Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole
PParticle on the sphere:group-theoretic quantizationin the presence of a magnetic monopole
Rodrigo Andrade e Silva ∗ and Ted Jacobson † Center for Fundamental Physics, University of MarylandCollege Park, MD 20742, USA
Abstract
The problem of quantizing a particle on a 2-sphere has been treated bynumerous approaches, including Isham’s global method based on unitary rep-resentations of a symplectic symmetry group that acts transitively on the phasespace. Here we reconsider this simple model using Isham’s scheme, enrichedby a magnetic flux through the sphere via a modification of the symplecticform. To maintain complete generality we construct the Hilbert space directlyfrom the symmetry algebra, which is manifestly gauge-invariant, using ladderoperators. In this way, we recover algebraically the complete classification ofquantizations, and the corresponding energy spectra for the particle. The fa-mous Dirac quantization condition for the monopole charge follows from therequirement that the classical and quantum Casimir invariants match. In anappendix we explain the relation between this approach and the more commonone that assumes from the outset a Hilbert space of wave functions that aresections of a nontrivial line bundle over the sphere, and show how the Casimirinvariants of the algebra determine the bundle topology. ∗ [email protected] † [email protected] a r X i v : . [ qu a n t - ph ] N ov ontents E algebra 28B Details on the construction of the Hilbert space 29C Wavefunctions and Chern numbers 32D Non-uniform magnetic fields 39 Canonical quantization is a magic wand, discovered by Dirac, that transmogrifiesa classical dynamical theory into a corresponding quantum theory, often in perfectagreement with observations. However, for most classical theories Dirac’s proceduredepends on the choice of phase space coordinates over which to wave the wand,so the resulting quantum theory is ambiguous. Moreover, a generic phase spacehas nontrivial topology, and does not even admit a global coordinate chart. Incomplete generality, the only recourse is to accept the ambiguity, and to explore allquantizations. But some classical dynamical systems possess symmetries that can beused to identify a restricted class of quantizations which preserve these symmetries1n the quantum theory. Such quantizations would obviously yield the best guess, ifindeed the original classical theory is the classical limit of some quantum theory.Isham provided a generalization of Dirac’s canonical quantization that is de-signed to preserve a chosen transitive group of phase space symmetries, and can beapplied to topologically nontrivial phase spaces [1, 2]. Our primary interest in this,as was Isham’s, is ultimately to restrict the possibilities for nonperturbative quanti-zation of general relativity. But to develop understanding of how the scheme works,the ambiguities that remain, and the relation to other quantization schemes, it isuseful to consider simpler systems. A particle on a 2-sphere is one of the simplestsuch systems, and has already been treated by many different approaches, includ-ing Isham’s [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Here we considerthis simple model, enriched by the inclusion of a magnetic flux through the sphere,with the aim of implementing Isham’s quantization scheme without making anychoices other than that of the group of canonical symmetries. In order to main-tain complete generality for the unitary representations of the quantized algebra,we construct the Hilbert space directly from the algebra, rather than adopting theframework of wave functions. In this way, we recover algebraically the completeclassification of quantizations, as well as the famous Dirac quantization conditionfor the monopole charge and the corresponding energy spectra. We also explain therelation between this approach and the more common one that assumes from theoutset a Hilbert space of generalized wave functions that are sections of a nontrivialline bundle over the sphere.Isham applied his quantization scheme to the phase space P = T ∗ S , the cotan-gent bundle of the 2-sphere, with the canonical symplectic form ω = dp i ∧ dq i .He found that the Hilbert space must carry some unitary irreducible (projective)representation of the 3-dimensional Euclidean group, E = R (cid:111) SO (3). FromMackey’s theory of induced representations, one obtains that each irreducible uni-tary representation U ( n ) of U (1), labeled by an integer n , yields a Hilbert spacerepresented by sections of a certain bundle, U ( n ) -associated to the Hopf bundle SU (2) → SU (2) /U (1) ∼ S . (See Appendix C for details.) These sections can beseen as “twisted” wavefunctions over the sphere, with the “twisting” described bythe first Chern number of the bundle (which is n for the U ( n ) -associated bundle).The choice of this integer n is equivalent to the assignment of an intrinsic spin tothe particle, and its value is fixed to be zero if one imposes, as a correspondence A similar approach appears to have been considered in [8], however we could not obtain accessto the relevant part of the document. Related work is also mentioned in [17] by the same author,but it refers to a preprint that we could not find. n in terms of the prod-uct of the monopole charge with the electric charge. The inclusion of the magneticmonopole field is thus equivalent to the assignment of an intrinsic spin to to theparticle.The usual prescription to include coupling to a magnetic field is to make thereplacement p → p − eA in the Hamiltonian, where p is the momentum, e is theelectric charge of the particle and A is the magnetic potential 1-form in some localgauge. If A is defined globally on the sphere then the magnetic field B = dA isan exact 2-form, so the net magnetic flux through the sphere must vanish. In thepresence of a nonzero net magnetic flux, therefore, A cannot be defined globallyon the sphere, and hence the Hamiltonian is not globally defined. This can be ac-commodated by defining the Hamiltonian in local gauge patches, and accompanyinga gauge transformation A → A + dλ with a corresponding symplectic (canonical)transformation p → p + edλ . In this way, however, the canonical momentum ceasesto be an observable, and a global description is lacking.Instead, we shall maintain manifest gauge invariance and a globally definedHamiltonian by incorporating the magnetic field into the symplectic structure [18,19]. That is, we replace p by p + eA in the symplectic form dp i ∧ dq i . This resultsin d ( p i + eA i ) ∧ dq i = d ( p i dq i ) + ed ( A i dq i ), which can be written covariantly as ω = dθ + eπ ∗ B . (1.1)Here θ is the canonical symplectic potential, defined by θ ( X ) = p ( π ∗ X ), where X is a tangent vector on the cotangent bundle T ∗ S , and π : T ∗ S → S is the bundleprojection map (with π ∗ and π ∗ the pull-back and push-forward of π ), and B is themagnetic 2-form on the sphere (whose integral over any region gives the magneticflux through it). Our problem is thus to quantize a phase space with topology T ∗ S and symplectic form given by (1.1).This paper is organized as follows. In section 2, we begin with a brief reviewof Isham’s quantization scheme. In section 3, we study the phase space T ∗ S withthe charged symplectic form (1.1), identifying the appropriate quantizing group. Insection 4.1, we proceed to the quantization, establishing the correspondence betweenclassical observables and quantum self-adjoint operators. In section 4.2, we showthat the Casimir invariants of the algebra play an important role in linking the clas-3ical and quantum worlds. In particular, this is how the magnetic monopole makesits way into the quantum theory. Next, in section 4.3, we study the representa-tions of the group by constructing ladder operators for J , the angular momentumsquared. From the assumption that the theory is free of negative-norm states, insection 4.4 we recover Dirac’s charge quantization condition. Finally, in section 4.5,we compute the energy spectrum for a (non-relativistic) particle on a geometricsphere. In Appendix A we establish the absence of nontrivial central extensions ofthe Euclidean algebra E , and in Appendix B we give some details omitted in thederivation of section 4.3. In Appendix C we show how the representation in termsof “twisted” wavefunctions can be recovered and, in particular, how the magneticmonopole is related to their twisting, and in Appendix D we consider non-uniformmagnetic fields. In the founding days of quantum mechanics, Dirac remarked that “
The correspon-dence between the quantum and classical theories lies not so much the limitingagreement when (cid:126) → as in the fact that the mathematical operations on the twotheories obey in many cases the same laws. ” [20]. This observation led him to pos-tulate the general canonical quantization scheme which replaces Poisson brackets ofclassical functions on phase space by quantum commutators of quantum operators,i.e., [ ˆ f , ˆ g ] = i (cid:126)(cid:92) { f, g } . More precisely, one seeks a linear homomorphism from thealgebra of real functions on the phase space, with product defined by the Poissonbracket { f, g } , to an algebra of self-adjoint operators on some Hilbert space, withproduct defined by the commutator i (cid:126) [ ˆ f , ˆ g ], satisfying certain conditions, such asmapping the constant function f = 1 to the identity ˆ1 and, for all functions φ ,mapping φ ( f ) to φ ( ˆ f ).It turns out that no such map exists in general, as a consequence of Groenewold–Van Hove obstructions. Hence one must be careful to select a relatively small setof observables that can be consistently quantized, but which is still large enough toallow for the construction of quantized versions of all other classical observables. Thetrivial example is that of P = R n , where one can canonically quantize the globalcoordinates q i and p i , and then carry along all other observables f ( q, p ) → f (ˆ q, ˆ p ) Strictly speaking, the Van Hove no-go theorem applies only for trivial phase spaces, P = R n .The result has been extended to other cases, and it is expected that this kind of obstruction isgeneric [21]. However, some examples have been found where a full, unobstructed quantization ispossible [22, 23].
