Quantum mechanics in magnetic backgrounds with manifest symmetry and locality
aa r X i v : . [ h e p - t h ] M a r Quantum mechanics in magneticbackgrounds with manifest symmetry andlocality
Joe Davighi, Ben Gripaios, and Joseph Tooby-Smith Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Wilberforce Road, Cambridge, UK Cavendish Laboratory, University of Cambridge, J. J. Thomson Ave, Cambridge, UKEmails: [email protected], [email protected] and [email protected]
Abstract:
The usual methods for formulating and solving the quantum mechanics of aparticle moving in a magnetic field respect neither locality nor any global symmetries whichhappen to be present. For example, Landau’s solution for a particle moving in a uniformmagnetic field in the plane involves choosing a gauge in which neither translation nor rotationinvariance are manifest. We show that locality can be made manifest by passing to a redundantdescription in which the particle moves on a U (1)-principal bundle over the original configurationspace and that symmetry can be made manifest by passing to a corresponding central extensionof the original symmetry group by U (1). With the symmetry manifest, one can attempt tosolve the problem by using harmonic analysis and we provide a number of examples where thissucceeds. One is a solution of the Landau problem in an arbitrary gauge (with either translationinvariance or the full Euclidean group manifest). Another example is the motion of a fermionicrigid body, which can be formulated and solved in a manifestly local and symmetric way via aflat connection on the non-trivial U (1)-central extension of the configuration space SO (3) givenby U (2). 1 ontents Consider a particle moving on a smooth, connected, manifold M in the presence of some back-ground magnetic field. Suppose furthermore that the dynamics is invariant under some, con-nected, Lie group G of global symmetries acting smoothly on M .The study of the quantum mechanics of such a system is complicated by two well-knownfacts. The first complication is that it is, in general, not possible to write down a term in thelagrangian representing the magnetic field that is valid globally on M . Instead, the best thatone can do is to cover M by overlapping patches and to use multiple lagrangians, each of whichis valid only locally on some patch. The most famous example, due to Dirac [1] and solvedby Tamm [2] (see also [3, 4]), is given by the motion of an electrically-charged particle in thepresence of a magnetic monopole, but we will see that there exists an example that is arguablyeven simpler (and certainly more prevalent in everyday life!), given by the motion of a rigidbody which happens to be a fermion.This latter example is interesting for another reason, which is that it shows that our set-upincludes systems in which there is no apparent magnetic field, but rather a vector potential isbeing used to encode a global topological effect – spin, in the case at hand – in a manifestlylocal way. Thus, we will be able to write a local term in the lagrangian that accounts for theextra factor of − G , but rather will shift by a total derivative. Perhaps the simplestexample, made famous by Landau [5], is given by the motion of a particle in a plane in thepresence of a uniform magnetic field, where there is no choice of gauge such that the lagrangianis invariant under translations in more than one direction.At the classical level, neither of these complications causes any problems, since they dis-appear once we pass from the lagrangian to the classical equations of motion. Indeed, theequations of motion are both globally valid and invariant (or rather covariant) under G . Thus,we can attempt to solve for the classical dynamics using our usual arsenal of techniques. Butthis is not the case at the quantum level. There, our usual technique is to convert the hamil-tonian into an operator on L ( M ) and to exploit the conserved charges corresponding to G tosolve, at least partially, the resulting Schr¨odinger equation. Here though, we do not have aunique hamiltonian, but rather several; even if we did have a unique hamiltonian, we would,in general, find that the na¨ıve operators corresponding to the conserved charges of G do notcommute with it. The last problem is often remedied by redefining the conserved charges, butthen one finds that the new charges do not form a Lie algebra, unless we add further charges.These two complications are apparently unrelated, at least as we have presented them. Butthey are related in the sense that neither could occur in the first place, were it not for a basictenet of quantum mechanics, namely that physical states are represented by rays in a Hilbertspace. Thus, the overall phase of a vector in a Hilbert space is not physical. This is what makesit possible, ultimately, to resolve the apparent paradox that, at a point in M where two patchesoverlap, we have multiple, distinct lagrangians, but each of them gives rise to the same physics.Similarly, it allows us to absorb extra phases that arise from boundary contributions in the pathintegral under a G transformation, when the lagrangian is not strictly invariant.In this work we show that, by exploiting this basic property, one can formulate and solve(or at least, attempt to solve) such quantum systems in a unified way, using methods fromharmonic analysis. In a nutshell, the idea is as follows. A magnetic field defines a connectionon a U (1)-principal bundle P over M . From G (which acts on M ), we can construct a centralextension ˜ G of G by U (1) (which depends on the connection and on P , and which acts on P ).We reformulate the original dynamical system on M in terms of an equivalent system (witha redundant degree of freedom) of a particle moving on P . This reformulation allows us tocircumvent both of the complications discussed above: not only do we have a unique, globally-valid, local lagrangian on P , but also the Hilbert space carries a bona fide representation of ˜ G (in contrast to the original theory, in which the Hilbert space carries a projective representationof G , corresponding to the fact that a quantum state is represented by a ray in a Hilbert space).As a result, we can attempt a solution using harmonic analysis, with respect to the group ˜ G .It should be remarked that neither the formulation nor the method of solution that wedescribe here can really be considered new. The formulation via central extensions has appearedin a number of places in the literature, mainly with applications to symplectic geometry andgeometric quantization (see e.g. , [6, 7]) and the use of harmonic analysis to solve quantumsystems in the absence of magnetic fields (and hence without the complications described above)was described in [8]. What is new, we hope, is the synthesis of these ideas, which leads to auniform approach to solving quantum-mechanical systems, including cases with magnetic fields(a type of topological interaction due to its independence from the worldvolume metric) or othernon-trivial topological terms.We remark in passing that our general formalism differs from that used in the study of in-tegrable systems. In an integrable system one requires there exist a set of mutually commutingcharges, while for us the charges are allowed to form any Lie algebra. Moreover, in our systems,the charges must correspond to the group action on the position space manifold. That said,3t is worth noting that a number of the quantum mechanics models we consider turn out tobe superintegrable, offering a complementary way of understanding their exact solvability. Forinstance, the Landau system is rendered maximally superintegrable by the fact that it is sym-metric under the full Euclidean group in two dimensions, providing a set of three independentconserved charges (which we may take to be the Hamiltonian and the two Johnson-Lippmanncharges [9]), two of which are in involution [10, 11, 12]. We exploit this same basic fact in § G acts transitively on M (meaningthat any point in M can be reached from any other via the action of G ) corresponding to aspecial case (0 + 1 spacetime dimensions) of the usual non-linear sigma model of quantum fieldtheory on a homogeneous space G/H . The constraint that G acts transitively is a strong one;it implies, in particular, that any potential term in the lagrangian must be a constant. Wethus have a ‘free’ particle, in the sense that, in the absence of the magnetic field (and ignoringpossible higher-derivative terms), the classical trajectories are given by the geodesics of some G -invariant metric. Despite the strong restrictions, one finds that a large class of interestingquantum mechanical models fall into this class and can be solved in this way. Examples discussedin the sequel include the systems considered by Landau (which, in contrast with Landau, wesolve by keeping a transitive group of symmetries - either translations or the full Euclidean group- manifest) and Dirac (where we constrain the particle to move on the surface of a sphere, sothat the rotation group acts transitively).In cases where G does not act transitively, the methods typically provide only a partialsolution, in that they allow us to reduce the Schr¨odinger equation to one on the space of orbitsof G . But even here we find interesting examples where a complete solution is possible.Since the existing literature underlying this work is somewhat arcane, and since we hopethat our results may be of interest to physicists and chemists who are not so mathematicallyinclined, we aim for a discussion that is both pedagogical and reasonably self-contained (inparticular, pertinent mathematical definitions are supplied in Appendix A). Thus, we start byillustrating the ideas with elementary (but incomplete) discussions of the examples of planarmotion in a uniform magnetic field ( § § §
