The Heisenberg-Weyl group on the Z N×Z N discretized torus membrane

Volume 228, number 3

PHYSICS LETTERS B

21 September 1989

T H E HEISENBERG-WEYL GROUP ON THE ~-NX~-ND I S C R E T I Z E D TORUS M E M B R A N E E.G. FLORATOS t CERN, CH-1211 Geneva 23, Switzerland Received 5 July 1989

We &screttze the torus membrane T 2 by the abehan latttce ZN×ZN in order to give a geometrical meaning of the S U ( N ) truncauon for the mfimte dlmenstonal dlffeomorphlsm group SDtff(T2), recently proposed by Falrhe, Fletcher and Zachos. We find that the S U ( N ) algebra defines, through the Heisenberg-Weyl group on ZN×ZN, the metapleetlc representation of the modular group SL (2, Z~.) acting as the area preserving symmetry of the &screUzed torus membrane

Recent attempts to study the relevance, to a unified description of elementary panicle physics, of relativistic extended objects other than strings, such as membranes (or more generally p-branes) meet serious difficulties, the reason being the intrinsic nonlinearities of these theories which cannot be gauged away, as it happens in the case of string theories [ 110]. On the other hand, intriguing connections w~th the S U ( N ) Yang-Mills [3] or SUSY Yang-Mills theories [ 11 ] have been found, pointing to new ways of quantizing membranes with a natural ultraviolet regulator. Recently, the spherical and the torus membrane have been in the focus of some interesting studies regarding thew symmetries and the connections with the S U ( N ) YM and SUSY-YM quantum mechanics [12-19]. In ref. [15], the S U ( N ) YM quantum mechanical system correspondmg to a toms membrane was found, which m the limit of N--,oo gives the toms membrane theory and no reason of the particular proposed realization was given (see though ref. [ 14 ] ). In this work we propose a simple and intuitive geometrical reason for the existence of the construcuon given in ref. [ 15 ]. We believe that this proposal opens a way for the study of representations of SDiff(T 2 ) and of its central extension. Also generahzations to membranes of higher genus of the On leave of absence from Physics Department, University of Crete and the F o u n d a u o n for Research and Technology, Irakhon, Crete, Greece

0370-2693/89/$ 03.50 © Elsevier Science Pubhshers B.V. ( North-Holland Physics Pubhshing Division )

S U ( N ) construction are possible in the presented framework. To begin with, the hamiltonian of the membrane in the light-cone gauge ~s, m appropriate units [2,3 ]

lff

2n

o

o

2n

H=~n z

da,

dt72[X'2+I{X',X'}21,

(1)

t,J=l,...,D--2, and the self-consistency constraint L(a,,oz)-{X',X'}=O,

(2)

where X ' = X ' ( r , at, a2) are the D - 2 coordinates of the membrane, 2n-periodic in a~ and oz for the toms membrane, X ' = ( d / d r ) X ' , the velocity of the membrane at the point a~, a2 and the bracket ~ m b o l is defined as OX' OXj {Xt'Xl}~- 00"2 0 a I

OX' OXj Oa I 0 a z "

(3)

The torus membrane has two addiuonal constraints: e'~= I d C J ( ' X ' = 0 ,

(4)

C,,

where C,, a = 1, 2, are two cycles of the toms which are non-homologous to zero and between themselves. The constraints (2) and (4) generate the area preserving transformations of the torus surface. SDfff(T2), which are the symmetries of the above hamdtonian system [3]. In ref. [ 12] the basis ofpe335

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PHYSICS LETTERSB

riodic exponential functions was used to define the generators 2n

L.= ~

2n

'If

dat

0

da2exp(in-~)L(~,.~),

21 September 1989

In the case r=2n/N, where NeZ, the authors of ref. [ 15] observed that the lattice Z×2~ splits into an infinite number of similar N × N blocks, so identifying

(5) T(m+pN, n2+qN ) p, qeT_ ,

Tim,n2),

0

in order to find the structure constants of SDiff(T 2)

a new algebra with N 2_ l elements appears (n 4: 0)

[L,,, L,,,] = (nxm)L,,+,,,

[J., arm] = - 2 i s i n ( 2 n / N ) ( n × m ) J . + . ,

[Po, L,,]=no, L,,,

n, m,~7/XZ,

(6a)

a=l,2,

(10)

(6b)

[P,,P2]=O.