4o the quantum theory (modulo operator-ordering issues). Isham’s proposal is togeneralize this by identifying a transitive group of symplectic symmetries of thephase space and using it to generate both a special set of classical observables andtheir associated quantum self-adjoint operators. We call this group the quantizinggroup . (It is sometimes referred to as the “canonical group”.) In the trivial case justmentioned, for example, it could be the group of coordinate translations, ( q, p ) → ( q + a, p + b ), where q and p are any global canonical coordinates. Typically, thedynamical system possesses other structure that would select a preferred canonicalgroup, such as a metric on the configuration space that appears in the Hamiltonian.We shall briefly review Isham’s scheme in this section. For more details, see [1, 2].Consider a phase space P with symplectic 2-form ω . Assume that the phasespace is a homogeneous space for some Lie group G of symplectic symmetries. Thatis, there is a transitive left action δ g : P → P of G on P such that δ ∗ g ω = ω forall g ∈ G . Each element ξ in the Lie algebra g ∼ T G of G induces a vector field X ξ on P defined by X ξ | p = φ p ∗ ( ξ ), where φ p : G → P is defined by φ p ( g ) = δ g ( p ).This map is an antihomomorphism from g into the algebra of vector fields on P , i.e.,[ X ξ , X η ] = X [ η,ξ ] . Because δ g preserves ω , X ξ is a (locally) Hamiltonian field, i.e., £ X ξ ω = 0. We therefore have d ( ı X ξ ω ) = £ X ξ ω − ı X ξ dω = 0, where ı denotes theinterior product. That is, ı X ξ ω is closed and thus locally exact, so that dQ ξ = − ı X ξ ω admits local solutions Q ξ , defined up to addition of a constant function on P . Sincewe want these charges Q ξ to play the role of the canonical observables, we requirethat G generates only globally Hamiltonian fields on P , meaning that the associatedcharges are all defined globally on P .The symplectic form endows the space of functions on the phase space with analgebraic structure, A C , where the product is given by the Poisson bracket . Inparticular, when the functions are taken to be the charges, we have { Q ξ , Q η } = − ω ( X ξ , X η ), and it happens that the map ξ (cid:55)→ Q ξ is a homomorphism from g into A C up to central charges, that is, { Q ξ , Q η } = Q [ ξ,η ] + z ( ξ, η ), where z ( ξ, η ) isconstant on P . In practice, we can assume that this is a true homomorphism, i.e., z = 0, since the group can always be extended by a central element to make thatso. We call charge any function generated in this way by the group of symplectic symmetries,regardless if they are (also) dynamical symmetries in the sense of Noether’s theorem. Because ω is non-degenerate, any function f on P can be associated with a unique vector field X f on P via the relation df = − ı X f ω . The Poisson bracket between two functions, f and f (cid:48) , isdefined by { f, f (cid:48) } := − ω ( X f , X f (cid:48) ). If the central charge z ( ξ, η ) is not trivial (i.e., it cannot be removed by a redefinition of thecharges Q ξ → Q ξ + f ( ξ ), for some f : g → R ), one can always extend the group by a centralelement so that the extended algebra, g ⊕ S R , has product law [( ξ, a ) , ( η, b )] = ([ ξ, η ] , z ( ξ, η )). The G to construct the quantum theory. Let U : G → Aut( H ) be anirreducible unitary representation of G on a Hilbert space H . Each element of thealgebra ξ ∈ g can be exponentiated to a one-parameter subgroup of G , exp( tξ ),and the corresponding one-parameter unitary group is generated by a self-adjointoperator (cid:98) Q ξ on H , as U (exp tξ ) = e t (cid:98) Q ξ /i (cid:126) . From the definition of a representation,the map ξ (cid:55)→ (cid:98) Q ξ is a homomorphism from g into A Q , where A Q is the algebraof self-adjoint operators on H with product given by i (cid:126) [ · , · ]. It follows that thequantization map, which associates to each classical charge Q ξ the correspondinggenerator (cid:98) Q ξ of the unitary representation, Q ξ (cid:55)→ (cid:98) Q ξ , (2.1)is a homomorphism from A C into A Q . The logic of this quantization scheme issummarized in Fig. 1. ξ ∈ g X ξ U (exp ξ ) Q ξ (cid:98) Q ξ Figure 1:
On the classical side, each element ξ of the Lie algebra of the quantizing groupinduces a Hamiltonian vector field X ξ (on the phase space), which in turn defines a Hamil-tonian charge Q ξ . On the quantum side, the group element exp ξ is represented by a unitarytransformation (on a Hilbert space), whose self-adjoint generator is (cid:98) Q ξ . Since the space of physical states is actually the ray space, R := H /U (1), corre-sponding to the quotient of the Hilbert space H by phases e iθ ∈ U (1), it is naturalto consider also projective representations of the group G . A projective representa-tion is a homomorphism from G into the group of projective unitary operators on R , P U ( H ), consisting of equivalence classes U ∼ e iθ U of unitary operators on H .In essence, including projective irreducible unitary representations of G amounts toconstructing the quantum theory based on irreducible unitary representations of new group has a natural action on P (where the central element acts trivially), and the new chargesare related to the old ones simply by Q ( ξ,a ) = Q ξ + a . Consequently, the map ( ξ, a ) (cid:55)→ Q ( ξ,a ) isa true homomorphism. Hence, we can always assume that G is already the extension of whatevergroup we started with. Here “unitary” means that the observables Q ξ are represented by self-adjoint operators (cid:98) Q ξ on Q ξ . The condition that the group acts transitively on the phase space ensures thatany function on P can (locally) be expressed in terms of the canonical observables Q ξ ’s. To see this, consider the momentum map, J , which is a function from P to g ∗ (the dual algebra of G ) defined by J ( p )( ξ ) := Q ξ ( p ), where p ∈ P and ξ ∈ g . If thismap were an embedding , then all real functions on P could be written as functionsof Q ξ ’s. More concretely, note that a basis { ξ i } of g induces a coordinate systemin g ∗ defined by coordinate functions w i ( σ ) := σ ( ξ i ), where σ ∈ g ∗ , and these havethe property that Q ξ i = J ∗ w i . Since any smooth function f : P → R could, in thiscase, be seen as the pull-back under J of some function F : g ∗ → R , then f could bewritten as function of the Q ξ i ’s. In general, although J may not be an embedding,transitivity of the group action guarantees that it is an immersion of P into g ∗ .Transitivity implies that, at any p ∈ P , any tangent vector V ∈ T p P is equal to X η for some η ∈ g . The non-degeneracy of ω then implies that dQ ξ ( V ) = − ω ( X ξ , X η ) isnonvanishing for at least one ξ . In other words, there is no direction V along whichall charges are (locally) constant, implying that any function on P can be locallywritten in terms of the charges . Therefore, the special set of observables generatedin this way is indeed not too “small”.This completes our review of Isham’s quantization scheme. In pursuit of gen-erality, the scheme refers only to minimal structure required to define a “canonicalquantization”, which associates to a certain chosen classical Poisson algebra of ob-servables a corresponding quantum algebra of observables. But in order to fullydefine a physical quantum theory, a particular representation of the algebra mustbe chosen, and the dynamics must be implemented via a quantization of the Hamil-tonian. This may require additional physical ingredients to be introduced in thequantization. In many cases the choice of a representation is restricted by whatwe shall call a Casimir correspondence principle . A classical Casimir invariant isan observable that Poisson commutes with the entire Poisson algebra. If that ob-servable admits a quantization (i.e., a choice of operator ordering) that commutes the Hilbert space. Technically, there would also be projective representations of G associated with non-trivialcentral extensions (by 2-cocycles) of its algebra, if those exist. However, unless such a centralextension appears already in the classical Poisson algebra, it will not be of interest to us here,because of the Casimir correspondence principle introduced below. To say there is no direction V along which all charges are (locally) constant is equivalent tosaying that the derivative J ∗ is injective, so that J is an immersion. A further analysis [1] revealsthat if the immersion fails to be an embedding, it is at worst a covering map, i.e., P is a coveringspace for its image under J . This limits the extent to which functions on P can fail to be globallyexpressible in terms of the charges. In this section we address the classical part of Isham’s quantization scheme for thecase of a particle on a 2-sphere in the presence of a magnetic monopole. The phasespace is P = T ∗ S but, as explained in the introduction, the symplectic form mustbe given by (1.1) if the Hamiltonian is to be a globally well-defined function. Ourgoal is to identify a suitable group of symplectic symmetries of this phase space andthen compute the associated Poisson charges. A magnetic field 2-form B on S admits an infinite dimensional symmetry groupthat acts transitively on S , provided that B is nowhere vanishing. This is thegroup of “area” preserving diffeomorphisms, where B defines the area element. Weare interested in quantizing an SO (3) subgroup of this group. There are infinitelymany such subgroups, which all lead to equivalent phase space quantizations. Wewrite the magnetic field as B = g(cid:15) , (3.1)where the 2-form (cid:15) is scaled so that (cid:82) S (cid:15) = 4 π , and g is a dimensionful coefficient.The total flux through the sphere is simply 4 πg , so g can be interpreted as themagnetic charge of a monopole “inside” . The kinetic energy term in the Hamilto-nian for a charged particle on S involves a particular metric on the S . In order If the magnetic field were not nowhere vanishing, we could still separate it as B = g(cid:15) + dA ,where A is a globally defined potential 1-form that enters the Hamiltonian, while the g(cid:15) is includedin the symplectic form.
8o naturally quantize this Hamiltonian, we will ultimately base the quantization onthe SO (3) group of this metric, however that choice plays no role until we come toquantizing the Hamiltonian.We denote by l R ( x ) the action of a rotation R ∈ SO (3) on a point x on thesphere. As with any action on the configuration space, there is a natural liftedaction to the cotangent bundle, defined by L R ( p ) = l − ∗ R p , (3.2)which maps the fiber over x to that over l R ( x ), i.e. it satisfies π ◦ L R = l R ◦ π . Thecanonical potential 1-form θ is invariant under the lift of any point transformation(diffeomorphism of the configuration space). In particular, we have L ∗ R θ = θ ,and therefore L ∗ R dθ = dL ∗ R θ = dθ . Moreover π ∗ B is invariant under rotations: L ∗ R ( π ∗ B ) = ( π ◦ L R ) ∗ B = ( l R ◦ π ) ∗ B = π ∗ B . Therefore the symplectic form (1.1) isinvariant, L ∗ R ω = ω . (3.3)That is, for all R , L R is a symmetry of the symplectic form.The quantizing group should be larger than just SO (3), since the rotations actonly “horizontally” on the phase space. For the quantizing group to act transitively,it should include elements that move points along the fibers of the cotangent bundle.The simplest “vertical” action is a translation of momentum, F α ( p ) = p − α , (3.4)where α is a 1-form field on S . (For notational simplicity we leave implicit thepoint π ( p ) at which α is evaluated.) This acts on the symplectic form (1.1) as F ∗ α ω = d ( F ∗ α θ ) + e ( π ◦ F α ) ∗ B . (3.5)The term π ∗ B is invariant since π ◦ F α = π . The symplectic potential, however,transforms non-trivially. For any V ∈ T p P , we have F ∗ α θ ( V ) = θ ( F α ∗ V ) = ( F α p )( π ∗ F α ∗ V )= ( p − α )( π ∗ V ) = ( θ − π ∗ α )( V ) , (3.6) Let φ be a diffeomorphism of the configuration space, and let Φ be its lift to the phase space,i.e., Φ( p ) := φ − ∗ p . If V ∈ T p P , then Φ ∗ θ ( V ) = θ (Φ ∗ V ) = (Φ( p ))( π ∗ Φ ∗ V ) = ( φ − ∗ p )( π ∗ Φ ∗ V ) = p ( φ − ∗ π ∗ Φ ∗ V ) = p ( π ∗ V ) = θ ( V ). Alternatively, using coordinates, p (cid:48) i dx (cid:48) i = p j ∂x j ∂x (cid:48) i ∂x (cid:48) i ∂x k dx k = p k dx k .
