Shifting Operators in Geometric Quantization

Richard Cushman; Jędrzej Śniatycki

doi:10.3390/axioms9040125

Cushman, Richard;Śniatycki, Jędrzej

2020-10-28 00:00:00

axioms Article Richard Cushman and Jedrzej ˛ Sniatycki * Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4 Canada; r.h.cushman@gmail.com * Correspondence: sniatycki@gmail.com Received: 21 September 2020; Accepted: 20 October 2020; Published: 28 October 2020 Abstract: The original Bohr-Sommerfeld theory of quantization did not give operators of transitions between quantum quantum states. This paper derives these operators, using the ﬁrst principles of geometric quantization. Keywords: Bohr-Sommerfeld; geometric quantization;shifting operator 1. Introduction Even though the Bohr–Sommerfeld theory was very successful in predicting some physical results, it was never accepted by physicists as a valid quantum theory in the same class as the Schrödinger theory or the Bargmann–Fock theory. The reason for this was that the original Bohr–Sommerfeld theory did not provide operators of transition between quantum states. The need for such operators in the Bohr–Sommerfeld quantization was already pointed out by Heisenberg [1]. The aim of this paper is to derive operators of transition between quantum states in the Bohr–Sommerfeld theory, which we call shifting operators, from the ﬁrst principles of geometric quantization. The ﬁrst step of geometric quantization of a symplectic manifold (P, w) is called prequantization. It consists of the construction of a complex line bundle p : L ! P with connection whose curvature form satisﬁes a prequantization condition relating it to the symplectic form w. A comprehensive study of prequantization, from the point of view of representation theory, was given by Kostant in [2]. The work of Souriau [3] was aimed at quantization of physical systems, and studied a circle bundle over phase space. In Souriau’s work, the prequantization condition explicitly involved Planck’s constant h. In [4], Blattner combined the approaches of Kostant and Souriau by using the complex line bundle with the prequantization condition involving Planck’s constant. Since then, geometric quantization has been an effective tool in quantum theory. We ﬁnd it convenient to deal with connection and curvature of complex line bundles using the theory of principal and associated bundles [5]. In this framework, the prequantization condition reads db = (p ) ( w), where b is the connection 1-form on the principal C -bundle p : L ! P associated to the complex line bundle p : L ! P, and C is the multiplicative group of nonzero complex numbers. The aim of prequantization is to construct a representation of the Poisson algebra (C (P),f , g,) of (P, w) on the space of sections of the line bundle L. Each Hamiltonian vector ﬁeld X on P lifts to a unique C -invariant vector ﬁeld Z on L that preserves the principal connection b on L . If the tX vector ﬁeld X is complete, then it generates a 1-parameter group e of symplectomorphisms tZ of (P, w). Then, the vector ﬁeld Z is complete and it generates a 1-parameter group e of connection preserving diffeomorphisms of the bundle (L , b), called quantomorphisms, which cover tX the 1-parameter group e . The term quantomorphism was introduced by Souriau [3] in the context Axioms 2020, 9, 125; doi:10.3390/axioms9040125 www.mdpi.com/journal/axioms Axioms 2020, 9, 125 2 of 33 of SU(n)-principal bundles and discussed in detail in his book [6]. The construction discussed here tX tZ f f follows [7], where the term quantomorphism was not used. In this case, e and e are 1-parameter tX tZ f f groups of diffeomorphisms of P and L , respectively. We refer to e and e as ﬂows of X and tZ tZ f f Z . Since L is an associated bundle of L , the action e : L ! L , induces an action b e : L ! L, tZ f tZ tX . f f which gives rise to an action on smooth sections s of L by push forwards, s 7! b e s = b e s e . tZ Although b e s may not be deﬁned for all sections s and all t, its derivative at t = 0 is deﬁned for all smooth sections. The prequantization operator d tZ P s = ih b e s, (1) dt t=0 where h is Planck’s constant h divided by 2p, is a symmetric operator on the Hilbert space H of square integrable sections of L. The operator P is self adjoint if X is complete. f f The whole analysis of prequantization is concerned with global Hamiltonian vector ﬁelds. Since every vector ﬁeld on (P, w) that preserves the symplectic form is locally Hamiltonian, it is of interest to understand how much of prequantization can be extended to this case. In particular, we are interested in the case where the locally Hamiltonian vector ﬁeld is the vector ﬁeld X of the integer angle variable J that is deﬁned up to an additive term n, where n 2 Z. For a globally Hamiltonian vector ﬁeld X , tZ t liftX f f 2pi t f /h b e s = e b e s, (2) t liftX where b e s is the horizontal transport of section s by parameter t along integral curves of X Replacing f by a multivalued function J, deﬁned up to an additive n, yields the multivalued f . expression tZ t liftX itJ/h J J b e s = e b e s. (3) We observe that, for t = h, Equation (3) gives a single valued expression h Z h liftX i J J J b e s = e b e s. (4) The shifting operator h Z h liftX J i J J a = b e = e b e (5) is an operator on H, which shifts the support of s 2 H by h along the integral curve of X . If the vector nhZ n J ﬁeld X is complete, then a = b e for every n 2 Z. Our results provide an answer to Heisenberg’s criticism that in Bohr–Sommerfeld theory there are not enough operators to describe transitions between quantum states [1]. i q h lift X Superﬁcially, the shifting operator a = e b e , see Equation (5), appears to be a iq quantization of an angle q = 2pJ. It depends on q and has the factor e considered by Dirac [8]. h liftX q - However, the factor b e , describing the parallel translation by h along integral curves of X , makes it nonlocal in the phase space. Therefore, a cannot satisfy local commutation relations with any local quantum variable that is described by a differential operator. Hence, it cannot be the canonical conjugate of the corresponding action operator, or any other operator, which is local in the phase space. In our earlier papers [9–12], we followed an algebraic analysis, similar to that used by Dirac [8], supplemented by heuristic guesses about the behaviour of the shifting operators at the points of singularity of the polarization. In particular, we assumed that a vanishes on the states concentrated t X on a set of limit points of e ( p) as t ! ¥. In the present paper, we derive shifting operators in the framework of geometric quantization, and extend our result to cases with a variable rank polarization. The second stage in geometric quantization consists of the choice of a polarization, which is an involutive complex Lagrangian distribution F on the phase space. Suppose that P is the cotangent bundle space of the conﬁguration space. In this case, the choice of F containing the vertical directions, leads the quantum mechanics of Schrödinger. If F leads to complex analytic structure on P, we have the Bargmann–Fock theory. If F is spanned by the Hamiltonian vector ﬁelds of a completely integrable Axioms 2020, 9, 125 3 of 33 system, we have Bohr–Sommerfeld theory. Each of these theories have speciﬁc structure, which is helpful in formulation and solving problems. In the following, we restrict our investigation to the Bohr–Sommerfeld theory in order to emphasize its membership in the class of quantum theories corresponding to different polarization. A common problem in arising in quantum theories is occurrence of singularities. Usually, one studies the geometric structure of the theory in the language of differential geometry of smooth manifolds, and then investigates the structure of singularities separately. The theory of differential spaces, introduced by Sikorski [13,14], is a powerful tool in the study of the geometry of spaces with singularities [15]. The main singularity encountered here corresponds to the fact that the polarization F spanned by the Hamiltonian vector ﬁelds of a completely integrable system does not have constant rank. This singularity is so well known that we do not have to use the language of differential spaces to get results. It should be noted that the results in [9,11] rely on the theory of differential spaces. In conclusion, it should be mentioned that the scientists, who used visual presentation of the Bohr–Sommerfeld spectra in terms of dots on the space of the action variables, are familiar with handling shifting operators. The line segments joining two dots corresponding to quantum states represent the shifting operators between these states. To make the paper more accessible to the reader, we have provided an introductory section with a comprehensive review of geometric quantization. Experts may omit this section and proceed directly to the next section on Bohr–Sommerfeld theory. 2. Elements of Geometric Quantization Let (P, w) be a symplectic manifold. Geometric quantization can be divided into three steps: prequantization, polarization, and unitarization. 2.1. Principal Line Bundles with a Connection We begin with a brief review of connections on complex line bundles. Let C denote the multiplicative group of nonzero complex numbers. Its Lie algebra c is isomorphic to the abelian Lie algebra C of complex numbers. Different choices of the isomorphism i : C ! c lead to different factors in various expressions. Here, to each c 2 C we associate the 2pi tc 1-parameter subgroup t 7! e of C . In other words, we take i : C ! c : c 7! i(c) = 2pi c. (6) The prequantization structure for (P, w) consists of a principal C bundle p : L ! P and a c -valued C -invariant connection 1-form b satisfying db = (p ) ( w), (7) where h is Planck’s constant. The prequantization condition requires that the cohomology class [ w] is integral, that is, it lies in H (P,Z), otherwise the C principal bundle p : L ! P would not exist. 