the uniqueness of the moyal algebra

Volume 248, number 3,4 PHYSICS LETTERS B 4 October 1990

The uniqueness of the Moyal algebra

Paul F le tcher 1,2 Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK

Received 22 June 1990; revised manuscript received 20 July 1990

It is shown that the most general Lie algebra of functions of two variables under the operation of a general bracket is the Moyal bracket algebra. Correspondingly all 2-index algebras are locally equivalent to the sine algebra, or one of its subalgebras. The many applications of this algebra in various bases are indicated.

1. Over the last 18 months much work has been done on infinite dimensional algebras, in particular the Poisson bracket algebra, which is the algebra o f area preserving diffeomorphisms of a 2-surface, and also corresponds to the N ~ limit o f S U ( N ) [ 11 ]. The Moyal bracket algebra is a deformation of this algebra involving a parameter; the algebra becoming the Poisson algebra i n the limit as this parameter tends to zero.

The Moyal bracket was introduced by Moyal [ 2 ] as an exemplification o f Weyl's correspondence prin- ciple [ 3 ] to formulate quantum mechanics in terms of Wigner distribution functions [4 ]. There is a detailed paper by Bayen et al. [ 5 ], which includes a statement that the Moyal bracket is the only deformation o f the Poisson bracket which may be used in this respect. The main assumption they make is that their posited bracket is a function o f the Poisson bracket. A more recent paper by Arveson [ 6 ] gives a more direct proof that the only function o f iterated Poisson brackets which satisfies the Jacobi identities is the Moyal bracket, and that this is the only structure which can be used as the phase space formulation o f quantum mechanics.

There are various ways o f writing the Moyal bracket of two func t ionsfand g;, in terms of a formal operator,

~ , g}Moyal = lim 1 s in(xV× V' ) f ( x )g (x ' ) , x ~ x ' 1(

l Supported by a Durham University Research Studentship. 2 Bitnet address: [email protected]

x a 2-vector, or its expansion,

(;) s=o ( 2 s + l ) ! j=~o ( - 1 ) s 2s 1

rW a:s+ [0~ ~+ X,~#~y ' - J f (x ,y ) ] -JO~g(x,y)] , (1.1)

or as a generalised convolution

1 f ~,, g}Moya! = ~ dx ' dx" f ( x ' )g(x" )

X sin x ( x X x ' +x' Xx" +x" X x ) .

In the limit x ~ 0 these become the Poisson bracket

Of Og of og {f,, g}Poi . . . . = 0x 0y 0y 0x" (1.2)

These are Lie algebras, as they are antisymmetric and satisfy the Jacobi identity

{ ~, g}, h} + c y c l i c = 0 . (1.3)

Another area of recent interest is that of infinite algebras of generators indexed by two integers, in general

[Lr,,, L,, ] =C~,Lp , (1.4)

where m, etc. are integral 2-vectors. There is in fact a correspondence between such algebras and the algebras o f functions under a bracket operation like the Moyal or Poisson examples above. The form of such brackets most easily dealt with is the sum of derivatives o f the functions, that o f (1.1) and (1.2), the most general such form being

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 323


~,g}

: ~ ~ ~r+s--2 ~ ~ brj,sk(OJxO~-jf)(Okx~Sy_kg), r = l s = l j = 0 k = 0

(1.5)

where the b's are arbitrary constants. The 2-index algebras may be thought of as algebras of the modes in the expansion of the algebra of derivatives of contin- uous coordinates on manifolds, the expansion being taken over the basis functions for that manifold. Given a bracket of two functions, it is possible to define generators Ly such that [Ly, g] = {f g}, which satisfy the algebra

[ Lf, Lg ] =L{f,g}. (1.6)

For example, in the Poisson case

L f = O f O Of O Ox Oy Oy Ox "

Now any choice of a complete set of functions as a basis for f, g may be substituted into the above to give a 2-index algebra. For example, choose a basis e= (x) = e ire'x, which is the basis for the torus, then for f= era, g = e~ in ( 1.6 ) we find SDiff(T 2 ), or SU ( oo ),

[Lm, L,, ] = (mXn)L,, ,+., (1.7)

where m X n = rn~n2- rn2n~, and any 0 t he r f g may be expanded in terms of the e., to give a linear combi- nation of the L m. A different choice of em(x) will lead to different structure constants, for example the choice e= (x) = emmx~ =- l, the cylindrical basis, leads to the algebra SDiff(S ~ X R),

[L=,L.]

