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INFINITE DIMENSIONAL ALGEBRAS

arXiv:hep-th/9111053v1 25 Nov 1991INFINITE DIMENSIONAL ALGEBRASIN CHERN-SIMONS QUANTUM MECHANICSRoberto FloreaniniIstituto Nazionale di Fisica Nucleare, Sezione di TriesteDipartimento di Fisica Teorica, Universit`a di TriesteStrada Costiera 11, 34014 Trieste, ItalyRoberto PercacciInternational School for Advanced Studiesvia Beirut 4, 34014 Trieste, ItalyandIstituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, ItalyandErgin SezginCenter for Theoretical PhysicsTexas A&M UniversityCollege Station, TX 77843-4242AbstractWe study a charged particle in an electromagnetic field without kineticor potential term. Although dynamically trivial, this system is interest-ing because it has an infinite dimensional symmetry group.

We discussthe way in which this group behaves under quantization.Ref. SISSA XX/91/EP (February 1991)1

It is a challenging task to quantize nonlinear sigma models with an action given only by aWess-Zumino-Witten term. We have already studied these models from a classical pointof view in [1].

Here we discuss the quantization of the 0 + 1-dimensional case. This isthe simplest version of such theories, describing a particle moving on an n-dimensionalmanifold N, with actionS =Zdt ˙qαAα(q) ,(1)where A is a fixed background electromagnetic potential.

This model is also related toChern-Simons theories and therefore has come to be known as “Chern-Simons quantummechanics” [2,3]. We will now recall the main features of the model, and refer the readerto [1] for more details.The Euler-Lagrange equations which follow from (1) are˙qαFαβ = 0(2)where Fαβ = ∂αAβ −∂βAα is the field strength.

These equations demand that ˙qα bea null eigenvector of F. Without loss of generality, in this paper we will consider onlythe case in which F is nondegenerate (i.e. it is a symplectic form).

In fact, if F wasdegenerate, there would be a nontrivial gauge group G and the model would be equivalentto a particle moving on N/G in a nondegenerate electromagnetic field. Note that whenF is nondegenerate, eq.

(2) states simply that ˙qα = 0. In this case the Hamiltonian isidentically zero.The action (1) is invariant under the group S(F) of symplectic diffeomorphisms of N,also called the canonical transformations of N. The Lie algebra of S(F), denoted X(F),is the algebra of vectorfields v on N such that the Lie derivative of F along v vanishes;equivalently, if N is simply connected,vαFαβ = ∂βΩv ,(3)for some (globally defined) real function Ωv on N (X(F) is called the algebra of hamilto-nian vectorfields on N).

The Noether charge correponding to an infinitesimal symmetrytransformation v is −Ωv. The symplectic diffeomorphisms are genuine symmetries and notgauge invariances.

Thus we are in a very unusual situation in which a finite dimensionalsystem possesses an infinite dimensional symmetry group.The phase space of this theory is N itself, with symplectic form F [1,4]. The Poissonbracket of two functions f, g on N is{f, g} = (F−1)αβ∂αf ∂βg .

(4)The Lie algebra of all smooth real functions on N with this bracket will be denoted byΓ. Given any Ω∈Γ we can construct a unique hamiltonian vectorfield v using eq.

(3).Since functions differing by a constant give the same v, Γ is a central extension of X(F).Conversely given a hamiltonian vectorfield v, eq. (3) defines Ωv only up to a constant.

If itis possible to fix this ambiguity in such a way that{Ωv, Ωw} = Ω[v,w] ,(5)2

then, as an algebra, Γ = R ⊕X(F). In this case the center R can be dropped and theNoether charges provide a realization of the abstract algebra X(F).

However, as we shallsee later, this is sometimes impossible and a nonremovable central term c(v, w) may appearon the r.h.s. of eq.(5).

Note that all this is still at the classical level.Since the dynamics of the system is trivial, the only interesting thing to discuss areits symmetries. Note further that the classical observables of a dynamical system are thefunctions on phase space; since in our case all functions are generators of symmetries, therefollows that the discussion of the algebra of observables coincides with the discussion ofthe symmetry algebra.Now, there is a well-known theorem, originally due to van Hove, which implies thatthere is no way of quantizing this system preserving the whole classical symmetry algebra.More precisely, there is no irreducible representation of the observables f ∈Γ as self-adjointoperators ˆf on a Hilbert space such that[ ˆf, ˆg] = i d{f, g} .

(6)For a clear exposition, see [5]. This impossibility of preserving a classical symmetry al-gebra at the quantum level is reminiscent of anomalies in gauge theories.

However, thesituation here is quite different. In gauge theories the anomalies manifest themselves asnon conservation of certain charges.

