eld theory is quantized by requiring the canonical commutators and constructing

the Fock space. There is also a corresp ondence in eld theories to a quantum me-

chanics on a nontrivial manifold. It is a nonlinear sigma mo del b ecause it has a

manifold-valued eld. For a review on nonlinear sigma mo dels, see the reference [2].

Originally the nonlinear sigma mo del is designed to describ e b ehavior of Nambu-

Goldstone (NG) b osons at low energy scale. NG b osons are massless excitations

asso ciated with SSB. When a continuous symmetry sp eci ed by a group G is broken

to a smaller symmetry sp eci ed by a subgroup H ,vacua form a manifold whichis

called a homogeneous space G=H . In this mo del NG b osons are describ ed by a eld

taking values in G=H . It is already known [3] that even the quantum mechanics

on G=H has inequivalent Hilb ert spaces. Therefore it is naturally exp ected that a

quantum eld theory with a manifold-valued eld may p ossess inequivalent Hilb ert

spaces. However in a usual approach, the nonlinear sigma mo del is quantized by the

canonical quantization as the scalar eld theory and is solved by the p erturbative

metho d. Thus the top ological nature of the theory is missed and only a Fock space

provides a representation.

If top ological prop erties of eld theories do not play an imp ortant role in physical

application, we could neglect them. However we know several mo dels whichhave

nontrivial top ology and in which top ology plays an imp ortant role. For instance,

the sine-Gordon mo del has top ological kinks [4]; some gauge theories have top olog-

ically nontrivial vacua, so-called  -vacua [5]; some nonlinear sigma mo dels have the

Wess-Zumino-Witten term, which re ects anomaly and top ology of vacua [6]; the

con guration space of nonab elian gauge elds mo dulo gauge transformations has an

extremely complicated top ology and causes the Grib ov problem [7], and so on. Hence

wewould like to develop quantum eld theories resp ecting top ological nature.

One of the aims of this pap er is to demonstrate that there are a lot of p ossibilities

in constructing of quantum eld theories, even if they are identical as classical theo-

ries. The second aim is to clarify the relation b etween top ology and quantization. In

1

the section 2 we will give a review of the quantum mechanics on S .We will discuss

physical implication of the existence of inequivalent representations. The section

3 is a main part of the present pap er. There we consider a simple but nontrivial

eld-theoretical mo del, the ab elian sigma mo del in (1+1) dimensions. We prop ose

a de nition of an algebra, which is quite di erent from the canonical one. Then we

will construct Hilb ert spaces and classify inequivalent ones. The section 4 is devoted

to discussion on the results and directions for future development. This pap er is a

detailed and extended sequel to the previous work [8].

1

2 Quantum mechanics on S

1

Here we shall give a review of the quantum mechanics on S [1]. Idea in it is useful

for the next step in the eld theory. More detailed consideration is found in the

reference [9].

2.1 Motivation

1

Let us consider a particle moving on a circle S . Its p osition is indicated by the angle

co ordinate  . The co ordinate  +2 indicates the same p ointas do es. Hence 

1

is multivalued function on S . Because of its nontrivial top ology, a continuous and

1

single-valued co ordinate do es not exist on S .

Now let us quantize it. We should de ne an algebra. If we take the canonical

^

^

approach, assume that  and P are self-adjoint op erators satisfying

^ ^

[ ; P ]=i: (2.1)

^

Next we should construct a representation space. If we de ne eigenstates of  by

^

 j i =  j i; (2.2)

we deduce that

@

^

h jP j i = i hj i (2.3)

@

^

for an arbitrary state j i. Notice that the eigenvalue of  ranges over from 1 to

+1;itisnot true that j +2i = ji. What wehave constructed is just the quantum

1

mechanics on R , not S .

^

What is wrong? It is wrong to use the multivalued co ordinate  as a generator of

the algebra. Wemust use a well-de ned single-valued generator from the b eginning.

i

As a substitute for  , Ohnuki and Kitakado prop osed to use U = e , whichis

complex-valued in the classical theory and a unitary op erator in the quantum theory.

From (2.1) we can deduce

^ ^

i i

^

[ e ; P ]=e : (2.4)

^

But they did not regard  as a generator. Instead they to ok existence of a unitary

^ ^

op erator U and a self-adjoint op erator P satisfying

^ ^ ^

[ U; P ]=U (2.5) as an assumption.

2.2 Representations

Nowwe see their construction of representations of the algebra de ned by the ab ove

^

commutation relation (2.5). Since P is a self-adjoint op erator, it has an eigenvector

with a real eigenvalue ;

^

P j i = j i; h j i =1: (2.6)

^ ^

U raises the eigenvalues of P ;

^ ^ ^ ^ ^ ^

P U j i = ([P;U ]+UP)j i

^ ^

= (U +U ) j i

^

= (1 + )U j i: (2.7)

1 y

^ ^

Inversely, U = U lowers them. Then we de ne

n

^

jn + i = U j i; (n =0;1;2;


) (2.8)

whichhave the following prop erties

^

P jn + i =(n+ )jn+ i; (2.9)

hm + jn + i =  : (2.10)

mn

^ ^

The latter follows from self-adjointness of P and unitarityof U. With a xed real

number ,we de ne a Hilb ert space H by completing the vector space of linear

combinations of jn + i (n :integer). Equation (2.9) with

^

U jn + i = jn +1+ i (2.11)

de nes an irreducible representation of the algebra (2.5) on H .

H and H are unitary equivalent if and only if the di erence ( )isan

integer. Therefore the classi cation of irreducible representations of the algebra

(2.5) has b een completed; the whole of inequivalent irreducible representation spaces

is fH g (0  <1). We call them Ohnuki-Kitakado representations.

2.3 Physical implications

Wave function

The physical meaning of the parameter seems obscure. To clarify it they [1] studied

^

eigenstates of the p osition op erator U .Ifwe put

1

X

in

j i = e jn + i; (2.12)

n = 1

it follows that

i

^

U j i = e j i; (2.13)

j +2i = ji; (2.14)

0 0

h j i =2(  ): (2.15)

In the last equation it is assumed that the  -function is p erio dic with p erio dicity

1

2 . Eq. (2.14) is a desired prop erty for the quantum mechanics on S . It is natural

^

to call U a p osition op erator due to this prop erty. On the other hand, if we de ne

^ ^

V () = exp(iP ) for a real number , (2.5) implies that

y i

^ ^ ^ ^

V () U V ()=e U: (2.16)

i( +)

^ ^ ^

Thus U V ()j i = e V ()j i is an immediate consequence. A direct calculation

shows that

i 

^

V ()j i = e j + i; (2.17)