2pi tc Let Y be the vector ﬁeld on L generating the action of e on L . In other words, tY the 1-parameter group e of diffeomorphisms of L generated by Y is tY 2pi tc e : L ! L : ` ! ` e . (8) The connection 1-form b is normalized by the requirement hbjY i = c. (9) c Axioms 2020, 9, 125 4 of 33 For each c 6= 0, the vector ﬁeld, Y spans the vertical distribution ver T L tangent to the ﬁbers of p : L ! P. The horizontal distribution hor T L on L is the kernel of the connection 1-form b, that is, hor T L = ker b. (10) The vertical and horizontal distributions of L give rise to the direct sum T L = ver T L hor T L , which is used to decompose any vector ﬁeld Z on L into its vertical and horizontal components, Z = ver Z + hor Z. Here, the vertical component ver Z has range in ver T L and the horizontal component has range in hor T L . If X is a vector ﬁeld on P, the unique horizontal vector ﬁeld on L , which is p -related to X, is called the horizontal lift of X and is denoted by lift X. In other words, lift X has range in the horizontal distribution hor T L and satisﬁes Tp lift X = X p . (11) Claim 1. A vector ﬁeld Z on L is invariant under the action of C on L if and only if the horizontal component of Z is the horizontal lift of its projection X to P, that is, hor Z = lift X and there is a smooth function k : P ! C such that ver Z = Y on L = (p ) ( p). k( p) p Proof. Since the direct sum T L = ver T L hor T L is invariant under the C action on L , it follows that the vector ﬁeld Z is invariant under the action of C if and only if hor Z and ver Z are C -invariant. However, hor Z is C invariant if Tp hor Z = X p for some vector ﬁeld X on P, that is, hor Z = lift X. However, this holds by deﬁnition. On the other hand, the vertical distribution ver T L is spanned by the vector ﬁelds Y for c 2 C. Hence, ver Z is C -invariant if and only if for every ﬁber L the restriction of ver Z to L coincides with the restriction of Y to L for some c 2 C, p p p that is, there is a smooth complex valued function k on P such that c = k( p). Let U be an open subset of P. A local smooth section t : U P ! L of the bundle p : L ! P gives rise to a diffeomorphism h : L = (p ) ( p) ! UC : ` 7! (p (` ), b) = ( p, b), jU p2U where b 2 C is the unique complex number such that ` = t( p)b. In the general theory of principal bundles the structure group of the principal bundle acts on the right. In the theory of C principal bundles, elements of L are considered to be one-dimensional frames, which are usually written on the right, see [2]. The diffeomorphism h is called a trivialization of L . It intertwines the action of jU C on the principal bundle L with the right action of C on U C , given by multiplication in C . If a local section s : U ! L of p : L ! P is nowhere zero, then it determines a trivialization h : L ! UC . Conversely, a local smooth section t such that h is a trivialization of L may be t t jU considered as a local nowhere zero section of L. In particular, for every c 2 C, which is identiﬁed with the Lie algebra c of C , Equation (7) gives t Y 2pi tc e t = e t. Differentiating with respect to t and then setting t = 0 gives Y t = 2pi c t. (12) For every smooth complex valued function k : P ! C, consider the vertical vector ﬁeld Y such that Y (` ) = Y for every ` 2 L . The vector ﬁeld Y is complete and the 1-parameter group k k k(p (` )) of diffeomorphisms it generates is t Y 2pi tk(p (` )) e : L ! L : ` 7! ` e . Axioms 2020, 9, 125 5 of 33 t Y 2pi tk For every smooth section t of the bundle p , we have e t = e t so that Y t = 2pi k t. (13) Let X be a vector ﬁeld on P and let lift X be its horizontal lift to L . The local 1-parameter t lift X group e of local diffeomorphisms of L generated by lift X commutes with the action of C on t lift X L . For every ` , e (` ) is called parallel transport of ` along the integral curve of X starting at t lift X p = p (` ). For every p 2 P the map e sends the ﬁber L to the ﬁber L tX . p e ( p) There are several equivalent deﬁnitions of covariant derivative of a smooth section of the bundle p in the direction of a vector ﬁeld X on P. We use the following one. The covariant derivative of the smooth section t of the bundle p : L ! P in the direction X is t lift X r t = (e ) t. (14) dt t=0 Claim 2. The covariant derivative of a smooth local section of the bundle p : L ! P in the direction X is given by r t = 2piht bjXi t. (15) Proof. For every p 2 P, we have d d t lift X t lift X t X r t( p) = (e ) t( p) = (e t e )( p) dt dt t=0 t=0 = lift X(t( p)) + Tt(X( p)) = lift X(t( p)) + hor (Tt)X( p) + ver (Tt)X( p) = ver (Tt)X( p). The deﬁnition of the connection 1-form b and Equation (13) yield ver (Tt(X( p)) = Y (t( p)) = 2pihbjTt Xit( p). hbjTt Xi Hence, r t = 2pihbjTt Xi t, (16) which is equivalent to Equation (15). 2.2. Associated Line Bundles The complex line bundle p : L ! P associated to the C principal bundle p : L ! P is deﬁned in terms of the action of C on (L C) given by F : C (L C) ! L C : b, (` , c) 7! (` b, b c). (17) Since the action F is free and proper, its orbit space L = (L C)/C is a smooth manifold. A point ` 2 L is the C orbit [(` , c)] through (` , c) 2 (L C), namely, ` = [(` , c)] = f(` b, b c) 2 L C b 2 C g. (18) The left action of C on C gives rise to the left action F : C L ! L : a, [(` , c)] 7! [(` , ac)], (19) Axioms 2020, 9, 125 6 of 33 1 1 which is well deﬁned because [(` , ac)] = [(` b, b (ac))] = [(` b, a(b c))] for every ` 2 L , every a, b 2 C and every c 2 C. The projection map p : L ! P induces the projection map p : L ! L/C = P : ` = [(` , c)] 7! p(`) = p([(` , c)]) = p (` ). Claim 3. A local smooth section s : U ! L of the complex line bundle p : L ! P corresponds to a unique mapping s : L ! C such that for every p 2 U and every ` 2 L jU s( p) = [(` , s (` ))], (20) ] 1 ] which is C -equivariant, that is, s (` b) = b s (` ). Proof. Given p 2 U there exists (` , c) 2 L C such that s( p) = [(` , c)]. Since the action of C on L is free and transitive, it follows that the C orbit f(` b, b c) 2 L C b 2 C g is the p p ] ] graph of a smooth function from L to C, which we denote by s . In particular, c = s (` ) so that p p p s( p) = [(` , c)] = [(` , s (` ))]. As p varies over U we get a map ] ] ] s : L ! C : ` 7! s (` ) = s (` ), jU p (` ) which satisﬁes Equation (20). For every b 2 C , Equations (18) and (20) imply that ] 1 ] ] s( p) = [(` , s (` ))] = [(` b, b s (` ))] = [(` b, s (` b))]. ] 1 ] ] Hence, s (` b) = b s (` ). Thus, the function s is C -equivariant. If t : U ! L is a local smooth section of the bundle p : L ! P, then for every p 2 P ] ] we have s( p) = [(t( p), s (t( p)))] or s = [(t, s t)] suppressing the argument p. The function y = s t : U ! C is the coordinate representation of the section t in terms of the trivialization h : L ! UC. jU Let Z be a C -invariant vector ﬁeld on L . Then, Z is p -related to a vector ﬁeld X on P, that is, tX tZ Tp Z = X p . We denote by e and e the local 1-parameter groups of local diffeomorphisms of P and L generated by X and Z, respectively. Because the vector ﬁelds X and Z are p -related, t Z t X t Z t X we obtain p e = e p . In other words, the ﬂow e of Z covers the ﬂow e of X. The local t Z group e of automorphisms of the principal bundle L act on the associated line bundle L by t Z t Z b e : L ! L : ` = [(` , c)] 7! [(e (` ), c)], (21) t Z which holds for all ` = [(` , c)] for which e (` ) is deﬁned. t Z Lemma 1. The map b e is a local 1-parameter group of local automorphisms of the line bundle L, which covers t X the local 1-parameter group e of the vector ﬁeld X on P. Proof. We compute. For ` = [(` , c)] 2 L we have (t+s)Z (t+s)Z (t+s)Z t Z s Z b e (`) = b e ([(` , c)]) = [(e (` ), c)] = [(e (e (` )), c)] t Z s Z t Z s Z t Z s Z = b e ([(e (` , c)] = b e b e ([(` , c)]) = b e b e (`). t Z Hence, b e is a local 1-parameter group of local diffeomorphisms. Moreover, t Z t Z t Z t X p b e (`) = p([(e (` ), c)]) = p (e (` )) = e (p (` )); while t X t X t X e p(`) = e (p([(` , c)])) = e (p (` )). Axioms 2020, 9, 125 7 of 33 t Z t X This shows that e covers e . Finally, for every ` = [(` , c)] 2 L and every b 2 C t Z t Z t Z t Z b b F (b e (`)) = F ([(e (` ), c)]) = [(F (e (` )), c)] = [(e (F (` )), c)], b b b b since Z is a C -invariant vector ﬁeld on L . Therefore, t Z t Z t Z t Z b b b F (b e (`)) = b e ([(F (` ), c)]) = b e F ([(` , c)]) = b e F (`). b b b b t Z This shows that b e is a local group of automorphisms of the line bundle p : L ! P. t lift X t X If Z = hor X, then e (` ) is parallel transport of ` along the integral curve e ( p) of X starting at p = p (` ). Similarly, if ` = [(` , c)] 2 L, then t lift X t lift X b e (`) = [(e (` ), c)] (22) t X is parallel transport of ` 2 L along the integral curve e ( p) of X starting at p. The covariant derivative of a section s of the bundle p : L ! P in the direction of the vector ﬁeld X on P is d d t lift X t lift X t X r s = (b e ) s = (b e s e ). (23) dt dt t=0 t=0 t lift X 1 t X 1 Since b e maps p (e ) onto p ( p), Equations (22) and (23) are consistent with the deﬁnitions in [5]. Theorem 1. Let s be a smooth section of the complex line bundle p : L ! P and let X be a vector ﬁeld on P. For every ` 2 L r s(p (` )) = [(` , L s (` ) )]. (24) lift X Here, L is the Lie derivative with respect to the vector ﬁeld X. Proof. Let p = p (` ). Equation (23) yields t lift X t X r s( p) = b e s e (s( p)) . dt t=0 Recall that s( p) = [(` , s (` ))]. Hence, t X t lift X ] t lift X s(e ( p)) = [(e (` ), s e (` ) )]. By Equation (22), t lift X t X t lift X t lift X ] t lift X b e s(e ( p)) = b e [(e (` ), s e (` ) )] t lift X t lift X ] t lift X ] t lift X = [(e e (` ), s e (` ) )] = [(` , s e (` ) )] Therefore, d d t lift X t X ] t lift X r s( p) = b e s(e ( p)) = [(` , s e (` ) )] dt dt t=0 t=0 ] t lift X ] = [(` , s (e (` )))] = [(` , L s (` ) )]. (25) lift X dt t=0 Axioms 2020, 9, 125 8 of 33 2.3. Prequantization Let p : L ! P be the complex line bundle associated to the C principal bundle p : L ! P. The space S (L) of smooth sections of p : L ! P is the representation space of prequantization. Since C C, we may identify L with the complement of the zero section in L. With this identiﬁcation, if s : U ! L is a local smooth section of p : L ! P, which is nowhere vanishing, then it is a section of the bundle p : L ! U. jU jL jU Theorem 2. A C -invariant vector ﬁeld Z on L preserves the connection 1-form b, on L if and only if there is a function f 2 C (P) such that Z = lift X Y , (26) f f /h where h is Planck’s constant. Proof. The vector ﬁeld Z on L preserves the connection 1-form, that is, L b = 0, which is equivalent to Z db = d(Z b). (27) Since hor Z b = 0, it follows that Z b = ver Z b. The C -invariance of Z and b imply the C -invariance of ver Z b. Hence, ver Z b pushes forward to a function p (ver Z b) 2 C (P). Thus, the right hand side of Equation (27) reads d(Z