On the other hand in our model all Noether chargesare conserved because of the vanishing Hamiltonian. Instead, the algebra obeyed by thesecharges is modified.In what follows we will consider various special cases for N and discuss the fate of thesymmetry group when the model is quantized.N = R2We take Aα = −12εαβqβ, with α, β = 1, 2 and ε12 = 1.

Then Fαβ = εαβ is the euclideaninvariant volume form on R2 and S(F) is the group SDiffR2 of volume-preserving dif-feomorphisms of R2. A constant factor g in front of F can be absorbed by a rescaling ofthe coordinates.

Thus the theory is independent of the strength of the magnetic field.The maximal finite dimensional subgroup of SDiffR2 is the semidirect product ofthe group R × R of translations (with generators v(1) = ∂1, v(2) = ∂2) and the groupSL(2, R) ∼Sp(1, R) (with generators v(0) = q2∂2 −q1∂1, v(+) = q2∂1 and v(−) = q1∂2).The nonvanishing Lie brackets of these vectorfields are [v(+), v(−)] = v(0), [v(0), v(±)] =±2v(±), [v(1), v(0)] = −v(1), [v(2), v(0)] = v(2), [v(1), v(−)] = v(2), [v(2), v(+)] = v(1). Notethat v(1), v(2) and v(−) −v(+) generate the Euclidean group.Already at the classical level, the Noether charges do not give a representation ofthe symmetry algebra X(F), but rather of its central extension Γ.

A basis for Γ is givenby the monomials f (m,n) = (q1)m(q2)n, with m, n = 0, 1, 2 . .

. .The functions whichgenerate the finite dimensional subgroup are Ω(1) = f (0,1), Ω(2) = −f (1,0), Ω(0) = −f (1,1),Ω(+) = 12f (0,2), Ω(−) = −12f (2,0).

The Poisson algebra of these generators is the sameas the Lie bracket algebra of the corresponding vectorfields, except for {Ω(1), Ω(2)} = −1.Clearly the center cannot be eliminated by redefining the generators. Therefore in thismodel the generators of translations fail to commute already at the classical level and thealgebra of translations is enlarged to the Heisenberg algebra h(1).3

The system can be quantized in the Schr¨odinger picture replacing q2 and q1 by theoperators ˆq2 = q and ˆq1 = −i ∂∂q acting on the Hilbert space of square integrable functionsof q. The quantization of all functions in Γ can be achieved by replacing each monomialf (m,n) by the symmetrically ordered operator ˆf (m,n) = S(m,n)α1,...,αm+n ˆqα1 .

. .

ˆqαm+n, whereS(m,n) is the totally symmetric tensor with componentsS(m,n)α1,...,αm+n =n m!n!/(m + n)!if αi = 1 for m values of i;0otherwise. (7)The algebra of these operators closes, but is different from the classical symmetry algebra.We begin by observing that the maximal finite dimensional subalgebra is not modifiedby quantization; in particular, the operators ˆΩ(0) and ˆΩ(±) give a representation of thealgebra sl(2, R) with Casimir operator 12(ˆΩ(+) ˆΩ(−) + ˆΩ(−) ˆΩ(+)) + 14(ˆΩ(0))2 =316.

This isknown as the metaplectic representation [6]. The complete quantum symmetry algebra isthe universal enveloping algebra of h(1) +☎✆sl(2, R).

The realization of this algebra on theHilbert space has a nontrivial kernel generated by the elements of the formˆΩ(0) −12(ˆΩ(1) ˆΩ(2) + ˆΩ(2) ˆΩ(1)),ˆΩ(+) −12(ˆΩ(1))2,ˆΩ(−) + 12(ˆΩ(2))2 . (8)The algebra which is realized faithfully is obtained by factoring out this kernel.It isisomorphic to the universal enveloping algebra U(h(1)).We note that any ordering prescription would give rise to a closed algebra of quantumoperators.

All these algebras are isomorphic. In fact, a choice of ordering prescription isequivalent to a choice of basis in the universal enveloping algebra.Another possibility is to introduce a Z2 grading in the space of the quantum operatorsˆf (m,n): we call them fermionic or bosonic depending on whether m + n is odd or even.Then, introducing a corresponding graded bracket, the operators ˆf (m,n) with m+n = 1, 2,generate the noncompact version of the superalgebra osp(1, 2).

Note that in this approachit is not necessary to introduce the constant multiples of unity to close the algebra. Thisrepresentation of osp(1, 2) is also known as the metaplectic representation [7].

With thisgraded bracket the entire algebra generated by the operators ˆf (m,n), with m + n > 0, isthe universal enveloping algebra of osp(1, 2). Note that this is also known as the higherspin algebra shs(2) [8].