1

^

whichsays that P is a generator of translation along S . It should b e noticed that

i 

an extra phase factor e is multiplied. These states j i (0  <2) de ne a

wave function ( ) for an arbitrary state j i2H by ()= hj i. This de nition

2 1

gives an isomorphism b etween H and L (S ) that is a space of square-integrable

1

functions on S . A bit calculation shows that the op erators act on the wave function

as

i

^ ^

U ()= hjUj i= e ( ); (2.18)

i 

^ ^

V () ( )=hjV()j i=e ( ); (2.19)

 

@

^ ^

P ()= hjPj i = i + (): (2.20)

@

In the last expression the parameter lo oks like the vector p otential for magnetic

1

ux  = 2 surrounded by S . It should b e noticed that cannot b e removed by

0 1

gauge transformation ( ) ! ( )=() (), where  is a function from S to

i 

U (1). If we could chose ( )=e ,

   

@ @

i  0

i + ()=e () (2.21) i

@ @

0 i 2 0 0

thus disapp eared. However, b ecause (2 )=e (0), is not a p erio dic

0 2 1

function. Hence do es not remain in L (S ). In picture of wave functions, it is

the b oundary condition (2 )= (0) what obstructs elimination of and therefore

causes inequivalent representations. In the quantum mechanics on R , there is no

such b oundary condition, thus such an extra term can b e wip ed awayby gauge transformation.

Sp ectrum

1

To see a physical e ect of the parameter let us consider a free particle on S .A

free particle is de ned by the hamiltonian

1

2

^ ^

H = P : (2.22)

2

Its eigenvalue problem is trivially solved by

1

2

^

(n + ) jn + i: (2.23) H jn + i =

2

Apparently, the sp ectrum dep ends on the parameter .For = m (m :integer),

all the eigenvalues but one of the ground state are doubly degenerate. While for

1

= m + , the all eigenvalues are doubly degenerate. For other values of , there

2

is no degeneracy. Itisshown in the previous work [9] that these degeneracies re ect

the parity symmetry.Asn+ =(n1)+( + 1), the sp ectrum on the Hilb ert

2 2

space H is same as that on H . Moreover, as (n + ) =(n ) , the sp ectrum

+1

of on H is same to that on H , to o. Therefore distinguishable values of range

1

. over 0  

2

Path integral

As another example to showa physical e ect of the parameter ,we shall see a path

1

integral expression of the quantum mechanics on S .We b orrow a result from [10 ].

When the hamiltonian is of the form

1

2 y

^ ^ ^ ^

P + V (U; U ); (2.24) H =

2

they derived a path integral expression of transition amplitude as

0 0

^

K ( ;;t) = h jexp(iHt)ji

Z

1

X

= D  exp(iS ); (2.25)

e

winding n times

n=1

where the e ective action is de ned as

" #

 

Z

2

1 d d

V ( ) S = dt : (2.26)

e

2 dt dt

1

In (2.25), the integration is p erformed over paths winding n times around S and

the summation is p erformed with resp ect to the winding numb er. Wewould like

to emphasize that the ab ove path integral expression is derived from the op erator

formalism. It should b e noticed that the global prop erty|winding numb er|app ears from the op erator formalism alone.

R

The last term in (2.26), d has no in uence on the equation of motion but

has an observable e ect on the amplitude. To see the role of this term we rewrite

(2.25) as

Z

1

X

0

0 i (  ) i 2n

K ( ;;t)= e e D  exp(iS ); (2.27)

0

winding n times

n=1

R

1

2

_

where S = dt [  V ( )]. An amplitude for a path winding n times is weighted by

0

2

the phase factor ! = exp(i 2n). This phase factor causes observable interference

n

e ect; this phenomenon is analogous to the Aharonov-Bohm e ect. Furthermore,

! 's ob ey comp osition rule; ! ! = ! , whichsays that m-times winding followed

n m n m+n

by n-times one is equal to (m + n)-times one. According to [11], ! can b e interpreted

n

1 1

as a unitary representation of the rst homotopy group  (S ).

We conclude this section by rep eating what has b een shown. To formulate the

1

quantum mechanics on S we should cho ose suitable generators to de ne the algebra.

We recognize that  is not suitable but U is suitable. The algebra is de ned by the

commutation relation (2.5). Representation spaces are constructed and inequivalent

ones are parametrized by a continuous parameter (0  <1). Inequivalent ones

give di erent solutions to physical problems. The role of resembles that of the

1

vector p otential. Top ology of S is an obstruction against elimination of sucha

vector p otential. In path integral picture, characterizes homotopy of a path of the

1 1

particle on S . Accordingly, existence of re ects top ology of S .

3 Ab elian sigma mo del

Here we shall consider the ab elian sigma mo del as a generalization of the quantum

1 1

mechanics on S . The ab elian sigma mo del has a eld which takes values in S .

This mo del is designed to describ e an NG b oson asso ciated with sp ontaneous break-

ing of U (1) symmetry.We shall de ne two classes of algebras of the eld theory;

1

the rst one is a natural generalization of the quantum mechanics on S , whichis

called an algebra without central extension; the second one is its nontrivial general-

ization, which is called an algebra with central extension. For b oth classes we shall

construct representation spaces combining the usual Fock representation with the

Ohnuki-Kitakado representation. Top ology of the mo del is carefully treated during

the construction.

3.1 Algebra

De nition

To motivate de nition of the algebra we will take three steps. First we start from the

n

quantum mechanics on a Euclidean space R .We shall give another expression to

P

^

the canonical commutation relations (1.1)-(1.3). If we put V (a) = exp(i a p^ )

j j

j

^

for real numb ers a =(a ;


;a ), V (a) is a unitary op erator and satis es

1 n

x^ x^ =^x x^ ; (3.1)

j k k j

y

^ ^

V (a)^x V(a)=x ^ +a ; (3.2)

j j j

^ ^ ^

V (a) V (b)=V(a+b): (3.3)

Geometrical meaning of the ab ove algebra is obvious. (3.1) says that co ordinates

x^ 's of con guration are simultaneously measurable. (3.2) implies that con guration

^

is movable by the displacement op erator V (a). (3.3) says that displacement op era-

tors satisfy the comp osition law. An irreducible representation is uniquely given by

n

2

L (R ).

Second we turn to the scalar eld theory in (1+1) dimensions. The usual canon-

ical commutation relations are

0 0

[^'();'^()]=0; (1 <; < +1) (3.4)

0 0

[^'();^()] = i(  ); (3.5)

0

[^();^( )]=0; (3.6)

where  is a co ordinate of the space. '^ and ^ are distributions valued in hermite

op erators. Weintro duce a unitary op erator

 

Z

1

^

f ( )^()d (3.7) V (f ) = exp i 1

for a real-valued test function f . Then they satisfy

0 0

'^( )^'()=' ^()^'(); (3.8)

y

^ ^

V (f )^'()V(f)=' ^()+ f(); (3.9)

^ ^ ^

V (f ) V (g )=V(f +g); (3.10)

where f and g are arbitrary real-valued test functions. The ab ove algebra is inter-

preted as follows. (3.8) means that the eld con gurations at separated p oints are

simultaneously measurable. (3.9) implies that a eld con guration is movable arbi-

trarily by displacement op erators. (3.10) is nothing but the comp osition prop ertyof

displacements. A representation is usually given in terms of the Fock space.