(6c)

which is identical to the S U ( N ) algebra in a basis of monomials generated by two, N × N, special unitary matrices (N odd)

This algebra has been studied first by Arnold [ 20,21 ] while in ref. [ 12 ] the unique non-trivial central extension of it was determined

J~ =ton'"2/2g"th"2,

[L., L,,,] = (n×m)L.+,,,+a.nS~+.,,.o,

g = d i a g ( 1, to..... coN-l ) ,

a ~ 2,

(7a)

[P,~, L.] =n,~L,,,

(7b)

[Pt,P2]=c.

(7c)

In ref. [22], generalizations of eqs. ( 7 a ) - ( 7 c ) to surfaces of higher genus were considered. The hamiltonian system, eqs. ( 1 ) and (2), has a striking similarity with the YM quantum mechanical system H = ½tr[.4'z+ ½[A', As ] 2] ,

(8a)

[,~', A'] =0,

(8b)

z = l ..... D - I .

which corresponds to the YM hamdtonian for a Lie algebra G in the gauge Ao= 0, restricted to potentials depending only on time A, = A , ( t ) ~ G . This similarity was proven to be identity for G = S U ( N ) and spherical topology of the membrane. The reason is that in an appropriate basis of SU(N), it is possible to show that SU(N)--.SDiff(S z) as N ~ o o and the commutator tends to the bracket symbol after appropriate normahzation [2,14]. An argument for the equivalence of the two theories independent of the basis of S U ( N ) was given in ref. [23 ] where the large N limit of the S U ( N ) YM lagrangian was given for spacetime dependent potentials. Only recently, the case of the torus membrane was shown to follow the same pattern of truncation to a SU(N)-YM quantum mechanical system [ 14]. In ref. [ 15 ] a new infinite algebra was proposed similar to eqs. ( 6 a ) - ( 6 c ) with the same graded structure but with trigonometric structure constants [Tn, T , ] =sin r (nxm)T,,+,,,. 336

(9)

n, m e Z x ,

h=

(0 0

1

to=exp(47ti/N) ,

.

( 11 )

(12)

1 The algebra (10) in the limit of N--.co (after multiplying both sides by N 2) leads to the torus algebra (6a). As mentioned by the authors of ref. [ 15 ], the matrices g, h were introduced by Weyl [24]. The new infinite dimensional algebra (9) [ 15], is related to a special bracket structure of hamiltonian mechanics introduced by Moyal [25 ] to adapt better the transition of the classical mechanics to quantum mechanics, to the structure of the algebra of classical observables [26 ]. It is interesting to notice that the Moyal bracket is the unique globally defined deformation of the flat symplectic structure for the toro~dal phase space [ 27 ]. To give a simple and intuitive geometrical reason for the existence of the race construction of ref. [ 15 ], we propose the following idea. Since the S U ( N ) approximation of the spherical or toroidal membrane is a truncation of the Fourier expansion of the membrane coordinates in spherical harmonics for the sphere or m periodic exponential functions for the torus, we see that this procedure is equivalent to approximating the membrane surface by a finite lattice (similar to the Regge calculus in gravity). For the approximate membrane lattice, we can ask the question, if we want to talk about the same hamiltonian problem, what would replace the SDiff(S 2) or

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3

SDiff(T*) symmetry. We show in this work that in the case of the torus membrane, the SU(N) algebra of ref. [ 15 ] is a representation of the SL( 2, HN) group acting as the discretized symmetry of the membrane lattice, which preserves the area. The area preserving symmetry of the torus membrane (7a)-( 7c) can be considered as the symmetry of canonical transformations for a classical hamiltonian system of one degree of freedom with the torus playing the role of the phase space [ 201. We note here for later use that for classical hamiltonian systems, there is a bigger symmetry including local time translations, the so-called contact transformations, generated by functions h(o,, 0~2)[21,28] t+t+h-a2 6,4,

ah/C&,

-ahlao,,

a2+f12 +ahiao,

,

LETTERS

2 1 September

B

, &= 3k,

k=O, l,..., N-l.