b) = (p ) d(p (ver Z b)) . (28) By deﬁnition Y b = c, for every c 2 c. This implies Y db = L b d(Y b) = 0. c Y c Thus, the left hand side of Equation (27) reads Z db = hor Z db. (29) The quantization condition (7) together with (28) and (29) allow us to rewrite Equation (27) in the form lift X (p ) ( w) = (p ) d(p (ver Z b)) . (30) Equation (30) shows that X is the Hamiltonian vector ﬁeld of the smooth function f = h p (ver Z b)) (31) on P. We write X = X . This implies that hor Z = lift X . (32) We still have to determine the vertical component ver Z of the vector ﬁeld Z. For each ` 2 L there is a c 2 c such that ver Z = Y . Since Y is tangent to the ﬁbers of the C principal bundle c c p : L ! P, the element c of c depends only on p (` ) = p 2 P. Therefore, (p (ver Z b))(` ) = (p (Y b))(` ) = c( p) = f ( p)/h c( p) by Equation (31). In other words, for every point ` 2 L we have ver Z(` ) = Y (` ), f ( p)/h where p = p (` ). Thus, we have shown that Z = Z = lift X Y . (33) f f f /h Axioms 2020, 9, 125 9 of 33 Reversing the steps in the above argument proves the converse. To each f 2 C (P), we associate a prequantization operator t Z ¥ ¥ - f P : S (L) ! S (L) : s 7! P s = ih (b e ) s, (34) f f dt t=0 t Z t Z f f where b e is the action of e : L ! L on L, see (22). Note that the deﬁnition of covariant t Z t Z f f derivative in Equation (23) is deﬁned in terms of the pull back (b e ) s of the section s by b e , t Z t Z f f while the prequantization operator in (34) is deﬁned using the push forward (b e ) s of s by b e . ¥ ¥ Theorem 3. For every f 2 C (P) and each s 2 S (L) P s = (ihr + f )s. (35) f X Proof. Since the horizontal distribution on L is C -invariant and the vector ﬁeld Y generates t lift X t Y t Y t lift X 2pi c f f /h f /h f multiplication on each ﬁber of p by e , it follows that e e = e e . Since f is constant along integral curves of X , t Z t(lift X Y ) t lift X t Y f f f /h f f /h e = e = e e t lift X 2pit f /h 2pi t f /h t lift X f f = e e = e e , (36) and d d t Z t lift X t Y - - f f f /h P s = ih (b e ) s = ih (e e ) s dt dt t=0 t=0 d d t lift X t Y - - f f /h = ih (b e ) s + ih (b e ) s. (37) dt dt t=0 t=0 t lift X t lift X f f Since (b e ) s = (b e ) s, Equation (23) gives d d t lift X t lift X - f - f - ih (b e ) s = ih (b e ) s = ihr s. (38) dt dt t=0 t=0 t Y f /h Since p b e = p id , where id is the identity map on P, it follows that P P tY t Y t Y f /h f /h f /h (b e ) s = b e s id = b e s. Let t : U P ! L be a smooth local section of p : L ! P, then s = [(t, s t)]. Thus, for every p 2 P t Y t Y t Y ] ] f /h f /h f /h b e s( p) = b e [(t( p), s (t( p)))] = [(e (t( p)), s (t( p)))] 2pi t f ( p)/h ] 2pi t f ( p)/h ] = [(t( p)e , s (t( p)))] = [(t( p), e s (t( p)))], since [(` b, c)] = [(` b, b (bc))] = [(` , bc)] for every ` 2 L , b 2 C and c 2 C. It follows that t Y 2pi t f ( p)/h ] 2pi t f ( p)/h f /h b e s( p) = [(t( p), e s (t( p)))] = e s( p). (39) Therefore, t Z t (lift X Y ) f f f /h (b e ) s = (b e ) s t lift X t Y 2pi t f ( p)/h t lift X f f /h f = (b e b e ) s = e b e s. (40) Axioms 2020, 9, 125 10 of 33 Since d t Y d f /h 2pi t f /h - - - ih b e s = ih (e s) = ih(2pi f /h)s = f s (41) dt dt t=0 t=0 Equations (37), (38) and (41) imply Equation (35). A Hermitian scalar product h j i on the ﬁbers of L that is invariant under parallel transport gives rise to a Hermitian scalar product on the space S (L) of smooth sections of L. Since the dimension of (P, w) is 2k, the scalar product of the smooth sections s and s of L is 1 2 (s js ) = hs js i w . (42) 2 2 1 1 The completion of the space S (L) of smooth sections of L with compact support with respect to the norm ksk = (sjs) is the Hilbert space H of the prequantization representation. Claim 4. The prequantization operator P is a symmetric operator on the Hilbert space H of square integrable sections of the line bundle p : L ! P and satisﬁes Dirac’s quantization commutation relations [P ,P ] = ihP . (43) f f f ,gg for every f , g 2 C (P). Moreover, the operator P is self adjoint if the vector ﬁeld X on (P, w) is complete. f f ¥ ¥ Proof. We only verify that the commutation relations (43) hold. Let f , g 2 C (P) and let s 2 S (L). We compute. i i i [r f ,r g]s = [r ,r ]s + r (gs) gr s X X X X X X - g - g - f h h f h f f r ( f s) fr s - X X g g = [r ,r ] + (L g L f ) s X X - X X g g f h f The quantization condition [r ,r ]r = w(X , X ) X X g [X ,X ] - f f g g f h yields i i i i [r f ,r g] = r w(X , X ) + (L g L f ) X - X - [X ,X ] - f - X X f g f g h h f g h h However, f f , gg = L f = w(X , X ). Thus, L g L f = fg, fg f f , gg = 2f f , gg. X f g X X g g Since X w = dg, it follows that [X , X ] w = L X w = L dg = dL g = df f , gg. g X g X X f f f Consequently, [X , X ] = X . Thus, f g f f ,gg i i i [r f ,r g] = r f f , gg. X - X - X - f h h f f ,gg h 2.4. Polarization Prequantization is only the ﬁrst step of geometric quantization. The prequantization operators do not satisfy Heisenberg’s uncertainty relations. In the case of Lie groups, the prequantization representation fails to be irreducible. These apparently unrelated shortcomings lead to the next step of geometric quantization: the introduction of a polarization. Axioms 2020, 9, 125 11 of 33 A complex distribution F T P = C T P on a symplectic manifold (P, w) is Lagrangian if for every p 2 P, the restriction of the symplectic form w to the subspace F T P vanishes identically, p p and rank F = dim P. If F is a complex distribution on P, let F be its complex conjugate. Let C C D = F\ F\ T P and E = (F + F)\ T P. A polarization of (P, w) is an involutive complex Lagrangian distribution F on P such that D and E are involutive distributions on P. Let C (P) be the space of smooth complex valued functions of P that are constant along F, that is, ¥ ¥ C (P) = f f 2 C (P) P hd fjui = 0 for every u 2 Fg. (44) The polarization F is strongly admissible if the spaces P/D and P/E of integral manifolds of D and E, respectively, are smooth manifolds and the natural projection P/D ! P/E is a submersion. A strongly admissible polarization F is locally spanned by Hamiltonian vector ﬁelds of functions in C (P) . A polarization F is positive if i w(u, u) 0 for every u 2 F. A positive polarization F is semi-deﬁnite if w(u, u) = 0 for u 2 F implies that u 2 D . Let F be a strongly admissible polarization on (P, w). The space S (L) of smooth sections of L that are covariantly constant along F is the quantum space of states corresponding to the polarization F. The space C (P) of smooth functions on P, whose Hamiltonian vector ﬁeld preserves the polarization F, is a Poisson subalgebra of C (P). Quantization in terms of the polarization F leads to quantization map Q, which is the restriction of the prequantization map ¥ ¥ ¥ P : C (P) S (L) ! S (L) : ( f , s) 7! P s = (ihr + f )s f X ¥ ¥ ¥ ¥ ¥ ¥ to the domain C (P) S (L) C (P) S (L) and the codomain S (L) S (L). In other words, F F F ¥ ¥ ¥ - Q : C (P) S (L) ! S (L) : ( f , s) 7! Q s = (ihr + f )s. (45) F F f X Quantization in terms of positive strongly admissible polarizations such that E\ E = f0g lead to unitary representations. For other types of polarizations, unitarity may require additional structure. 3. Bohr–Sommerfeld Theory 3.1. Historical Background Consider the cotangent bundle T Q of a manifold Q. Let p : T Q ! Q be the cotangent bundle projection map. The Liouville 1-form a on T Q is deﬁned as follows. For each q 2 Q, p 2 T Q and Q q u 2 T (T Q), p p ha ju i = h pjTp (u )i. (46) p p Q Q The exterior derivative of a is the canonical symplectic form da on T Q. Q Q Let dim Q = k. A Hamiltonian system on (T Q, da ) with Hamiltonian H is completely integrable Q 0 if there exists a collection of k 1 functions H , . . . , H 2 C (T Q), which are integrals of X , 1 k1 H that is, f H , H g = 0 for i = 1, . . . , k 1, such that f H , H g = 0 for i, j = 1, . . . , k 1. Assume 0 i i j that the functions H , . . . , H are independent on a dense open subset of T Q. For each p 2 T Q, 0 k1 let M be the orbit of the family of Hamiltonian vector ﬁelds fX , . . . , X g passing through p. H H 0 k1 This orbit is the largest connected immersed submanifold in T Q with tangent space T 0( M ) equal to t X 0 0 H span fX ( p ), . . . , X ( p )g. The integral curve t 7! e ( p) of X starting at p is contained in H H H R 0 k1 0 M . Hence, knowledge of the family f M p 2 T Qg of orbits provides information on the evolution p p of the Hamiltonian system with Hamiltonian H . Bohr–Sommerfeld theory, see [16,17], asserts that the quantum states of the completely integrable system ( H , . . . , H , T Q, da ) are concentrated on the orbits M 2 f M p 2 T Qg, which satisfy the 0 p k1 Q Axioms 2020, 9, 125 12 of 33 Bohr–Sommerfeld Condition: For every closed loop g : S ! M T Q, there exists an integer n such that g (a ) = n h, (47) where h is Planck’s constant. This theory applied to the bounded states of the relativistic hydrogen atom yields results that agree exactly with the experimental data [17]. Attempts to apply Bohr–Sommerfeld theory to the helium atom, which is not completely integrable, failed to provide useful results. In his 1925 paper [1], Heisenberg criticized Bohr–Sommerfeld theory for not providing transition operators between different states. At present, the Bohr–Sommerfeld theory is remembered by physicists only for its agreement with the quasi-classical limit of Schrödinger theory. Quantum chemists have never stopped using it to describe the spectra of molecules. 