1

As the third step we generalize the quantum mechanics on S to the one on n-

n 1 n

^

dimensional torus T =(S ) .Weintro duce unitary op erators U and self-adjoint

j

P

n

^ ^ ^

op erators P (j =1;


;n). Put V () = exp(i  P ) for  =( ;


; ) 2 R .

j j j 1 n

j

Naive generalization of (2.5) or (2.16) leads the following relations

^ ^ ^ ^

U U = U U ; (3.11)

j k k j

y i

j

^ ^ ^ ^

V () U V ()=e U ; (3.12)

j j

^ ^ ^

V () V ( )=V(+): (3.13)

Geometrical meaning is so obvious that explanation is not rep eated. We only p oint

n

n

out that (3.12) expresses action of R on T by displacement. Representations of

this algebra are constructed by tensor pro ducts of Ohnuki-Kitakado representations

H


H . Therefore irreducible representations are parametrized by n-tuple

n

1

parameter =( ;


; ).

1 n

Finally we turn to the ab elian sigma mo del in (1+1) dimensions. The space-time

1

is assumed to b e S  R . On an equal-time space-slice, the classical eld variable is a

1 1 1 1

map from S to S . So the con guration space of the mo del is Q = Map(S ; S ). On

1

the other hand = Map(S ; U (1)) b ecomes a group by p ointwise multiplication. The

1

group U (1) acts on S by displacement. Thus the group acts on the con guration

space Q by p ointwise action, that is to say, for 2 and  2 Q let us de ne
 2 Q

by

1

(
)( )= ()
() ( 2 S ); (3.14)

where  denotes a p oint of the base space. In the right-hand side the multiplication

1

indicates the action of U (1) on S .

To clarify geometry of the classical theory,we shall decomp ose the degrees of

1 1



freedom of  2 Q and 2 . In the classical sense wemay rewrite  : S ! S

=

U (1) by

i(N+'())

( )= Ue ; (3.15)

where U 2 U (1), N 2 Z . ' satis es the no zero-mo de condition;

Z

2

1 1 1

Map (S ; R)=f':S !RjC ; '()d =0g: (3.16)

0

0

1 1



The decomp osition (3.15) says that Q S  Z  Map (S ; R). Geometrical meaning

=

0

of this decomp osition is apparent; U describ es the zero-mo de or collective motion of

the eld ; N is nothing but the winding numb er; ' describ es uctuation or lo cal

degrees of freedom of .Top ologically nontrivial parts are U and N .

1

Similarly : S ! U (1) is also rewritten as

i(+m +f ( ))

( )=e ; (3.17)

1

where  2 R , m 2 Z and f 2 Map (S ; R). The action (3.14) of on  is

0

decomp osed into

i

U ! e U; (3.18)

N ! N + m; (3.19)

'( ) ! '( )+ f(); (3.20)

according to (3.15) and (3.17). Thus the rst comp onentof (3.17) translates

the zero-mo de; the second one changes the winding numb er; the third one gives a

homotopic deformation.

^

To quantize this system let us assume that ( ) is a unitary op erator for each

1

^

p oint  2 S and V ( ) is a unitary op erator for each element 2 . Moreover we

^

^

de ne an algebra generated by ( ) and V ( ) with the following relations

0 0 0 1

^ ^ ^ ^

( ) ( )=()(); (; 2S ) (3.21)

y

^ ^ ^ ^

V ( )  ( ) V ( )= ()(); (3.22)

ic( ; )

1 2

^ ^ ^

V ( ) V ( )=e V ( ) ( ; 2): (3.23)

1 2 1 2 1 2

At the last line a function c : ! R is called a central extension, which satis es

the co cycle condition

c( ; )+c( ; )=c( ; )+c( ; ) (mo d 2 ) (3.24)

1 2 1 2 3 1 2 3 2 3

^ ^ ^ ^ ^ ^

to ensure asso ciativity(V( )V( ))V ( )=V( )(V ( )V ( )). If c  0, the al-

1 2 3 1 2 3

gebra (3.21)-(3.23) is a straightforward generalization of (3.11)-(3.13) to a system

with in nite degrees of freedom. We call the algebra de ned by (3.21)-(3.23) the

fundamental algebra of the ab elian sigma mo del.

ic( ; )

1 2

^

is intro duced. V ( ) We should explain why such an extra phase factor e

^

acts on ( )by adjoint action as shown in (3.22). This action expresses the action

of the group on Q in terms of quantum op erators. It satis es the comp osition law

y y y

^ ^

^ ^ ^ ^ ^ ^

V ( ) V ( ) ( ) V ( )V ( )= V ( )()V( ); (3.25)

2 1 1 2 1 2 1 2

which is demanded from geometrical viewp oint. However the ab ove comp osition law

^ ^ ^ ^

do es not imply that V ( ) V ( )=V( ). In other words, V is not necessarily a

1 2 1 2

genuine unitary representation of the group . We always have a p ossibility to insert

^

a phase factor as done in (3.23). In other words, V may b e a pro jective unitary

representation. If we can nd a function b : ! R such that

c( ; )=b( )+b( )b( ); (mo d 2 ) (3.26)

1 2 1 2 1 2

ib( )

~ ^

by de ning V ( )=e V ( ), (3.23) results in a genuine unitary representation

~ ~ ~

V ( ) V ( )= V( ). In that case c is called a cob oundary of b and denoted by

1 2 1 2

c = b. A cob oundary c = b identically satis es the condition (3.24), namely a

cob oundary is always a co cycle. We usually demand b oth b and c to b e continuous

functions from to R =2 Z , since is continuous. A class of co cycles mo dulo

cob oundaries is called a cohomology. Existence of a nontrivial cohomology dep ends

on top ology of the group . The group considered now is a so-called lo op group =

1

Map(S ; G) with G = U (1). Its nontrivial top ology allows for existence of a nontrivial

n

cohomology. On the other hand, when = R , all co cycles are cob oundaries. Thus

there is no need to insert such a central extension into (3.3) when we considered the

n

quantum mechanics on R .