9o that F ∗ α ω = ω − π ∗ dα . (3.7)In order for F α to be a symplectic symmetry, we must therefore require α to beclosed. We will restrict further to exact α = df , with f globally defined on S to ensure that the associated charges will be globally defined.The (infinite-dimensional) space of all exact 1-form fields is unnecessarily large,so we look for a “minimal” set of momentum translations that act transitively alongthe fibers and are consistent with the spherical symmetry, in the sense that theyextend the chosen SO (3) into a larger group. As observed by Isham (in a moregeneral setting), a suitable set can be generated by realizing the configuration space S as an orbit of a representation of SO (3) in a vector space, and defining themomentum translations as the pullback to the orbit of the “constant” 1-forms onthat vector space. In particular, we can choose the fundamental representation on R , and identify the S with the orbit passing through u = (0 , ,
1) in R , that is,with the set of unit vectors x ∈ R such that x = Ru for some R ∈ SO (3). (Thenotation “ x ” for these vectors coincides with that we used already to label the pointsin S .) Any dual vector α ∈ R ∗ , can naturally be seen as a 1-form field on R .Moreover, this 1-form field is exact, for it can be written as df α , where the function f α : R → R is defined by f α ( x ) = α ( x ) . (3.8)These 1-form fields can be pulled-back to S to define the corresponding actionalong the fibers of P .Combining the L R and F α transformations, we get a transitive group of sym-plectic symmetries of the phase space: the semidirect product G = R ∗ (cid:111) SO (3),acting on P as Λ ( α,R ) ( p ) = l − ∗ R p − α , (3.9)which satisfies the product rule( α, R )( α (cid:48) , R (cid:48) ) = ( α + l − ∗ R α (cid:48) , RR (cid:48) ) . (3.10)Since R ∗ ∼ R and the co-representation of SO (3) in R ∗ is equivalent to itsrepresentation in R , this group is isomorphic to the Euclidean group, E = R (cid:111) This identification of the configuration space with the unit sphere in the abstract R shouldnot be confused with the physical sphere, which may have its own geometry. The orbit is a “unitsphere” with respect to the inner product (cid:104) v, u (cid:105) = (cid:80) i =1 v i u i on the abstract R . In a matrix realization, R − ∗ α = ( R − ) T α = Rα . O (3).We take this group, G , to be the quantizing group. Note that it is independentof the magnetic term in the symplectic form. Since its algebra does not admit anynon-trivial central extension by 2-cocycles z ( ξ, η ), no (non-trivial) central chargecan appear in the associated Poisson algebra. The quantizing algebra is thus thesame as in the uncharged case. However, as we shall see in Sec. 4.2, the magneticterm makes itself felt through the value of a Casimir invariant of the correspondingPoisson bracket algebra, which carries over to the quantum theory. We next compute the classical charges associated with the quantizing group G ,beginning with the SO (3) generators. Let n be an element of the algebra so (3), anddenote its exponential by R n = exp( n ). (Here exp : g → G is the usual Lie groupexponential map.) Let X n be the vector field induced by n through the action of SO (3) on P defined in (3.2), and let X n be the corresponding vector field on S ,i.e. the projection of X n .The corresponding charge P n is defined by dP n = − ı X n ω = − ı X n ( dθ + eg π ∗ (cid:15) )= d [ θ ( X n )] − eg π ∗ ı X n (cid:15) . (3.11)The first term in the last line follows from 0 = £ X n θ = ı X n dθ + dı X n θ , and θ isinvariant under any point transformation, as discussed in the paragraph leading to(3.3). Like the first term, the second term is also an exact 1-form: it is closed since d ( π ∗ ı X n (cid:15) ) = π ∗ dı X n (cid:15) = π ∗ L X n (cid:15) = 0, and since S is simply connected it is thereforealso exact, i.e., ı X n (cid:15) = d Γ n , for some Γ n : S → R . The charge P n is thus given by P n = p ( X n ) − eg Γ n ◦ π (3.12)up to an additive constant. The use of the symbol “ P ” is motivated by the fact thatthe charges associated with spatial transformations are the analogue of momentumcoordinates. The term p ( X n ) alone is the usual orbital angular momentum asso-ciated with the rotation Killing vector field X n , while P n is the canonical angular Since we found no demonstration of this statement in the literature, we include one in AppendixA for completeness. This formal proof is not really necessary for our purposes, however, as in thenext section we explicitly compute the Poisson algebra and show that no central charges arise. ω (1.1). On a phase space with the “usual” symplectic form dθ , thecanonical angular momentum would have been simply p ( X n ).Next we compute the charges associated with the R ∗ part of the group. Sincethe group R ∗ is a vector space, it can be naturally identified with its Lie algebra.Let Y α be the momentum translation vector field on P induced by an element α ofthe Lie algebra of R ∗ . The corresponding charge Q α is defined by dQ α = − ı Y α ω = − ı Y α dθ = − £ Y α θ = − ddt F ∗ tα θ (cid:12)(cid:12)(cid:12)(cid:12) t =0 = π ∗ α = π ∗ df α = d ( f α ◦ π ) , (3.13)where in the second line we used that ı Y α (cid:15) = 0; in the third line that ı Y α θ = 0; inthe fourth line that the flow induced by α is p (cid:55)→ F tα ( p ); in the fifth line we used(3.6); and in the sixth line we used that df α = α , as defined in (3.8). The chargeassociated with α is therefore Q α = f α ◦ π (3.14)up to an additive constant. The use of the symbol “ Q ” here is motivated by thefact that the charges associated with momentum translations are the analogue ofposition coordinates.To be more concrete, it is convenient to use the realization of S as the unit spherein R , which was introduced in the previous subsection. We identify n ∈ so (3) withthe vector in R whose direction is the corresponding axis of rotation and whosemagnitude | n | gives the angle of rotation of exp( n ), according to the right-hand rule.Then, using adapted spherical coordinates in which n is aligned with θ = 0, we have X n = | n | ∂ φ , hence ı X n (cid:15) = | n | ı ∂ φ sin θ dθ ∧ dφ = | n | d (cos θ ) = d ( n · x ), so we canchoose Γ n = n · x . Therefore, the canonical charges are given, up to an additive Alternatively, in local coordinates adapted to the bundle structure of T ∗ S , the flow generatedby α = α i dq i is given by F tα ( q i , p i ) = ( q i , p i − tα i ), so Y α = − α i ∂∂p i , and the charge differential is dQ α = − i Y α ( dp i ∧ dq i + eg (cid:15) ij dq i ∧ dq j ) = α i dq i = π ∗ α . In the last step the π ∗ appears because, inthis equation, q i are coordinates on T ∗ S , while in the definition of α they are coordinates on S . P n = p ( X n ) − eg n · x (3.15) Q α = α ( x ) . (3.16)The notation is somewhat abbreviated here. Strictly speaking, P n and Q α arefunctions on the phase space P , which is specified above by giving their values at apoint p ∈ P . The vector field X n is implicitly evaluated at π ( p ), and x is the unitvector representative of π ( p ) in the embedded realization of S ⊂ R .We next consider the Poisson bracket algebra of the charges. By construction,this algebra matches the Lie algebra of the canonical group that defined the charges,up to a possible central extension. If a central extension appears in such an algebra,in general it may or may not be removable using the freedom to shift the charges byaddition of constants. As mentioned above, the Euclidean algebra in itself (i.e. apartfrom any canonical realization) does not admit any non-trivial central extension (by2-cocycles), so that it must be possible to choose the additive constants such thatthe Poisson algebra matches the Lie algebra. In fact, the choices we have made in(3.15) and (3.16) satisfy this criterion, and the Poisson algebra takes the form { P n , P n (cid:48) } = P [ n,n (cid:48) ] { Q α , P n } = Q L Xn α { Q α , Q α (cid:48) } = 0 , (3.17)which matches the semi-direct product structure of the algebra g = R ∗ ⊕ S so (3) of G without central charges. That is, denoting elements of g by ( α, n ) ∈ R ∗ ⊕ S so (3),the product rule reads [(0 , n ) , (0 , n (cid:48) )] = (0 , [ n, n (cid:48) ])[( α, , (0 , n )] = ( L X n α, α, , ( α (cid:48) , , (3.18)revealing how the linear association ( α, n ) (cid:55)→ P n + Q α is a (true) homomorphism.To verify that the choices (3.15) and (3.16) lead to no central charges, and forlater purposes, it is convenient to introduce a basis for g . Using the identification so (3) ∼ R , choose an orthonormal basis { e i } ( i = 1 , ,
3) in R . (Note that exp( e i )implements a right-handed rotation by the angle 1 around the i -axis.) It is straight-13orward to check that [ e i , e j ] = ε ijk e k , where ε ijk is the Levi-Civita symbol. Let { e i } denote the dual basis, satisfying e i ( e j ) = δ ij . We define J i := P e i N i := Q e i , (3.19)which satisfy the algebra { J i , J j } = ε ijk J k { J i , N j } = ε ijk N k { N i , N j } = 0 . (3.20)This is the algebra of the Euclidean group, presented in terms of a basis of generatorsof rotation and translation.We can express (3.15) and (3.16) in this basis. If x ∈ S ⊂ R , we can write X e i = e i × x . Also, if p is a co-vector on S at x , we can (abusing the notation)associate it with a vector p in R , tangent to S at x , such that p · v = p ( v ), where v is any vector on R tangent to S at x . In this way, we have p ( X e i ) = p · ( e i × x ) = e i · ( x × p ), which is the familiar orbital angular momentum about the axis e i in R .Thus, J i = e i · ( x × p − eg x ). Also, N i , evaluated at any point in the fiber over x ,can be written as N i = e i ( x ) = e i · x . In a 3-vector notation, J = x × p − eg x (3.21) N = x , (3.22)so we have J i = e i · J and N i = e i · N .To establish (3.20), i.e., to verify that indeed there are no missing central terms,we may evaluate the brackets at points in the phase space where both sides of theequation vanish. For example, recall that { J , J } = − ω ( X e , X e ). The vector field X e vanishes at the points in phase space with zero momentum and located at therotation axis e on the S (i.e., the two points in the intersection of the zero sectionwith the fibers over x = ± e ). Hence { J , J } vanishes there. On the other hand,according to (3.21), at the same point the function J is equal to − eg e · e = 0.Any constant added to J would spoil this agreement. This argument works for all As SO (3) acts on R from the left, the algebra element n ∈ R induces the vector field X n (cid:12)(cid:12) x = n × x , where x ∈ R . The Lie bracket of two such vector fields is given by [ X n , X n (cid:48) ] = − X n × n (cid:48) .Together with [ X ξ , X η ] = X [ η,ξ ] (see third paragraph of section 2) this yields [ n, n (cid:48) ] = n × n (cid:48) .
14f the J i brackets, so we conclude that no central term need be added on the righthand side of the first bracket in (3.20). The argument just given also implies that { J , N } = − ω ( X e , Y e ) vanishes at the axis point e , while N equals e ( e ) = 0at that same point. Hence no central term appears in the second bracket either. Asfor the last brackets in (3.20), since the right hand side vanishes, it is unaffectedby addition of a constant to any charge. In conclusion, as we claimed, the chargesdefined in (3.15) and (3.16) provide a realization of the quantizing algebra withoutcentral charges. Quantization of the theory amounts to constructing some unitary irreducible pro-jective representation of the canonical group G ∼ E . Since the Euclidean algebradoes not admit (non-trivial) central extensions, the projective representations of E are in one-to-one correspondence with true representations of its universal cover, (cid:98) E ∼ R (cid:111) SU (2). Equivalently, this means that we should look for irreducibleunitary representations of the Euclidean algebra g = R ⊕ S so (3). Moreover, weshould satisfy the Casimir correspondence principle. That is, we should select theirreducible representation(s) in which the value of any classical Casimir invariantcarries over to the quantum theory, modulo possible operator ordering ambiguitythat might arise in quantizing the Casimir invariant.