3.2. Geometric Quantization in a Toric Polarization To interpret Bohr–Sommerfeld theory in terms of geometric quantization, we consider a set P T Q consisting of points p 2 T Q where X ( p), . . . , X ( p) are linearly independent and the H H 0 k1 orbit M of the familyfX , . . . , X g of Hamiltonian vector ﬁelds on (T Q, da ) is diffeomorphic p H H T Q k1 k k k to the k torus T = R /Z . We assume that P is a 2k-dimensional smooth manifold and that the set B = f M p 2 Pg is a quotient manifold of P with smooth projection map r : P ! B. This implies that the symplectic form da on T Q restricts to a symplectic form on P, which we denote by w. Let D be the distribution on P spanned by the Hamiltonian vector ﬁelds X , . . . , X . Since f H , H g = 0 H H i j 0 k1 for i, j = 0, 1, . . . , k 1, it follows that D is an involutive Lagrangian distribution on (P, w). Moreover, F = D is a strongly admissible polarization of (P, w). Since the symplectic form da on T Q is exact, we may choose a trivial prequantization line bundle p : L = C T Q ! T Q : b, (q, p) 7! (q, p) T Q db 1 1 with connection 1-form b = + a . Let L be the restriction of L to P and let a be the Q Q 2pi h T Q 1-form on P, which is the restriction of a to P, that is, a = a . Then, L = C P is a principal C Q Q jP bundle over P with projection map p : L = C P ! P : (b, p) 7! p db 1 1 and connection 1-form b = + a. The complex line bundle 2pi h p : L = C P ! P : (c, p) 7! p associated to the principal bundle p is also trivial. Prequantization of this system is obtained by adapting the results of Section 2. Since integral manifolds of the polarization D are k-tori, we have to determine which of them admit nonzero covariantly constant sections of L. Theorem 4. An integral manifold M of the distribution D admits a section s of the complex line bundle L, which is nowhere zero when restricted to M, if and only if it satisﬁes the Bohr–Sommerfeld condition (47). Proof. Suppose that an integral manifold M of D admits a nowhere zero section of L . Since s is j M 1 1 nowhere zero, it is a section of L . Let g : S ! M be a loop in M. For each t 2 S , let g(t) 2 T M g(t) j M be the tangent vector to g at t. Since s is covariantly constant along M, Claim 2 applied to the section s : M ! L = C M : p 7! (b( p), p) j M Axioms 2020, 9, 125 13 of 33 gives r s( p) = 2pihs (b)( p)jX( p)i s( p) = 0 X( p) for every p 2 P and every X( p) 2 T M. Taking p = g(t) and X( p) = g ˙ (t) gives 2pihs b(g(t))jg ˙ (t)i s(g(t)) = 0. (48) 1 db 1 Since b = + a and (s g)(t) = (b(g(t), g(t))), we get 2pi b h ˙ ˙ 2pihs b(g(t))jg(t)i = 2pihb(s(g(t)))jg(t)i 1 db(g(t)) 2pi = + hajg ˙ (t)i b(g(t)) dt h d 2pi = ln b(g(t)) + ha(g(t))jg(t)i. dt h Hence, Equation (48) is equivalent to d 2pi ln b(g(t)) + ha(g(t))jg ˙ (t)i = 0, dt h which integrated from 0 to 2p gives Z I 2p 2pi 2pi ln b(g(2p)) ln b(g(0)) = ha(g(t))jg ˙ (t)i dt = g a. h h If g bounds a surface S M, then Stokes’ theorem together with Equation (47) and the quantization condition (7) yield I Z Z 2pi 2pi 2pi g a = da = w = 0, h h h S S because M is a Lagrangian submanifold of (P, w). Thus, ln b(g(2p)) = ln b(g(0)), which implies that the nowhere zero section s is parallel along g. If g does not bound a surface in M, but does satisfy the Bohr–Sommerfeld condition g a = nh (47) with a replaced by its pull back a to P, then Q Q b(g(2p)) 2pi 2pi ln = g a = nh = 2pi n, b(g(0)) h h so that b(g(2p)) 2pi n = e = 1. b(g(0)) Hence, b(g(2p)) = b(g(0)) and the nowhere zero section s is parallel along g. Note that the manifolds M that satisfy Bohr–Sommerfeld conditions (47) are k-dimensional toric submanifolds of P. We call them Bohr–Sommerfeld tori. Since Bohr–Sommerfeld tori have dimension k = dim P, there is no non-zero smooth section s : P ! L that is covariantly constant along D. However, for each Bohr–Sommerfeld torus M, Theorem 4 guarantees the existence of a non-zero, covariantly constant along D , smooth section s : M ! L , where L denotes the restriction of L j M j M j M to M. Let S = f Mg be the set of Bohr–Sommerfeld tori in P. For each M 2 S , there exists a non-zero, covariantly constant along D , smooth section s of L restricted to M determined up to a factor in j M C . The direct sum S = fC s g (49) M2S is the the space of quantum states of the Bohr–Sommerfeld theory. Thus, each Bohr–Sommerfeld torus M represents a 1-dimensional subspacefC s g of quantum states. Moreover,fC s g\fC s 0g = f0g M M M Axioms 2020, 9, 125 14 of 33 if M 6= M because Bohr–Sommerfeld tori are mutually disjoint. Hence, the collection fs g is a basis of S. For our toral polarization F = D , the space of smooth functions on P that are constant along ¥ ¥ ¥ F, see Equation (44), is C (P) = r (C (B)), see Lemma A3. For each f 2 C (P), the Hamiltonian F F vector ﬁeld X is in D, that is, r s = 0 for every basic state s 2 S. Hence, the prequantization X M M and quantization operators act on the basic states s 2 S by multiplication by f , that is, Q s = P s = f s = f s . (50) f M f M M j M M Note that f is a constant because f 2 C (P). For a general quantum state s = å c s 2 S, M M j M M2S Q s = Q c s = c Q s = c f s . f f å M M å M f M å M M j M M2S M2S M2S We see that, for every function f 2 C (P), each basic quantum state s is an eigenstate of Q corresponding to the eigenvalue f . Since eigenstates corresponding to different eigenvalues of the j M same symmetric operator are mutually orthogonal, it follows that the basis fs g of S is orthogonal. This is the only information we have about scalar product in S. Our results do not depend on other details about the scalar product in S. 3.3. Shifting Operators 3.3.1. The Simplest Case P = T T We begin by assuming that P = T T with canonical coordinates (p, q) = ( p , ..., p , q , ..., q ) 1 k 1 k where, for each i = 1, ..., k, q is the canonical angular coordinate on the ith torus and p is the conjugate i i momentum. The symplectic form is k k w = d p dq = d p ^ dq . å i i å i i i=1 i=1 In this case, action–angle coordinates (j, J) = (j , . . . , j , J , . . . , J ) are obtained by rescaling the 1 k 1 k canonical coordinates so that, for every i = 1, ..., k, we have j = 2p p and J = q /2p. Moreover, i i i i the rescaled angle coordinate J : T T ! T = R/Z is interpreted as a multi-valued real function, the symplectic form w = dj ^ dJ , (51) å i i i=1 n o ¶ ¶ and the toric polarization of (P, w) is given by D = span , . . . , . ¶J ¶J 1 1 In terms of action–angle coordinates, the Bohr–Sommerfeld tori in T T are given by equation j = (j , ..., j ) = (n h, . . . , n h) = nh, (52) 1 k 1 k k k k where n = (n , ..., n ) 2 Z . For each n 2 Z , we denote by T the corresponding Bohr–Sommerfeld 1 k n 1 db 1 k torus in B. If b = + j dJ is the connection form in the principal line bundle L = i i i=1 2pi b h k k C T ! T , then sections n n k 2pi(n J +...+n J ) 1 1 k k s : T ! L : (J , . . . , J ) 7! e , (53) n 1 n k form a basis in the space S of quantum states. Axioms 2020, 9, 125 15 of 33 ¶ ¶ ¶ For each i = 1, ..., k, the vector ﬁeld is transverse to D and w = dJ , so that is the ¶j ¶j ¶j i i i Hamiltonian vector ﬁeld of J . In the following, we write X = = X (54) i J ¶j to describe the actual vector ﬁeld X without referring to its relation to the action angle coordinates (j, J). Equation (36) in Section 2.1, for f = J , is multi-valued because the phase factor is multi-valued, and tZ 2pitJ /h t lift X J i i e = e e . (55) Claim 5. If t = h, then h Z 2piJ h lift X i i e = e e . (56) is well deﬁned. Proof. For every i = 1, ..., k, consider an open interval (a , b ) in R such that 0 < b a < 1. Let i i i i 1 1 1 W = J (a , b )\ J (a , b )\ ...\ J (a , b ). (57) 2 2 1 1 k k 1 2 k Since the action–angle coordinates (j , . . . , j , J , . . . , J ) are continuous, W is an open subset of P. 1 1 k k Let h be a unique representative of J with values in (a , b ). With this notation, i ijW i i w = dj ^ dJ . (58) jW å ijW i i=1 The restriction to W of the vector ﬁeld X is the genuinely Hamiltonian vector ﬁeld of h , namely, J i X = X . (59) J h ijW The vector ﬁeld Z = lift X Y (60) h h h /h i i i t Z t lift X h 2pi h /h h i i is well deﬁned. Equation (36) yields e = e e . Hence, h Z 2pi h h lift X h h i i e = e e . (61) 0 0 0 0 0 If we make another choice of intervals (a , b ) in R such that 0 < b a < 1 and let W = i i i i k 0 0 0 0 0 0 \ J (a , b ). Then, h with values in (a , b ) differs from h by an integer, so that h = h + n , and, i i i i=1 i i i i i i i in W \ W , we have 2pi h 2pi (h +n ) 2pi h i i i e = e = e . Moreover, X 0 = X 0 0 = X 0 , so that q jW\W h jW\W ijW\W h Z h Z h Z h X h i i i (e ) = (e ) = (e ) . jL jL jL 0 0 0 jW\W jW\W jW\W h Z Since we can cover P by open contractible sets deﬁned in Equation (57), we conclude that e is well deﬁned by Equation (56) and depends only on the vector ﬁeld X . Consequently, there exists a connection preserving automorphism A : L ! L such that, if ` 2 L , where W P is given by Equation (57), then jW h Z A (` ) = e (` ). (62) i Axioms 2020, 9, 125 16 of 33 Claim 6. The connection preserving automorphism A : L ! L , deﬁned by Equation (62) depends only on the vector ﬁeld X and not the original choice of the action–angle coordinates. 0 0 0 0 Proof. If (j , . . . , j , J , . . . , J ) is another set of action–angle coordinates then 1 k 1 k k k 0 0 j = a j and J = b J , (63) i i å il å il l l l=1 l=1 1 T where the matrices A = (a ) and B = (b ) lie in Sl(k,Z) and B = ( A ) . In the new coordinates, il il k k ¶ ¶ X = = a = b X 0 = X 0 0 . J å il å il J (b J +...+b J ) i 0 i1 ik l 1 k ¶j ¶j l=1 l=1 Clearly, t lift X 0 0 t lift X (b J ++b J ) i1 ik i 1 k e = e . (64) To compare the phase factor entering Equation (55), we consider an open contractible set W P. 0 0 As before, for each i = 1, ..., k, choose a single-valued representative h of (J ) . Then, jW i i k k k k 0 0 0 h = b (h + l ) = b h + b l = b h + l, (65) i å i j j å i j å i j j å i j j j j j=1 j=1 j=1 j=1 where each l is an integer and thus l = b l is also an integer. Hence, j i j j j=1 0 0 0 0 2pi h 2pi (b h +...+b h +l) 2pi (b h +...+b h ) i i1 ik i1 ik 1 k 1 k e = e = e , (66) where b , . . . , b are integers. Since l is constant, i1 ik X = X = X 0 0 J (b h +...+b h +l) i i jW i1 ik 1 k = X 0 0 = X 0 0 . (67) (b h +...+b h ) (b J +...+b J ) i1 ik i1 ik 1 k 1 k jW Therefore, h lift X 0 0 0 0 h Z 2pi h h lift X 2pi (b h +...+b h ) (b J +...