We add a comment. In quantum eld theories, wehave met such extensions of

algebras when we study anomalous gauge theories and conformal eld theories. In

a gauge theory in (3+1) dimensions with a gauge group G, gauge transformations

3

form a group = Map(S ;G). Anomaly is deeply related to top ology of . When a

nontrivial cohomology exists, extra terms are added to the commutators of the Gauss

law constraints [12 ]. Similarly in two dimensional conformal eld theories [13], the

algebra of energy-momentum tensor is deformed by a central extension due to the

conformal anomaly and b ecomes the Virasoro algebra. In the two dimensional Wess-

Zumino-Witten (WZW) mo del [6], the current algebra is also deformed by a central

extension and b ecomes the Kac-Mo o dy algebra. Its deformation is caused by the

WZW term broughtinto the lagrangian of a nonlinear sigma mo del. A necessary

3

condition for existence of the WZW term is that G has nontrivial cohomology H (G).

But U (1) do es not satisfy it. We do not yet knowa physical reason whywemust

intro duce a central extension into the ab elian sigma mo del. Even a reason to chose

a sp eci c central extension is not clear. In the case of the WZW mo del, QCD as an

underlying theory tells the reason to bring the anomaly. But the physical meaning

of the central extension in our mo del still remains obscure.

Now let us return to the fundamental algebra (3.21)-(3.23). As a nontrivial

central extension for given by (3.17) and

1

i(+n +g ( ))

( )= e ; (3.27) 2

we de ne

( )

 

Z

2

1 df dg

c( ; )=k m n + g ( ) f ( ) d ; (3.28)

1 2

4 d d

0

where k is an integer. This central extension is the simplest but nontrivial one which

is invariant under the action of the group of orientation-preserving di eomorphims

+ +

1 1

Di (S ); c(  !;  !)=c( ; ) for any ! 2 Di (S ). The group asso ciated

1 2 1 2

with suchaninvariant central extension is called a Kac-Mo o dy group of rank k . The

^

relation (3.23) says that V is a unitary representation of the Kac-Mo o dy group. For

classi cation of central extensions see the literature [14].

Algebra without central extension

According to decomp osition of classical variables (3.15) and (3.17), quantum op era-

tors are also to b e decomp osed. For simplicitywe consider the fundamental algebra

(3.21)-(3.23) without the central extension, that is, here we restrict c  0.

^

Corresp onding to (3.15), weintro duce a unitary op erator U , a self-adjoint op er-

^

ator N satisfying

^

exp(2iN )=1; (3.29)

^

which is called the integer condition for N , and a distribution' ^( )valued in hermite

op erators and constrained by

Z

2

'^ ( ) d =0: (3.30)

0

^

We demand that the quantum eld ( ) is expressed in terms of these as

^

i(N+^'())

^

^

( )= Ue : (3.31)

Actually this equation is to o naive. Because' ^( ) is an op erator-valued distribu-

tion, its exp onentiation exp(i '^( )) is ill-de ned. To makeitwell-de ned we should

regularize its divergence. This problem is p ostp oned until discussion on the normal

ordering.

^

Next, corresp onding to (3.17) weintro duce a self-adjoint op erator P , a unitary

^

op erator W , and a distribution ^ ( )valued in hermite op erators and constrained by

Z

2

^ ( ) d =0; (3.32)

0

^

When is given by (3.17), the op erator V ( ) is de ned by

 

Z

2

^

iP m

^ ^

V ( )=e W exp i f ( )^()d : (3.33) 0

Using these op erators the relation (3.22) is now rewritten as

^ ^

iP iP i

^ ^

e Ue = e U; (3.34)

y

^ ^ ^ ^

W N W = N +1; (3.35)

   

Z Z

2 2

exp i f ( )^()d '^( ) exp i f ( )^()d =^'()+ f():(3.36)

0 0

These represent the action of on Q by (3.18)-(3.20). Observing the relation (3.35),

^ ^

we call N and W the winding numb er and the winding op erator, resp ectively. They

are also rewritten in terms of commutation relations

^ ^ ^

[ P; U ]= U; (3.37)

^ ^ ^

[ N; W ]=W; (3.38)

 

1

0 0

[^'();^()] = i (  ) ; (3.39)

2

with all other vanishing commutators. In (3.39) it is understo o d that the  -function

1

is de ned on S .

Algebra with central extension

Before constructing representations, we reexpress the fundamental algebra with the

central extension (3.28) resp ecting the decomp osition (3.15) and (3.17). The decom-

^

p osition (3.31) of  do es not need to b e changed. On the other hand the decomp o-

^ ^

sition (3.33) of V should b e mo di ed a little. We formally intro duce an op erator

by

^

i

^

W = e : (3.40)

^ ^ ^

Although W itself is well-de ned, is ill-de ned. If exists, (3.38) would imply

^ ^

[ N; ] = i, which is nothing but the canonical commutation relation. Therefore

^

N should have a continuous sp ectrum, that contradicts the integer condition (3.29).

^

Consequently must b e eliminated after calculation. Bearing the ab ove remark in

mind, we replace (3.33) by

 

Z

 

2

^ ^ ^

: (3.41) V ( ) = exp i P + m + f()^()d

0

For the central extension (3.28) of rank k , addition of the following commutation

relations to (3.37)-(3.39) is enough to satisfy the fundamental algebra;

^ ^

[ P; ] = 2ik ; (3.42)

ik

0 0 0

[^();^( )] =  (  ): (3.43)



We assume that all other commutators vanish.

Belowweverify that they are sucient for the fundamental algebra. A useful

formula is

1

^ ^ ^ ^ ^ ^

Y X X+Y [ X;Y ]

2

= e e ; (3.44) e e

^ ^ ^ ^ ^ ^

whichisvalid when [ X; [ X; Y ]] = [Y;[X; Y ] ] = 0. This formula yields

1

^ ^ ^ ^ ^ ^ ^ ^

i(P +m ) i( P +n ) (m n)[ ;P ] i((+ )P +(m+n) )

2

e e = e e : (3.45)

If we substitute (3.42), we obtain

^ ^ ^ ^ ^ ^

i(P +m ) i( P +n ) ik (m n) i((+ )P +(m+n) )

e e = e e ; (3.46)

which is coincident with (3.23) for the central extension (3.28). Similarly the formula

(3.44) implies

R R

2 2

0 0 0

i f ( )^()d i g ( )^( )d

0 0

e e

R R R

2 2 2

1

0 0 0

f ( )g ( )[ ^ ( );^ ( )]d d i (f ( )+g ( )) ^( )d

2

0 0 0

= e e : (3.47)

Substitution of (3.43) yields

Z Z

2 2

1

0 0 0

f ( )g ( )[ ^ ( ); ^ ( )]d d

2

0 0

Z Z

2 2

ik

0 0 0 0

= f ( )g ( )  (  )d d

2

0 0

Z

2

ik

0 0

= (f ( )g ( )+f()g ())d: (3.48)

4

0

Hence (3.47) coincides with (3.23) for the central extension (3.28).