The quantum version of the classical canonical observables J i and N i are the self-adjoint generators of the corresponding unitary transformations in some Hilbertspace H . That is, given some unitary irreducible representation U of G (or itsuniversal cover), and denoting elements of g by ( α, n ), we define the operators ˆ J i and ˆ N i by U [exp(0 , λe i )] =: e − iλ ˆ J i / (cid:126) U [exp( λe i , e − iλ ˆ N i / (cid:126) , (4.1) See [24]; or, for an informal discussion, [25]. λ is an arbitrary real parameter. It follows from the group structure that thequantized algebra satisfies [ ˆ J i , ˆ J j ] = i (cid:126) ε ijk ˆ J k [ ˆ J i , ˆ N j ] = i (cid:126) ε ijk ˆ N k [ ˆ N i , ˆ N j ] = 0 . (4.2)The quantization map is J i → ˆ J i and N i → ˆ N i , and (4.2) is the quantization of thePoisson algebra (3.20). For notational simplicity we shall henceforth omit the “hat”symbol over quantum operators, since it should be clear from the context whetherwe are referring to the classical or the quantum observables. Casimir operators, by definition, commute with all elements of the algebra, and theireigenvalues can therefore be used to label its irreducible representations . Thereare two independent Casimir operators associated with the algebra g , N = (cid:88) i ( N i ) N · J = (cid:88) i N i J i . (4.3)Their classical correspondents Poisson-commute with everything in the classical al-gebra and, using (3.22) and (3.21), we have N = x = 1 N · J = − eg x = − eg , (4.4)revealing that these two quantities are constant classical observables. Since nooperator ordering ambiguities arise in quantizing N and N · J (= J · N ), we takeas part of the quantization prescription that the quantum theory must carry therepresentation for which the values of the two Casimirs (4.3) are given by preciselythe corresponding classical values (4.4). Note that the value of N · J is the only waythe presence of the magnetic monopole is felt in the quantum theory. If an operator commutes with all other operators in an irreducible representation, then accord-ing to Schur’s lemma it must be proportional to the identity operator. .3 Representations of the algebra Although the representation theory of the Euclidean group is well known accordingto the Mackey theory of induced representations on a space of wavefunctions [26],we will present it here using an abstract ladder-operator approach. The methodis reminiscent of the usual derivation of the unitary irreducible representations of SU (2). Some of the details are left to Appendix B. In Appendix C we discuss analternative derivation based on Mackey theory, which provides a construction of theHilbert space based on wave functions on S .Note first that the Euclidean algebra (4.2) is invariant under rescaling of N .Thus, without loss of generality, we can particularize to the representation with N = 1. The value of the other Casimir is left arbitrary, N · J = s (cid:126) , (4.5)with s some real parameter.Let us start with a basis of simultaneous eigenvectors of J and J , denoted as | j, m (cid:105) , defined by J | j, m (cid:105) = j ( j + 1) (cid:126) | j, m (cid:105) J | j, m (cid:105) = m (cid:126) | j, m (cid:105) . (4.6)At this point it is not clear which values of j are allowed, for a given s , nor if thereis more than one state with a given value of j and m . Note that J ± = J ± iJ act asraising and lowering operators for m and, from the standard analysis of the angularmomentum algebra, we know that, for a given j , m varies from − j to j in integersteps. Also, j can only be a non-negative integer or half-integer (i.e., j ∈ Z ≥ ), butit may be that only a subset of that is included.Before systematically deriving the properties of the irreducible representations,it is enlightening to guess, by a simple but non-rigorous reasoning, which values of s and j are included. Supposing that there is (in a limiting sense) a state localizedat the north pole of the sphere, the operator N · J acts on such a state as J (see(4.18) for details). This implies that 2 s must be integer valued in order for therepresentation to be non-trivial. Since it has the J eigenvalue s (cid:126) , such a state mustbe constructed from states with j ≥ s . By virtue of rotational symmetry, the samecan be said about states localized at any other point on the S , and one would After completing this work we found that a similar realization of the representation appears in[9], although no explicit derivation is presented there. j ≥ s . Moreover, since N is a vector operator (in the way it transforms undercommutation with the J i ), its action can change the value of j by plus or minusunity, which suggests that repeated action of N will both raise the j values withoutbound and lower them until they reach a floor, which presumably lies at j = s , since j ≥ s is the only apparent constraint. That is, the representation must include allvalues j = s + n , for non-negative integers n . Indeed this is the correct spectrum,as we now show by explicit construction of the representation. We now present the rigorous derivation of the irreducible representations, ana-lyzing how certain operators in the algebra act as “ladder operators” to shift thevalues j and m . From the Wigner-Eckart theorem we know that when the vectoroperator N i acts on a state, it can only change the value of j by −
1, 0 or 1. And,since acting on a state with N ± = N ± iN changes the m value by ±
1, it followsthat N + | j, j (cid:105) ∝ | j + 1 , j + 1 (cid:105) . Thus N + acts as a raising operator for edge states | j, j (cid:105) , i.e. states with the maximal m = j for a given j . For now, let us assume thatif | j, j (cid:105) is in the Hilbert space then so is N + | j, j (cid:105) , i.e., that its norm is positive. Thisassumption will be justified later.Now let | j , j (cid:105) be the ground state, in the sense that there are no states with j < j in the representation being constructed. (We know that there must be sucha lowest j state, since j is non-negative.) Since we are constructing an irreduciblerepresentation, the whole Hilbert space H must be generated by acting with allelements of the algebra on any given state, in particular the ground state. Considerthen the set of states | j, m (cid:105) := ( J − ) j − m ( N + ) j − j | j , j (cid:105) , (4.7)where j − j ∈ Z ≥ and − j ≤ m ≤ j . Since these states have distinct eigenvaluesfor the the self-adjoint operators J and/or J , they are necessarily orthogonal(although not normalized). We will show that a representation exists only if s ∈ Z ,in which case there is a unique irreducible representation, spanned by the states (4.7)with j = | s | . To establish this, we first prove that these states are closed underthe action of the entire algebra, and then we show that that they all have positivenorm, provided the s quantization condition holds and j has the required value.It is clear how to define the action of J i on this basis, using just the angularmomentum algebra. That is, J i | j, m (cid:105) can be written as a linear combination of The earliest mention of this spectrum that we have found appears in [27], although no derivationwas given there. | j, j (cid:105) and theaction of the operators N + , N , N − and J − on it — the image of | j, j (cid:105) is in a linearsuperposition of states at the endpoints of the corresponding arrows; note that J keeps | j, j (cid:105) fixed and J + annihilates it. The diagram on the right shows the exampleof the representation with s = 3 /
2, indicating the states that are present in it. | j, m − (cid:105) , | j, m (cid:105) and | j, m + 1 (cid:105) with coefficients determined by the algebra. So let usfocus on defining the action of the N ’s. It is easy to see that, since the commutatorof an N with a J gives an N , the action of N i on any state is well-defined providedthat the N ’s have a well-defined action on the edge states | j, j (cid:105) . Similarly, if theedge states have positive norm then so do the rest of the states.By the definition of the basis states (4.7) we have N + | j, j (cid:105) = | j + 1 , j + 1 (cid:105) . (4.8)Next, using only the algebra and the Casimir invariants, we show in Appendix Athat N | j, j (cid:105) = s ( j + 1) | j, j (cid:105) − j + 1) | j + 1 , j (cid:105) (4.9)and, for j > j , N − | j, j (cid:105) = 2 j j + 1 (cid:18) − s j (cid:19) | j − , j − (cid:105) ++ s | j, j − (cid:105) j ( j + 1) − | j + 1 , j − (cid:105) j + 1)( j + 1) . (4.10)(The action of the algebra on the edge state | j, j (cid:105) is depicted on the left in Fig. 2.) Itthen remains to determine N − | j , j (cid:105) , to establish that the states | j, j (cid:105) with j ≥ j have positive norm, and to show that the algebra is consistent with the assumptionthat j cannot be lowered below j . 19bserve that the form of (4.10) indicates that the operator L ( j ) = N − − sj ( j + 1) J − + 12(2 j + 1)( j + 1) ( J − ) N + (4.11)acts as a j -lowering operator for | j, j (cid:105) , i.e. , L ( j ) | j, j (cid:105) ∝ | j − , j − (cid:105) , modulo theassumption that j > j . In fact, using only the algebra it can be established for any j , including j , that [ J , L ( j ) ] | j, j (cid:105) = − j (cid:126) L ( j ) | j, j (cid:105) , (4.12)hence J L ( j ) | j, j (cid:105) = j ( j − (cid:126) L ( j ) | j, j (cid:105) .The squared norms of the raised and lowered edge states can be computed usingonly the algebra and Casimir invariants. Doing so, we find (cid:107) N + | j, j (cid:105)(cid:107) = 2( j + 1)2 j + 3 (cid:18) − s ( j + 1) (cid:19) (cid:104) j, j | j, j (cid:105) (4.13) (cid:107) L ( j ) | j, j (cid:105)(cid:107) = 2 j j + 1 (cid:18) − s j (cid:19) (cid:104) j, j | j, j (cid:105) . (4.14)(See Appendix B for more details.) Starting with a state | j, j (cid:105) , we can apply asequence of L ( j ) operators to lower j successively. This process stops, preventing j from becoming negative, only if j differs from j by an integer and L ( j ) | j , j (cid:105) = 0. Itthus follows from (4.14), and the non-degeneracy of the Hilbert space, that j = | s | .Moreover, the condition L ( j ) | j , j (cid:105) = 0, together with (4.11), defines the action of N − on the ground state, N − | j , j (cid:105) = s | j , j − (cid:105) j ( j + 1) − | j + 1 , j − (cid:105) j + 1)( j + 1) . (4.15)This happens to agree with (4.10) particularized to j = j = | s | , although thederivation of (4.10) applied only for j > j . Finally, (4.13) shows that, since j ≥ | s | , N + always creates a positive-norm state. This establishes that, if | j , j (cid:105) has positive(squared) norm, then so do all of the other edge states. It follows that all statesdefined in (4.7) have positive norm and thus must be in H . (This array of states isillustrated on the right in Fig. 2, for the case j = 3 / N , N · J ) =(1 , s (cid:126) ), is indeed spanned by the states defined in (4.7), with j = | s | . As a conse-quence, we see that the Hilbert space is non-trivial if and only if s ∈ Z , (4.16)20ince j can only take values in Z ≥ . According to the matching condition for the Casimir (4.4), and the definition of s (4.5), the quantum theory is based on the representation with s = − eg/ (cid:126) . Therequirement for the theory to be non-trivial, (4.16), thus implies eg = n (cid:126) , n ∈ Z , (4.17)which is precisely Dirac’s charge quantization condition.It is interesting to note how the more restrictive Schwinger condition [28, 29, 30], eg = n (cid:126) , would appear in this approach. In