: [ ( n 2 - - 1 )ml - ( m 2 - 1 )nl ] L ( m i + n i , m 2 + n 2 _ 2 ) •

Similarly for the Moyal bracket; with the torus basis functions, the algebra obtained is

[Lm, L,,] = ls in(xm×n)Lm+,, . (1.8)

Thus for a given bracket of functions, there are many choices of basis which lead to seemingly different 2-index algebras.

Conversely it may be seen that any 2-index infinite algebra may be written as a bracket algebra of functions. This has also been shown by Dorfman and

Gelfand [ 7]; here I give a simpler argument with a slight restriction on the structure constants.

The general 2-index infinite algebra (1.4) may be rewritten as an algebra of functions. Let era(x) be a basis of functions, and define Le, =L=. Then any Ly is defined through the composition f ( x ) = ~mf=em(X) to be Y~mfmLe,. The commutation relations become

[Le,., Le,,] = C~nLp =Lc~.ep,

the last step by linearity. This is equivalent to a bracket algebra of the above

type if there is some bracket such that {era, e.}= C~,,ep. Now ife ' s are powers, era(x)_ ..m~+s,v,-2+s2 ---'~" 1 -4"2 ,

for sufficiently large integers &, s2, and the structure constants C~, tend to zero a s p - m - n tends to infin- ity, then it is possible to write

Y~ C~.ep= Y~ b,,iOJxemO~-Je., p r , j

this expression determining the b's in terms of the (given) Cs, and so the algebra may be written as a bracket algebra of the form ( 1.5 ).

Other algebras of current interest are the Zamolod- chikov Wy-algebras [ 8 ], the conformal algebras of spin ~< N, which are not Lie algebras, as they are not closed under commutation, but are non-linear. Re- cent work of Bakas [ 9 ], building on results of Bilal [ 10 ], demonstrates that the infinite limit of these algebras is a Lie algebra, explicitly, the Poisson algebra on the cylinder. Further work by Pope et al. [ 11 ] finds a deformation of this which has central extensions at all spins; this is also a Lie algebra, in fact, as shown by Faiflie and Nuyts [ 12 ], the Moyal algebra in some special basis.

The aim of this paper is to provide a straightfor- ward proof that all 2-index infinite Lie algebras correspond to the Moyal algebra (or its special case, the Poisson algebra) in some basis. This is done by showing that any bracket algebra satisfying the Jacobi identities may be transformed to the Moyal bracket algebra. This also means that the only associative product allowed as a deformation of the Poisson is that corresponding to the Moyal bracket, the expo- nential bracket ,l. This basis-independent formulation of 2-index algebras in terms of brackets does not

at Which is an associative product, not a Lie bracket.

324


allow the discussion of central extensions, the exis- tence and form of which depend on the basis used, and so for physical applications the basis is of great importance, but in all cases the fundamental algebra and its underlying associative product are the same.

The many applications may be summarised in terms of the basis used. The Poisson algebra is the algebra of area preserving diffeomorphisms of a 2- surface [ 13 ], as the Poisson bracket preserves the in- finitesimal area element. It is realised for different manifolds in terms of their basis functions.

For the toms, exp[i(mlx+rnzy)], the algebra SDiff(T 2 ) is ( 1.7 ) [ 13 ], this is also SU (oo) [ 1 ] and has been used in membrane physics [ 14 ]. It allows a linear central extension [ 15 ].

On the sphere, Y,,,,,,,,(x, y), there are applications in membrane physics [ 16 ] and also in atmospheric physics [ 17 ].

On the plane, x ' ' + ~ym2+ 1, the Poisson algebra was studied by Fuks [ 18 ] and Bakas [ 9 ] found that the algebra woo was equivalent to this algebra on the cylinder, x m'- l emW.

On the torus the Moyal algebra is the sine algebra [ 1 ] ( 1.8 ), which contains all the classical finite Lie algebras as special cases. Saveliev and Vershik [ 19 ] have studied these algebras in a different formalism, and some of their applications.

On the plane Bender and Dunne [ 20 ] use this algebra in a formulation of quantum mechanics.