Using (3.40) Eq. (3.42) implies

^ ^ ^

[ P; W ]=2kW; (3.49)

^

whichsays that the zero-mo de momentum P is decreased by2k units when the

^ ^

winding number N is increased by one unit under the op eration of W . This is an

inevitable consequence of the central extension. We call this phenomenon \twist".

Using (3.42) and (3.44), the decomp osition (3.41) results in

 

Z

 

2

ik m m

^ ^ ^

V ( )=e exp i P + f ( )^()d W : (3.50)

0

Here we summarize a temp oral result. Generators of the fundamental algebra

are decomp osed as (3.31) and (3.50) considering top ological nature of the mo del.

They are constrained by the no-zero-mo de condition (3.30), (3.32) and the integer

condition (3.29). The commutation relations are also decomp osed into (3.37), (3.38),

(3.39), (3.43) and (3.49). Noticeable e ects of the central extension are the anoma-

lous commutator (3.43) and the twist relation (3.49). These features also a ect

representation of the algebra as seen in the following sections.

3.2 Representations

Without the central extension

Nowwe pro ceed to construct representations of the algebra de ned by (3.37)-(3.39)

and other vanishing commutators with the constraints (3.29), (3.30) and (3.32).

^ ^ ^ ^

Rememb er that P and N are self-adjoint and that U and W are unitary. Both of

the relations (3.37) and (3.38) are isomorphic to (2.5). Hence the Ohnuki-Kitakado

^ ^

representations provide representations for them, to o. P and U act on the Hilb ert

^ ^

space H via (2.9) and (2.11). N and W act on another Hilb ert space H via

^

N jn + i =(n+ )jn+ i; (3.51)

^

W jn + i = jn +1 + i: (3.52)

The value of is arbitrary.However is restricted to b e an integer if we imp ose the

condition (3.29).

y

For' ^ and ^ the Fock representation works. We de ne op eratorsa ^ anda ^ by

n

n

s

X

1 

in y in

'^( )= (^a e +^a e ); (3.53)

n

n

2 jnj

n6=0

q

X

i

in y in

^ ( )=  jnj (a^ e +^a e ): (3.54)

n

n

2

n6=0

In the Fourier series the zero-mo de n = 0 is excluded b ecause of the constraints

(3.30) and (3.32). It is easily veri ed that the commutator (3.39) is equivalentto

y

[^a ;^a ]=  (m; n = 1; 2;


) (3.55)

m mn

n

with the other vanishing commutators. Hence the ordinary Fock space F gives a

y

representation ofa ^ 's and ^a's, which are called creation op erators and annihilation

op erators, resp ectively.

Consequently the tensor pro duct space H H F gives an irreducible represen-

0

tation of the fundamental algebra without the central extension. The inequivalent

ones are parametrized by (0  <1).

A remark is in order here; the co ecients in frontof^a's in (3.53) and (3.54) are

chosen to diagonalize the hamiltonian of free eld

 

Z

   

2

2 2

1 1

^ ^ ^

d + @'^()+N P +^() H =

2 2

0

   

X

1 1 1

2 2 y

^ ^

= P +2N + jnj a^ ^a + : (3.56)

n

n

2 2 2

n6=0

This hamiltonian corresp onds to the lagrangian density

1

y 

L = @  @ ; (3.57)

 2

where  is given by (3.15). This lagrangian describ es a massless b oson. Although

Coleman's theorem [15] forbids existence of massless b osons in (1+1) dimensions due

to infrared catastrophe, our mo del is still p ermitted. In our mo del the base space

1

S is compact, hence there is no infrared divergence. Interacting eld theory will b e

brie y discussed later.

Another remark is added. The Kamefuchi-O'Raifeartaigh-Salam theorem [16]

states that the S -matrix in quantum eld theories remains unchanged under any

p oint transformation of eld variables. Their theorem is proved within the framework

of the conventional canonical formalism. Thus this theorem is not applicable to our

i'( )

mo del. If we take a real scalar eld ' such that ( )= e , the lagrangian (3.57)

b ecomes a genuine free scalar theory,

1



L = @ '@ ': (3.58)



2

In this case the Fock space is a unique representation, thus there is no ro om for

such an undetermined parameter . They considered only eld theories with a

trivial top ology, then they derived the equivalence theorem for S -matrix. On the

other hand, we considered a eld theory with a nontrivial top ology, then we reached

existence of inequivalent representations. However the S -matrix of our mo del is not

yet calculated, hence it is left undetermined whether S -matrix do es dep end on the

parameter or not.

With the central extension

Next we shall construct representations of the algebra de ned by (3.37), (3.38),

(3.39), (3.43), (3.49) and other vanishing commutators with the constraints (3.29),

(3.30) and (3.32). The way of construction is a bit mo di ed from to the previous

one.

^ ^ ^

Taking account of the twist relation (3.49), the representation of P , U , N and

^

W are given by

^

P j p + ; ni =(p+ )jp+ ; ni; (3.59)

^

U j p + ; ni = j p +1+ ; ni; (3.60)

^

N j p + ; ni = nj p + ; ni; (3.61)

^

W j p + ; ni = j p 2k + ; n +1i: (3.62)

The inner pro duct is de ned by

hp + ; m j q + ; ni =   (p; q ; m; n 2 Z ): (3.63)

pq mn

The Hilb ert space formed by completing the space of linear combinations of j p + ; ni

is denoted by T .(T indicates \twist".)

Let us turn to' ^ and ^ . Considering the anomalous commutator (3.43), after a

tedious calculation we obtain a Fourier expansion

X

1

in y in

q

'^( )= (^a e +^a e ); (3.64)

n

n

2jknj

n6=0

) (

P

q

1

i

(k>0)

in y in

n=1

P

^ ( )= 2jknj(a^ e +^a e ); (3.65)

n

1

n

(k<0)

2

n=1

y

wherea ^ 's and ^a 's ob ey the same commutation relations (3.55). The derivation of the

ab ove expansion is shown in the app endix. It should b e noticed that only p ositive

n's app ear in the expansion of ^ when k>0, while only negative n's app ear when

k<0. However b oth p ositive and negative n's app ear in' ^.Physical implication of

lack of half of mo des in ^ ( ) is still unclear but will b e discussed later.

The algebra de ned by (3.55) is also represented by the Fock space F . Hence the

tensor pro duct space T F gives an irreducible representation of the fundamental

algebra with the central extension for eachvalue of (0  <1).

3.3 Normal ordering

Although most of our main sub jects are nished, a subtle problem is still left. At

(3.31) exp onential of the lo cal op erator' ^( )isintro duced. As mentioned there, it

contains divergence thus it is ill-de ned. Nowwe shall consider this problem closely.