Again, there is a nontrivial kernel given by (8).Finally, we observe that all that has been said so far for the case N = R2 can begeneralized in a straightforward manner to the case N = R2n.We choose F2i−1,2i =−F2i,2i−1 = 1, for i = 1, . .

., n. Then, the quantum symmetry algebra is the universalenveloping algebra of h(n) +☎✆sp(n, R) if only commutators are used, and of osp(1, 2n) ifthe graded brackets are used.N = S2 = SO(3)/O(2)In this case we take F to be the field strength of a monopole with magnetic charge g. Unlikethe previous case, here it is not possible to eliminate the constant g from the discussionby redefining coordinates. In polar coordinates θ and ϕ, Aθ = 0, Aϕ = −g(1 + cos θ), andFθϕ = g sin θ.

The symmetry group S(F) is now the group SDiffS2 of volume-preservingdiffeomorphisms of the sphere. Its maximal finite dimensional subgroup is SO(3).4

The Noether charges can be uniquely defined by requiring that their integral over S2 bezero. In this way no center arises [1].

This gives a realization of the Lie algebra of SDiffS2as the Poisson algebra of functions on S2 with vanishing integral. A suitable basis for thesefunctions are the spherical harmonics Y mlwith l = 1, 2 .

. ..

The structure constants of theLie algebra of SDiffS2 have been computed in this basis in [9].We note for futurereference that the spherical harmonics can be thought of as homogeneous polynomials inthe variables x1, x2, x3, if S2 is embedded in R3 by the equation x21 + x22 + x23 = 1. Thethree harmonics with l = 1 (corresponding to the polynomials x1, x2, x3) generate thegroup SO(3).

In fact, definingJ1 = −gx1 = −g sin θ cos ϕ = gr2π3 (Y 11 + Y −11) ,(9a)J2 = −gx2 = −g sin θ sin ϕ = −igr2π3 (Y 11 −Y −11) ,(9b)J3 = −gx3 = −g cos θ= −gr4π3 Y 01 ,(9c)and using the Poisson bracket (4), we get {Jα, Jβ} = εαβγJγ.For the quantization of a classical system with compact phase space the best approachis that of geometric quantization [10]. This method yields a Hilbert space and a set ofquantum observables whose algebra is isomorphic to the Poisson algebra of the correspond-ing classical observables.

Because of van Hove’s theorem, this procedure can not work forall observables. In the case of S2 it works successfully only for the functions Jα.Geometric quantization requires that 2g be an integer, i.e.

that the Dirac conditionbe satisfied. Then, the Hilbert space Hg is the space of polynomials of order ≤2g in thecomplex variable z = eiϕ cot(θ/2), with inner product [11]⟨Ψ, Φ⟩= 2g + 12πiZS2dz d¯z(1 + z¯z)2g+2 Ψ∗(z)Φ(z) .

(10)The dimension of Hg is 2g + 1 and a basis is given by the functions 1, z, z2, . .

. , z2g.The operators corresponding to J1, J2 and J3 areˆJ1 = 12(1 −z2)∂z + gz ,(11a)ˆJ2 = i2(1 + z2)∂z −igz ,(11b)ˆJ3 = z∂z −g ,(11c)obeying [ ˆJα, ˆJβ] = iεαβγ ˆJγ.

Notice that the value of the Casimir operator ˆJ2 in the rep-resentation (11) is g(g + 1). Therefore, the Hilbert space Hg is the familiar representationspace of angular momentum spanned by kets |j, m⟩, with j = g. The function zn in Hgcorresponds to the state |g, n −g⟩.As we have mentioned, the remaining functions on the sphere cannot be quantized insuch a way that their classical Poisson algebra is preserved.

Nevertheless, as in the case ofR2, one could envisage a more general scheme.5

It follows from previous remarks that a basis for all smooth functions on the sphere isgiven by polynomials in the three harmonics with l = 1. In order to achieve a quantizationof the whole set of classical observables it is therefore sufficient to quantize powers of J’s.We associate to each monomial f (n1,n2,n3) = (J1)n1(J2)n2(J3)n3 an operator ˆf (n1,n2,n3),in which the factors ˆJα have been ordered according to some given prescription.

In thisway the whole algebra Γ of functions on S2 is turned under quantization into the universalenveloping algebra U(so(3)). The quantum symmetry algebra consists of the operators inHg corresponding to functions with vanishing integral.

These form the algebra U(so(3))/R,where R are the constant multiples of unity.Since the Hilbert space is finite dimensional, this representation will have an infinitedimensional kernel. A classical theorem of Burnside [12] says that factoring out this kernelwe remain with the finite dimensional algebra of linear transformations in Hg.