The algebras studied by Pope et al. [ 11 ] and Fairlie and Nuyts ( 12 ] cases 1,2, 3 correspond to the Moyal algebra v,~th basis functions em'X/yy 2(m2+ 1 ), a cone, for Pope and for Fairlie's case 1, the cylinder for Fair- lie's case 2 and a hypergeometric function for case 3.

Pope and Romans [ 21 ] have also studied this algebra on K 2 and p2; in fact this basis allows only a subalgebra of the Moyal algebra, which has finite special cases of the subalgebras of SU (N).

2. First consider a bracket of the following form:

{f,g}._:.. ~ 2r--l ~ ~ j r--j k r--k bo~(OxO~ f ) (O~or~ g). r = I j = 0 k = 0

(2.1)

When 2 = 0 this reduces to the Poisson bracket, an- t isymmetry forcing bloo = b111 = 0, bl to = - biol. Here

there is an overall normalisation, blol, which may be set to 1. In general, antisymmetry requires b~jk= -brkj, and there will be one overall normalisation, corresponding to a choice of biol. The parameter 2 may be absorbed into the b's, and henceforth this is done.

It is also necessary to factor out by transformations of the independent variables x, y, such as

oy/ \c dJ\Oy]"

This transformation changes the b's but does not in- dicate a genuinely different bracket. The coefficient of (OefOxg- OArOyg) transforms from blol to (ad- bc)blol so ad-bc is chosen to make the new b~01 = 1. This is merely a choice of overall normalisation. There is still some freedom; that of transformations of the above type with ad-bc= 1, which leave this choice unaffected. Given a bracket of the above form, with arbitrary b's, a, b, c, d may be chosen to transform the b2's to a simple form. Under the above change of variables they are mapped

b20z~--,|b'202l:lcd bc+ad abllb202 ~. b212] \b'21:] \c 2 2ac a2J\b21:J

Note that this mapping preserves b22o2-b2ol b212 = q2. The matrix

-b2ol/2rl 1 b2olb212/2q(b2o2+q)-(b2o2+q)/bzol)(l: b Ob)

for b2ol, q ~ 0

makes b ~Ol = b~12 = 0, b~o2 = q. When b2o~ = 0 the corresponding matrix is

(o 1

When t/=0 there are certain subtleties, but detailed analysis shows that this is equivalent to the case b~o~ =b~12 =0, b~oz =t l=0 . Thus any bracket of the form (2.1) can be transformed to this case with general q.

Further constraints on the b's are given by the Jacobi identity ( 1.3 ),

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{{f ,g},h}= ~ ~ ~ ba,~(OlxOTt(f,,g}q)(Omot-mh) t= 1 I=0 m=O

= ~ ~ ~ ~ t - - l l ) ( t - - ' )

t=l l=Om=Or=lj=Ok=On=Op~----o(n P

]~ k I ~J+n~r--jWPl~t i~k+l--n~r--k+t--l--p~ X ~tlmVrjk t, Vx Uy j ] ~u x ,ay 151

x (~07mh) .

This plus cyclic permuta t ions must be zero for ar- bi trary functions f, g, h, so all the derivatives of these functions must have zero coefficients. Let the coefficient of

fx fr gx gy hx hy (OxOfJ)(Ox Oy g)(Ox Oy h)

be T(f i , fy, gx, gy, hx, he), and then T + cyclic (f, g, h) = 0.

l t - l T(fx,L, gx, gy, hx, hy)= E ( ) ( )btlmbrj k,

t,l,m,r,j,k,n,p \ n / \ p /

summing under the constraints

f ~ = j + n , O<~l, m, f i k, n, p ,

f y = r - j + p , 1 <t, r ,

g x = k + l - n , l , m < t ,

g y = r - k + t - l - p , .h k <~r ,

h x = m , n<~l ,

h e = t - m , p<<.t-l ,

which may be rewritten as

m = h x ,

t=hx + h e ,

r= ( f x + f y + g x + g y - h x - h y ) / 2 ,

j = f x + g x - k - l ,

p = r + t - g y - k - l ,

n = k + l - g x ,

O, r - g y <~k <~r, gx ,

O <~ l <~ t ,

g x , f x + g x - r < k + l < ~ f ~ + g x , r + t - g y ,

f rom which any T c a n be calculated by summing over allowed k, l.