Without the central extension

i'^( )

e should b e expressed in terms of creation and annihilation op erators to act on

the Fock space. First we study the case without central extension. If we de ne

s

X

1 

in

^( )= a^ e ; (3.66)

n

2 jnj

n6=0

y

(3.53) is decomp osed into' ^( )= ^()+ ^ (). A bit calculation shows that

 

1 1

X X

1 1 1

0 0

y 0 in(  ) in(  )

[^ (); ^ ()] = e + e

4 n n

n=1 n=1

1

0 0

i(  ) i(  )

= flog(1 e ) + log (1 e )g; (3.67)

4

0

which diverges when  closes to  ,

y 0

lim [^ (); ^ ()] = 1: (3.68)

0

 !

Thus application of the formula (3.44) tells that

1

y y y

i'^( ) i ^( )+i ^ ( ) [^ (); ^ ()] i ^ () i ^()

2

e = e = e e e (3.69)

diverges. To eliminate the divergence weintro duce the normal ordering pro cedure,

which is a rule to rearrange creation op erators to the left and annihilation op erators

to the right for each term. This pro cedure is denoted by sandwiching by double

colons, for example

y

i'^( ) i ^ () i ^()

: e := e e : (3.70)

^ ^ ^ ^ ^ ^

X Y [ X;Y ] Y X

As (3.44) implies that e e = e e e , (3.67) gives

0

i'^( ) i'^( )

: e :: e :

y y 0 0

i ^ ( ) i ^( ) i ^ ( ) i ^( )

= e e e e

y 0 y y 0 0

[^ (); ^ ( )] i ^ () i ^ ( ) i ^() i ^( )

= e e e e e

 

1

0 0 0

i(  ) i(  ) i'^( ) i'^( )

= exp log (1 e )(1 e ) : e e : (3.71)

4

0

whichiswell-de ned except for  =  and invariant under p ermutation of  with

0

 . Therefore we conclude that

0 0

i'^( ) i'^( ) i'^( ) i'^( )

: e :: e :=: e :: e :; (3.72)

then (3.21) is satis ed.

The other relation (3.22) also must b e satis ed after the normal ordering pro ce-

dure. Let us check it. We substitute (3.54) and

X

in 

f ( )= f e (f = f ) (3.73)

n n

n

n6=0

into (3.33) to obtain

2 3

 

Z

q

2

X

y

^

4 5

V (f ) = exp i f ( )^()d = exp  jnj (f a^ + f a^ ) : (3.74)

n n n

n

0

n6=0

Then it is easily seen that

q

y

^ ^

jnjf ; (3.75) V (f )^a V (f)=^a +

n n n

q

y y y

^ ^

V (f )^a V (f)=^a + jnjf : (3.76)

n

n n

The de nitions (3.66) and (3.73) with the ab ove equations yield

s

q

X

 1

in y

^ ^

 jnj f )e V (f )^ ()V(f) = (^a +

n n

2 jnj

n6=0

1

= ^ ( )+ f(); (3.77)

2

s

q

X

1 

y y y in

^ ^

V (f )^ ()V(f) = (^a +  jnj f )e

n

n

2 jnj

n6=0

1

= ^ ( )+ f(); (3.78) 2

therefore

y

y i'^() y i ^ () i ^()

^ ^ ^ ^

V (f ):e :V(f) = V (f)e e V(f)

y

if ( ) i ^ ( ) i ^( )

= e e e

if ( ) i'^( )

= e : e : : (3.79)

This result coincides with (3.22). Consequently wehavechecked that the funda-

mental relations (3.21) and (3.22) are preserved by the normal ordering pro cedure.

i'^()

^ ^

Although ( ) is claimed to b e a unitary op erator ab ove (3.21), ( )=: e :is

y

^ ^

not unitary.Ifitwere unitary, ( )  ( )must b e equal to identity. But actually

y y 0 0

y 0 i ^ ( ) i ^( ) i ^ ( ) i ^( )

^ ^

lim ( )  ( ) = lim e e e e

0 0

 !  !

y 0 y y 0 0

[^ (); ^ ( )] i ^ () i ^ ( ) i ^() i ^( )

= lim e e e e e

0

 !

y

[^ (); ^ ()]

^

= lim e 1

0

 !

= 0 (3.80)

i'^()

^

due to (3.68), hence ( )=: e : is not unitary.

With the central extension

Similarly we can verify the case of k 6= 0. The wayofveri cation is almost identical

to the previous one but a bit changed. We de ne

X

1

in

q

^ ( )= ^a e ; (3.81)

n

2jknj

n6=0

y

to decomp ose (3.64) into' ^( )= ^()+ ^ (). A bit calculation yields

 

1 1

X X

1 1 1

0 0

in(  ) in(  ) y 0

e + e [^ (); ^ ()] =

2jkj n n

n=1 n=1

1

0 0

i(  ) i(  )

flog (1 e ) + log (1 e )g; (3.82) =

2jk j

0

which is also divergent when  =  .To eliminate the divergence we use the normal

ordering pro cedure again;

y

i'^( ) i ^ () i ^()

: e := e e : (3.83)

A calculation similar to (3.71) yields

" #

1

0 0 0 0

i'^( ) i'^( ) i(  ) i(  ) i'^( ) i'^( )

: e :: e : = exp log (1 e )(1 e ) : e e : :

2jk j

(3.84)

Thus the commutativity (3.21) is ensured again.

Veri cation of the other relation (3.22) is a bit complicated. We de ne

1

X

(+) in

f ( )= f e ;

n

n=1

1

X

() in

f ( )= f e ;

n

n=1

(+) () 

f ( )=f ( )+ f (): (f = f ) (3.85)

n

n

Substitution of (3.65) into (3.50) gives

 

Z

2

^

V (f ) = exp i f ( )^()d

0

# "( )

P

q

1

(k>0)

y

n=1

P

2jknj(f ^a + f ^a : (3.86) ) = exp

n n n

1

n

(k<0)

n=1

It follows that

q

y

^ ^

2jknj f ; (3.87) V (f )^a V (f)=^a +(kn)

n n n

q

y y y

^ ^

V (f )^a V (f)=^a +(kn) 2jknj f ; (3.88)

n

n n

where  (x) is 1 when x>0 and 0 when x<0. The de nitions (3.81) and (3.85)

with the ab ove equations yield

q

X

1

y in

^ ^

q

V (f )^ ()V(f) = (^a + (kn) 2jknj f )e

n n

2jknj

n6=0

()

= ^ ( )+ f (); (3.89)

q

X

1

y y y in

^ ^

q

V (f )^ ()V(f) = (^a + (kn) 2jknj f )e

n

n

2jknj

n6=0

y ()

= ^ ( )+ f (); (3.90)

where the alternative sign is chosen according to the sign of k . Again we arriveat

the same result

y

y i'^() y i ^ () i ^()

^ ^ ^ ^

V (f ):e :V(f) = V (f)e e V(f)

(+) () y

if ( )+if ( ) i ^ ( ) i ^( )

= e e e

if ( ) i'^( )

= e : e : : (3.91)

4 Summary and discussion

4.1 Summary

Now let us summarize what have b een done in this pap er. Wehave reviewed the

1

quantum mechanics on S originally formulated by Ohnuki and Kitakado. They

1

chose generators resp ecting top ology of S . They de ned the algebra and classi ed

its irreducible representations. Inequivalent representations are characterized bya

continuous parameter (0 < 1). Elimination of is obstructed by nontrivial

1

top ology of S .

