Monopole-based quantization: a programme
J.F. Carinena, J.M. Gracia-Bondia, Fedele Lizzi, Giuseppe Marmo, Patrizia Vitale
aa r X i v : . [ m a t h - ph ] D ec DSF/18/2009ICCUB-09-426
Monopole-based quantization: a programme
Jos´e F. Cari˜nena a , J.M. Gracia-Bond´ıa a , Fedele Lizzi b,c , Giuseppe Marmo b and Patrizia Vitale ba Departamento de F´ısica Te´orica, Facultad de Ciencias,Universidad de Zaragoza, 50009 Zaragoza, Spain [email protected],[email protected] b Dipartimento di Scienze Fisiche, Universit`a di Napoli
Federico II and INFN, Sezione di Napoli, Monte S. AngeloVia Cintia, 80126 Napoli, Italy [email protected], [email protected],[email protected] c High Energy Physics Group, Dept. Estructura i Constituents de la Mat`eria,Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Catalonia, Spainand Institut de Ci`encies del Cosmos, UB, Barcelona
Abstract
We describe a programme to quantize a particle in the field of a (three dimen-sional) magnetic monopole using a Weyl system. We propose using the mapping ofposition and momenta as operators on a quaternionic Hilbert module following thework of Emch and Jadczyk.
Contribution to the volume:
Mathematical Physics and Field Theory, Julio Abad, InMemoriam
M. Asorey, J.V. Garc´ıa Esteve, M.F. Ra˜nada and J. Sesma Editors, Prensas Universitaria deZaragoza, (2009)
Introduction: several birds with a stone
Quantum kinematics in the field of a magnetic monopole allows for angular momentum-isospin coupling, in particular spin-isospin and orbit-isospin couplings. However, toour knowledge the rich conceptual and mathematical structures associated to angularmomentum-isospin coupling have gone unnoticed so far in deformation quantization the-ory.The aim of this contribution is to set out the basis for a rigorous investigation ofthose structures in canonical quantization, up to defining the pertinent monopole-basedWeyl systems and star products. We note that angular momentum-isospin coupling is afeature of high physical interest; for a current example of practical application, we referto graphene edge model [1]. Rather than presenting novel results we will outline, withthe detail permitted by the total length limit, a general framework in which it will bepossible to use the monopole for a variety of investigations both from the physical andmathematical point of view.A classical particle in the field of a magnetic monopole of unitary charge is described,in proper units, by the Poisson structure: { p i , x j } = δ ji { x i , x j } = 0 { p i , p j } = − ǫ ijk x k k x k (1.1)This Poisson structure is position dependent and therefore its quantization is not triv-ial, but extremely rich! The right mathematical framework for the quantization of thisstructure is the formulation of quantum mechanics on a quaternionic Hilbert space givenby Emch and Jadcyzk over 10 years ago [2]. Since this does not seem to be commonknowledge, at some point we summarize the findings of that paper, insofar they are usefulfor our purposes.Prior to doing the above, we exhibit another problem of principle, in some sensedual to quantization in the field of a monopole, whose resolution hangs from the samemathematical thread. Thus the article is organized as follows. An ab initio calculation,using the Kirillov coadjoint picture [3] allows to regard the photon as a classical elementaryphysical system for the inhomogeneous Lorentz group P . On the arena of phase space,this turns out to be formally dual (exchanging position and momenta) to the orbit-isospincoupling system. Section 3 deals with the structure of the gCCR (generalized canonicalcommutation relations) on the quaternionic Hilbert space. In Section 4, we show howthe Emch–Jadcyzk (EJ) calculus provides us the necessary tools for quantization of theabove indicated systems. Quantization and dequantization proper are sketched in thenext section. In Section 6 we give the conclusion and outlook for construction of the starproduct describing the scalar particle-monopole system.1able 1: The coadjoint action Coad (exp X ) yX − a H a · P α m · J ζ n · K h h h h (cosh ζ ) h − (sinh ζ ) n · pp p p R α m p p − (sinh ζ ) h n + (cosh ζ − n · p ) nj