3, we give full mathematical details of the method. We then complete thediscussion of rigid body rotation ( § § § § § § G on M is not transitive. All the examples considered in this paperare summarised in Table 1.In §
5, we discuss one further subtlety: it has long been known [13, 14] that only a subgroupof the symmetry of the classical equations of motion will be well-defined at the quantum level, sowe discuss what happens in such cases. Such anomalies can occur in the presence of a magneticbackground, dispite the absence of chiral fermions. Our conclusions are presented in § Our first example is one made famous by Landau, in which a particle moves in the xy -plane witha uniform magnetic field B ∈ R in the z -direction. In this example, the subtleties are entirely4ue to the presence of the magnetic field. In particular, no matter what gauge is chosen, theusual lagrangian shifts by a non-vanishing total derivative under the action of the symmetrygroup, which for the purposes of the present discussion we take to be translations in R . As aresult, the usual quantum hamiltonian does not commute with the momenta and one cannotsolve via a Fourier transform (which corresponds to harmonic analysis with respect to the group R ).To circumvent this we write the action, contributing to the action phase e iS , as S = Z dt (cid:18)
12 ˙ x + 12 ˙ y − ˙ s − By ˙ x (cid:19) , (2.1)with an additional degree of freedom s ∈ R , with s ∼ s + 2 π , which shall be redundant. Theadvantage of doing so is that, unlike the lagrangian without s , which shifts by a total derivativeproportional to B ˙ x under a translation in y , the lagrangian in (2.1) is genuinely invariant undera central extension by U (1) of the translation group.This central extension is the Heisenberg group, Hb, defined as the equivalence classes of( x, y, s ) ∈ R under the equivalence relation s ∼ s + 2 π , with multiplication law[( x ′ , y ′ , s ′ )] · [( x, y, s )] = [( x + x ′ , y + y ′ , s + s ′ − By ′ x )] , (2.2)and corresponding to R × S as a manifold. Notice that the group R of translations appearsnot as a sub group of Hb, but rather as the quotient group of Hb with respect to the central U (1) subgroup { [(0 , , s )] } . Thus we have a homomorphism Hb → R , given explicitly by[( x, y, s )] ( x, y ), whose kernel is the central U (1). Notice that our definition of the groupmultiplication law depends on B ∈ R , reflecting the fact that even though the groups withdistinct values of B are isomorphic as groups, they are not isomorphic as central extensions.Given (2.1), the momentum p s conjugate to s satisfies the constraint p s + 1 = 0. We takecare of this in the usual way, by forming the total hamiltonian (see e.g. [15]) H = 12 ( p x + By ) + 12 p y + v ( t ) ( p s + 1) , (2.3)with p x and p y being the momenta conjugate to x and y respectively, and with v ( t ) being aLagrange multiplier. Upon quantizing (something we will later define formally), we obtain thehamiltonian operator ˆ H = 12 (cid:18) − i ∂∂x + By (cid:19) − ∂ ∂y + v ( t ) (cid:18) − i ∂∂s + 1 (cid:19) , (2.4)which has a natural action on the space of square integrable functions on the Heisenberg group, L (Hb). The physical Hilbert space H must take account of the constraint (or, equivalently,the redundancy in our description), so we define it to be not L (Hb), but rather the subspace H = (cid:26) Ψ( x, y, s ) ∈ L (Hb) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i ∂∂s + 1 (cid:19) Ψ( x, y, s ) = 0 (cid:27) . (2.5)Note that this subspace of L (Hb) is closed under the action of the Heisenberg group and underthe action of ˆ H , implying that it is also closed under time evolution.We then want to solve the time-independent Schr¨odinger equation (from hereon ‘SE’) ˆ H Ψ = E Ψ. To solve the SE, we decompose Ψ into unitary irreducible representations (henceforth‘unirreps’) of Hb: Ψ( x, y, s ) = Z drdt | B | π π B ( r, t ; x, y, s ) f ( r, t ) , (2.6) To say we are ‘decomposing Ψ into unirreps of Hb’ is a slight abuse of terminology; what we mean, precisely,is discussed in § r, t ∈ R are real numbers. Here, π k ( r, t ; x, y, s ) = e ik ( xr − s/B ) δ ( r + y − t ) , k/B ∈ Z , (2.7)which denote the matrix elements of the infinite-dimensional unirreps of Hb, which act on thevector space L ( R , dt ). The fact that only the unirrep with k = B appears in the decomposition(2.6) follows from enforcing the constraint in (2.5), as we show in Appendix B.Notice that with this decomposition Ψ( x, y, s ) may not be square integrable (as the matrixelements of π B themselves are not). As such, once we have found our ‘solutions’ to the SEwith this decomposition we must check that they are square integrable (or more generally thelimit of a Weyl sequence). This subtlety will be omitted here due to the familiar form our finalsolutions will take.Substituting the decomposition (2.6) into the SE, and using the constraint to eliminate theLagrange multiplier, yields | B | π Z drdt (cid:18) − i ∂∂x + By (cid:19) − ∂ ∂y − E ! f ( r, t ) e i ( Bxr − s ) δ ( r + y − t ) = 0 . (2.8)After some straightforward manipulation, this reduces to (cid:18) B t − ∂ ∂t − E (cid:19) f ( r, t ) = 0 . (2.9)This differential equation, which we recognise as the SE for the simple harmonic oscillator, hasthe solutions f ( r, t ) = H n (cid:16)p | B | t (cid:17) e −| B | t / g ( r ) , E = | B | ( n + 1 / , (2.10)where H n ( x ) are the Hermite polynomials and g ( r ) is an arbitrary function of r . The corre-sponding eigenfunctions are thusΨ n ( x, y, s ) = | B | π Z drdtH n (cid:16)p | B | t (cid:17) e −| B | t / g ( r ) e i ( Bxr − s ) δ ( r + y − t ) . (2.11)We can of course eliminate our redundant degree of freedom, by setting s = 0 for example,to obtain corresponding wavefunctions living in L ( R ) (more precisely, the wavefunction isdescribed by a section of a Hermitian line bundle). In the above expression g ( r ) accounts forthe degeneracy in the Landau levels. On choosing g ( r ) = δ ( r − α/B ) for α ∈ R (and setting s = 0) we arrive at familiar solutions to this system, of the formΨ n,α ( x, y ) = e iαx H n (cid:16)p | B | ( y + α/B ) (cid:17) e − | B | ( y + α/B ) . (2.12)Now let us now recap what we have achieved. Certainly, our result for the spectrum is notnew; nor are our observations regarding the momentum generators. Rather, what is new is theobservation that we can reformulate the problem via a redundant description, in which a centralextension of G by U (1) acts on the configuration space of that redundant description, in a waythat allows us to solve for the spectrum using methods of harmonic analysis. While this mayseem like overkill, it is important to realise that Landau’s original method of solution [5] onlyworks for this specific system of a particle on R in a magnetic background, and moreover worksonly in a particular gauge (the ‘Landau gauge’). It is not at all clear how such an approachcould be generalised to other target spaces (or gauges). In contrast, as we shall soon see in § G acting onany target space manifold M , since it exploits the underlying group-theoretic structure of thesystem. 6 .2 Bosonic versus fermionic rigid bodies Our second prototypical example illustrates the approach in a case where one cannot form aglobally-defined lagrangian without extending the configuration space by a redundant degree offreedom. This prototype also provides an example where the relation to magnetic fields is notimmediately apparent.To wit, we consider the quantum mechanics of a rigid body in three space dimensions, whoseconfiguration space is SO (3), with dynamics invariant under the rotation group. Evidently, sucha rigid body could be either a boson or a fermion (it could, for example, be a composite madeup of either an even or odd number of electrons and protons). If it is a fermion, then itswavefunction should acquire a factor of − SO (3) invariant and topological. It is thus reasonable to guess that it can be written in termsof a magnetic field, or more precisely, a connection on some U (1)-principal bundle over SO (3). Confirmation that this is indeed the case comes from the fact that (up to equivalence), there arejust two U (1)-principal bundles over SO (3) (to see this, note that such bundles are classified bythe first Chern class, which is a cohomology class in H ( SO (3) , Z ) ∼ = Z / SO (3) × U (1) and a non-trivial bundle, which we may take to be U (2), the groupof 2 × U (1)-principal bundles, but also they havethe structure of central extensions of SO (3) by U (1), which we need for our construction. Thetrivial bundle admits the zero connection and describes the boson, while the non-trivial bundleadmits a non-zero (but nevertheless flat) connection, which accounts for the fermionic phase.Let us now see this more clearly by means of an explicit construction. An element U ∈ U (2)projects down to an element O ∈ SO (3) by projecting out its ( U (1)-valued) overall phase. Weparameterize a matrix U ∈ U (2) by U = e iχ (cid:18) e i ( ψ + φ ) / cos( θ/ e − i ( ψ − φ ) / sin( θ/ − e i ( ψ − φ ) / sin( θ/ e − i ( ψ + φ ) / cos( θ/ (cid:19) , (2.13)where θ ∈ [0 , π ], φ ∈ [0 , π ), ψ ∈ [0 , π ) and χ ∈ [0 , π ) with the equivalence relation( θ, φ, ψ, χ ) ∼ ( θ, φ, ψ + 2 π, χ + π ). Now, consider the curve γ ′ ( t ) in U (2) defined by γ ′ ( t ) = (cid:18) e it e − it (cid:19) , t ∈ [0 , π ] , (2.14)and define the curve γ ( t ) to be the projection of γ ′ ( t ) to SO (3), which one might think of asthe particle worldline in the original configuration space. The curve γ ′ ( t ) is a horizontal lift of γ ( t ) with respect to the connection, which in our coordinates can be represented by A = dχ .For our purposes here, this simply means that the tangent vector X γ ′ to the curve γ ′ ( t ) satisfies A ( X γ ′ ) = 0, i.e. it has no component in the χ direction.Notice that in U (2) we have γ ′ (0) = I and γ ′ ( π ) = − I , and that these two points, whiledistinct in U (2), both project to the identity in SO (3). The relative phase of π between γ ′ (0)and γ ′ ( π ) is called the holonomy of γ ( t ). This implies that the rigid body is in this case afermion, because the loop γ ( t ) in SO (3) corresponds to a 2 π -rotation about the z -axis in R .If we had instead equipped the rigid body with the trivial choice of bundle SO (3) × U (1),instead of U (2), then the phase returns to zero upon traversing any closed loop in SO (3), thuscorresponding to a boson.This fermionic versus bosonic nature is furthermore manifest in the differing representationtheory of the Lie groups U (2) and SO (3) × U (1). This shall be important when we solve for the For those readers unfamiliar with principal bundles, we note that a technical understanding should not benecessary to follow the discussion in this Section. Nonetheless, since the notion of a principal bundle shall becentral to the general formalism which we shall set out in §
3, we provide a more-or-less self-contained introductionto the relevant concepts in Appendix A. § SO (3) × U (1) areall odd-dimensional (as we would expect for the integral angular momentum eigenstates of abosonic rigid body), U (2) also contains unirreps of even dimension (for example, the defining 2-drepresentation), leading to the possibility of eigenstates with half-integral angular momentum,which is exactly what we expect for a fermionic rigid body, via the spin-statistics theorem.For our purposes, it will be useful to consider a different path ˜ γ ( t ) in U (2) that also projectsdown to γ in SO (3), defined by ˜ γ ( t ) = (cid:18) e it
00 1 (cid:19) , t ∈ [0 , π ] . (2.15)While this path ˜ γ is not a horizontal lift of the worldline γ , it nonetheless still projects downto γ , but is now a closed loop in U (2) with the property that the exponential of the integralover ˜ γ of the connection A = dχ is equal to the holonomy, viz. e − i R ˜ γ A = e − i R π dt = −
1. Thismeans that we can represent the holonomy (which is the contribution to the action phase fromthe topological term) in terms of a local action, namely the integral of the connection over anappropriately chosen loop ˜ γ . Given the existence of the horizontal lift, the fact that U (1) isconnected means such a loop always exists. As we might expect from the fact that there is aredundancy in our description, the choice of loop is, however, not unique. Nevertheless, theintegral is of course independent of this choice.The upshot is that this topological phase, which results in fermionic statistics of the rigidbody, can be obtained from the integral of a lagrangian (the connection) on the principal bundle,here U (2), which is both globally-defined and manifestly local. Due to the topological twistingof the bundle, there is no corresponding globally-defined lagrangian on the original configurationspace, here SO (3).In this Section we have discussed two quantum mechanical prototypes, which are at firstsight very different from a physical perspective. What both examples have in common is thepossibility of a topological term in the action phase. In our first example of quantum mechanicson the plane ( § U (1)-principal bundle P over the configurationspace M . Such a topological term may not correspond to any globally-defined lagrangian on M (as in § G which acts on M (asin § P ) with an action by a central extension of G , weare now ready to explain the general formalism. We shall consider quantum mechanics of a point particle whose configuration space is a smooth,connected manifold M . This can be described by an action whose degrees of freedom are maps φ from the 1-dimensional worldline, Σ, to the target space M , viz. φ : Σ → M . We considerthe smooth action α : G × M → M of a connected Lie group G on M , which shall define the8global) symmetries of the system. Since, in the path integral approach to quantum mechanics,it is only the relative action phase between pairs of worldlines that is physical, we are free areto consider only worldlines which are closed, without loss of generality. We will now define the dynamics of the particle on M by specifying a G -invariant action phase, e iS [ φ ] , defined on all closed worldlines, or equivalently on all piecewise-smooth loops in M .The action consists of two pieces (ignoring potential and higher-derivative terms). The firstpiece is the kinetic term, constructed out of a G -invariant metric on M . The second piece inthe action couples the (electrically charged) particle to a background magnetic field. This is atopological term in the action phase (in the sense that it does not require the metric), equalto the holonomy of a connection A on a U (1)-principal bundle P over M (see Appendix A),evaluated over the loop φ . It is shown in [16] that for this term in the action phase to beinvariant under the action α of the Lie group G , we require that the contraction of each vectorfield X generating α with the curvature 2-form ω is an exact 1-form. That is, we require ι X ω = df X ∀ X ∈ g , (3.1)where each f X is a globally-defined function (equivalently, a 0-form) on M . This condition,which we shall refer to as the Manton condition, is necessary for the G -invariance of the topo-logical term evaluated on all piecewise-smooth loops in M (provided that G is connected, as weare assuming). This Manton condition is analogous to the moment map formula for a groupaction to be hamiltonian with respect to a given symplectic structure. The difference here,mathematically, is that the field strength ω need not be a non-degenerate 2-form.It will be of use later, when we end up constructing an equivalent action on P , to specify alocal trivialisation of P over a suitable set of coordinate charts { U α } on M . We let s α ∈ [0 , π )be the U (1)-phase in this local trivialisation and define the transition functions t αβ = e i ( s α − s β ) .Technically speaking, we need two coordinate charts on P , denote them V α, ( s α = π ) and V α, ( s α = 0), for each U α , to cover the S fibre. In what follows, we will often gloss over thistechnicality; from hereon, s α should be assumed to be written locally in one of these coordinatecharts, which we shall denote collectively by V α to avoid drowning in a sea of indices. Followingthis ethos, we will also tend to drop the α subscript on s α when we turn to solving the examplesin § G -invariant quantum mechanics, whichwe shall ultimately achieve by passing to a central extension of G by U (1), and using harmonicanalysis on that central extension.To motivate our method, we shall first review how harmonic analysis can be used to solve thecorresponding (time-independent) SE in the absence of the magnetic background, by exploitingthe group-theoretic structure of the system [8]. Solving the SE amounts to finding the spectrumof an appropriate hamiltonian operator ˆ H , which in this case can be quantized as the Laplace-Beltrami operator corresponding to the choice of G -invariant metric on M , on an appropriateHilbert space. In the absence of a magnetic field, the Hilbert space can be taken to be L ( M ).We can endow this Hilbert space with a highly reducible, unitary representation of G , namelythe left-regular representation defined by ρ ( g )Ψ( m ) := Ψ( α g − m ) for m ∈ M , g ∈ G , and Ψ ∈ L ( M ) . (3.2)The action of ρ allows us to decompose the vector space L ( M ) into a direct sum (or, moregenerally, a direct integral) of vector spaces V λ,t , such that the restriction of ρ to each V λ,t yield a unirrep of G , which we label by its equivalence class λ ∈ Λ. Each unirrep may, of course,appear more than once in the decomposition of L ( M ) and so we index these by t ∈ T λ . We9ill fix a basis for each vector space V λ,t , which we denote by e λ,tr , where r ∈ R λ indexes the(possibly infinite-dimensional) basis, which does not depend on t .In our examples we often specify the operator in the unirrep λ by its form in the chosenbasis, which we denote π λ ( s, q ), where s and q index the basis. In many cases, as in § e λ,tr = π λ ( r, t ). In other instances were this is not the case, one cannonetheless infer a suitable form for the e λ,tr from π λ ( s, q ).It is then a consequence of Schur’s lemma that ifˆ Hρ ( g ) f ( m ) = ρ ( g ) ˆ H f ( m ) , (3.3)then the operator ˆ H will be diagonal in both λ and r , and can only mix e λ,tr in the index t and not r or λ , i.e. it only mixes between equivalent unirreps. In most cases this simplifies theSE by reducing the number of different types of partial derivatives present, often resulting in afamily of ODEs [8]. Interestingly, coupling our particle on M to a magnetic background, in the manner described in § §