the previous section we derived theconstraint on s by analyzing the representations of the algebra generated by J ’sand N ’s. However, not all of these can be “integrated” to representations of thegroup G = R ∗ (cid:111) SO (3). As mentioned before, the representations of the algebraare in correspondence with representations of the universal cover of the group, (cid:101) G = R ∗ (cid:111) SU (2). In order to have a true representation of SO (3), rather than justa projective representation (i.e. a representation up to a phase), one must imposethat a rotation by 2 π corresponds to the identity operator, which implies that onlyinteger spins, j ∈ Z ≥ , are allowed. Were one to insist that the quantum theory bebased on true representations of the quantizing group, G , this would thus imposethe constraint s ∈ Z , which leads to Schwinger’s condition. However, there is nofundamental reason within quantum mechanics to exclude projective representationsas realizations of a symmetry.Let us now make a comment about intrinsic spin. In addition to the interpreta-tion given in Section 4.2 for the Casimir N · J , as − eg , there is another interpretation,as a measure of the intrinsic spin of the particle. To see this, consider a basis ofsimultaneous eigenvectors of N = ( N , N , N ), denoted by | n , n , n (cid:105) . In the rep-resentation with N = 1 we must have n + n + n = 1. This can be interpreted as a(non-normalizable) state localized at the point n = ( n , n , n ) on the sphere. With-out loss of generality, consider such a state localized at the north pole | u (cid:105) = | , , (cid:105) .We have J | u (cid:105) = J · N | u (cid:105) = s (cid:126) | u (cid:105) , (4.18)so that | u (cid:105) is an eigenstate of J with eigenvalue s ∈ Z . But | u (cid:105) is localized atthe north pole, so this angular momentum must be an intrinsic spin of the particle.Hence, there is an equivalence between a spinless particle with electric charge e in21he presence of a magnetic monopole of charge g inside the sphere, and a particlewith spin s (cid:126) = − eg , with no magnetic monopole [31, 32]. Interestingly, the sameequivalence occurs in classical physics: a free particle with spin would precess ona circle smaller than the great circle, conserving angular momentum, exactly as ifthere were a magnetic field curving the orbit of a spinless charged particle. In fact,if it is not possible to turn off the magnetic monopole (or modify its charge), andthe particle is truly living on a sphere (without access to higher dimensions), thenthere is no way to distinguish a magnetic monopole from an intrinsic spin. In this section we compute the energy spectrum for a non-relativistic particle ofmass m living on an Euclidean sphere of radius r , in the presence of a magnetic fieldgiven by (3.1). We assume here that the magnetic field is uniform with respect tothe metric on the sphere. The classical time-evolution Hamiltonian is H = 12 m h ab p a p b , (4.19)where p a is the canonical momentum—a cotangent vector on the sphere—and h ab is the metric on the sphere (using abstract tensor index notation). Note that themagnetic field does not appear in the Hamiltonian for, in our approach, it is fullyencoded in the symplectic form. From Hamilton’s equations, we see that the kine-matical velocity is simply related to the momentum by˙ x α = { x α , H } = 1 m h αβ p β , (4.20)where x α = ( x , x ) are any coordinates for the sphere. Thus, on a solution, thevalue of the Hamiltonian is the kinetic energy, mh αβ ˙ x α ˙ x β .Up to this point only the topology of the sphere, and the choice of an SO (3)subgroup of the diffeomorphisms preserving B , has played a role in identifying andquantizing the canonical group. Now we identify this subgroup with the symmetrygroup of the metric h ab , and use that identification to express the Hamiltonian interms of the canonical observables J and N . To this end, we first express the inversemetric of the sphere in terms of the rotation Killing vector fields X i := X n = e i (cf.footnote 15) and the geometrical radius r , h ab = 1 r X ai X bi , (4.21)22ith implicit summation over the repeated index i . Inserting (4.21) for h ab in theHamiltonian (4.19) yields H = 12 mr ( X ai p a )( X bi p b )= 12 mr ( J i + egN i )( J i + egN i )= 12 mr [ J − ( eg ) ] , (4.22)where we used (3.15), (3.16), and (3.19) in the second line, and (4.4) in the thirdline. To quantize, we just replace the function J by the quantum operator J . TheHamiltonian is clearly diagonal in the basis | j, m (cid:105) , and the energy eigenvalues aredefined by H | j, m (cid:105) = (cid:126) mr (cid:2) j ( j + 1) − j (cid:3) | j, m (cid:105) , (4.23)where j = | s | = | eg | / (cid:126) , and j − j ∈ Z ≥ . Equivalently, in terms of the non-negativeintegers k = j − j , the energies can be enumerated as E km = (cid:126) mr [ k ( k + 1) + | s | (2 k + 1)] . (4.24)The degeneracy of these “Landau levels” arises only from m , so there are 2 j + 1 =2( k + | s | )+1 states at level j . In particular, the ground state has degeneracy 2 | s | +1,corresponding to one additional state for each magnetic flux quantum. This agreeswith other derivations of the energy spectrum [14, 33, 27, 34].The case where the magnetic field is not uniform with respect to the metricis slightly more subtle. In principle, we could include it in the symplectic formand proceed as before. However, unless B is uniform, no SO (3) subgroup of the B -preserving diffeomorphisms of the sphere would be an isometry of the metric.Consequently, there would be no preferred form for the Hamiltonian when writtenin terms of the canonical observables, leading to operator-ordering ambiguities inthe quantization. Instead we can split the magnetic field into its monopole andhigher multipole parts, continuing to include the monopole part in the symplecticstructure, but incorporating the higher multipole part into the Hamiltonian via theusual minimal coupling. By doing so, we obtain a globally defined Hamiltonian,which admits a standard application of canonical quantization. We discuss this case To verify that this sum yields the inverse metric, note first that it is invariant under rotations.The normalization constant can be determined by looking at a specific point, namely ( θ, φ ) =( π/ , X = 0, X = ∂ θ and X = ∂ φ , and the inverse metric thus reads r − ( ∂ θ + ∂ φ ), as desired.
23n Appendix D, where we also address the issue of gauge invariance in Isham’s grouptheoretic quantization scheme.
In this paper we have studied the problem of quantizing a particle on a 2-sphere inthe presence of a magnetic monopole using Isham’s group-theoretic scheme. Basedon the principles of canonical quantization, the quantum theory is constructed fromunitary irreducible (projective) representations of a transitive group of symplecticsymmetries of the phase space. Our goal was to analyze the problem in a rigor-ous manner, emphasising the role of Casimir invariants in connecting the classicaland the quantum worlds. To ensure robustness and generality of the quantization,we referred only to intrinsic properties of the system, adopted a gauge-invariantapproach, and did not make any a priori assumptions about the Hilbert space.In order to formulate the problem in a gauge-invariant language, with globallydefined objects, we described the magnetic monopole as a “flux term” in the sym-plectic form on the phase space. A natural transitive group of symmetries of thephase space is the Euclidean group, E = R ∗ (cid:111) SO (3), where SO (3) implementsspatial rotations of the sphere and R ∗ corresponds to momentum translations (i.e.,boosts). The canonical observables consist of three angular momentum “coordi-nates”, J , generating the spatial rotations, and three position “coordinates”, N ,generating the momentum translations. Although the group structure is indepen-dent of the magnetic monopole, the presence of the latter affects the expressions ofthe canonical observables, J and N , as functions on the phase space. To constructthe Hilbert space, we employed an algebraic method involving ladder operators thatraise and lower the eigenvalue of the J operator. This method for deriving theunitary irreducible representations of the Euclidean group resembles the usual ap-proach for SU (2). By imposing that the theory is free of negative-norm states, onegets a constraint on the possible values of the Casimir invariants N and J · N . Ifwe set N = 1 (which can always be done without loss of generality), then J · N must be an integer multiple of (cid:126) /
2. Insisting that the values of these quantum in-variants must be the same as their classical counterparts, we obtain Dirac’s chargequantization condition, eg = n (cid:126) / SO (3) subgroup of the (infinite-dimensional) group of volume-preserving diffeo-morphisms of the sphere, whose lift to the phase space was used to construct the24uantizing group. Although we argued that this choice does not affect the Hilbertspace, it is relevant for the dynamics of the quantum theory. Indeed, the form ofthe Hamiltonian as a function of the canonical observables depends on which SO (3)subgroup is chosen. For a generic metric, no such a choice is preferred and thusthe Hamiltonian would have no preferred form, leading to operator-ordering am-biguities. We contrast this with the case of the round sphere, where the group ofisometries of the metric provides a preferred SO (3) subgroup of symmetries, andleads to a simple (free) Hamiltonian that can be quantized unambiguously. Savefor a few other highly symmetric cases (like the ellipsoid), some additional princi-ple would be needed to resolve the ambiguities associated with a generic geometry.Another, more fundamental, ambiguity that is resolved by the metric is the choiceof the quantizing group itself. In fact, there are many finite-dimensional groupsthat act transitively on the sphere (e.g., the Lorentz group), so it may be that theunderlying justification for the choice of SO (3) comes down to the symmetries ofthe metric. In particular, a selection criterion like the fact that SO (3) is the small-est transitive group on the configuration space S does not apply in general. Forexample, for a particle on S one could choose SU (2) ∼ S itself as the group of“spatial translations”, while the (larger) group SO (4) seems more appropriate for around metric.It is important to stress that this procedure is predicated on the assumption thatthe classical theory, and in particular all of the observables, are defined globally onphase space, since the quantum description is inherently global. To illustrate thispoint, suppose that, instead of incorporating the magnetic monopole field in thesymplectic form, we took the more common route of using the canonical sympleticform, and including the vector potential A in the Hamiltonian, H = m ( p − eA ) .It would still be natural to choose the Euclidean group for the quantizing group. Ifwe did so, the Casimir J · N would vanish classically and, if the Casimir matchingprinciple were to apply, this would mean that J · N would also vanish quantummechanically. But the quantum theory should not depend on the particular way onedecides to describe the classical system, and we have already established that in thequantum theory J · N should be equal to − eg . This failure of the Casimir matchingprinciple can be attributed to the failure of the classical phase space description tobe global. Since