j j + a × p R α m j (cosh ζ ) j + (sinh ζ ) n × k − (cosh ζ − n · j ) nk k − a p k + h a R α m k (cosh ζ ) k − (sinh ζ ) n × j − (cosh ζ − n · k ) n The unique Poincar´e-invariant Stratonovich–Weyl (de)quantizer and Moyal product onthe phase space T ∗ R × S ≃ R × S for massive relativistic particles with spin (degener-ating to T ∗ R for spinless particles) was constructed in [4] with help of results in [5]. Itspractical interest is nil since (contrary to the Galilean case) it breaks down as soon as oneintroduces interaction. However, this construction was an important matter of principle:the formalism based on this Moyal product is equivalent to relativistic quantum mechanics(and of course participates in its flaws). In particular, it gave geometric quantization onphase space and relativistic Wigner functions, establishing the bridge between the Kirillovcoadjoint orbit picture and the Wigner theory of unitary irreps for the Poincar´e group P .For massless particles, although we knew the arrival point as well (see [6] for a modern,streamlined treatment), we were stumped. The time has come to revisit the question.We describe coadjoint for the splitting group e P of the Poincar´e group; this is just theuniversal covering T ⋉ SL (2 , C ), without nontrivial extensions [7]. The Lie algebra of e P is generated by ten elements H, P i , J i , K i (for i = 1 , ,
3) corresponding respectively totime translations, space translations, rotations and pure boosts. Write elements of e P inthe standard form g = exp( − a H + a · P ) exp( ζ n · K ) exp( α m · J ) , where a = ( a , a ) ∈ T , n and m are unit 3-vectors, ζ ≥ ≤ α ≤ π , with theunderstanding that exp(2 π m · J ) = − in SL (2 , C ) for all m . The coadjoint action of e P on p ∗ can be derived from the well-known commutation relations for the generators. Let h be the linear coordinate on p ∗ associated to H ∈ p , and similarly let p i , j i , k i be thecoordinates associated to P i , J i , K i ( i = 1 , , Casimir functions C , C on p ∗ ,which are easy to obtain explicitly (or to guess from other treatments). Let p = ( h, p ) bethe energy-momentum w = ( w , w ) the Pauli–Luba´nski w = j · p , w = p × k + h j . From Table 1 one verifies that w transforms like h and w like p under the coadjoint2ction; in particular, under Coad (cid:0) exp( ζ n · K ) (cid:1) : w (cosh ζ ) w − (sinh ζ ) n · w , w w − (sinh ζ ) w n + (cosh ζ − n · w ) n . Thus the Casimir functions we seek are C := ( pp ) = − h + p · p , C := ( ww ) = − ( j · p ) + k p × k + h j k . Notice that p and w are orthogonal in the Minkowski sense: ( pw ) = 0. Let us focuson the shape of light-like coadjoint orbits, for which C = 0. For physical reasons (no‘continuous-spin’ representations) we take the momentous decision of stipulating that w is parallel to p . Clearly p ∈ R \ { } (the origin is an orbit). We can postulate q := k /h ,which is well defined, and takes all values in R , and everything is determined: h = k p k , p = p , j = λ p k p k + q × p , k = k p k q , where λ p / k p k plays the role of the spin, with the helicity λ = j · p / k p k being the projectionof the total angular momentum j on the momentum. Therefore the orbit is 6-dimensional,and isomorphic to R × ( R \{ } ) ≃ R × R + × S . This non-trivial topology has non-trivialconsequences.By the general theory, the Poisson bracket is given by { f, g } = c kij ∂f∂x i ∂g∂x j x k . The commutation relations for the generators yield: { p i , q j } = { p i , h − k j } = h − { p i , k j } = − δ ij . (2.1)On the other hand, { q i , q j } = { h − k i , h − k j } = h − { k i , k j } + h − k j { k i , h − } + h − k i { h − , k j } = h − ( − ǫ ijk j k − q j p i + q i p j ) = − λǫ ijk p k k p k ; (2.2)which is dual to the Poisson structure (1.1) upon the exchange p ↔ q .The coordinates in (2.2) are not canonical coordinates (Darboux coordinates do notexist globally, but d q d p is a global Liouville measure). All this agrees nicely with theanalysis in [8]. We can recover from Table 1 the expression of the coadjoint action of e P on the orbit in terms of the coordinates ( p , q ). There is no need to rewrite the actionon p . Also, we readily obtain: exp( − a H ) ⊲ q = q − a p k p k exp( a · P ) ⊲ q = q + a exp( α m · J ) ⊲ q = R α m q . Pace the founding fathers:in the old paper [9] Wightman writes that no such a realization can exist; but he assumescommuting coordinates.) The symplectic form corresponding to (2.1) and (2.2) is givenby: ω = dq i ∧ dp i − λǫ ijk p k dp i ∧ dp j k p k . This is exactly the one of the magnetic monopole, with the roles of q and p interchanged:see further below. That is to say, the ‘photon’ and monopole phase spaces are dualsystems.The stability subgroup G giving rise to our coadjoint orbit O is a torus extension ofthe standard little group for massless particles E , so e P /H ≃ R × R + × S . However e P /E ≃ R × R + × S , and this S sits over S like in the Hopf fibration. Weyl systems on a complex Hilbert space H are usually presented starting with a (real)symplectic vector space, say ( V, ω ), and a strongly continuous map V → U ( H ) to theunitary group on it, required to satisfy W ( v ) W ( v ) W † ( v ) W † ( v ) = e iω ( v ,v ) I . (3.1)Strong continuity, by means of the Stone-von Neumann theorem, implies that there existsa selfadjoint operator R ( v ) such that W ( v ) = e iR ( v ) . (3.2)From the commutator relation we have R (cid:0) [ v , v ] (cid:1) − [ R ( v ) , R ( v )] = iω ( v , v ) I . (3.3)Another theorem by von Neumann says that Weyl systems exist for any finite dimensionalsymplectic vector space. They can be defined on a linear space of square integrablefunctions on any Lagrangian subspace of ω in V . If we denote by L such a Lagrangiansubspace, we may write ( V, ω ) ⇋ ( L ⊕ L ∗ , ω ) , that is to say ( V, ω ) is symplectically isomorphic with ( T ∗ L, dθ ≡ ω ), where θ is theLiouville 1-form on T ∗ L . By denoting v ∈ V as ( y, α ) ∈ T ∗ L , we have a Weyl systemrealized by [ W (0 , α ) ψ ]( x ) = e i h α,x i ψ ( x );[ W ( y, ψ ]( x ) = ψ ( x + y ) . (3.4)4n defining either W ( y, α ) = W ( y, W (0 , α ) or as W ( y, α ) = W (0 , α ) W ( y, , (3.5)we find an ordering phase factor ambiguity. At the infinitesimal level we have a realizationin terms of differential operators iR ( y,
0) = ∂∂x ; iR (0 , α ) = ˆ x. (3.6)The symplectic structure evaluated on vectors ( y,
0) and (0 , α ) amounts to the commutatorof the differential operators ∂/∂x and ˆ x . In general, even though differential operatorsare unbounded, one prefers to see the algebra of operators acting on H realized as thealgebra of differential operators acting on functions defined on L . In this framework itis more convenient to deal with square integrable functions considered as sections of anassociated U (1)-bundle P over L . It is possible to write a function f on L as a functionon P by setting ˜ f ( x, t ) = f ( x ) e it . With this choice functions on L are associated withfunctions on P satisfying the equation − i ∂∂t ˜ f = ˜ f ; (3.7)then our algebra of differential operators may be realized in terms of vector fields, towit ∂/∂x, − ix∂/∂t, − i ∂/∂t . These vector fields close the Lie algebra of the Heisenberg–Weyl group, with − i ∂/∂t generating the linear space of central elements. While sectionsof a line bundle are appropriate to describe spinless particles, to describe particles withan inner structure it is necessary to consider sections of some Hermitian complex vectorbundle. Usually we consider f : L → C s +1 as functions associated with a trivializationof the U (2 s + 1) Hermitian bundle P over L . In this setting the generators of our Weylsystems will be matrix valued differential operators.The approach to quaternionic Quantum Mechanics undertaken more than forty yearsago, may be considered from this perspective. Let H denote the field of quaternionicnumbers H = ( q = X µ =0 q µ e µ | q µ ∈ R ) , with their ordinary multiplication and involution. We will use the notations 1 = e and e = ( e , e , e ), so that x · e = P i =1 x i e i for x ∈ R . Note that q ∗ q = k q k H definesthe quaternion norm, and that ( x · e ) ∗ ( x · e ) = k x k . Then H ≡ L ( R , d x ; H ) is aHilbert space of quaternion-valued functions. We consider wave functions realized on aHilbert space (module) of quaternionic valued functions. By using the