2, there are two obstructions to this method.Firstly, as demonstrated by our prototypical example ( § M . Secondly, as demonstrated by our prototypical example ( § i.e. when ω , the magneticfield strength, is the exterior derivative of a globally-defined 1-form), the lagrangian may varyby a total derivative under the action of G . This means that (3.3) will fail to hold, and thehamiltonian will not act only between equivalent unirreps of G .It is possible to overcome both these problems by considering an equivalent dynamics onthe principal bundle π : P → M , instead of on M , as we shall now explain.The topological term, which is just the holonomy of the connection A on P , can be writtenas the integral of A over any loop ˜ φ in P which projects down to our original loop φ on M , i.e. one that satisfies π ◦ ˜ φ = φ (see Appendix A). Pulling back A to the worldline using ˜ φ , weobtain on a patch V α of P ˜ φ ∗ A = (cid:16) ˙ s α ( t ) + A α,i (cid:16) x k ( t ) (cid:17) ˙ x i ( t ) (cid:17) dt, (3.4)where x i ( t ) ≡ x i ( π ◦ ˜ φ ( t )) denote local coordinates in M (with i = 1 , . . . , dim M ), s α ( t ) ≡ s α ( ˜ φ ( t )), ˙ s α ≡ ds α /dt & c , and A | V α ≡ ds α + A α,i dx i is the connection restricted to the patch V α . Given that we can also pull back the metric, and thus the kinetic term, from M to P , we can‘lift’ our original definition of the action from M to the principal bundle P . The contributionto the action from a local patch V α is then S [ ˜ φ ] (cid:12)(cid:12)(cid:12) V α = Z dt (cid:8) g ij ˙ x i ˙ x j − ˙ s α − A α,i ˙ x i (cid:9) , (3.5)where g ij dx i dx j will henceforth denote the pullback of the metric to P .As we have anticipated, this reformulation of the dynamics on P has two important virtues.Firstly, there is a globally-defined lagrangian 1-form on P for the topological term, namelythe connection A . Secondly, this lagrangian is strictly invariant under the Lie group centralextension ˜ G of G by U (1), defined to be the set˜ G = { ( g, ϕ ) ∈ G × Aut(
P, A ) | π ◦ ϕ = α g ◦ π } , (3.6)endowed with the group action ( g, ϕ ) · ( g ′ , ϕ ′ ) = ( gg ′ , ϕ ◦ ϕ ′ ) [17, 7], which as a manifold isthe pullback bundle of π : P → M by the orbit map φ m : G → M , g g · m , for any10 ∈ M [17]. Here, Aut( P, A ) denotes the group of principal bundle automorphisms of P ( i.e. diffeomorphisms which commute with the right action of the structure group on P ) whichpreserve A , i.e. for ϕ ∈ Aut(
P, A ) we have ϕ ∗ A = A . There is a short exact sequence0 U (1) ˜ G G , ι π ′ (3.7)with the subgroup Im( ι ) central in ˜ G , thus exhibiting ˜ G as a central extension of G by U (1).Here ι : U (1) ∋ e iθ (id , R e iθ ) ∈ ˜ G , where R g ∈ Aut(
P, A ) indicates the right action of U (1)on the bundle P , and π ′ : ˜ G ∋ ( g, φ ) g ∈ G . This group has a natural action on the principalbundle P , which we denote by ˜ α : ˜ G × P → P , defined by ˜ α ( g,ϕ ) p = ϕ ( p ), for p ∈ P .The price to pay for these two virtues is that we have introduced a redundancy (whichlocally comes in the form of an extra coordinate s α ) into our description. We must account forthis redundancy with an appropriate definition of the Hilbert space, to which we turn in thenext subsection. Equipped with this reformulation of the dynamics on P , and the extended Lie group ˜ G , weare now in a position to construct a local hamiltonian operator and solve for its spectrum bydecomposing into unirreps of ˜ G .To do this, we first form the classical hamiltonian by taking the Legendre transform ofthe lagrangian, defined on the ‘extended phase space’ T ∗ P . At this stage the redundancyin our description becomes apparent, with the momentum p s α conjugate to the (local) fibrecoordinate s α being constant, viz. p s α + 1 = 0, as we saw in § H | V α = 12 ( p i + A α,i ) g ij ( p j + A α,j ) + v ( t )( p s α + 1) , (3.8)where p i is the momentum conjugate to the coordinate x i , and v ( t ) is an arbitrary function of t which plays the role of a Lagrange multiplier. This hamiltonian is naturally quantized as themagnetic analogue of the Laplace-Beltrami operator, in which the covariant derivative ∇ on M is replaced by ∇ + A , givingˆ H (cid:12)(cid:12)(cid:12) V α = 12 (cid:18) − i √ g ∂∂x i √ g + A α,i (cid:19) g ij (cid:18) − i ∂∂x j + A α,j (cid:19) + v ( t ) (cid:18) − i ∂∂s α + 1 (cid:19) , (3.9)which is a Hermitian operator acting on the Hilbert space H = (cid:26) Ψ ∈ L ( P, ˜ µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i ∂∂s α + 1 (cid:19) Ψ = 0 on V α (cid:27) (3.10)where locally the measure is given by ˜ µ = √ g dsdx . . . dx n . The Hilbert space H is isomorphicto the space of square integrable sections on the hermitian line bundle associated with P withrespect to the measure µ = √ g dx . . . dx n [18, 4]. Because the local hamiltonian commutes with the left regular representation of ˜ G , we expectto be able to use harmonic analysis on ˜ G (when it exists!) to solve for the spectrum of (3.9).The Hilbert space H is endowed with the left-regular representation ρ of ˜ G , under which awavefunction Ψ ∈ H transforms as˜ ρ (˜ g )Ψ( p ) ≡ Ψ( ˜ α ˜ g − p ) ∀ p ∈ P, ˜ g ∈ ˜ G. (3.11)11e use harmonic analysis to decompose this representation into unirreps of ˜ G , in analogy withhow we decomposed into unirreps of G in the absence of a magnetic background, above. Thus,let e λ,tr ( p ∈ P ) now denote a basis for this decomposition, which schematically takes the formΨ = X λ Z µ ( λ, r, t ) f λ ( r, t ) e λ,tr ( p ) ∈ L ( P, ˜ µ ) (3.12)for an appropriate measure µ ( λ, r, t ). Note that the basis functions may not be square integrable;if this is not the case one may check that the solutions are the limit of an appropriate Weylsequence (see e.g. [8]). In the presence of the magnetic background, we have passed to aredundant formulation of the dynamics on P , and the crucial difference is that we must nowaccount for this redundancy when using harmonic analysis. It turns out (see Appendix B)that this redundancy can often be accounted for by restricting the decomposition in (3.12)to the subspace of unirreps which satisfy the constraint ( − i∂ s + 1) e λ,tr ( p ) = 0, which we canmoreover equip with an appropriate completeness relation. In the examples that follow in § G , not G ) means that the action of ˆ H will only mix equivalent representations(that is, it can mix between different values of the t index, but not the r index or λ label). Thus,the SE will be simplified, often to a family of ODEs, as we shall see explicitly in a plethora ofexamples in the following Section.It is important to acknowledge that performing harmonic analysis in the manner we havedescribed, for the general setup of interest in which a (possibly non-compact) general Lie groupacts non-transitively on the underlying manifold, is far from being a solved problem in math-ematics. For example, it is not known under what conditions the integrals denoted in (3.12)actually exist, and whether the functions f λ ( r, t ) can be extracted from Ψ by appropriate in-tegral transform methods. Thus, much of what has been said should be taken with a degreeof caution. Fortunately, in the examples that we consider in §
4, all of the required propertiesfollow from properties of the usual Fourier transform, and in all cases the method that we haveoutlined in this section works satisfactorily. In §§ § U (2). After this we will look at a series of other examples whereour method is of use. Some of these are well known systems, e.g. charged particle motion in thefield of a Dirac monopole, whilst others are new, e.g. the motion of a particle on the Heisenbergmanifold. The results of all the examples considered in this paper are summarised in Table 1. We resume the example discussed in § P = U (2), we definea U (2)-invariant action incorporating a kinetic term by S = Z dt (cid:18)
12 ˙ θ + 12 ˙ φ sin θ + 12 (cid:16) ˙ ψ + ˙ φ cos θ (cid:17) − ˙ s (cid:19) . (4.1)The total hamiltonian on this patch is H = 12 p θ + 12 sin θ (cid:0) p φ + p ψ − θ p φ p ψ (cid:1) + v ( t )( p s + 1) , (4.2)12 M [ G ] P [ ˜ G ] Lagrangian on P Spectrum2.1Landaulevels R [ R ] R × U (1)[Hb] ˙ x + ˙ y − ˙ s − By ˙ x | B | ( n + 1 / n ∈ N R P [ SO (3)] U (2)[ U (2)] (cid:18) ˙ θ + ˙ φ sin ( θ ) + (cid:16) ˙ ψ + ˙ φ cos( θ ) (cid:17) (cid:19) − ˙ s j ( j + 1) / j ∈ N + 1 / S [ SU (2)] L ( g, SU (2) × U (1)] (cid:16) ˙ θ + sin ( θ ) ˙ φ (cid:17) − ˙ χ − g cos( θ ) ˙ φ (4 j + 4 j − g ), j ∈ N + g/ R + × S [ SU (2)] R + × L ( g, SU (2) × U (1)] (cid:16) ˙ θ + sin ( θ ) ˙ φ (cid:17) − qr − ˙ χ − g cos( θ ) ˙ φ − q / (2( n + a )), n ∈ N > , a = (1 + ((2 j +1) − g ) / )4.4Landaulevels R [ ISO (2)] R × U (1)[ g ISO(2)] ( ˙ x + ˙ y ) − ˙ s − ∂ x h ( x, y ) ˙ x − ∂ y h ( x, y ) ˙ y − By ˙ x | B | ( n + 1 / n ∈ N R [Hb] R [ f Hb] ( ˙ x + ˙ y + ( ˙ z − x ˙ y ) ) − ˙ s − x ˙ z + x ˙ y Anharmonicoscillator4.6 R [ R ] R × U (1)[Hb] (cid:16) a + z ˙ x + a + z ˙ y + ˙ z (cid:17) − ˙ s − By ˙ x p | B | (2 n + 1)( m +1 /
2) + a | B | ( n +1 / n, m ∈ N Table 1: Summary of examples presented in this paper. The particle lives on the manifold M ,with dynamics invariant under G . Coupling to a magnetic background defines a U (1)-principalbundle π : P → M , on which we form a lagrangian strictly invariant under a U (1)-centralextension of G , denoted ˜ G . 13hich we quantize as the operatorˆ H = −
12 sin θ ∂∂θ (cid:18) sin θ ∂∂θ (cid:19) −