(17) The particle states -k k=O, 1, .... N- 1 ,

,

4x Q= -diag(O, N

(13b)

The particle states

(13c)

1, .... N- 1) .

qWkJ,eJ),

,k)=

(19)

k=O, 1, .... N- 1

(15)

The role of this special three-diffeomorphism algebra in the membrane theory has not been investigated yet. We truncate the parameter space of the torus T *= S’ x S’ by two discrete circles &={O,

1, .... N- l},

are the eigenstates of the momentum operator which must be replaced by a finite shift in position

Acting on the state (17) 0

1 0

h=

[or (20)]

(hw)(&)=

1 0. *. *.

i

which represent the momenta and the positions of a particle on a circle with N points at angles 0, 47r/N, .... 4n(N- 1 )/N. Weyl introduced in Chapter 4 of his famous book [ 241 a deep connection of the non-commutativity of quantum mechanical observables with the phase ambiguity of quantum mechanical states. We shall interpret his discussion with a model for a free quantum mechanical particle on the discrete circle Sk. The states of this system are complex functions w: Sh-+@ which form an N-dimensional space of states

(21)

w( &+ l ) we find

Nodd prime, (16)

(20)

JNJdl

hlk)=o“lk). [ Vh, V&l= V{h,h.).

(18)

are the eigenstates of the position operator Q

(13a)

which preserve the element of the action, the phasespace area and satisfy the same algebra (7a)-( 7c)

1989

(22)

.

1 0 .i

We point out that the NX N matrix U, F,=

-&cd,

1,j=O, 1, .... N-l

,

(23)

is the Fourier transform operator which transforms position eigenstates to momentum eigenstates Ik)=FI&),

k=O ,..., N-l,

(24)

and hF= Fg ,

(25)

so h is the shift operator in position space Sk and g is the shift operator in momentum space ZN. The Hei-

337

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senberg commutation relations are replaced by the group commutator hgh-lg-1=og.

(26)

The two generators of the abelian groups

21 September 1989

berg commutation relation for P and Q and its universal enveloping algebra is the quantum version of the contact transformations [ 28,29 ]. In our case, the group law is (n~, position, n2, momentum, n 3, time): ( h i , r/2, 1"/3)"(ml, m2, m3)

Z f = { h k,

k = 0 , 1 ..... N - l } ,

(27a)

Z,~-{g k,

k=O, 1.... , N - l } ,

(27b)

(34)

g, h were used by the authors ofref. [ 15 ] to construct the truncation of SDlff(T 2), ( 7 a ) - (7c)

n,, m,e2~, ~= 1, 2, 3, where all operations are modulo N. The centre of H~ is

L. --, J. = oJ~'"2/2g"'h"2 .

(28)

The role of P~ and P2 is taken by the operators (19) and (21) Pi-'Q,

(29a)

4n

P2 ~ -ff P = F - t Q F .

(29b)

The generators (28) define a projective representation of the abelian translation group of the discretized torus S),,×Z,,, (16). By definition the projective unitary representations of a group G are unitary matrix representations such that T¢g, ) T(g2) = exp [ ict (gt, g2 ) ] T~x,~)

(30)

[29,30]. The phase c~(g~, g2) is constrained to satisfy the associativity properly (co-chain rule) [29,30] a(glg2,

(31)

We easily check that the phase appearing in J,,J,,, =oJ"x"/2Jm+., a(n,m)=-

2n -~ (n×m) ,

(35)

We give below all the possible representations of H [31 ], adapted to discrete phase space Z,~,×Zx and time 2VN.Define the matrix which on any function acts as follows: [ U(n, n 3 ) j q ( m )

=exp{i2[ ~n3 +nl ( m + ~n2) ] } f ( m + n 2 ) , f : 2v,v-oC

(36)

It is easy to check that this is a r~presentation of the Heisenberg group for any 2, U(n; n3)U(m; m3) = U ( n + m ; n3 + m 3 + n × m ) .

(37)

Since the element (0; 1) is cyclic,

(38)

we find that exp(½i2N) = 1

(32) (33)

satisfies the co-chain rule. The projective representations of a group Go are usually constructed as irreducible representations of an abelian extension of Go by a subgroup H of the centre of a bigger group G such that G o = G / H [29,30]. So it is natural to ask if there is such a group for the case of the projective representation of the abelian translation group Z~'XZN. The answer is yes and the corresponding group is the Heisenberg-Weyl group H~, o v e r Z N ~ ZNX Z u [ 29 ]. The HeisenbergWeyl group is the group with Lie algebra, the Heisen338

Tt = {(0, 0, n)l n~Zu} •

(0; l)U= (0; 0),

g3) + ct(gl , g2)

= a ( g ~ , g2g3) + ot (gz, g3) •

= (nl + m r , n2 + m 2 , n3 + m 3 + n t m2 - - n 2 m ~ ) ,

4n ~2=~-k,

k = 0 , 1 ..... N - l ,

(39)

and finally (choose k = 1 ), U(0; n 3 ) = c o ~/2,

to=exp(4ni/N).