+b J ) h h i i1 ik i1 ik i i 1 k 1 k e = e e = e e , (68) which shows that the automorphism A : L ! L depends on the vector ﬁeld X and not on the J i action angle coordinates in which it is computed. hX Claim 7. For each i = 1, ..., k, the symplectomorphism e : P ! P, where h is Planck’s constant, preserves the set B of Bohr–Sommerfeld tori in P. tX Proof. Since X is complete, e : P ! P is a 1-parameter group of symplectomorphisms of (P, w). h X Hence, e : P ! P is well deﬁned. By Equation (52), j = n h for every Bohr–Sommerfeld torus i i jT T , where n = (n , . . . , n ). n 1 k Since X = , ¶j L (j dJ ) = X dj ^ dJ + d(X j dJ ) = dJ , X i i i i i i i i i L (j dJ ) = X dj ^ dJ + d(X j dJ ) = 0 for l 6= i. X l l i l l i l l tX tX i i This implies that, for every l 6= i, (e ) (j dJ ) = j dJ and (e ) (j dJ ) = (j t)dJ . Therefore, l l l l i i i i h X tX i i if j = nh, then (e ) j = j = n , if l 6= i, and (e ) j = (j h) = (n 1)h if ` = i. This implies l l l i i i hX J k that e (T ) is a Bohr–Sommerfeld torus. n Axioms 2020, 9, 125 17 of 33 b b We denote by A : L ! L the action of A : L ! L on L. The automorphism A acts on X X X i i i sections of L by pull back and push forward, namely, h Z h Z h X i i i (A ) s = (b e ) s = b e s e , (69) h Z h Z h X i i i (A ) s = (b e ) s = b e s e . Since A : L ! L is a connection preserving automorphism, it follows that, if s satisﬁes the b b Bohr–Sommerfeld conditions, then (A ) s and (A ) s also satisfy the Bohr–Sommerfeld conditions. X X i i b b In other words, (A ) and (A ) preserve the space S of quantum states. The shifting operators a X X X i i i b b and b , corresponding to X , are the restrictions to S of (A ) and (A ) , respectively. For every X i X X i i i n = (n , . . . , n ) 2 Z , Equations (53) and (56) yield 1 k 2pi( n J +(n 1)J ) h Z h X å j6=i j j i i i i a s = b e s b e = s = e s X n n n i n (70) 2pi(å n J +(n +1)J ) h Z h X j j i i i i j6=i b s = b e s b e = s + = e s . X n n n i n For each i = 1, . . . , k, a b = b a = id . In addition, the operators a , b , for i, j = 1, . . . , k, X X X X S X X i i i i i j generate an abelian group A of linear transformations of S into itself, which acts transitively on the space of one-dimensional subspaces of S. Given a non-zero section s 2 S supported on a Bohr–Sommerfeld torus, the family of sections n n k 1 f(a a s) 2 S n , ...n 2 Zg (71) 1 k X X k 1 is a linear basis of S, invariant under the action of A. Since A is abelian, there exists a positive, deﬁnite Hermitian scalar product h j i on S, which is invariant under the action of A, and such that the basis in (71) is orthonormal. It is deﬁned up to a constant positive factor. The completion of S with respect to this scalar product yields a Hilbert space H of quantum states in the Bohr–Sommerfeld quantization of T T . Elements of A extend to unitary operators on H. 3.3.2. General Case of Toral Polarization Hilbert Space and Operators Let (P, w) be a symplectic manifold with toroidal polarization D and a covering by domains of action–angle coordinates. If U and U are the domain of the angle-action coordinates (j, J) = 0 0 0 0 0 0 0 (j , . . . , j , J , . . . , J ) and (j , J ) = (j , . . . , j , J , . . . , J ), respectively, and U\ U 6= ?, then in U\ U 1 k 1 k 1 1 k k we have k k 0 0 j = a j and J = b J , (72) i å il i å il l l l=1 l=1 1 T where the matrices A = (a ) and B = (b ) lie in Sl(k,Z) and B = ( A ) . il il Consider a complete locally Hamiltonian vector ﬁeld X on (P, w) such that, for each angle-action coordinates (j, J) with domain U, (X w) = d(c J) = d(c J + . . . + c J ), (73) 1 1 jU k k k 0 for some c = (c , ..., c ) 2 Z . Equation (72) shows that in U\ U , we have 1 k 0 0 0 0 c J + . . . + c J = c J + . . . + c J , 1 1 k k 1 1 k k 0 k where c = c b 2 Z, for i = 1, . . . , k. As in the preceding section, Equation (36) with f = c J = j ji i j=1 c J + . . . + c J , which is multi-valued, gives 1 1 k k tZ 2pi t cJ/h t liftX cJ e = e e , (74) Axioms 2020, 9, 125 18 of 33 which is multivalued, because the phase factor is multi-valued. As before, if we set t = h, we would h Z 2picJ h liftX cJ get a single-valued expression e = e e because c , . . . , c 2 Z. This would work along 1 k t X all integral curves t 7! e (x) for t 2 [0, 1], which are contained in U. hX 0 Now, consider the case when, for x 2 U, e (x) 2 U and there exists t 2 (0, h) such that 0 1 t X 0 0 0 0 x = e (x ) 2 U \ U , where U and U are domains of action–angle variables (j, J) and (j , J ), 1 0 tX tX 0 respectively. Moreover, assume that e (x ) 2 U for t 2 [0, t ] and e (x ) 2 U for t 2 [0, h t ]. 0 1 1 1 Using the multi-index notation, for l 2 L , we write (ht )Z t Z 1 0 0 1 cJ c J A (` ) = e (e (` )) 2pi(ht )c J /h (ht )liftX 2pit cJ/h t liftX 1 1 1 1 = e e (e e (` )) (75) 0 0 2pi(ht )c J /h 2pit cJ/h (ht )liftX t liftX 1 1 1 1 = (e e )e (e (` )) 0 0 0 2pit (cJc J )/h 2pic J h liftX = e e e (` ). 0 0 Let W be a neighborhood of x in P such that U\ W and U \ W are contractible. For each i = 1, ..., k, let q be a single-valued representative of J as in the proof of Claim 5. Similarly, we denote by h i i 0 0 a single-valued representative of J . Equation (73) shows that in U \ U \ W, the functions c h + 1 1 0 0 0 0 + c h and c h + + c h are local Hamiltonians of the vector ﬁeld X and are constant along the k k 1 1 k k integral curve of X . Hence, we have to make the choice of representatives h and h so that jW 0 0 c h (x ) + + c h (x ) = c h (x ) + + c h (x ). (76) 1 1 1 k k 1 1 1 k 1 1 k 0 0 2pit (cJc J )/h With this choice, e = 1, and 0 0 2pic J h liftX A (l ) = e e (l ) (77) is well deﬁned. It does not depend on the choice of the intermediate point x in U\ U . In the case when m + 1, action–angle coordinate charts with domains U , U ..., U are needed 0 1, h X t X t X 1 2 to reach x = e (x ) 2 U from x 2 U ; we choose x = e (x ) 2 U \ U , x = e (x ) 2 m 0 m 0 0 1 0 0 1 2 1 t X (ht ...t )X m1 1 m1 U \ U , . . . , x = e (x ) 2 U and end with x = e (x ) 2 U . At each m m 1 2 m1 m2 m1 m1 intermediate point x , . . . , x , we repeat the the argument of the preceding paragraph. We conclude 1 m1 that there is a connection preserving automorphism A : L ! L well deﬁned by the procedure given here, and it depends only on the complete locally Hamiltonian vector ﬁeld X satisfying condition (73). The automorphism A : L ! L of the principal bundle L leads to an automorphism A of X X the associated line bundle L. As in Equation (69), the shifting operators corresponded to the complete locally Hamiltonian vector ﬁeld X are a : S ! S : s 7! (A ) s, X X (78) b : S ! S : s 7! (A ) s. X X In absence of monodromy, if we have k independent, complete, locally Hamiltonian vector ﬁelds X on (P, w) that satisfy the conditions leading to Equation (73), then the operators a , b for i X X i j i, j = 1, ..., k generate an abelian group A of linear transformations of S. If the local lattice S of Bohr–Sommerfeld tori is regular, then A acts transitively on the space of one-dimensional subspaces of S. This enables us to construct an A-invariant Hermitian scalar product on S, which is unique up to an arbitrary positive constant. The completion of S with respect to this scalar product yields a Hilbert space H of quantum states in the Bohr–Sommerfeld quantization of (P, w). Axioms 2020, 9, 125 19 of 33 Local Lattice Structure The above discussion does not address the question of labeling the basic sections s in H by the quantum numbers n = (n , . . . , n ) associated to the Bohr–Sommerfeld k-torus T = M , k b the support of s . 0 0 k These quantum numbers do depend on the choice of action angle coordinates. If (j , J ) 2 V T 0 0 0 is another choice of action angle coordinates in the trivializing chart (U , y ), where T U , then the 0 0 0 quantum numbers n of T in (j , J ) coordinates are related to the quantum numbers n of T in (j, J) coordinates by a matrix A 2 Gl(k,Z) such that n = A n, because by Claim A2 in Appendix A on 0 0 U\ U the action coordinates j is related to the action coordinate j by a constant matrix A 2 Gl(k,Z). Let L = fn 2 Z T Ug. Then, L is the local lattice structure of the Bohr–Sommerfeld tori T , n n jU jU 0 0 which lie in the action angle chart (U, y). If (U, y) and (U , y ) are action angle charts, then the set of 0 0 Bohr–Sommerfeld tori in U\ U are compatible. More precisely, on U\ U the local lattices L and jU L 0 are compatible if there is a matrix A 2 Gl(k,Z) such that L 0 = AL . Let U = fU g be a jU jU jU i i2 I good covering of P, that is, every ﬁnite intersection of elements of U is either contractible or empty, such that for each i 2 I we have a trivializing chart (U , y ) for action angle coordinates for the toral i i bundle r : P ! B. Then, fL g is a collection of pairwise compatible local lattice structures for the u i2 I collection S of Bohr–Sommerfeld tori on P. We say that S has a local lattice structure. h Z The next result shows how the operator (b e ) of Section 3.3 affects the quantum numbers of the Bohr–Sommerfeld torus T = T . Claim 8. Let (U, y) be a chart in (P, w) for action angle coordinates (j, J). For every Bohr–Sommerfeld torus h X T = T in U with quantum numbers n = (n , . . . , n ), the torus e (T) is also a Bohr–Sommerfeld torus n 1 0 0 T , where n = (n , . . . , n , n 1, n , . . . , n ). 0 1 `1 ` `+1 k Proof. For simplicity, we assume that ` = 1. Suppose that the image of the curve g : [0, h] ! B : t 7! e (r(x )) lies in V = y(U), where x 2 T = T . For x 2 T and t 2 [0, h] we have 0 0 n X j = j = 1, if ` = 1 1 1 1 ¶j X j = J ` 1 ¶ X j = j = 0, if ` = 2, . . . , k J ` ` 1 ¶j and X J = J = 0. Since x 2 T has action angle coordinates (j (x), . . . , j (x), J (x), . . . , J (x)) 1 1 J ` ` k k 1 ¶ p t J in U, the point e (x) has action angle coordinates (j (x) t, . . . , j (x), J (x), . . . , J (x)). In particular, 1 k 1 k t X the point e (x ) has action angle coordinates (j (x ) t, . . . , j (x ), J (x ), . . . , J (x )). Thus, 0 1 0 k 0 1 0 k 0 j h, if ` = 1 h X (e ) j = j , if ` = 2, . . . , k h X and (e ) J = J for ` = 1, 2, . . . , k. Since T is the Bohr–Sommerfeld torus T , we have ` ` j = j dJ = n h. Then, ` ` ` ` Z Z 1 1 h X h X J J 1 1 (e ) j d (e ) J = (j h)dJ 1 1 1 1 0 0 = j h = (n 1)h. 1 1 h X Thus, the torus e (T) is a Bohr–Sommerfeld torus T 0 with n = (n , . . . , n , n 1, n , . . . , n ). n 1 `1 ` `+1 k t X Now, consider the case when the image of the curve g : [0, h] ! B : t 7! e (r(x )) is not t X t X J J 1 1 contained in V. This means that e (U), where U = r (V), does not contain the torus T. Since e t X is a 1-parameter group of symplectomorphisms of (P, w), for every t 2 R, the functions (e ) j , t X t X J J 1 1 with ` = 1, . . . , k and (e ) J , ` = 1, . . . , k are action angle coordinates on (e ) (U). Choose ` Axioms 2020, 9, 125 20 of 33 t X t > 0 so that e (T) U. Suppose that h = t + h, where h 2 [0, t). Observe that for t 2 [0, t) the action angle coordinates (j , . . . , p , J , . . . , J ) in U satisfy 1 k 1 k j t if ` = 1 tX 1 t X J J 1 1 (e ) j = and (e ) J = q . ` ` ` j if ` = 2, 3, . . . , k t X Hence, (e ) j = j t and 1 1 h X (t+h)X t X h X J J J J 1 1 1 1 (e ) j = (e ) j = (e ) (e ) j 1 1 1 t X t X J J 1 1 = (e ) (j h) = (e ) (j ) h, 1 1 because h is constant. Moreover, Z Z Z 1 1 1 t X t X J J 1 1 e ) j d e ) J = (j t) dJ = j dJ t = j t. 1 1 1 1 1 1 1 0 0 0 Similarly, Z Z 1 1 h X h X t X t X J J J J 1 1 1 1 e ) j d e ) J = e ) j h d e ) J 1 1 1 1 0 0 Z Z 1 1 t X t X t X J J J 1 1 1 = e ) j d e ) J h d e ) J 1 1 1 0 0 Z Z 1 1 = p dJ t h = p dJ h = (n 1)h, 1 1 1 1 1 0 0 h X because T is a Bohr–Sommerfeld torus T with quantum numbers (n , . . . , n ). Thus, e (T) is a n 1 Bohr–Sommerfeld torus corresponding to the quantum numbers (n 1, n , . . . , n ). This argument 1 2 k may be extended to cover the case where h= kt + h for any positive integer k and h 2 [0, t). 