12 sin θ (cid:18) ∂ ∂ψ + ∂ ∂φ − θ ∂ ∂φ∂ψ (cid:19) + v ( t ) (cid:18) − i ∂∂s + 1 (cid:19) , (4.3)acting on wavefunctions Ψ( θ, φ, ψ, s ) ∈ L ( U (2)) satisfying (cid:0) − i ∂∂s + 1 (cid:1) Ψ = 0. The unirrepswhose matrix elements satisfy this condition when considered as functions on U (2), are givenby π jm,m ′ ( θ, φ, ψ, s ) = e − is D jm ′ m ( θ, φ, ψ ) , (4.4)where j is a positive half-integer, m , m ′ ∈ {− j, − j + 1 , . . . , j } , and D jm ′ m is a Wigner D-matrix,defined (in our local coordinates) by D jm ′ m ( θ, φ, ψ ) = (cid:18) ( j + m )!( j − m )!( j + m ′ )!( j − m ′ )! (cid:19) / (sin( θ/ m − m ′ (cos( θ/ m + m ′ P ( m − m ′ ,m + m ′ ) j − m (cos θ ) e − im ′ ψ e − imφ . (4.5)These are matrix elements of an unirrep of U (2) and, as was the case in § X m ′ ∈ Z +1 / X m ∈ Z +1 / ∞ X j =max( | m | , | m ′ | ) j + 18 π (cid:16) D jm ′ m ( θ ′ , φ ′ , ψ ′ ) (cid:17) ∗ D jm ′ m ( θ, φ, ψ )= δ π ( φ − φ ′ ) δ π ( ψ − ψ ′ ) δ (cos θ − cos θ ′ ) , (4.6)where δ π ( · · · ) represents a Dirac delta comb with periodicity 2 π , and the sum over j is overhalf-integers.Following the formalism set out in §
3, we decompose Ψ into a basis { e j,m ′ m } for L ( U (2)),which in this case can be chosen to be e j,m ′ m = π jm,m ′ , the matrix elements of unirreps of U (2)introduced above, giving usΨ = X m ′ ∈ Z +1 / X m ∈ Z +1 / ∞ X j =max( | m | , | m ′ | ) j + 18 π e − is D jm ′ m ( θ, φ, ψ ) f jm ′ m , (4.7)with inverse f jm ′ m = Z d (cid:0) cos( θ ′ ) (cid:1) dψ ′ dφ ′ (cid:16) D jm ′ m ( θ ′ , φ ′ , ψ ′ ) e − is (cid:17) ∗ Ψ( θ ′ , φ ′ , ψ ′ , s ) . (4.8)The SE then reduces to X m ′ ∈ Z +1 / X m ∈ Z +1 / ∞ X j =max( | m | , | m ′ | ) j + 18 π (cid:26) j ( j + 1)2 − E (cid:27) e − is D jm ′ m ( θ, φ, ψ ) f jm ′ m = 0 , (4.9)yielding the energy levels E jm ′ m = 12 j ( j + 1) , for j half-integer . (4.10)The corresponding wavefunctions, on our local coordinate patch, can be writtenΨ jm ′ m ( θ, φ, ψ, s ) = e − is D jm ′ m ( θ, φ, ψ ) . (4.11)14etting the fibre coordinate s to zero defines, a section on the hermitian line bundle associatedwith the principal bundle U (2), in other words a physical wavefunction. On traversing a doubleintersection of coordinate charts on SO (3), the above expression for the section will shift by atransition function.We note in passing that on setting s = 0 the U (2) representations appearing in this decom-position reduce to representations of SU (2). This occurs due to a well-known happy accident,namely that the projective representations of a Lie group G (here SO (3)) whose second Liealgebra cohomology vanishes (as is the case for every semi-simple Lie group) in fact correspondto bona fide representations of the universal cover of G (here SU (2)). That is, under theseconditions, familiar to most physicists, we may decompose the Hilbert space into unirreps ofthe universal cover of G , without technically needing to pass to a central extension. It is, how-ever, important to point out that even in an example such as this, one cannot write down alocal action for the topological term on the universal cover SU (2), but must pass to the centralextension, U (2). Here we consider the G = SU (2)-invariant dynamics of a particle moving on the 2-sphere. Wemay embed M = S in R , parametrized by the standard spherical coordinates ( θ ∼ θ + π, φ ∼ φ + 2 π ). We cover S with two charts U + and U − , which exclude the South and North polesrespectively. At the centre sits a magnetic monopole of charge g ∈ Z . This background magneticfield specifies a particular U (1)-principal bundle P g over S with connection A , which we maywrite in our coordinates as A | U + = ds + − g − cos θ ) dφA | U − = ds − − g − − cos θ ) dφ, (4.12)where s ± denotes a local coordinate in the U (1) fibre. This can be conveniently written as A = 12 dχ + g θdφ, (4.13)where χ = s + − g φ on U + and χ = s − + g φ on U − . The transition functions over atrivialisation on { U + , U − } are specified via the choice( p, e iδ ) ∈ U + × U (1) ( p, e iδ e igφ ) ∈ U − × U (1) . (4.14)For general g , this bundle P g is in fact the lens space L ( g, S by a Z /g Z action. When g = 1, the bundle is simply P ∼ = S , described via the Hopffibration and when g = 2, the bundle is simply R P . As was the case in the previous example, it is here not possible to write down a global 1-formlagrangian on S . Rather, as was first demonstrated by Wu & Yang [3], one must write theaction on S as a sum of line integrals on different charts, together with the insertion of 0-forms(the transition functions) evaluated at points in double intersections of charts. Thus, it is notpossible to use the usual hamiltonian formalism to solve for the spectrum of the correspondingquantum mechanics problem.Following our formalism, we should instead reformulate the problem by writing down anequivalent, globally-defined lagrangian on the U (1)-principal bundle P g = L ( g,
1) defined above.The action is S = Z dt (cid:26) (cid:16) ˙ θ + sin θ ˙ φ (cid:17) −
12 ˙ χ − g θ ˙ φ (cid:27) . (4.15) The lens spaces L ( g,
1) make another appearance in physics as the possible vacuum manifolds for the elec-troweak interaction [19]. G = SU (2) × U (1), the unique (up to Lie group isomorphisms) U (1)-central extension of SU (2), with uniqueness following from the fact that SU (2) is a simpleand simply-connected Lie group [7]. We parametrize an element ˜ g ∈ ˜ G by˜ g = e i ( ψ + φ ) / cos θ e − i ( ψ − φ ) / sin θ − e i ( ψ − φ ) / sin θ e − i ( ψ + φ ) / cos θ , e i ( gψ − χ ) / ∈ SU (2) × U (1) . (4.16)The corresponding total hamiltonian isˆ H = 12 p θ + 12 sin θ (cid:16) p φ + g θ (cid:17) + v ( t ) (cid:18) p χ + 12 (cid:19) , (4.17)which when quantized givesˆ H = −
12 sin θ ∂∂θ (cid:18) sin θ ∂∂θ (cid:19) + 12 sin θ (cid:18) − i ∂∂φ + g θ (cid:19) + v ( t ) (cid:18) − i ∂∂χ + 12 (cid:19) , (4.18)where the Hilbert space H is the subspace of square integrable functions on L ( g,
1) for whichthe last term in (4.18) vanishes.We now wish to solve for the spectrum of this hamiltonian using harmonic analysis on theLie group ˜ G = SU (2) × U (1). Matrix elements of unirreps of SU (2) × U (1) which are annihilatedby the constraint (cid:16) − i ∂∂χ + (cid:17) π jm,m ′ = 0 are given by π jm,m ′ ( θ, φ, ψ, χ ) = e i ( gψ − χ ) / D jm ′ m ( θ, φ, ψ ) . (4.19)Here D jm ′ m ≡ e − im ′ ψ − imφ d jm ′ m ( θ ) are the same Wigner D -matrices as defined in (4.5), andthe matrices d jm ′ m ( θ ) are conventionally referred to as ‘Wigner d -matrices’. The subspace ofthese unirreps with m ′ = g/ ψ , and provide a suitable basisfor decomposing square-integrable functions on the lens space L ( g, e j,g/ m ( θ, φ, χ ) = π jm,g/ ( θ, φ, ψ, χ ), which satisfy the constraint condition and whichtransform as unirreps of SU (2) × U (1). This subspace of H carries the completeness relation X m + g/ ∈ Z ∞ X j =max( | m | ,g/ j + 14 π (cid:16) e j,g/ m ( θ ′ , φ ′ , χ ′ ) (cid:17) ∗ e j,g/ m ( θ, φ, χ )= e − i ( χ − χ ′ ) / δ π ( φ − φ ′ ) δ (cos θ − cos θ ′ ) , (4.20)which allows us to decompose any wavefunction in Ψ ∈ H into unirreps as followsΨ( θ, φ, χ ) = e − iχ/ X m + g/ ∈ Z ∞ X j =max( | m | ,g/ j + 14 π f jm e − imφ d jg/ ,m ( θ ) , (4.21)where f jm = Z d (cos θ ′ ) dφ ′ e imφ ′ + iχ ′ / d jg/ ,m ( θ ′ )Ψ( θ ′ , φ ′ , χ ′ ) . (4.22)If we now substitute the decomposition (4.21) into the SE, after simplification, we get X m + g/ ∈ Z ∞ X j =max( | m | ,g/ j + 14 π (cid:18)
18 (4 j + 4 j − g ) − E (cid:19) e − iχ/ e − imφ d jg/ ,m ( θ ) = 0 . (4.23)Thus the solution to the SE isΨ jm ( θ, φ, χ ) = e − iχ/ − imφ d jg/ ,m ( θ ) , E jm = 18 (4 j + 4 j − g ) . (4.24)16otice that the eigenstates are labeled by two quantum numbers j and m , but that for a given j the eigenstates with different values of m are degenerate in energy due to the rotationalinvariance of the problem.To write our solution in terms of a section on a hermitian line bundle associated with P g ,we set s + = 0 on U + and s − = 0 on U − , corresponding to χ = − gφ and χ = gφ respectively.This yields Ψ jm, + ( θ, φ ) = e i g φ − imφ d jg/ ,m ( θ ) , Ψ jm, − ( θ, φ ) = e − i g φ − imφ d jg/ ,m ( θ ) . (4.25)These solutions agree with the solutions of Wu and Yang [4], who solved this system by consid-ering local hamiltonians on U + and U − separately. In the previous Section we found the spectrum of an electrically charged particle in the presenceof a magnetic monopole. Within our formalism, it is straightforward to generalize this to studyan electrically charged particle in the background field of a dyon, and use harmonic analysis toreduce the corresponding SE to an ODE.The required modification is to include an r -dependent kinetic term, where r is the radialdistance from a dyon located at the origin, together with an r -dependent potential term, in theaction (4.15). We have S = Z dt (cid:26) (cid:16) ˙ r + r ˙ θ + r sin θ ˙ φ (cid:17) − qr −
12 ˙ χ − g θ ˙ φ (cid:27) . (4.26)where q is the electric charge of the dyon, and g ∈ Z is the (quantized) magnetic charge of thedyon as before. The original configuration space M of the system is R + × S , whilst this actionis written on the U (1)-principal bundle P q,g = R + × L ( g,
1) where L ( g,
1) is the lens space as in § non-transitive action of SU (2) × U (1), as defined in theprevious Section.The quantized total hamiltonian corresponding to (4.26) is given byˆ H = − r ∂∂r (cid:18) r ∂∂r (cid:19) −