representation ofquaternions by means of 2 × e = σ , e i = − iσ i , and F ( x ) = f ( x ) e + f i ( x ) e i . (3.8)The group SU (2) acts on these fibres by conjugation and the vector bundle may beconsidered as an associated bundle with structure group SU (2). If we identify L ≡ R ,5e may repeat our construction and lift vector fields from R to the total space of ourvector bundle. To this aim we have to introduce a connection, that is, a procedure to liftvector fields to horizontal vector fields. We use the gauge potential A = k [ e · x , e · d x ] k x k . (3.9)The origin of this choice may be traced back to the Hopf fibration π : SU (2) −→ S : ifwe consider s ∈ SU (2) we may define [11]: π ( s ) = s − σ s = x · σ (3.10)and σ · d x = − s − dss − σ s + s − σ ds = − s − ds ( x · σ ) + ( x · σ ) s − ds = [ s − ds, x · σ ] . Moreover, since S × R + = R − { } , we may define a lifting which would consider wavefunctions as fields transforming covariantly under the rotation group, whose action in theinner space is by means of SU (2). Given any u ∈ S , the translation u · ∂/∂ x is lifted to ∇ u = e u · ∂∂ x + 12 [ e · x , e · u ] k x k , (3.11)considered as quaternionic-valued differential operator. Clearly ∇ u ∇ u − ∇ u ∇ u = Ω( u , u ) (3.12)because h u · ∂∂ x , u · ∂∂ x i = 0 . (3.13)The curvature Ω is quaternion-valued and we may define the three presymplectic forms e Ω = ω , e Ω = ω , e Ω = ω , (3.14)giving us the gCCR. The EJ model is an appropriate quantum framework for orbit-isospin coupling. In orderto endow H with a complex linear structure we introduce: for every x = 0, let j ( x ) bethe imaginary unit quaternion j ( x ) = e · x k x k . (4.1)Then the linear operator J given by ( J ψ )( x ) = j ( x ) ψ ( x ) satisfies the relations J ∗ J = I = J J ∗ and J ∗ = − J , that is, it is unitary and skew–Hermitian; clearly, we also have J = − I .Remark that the choice (4.1) defines a J invariant under rotations that commutes with theposition operators; this is an easy calculation using L i = ǫ ijk x j ∂ k − e i for the generatorof rotations. 6 .1 Noncommutative translations On H one usually defines the translation by a by the operator V ( a ) such that[ V ( a ) ψ ]( x ) = ψ ( x − a ) , but taking into account the character of rays rather than vectors of states one can alsoadmit a phase factor. Here such translation is realized by the operator U ( a ) defined by:[ U ( a ) ψ ]( x ) = w ( a ; x − a ) ψ ( x − a ) , (4.2)with a ∈ R . Here, for every a , w ( a ; x ) is the quaternion w ( a ; x ) = cos( α/
2) + j ( x × a ) sin( α/
2) = exp [ j ( x × a ) α/ , (4.3)with α being the angle between x and x + a . If we use w to define the linear operator W ( a ) by [ W ( a ) ψ ]( x ) = w ( a ; x ) ψ ( x ) , a . e . (4.4)and then U ( a ) = V ( a ) W ( a ) . (4.5)Some properties of w which will be useful below are: • w (0; x ) = 1. • w ( a ; x ) w ∗ ( a ; x ) = 1. • w ( a ; x − a ) = w ∗ ( − a ; x ). • w ( t a ; x + s a ) w ( s a ; x ) = w (( s + t ) a ; x ).Let now u ∈ S . We can define generators for the continuous unitary representation U ( s u ) by ∇ u = lim t ↓ " ddt U ( t u ) ψ ( x ) = lim t ↓ " ddt w ( t u ; x − t u ) ψ ( x − t u ) . (4.6)One obtains ∇ u = (cid:18) u · ∂ + 12 e · u × x || x || (cid:19) , whereupon we recognize the sum of the (non-commuting) infinitesimal generators of V and W , respectively. Thus, with the obvious meaning for the X i , we readily compute thefollowing commutation relations[ ∇ i , X j ] = δ ij , [ ∇ i , ∇ j ] = − J ǫ ijk x k k x k , [ X i , X j ] = 0 , U ( a ) actually definea locally operating projective representation of the translation group [10], i.e. U ( a ) U ( b ) = U ( a + b ) M ( a , b ) . Here M ( a , b ) is a phase factor multiplication[ M ( a , b ) ψ ]( x ) = m ( a , b , x ) ψ ( x )with m ( a , b , x ) being given by m ( a , b , x ) = exp (cid:0) J S ( a , b , x ) (cid:1) , where S denotes the (product of the monopole strength) by the area of the trianglespanned by x , x + a , x + a + b . This guarantees associativity: U ( a ) (cid:2) U ( b ) U ( c ) (cid:3) =[ U ( a ) U ( b )] U ( c ) (see next section).Except for the presence of the quaternionic complex structure J , this is similar to theMoyal product, which bodes well for the quantization/dequantization procedure. In the usual case Weyl systems are represented as exponential as in (3.2). It is useful toexpress also the quaternionic Weyl system as an exponential. A first problem is that inthe quaternionic context there is no single imaginary unit. Nevertheless EJ have providedthe solution of the problem in the function j ( x ) defined in (4.1), and its operatorialcounterpart J . We can therefore define the operator P i = J ∇ i = − J (cid:18) ∂ i + 12 ǫ ijk x j e k k x k (cid:19) (5.1)it is possible to prove that J commutes with ∇ i and therefore [ P i , P j ] = (1 / ǫ ijk ( x k / k x k ).Therefore P i is the generator of translations in the quaternionic Hilbert space. Notice thatthe two summands in P i do not commute.Finally, let us consider the product of two finite translations( U ( a ) U ( b ) ψ )( x ) = ( U ( a ) ψ ′ )( x ) = w ( a ; x − a ) ψ ′ ( x − a ) = w ( a ; x − a ) w ( b ; x − a − b ) ψ ( x − a − b ) . On the other hand,( U ( a + b ) M ( a , b ) ψ )( x ) = w ( a + b ; x − a − b )( M ( a , b ) ψ )( x − a − b ) , (5.2)with M ( a , b ) defined as( M ( a , b ) ψ )( x ) = m ( a , b ; x ) ψ ( x ) = w ∗ ( a + b , x ) w ( a ; x + b ) w ( b ; x ) ψ ( x ) . (5.3)8ince w ( a ; x ) = 1 and w (0; x − a ) = w ∗ ( a ; x ) we have that m ( a , b ; x ) satisfies m ( a , − a ; x ) = 1 . (5.4)We obtain then U ( a ) U ( b ) = U ( a + b ) M ( a , b ) . (5.5)We now construct a Weyl system from P and X . Consider the operator T ( α ) = e J [ a · P + a ′ · X ] = e J a · P e J a ′ · X e a i a ′ j [ P i ,X j ] = e J a · P e J a ′ · X e − J a · a ′ = e J a ′ · X e J a · P e J a · a ′ (5.6)with α = ( a , a ′ ). Remember that exp( J a · P ) ≡ U ( a ). We have then( T ( α ) ψ )( x ) = ( e J a · P e J a ′ · X e − J a · a ′ ψ )( x ) = w ( a ; x − a ) e j ( x − a ) a ′ · ( x − a ) e − j ( x − a ) a · a ′ ψ ( x − a ) , (5.7)but also( T ( α ) ψ )( x ) = ( e J a ′ · X e J a · P e J a · a ′ ψ )( x ) = e j ( x ) a ′ x w ( a ; x − a ) e j ( x − a ) a · a ′ ψ ( x − a ) . (5.8)We compute T ( α ) T ( β ) = e J [ a · P + a ′ · X ] e J [ b · P + b ′ · X ] = e J a · P e J a ′ · X e J b · P e J b ′ · X e − J ( a · a ′ + b · b ′ ) . (5.9)On using (5.5) and (5.6) we arrive at the Weyl system T ( α ) T ( β ) = T ( α + β ) M ( a , b ) exp (cid:16) J ( a · b ′ − b · a ′ ) (cid:17) . (5.10)We see that this Weyl system provides as usual, but there are two phases. The firstis the antisymmetric product of the two vectors in R . This would be present also inthe absence of the monopole, it gives the noncommutativity of position and momenta,however with the ‘imaginary’ unit given by the quaternionic radial functions j ( x ). Thefactor M instead is the one which contains the information on the noncommutativity ofthe translations. Both phases are of course crucial for the description of the particle inthe field of a monopole. Quaternions are well suited to describe the quantization of classical particles in the pres-ence of the field of a magnetic monopole. We have laid the basis for this quantization. Itis in principle possible (and will be presented elsewhere) to construct a full deformationquantization. One can built a full Weyl map which associates functions on phase spacewith operators on the quaternionic Hilbert space. The quantized functions belong to asubalgebra, in such a way that the map is (at least formally) invertible, and therefore9rovides a star product. Possible colour-breaking phenomena —see [12, 13]— are to befit in the formalism; indeed, there are plenty of questions here we should have answeredlong ago. But “the gaps in the knowledge of the wise has been filled even so slowly” [14].
Note Added
After this work had appeared we constructed the star product quantizingthe motion of particle in a monopole field in [15].
Acknowledgments:
FL would like to thank the Department of Estructura i Constituentsde la materia, and the Institut de Ci`encies del Cosmos, Universitat de Barcelona forhospitality. His work has been supported in part by CUR Generalitat de Catalunyaunder project 2009SGR502.
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