no globally defined vector potential A can describe the monopole,one must cover the sphere in at least two overlapping gauge patches. Classically,one can focus on phase space trajectories that are temporarily confined to one or theother patch, switching descriptions in the overlap region, but that precludes a globalmap between the classical and quantum observables. This is not to say that one25annot describe the quantum theory using the vector potential for the monopole,but only that it cannot be done according to the standard framework of globalcanonical quantization. Instead, one can ensure that the quantum description itselfis globally well defined. As explained in Appendix C, in such a global descriptionthe wave functions are sections of a complex line bundle over the sphere carryinga representation of the Euclidean group. In such a representation, p − eA can berealized as a covariant derivative provided that the product of the charges satisfiesthe Dirac quantization condition eg = n/ n , and that the Chernnumber of the bundle is n . In this formulation, the quantum Casimir J · N takesthe correct value, − eg .As the previous paragraph makes clear, when quantizing a classical system thedetermination of the correct Hilbert space can depend not only on the intrinsicstructure of the phase space, but on the dynamics as well. For the particle onthe sphere in a monopole field, the impact of the dynamics on the Hilbert spaceis felt either from the Casimir matching principle (when the dynamical effects ofthe magnetic monopole are encoded in the symplectic form), or from the Chernclass selection (when the monopole is encoded in the Hamiltonian via the gaugepotential). This role of the dynamics is absent in quantum mechanics on R n since (bythe Stone-von Neumann theorem) the canonical algebra of position and momentumcoordinates has a unique representation, so the Hamiltonian plays no role in selectingthe Hilbert space. On the other hand, it is ubiquitous in quantum field theory, wherethe infinite dimensionality of the algebra allows for many representations of thecanonical algebra, and the selection of a representation depends on other physicalobservables, such as the Hamiltonian. For example, as shown by Haag’s theorem [35,36], the representations containing a translation invariant vacuum state in infinitevolume differ even for a free scalar field with different masses [37]. The problem ofa particle on a sphere provides a simple example where the non-uniqueness of therepresentation, resolved only by the dynamics, comes from the non-trivial topologyof the phase space, rather than from the infinite dimensionality of the algebra.The difference between the quantizations on a plane and on a sphere can beunderstood from a group-theoretic perspective as follows, in terms of the “planarlimit” of the sphere. We expect that a particle that remains near the north pole ofthe sphere at all times should not be able to “feel” the global structure of the sphere.Thus, in some limit, the quantum mechanics on a sphere must reduce to the usual oneon a plane. To see how this works, consider a “sector” of the Hilbert space in which X := N ∼ θ , X = N ∼ θ and N ≈
1, where θ (cid:28) P ∼ θ | P |
26s small in this sector, so we can approximate J ≈ − P − egX , J ≈ P − egX and J ≈ X P − X P − eg , where terms of order θ were neglected. In thissector, the algebra reduces to [ X i , P j ] = i (cid:126) δ ij N , J behaves as the generator ofrotations for X i and P i , and N becomes a central element (taking the value 1 inthe relevant representation). This deformation of the algebra is known as the In¨on¨u-Wigner contraction. At the group level, this corresponds to a deformation of theEuclidean group E = R (cid:111) SO (3) into ( R × R ) (cid:111) ( R (cid:111) SO (2)), where the firstfactor is generated by ( X , X ; N ) and the second by ( P , P ; J ). This contractedgroup can be reexpressed as H (2) (cid:111) SO (2), where H (2) is the Heisenberg group intwo spatial dimensions, generated by ( X , P , X , P ; N ), and SO (2) is the rotationgroup around the origin, generated by J . Note that S := J − ( X P − X P ) = − eg is a Casimir operator, interpreted as the intrinsic spin of the particle. Since J differs from X P − X P only by a Casimir operator, it follows that the irreduciblerepresentations of H (2) (cid:111) SO (2) are also irreducible when restricted to H (2). As H (2) has a unique irreducible unitary representation, this confirms that we do in factrecover the quantum mechanics on a plane (for any value of the intrinsic spin Casimir S ). It is interesting to note an important difference between the plane and thesphere: the subgroup SO (2) of E was “pulled out” of SO (3) during the deformation,so it appears as a factor in H (2) (cid:111) SO (2) rather than as a subgroup of SO (3). Thespin is therefore not quantized on a plane, because the SO (2) gets “unwrapped”to R when considering projective representations (that is, when considering theuniversal cover of the group). Thus the quantization of the spin on a sphere is atruly topological effect.In conclusion, we have seen that the problem of quantizing a particle on a sphereis quite effective in revealing some of the subtleties associated with a non-trivialphase space topology. It also serves to illustrate how Isham’s scheme, which providesa general class of quantum theories compatible with the classical kinematics, must bepaired with additional principles (e.g., Casimir matching) and considerations aboutdynamics in order to single out a preferred quantum theory. The lesson that oneshould be mindful about global aspects of quantization applies as well to quantumfield theories with non-trivial phase space topology, for which there may remainthings to be learned about its implications. In particular, the longstanding problemof quantizing general relativity, as a candidate theory of quantum gravity, is a primeexample: even before imposing the Hamiltonian constraints, the configuration spaceis given by metrics (or co-frame fields) on a spatial slice, and the condition that theseare non-degenerate leads to a phase space with non-trivial topology. This suggeststhat standard canonical commutation relations are not appropriate, and that instead27he quantization should be based on an affine algebra [38, 39, 40, 41, 42], a conclusionthat is also reached by the application of Isham’s global, group theoretic quantizationscheme [43, 44]. Acknowledgements
We are grateful to M. Greiter for helpful correspondence. This research was sup-ported in part by the National Science Foundation under Grants PHY-1708139 atUMD and PHY-1748958 at KITP.
A No central extensions for the E algebra In this section we offer a simple derivation of the fact that the algebra E (3.20) ofthe Euclidean group E does not admit non-trivial central extensions by 2-cocycles.Although we believe that this result is well-known, we have not found a referencefor it.Let g be a Lie algebra and (cid:101) g = g ⊕ S R be a central extension. If ξ ∈ g and r ∈ R ,we generically define the product on (cid:101) g as (cid:2) ( ξ, r ) , ( ξ (cid:48) , r (cid:48) ) (cid:3) = (cid:0) [ ξ, ξ (cid:48) ] , θ ( ξ, ξ (cid:48) ) (cid:1) , (A.1)for some function θ : g → R . If this is to form a Lie algebra, θ must be linear,antisymmetric and, due to Jacobi identity, satisfy θ ( ξ, [ ξ (cid:48) , ξ (cid:48)(cid:48) ]) + θ ( ξ (cid:48) , [ ξ (cid:48)(cid:48) , ξ ]) + θ ( ξ (cid:48)(cid:48) , [ ξ, ξ (cid:48) ]) = 0 , (A.2)which is called the 2-cocycle condition. The extension is said to be trivial if θ ( ξ, ξ (cid:48) ) = f ([ ξ, ξ (cid:48) ]) for some linear f : g → R . (In this case, note that ξ (cid:55)→ ( ξ, f ( ξ )) is ahomomorphism from g to (cid:101) g .)Consider now the E algebra. Define θ ij = θ ( J i , J j ) θ αβ = θ ( N α , N β ) θ αi = θ ( N α , J i ) , (A.3)where Latin and Greek indices are used to distinguish these components (i.e., θ ij represents a different set of numbers than θ αβ ). These components can be rewritten28s θ ij = (cid:15) ijk w k θ αβ = (cid:15) αβγ h γ θ αi = (cid:15) αiβ w β , (A.4)for the nine numbers w i , h α and w α .For a trivial extension, we must have θ ij = θ ( J i , J j ) = f ([ J i , J j ]) = f ( (cid:15) ijk J k ) = (cid:15) ijk f k θ αβ = θ ( N α , N β ) = f ([ N α , N β ]) = 0 θ αi = θ ( N α , J i ) = f ([ N α , J i ]) = f ( (cid:15) αiβ N β ) = (cid:15) αiβ f β , (A.5)where f i = f ( J i ) and f α = f ( N α ). We conclude that the extension is trivial if andonly if h α = 0, for in this case we can always define f i = w i and f α = w α .Consider the cocycle condition for two N ’s and one J , θ ( N α , [ N β , J i ]) + θ ( N β , [ J i , N α ]) + θ ( J i , [ N α , N β ]) = 0 , (A.6)which gives (cid:15) βiγ θ αγ + (cid:15) iαγ θ βγ = 0. Or, in terms of h α , δ iα h β − δ iβ h α = 0. Contracting i and α , we get h β = 0, proving that the E algebra admits only trivial centralextensions. B Details on the construction of the Hilbert space
In this appendix we explain some of the details involved in the construction of theHilbert space of the theory, presented in section 4.3. In particular, we want to deriveequations (4.9)-(4.15). For simplicity, we use (cid:126) = 1 in this section.Note first that the angular momentum algebra,[ J , J ± ] = ± J ± , [ J + , J − ] = 2 J , (B.1) We do not know whether this very three-dimensional proof can be generalized to higher dimen-sions. J i on the basis states (4.7): J | j, m (cid:105) = m | j, m (cid:105) , (B.2) J − | j, m (cid:105) = | j, m − (cid:105) , (B.3) J + | j, m (cid:105) = [ j ( j + 1) − m ( m + 1)] | j, m + 1 (cid:105) . (B.4)Moreover, the norms of all the states | j, m (cid:105) are related to the norm of | j, j (cid:105) recur-sively, via (cid:104) j,m − | j, m − (cid:105) = 2 m (cid:104) j, m | j, m (cid:105) ++ [ j ( j + 1) − m ( m + 1)] (cid:104) j, m + 1 | j, m + 1 (cid:105) , (B.5)hence they all have positive norm provided the edge states | j, j (cid:105) do. Thus we needonly consider the action on and norms of the edge states.For the rest, we use the algebra relations[ J , N ± ] = ± N , [ J ± , N ] = ∓ N ± , [ J ± , N ∓ ] = ± N , (B.6)(and [ J ± , N ± ] = 0). Using the last of these, the Casimir N · J = s can be written as N · J = N ( J + 1) + J − N + + N − J + , (B.7)and applying this on | j, j (cid:105) yields N | j, j (cid:105) = sj + 1 | j, j (cid:105) − j + 1) | j + 1 , j (cid:105) . (B.8)To find N − | j, j (cid:105) for j > j , we can write N − | j, j (cid:105) = N − N + | j − , j − (cid:105) = (cid:0) − N (cid:1) | j − , j − (cid:105) , (B.9)in which we set N = 1. Using formula (B.8) we have N | j − , j − (cid:105) = N (cid:18) sj | j − , j − (cid:105) − j J − | j, j (cid:105) (cid:19) = sj N | j − , j − (cid:105) − j ( J − N − N − ) | j, j (cid:105) , (B.10)30hich together with (B.8) and (B.9) yields N − | j, j (cid:105) = 2 j j + 1 (cid:18) − s j (cid:19) | j − , j − (cid:105) ++ sj ( j + 1) | j, j − (cid:105) − | j + 1 , j − (cid:105) j + 1)( j + 1) , (B.11)provided that j > j .Next we compute the norms (cid:107) N + | j, j (cid:105)(cid:107) and (cid:107) L ( j ) | j, j (cid:105)(cid:107) , where L ( j ) is the j -lowering operator defined in (4.11), L ( j ) = N − − sj ( j + 1) J − + 12(2 j + 1)( j + 1) ( J − ) N + . (B.12)In what follows, we use ≈ to denote operator identities that are valid only within (cid:104) j, j | · · · | j, j (cid:105) . For the raised state we compute N − N + = 1 − N ≈ − (cid:18) sj + 1 − j + 1) N − J + (cid:19) (cid:18) sj + 1 − j + 1) J − N + (cid:19) ≈ − s ( j + 1) − j + 1) N − N + , (B.13)and, solving for N − N + , N − N + ≈ j + 1)2 j + 3 (cid:18) − s ( j + 1) (cid:19) , (B.14)which yields (4.13) for the squared norm. For the lowered state we have L ( j ) † L ( j ) ≈ j (2 j + 3)( j + 1)(2 j + 1) N − N + − s j ( j + 1) , (B.15)and using the result for N − N + we get L ( j ) † L ( j ) ≈ j j + 1 (cid:18) − s j (cid:19) , (B.16)which yields (4.14) for the squared norm.31 Wavefunctions and Chern numbers