12 sin θ ∂∂θ (cid:18) sin θ ∂∂θ (cid:19) − r sin θ (cid:18) − i ∂∂φ + g θ (cid:19) + qr + v ( t ) (cid:18) − i ∂∂χ + 12 (cid:19) (4.27)which acts on the physical Hilbert space. The decomposition of a wavefunction Ψ( r, θ, φ, χ ) inthis Hilbert space is completely analogous to the decomposition in (4.21), however this time the f jm , which where previously constants, should be replaced with functions f jm ( r ). On substitutingthis decomposition into the SE, we arrive at the following differential equation for f jm ( r ), (cid:18) − r ∂∂r (cid:18) r ∂∂r (cid:19) + 18 r (4 j + 4 j − g ) + qr − E (cid:19) f jm ( r ) = 0 . (4.28)The bounded solutions to this ODE were derived in [20], giving the spectrum E n = − q n + a ) , n ∈ N > , (4.29)where a = (cid:16) (cid:0) (2 j + 1) − g (cid:1) / (cid:17) . 17 .4 Planar motion in a uniform magnetic field (take two) In § R in the presence of a uniform magneticfield perpendicular to the plane, by considering the group R of translations in the plane, andpassing to its central extension, the Heisenberg group Hb. Of course, the symmetry group ofthis system is larger than R , because both the kinetic term and the magnetic coupling areinvariant not just under translations, but also under rotations. Thus, in this Section, we revisitthis problem (and solve it again) using a different implementation of our general method, byinstead considering the particle as living on the quotient space M = ISO(2) /SO (2) ∼ = R , with G = ISO(2) being the Euclidean group in two dimensions. Thus, our solution here shall involvethe representation theory of a central extension of G = ISO(2), which will be a four-dimensionalgroup, rather than the representation theory of Hb which was used in § U (1)-principal bundle P over the target space M =ISO(2) /SO (2) ∼ = R . Using coordinates ( x, y, s ), where ( x, y ) ∈ R provide global coordinateson the base space, and s denotes a local coordinate in the U (1) fibre, the action is S = Z (cid:18)
12 ( ˙ x + ˙ y ) − ˙ s − ∂h∂x ˙ x − ∂h∂y ˙ y − By ˙ x (cid:19) dt, (4.30)where h ( x, y ) is an arbitrary smooth function of x and y , which corresponds to a choice ofgauge for the magnetic vector potential. Note that in all the examples in this paper, there is achoice of gauge made in writing down the magnetic vector potential which appears in the action.While different choices of gauge will in general result in different central extensions ˜ G , gauge-equivalent vector potentials nonetheless correspond to central extensions which are isomorphicas Lie groups. In this sense, the choice of gauge has little affect on the representation theoryused in our calculations. For this example, we have chosen to make this gauge-dependence (or,rather, independence) explicit, by formulating the action in a general gauge from the outset.As usual, the lagrangian is not invariant under the isometry group G = ISO(2), but ratherit shifts by a total derivative under the translation subgroup. The lagrangian is, however,genuinely invariant under a U (1)-central extension of ISO(2), which we will denote by g ISO(2),which is a four-dimensional group defined by n ξ ′ x , ξ ′ y , ξ ′ c , ξ ′ s o · n ξ x , ξ y , ξ c , ξ s o = n ξ ′ x + ξ x cos ξ ′ c + ξ y sin ξ ′ c , ξ ′ y + ξ y cos ξ ′ c − ξ x sin ξ ′ c , ξ c + ξ ′ c ,ξ s + ξ ′ s − B (cid:0) ( ξ x cos ξ ′ c + ξ y sin ξ ′ c ) ξ ′ y − ( ξ y cos ξ ′ c − ξ x sin ξ ′ c ) ξ ′ x (cid:1) o . (4.31)This group acts on the principal bundle P via˜ α ( ξ ′ x ,ξ ′ y ,ξ ′ c ,ξ ′ s ) · ( x, y, s ) = n x ′ , y ′ ,ξ s + ξ ′ s − B (cid:0) ( x cos ξ ′ c + y sin ξ ′ c ) ξ ′ y − ( y cos ξ ′ c − x sin ξ ′ c ) ξ ′ x (cid:1) + (cid:18) B xy − B x ′ y ′ (cid:19) +( h ( x, y ) − h ( x ′ , y ′ )) o , (4.32)where x ′ = ξ ′ x + x cos ξ ′ c + y sin ξ ′ c and y ′ = ξ ′ y + y cos ξ ′ c − x sin ξ ′ c .The corresponding total hamiltonian is H = 12 (cid:18) p x + ∂h∂x + By (cid:19) + 12 (cid:18) p y + ∂h∂y (cid:19) + v ( t )( p s + 1) , (4.33)which we quantize as the Hermitian operatorˆ H = 12 (cid:18) − i ∂∂x + ∂h∂x + By (cid:19) + 12 (cid:18) − i ∂∂y + ∂h∂y (cid:19) + v ( t ) (cid:18) − i ∂∂s + 1 (cid:19) . (4.34)18he Hilbert space H is the subspace of square integrable functions on the bundle P which areannihilated by the constraint (cid:0) − i ∂∂s + 1 (cid:1) = 0. We shall now solve the SE for this system bydecomposing this Hilbert space into unirreps of the group g ISO(2) defined above. We start fromthe following unirreps [21] π λm ≥ n ( ξ x , ξ y , ξ c , ξ s ) = e − i (Sgn( B ) n + λ +˜ δ ) ξ c e − iξ s (cid:18) n ! m ! (cid:19) e i Sgn( B )( m − n ) tan − (cid:16) ξyξx (cid:17) e − | B | ( ξ x + ξ y )4 − i q ξ x + ξ y (cid:12)(cid:12)(cid:12)(cid:12) B (cid:12)(cid:12)(cid:12)(cid:12) / ! m − n L m − nn (cid:18) | B | ξ x + ξ y ) (cid:19) , (4.35) π λm ≤ n ( ξ x , ξ y , ξ c , ξ s ) = e − i (Sgn( B ) n + λ +˜ δ ) ξ c e − iξ s (cid:18) m ! n ! (cid:19) e i Sgn( B )( m − n ) tan − (cid:16) ξyξx (cid:17) e − | B | ( ξ x + ξ y )4 − i q ξ x + ξ y (cid:12)(cid:12)(cid:12)(cid:12) B (cid:12)(cid:12)(cid:12)(cid:12) / ! n − m L n − mm (cid:18) | B | ξ x + ξ y ) (cid:19) , (4.36)where λ ∈ Z , m, n ∈ N , ˜ δ = 1 if B > δ = 0 otherwise, and L m − nn are the associatedLaguerre polynomials. A set of functions in the Hilbert space which transform under theserepresentations can be inferred by comparing the multiplication rule in g ISO(2) with the groupaction on the principal bundle P . We thus obtain the following basis of functions on P : e λ ,mn | m ≥ n ( x, y, s ) = e − i ( s + h + B xy ) (cid:18) n ! m ! (cid:19) e i Sgn( B )( m − n ) tan − ( yx ) e − | B | ( ξ x + ξ y )4 − i q ξ x + ξ y (cid:12)(cid:12)(cid:12)(cid:12) B (cid:12)(cid:12)(cid:12)(cid:12) / ! m − n L m − nn (cid:18) | B | ξ x + ξ y ) (cid:19) , (4.37) e λ ,mn | m ≤ n ( x, y, s ) = e − i ( s + h + B xy ) (cid:18) m ! n ! (cid:19) e i Sgn( B )( m − n ) tan − ( yx ) e − | B | ( x y − i p x + y (cid:12)(cid:12)(cid:12)(cid:12) B (cid:12)(cid:12)(cid:12)(cid:12) / ! n − m L n − mm (cid:18) | B | x + y ) (cid:19) . (4.38)where λ = − Sgn( B ) − ˜ δ . When acted on by the left regular representation of g ISO(2) thesefunctions transform under the unirrep corresponding to the conjugate of the λ = λ unirrepdefined in (4.35, 4.36) above. We know it is sufficient to consider only these unirreps since theysatisfy a completeness relation given by | B | π X m,n (cid:16) e λ ,mn ( x ′ , y ′ , s ′ ) (cid:17) ∗ e λ ,mn ( x, y, s ) = e − i ( s − s ′ ) δ ( x − x ′ ) δ ( y − y ′ ) . (4.39)Thus, we can decompose a wavefunction in our Hilbert space into unirreps of g ISO(2) asΨ( x, y, s ) = | B | π X m,n e λ ,mn ( x, y, s ) f m,n , (4.40)where the inverse transform is given by f m,n = Z dxdy ( e λ ,mn ( x ′ , y ′ , s ′ )) ∗ Ψ( x, y, s ) . (4.41)19fter substituting the decomposition (4.40) into the SE, we obtain | B | π X m,n ( | B | ( n + 1 / − E ) e λ ,mn ( z, ys ) f m,n = 0 . (4.42)Thus, we arrive at the familiar Landau level spectrum E m,n = | B | ( n + 1 / , Ψ m,n = e λ ,mn ( x, y, s ) , (4.43)where setting s = 0 in e λ ,mn gives us a suitable set of eigenfunctions on R . In this Section, we turn to a new example not previously considered in the literature, of particlemotion on the Heisenberg group. We equip M = Hb with a left-invariant metric, and thus take G = Hb also. We shall couple the particle to a background magnetic field, corresponding toan Hb-invariant closed 2-form on Hb, for which the magnetic vector potential which appears inthe lagrangian shifts by a total derivative under the action of the group Hb on itself.While a version of the Heisenberg group appeared in § R ), for our purposes in this Section we shall redefine the Heisenberg groupto be the set of triples ( x, y, z ) ∈ R equipped with multiplication law( x ′ , y ′ , z ′ ) · ( x, y, z ) = ( x + x ′ , y + y ′ , z + z ′ + yx ′ ) . (4.44)To avoid any possible confusion, we emphasise that in this Section the Heisenberg group is takenas the original configuration space of our particle dynamics, which we shall reformulate as anequivalent dynamics on a central extension of the Heisenberg group . This central extension willbe a four-dimensional Lie group which we shall denote f Hb.Before we proceed with writing down the action for this system (and eventually solving forthe spectrum using harmonic analysis on f Hb), we first pause to offer a few words of motivationfor considering this system, since it does not correspond to any physical quantum mechanicssystem (although there are indirect links to the anharmonic oscillator, see e.g. [22]). In anycase, our motivation is entirely mathematical. Firstly, we wanted a new example where thecentral extension of Lie groups 0 → U (1) → ˜ G → G is non-trivial, i.e. ˜ G is not just a directproduct, and moreover that it corresponds to a non-trivial central extension of Lie algebras0 → R → ˜ g → g . The requirement that a Lie algebra g admits a non-trivial central extensionrequires, by a theorem of Whitehead [23, 24], that the Lie algebra g cannot be semi-simple. Ofcourse, abelian Lie groups provide a source of such non-trivial central extensions, because theirLie algebra cohomology is in a sense maximal (noting that the second Lie algebra cohomology of g is isomorphic to the group of inequivalent (up to Lie algebra isomorphisms) central extensionsof g ). However, we sought a more interesting example where the original group G is non-abelian. To that end, non-abelian nilpotent Lie groups provide a richer source of suitablecentral extensions, because the second Lie algebra cohomology of any nilpotent g is at least two-dimensional [25]. The Heisenberg Lie algebra, and the corresponding Lie group Hb, providesthe simplest such example.Since we are taking the Heisenberg group to be topologically just R , we