3.4. Singularity of Toral Polarization in Completely Integrable Hamiltonian Systems A completely integrable Hamiltonian system on a symplectic manifold (P, w) of dimension 2k is given by k functions H , ..., H 2 C (P), which Poisson commute with each other, and are independent 1 k on the open dense subset P of P. We assume that, for every i = 1, ..., k, and each x 2 P , the maximal 0 0 integral curve of X through x is periodic with period T (x) > 0. The complement PnP of P in P is H 0 0 the set of singular points of the real polarization D = spanfX , ..., X g of (P, w). H H 1 k Applying the arguments of Section 3.1 and the beginning of Section 3.2, we obtain the set S = f Mg of Bohr–Sommerfeld tori M in P. Each M is an integral manifold of D, which admits a covariantly constant section s : M ! L . The section s is determined up to a non-zero constant. M j M M The direct sum S = fCs g M2S is the space of quantum states of the Bohr–Sommerfeld theory. Each Bohr–Sommerfeld torus M represents a one-dimensional subspace of quantum states. The collection fs g is a basis of S, and Q s = H s . H M ij M M Let S = f M 2 S j M P g be the set of the Bohr–Sommerfeld tori in P . Then, 0 0 0 S = fCs g 0 M M2S 0 Axioms 2020, 9, 125 21 of 33 is the space of quantum states of the system, which are described by the Bohr–Sommerfeld quantization of P . The collection fs j M P g is a basis of S , and 0 M 0 0 Q s = H s H M ij M M for every M P . The restriction D of D to P is a toral polarization of (P , w ) discussed earlier. The functions 0 0 jP jP 0 0 H , . . . , H 2 C (P), which deﬁne the system, give rise to action–angle coordinates (j, J) on P , 0 0 k1 where for each i = 0, . . . , k 1, j = H jT and J is the multivalued angle coordinate corresponding i i ijP ijP 0 0 to j . Since we deal with the single set of action–angle coordinates, most of the analysis of Section 3.3.1 applies to this problem. As in Section 3.3.2, Equation (54), for i = 1, . . . , k we introduce the notation X = = X . i J ¶j Each X is a locally Hamiltonian vector ﬁeld on P . However, since P 6= T T , we cannot assume that i 0 0 the vector ﬁeld X is complete. In terms of action–angle coordinates (j, J) on P , the Bohr–Sommerfeld tori in P are given by 0 0 equation j = (j , . . . , j ) = (n h, . . . , n h) = nh, 1 k 1 k k k where n = (n , . . . , n ) 2 Z . For n 2 Z , 1 k M = M = fx 2 P j j(x) = nhg (79) n 0 (n ,...,n ) 1 k denotes the Bohr–Sommerfeld torus in P corresponding to the eigenvalue n of j. If nh is not in the spectrum of j, then M = ?. In a trivialization L = C P of the complex line bundle L restricted to n 0 jP P , for each M 6= Æ, we can choose 2pi(n J +...+n J ) 1 1 k k s : M ! L : (J , . . . , J ) 7! (J , . . . , J ), e , (80) n n 1 k 1 k form a basis in the space S of quantum states in P . 0 0 Claim 5 implies the following Corollary 1. If, for every x 2 P and each i = 1, ..., k, Planck constant h is in the domain of the maximal hZ tX 2piJ h lift X i i i integral curve e (x) of X starting at x, then e = e e is well deﬁned. Under the assumptions of Corollary 1, we may follow the arguments of Section 3.3.1 leading to Equation (70). Applied to the case under consideration, it may be rewritten as follows. For every n = (n , ..., n ) 2 Z , such that M P , one has 1 n 0 a s = s , (n ,...,n ) (n ,...,n ,n 1,n ,...n ) i 1 k 1 i1 i i+1 k if M P , and (n ,...,n ,n 1,n ,...n ) 0 1 i1 i i+1 k b s = s , X (n ,...,n ) (n ,...,n ,n +1,n ,...n ) 1 k 1 i1 i i+1 k if M P . (n ,...,n ,n +1,n ,...n ) 1 i1 i i+1 k It remains to extend the action of a and b given above to all states in S. This involves a study X X i i of the integral curves of X on P, which originate or end at points in the singular set PnP . i 0 Axioms 2020, 9, 125 22 of 33 Suppose we manage to extend the action of the shifting operators to all states in S. Monodromy occurs when, there exist loops in the local lattice of Bohr–Sommerfeld tori such that for some a , . . . , a 2 f1, . . . , kg the mapping 1 m hX hX a a m 1 e e : M ! M n n need not be the identity on M . In this case shifting operators are multivalued, and there exists a phase ij factor e such that ij (a a )s = e s . X X n n a a m 1 Given a non-zero section s 2 S supported on a Bohr–Sommerfeld torus M. Any maximal family n n k 1 B = f(a a s) 2 S n , ...n 2 Zg (81) 1 k X X k 1 of sections in S, such that no two sections in B are supported on the same Bohr–Sommerfeld torus, is a linear base of S. We can deﬁne a scalar producth j i on S as follows. First, assume that basic sections supported on different Bohr–Sommerfeld tori are perpendicular to each other. Then, assume that for n n k 1 every a a s 2 B, X X D E n n n n k 1 k 1 a a s j a a s = hs j si . (82) X X X X k 1 k 1 This deﬁnition works even in the presence of monodromy. The completion of S with respect to this scalar product yields a Hilbert space H of quantum states in the Bohr–Sommerfeld quantization of the completely integrable Hamiltonian system under consideration. Example: The 2-d Harmonic Oscillator We consider the harmonic oscillator with 2 degrees of freedom, see [9]. Its conﬁguration 2 2 2 2 space is R with coordinates q = (q , q ). Its phase space P = T R = R R has coordinates (p, q) = (q , q , p , p ) with symplectic form w = d p ^ dq + d p ^ dq . The 2-d harmonic oscillator is 1 2 1 2 1 1 2 2 1 2 2 2 2 e e completely integrable with integrals the Hamiltonian H with H(p, q) = ( p + p + q + q ) and the 2 1 2 1 2 e e angular momentum J with J(p, q) = q p q p . 1 2 2 1 The change of variables 0 1 0 1 0 1 0 1 q x 1 0 0 1 q 1 1 1 B C B C B C B C q x 1 0 0 1 q B 2C B 2C B C B 2C y : 7! = p B C B C B C B C @ p A @h A @ 0 1 1 0A @ p A 1 1 1 p h 0 1 1 0 p 2 2 2 is symplectic, that is, w = dh ^ dx + dh ^ dx , preserves the diagonal form of the Hamiltonian H = 1 1 2 2 1 2 2 2 2 1 2 2 2 2 e e (x + h + x + h ) = y H, and diagonalizes the angular momentum J = (x + h x h ) = y J. 2 1 1 2 2 2 1 1 2 2 The functions 1 2 2 A = (x + h ) = H + J, 2 1 1 (83) 1 2 2 A = (x + h ) = H J 2 2 2 are action variables for the two-dimensional harmonic oscillator. The corresponding angle variables are q and q , respectively. In the variables (A, q) = ( A , A , q , q ), the symplectic form w is dA ^ dq + 1 2 1 2 1 2 1 1 d A ^ dq . The rescaled action angle coordinates (j, J) = (j , j , J , J ), used previously, are given by 2 2 1 2 1 2 j = 2p A and J = q /2p for i = 1, 2. i i i i The Bohr–Sommerfeld tori M = fx 2 T R j j = nh, j = mhg are parameterized by two n,m 1 2 integers n, m. The corresponding basic sections are 2pi(nJ +mJ ) 1 2 s : M ! L : (J , J ) 7! e , n,m n,m 1 2 Axioms 2020, 9, 125 23 of 33 see Equation (80). Equations (83) yield 1 1 H = ( A + A ) = (j /2p + j /2p), 1 2 1 2 2 2 1 1 J = ( A A ) = (j /2p j /2p). 2 2 1 1 2 2 Hence, the quantum operators Q and Q act on s as follows. H J n,m Q s = H s = (j /2p + j /2p) s H n,m n,m 1 2 n,m j M j M n,m 2 n,m 1 1 = (nh/2p + m/2p)s = (n + m)h ¯ s , n,m n,m 2 2 where h ¯ = h/2p and Q s = J s = (n m)h ¯ s . J n,m n,m n,m j M n,m 2 The regular part of P = T R is P = fx 2 T R j j (x) > 0 and j (x) > 0g. 0 1 2 The singular part of P = T R consists of three strata S = fx 2 T R j j (x) > 0 and j (x) = 0g, 1 1 2 S = fx 2 T R j j (x) = 0 and j (x) > 0g, 2 1 2 S = fx 2 T R j j (x) = 0 and j (x) = 0g. 0 1 2 S is the origin of T R , while S and S are cylinders parameterized by (j , J ) and (j , J ), 0 1 2 1 1 2 2 respectively. As before, for i = 1, 2, we consider the locally Hamiltonian vector ﬁelds X = = X . i J ¶j The conditions of Corollary 1 are satisﬁed. Hence, in P , we get shifting operators a s = s , provided n > 1 and m > 0, n,m X n1,m b s = s , provided n > 0 and m > 0, X n,m n+1,m a s = s , provided n > 0 and m > 1, n,m X n,m1 b s = s , provided n > 0 and m > 0. X n,m n,m+1 tX Next, we have to consider limits as integral curves of X . Note that the integral curve e (x ) 1 0 hX of X , originating at x 2 M , after time t = h reaches x = e (x ) 2 M . Moreover, the integral 1 1,m 1 0 0,m tX curve e (x ) of X originating at x 2 M , for n > 0, after time t = h reaches M and after time 0 1 n,0 n1,0 tX t = nh reaches the origin M . Similarly, the integral curve e (x) of X originating at x 2 M 0,0 1 n,0 after time t = h reaches M and after time t = kh it reaches M for every k > 0. This argument n+1,0 n+k also applies to X . It enlarges the above table of shifting operators as follows. a s = s , provided m 0, X 1,m 0,m a s = s , provided n 0. X n,1 n,0 Since X (x) is unbounded as j ! 0 , it is not possible to discuss integral curves of X starting at 1 1 1 points in M . However, for n > 0, 0,n b s = s and a s = s . n,m n,m X n+1,m X n+1,m 1 1 Axioms 2020, 9, 125 24 of 33 Thus, b shifts in the opposite direction to a . Similarly, b shifts in the opposite direction to a . X X X X 1 1 2 2 It is natural to extend these relations to the boundary and assume that b s = s , provided m 0, 0,m X 1,m b s = s , provided n 0. X n,0 n,1 The actions of the lowering operators a on states s and a on states s not deﬁned, but they X 0m X m0 1 2 never occur in the theory. Therefore, we may assume that a s = 0, and a s = 0. X 0,m X n,0 1 2 3.5. Monodromy Suppose that U = fU g is a good covering of P such that for every i 2 I the chart (U , y ) is i i i i2 I the domain of a local trivialization of the toral bundle r : P ! B, associated to the ﬁbrating toral polarization D of P, given by the local action angle coordinates k i i i i i i r : U ! V T : p 7! y ( p) = (j , J ) = (j , . . . , j , J , . . . , J ) jU i i i 1 k 1 k k i i with (r ) (w ) = dj ^ dJ . We suppose that the set S of Bohr–Sommerfeld tori on P has the jU jU `=1 i i ` ` local lattice structure fL g of Section 3.3. i i2 I 0 0 Let p and p 2 P and let g : [0, 1] ! P be a smooth curve joining p to p . We can choose a ﬁnite good subcovering fU g of U such that g([0, 1]) [ U , where g(0) U and g(1) 2 U . 