In this appendix we review Mackey’s approach, and apply it to the constructionof the irreducible unitary representations of the Euclidean group E (and of itsuniversal cover), which recovers the usual concept of wavefunctions as sections ofcomplex vector bundles. We show in particular that the value of the Casimir N · J , which is related to the magnetic charge, determines the first Chern numberof the bundle via the Casimir matching requirement. Interestingly, in geometricquantization [6, 14], where quantum states are also given by sections of a line bundle,the same relation between the magnetic charge and the bundle topology arises,albeit in a different manner: by construction, the line bundle carries a connectionwhose curvature is required to coincide with the symplectic form (1.1). We willfollow closely Isham’s presentation of Mackey’s theory, using the language of fiberbundles; for details, see [1]. For Mackey’s original presentation, using the languageof measure theory, see for example [45].As motivation, let us first consider only the SO (3) part of the group. Givenits natural action on S , l R ( x ) =: Rx , we can construct a representation withwavefunctions ψ : S → C d defined by( U ( R ) ψ ) ( x ) = ψ ( R − x ) , (C.1)where R ∈ SO (3) and x ∈ S . This representation is unitary with respect tothe inner product (cid:104) ψ, φ (cid:105) = (cid:82) dµ ψ ∗ φ , where dµ is the Euclidean measure on thesphere, but it is not irreducible. Also, it is not exhaustive (i.e., not all unitaryrepresentations have this form).To generalize this, consider an Hermitian vector bundle over the sphere, C d → B → S . We want to take sections of this bundle, Ψ : S → B , as the vectors ofthe representation. Since the analogue of (C.1) would involve a mapping betweendistinct fibers, we must require that the bundle admits a lift of the group action, L R : B → B , satisfying τ ◦ L R = l R ◦ τ (compatibility with fiber structure) and L R ◦ L R (cid:48) = L RR (cid:48) (compatibility with group structure). In that case, we define( U ( R )Ψ) ( x ) = L R (Ψ( R − x )) . (C.2) For example, in the case d = 1, the space of complex functions on the sphere can be expandedin spherical harmonics, but since each l -subspace is invariant the representation is not irreducible. In the notation F (fiber) → E (total space) → M (base space), the second arrow represents thebundle projection map from the total space to the base space, while the first arrow merely indicatesthat each fiber of the bundle is an embedded copy of F . L R maps the point Ψ( R − x ), on the fiber over R − x , to a point on the fiber over x . The Hermitian structure of B , with innerproduct on each fiber denoted by ( , ), gives rise to an inner product on the space ofsections defined by (cid:104) Ψ , Φ (cid:105) = (cid:90) S dµ ( x ) (Ψ( x ) , Φ( x )) . (C.3)In order for the representation to be unitary with respect to this inner product, wemust require that the group lift is compatible with the hermitian structure of thebundle, i.e., ( L R z, L R z (cid:48) ) = ( z, z (cid:48) ), where z and z (cid:48) are points on the same fiber of B . Note that (C.1) is the special case where both the bundle and the group lift aretrivial.To find the most general wavefunction representation of SO (3) on S , we mustclassify all the bundles C d → B → S that admit such a lift of SO (3). It ispossible to show [2] that any such a bundle must be associated to the “masterbundle” SO (2) → SO (3) → SO (3) /SO (2) ∼ S , with projection map equal to thequotient q : SO (3) → SO (3) /SO (2), via some homomorphism U : SO (2) → U ( d ).More precisely, B is necessarily isomorphic to SO (3) × U C d , the bundle definedby the equivalence classes [ R, z ] = [
Rh, U ( h − ) z ], for R ∈ SO (3), z ∈ C d and h ∈ SO (2) ⊂ SO (3), with projection map q U ([ R, z ]) = q ( R ). Note that U is aunitary representation of SO (2), and we will later be interested in the irreducibleones. Since SO (2) is abelian, its irreducible representations are one-dimensional, soonly the case d = 1 is relevant. These representations are given by U ( n ) ( θ ) = e − inθ , (C.4)where n ∈ Z . The choice of this representation is the only discrete choice that entersthe construction of the general representation of E using Mackey theory, hence onecan anticipate that n must be related to the magnetic monopole index in (4.17).We are now ready to consider the full group, R ∗ (cid:111) SO (3). For more transparency,let us consider first a generic group of the form V (cid:111) K , where V is a vector spaceand K is a Lie group. The product rule is given by( v, k )( v (cid:48) , k (cid:48) ) = ( v + ρ k v (cid:48) , kk (cid:48) ) , (C.5)where v ∈ V , k ∈ K and ρ : K → Aut( V ) is a left K -action on V . Later we shall Mackey’s theory also applies, with a minor modification, if V is abelian and K is a separable,locally compact group. When V is not a vector space, we just need to replace below the dual space V ∗ by the space of unitary characters Char( V ). K = SO (3) and V = R ∗ , in the dual representation ρ R = l ∗ R − (see(3.10)). Since a generic element ( v, k ) can be decomposed as ( v, e )(0 , k ), where e isthe identity element of K , the operators representing V (cid:111) K on a Hilbert space H will factorize accordingly, U ( v, k ) = U ( v, e ) U (0 , k ). We can define A ( v ) := U ( v, e )and D ( k ) := U (0 , k ), so that U ( v, k ) = A ( v ) D ( k ) . (C.6)Thus, in classifying the representations of V (cid:111) K , we can study the representationsof V and K separately, in the following manner.Starting with V , define the self-adjoint generators N ( v ) by A ( λv ) = e − iλN ( v ) , (C.7)where λ ∈ R . Since V is abelian, we have [ N ( v ) , N ( v (cid:48) )] = 0, and N ( v + λv (cid:48) ) = N ( v ) + λN ( v (cid:48) ), meaning that N is a linear map from V into a set of commuting,self-adjoint operators on H . Accordingly, a simultaneous eigenvector | χ (cid:105) of N ( v ),for all v , determines an element w ∈ V ∗ , such that N ( v ) | χ (cid:105) = w ( v ) | χ (cid:105) . (C.8)Nothing requires the eigenvalues of N ( v ) to be non-degenerate, so each w may labela Hilbert (sub)space S w ⊂ H . It follows from the group structure that the operator D ( k ) maps S w unitarily onto S (cid:101) ρ k w , where (cid:101) ρ k is the dual action of K on V ∗ , definedas (cid:101) ρ k w ( v ) = w ( ρ k − v ) for all v ∈ V . The Hilbert space H will be given by a “directsum” (or rather, “direct integral”) of S w over w ’s in some region of V ∗ . If D ( k ) isto act in a closed fashion, such a region must consist of one or more orbits of K . Toensure irreducibility of the V (cid:111) K representation, we must take this region to be a single K -orbit O (or its closure) in V ∗ . Roughly speaking, H ∼ “ ⊕ w ∈O S w ” . (C.9)More precisely, H will be the space of sections of a vector bundle over O , with fibers S w ∼ C d (for some dimension d ). To classify these representations we must thereforeclassify the corresponding vector bundles.For such a vector bundle C d → B → O over O ∼
K/H (where H is thelittle group corresponding to the orbit O ) to carry a representation of K , it mustadmit a lift of the K -action, and thus must be associated to the master bundle34 → K → K/H via a unitary irreducible representation U : H → U ( d ) of H (see earlier discussion about the case K = SO (3)). Cross sections of this bundle,Ψ : O → B = K × U C d , form a linear space which carries a representation of V (cid:111) K .The element ( v, k ) is represented by( U ( v, k )Ψ)( w ) = e − iw ( v ) (cid:115) dµ k dµ ( w ) L k (Ψ( (cid:101) ρ k − w )) , (C.10)where w ∈ O and L k is the lift of K to K × U C d defined by L k [ k (cid:48) , z ] = [ kk (cid:48) , z ].The phase factor on the right-hand side is A ( v ), and the rest is the factor D ( k ), asin (C.6). Note that D ( k ) is analogous to (C.2), except for the Jacobian-like factor dµ k /dµ , which deserves a few words. In the case of SO (3), it is possible to definethe inner product (C.3) using the Euclidean measure on the spherical orbit, andthe invariance of this measure under SO (3) implies that U ( R ) is unitary, so thisJacobian-like factor is not needed. In general, however, the orbit O may not admitan invariant measure, but fortunately it always admits a measure µ that is quasi-invariant under K . That is, µ and its push-forward µ k through (cid:101) ρ k ∗ have the samesets of measure zero. In order to make U ( v, k ) unitary under such a measure, wemust introduce the Jacobian-like factor dµ k /dµ , called the Radon-Nikodym deriva-tive of µ k with respect to µ , which is a positive ( µ -almost-everywhere) continuousfunction on O satisfying µ k [ B ] = (cid:82) B dµ k dµ dµ for all Borel sets B ⊂ O . Representa-tions defined for equivalent measures (i.e., having the same sets of measure zero) areunitarily equivalent, and since there is only one quasi-invariant measure on K/H (up to equivalence), the measure in (C.10) is determined by O (up to equivalence).Note that these representations are labeled by the choice of the orbit O and thelittle group representation U . These representations are irreducible as long as U is irreducible. If V ∗ decomposes into regular K -orbits, meaning that there existsa Borel map ζ : V ∗ /K → V ∗ that associates a dual vector to each orbit, then allunitary irreducible representations are generated in this way. This is Mackey’s mainresult.In the case of interest, R ∗ (cid:111) SO (3), the space where the orbits live is ( R ∗ ) ∗ ,which can be naturally identified with R . Thus (cid:101) ρ R , which acts on w ∈ R ∗∗ as w (cid:55)→ ρ ∗ R − w = l ∗∗ R w , acts on x ∈ R as x (cid:55)→ l R x = Rx . That is, SO (3) acts on( R ∗ ) ∗ ∼ R in just the standard way. The orbits O decompose into two classes: In many cases, such as when K is locally compact and H is compact, the Haar measure on K can be pushed down to K/H , defining an invariant measure on O . Given a measurable map f : X → Y and a measure µ on X , its push-forward to Y is definedas f ∗ µ [ B ] = µ [ f − ( B )], where B is any Borel subset of Y and f − denotes the pre-image under f . SO (2), while for the second it is SO (3). Since the orbits are regular, theirreducible unitary representations are labeled by the radius a ∈ R + ∪ { } of theorbit and, for a > n ∈ Z specifyingthe irreducible unitary representation U ( n ) ( θ ) = e − inθ of the little group SO (2).For a given value of n , the Hilbert space consists of sections of the line bundle SO (3) × U ( n ) C .How are the basic operators, N and J , realized on this Hilbert space? Since theEuclidean measure on S is invariant under SO (3), we do not have the Jacobianfactor in (C.10), which simplifies to( U ( α, R )Ψ)( x ) = e − iα ( x ) / (cid:126) L R Ψ( R − x ) , (C.11)Note that we have introduced an (cid:126) in the phase factor, for notational convenience.In analogy with (4.1), we define more generally the generating operators J η and J α via U (exp(0 , λη )) =: e − iλJ η / (cid:126) U (exp( λα, e − iλN α / (cid:126) , (C.12)where η ∈ so (3) ∼ R and α ∈ R ∗ . It follows that J η = i (cid:126) ddλ U (exp(0 , λη )) (cid:12)(cid:12)(cid:12)(cid:12) λ =0 N α = i (cid:126) ddλ U (exp( λα, (cid:12)(cid:12)(cid:12)(cid:12) λ =0 . (C.13)Hence, ( J η Ψ)( x ) = − i (cid:126) D η Ψ( x )( N α Ψ)( x ) = α ( x )Ψ( x ) , (C.14)where D is a derivative operator defined by D η Ψ( x ) = − ddλ L R λη Ψ (cid:16) R − λη x (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) λ =0 . (C.15)It satisfies D η ( f Ψ) = f D η Ψ + X η ( f )Ψ, where f : O → C and X η is the vector field(tangent to O ∼ S ) generated by η . 