Thus onecan regard the quantum symmetry algebra to be sl(2g + 1, C) or sl(2g + 1, R), dependingon whether the representation of so(3) is complex or real.N = S(1,1) = SO(1, 2)/O(2)The hyperboloid S(1,1) is the surface in R3 defined by the equation x21 + x22 −x20 = −1.On this surface we define coordinates χ, ϕ by x1 = sinh χ cos ϕ, x2 = sinh χ sin ϕ andx0 = cosh χ. As electromagnetic field we take Aχ = 0, Aϕ = g cosh χ and Fχϕ = g sinh χ.The symmetry group S(F) = SDiffS(1,1) consists of volume-preserving diffeomorphismsof S(1,1).

Its maximal finite dimensional subgroup is SO(1, 2). As in the case of the sphere,a basis for functions on the hyperboloid is given by the homogeneous polynomials in thevariables x1, x2, x0.

The polynomials of degree ≥1 clearly give rise (through eq. (3)) toall hamiltonian vectorfields, and close under Poisson bracket.

Therefore also in this casethe Poisson algebra of the Noether charges does not get a central extension.The maximal compact subalgebra so(1, 2) is generated by the functionsK1 = gx1 = g sinh χ cos ϕ ,(13a)K2 = gx2 = g sinh χ sin ϕ ,(13b)K0 = gx0 = g cosh χ ,(13c)obeying {K1, K2} = −K0, {K0, K1} = K2, {K0, K2} = −K1. This subalgebra can bequantized without modification.

It is convenient to define the complex coordinate z =tanh(χ/2)eiϕ (|z| < 1).For g ≥1/2 we define the Hilbert space Hg as the space ofholomorphic functions on the unit disk with inner product⟨Ψ, Φ⟩= (2g −1)2πiZ|z|<1dz d¯z(1 −z¯z)2g−2Ψ∗(z)Φ(z) . (14)The condition on g is dictated by the requirement that the inner product be well defined(similar conditions have been discussed in [13]).

For g < 1/2 one can use other represen-tations of so(1, 2)[14], but we will not discuss this here. The operators corresponding toK1, K2 and K3 are [14,11]ˆK1 = 12(1 + z2)∂z + gz ,(15a)6

ˆK2 = i2(1 −z2)∂z −igz ,(15b)ˆK3 = z∂z + g . (15c)The value of the Casimir operator ˆK21 + ˆK22 −ˆK20 in this representation is g(1−g).

For non-integer values of 2g, the above representation of the algebra so(1, 2) leads to multi-valuedrepresentations of the group SO(1, 2) [14].Proceeding as in the case of the sphere, we now associate to each monomial f (n1,n2,n3) =(K1)n1(K2)n2(K3)n3 the operator ˆf (n1,n2,n3) with a given ordering of the factors ˆKα. Thenthe quantum algebra of the Noether charges will be U(so(1, 2))/R.

It is represented faith-fully on Hg.The examples we have considered suggest that the quantization of Chern-Simons quan-tum mechanics will follow the same general pattern, at least for a wide class of manifoldsN. The classical symmetry group of canonical transformations has a maximal finite dimen-sional subgroup G. One can build a Hilbert space H carrying a realization of this group;thus the algebra of the corresponding Noether charges can be quantized in accordance witheq.(6).

We assume that all the symmetry charges (all functions on N) can be expanded inpolynomials in the generators of G (this is called the “strong generating principle” in [15]).Then, choosing any ordering, all these charges can be realized as quantum operators onH. However, their algebra will now be different from the classical one: it is the universalenveloping algebra of the Lie algebra of G. This realization of the universal envelopingalgebra might have a nontrivial kernel, so to obtain a faithfully realized symmetry algebraone has to factor it out.

If N is compact, the resulting quantum symmetry algebra is finitedimensional.We conclude by observing that this way of quantizing a classical dynamical system isnot restricted to Chern-Simons quantum mechanics, but may be used also in the presenceof a kinetic term. Our approach allows to quantize all functions on phase space, i.e.

toconstruct a quantum operator acting on Hilbert space for each classical observable. Theobstruction to quantization given in van Hove’s theorem is circumvented by relaxing thecondition that (6) holds for all f, g. As a consequence the algebra of quantum observablesis a deformation of the classical Poisson algebra and only reduces to the latter in theclassical limit.

This is different from the usual point of view on quantization, where oneinsists on (6) but does not seek to quantize all functions on phase space. Typically, onetries only to quantize those functions which generate the symmetries of the theory.

Whena kinetic or potential term is present, the symmetry group is finite dimensional and, as wehave observed, the algebra of its generators is quantized without deformation. It is onlyfor “topological” theories with vanishing Hamiltonian that every function on phase spaceis a symmetry generator, and the necessity of quantizing all observables arises.7

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