1 N k k=1 k k=1 Using the fact that the local lattices fL g are compatible, we can extend the local action functions k k=1 1 N j on V = y (U ) B to a local action function j on V B. Thus, using the connection E 1 1 1 N (see Corollary A1), we may parallel transport a Bohr–Sommerfeld torus T U along the curve g to n 1 a Bohr–Sommerfeld torus T 0 U (see Claim 7). The action function at p , in general depends on the n N path g. If the holonomy group of the connectionE on the bundle r : P ! B consists only of the identity element in Gl(k,Z), then this extension process does not depend on the path g. Thus, we have shown Claim 9. If D is a ﬁbrating toral polarization of (P, w) with ﬁbration r : P ! B and B is simply connected, then there are global action angle coordinates on P and the Bohr–Sommerfeld tori T 2 S have a unique quantum number n. Thus, the local lattice structure of S is the lattice Z . If the holonomy of the connection E on P is not the identity element, then the set S of Bohr–Sommerfeld tori is not a lattice and it is not possible to assign a global labeling by quantum numbers to all the tori in S . This difﬁculty in assigning quantum numbers to Bohr–Sommerfeld tori has been known to chemists since the early 1920s. Modern papers illustrating it can be found in [18,19]. We give a concrete example where the connectionE has nontrivial holonomy, namely, the spherical pendulum. Example: Spherical Pendulum The spherical pendulum is a completely integrable Hamiltonian system ( H, J, T S , da ), T S 2 3 2 where T S = f(q, p) 2 T R hq, qi = 1 & hq, pi = 0g is the cotangent bundle of the 2-sphere S with h , i the Euclidean inner product on R , see [20]. The Hamiltonian is 2 1 H : T S ! R : (q, p) 7! h p, pi +hq, e i, T 3 where e = (0, 0, 1) 2 R and the e -component of angular momentum is J : T S ! R : (q, p) 7! q p q p . 1 2 2 1 Axioms 2020, 9, 125 25 of 33 The energy momentum map of the spherical pendulum is 2 2 EM : T S ! R R : (q, p) 7! H(q, p), J(q, p) . Here, R is the closure in R of the set R of regular values of the integral map EM. The point (1, 0) 2 R is an isolated critical value of EM. Thus, the set R has the homotopy type of S and is not simply connected. Every ﬁber of 1 2 EM 1 : EM (R) ! R R jEM (R) 2 2 over a point (h, j) 2 R is a smooth 2-torus T , see chapter V of [21]. At every point of T S n h,j 1 1 (EM (1, 0)[EM ¶R) there are local action angle coordinates ( A , A , J , J ). The actions are 1 2 1 2 A = A EM and A = A EM. Here, 1 1 2 2 Z + 2 p 2(h p )(1 p ) j 1 1 1 A (h, j) = dp , 1 1 p 1 p 2 2 where p 2 [1, 1] and 2(h p )(1 (p ) ) j = 0, and 1 1 1 A (h, j) = j; e e while the angles are J = J EM and J = J EM, where 1 1 2 2 Z + j 1 1 J (h, j) = dp 1 1 p 2 2 2 (1 p ) 2(h p )(1 p ) j 1 1 and J (h, j) = , 2p where t is the time parameter of the integral curves of the vector ﬁeld X on the 2-torus T , which are h,j periodic of period 2p, see Section 2.4 of [20]. The action map A : R R ! R R : (h, j) 7! A (h, j),A (h, j) 0 2 is a homeomorphism of Rnf(1, 0)g onto (R R)nf( , 0)g, which is a real analytic diffeomorphism of Rnfj = 0g onto R (Rnf0g , see Fact 2.4 in [20]. >0 For every (n, m) 2 Z Z, the Bohr–Sommerfeld tori are 2 2 T = f(q, p) 2 T S A (q, p) = n h & A (q, p) = m hg. 1 2 m,n The ﬁbers of EM corresponding to the dark points in Figure 1 are the Bohr–Sommerfeld tori. The basic sections of the quantum line bundle p : L ! T S are 2 2 2pi(nJ +mJ ) 1 2 s : T T S ! L : (J , J ) 7! J , J , e . n,m 1 2 1 2 n,m The family of sections B = fs (n, m) 2 Z Zg forms a basis of quantum states of the n,m 0 Bohr–Sommerfeld theory of the spherical pendulum. Let H be the Hilbert space of quantum states for Axioms 2020, 9, 125 26 of 33 Figure 1. The Bohr–Sommerfeld quantum states of the spherical pendulum in R. which B is an orthogonal basis. The Bohr–Sommerfeld energy momentum spectrum S of the spherical pendulum is the range of the map A : Z Z R R ! R : 0 0 jZ Z - - (n, m) 7! h(n h , m h), j(n h, m h) = h (n) h, m h . (n, m) 2 Z Z are the quantum numbers of the spherical pendulum. In terms of actions A and A , we may write H = H( A , A ). Hence, the quantum operators Q 1 2 1 2 H and Q act on the basic sections s as follows m.n Q s = H 2 s = h (n) hs H n,m n,m m n,m jT n,m and Q s = J s = m hs . J n,m n,m n,m jT n,m The regular part of T S is 1 1 S = EM (R) = EM A (fA > 0 & A 6= 0g) . 0 1 2 The singular part of T S consists of six strata: 1 1 S = EM A (f(0, 0)g , S = A (f(4/p, 0)g , S = A (f(A , 0) 0 < A < 4/pg), 3 1 1 S = A (f(A , 0) 4/p < A g), 4 1 1 S = A (f(0,A ) A > 0g), 5 2 2 S = A (f(0,A ) A < 0g). 6 2 2 1 Axioms 2020, 9, 125 27 of 33 The stratum S is the point (0, 0,1, 0, 0, 0) 2 T S ; while the stratum S is the point (0, 0, 1, 0, 0, 0). 1 2 The stratum S is the subset of T S , where A 2 (0, 4/p) and A = 0, which is a cylinder 3 1 2 parameterized by ( A , J ); while S is the subset where A 2 (4/p, ¥) and A = 0, which is a 1 1 4 1 2 cylinder parameterized by ( A , J ). The stratum S is the subset of T S where A = 0 and A > 0, 1 1 5 1 2 which is a cylinder parameterized by ( A , J ); while S is the subset where A = 0 and A < 0, 2 2 6 1 2 which is a cylinder parameterized by ( A , J ). 2 2 The conditions of Corollary 1 are satisﬁed. For i = 1, 2 let X = X . In the regular stratum S we i J 0 get the shifting operators a s = s , provided n > 1 and m 6= 0 n,m X n1,m b s = s , provided n > 0 and m 6= 0 X n,m n+1,m a s = s , provided n > 0 and m 6= 1 n,m X n,m1 b s = s , provided n > 0 and m 6= 0 X n,m n,m+1 Arguing as in the example of the 2-d harmonic oscillator, we can extend the above relations to b s = s , provided m 6= 0 X 0,m 1,m b s = s , provided m 6= 0. X n,0 n,1 In addition, we may assume that a s = 0 and a s = 0. X 0,m X n,0 1 2 Since the are no global action angle coordinates, the action function A on R is multi-valued. After encircling the point (1, 0), the quantum number of the Bohr–Sommerfeld torus represented by the upper right hand vertex of the rectangle on the h-axis, see Figure 2, becomes the quantum number T 1 1 of the upper right hand vertex of the parallelogram formed by applying M = to the original 0 1 rectangle, which is the transpose of the monodromy matrix M of the spherical pendulum. −1 Figure 2. Using the shifting operators to show that the quantized spherical pendulum has monodromy. The holonomy of the connection E is called the monodromy of the ﬁbrating toral polarization D on (P, w) with ﬁbration r : P ! B. 1 Axioms 2020, 9, 125 28 of 33 e e Corollary 2. Let B be the universal covering space of B with covering map P : B ! B. The monodromy map M, which is a nonidentity element holonomy group of the connection E on the bundle r sends one sheet of the universal covering space to another sheet. Proof. Since the universal covering space B of B is simply connected and we can pull back the symplectic manifold (P, w) and the ﬁbrating toral distribution D by the universal covering map to a e e e e symplectic manifold (P, we) and a ﬁbrating toral distribution D with associated ﬁbration re : P ! B. The connection E on the bundle r pulls back to a connection E on the bundle re. Let g be a closed curve on B and let M be the holonomy of the connection E on B along g. Then, g lifts to a curve ge k k on B, which covers g, that is, re ge = g. Thus, parallel transport of a k-torus T = R /Z , which is an integral manifold of the distribution D, along the curve ge gives a linear map M of the lattice Z deﬁning the k-torus M(T). The map M is the same as the linear map M of Z into itself given by parallel transporting T, using the connectionE , along the closed g on B because the connectionE is the pull back of the connection E by the covering map r. The closed curve g in B represents an element of the fundamental group of B, which acts as a covering transformation on the universal covering space e e B that permutes the sheets (= ﬁbers) of the universal covering map P. In the spherical pendulum, the universal covering space R of Rnf(1, 0)g is R . If we cut R by the line segment ` = f(h, 0) 2 R h > 1g, then R = Rn ` is simply connected and hence represents one sheet of the universal covering map of R. For more details on the universal covering map, see 1 1 [22]. The curve chosen in the example has holonomy M = . It gives a map of R into itself, 0 1 which sends R to the adjacent sheet of the covering map. Thus, we have a rule how the labelling of the Bohr–Sommerfeld torus T , corresponding to (h, j) 2 R , changes when we go to an (n ,n ) 1 ! adjacent sheet, which covers R , namely, we apply the matrix M to the integer vector . Since our chosen curve generates the fundamental group of Rnf(1, 0)g, we know what the quantum numbers of Bohr–Sommerfeld are for any closed curve in Rnf(1, 0)g, which encircles the origin. Author Contributions: The author R.C. conceptualized and wrote Section 2, the appendix, and the spherical pendulum example. The author J.S. conceptualized and wrote the rest. All authors have read and agreed to the published version of the manuscript. Funding: This reasearch received no extenal funding. Conﬂicts of Interest: The authors declare no conﬂict of interest. Appendix A We return to study the symplectic geometry of a ﬁbrating toral polarization D of the symplectic manifold (P, w) in order to explain what we mean by its local integral afﬁne structure, see [23]. We assume that the integral manifolds f M g of the Lagrangian distribution D on P form a p2P smooth manifold B such that the map r : P ! B : p 7! M is a proper surjective submersion. If the distribution D has these properties we refer to it as a ﬁbrating polarization of (P, w) with associated ﬁbration r : P ! B. Lemma A1. Suppose that D is a ﬁbrating polarization of (P, w). Then, the associated ﬁbration r : P ! B has an Ehresmann connection E with parallel translation. Thus, the ﬁbration r : P ! B is locally trivial bundle. Proof. We construct the Ehresmann connection as follows. For each p 2 P let (U, y) be a Darboux chart for (P, w). In other words, (y ) (w ) is the standard symplectic form w on TV, jU 2k 2k where V = y(U) R with y( p) = 0. In more detail, for every u 2 U there is a frame #(u) of Axioms 2020, 9, 125 29 of 33 ¶ ¶ ¶ ¶ 2k P at u, whose image under T y is the frame #(v) = , . . . , , , . . . , of T V = R , u v ¶x ¶x ¶y ¶y 1 v k v 1 v k v where v = y(u), such that ¶ ¶ ¶ ¶ w (v) , = w (v) , = 0 2k 2k ¶x ¶x ¶y ¶y i v j v i v j v and ¶ ¶ w (v) , = d . 