36e now wish to relate the Casimirs N and N · J with the labels a and n of thewavefunction representations. In an orthonormal basis e i for R , and dual basis e i for R ∗ , we have N = (cid:80) i =1 ( N e i ) , and N e i Ψ( x ) = e i ( x )Ψ( x ) = x i Ψ( x ), so N Ψ( x ) = x Ψ( x ) = a Ψ( x ) . (C.16)Thus, not surprisingly, N corresponds to the radius squared, a , of the sphere.The representation scales trivially with N , and the choice we made previously was N = 1, so we here consider also the case a = 1.Next we consider N · J , which as we now show is related with the index n ofthe bundle SO (3) × U ( n ) C . We establish this in two independent ways. First, weshow how N · J is related to the little group representation (C.4), which is labeledby n . Second, we show how N · J can be directly related to the integrated curvatureof a suitably constructed connection, which directly yields the Chern number ofthe bundle, and thus the index n . In this second way, we will not need to invokeSchur’s lemma, but rather it will be a consequence of the construction that N · J isproportional to the identity, similarly to what happened with N above.First, we note that since N · J is a Casimir in the algebra, it must according toSchur’s lemma be proportional to the identity, so it suffices to evaluate its actionon a single state Ψ( x ) at any given point x . Let us take x to be the north pole u = (0 , , SO (3) × U ( n ) C using the SO (2) little groupof u . Then we have N · J Ψ( u ) = J Ψ( u ) and, since u is a fixed point of R λe , thevalue of J Ψ( u ) depends only on the value of Ψ at u (as opposed to in a neighborhoodof u ). Denoting Ψ( u ) = [1 , z ] ∈ SO (3) × U ( n ) C , where 1 is the identity element of SO (3) and z ∈ C , we have J Ψ( u ) = − i (cid:126) D e Ψ( u )= i (cid:126) ddλ L R λe Ψ( u )= i (cid:126) ddλ [ R λe , z ]= i (cid:126) ddλ [1 , U ( R λe ) z ]= i (cid:126) ddλ e − inλ [1 , z ]= (cid:126) n Ψ( u ) , (C.17)where d/dλ is evaluated at λ = 0 at every step. In the second line we used that37 − λe u = u ; in the third line we used the definition of the group lift to the associatedbundle, L R [ R (cid:48) , z ] = [ RR (cid:48) , z ]; in the fourth line we used the defining property of theassociated bundle SO (3) × U ( n ) C , [ R, z ] = [
Rh, U ( h − ) z ], with h = R − λe ; and in thefifth line we used (C.4). Therefore J · N Ψ( x ) = (cid:126) n Ψ( x ), so we conclude that thevalue of the Casimir invariant J · N is determined by the little group representation U ( n ) . Given the identification J · N = − eg , we obtain the Schwinger condition eg = − n (cid:126) , which is more restrictive than Dirac’s condition (4.17).In order to include also projective representations of the quantizing group, wemust extend SO (3) to SU (2) and repeat the same analysis. The relevant masterprincipal bundle is then the Hopf bundle U (1) → SU (2) → SU (2) /U (1) ∼ S . Be-cause the little group is U (1), the associated bundles are again constructed withthe representations U ( n ) , so the quantum states are represented by sections of SU (2) × U ( n ) C . The group SU (2) acts on the sphere as e iv · σ ( x · σ ) e − iv · σ = ( R v x ) · σ , (C.18)where v ∈ R , x ∈ S ⊂ R and σ = ( σ , σ , σ ) are the Pauli matrices. To obtain theusual normalization for the su (2) algebra (i.e., with structure constants f ijk = ε ijk )we must take the basis { σ / , σ / , σ / } , and this means that we have J = J η = σ / .Repeating the steps of the previous derivation, the only difference is that the littlegroup phase factor is U ( e i ( σ / λ ) = e − inλ/ , which leads to J Ψ( x ) = (cid:126) n/
2. Thatis, s = n/
2, matching Dirac’s condition.In the second way of evaluating N · J , we note that the Chern number is atopological property of the bundle, as it does not depend on the connection used toevaluate it, so we shall just use the structures available to construct some arbitraryconnection. The natural ingredient to use is the derivative operator D , which isdefined in (C.15) in terms of the group lift L R . A possible definition for a covariantderivative is ∇ V Ψ( x ) = D x × V Ψ( x ) , (C.19)where V ∈ T x S ⊂ T x R . One can check that this ∇ satisfies all properties of acovariant derivative, so it defines a connection on the bundle. Using (C.14) we canwrite it as ∇ V Ψ = − i (cid:126) V · ( N × J )Ψ . (C.20) From (3.21) we see that − i (cid:126) ∇ V is nothing more than the “natural” quantization of the classicallinear momentum along V , since V · p = − V · ( N × J ), recovering the picture that p acts as a derivativeon wave functions. F of this connection is defined by( ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X,Y ] )Ψ = F ( X, Y )Ψ , (C.21)for vector fields X and Y on S . Treating X and Y as vectors on R , tangent tothe sphere, and using (C.20), we obtain F ( X, Y ) = − (cid:126) X i Y j [( N × J ) i , ( N × J ) j ] . (C.22)Using the algebra (4.2) of N and J we get F ( X, Y ) = 1 i (cid:126) N ( X × Y ) · J . (C.23)Since X × Y is normal to the sphere, it acts on wavefunctions (in the representationwith N = 1) as (cid:15) ( X, Y ) N , where (cid:15) = sin θdθ ∧ dφ is the area form on the unitsphere. Thus the curvature 2-form can be expressed as F = 1 i (cid:126) (cid:15) N · J . (C.24)We see that N · J must act on sections as a function, i.e., N · J Ψ( x ) = (cid:126) s ( x )Ψ( x ),for some s : S → R . Because the connection was constructed in a rotation invariantway, the curvature must also be rotation invariant. This implies that s is actuallya constant, which is consistent with the fact that N · J is a Casimir operator in anirreducible representation. The corresponding first Chern number is then C = (cid:90) S i π F = s π (cid:90) S (cid:15) = 2 s , (C.25)which shows that the Casimir N · J is directly related to this topological number, asanticipated. The possible values of s are quantized, since the bundle SO (3) × U ( n ) C has first Chern number 2 n , while the extended bundle SU (2) × U ( n ) C has first Chernnumber n . Note that this is consistent with our previous result, where s = N · J/ (cid:126) was shown to be related to the bundle index n . D Non-uniform magnetic fields
In this appendix we discuss the case where the magnetic field is not uniform withrespect to the metric on the sphere. In principle, we could proceed as before,including the magnetic field in the symplectic form, so maintaining gauge invariance39xplicitly. However, unless B is uniform, no SO (3) subgroup of the B -preservingdiffeomorphisms of the sphere would be an isometry of the metric. Consequently,there would be no preferred form for the Hamiltonian, leading to operator-orderingambiguities in the quantization. Thus, it convenient to split B into its monopole andhigher-pole parts, B = g(cid:15) + dA , where (cid:15) is the normalized volume form (i.e., whoseintegral is 4 π ) invariant under the isometries of the metric and A is a globally-definedpotential 1-form, and to include only the monopole term, g(cid:15) , in the symplectic form,as in (1.1), while including the higher-pole part, dA , in the Hamiltonian via the usualminimal coupling. In this way, the quantizing group and the associated canonicalobservables, J and N , depend only on the monopole term, while the higher-poleterm affects only the Hamiltonian.In a given global gauge the Hamiltonian reads H = 12 m ( p − eA ) , (D.1)in which ( p − eA ) = h ab ( p − eA ) a ( p − eA ) b , where h is the metric on the sphere.Assuming a round metric, and using expression (4.21) in terms of the Killing vectorfields X i (generated by J i ), the Hamiltonian becomes H = 12 mr [ J i − egN i − eA ( X i )] , (D.2)where a summation over i ∈ { , , } is implicit. Since A ( X i ) is a function of theposition only, it can be written in terms of N unambiguously. In the wavefunctionrealization of Appendix C, it acts simply by multiplication.Now we must worry about gauge-invariance. In particular, we must ensure thatthe same theory is obtained if one uses another choice of (global) gauge A (cid:48) . Sincethe sphere is simply connected, we have A (cid:48) = A + dσ , (D.3)for some function σ : S → R . Note that H (cid:48) = 12 mr ( p − eA (cid:48) ) = 12 mr ( p − edσ − eA ) , (D.4)which has the same form as the original Hamiltonian if we define a new momentumvariable p (cid:48) = p − d ( eσ ). This corresponds to a momentum translation, defined as K σ ( p ) = p − d ( eσ ) , (D.5)40hich is a symplectomorphism of the phase space satisfying H (cid:48) = K ∗ σ H . Note thatit maps the vector flow of H (cid:48) into that of H and, being vertical on the phase space,it leaves unchanged the projection of the dynamical trajectories to the configurationspace. Therefore the two gauge-related Hamiltonians produce equivalent classicaldynamics. As to the quantization, note first that H (cid:48) = K ∗ σ H implies that H (cid:48) hasthe same functional form when written in terms of transformed charges Q (cid:48) = K ∗ σ Q as H written in terms of Q . That is, if H = f ( Q ) then H (cid:48) = f ( Q (cid:48) ). Since thePoisson brackets is defined from the symplectic structure, which is invariant under K σ , we have that the algebra of charges is preserved under such a transformation,i.e., { K ∗ σ Q i , K ∗ σ Q j } = K ∗ σ { Q i , Q j } . Thus the charges Q (cid:48) satisfy the same algebraas Q . Moreover the Casimirs are functionals of the charges, with form dependingonly on the algebra, so it should be that C (cid:48) = K ∗ C . But since C is constant onthe phase space, we have C (cid:48) = C . Consequently, as the charges satisfy the samealgebra, with the same Casimir values, the quantizations are equivalent (providedthe same ordering prescription is applied to the Hamiltonian).At the group level, the charges Q and Q (cid:48) generate the same group G , but realizeddifferently on the phase space. Namely, if the charges Q generate a realizationΛ : G → Diff( P ) of G as symplectomorphisms of P , then it can be shown that Q (cid:48) generate the transformed realization, Λ (cid:48) , defined byΛ (cid:48) g = K − σ ◦ Λ g ◦ K σ . (D.6)Therefore, we see that a change of gauge in the Hamiltonian can be “reversed”by simply changing the way that the quantizing group acts on the phase space.Since it is the same group (with the same Casimir values) which is undergoingquantization, the same quantum theory is obtained. At the quantum level, thischange of realization corresponds to a unitary transformation on the Hilbert space.Mirroring (D.6), the K -transformation is implemented as U (Λ (cid:48) g ) = T ( K σ ) † U (Λ g ) T ( K σ ) , (D.7)where T ( K σ ) is a unitary transformation, ensuring that the representation U (Λ (cid:48) g ) isequivalent to U (Λ g ). 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