can cover the targetspace with a single patch and write the lagrangian using globally-defined coordinates ( x, y, z ).The action on Hb, including the topological term, is S = Z dt (cid:18) (cid:0) ˙ x + ˙ y + ( ˙ z − x ˙ y ) (cid:1) − x ˙ z + x y (cid:19) . (4.45)The kinetic term corresponds to a left-Hb-invariant metric on Hb, as mentioned above, and wehave chosen a normalization for the (real-valued) coefficient of the topological term − x ˙ z + x ˙ y . Note that this is not the most general Hb-invariant topological term we can write down. U (1)-principal bundle P over Hb, on which s provides a local coordinate in the fibre. The action on P is written S = Z dt (cid:18) (cid:0) ˙ x + ˙ y + ( ˙ z − x ˙ y ) (cid:1) − ˙ s − x ˙ z + x y (cid:19) , (4.46)where the only difference is the ˙ s term. By adding this redundant degree of freedom to theaction it becomes strictly invariant under the U (1)-central extension of Hb defined by themultiplication law( x ′ , y ′ , z ′ , s ′ ) · ( x, y, z, s ) = (cid:18) x + x ′ , y + y ′ , z + z ′ + yx ′ , s + s ′ − zx ′ − y x ′ (cid:19) , (4.47)which we denote by ˜ G = f Hb.The total hamiltonian corresponding to the action (4.45) is given by H = 12 p x + 12 ( p z + x ) + 12 (cid:18) p y − x x ( p z + x ) (cid:19) + v ( t ) ( p s + 1) , (4.48)which quantizes toˆ H = − ∂ ∂x + 12 (cid:18) − i ∂∂z + x (cid:19) + 12 (cid:18) − i ∂∂y − x x (cid:18) − i ∂∂z + x (cid:19)(cid:19) + v ( t ) (cid:18) − i ∂∂s + 1 (cid:19) . (4.49)acting on the Hilbert space of square integrable functions on f Hb that are annihilated by (cid:0) − i ∂∂s + 1 (cid:1) .Because the group f Hb defined in (4.47) has a nilpotent Lie algebra, its representation theorycan be found via Kirillov’s orbit method [26]. The unirrep matrix elements that we are interestedin, which in this case are functions on f Hb, are infinite-dimensional, given by π q ( r, t ; x, y, z, s ) = δ ( t − r − x ) e i ( − s + zr + yr )+ q/ y , (4.50)which satisfy the completeness relation Z dqdrdt π ) (cid:0) π q ( r, t ; x ′ , y ′ , z ′ , s ′ ) (cid:1) ∗ π q ( r, t ; x, y, z, s ) = e − i ( s − s ′ ) δ ( x − x ′ ) δ ( y − y ′ ) δ ( z − z ′ ) . (4.51)We thus decompose a wavefunction into unirreps using these functions as our basis elements, e q,tr ( x, y, z, s ) = π q ( r, t ; x, y, z, s ), giving usΨ( x, y, z, s ) = Z dqdrdt π ) e q,tr ( x, y, z, s ) f q ( r, t ) , (4.52)where f q ( r, t ) = Z dx ′ dy ′ dz ′ (cid:0) e q,tr ( x ′ , y ′ , z ′ , s ′ ) (cid:1) ∗ Ψ( x ′ , y ′ , z ′ , s ′ ) . (4.53)Using this decomposition, and the expression (4.49) for the hamiltonian, the SE reduces to − π ) Z dqdrdt e q,tr ( x, y, z, s ) (cid:18) ∂ f q ( r, t ) ∂t + 2 Ef q ( r, t ) − (cid:0) ( t + q ) + 4 t (cid:1) f q ( r, t ) (cid:19) = 0 . (4.54)The ODE in the parentheses coincides with the SE for an anharmonic oscillator. This differentialequation can be solved order-by-order in perturbation theory (in the parameter q ), as is discussedin numerous sources, for example [27]. If the SE of this problem could be solved using othermeans, this decomposition would allow one to study the eigenstates of the anharmonic oscillator.21 .6 Trapped particle in a magnetic field Our last example will demonstrate our method in a case where the group action α : G × M → M is non-transitive (we saw another such non-transitive example, that of a particle orbiting a dyon,in § M = R , invariant under the actionof a subgroup G = R ⊂ R corresponding to translations in x and y . We will begin this Sectionby formulating the problem, and introducing the necessary representation theory, to describea generic such action. We will then consider a special case, in which the components of theinverse metric on R vary quadratically in the z direction. This corresponds, physically, toa z -dependent effective mass. In this special case, we shall find that the solutions to the SEbecome localized (or ‘trapped’) around the z = 0 plane.Consider the action S = Z dt (cid:18) (cid:0) a x ( z ) ˙ x + a y ( z ) ˙ y + a z ( z ) ˙ z (cid:1) + V ( z ) − By ˙ x − yf ′ ( z ) ˙ z (cid:19) , (4.55)for a particle moving on R . Here a x ( z ), a y ( z ), a z ( z ), V ( z ), and f ( z ) are (for now) arbitrarysmooth functions of z , with a x ( z ), a y ( z ), and a z ( z ) necessarily non-vanishing. This action isquasi-invariant under the non-transitive action of translations in x and y , but is not invariantunder translations in the z direction. We thus consider an equivalent action on a U (1)-principalbundle over R , which has to be the trivial one, P = R × U (1), with coordinates ( x, y, z, s ∼ s + 2 π ). The action is given by S = Z dt (cid:18) (cid:0) a x ( z ) ˙ x + a y ( z ) ˙ y + a z ( z ) ˙ z (cid:1) + V ( z ) − ˙ s − By ˙ x − yf ′ ( z ) ˙ z (cid:19) , (4.56)which is strictly invariant under ˜ G = Hb, the Heisenberg group (the unique U (1)-central ex-tension of R up to isomorphism), which in this Section we parametrize by ( ζ x , ζ y , ζ s ), with itsgroup action on the bundle R × U (1) defined by˜ α ( ζ ′ x ,ζ ′ y ,ζ ′ s ) ◦ ( x, y, z, s ) = ( x + ζ ′ x , y + ζ ′ y , z, s + ζ ′ s − ζ ′ y ( Bx + f ( z ))) . (4.57)The total hamiltonian corresponding to the above action is given by H = 12 a x ( z ) ( p x + By ) + 12 a y ( z ) p y + 12 a z ( z ) (cid:0) p z + yf ′ ( z ) (cid:1) + V ( z ) + v ( t )( p s + 1) , (4.58)which we quantize as the operatorˆ H = 12 a x ( z ) (cid:18) − i ∂∂x + By (cid:19) − a y ( z ) ∂ ∂y + 12 a z ( z ) (cid:18) − i ∂∂z + yf ′ ( z ) (cid:19) + V ( z )+ v ( t ) (cid:18) − i ∂∂s + 1 (cid:19) . (4.59)We decompose a wavefunction into unirreps of Hb, exactly as in § z , viz. Ψ( x, y, z, s ) = 2 π | B | Z drdte B,tr ( x, y, s ) f ( r, t ; z ) , (4.60)where as before e B,tr ( x, y, s ) = e iBxr − is δ ( r + y − t ) . (4.61)This however, now transforms under the unirrep of Hb defined by˜ π − B ( r, t ; ζ x , ζ y , ζ z ) = (cid:0) exp ( if ( z ) ζ y ) e B,tr ( ζ x , ζ y , ζ s ) (cid:1) ∗ , (4.62)22hich takes account of the transformation of s which is not the same as ζ s , as was the case inour previous examples. This can be seen from ρ (( ζ ′ x , ζ ′ y , ζ ′ s )) · e i ( Bxr − s ) δ ( r + y − t ) = e i ( B ( x − ζ ′ x ) r − i ( s − ( ζ ′ s + Bζ ′ y ζ ′ x )+ ζ ′ y ( Bx + f ′ ( z )) δ ( r + y − ζ ′ y − t ) , = Z dq (cid:16) e if ′ ( z ) ζ y e i ( Bζ x q − ζ s ) δ ( q + ζ y − r ) (cid:17) ∗ e i ( Bxq − s ) δ ( q + y − t ) . (4.63)Upon this decomposition, the SE reduces to the following PDE B t a x ( z ) − ∂ t a y ( z ) + ( − i∂ z + ( t − r ) f ′ ( z )) a z ( z ) + V ( z ) ! f ( r, t ; z ) = Ef ( r, t ; z ) . (4.64)Even in this case where G acts non-transitively on M , we see that using harmonic analysis (ona central extension) has removed derivatives with respect to the two variables x and y , andreplaced them with derivatives with respect to the single variable t , which labels distinct copiesof the unirrep (4.62) that appears in the Hilbert space.As a specific example where this PDE can be solved analytically, we take f ′ ( z ) = 0, V ( z ) = 0, a z ( z ) = 1, and a x ( z ) = a y ( z ) = ( a + z ) − with a ∈ R + . That is, we do not consider the additionof a z -dependent potential, but we do consider a (specific) z -dependent metric on R . Thisequation admits solutions by separation of variables, viz. f ( r, t ; z ) = f ( r, t ) g ( z ), after which f ( r, t ) is found to satisfy a simple harmonic oscillator equation (with quantum number n ∈ Z )analogous to (2.9). Likewise, g ( z ) is then found to satisfy (cid:18) − ∂ ∂z g ( z ) + | B | ( n + 1 / g ( z )( a + z ) (cid:19) = Eg ( z ) , n ∈ Z , (4.65)which is simply the harmonic oscillator equation again. As such the z -dependence may bewritten in the form g ( z ) = H m (cid:16) ( | B | (2 n + 1)) / z (cid:17) e − √ | B | (2 n +1) z / , m ∈ Z . (4.66)We can obtain an expression for the eigenstates by inverting the decomposition in (4.60) andsetting s = 0, to obtain functions on R . Following a similar procedure to that in § m,n,α ( x, y, z ) = H m (cid:16) ( | B | (2 n + 1)) / z (cid:17) e − √ | B | (2 n +1) z / e iαx H n ( p | B | ( y + α/B )) e − | B | ( y + α/B ) , (4.67)where α ∈ R . The energy levels depend only on the two quantum numbers n and m , both in Z , and are given by E m,n,α = p | B | (2 n + 1)( m + 1 /
2) + a | B | ( n + 1 / . (4.68)Thus, interestingly, the eigenstates for this system appear to be trapped in the z -direction (eventhough na¨ıvely one may expect the opposite). Back in §