Now the Jacobi identities are

CTCfx, f~, gx, ge, hx, he)

= T( fx , fy , gx,gy, hx, hy) + c y c l i c ( f g, h ) = 0 .

The b's can be determined by looking at certain of these, in particular

CT(fl, a - 3 , Y, a - y, 1, 1 )

= 2b2ol b~_ l,p,r - 2b212bc~_ 1,~- l,y-- 1

+ (27--a)b~,p,y + (2fl-- a)b.,p,~

= 0 ,

CT(~, a - ~ , a - l + 1, ~ - 1, 0, 2)

= (b~,,¢,,~_ ¢ - b 2 o l b ._ 1,¢,,~-¢ - 2b202 b,~_ 1,¢- l,,~-¢

-- b2olba-l,¢-l,a-¢+l + (o l -~+ 1 )ba,¢-l,a-¢+l

=0 ,

C T ( a , 0, 0, a - 1 , 1 ,2)

--- b312ba-z,o,a-2 + ½ 0 t ( a - 1 )b..o.~

- b . _ l , o , ~ - 2 b212 - ( ot - 1 ) b ~ _ I , O , . - I b2o2

= 0 ,

which may be solved to give the following recurrence relations for the b's:

1 b'~aY = o r - f l - 7 (b2oi b=_ t,p,y - b212 bc,- l,#- l,r- 1 )

for a # fl+ y ,

1 b,,,¢,,~_¢ = ~ [ ( ( - 1 - ot)b~.¢_l,~_¢+ 1 +b2ol b~- l ,c~-¢

+ 2b2o2 b,,_ ~z- i,~- ¢ + b2o ~ b~_ ~,¢_ ~,._ ¢+ ~ ]

for ( # O,

2 , , - - 2 2 b ~ o ~ - a ( a - 1 ) [ (3b3o3 - b2o2)b~_2,o,._2

+ ( a -- 1 ) b2o2 ba- l,o,a- 1 ] •

There is a unique solution to these recurrence relations in closed form in the case bzol =b212 = 0 and b2o2 = ~]"

0 / _ p ) ¢ ( q + p ) ' ~ - ¢ - 01+ p )¢( q - p ) '~-¢ b,~,¢,,~_ ¢ = 2p~! ( a - ~) ! '

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Volume 248, number 3,4

where p = x/6b3o3 - 3q 2 and b ~ = 0 if a ~ f l+ 7. I f t /= 0 this is the Moyal bracket, where b~,¢,,~_¢ =

p " - ~/(! ( a - ()! if ot is odd, zero otherwise, so

(--I)SK2S2s+l ( ; )

Z (0Jx02*+'-g) ~a 2*+l-jaa ,,~

where s = ( r - 1 ) /2 , x= ip . In fact the q dependence may be removed as there

is a convolution o f the arbitrary functions that transforms this from any q to rl=O. This corresponds to a change of origin for the independent variables, where f (x , y) is replaced by a convolution:

(27t)2 exp - r/

×f(x ' , y ' ) dx' dy' .

The iteration o f this expression with a second parameter q' just reproduces the same formula with parameter ~/+ ~/'.

This convolution takes any bracket o f the above type to Moyal form, and so all brackets o f the form (2.1) which satisfy the Jacobi identities are equivalent to the Moyal bracket.

3. There is a more general bracket than that consid- ered above, one in which the functions f g are not always o f the same order in derivatives, that is r and s are not necessarily the same in the following expression:

r = 1 s = I j = O k = O

X j r-j ks-k. (OxOy f)(OxOy g) . (3.1)

The analysis o f the previous section may be applied to this more general case, and in a similar way pro- duces recurrence relations defining the b's in terms of those at the lowest level. In particular, all the b's with r~s are defined in terms o f boo, m and booa~. But there are transformations which take the general bracket above to one with these particular b's vanish- ing, and so the Jacobi identities imply that all the b's with r e s must vanish.