(40)

We can check now that the elements (n; 0) are represented by the matrices W ( n ) = U( n; O) =oam~2/2gmh ~2 .

(41)

Indeed, W(1,O)=g, W(n,,O)=g~',

W(O, 1 ) = h , W(O, n 2 ) = h ~2,

W(nt, 0) W(0, n2) = w . . . . 2/2W(nl, n2) •

(42a) (42b) (43)

Volume 228, number 3

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21 September 1989

We proved finally that the SU (N) algebra of ref. [ 15 ] is the projective representation of H~/C~. It is easy to see that the full H ~satisfies the same algebra

U(n; n3)= U(Ano; n3)

U(n; n3)U(m; m3)

where A is constructed out of products of the matrices Tand J. This defines what is called the metaplectic representation ofSp(2, ZN) =LS(2, Zx) [31 ]. The representation of the Heisenberg group constructed above has a corresponding group algebra [24], which becomes a Lie algebra with bracket, the usual commutator of matrices. This Lie algebra is then the SU(N) algebra of ref. [15]. With this observation we easily extend the construction to the case N--, oo. Consider the space of functions fe L2 (Z),

= c o " × " / Z U ( n + m ; n3 + m 3 ) •

(44)

= i t ( A ) U ( n o , n3)it-l (A) ,

We close our discussion with the observation that the constructed representation of the Heisenberg-Weyl group can be described also as the metaplectic representation of SL(2, Z:,,) the discretized area preserving group of the membrane lattice Z~.× Zu. Consider the stability group of the point no =- (0, 1 ) on the lattice Z,~×ZN, this is the abelian subgroup of SL(2, Zx), Tx.

f : Z--*C

TN={/In

such that

01):n~Z~. } .

(45)

The number of classes SL(2, ZN/TN is (N 2 - 1 ) [ 32 ]. Any other point on Z~,× Z:,.can be reached by the action of products of the generators of SL(2, ZN) 11), J = (01 - ; ) .

(46a,b)

keZN

(47)

then we can check that It ( T ) U( n; n3 ) It - l ( T ) = U( Tn; n3 ) •

and with inner product

(49)

h F - t g -l ,

(50a,b)

FU(n; n3)F -~ = U ( J n ; n3) •

(51)

so It(J) = F. Since any element of the lattice n can be reached by an appropriate element ofSL(2, ZN)

then

(56a)

( Pf) ( n ) = e x p ( i2n )f( n ) ,

(56b)

where 2 # 0 is a fixed real number. These matrices are unitary and satisfy the basic relation TP=exp(i2 )PT.

no = (10),

(57)

J~ =exp( ½12nl n 2 ) P " T "2 , n = ( n l , n2)~Z×Z

we find

N=Ano,

neZ,

(Tf)(n)=f(n+l),

Then the generators J., defined as

From the relations (tn--,tn "-t ) hF=Fg,

(55)

~ f*(n)g(n). n~Z

(48)

Consider also the action of the Fourier transform matrix (27) FU(n; n3)F -I =to~'3+"'~)/ZFg"h'~F -I

(54b)

rleZ

Define two operators (infinite dimensional matrices Tand P),

Define now the matrix [ 33 ] [it(T)f] ( k ) = c o k ' / 2 f ( k ) ,

(54a)

I~1- ~ If(n)l 2 < + ~

(f,g)-

T= (~

(53)

(52)

(58)

form an irreducible unitary, infinite dimensional representation of the algebra, [J., arm]= - 2 i sin( ½2.n×m)J,,+,.

(59)

the irreducibility follows from the lrreducibdity of the representation of the Heisenberg group which eq. (58) define [31 ]. We conclude by noticing that the representation theory of the Heisenberg group has applications to 339

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PHYSICS LETTERS B

methods of fast Fourier transform [34 ] and the theory of classical theta functions [ 35 ]. I w o u l d like to t h a n k G . L e o n t a n s for h~s c o n t r i b u t i o n in t h e i n i t i a l stage o f t h i s w o r k .

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