1

As a generalization of the quantum mechanics on S ,wehave prop osed the

de nition of the algebra for the ab elian sigma mo del in (1+1) dimensions. The

central extensions are also intro duced into the algebra.

The degrees of freedom of the eld variable are separated as

1 1 1 1



S  Z  Map (S ; R): (4.1) Map(S ; S )

=

0

1 1 1

The separation is done as follows. Identify  : S ! S with  : S ! U (1).

Take a branch of its logarithm and put' ~( )=ilog ( ). De ne N 2 Z by

R

2

i 1

2N =~'(2 ) '~(0). Next de ne e 2 S by2 = (~'() N)d . Finally

0

R

2

1

de ne ' : S ! R by '( )=' ~()N.Thus '(2 )='(0) and '( )d =0.

0

Then

i(+N+'())

( )= e (4.2)

i

is a decomp osition according to (4.1) and results in (3.15) by putting U = e .

What wehave done is to de ne a co ordinate system in the in nite dimensional

1 1

manifold Q = Map(S ; S ). In the same waywe can de ne a co ordinate in the group

1

= Map(S ; U (1)) as given by (3.17). These co ordinates are convenient; they are

direct pro duct decomp ositions of the manifold Q, the group and the action of

on Q. In other words, these decomp ositions are preserved under group op eration of

and are also preserved under the action of on Q as shown in (3.18)-(3.20).

Existence of such co ordinates is crucial for construction of the quantum theory.

The fundamental relations (3.21)-(3.23) are easy to understand intuitively,however

to o complicated to construct its concrete representation. The co ordinates reduce

them to simpler relations (3.37)-(3.39). Even if the central extension exists, other

complication is only addition of the anomalous commutator (3.43) and the twist

relation (3.49). Thus wehave noticed that the Ohnuki-Kitakado representations and

the Fock representation provide representations for our mo del.

We conclude that inequivalent irreducible representations are parametrized again

by (0  <1). When there is the central extension, the action of the op erator

^

W is changed as (3.62) and half of mo des in ^ ( ) is removed as (3.65).

Exp onential of the lo cal op erator' ^( )must b e regularized by the normal ordering

pro cedure. Wehave shown that the pro cedure preserves the fundamental relations

i'^()

but violates unitarityofe .

4.2 Discussion

For what kind of physics is our theory applicable? What wehave done is just

formulation of a rather ideal mo del. It gives a lesson; when a model has nontrivial

topology, it is possible to construct inequivalent quantum theories, even if they are

equivalent as classical theories. Wewould like to p oint out some mo dels whichhave

such p ossibilities.

The rst example is still an ideal mo del in (1+1) dimensions; it is the sine-Gordon

mo del, whose lagrangian is

1

 2

L = @ '~(x)@ '~(x)+ cos(' ~(x)): (4.3)



2

It is a mo del which has interaction. It can b e rewritten by the variables of the

i'~

ab elian sigma mo del without the central extension by identifying  with e . Then

the corresp onding hamiltonian is de ned by

 

Z

 

2

2

1 1

y 2 y

^ ^ ^ ^ ^ ^

H = P +^ +@ @ (+ ) d

2 2

0

   

X

1 1 1

y 2 2

^ ^

+ jnj a^ ^a + = P +2N

n

n

2 2 2

n6=0

Z

2

 

2



^ ^

i(N+^'()) y i(N+^'())

^ ^

d: (4.4) Ue + U e

2

0

The last term contains highly nonlinear complicated interaction. It also contains

^

interaction b etween the zero-mo de U and the uctuation mo de' ^. This hamiltonian

^

commutes with N , hence the winding numb er is conserved. If the hamiltonian in-

^

cludes the winding op erator W ,change of the winding numb er can o ccur. Sucha

jumping motion is not allowed in classical theory but is p ossible in quantum theory.

The winding numb er is sometimes called soliton or kink numb er. It is known [4] that

this mo del has a top ological soliton, which b ehaves like a fermion. However it is still

obscure whether our formulation is relevant to soliton physics or not. It is exp ected

that our mo del may give an insight to quantum theory of solitons.

Other examples are found in b oth eld theories and string theories. From b oth

p oints of view, it is hop ed to extend our mo del to nonab elian groups and to higher

1 1

dimensions. Our mo del has the eld con guration space Q = Map(S ; S ). The

1

group = Map(S ; U (1)) acts on Q simply transitively. The most general mo del has

Q = Map(X ; M ), in which X is called a base space and M is a target space. When

dim X>1, we call it a higher dimensional mo del. If a group G acts on M transitively,

M is called a homogeneous space G=H . Then an in nite dimensional group =

Table 1: Possible extensions

ab elian sigma mo del higher dimensions nonab elian

n

1 n n

base space X S S ;T ;R

1 n

target space M S G; G=H ; T =Z

group G U (1) SU (n);SO(n) etc

Map(X ; G) acts transitively on Q = Map(X ; M )by p ointwise multiplication. When

the group G is nonab elian, it is called a nonlinear sigma mo del. When the base space

1

X is S , it is called a b osonic string mo del. In addition, when the target space M

n

is an orbifold, for example a toroidal orbifold T =Z , it is called an orbifold string

mo del. Directions for extensions are summarized in the table.

We should refer to the known results on nonlinear sigma mo dels in two dimen-

sions. Some of them are exactly solved by the metho d of factorization theory and the

Bethe ansatz [17 ]. Here \exactly solved" means that the exact S -matrix is obtained

and therefore the mass sp ectrum de ned by p oles of the S -matrix is also calculated.

Wiegmann et al [18] have already obtained exact solutions of nonlinear sigma mo dels

for the algebras SO(n +2);SU(n+1);Sp(2n)(n =1;2;


). All of them exhibit

massive sp ectra.