The 2 = 0 case o f this bracket is

PHYSICS LETTERS B 4 October 1990

( f g}o = boo,,o (fOyg- gOyf)

+ boo, l l (fOxg-gO~f) + blo,1, ( OyfOxg-- OygO~f),

the last term of this being the Poisson bracket. I f the basis functions are multiplied by a factor and simul- taneously the coordinates are changed,

f(x, y) ~ e x p (boo, l o x - boo, l IY) F(x, y) ,

g(x, y) -~exp (booaoX- boo, l 1Y) G(x, y) ,

1 (x, y) ~ (X, Y) = 2v/~

×(, , ) boo,11 exp(-2bOO, l ly) , boo, lo exp(2booaox) ,

this reduces to the Poisson bracket,

{ f g)o = blo, j2 ( OrF OxG- OrG OxF) .

Thus all brackets o f the form (3.1) may be transformed to a bracket with boo, m = boo,~ = 0, and any such bracket satisfying the Jacobi identity must be o f the form of (2.1), and so these, too, are all equivalent to Moyal.

I would like to thank David Fairlie, Cosmas Zachos and Jean Nuyts for many useful discussions. I am also grateful to the Argonne Quantum Group Institute, at which this work was completed, for its hospitality.

References

[ 1 ] D.B. Fairlie, P. Fletcher and C.K. Zachos, Phys. Lett. B 218 (1989) 203; D.B. Fairlie and C.K. Zachos, Phys. Len. B 244 (1989) 101; D.B. Fairlie, P. Fletcher and C.K. Zachos, J. Math. Phys. 31 (1990) 1085.

[ 2 ] J. Moyal, Proc. Camh. Phil. Soc. 45 ( 1949 ) 99. [3 ] H. Weyl, The theory of groups and quantum mechanics

(Dover, 1931 ). [4] E.P. Wigner, Phys. Rev. 40 (1932) 749. [5] E Bayen, M. Flato, C. Fronsdahl, A. Lichnerowicz and D.

Sternheimer, Ann. Phys. (NY) 111 (1978) 61. [61W. Arveson, Commun. Math. Phys. 89 (1983) 77. [711.M. Gelfand and I.Ya. Dorfman, Funk. Anal. Pril. 13

(1980) 248; 14 (1980) 248; 15 (1981) 173; 16 (1982) 241. [8] A.B. Zamolodchikov, Teor. Mat. Fiz. 65 (1985) 347. [9 ] I. Bakas, The structure of the W~ algebra, preprint PP#90-

085. [10] A. Bilal, Phys. Lett. B 227 (1989) 406.

327


[ 11 ] C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B 236 (1990) i73;B242 (1990) 401.

[12]D.B. Faidie and J. Nuyts, Renormalisations and deformations of W~o, CERN preprint CERN-TH-5691/90.

[13]V. Arnold, Ann. Inst. Fourier XVI:I (1966) 319; Mathematical methods of classical mechanics (Springer, Berlin, 1978), Appendix 2.K, p. 339.

[ 14 ] E. Floratos and J. Iliopoulos, Phys. Lett. B 201 ( 1988 ) 237. [ 15 ] I. Bars, C. Pope and E. Sezgin, Phys. Lett. B 210 (1988) 85. [ 16 ] J. Hoppe, MIT Ph.D. Thesis ( 1982); in: Constraints theory

and relativistic dynamics (Florence 1986 ), eds. G. Longhi and L. Lusanna (World Scientific, 1987 ) p. 267; Phys. Lett. B215 (1988) 706.

[ 17 ] J.S. Dowker and M. Wei, Area preserving diffeomorphisms and the stability of the atmosphere, preprint MUTP-90/9; V.Yu. Zeitlin, Algebraization of 2-dimensional ideal fluid hydrodynamical systems and their finite mode approximation, Kiev preprint.

[ 18 ] D. Fuks, Funct. Anal. Appl. 19 ( 1985 ) 305. [ 19 ] M.V. Saveliev and A.M. Vershik, Commun. Math. Phys. 126

(1989) 367; Phys. Lett. A 143 (1990) 121; R.M. Kasaev, M.V. Saveliev, S.A. Savelieva and A.M. Vershik, On non-linear equations associated with Lie algebras of diffeomorphism groups of 2D manifolds, prepfint IHEP-90-1.

[20] C.M. Bender and G. Dunne, Phys. Rev. D 40 (1989) 3504. [21]C.N. Pope and L.J. Romans, Class. Quantum Grav. 7

(1990) 97.

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the uniqueness of the moyal algebra

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