2k i j ¶x ¶y i j v v For u 2 M \ U, we see that l = T y(T M ) is a Lagrangian subspace of the symplectic vector p v u u p space T V, w (v) . Letf g be a basis of l withfdz (v)g the corresponding dual basis of l . v v 2k j v ¶z j=1 j=1 Extend each covector dz (v) by zero to a covector dZ (v) in T V, that is, extend the basis fdz (v)g j j j j=1 < dZ (v) = dz (v), for j = 1, . . . , k j jl j # of l to a basisfdZ (v)g of T V, where Since w (v) : T V ! T V is a linear j v v v v : 2k j=1 dZ (v) = 0, for j = k + 1, . . . , 2k. j jl isomorphism with inverse w (v) for every v 2 V, we see that the collection 2k ¶ k f = w (v)(dZ (v))g 2k ¶w j=1 j v of vectors in T V spans an k-dimensional subspace m . We now show that m is a Lagrangian subspace v v v of T V, w (v) . By deﬁnition v 2k ¶ ¶ ¶ ¶ ¶ w (v) , = w (v) = dZ (v) = 0. 2k i 2k ¶w ¶w ¶w ¶w ¶w i v j v i v j v j v The last equality above follows because 2 / l . To see this we note that ¶w ¶ ¶ ¶ ¶ w (v) , = dZ (v) = dz (v) = 1. 2k j j ¶w ¶z ¶z ¶z j v j v j v j v The Lagrangian subspace m is complementary to the Lagrangian subspace l , that is, T V = l m v v v v v for every v 2 V. Consequently, hor = T y m is a Lagrangian subspace of T U, w(u) , which is complementary u v v u to the Lagrangian subspace T M . Since the mapping hor : U ! TU : u 7! hor is smooth u p u jU and has constant rank, it deﬁnes a Lagrangian distribution hor on U. Hence, we have a jU Lagrangian distribution hor on (P, w). Since T M is the tangent space to the ﬁber r r( p) = M , u p p the distribution ver : U ! TU : u 7! ver = T M = l deﬁnes the vertical Lagrangian distribution u u p v jU ver on P. Because ver = ker T r, it follows that T r(hor ) = T B. Hence, the linear mapping u u u u r(u) T r : hor ! T B is an isomorphism. Since T P = hor ver for every p 2 P and the u u p p p jhor r(u) mapping T r : hor ! T B is an isomorphism for every p 2 P, the distributions hor and ver on p p jhor r( p) P deﬁne an Ehresmann connection E for the Lagrangian ﬁbration r : P ! B. tX Let X be a smooth complete vector ﬁeld on B with ﬂow e . Because the linear mapping T r : jhor hor ! T B is bijective, there is a unique smooth vector ﬁeld liftX on P, called the horizontal lift of X, r( p) t liftX which is r-related to X, that is, T r liftX( p) = X r( p) for every p 2 P. Let e be the ﬂow of liftX. t liftX tX Then, r(e ) = e (r( p)). Let s : W B ! P be a smooth local section of the bundle r : P ! B. Deﬁne the covariant derivative r s of s with respect to the vector ﬁeld X by t lift X t X (r s)(w) = e s(e (w)) dt t=0 Axioms 2020, 9, 125 30 of 33 for all w 2 W. Because the bundle projection map r is proper, parallel transport of each ﬁber of the bundle r : P ! B by the ﬂow of liftX is deﬁned as long as the ﬂow of X is deﬁned. Because the Ehresmann connection E has parallel transport, the bundle presented by r is locally trivial, see pp. 378–379, [21]. Claim A1. If D is a ﬁbrating polarization of the symplectic manifold (P, w), then for every p 2 P the integral manifold of D through p is a smooth Lagrangian submanifold of P, which is an k-torus T. In fact T is the ﬁber over r( p) of the associated ﬁbration r : P ! B. We say that D is a ﬁbrating toral polarization of (P, w) if it satisﬁes the hypotheses of Claim A1. The proof of Claim A1 requires several preparatory arguments. ¥ ¥ Let f 2 C (B). Then, r f 2 C (P). Let X be the Hamiltonian vector ﬁeld on (P, w) with r f Hamiltonian r f . We have Lemma A2. Every ﬁber of the locally trivial bundle r : P ! B is an invariant manifold of the Hamiltonian vector ﬁeld X . r f Proof. We need only show that for every p 2 P and every q 2 M , we have X (q) 2 T M . Let Y be p r f q p tY a smooth vector ﬁeld on the integral manifold M with ﬂow e . Then, tY tY r f e (q) = f r(e (q)) = f r( p) , tY since e maps M into itself. So tY 0 = r f e (q) = L (r f )(q) = d r f (q)Y(q) dt t=0 = w(q) X (q), Y(q) . r f However, T M is a Lagrangian subspace of the symplectic vector space (T P, w(q)). Consequently, q p q X (q) 2 T M . r f q p Since the mapping r : P ! B is surjective and proper, for every b 2 B the ﬁber r (b) is a smooth t X r f compact submanifold of P. Hence, the ﬂow e of the vector ﬁeld X is deﬁned for all t 2 R. r f Lemma A3. Let f , g 2 C (B). Then, fr f , r gg = 0. Proof. For every p 2 P and every q 2 M from Lemma A2 it follows that X (q) and X (q) lie in r f r g T M . Because M is a Lagrangian submanifold of (P, w), we get q p p 0 = w(q) X (q), X (q) = fr f , r gg(q). (A1) r g r f Since P = q M , we see that (A1) holds for every p 2 P. p2P Proof of Claim A1. From Lemma A3 it follows that r (C (B)),f , g, is an abelian subalgebra t of the Poisson algebra (C (P),f , g,). Since the bundle projection mapping r : P ! B is surjective and dim B = k, the algebra t has k generators, say, fr f g , whose differentials at q span T (r (b)) i q i=1 t X 1 r f for every b 2 B and every q 2 r (b). Using the ﬂow e of the Hamiltonian vector ﬁeld X on r f (P, w) deﬁne the R -action t X t X k 1 k r f r f 1 k F : R P ! P; t = (t , . . . , t ), p 7! e ( p), . . . , e ( p) (A2) 1 k Since span fX (q)g = T (r (b)) and each ﬁber is connected, being an integral manifold of the r f 1ik i k 1 distribution D, it follows that the R -action F is transitive on each ﬁber r (b) of the bundle r : P ! B. 1 k k Thus, r (b) is diffeomorphic to R /P , where P = ft 2 R F (q) = qg is the isotropy group at q. q q t Axioms 2020, 9, 125 31 of 33 1 k k If P = f0g for some q 2 P, then the ﬁber r r(q) would be diffeomorphic to R /P = R . However, q q this contradicts the fact the every ﬁber of the bundle r : P ! B is compact. Hence, P 6= f0g for every k 1 q 2 P. Since R /P is diffeomorphic to r (b), they have the same dimension, namely, k. Hence, P is q q k k 1 k k a zero dimensional Lie subgroup of R . Thus, P is a rank k lattice Z . Thus, the ﬁber r (b) is R /Z , which is an afﬁne k-torus T . We now apply the action angle theorem, see chapter IX of [21], to the ﬁbrating toral Lagrangian polarization D of the symplectic manifold (P, w) with associated toral bundle r : P ! B to obtain a more precise description of the Ehresmann connection E constructed in Claim A1. For every p 2 P there is an open neighborhood U of the ﬁber r r( p) in P and a symplectic diffeomorphism 1 k k k y : U = r (V) P ! V T R T : u 7! (j, J) = (j , . . . , j , J , . . . , J ) 1 k 1 k such that r : U P ! V R : u 7! (p y)(u) = j, jU k k is the momentum mapping of the HamiltonianT -action on (U, w ). Here, p : VT ! V : (j, J) ! jU 1 k 1 j. Thus, the bundle r : P ! B is locally a principal T -bundle. Moreover, we have (y ) w = jU dj ^ dJ . i i i=1 Corollary A1. Using the chart (U, y) for action angle coordinates (j, f), the Ehresmann connection E gives jU an Ehresmann connection E on the bundle p : V T ! V deﬁned by jVT ¶ ¶ ver = span f g and hor = span f g. v v 1ik 1ik ¶J ¶j v=y(u) v=y(u) i i Proof. This follows because ¶ ¶ T y ver = span f g and T y hor = span f g, u u p u 1ik 1ik ¶J ¶j v=y(u) v=y(u) i i for every u 2 U. From the preceding equations for every u 2 U we have ver = span fX (u)g r (j ) 1ik k k and hor = span fX (u)g. Here, p : V T ! T : (j, j) 7! j. u 2 (p y) (J ) 1ik 2 i Corollary A2. The Ehresmann connection E on the locally trivial toral Lagrangian bundle r : P ! B is ﬂat, that is, r s = 0 for every smooth vector ﬁeld X on B and every local section s of r : P ! B. Proof. In action angle coordinates a local section section s of the bundle r : P ! B is given by k ¶ t X s : V ! V T : j 7! j, s(j) . Let X = for some 1 ` k with ﬂow e . Let liftX be the ¶j horizontal lift of X with respect to the Ehresmann connection E on the bundle p : V T ! V. VT Thus, for every j 2 V we have t liftX tX (r s)(j) = e s(e (j)) dt t=0 t liftX tX = e s(j(t)) , where e (j) = j(t) dt t=0 t liftX = e j, s(j) , since j for 1 i n are integrals of X dt t=0 t liftX tX = j(t), s(j(t)) , since p e (j, s(j)) = e (j) dt t=0 = 0. Axioms 2020, 9, 125 32 of 33 This proves the corollary, since every vector ﬁeld X on W B may be written as c (j) for some i=1 ¶j ¥ i ¶ c 2 C (W) and the ﬂow fj g of f g on V pairwise commute. t i=1 ¶j i=1 Claim A2. Let r : P ! B be a locally trivial toral Lagrangian bundle, where (P, w) is a smooth symplectic manifold. Then, the smooth manifold B has an integral afﬁne structure. In other words, there is a good open covering fWg of B such that the overlap maps of the coordinate charts (W , j ) given by i i i i2 I 1 k k j = j j : V \ V R ! V \ V R , i` ` i ` i ` where j (W ) = V , have derivative Dj (v) 2 Gl(k,Z), which does not depend on v 2 V \ V . i i i i` i ` Proof. Cover P by U = fU g , where (U , y ) is an action angle coordinate chart. Since every open i i i i2 I covering of P has a good reﬁnement, we may assume that U is a good covering. Let W = r(U ). i i Then, W = fWg is a good open covering of B and (W , j = p y ) is a coordinate chart for B. i i i i i2 I By construction of action angle coordinates, in V \ V the overlap map j sends the action coordinates i ` i` i ` j in V \ V to the action coordinates j in V \ V . The period lattices P 1 and P 1 are equal i ` i ` i ` y (j ) y (j ) i ` i ` since for some p 2 W \ W we have y ( p) = j and y ( p) = j . Moreover, these lattices do not depend i ` i ` k ¶ on the point p. Thus, the derivative Dj (j) sends the lattice Z spanned by f g into itself. i` i ¶j i=1 Hence, for every j 2 W \ W the matrix of Dj (j) has integer entries, that is, it lies in Gl(k,Z) and i ` i` the map j 7! Dj (j) is continuous. However, Gl(k,Z) is a discrete subgroup of the Lie group Gl(k,R) i` and W \ W is connected, since W is a good covering. Thus, Dj (j) does not depend on j 2 W \ W . i ` i` i ` Corollary A3. Let g : [0, 1] ! B be a smooth closed curve in B. Let P : [0, 1] ! P be parallel translation along g using the Ehresmann connection E on the bundle r : P ! B. Then, the holonomy group of the k-toral k k k ﬁber T = T is induced by the group Gl(k,Z)nZ of afﬁne Z-linear maps of Z into itself. g(0) References 1. Heisenberg, W. Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Phys. 1925, 33, 879–893. 2. Kostant, B. Quantization and unitary representations. I. Prequantization. In Lectures in Modern Analysis and Applications III; Lecture Notes in Mathematics; Springer: Berlin, Germany, 1970; Volume 170, pp. 87–208. 3. Souriau, J.-M. Quantiﬁcation géométrique. In Physique Quantique et Géométrie; Hermann: Paris, France, 1988; pp. 141–193. 4. Blattner, R.J. 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Global properties of integrable Hamiltonian systems. Reg. Chaotic Dyn. 2008, 13, 602–644. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Shifting Operators in Geometric Quantization

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DOI: 10.3390/axioms9040125
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