Their approach is quite di erent from ours. In fact they use neither eld vari-

ables nor lagrangians. They demand some reasonable prop erties to b e satis ed by the

S -matrix; unitarity, factorizability, crossing symmetry, analyticity and other symme-

tries sp eci ed by a Lie algebra. In (1+1) dimensions such a requirement determines

the S -matrix directly and unambiguously. Actually what they have constructed is

a realization of symmetries in terms of S -matrix. But they do not pay attention to

top ology. Although the mo del whichwehave considered is quite simple, that is the

SO(2) sigma mo del, wehave shown existence of inequivalent quantizations as a con-

sequence of the nontrivial top ology.At presentwe do not knowhow to incorp orate

top ology into the algebraic approach of Zamolo dchikov and Wiegmann et al.

We shall brie y comment up on the known results on orbifold string mo dels [19 ].

Sakamoto et al [20] investigated theories of closed b osonic strings on orbifolds in

op erator formalism. He have shown that commutators among the zero-mo de and

winding-mo de variables are left ambiguous and that these variables ob ey nontrivial

quantization. The relation b etween his result and ours is still left obscure.

Finally wewould like to suggest a way to explore nonab elian and higher dimen-

sional theories. The decomp osition (4.1) heavily relies on the ab elian nature of the

group U (1). As a generalization to a nonab elian group G,we exp ect a decomp osition

n n



= Map (S ; G) G   (G)  Map (S ; Lie(G)); (4.5)

=

n 0

where  denotes the n-th homotopy group and

n

Z

n n 1

Map (S ; Lie(G)) = f g : S ! Lie(G) j C ; g =0g: (4.6)

0

n

S

In the decomp osition (4.5), the rst comp onent describ es the zero-mo de, the second

one do es the top ologically disconnected mo de, and the third one do es the uctuation

mo de. Unfortunately such a decomp osition do es not exist, b ecause the nonab elian

nature severely entangles degrees of freedom of . It seems hop eless to nd a con-

venient co ordinate in generally. We should take a rather abstract approachto

construct a representation for such a complicated group. Gelfand et al [21] has in-

vestigated the representation theory of the group = Map(X ; G) for a base space

X (dim X  2) and a compact semisimple G. They do not rely on any co ordi-

nates but take a quite abstract approach. However they do not consider a manifold

Q = Map(X ; G=H ) on which acts. We do not knowhow to incorp orate sucha

con guration space Q into their representation theory.

Acknowledgments

I am indebted to so many p eople that I cannot record all of their names here. I grate-

fully acknowledge helpful discussions with T.Kashiwa and S.Sako da (Kyushu Uni-

versity), K.Higashijima (Osaka University), M.Sakamoto and M.Tachibana (Kob e

University), I.Tsutsui and S.Tsujimaru (Institute for Nuclear Study), K.Fujii (Yoko-

hama City University) and Y.Igarashi (Niigata University). I wish to express my

gratitude to Y.Ohnuki, S.Kitakado and H.Ikemori, who discussed with me and en-

couraged me continuously. I thank all memb ers of E-lab oratory, who provided me

with encouraging environment during the whole p erio d of my studying.

A App endix

Here we give an explicit calculation of the Fourier expansion of (3.39) and (3.43) to

derive (3.64), (3.65) and (3.55). We rep eat the assumptions;

0

[^'();'^()]=0; (A.1)

 

1

0 0

[^'();^()] = i (  ) ; (A.2)

2

ik

0 0 0

[^();^( )] =  (  ); (A.3)



with constraints (3.30) and (3.32). k is assumed to b e a non-zero integer. We de ne

Z

2

1

im

'^ = '^( )e d; (A.4)

m

2

0

Z

2

1

in

^ = ^ ( )e d; (A.5)

n

2

0

y y

for integers m; n.Obviously,^' =^' ,^ =^ and' ^ =^ = 0. Multiplying

m n 0 0

m n

R R

0

2 2

2 im in 0

(2 ) e to (A.1)-(A.3) and integrating d d ,we obtain

0 0

[^' ;'^ ]=0; (A.6)

m n

i

[^' ;^ ]= (   ); (A.7)

m n m+n;0 m;0 n;0

2

k

m : (A.8) [^ ;^ ]=

m+n;0 m n

2

2

We put

v v

u u

2 2

u u

2 2

y

t t

^ ^

b = i ^ ; b = i ^ ; (n>0) (A.9)

n n n

n

jk jn jk jn

then (A.8) implies

^ ^

[ b ; b ]= 0; (A.10)

m n

y

^ ^

[ b ; b ]= (k) ; (m; n > 0) (A.11)

m mn

n

where (k )=k=jkj. By de nition wehave

s s

jkjn jkjn

y

^ ^

^ =i ; (n>0) (A.12) b ; ^ =i b

n n n

n

2 2

2 2

hence

X

in

^ ( ) = ^ e

n

n6=0

1

q

X

i

in y in

^ ^

= 2jk jn (b e + b e ): (A.13)

n

n

2

n=1

On the other hand, for non-zero m and n, (A.6) and (A.7) yield

km i i i

[^' ^ ; ^ ]=   =0 (A.14)

m m n m+n;0 m+n;0

2

km 2 km 2

and also

i i 1

[^' ^ ; '^ ^ ]=  : (A.15)

m m n n m+n;0

km kn 2kn

Therefore if we put

8

q

i

<

^

^ ); b = (k ) 2jk jn (^' +

n n n

kn

q

(n>0) (A.16)

y

i

:

^

b = (k ) ^ ); 2jk jn (^'

n n

n

kn

they satisfy

^ ^

[ b ; b ]= 0; (A.17)

m n

y

^ ^

[ b ; b ]= (k) : (m; n 6=0) (A.18)

m mn

n

It is easily seen that

(k) (k ) i

y y

^ ^ ^

q q

^ = '^ = b (b + b + ); (A.19)

n n n

n n

kn

2jk jn 2jkjn

(k) i (k )

y

^ ^ ^

q q

^ = (b + b ); (n>0) (A.20) b '^ =

n n n n

n

kn

2jk jn 2jkjn

using (A.16) with (A.12). Thus wehave

X

in

'^() = '^ e

n

n6=0

X

1

in y in

^ ^

q

= (k ) (b e + b e ): (A.21)

n

n

2jknj

n6=0

Finally we de ne

( (

y

^ ^

b (k>0) b

n

y

n

= (A.22) a^ ^a =

n

y

n

^ ^

b (k<0) b

n

n

to get

y

[^a ;^a ]=  ; (A.23)

m mn

n

X

1

in y in

q

'^( )= (^a e +^a e ); (A.24)

n

n

2jknj

n6=0

( )

P

q

1

i

(k>0)

in y in

n=1

P

2jknj(a^ e +^a e ); (A.25) ^ ( )=

n

1

n

(k<0)

2

n=1

which are the desired results (3.55), (3.64) and (3.65).

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