3, we claimed that a certain condition (3.1) on the field strength 2-form ω , which wecalled the Manton condition, must be satisfied in order for particle motion in that magneticbackground to result in a G -invariant quantum mechanics. Specifically, this condition, whichwas proven (in the context of sigma models in any dimension) in [16], demands that the23ontraction of ω with each vector field generating the G action on M must be an exact 1-form.In all the examples considered so far in this paper, that condition has been satisfied, and thus,while there might not necessarily have existed a G -invariant lagrangian corresponding to thattopological term, we saw that there nevertheless always existed a G -invariant action .When the Manton condition is violated, however, there will exist non-contractible word-lines in M on which a G -invariant action cannot be written down at all (the necessity ofnon-contractible cycles in M for the Manton condition to fail makes manifest the topologicalcharacter of this condition). In that sense, the symmetry group of a particle on M in thepresence of such a Manton condition-violating magnetic background is reduced from G down tosome subgroup K ⊂ G on which the Manton condition holds, which one may determine. Sincethe classical equations of motion nevertheless retain invariance under all of G , this symmetrybreaking due to the magnetic background may be interpreted as an anomaly of the quantumtheory, albeit of a kind that might be unfamiliar to many readers. In particular, this kind ofanomaly does not derive from an inability to appropriately regularize the path integral measurefor fermions in a way that is compatible with the symmetry; indeed, this anomaly is not relatedto fermions at all, but follows only from topological considerations. Furthermore, the lagrangian may still shift by a total derivative under K , in which case we should pursue a similar strategyas in the rest of this paper and write an equivalent dynamics which is invariant under a U (1)central extension ˜ K of K .In this Section, we elucidate in more detail how this type of anomaly can arise, by discussingtwo examples. Firstly, we review quantum mechanics on a torus, which was discussed in [16](in fact, this example was considered by Manton [13, 14], where this type of anomaly was firstobserved). We then turn to a new example where the Manton condition is violated, which isquantum mechanics on the compact Heisenberg manifold. In both cases, it is not our goal inthis Section to actually solve for the spectrum of these systems using harmonic analysis; rather,here, we content ourselves with a careful analysis of the symmetries that are preserved in thequantum theory, i.e. with the determination of the unbroken subgroup K in both examples. We start with the simpler example of quantum mechanics on the 2-torus [13, 14], M = ( R / Z ) ,parametrized by two periodic coordinates x ∼ x + 1 and y ∼ y + 1, with translation symmetry G = U (1) × U (1). We define a magnetic background corresponding to the translation invariantfield strength 2-form ω = 2 πBdx ∧ dy , for B ∈ Z (where this quantization condition on B ensures that ω is the curvature of a well-defined U (1)-principal bundle over T , for which thefirst Chern class must of course be an integer). However, contracting this 2-form with the vectorfield generating translations, X = a x ∂ x + a y ∂ y , yields ι a x ∂ x + a y ∂ y (2 πBdx ∧ dy ) = 2 πa x Bdy − πa y Bdx, (5.1)which is a closed but not an exact 1-form on T and thus violates the Manton condition (unless a x = a y = 0 or B = 0).To see that one cannot indeed write down a G -invariant action (or, more precisely, actionphase), consider a loop γ on the torus at constant x = x which wraps around the y -direction.On such a loop, we may introduce the vector potential A = 2 πBxdy such that ω = dA , andfrom here evaluate the action phase, which is the holonomy over this loop. It is here sufficientto integrate A over γ , yielding the action phase e i πBx . Note that the value of the holonomy In [28], we discussed a number of analogue examples from field theory in which the Manton condition isviolated in a similar way, namely in four-dimensional Composite Higgs models (in which the rˆole of the magneticbackground is replaced by a Wess-Zumino term). In these examples, and indeed for sigma models in any numberof dimensions ( i.e. not just the (0 + 1)-dimensional version that is the subject of the present paper), the resultof violating the Manton condition is the same; namely, there is a reduction in the symmetries of the quantumsystem.
24f a connection (evaluated over a given loop) only depends on the curvature ω and on itscharacteristic class, which may contain torsion information. Thus, the action phase that weevaluate does not depend on our particular choice of A , for fixed ω and characteristic class.This is sure enough not invariant under generic translations in the x direction, but only underdiscrete translations x → x + a/B for a ∈ Z B . Similarly, we may conclude (from evaluating theholonomy over a loop in the x direction at constant y ) that the action phase is only invariantunder discrete translations in the y direction also, y → y + a/B for a ∈ Z B . Thus, the symmetrygroup is here reduced from G = U (1) × U (1) to the discrete group K = Z B × Z B . This fact wasfirst derived by explicitly solving the SE for this system, and finding that the correspondingeigenfunctions do not respect the continuous translation invariance of the classical equationsof motion. Rather, the eigenfunctions of the hamiltonian become localized when the magneticfield is switched on, preserving only the discrete Z B × Z B symmetry [14]. Our second example of this type of anomaly is new, and is that of quantum mechanics on theHeisenberg manifold. The Heisenberg manifold , to be contrasted with the Heisenberg group discussed in §§ x , y , and z are all integers. Thus, the Heisenberg manifold,which we can denote by the coset space M = Hb( R ) / Hb( Z ), is parametrized by ( x, y, z ) ∈ R with the equivalence relation x ∼ x + p, (5.2) y ∼ y + m, (5.3) z ∼ z + n + xm, (5.4)where ( p, n, m ) ∈ Z . We shall consider quantum mechanics on this space in the presence of amagnetic background, with symmetry group G = Hb( R ), which acts on [( x, y, z )] ∈ M by lefttranslation (4.44).In particular, we consider a topological term in the action for which the curvature 2-form is ω = Bdx ∧ dy, B ∈ Z , (5.5)which is the unique topological term on M = Hb( R ) / Hb( Z ), as it is the only closed left-invariant2-form on Hb which is constant on the equivalence classes defined in (5.2-5.4). The quantizationcondition on the coefficient B ensures that ω is an integral 2-form on M (meaning its integralover any 2-cycle in M evaluates to an integer), and thus the U (1)-principal bundle over M ,which defines the background magnetic field, is well-defined.Despite being invariant under the action of G = Hb, the 2-form ω does not, however, satisfythe (stronger) Manton condition. In our coordinates, a basis for the right-Hb-invariant vectorfields (which generate left translations on M ) is { X , X , X } = { ∂ x + y∂ z , ∂ y , ∂ z } . (5.6)When a linear combination of these vector fields is contracted with ω , we obtain ι α X + α X + α X ( Bdx ∧ dy ) = B ( α dy − α dx ) . (5.7)Just as the 1-form dθ on a circle is closed but not exact because θ ∼ θ + 2 π , so dx and dy areclosed but not exact 1-forms on the Heisenberg manifold because of the identifications in (5.2-5.4). Thus, the Manton condition is only satisfied for X , hence the topological term remainsinvariant on the 1-parameter subgroup that corresponds to the integral curves of X . Indeed,it is not surprising that the Manton condition is satisfied for X , but not for X or X , because25t was proven in [16] that the Manton condition is necessarily satisfied for any element in [ g , g ],which in this case is just X .Nonetheless, the continuous symmetries that are generated by X and X are not brokencompletely; as in the case of quantum mechanics on the torus discussed above, a discrete sub-group of the R subgroup generated by X and X remains unbroken. The unbroken symmetrygroup K turns out to be the subgroup K = n(cid:16) nB , mB , b (cid:17) ∈ Hb | b ∈ R , ( n, m ) ∈ Z B × Z B o . (5.8)This group is a (non-trivial) central extension (by R ) of the discrete subgroup Z B × Z B , definedby the exact sequence 0 R K Z B × Z B , (5.9)where the group homomorphisms involved should be obvious given (5.8). The lagrangian,including both the kinetic energy and this topological term, is in this case strictly invariantunder this subgroup K , so there is no need to pass to a U (1)-central extension. We have formulated the quantum mechanics of a particle moving on a manifold M , with dy-namics invariant under the action of a Lie group G , in the presence of a background magneticfield. The coupling to a magnetic background, which is included via a topological term in theaction, defines a U (1)-principal bundle P over M with connection. We suggest that such adynamics should be recast using an equivalent action on this principal bundle P , for two rea-sons. Firstly, a globally-defined lagrangian is guaranteed to exist only on P , but not on M itself. Secondly, even if a lagrangian were to be defined (locally) on M , this lagrangian wouldnot in general be invariant under the action of G ; rather, due to the presence of the topologicalterm, it might shift by a total derivative. Once reformulated on P , we have shown that thelagrangian will be strictly invariant, not under G , but under a larger symmetry group ˜ G , whichis a U (1)-central extension of G . We show how to construct this central extension ˜ G , which isa bona fide symmetry group of the system, in the general case.We have discussed a plethora of examples in which these two (related) complications arise incoupling a particle to a magnetic background, and in every case show explicitly how reformulat-ing the dynamics on the principal bundle P remedies the issues. To highlight just one example,we have revisited the seemingly humble problem of quantizing a rotating rigid body in threedimensions, a system that is familiar from every undergraduate quantum mechanics course,which is equivalent to particle motion on the configuration space SO (3). What is perhaps lessfamiliar, and which is of interest to us in this paper, is that there is in fact a topological term inthis theory. This topological term, whose existence stems from the non-vanishing cohomologygroup H ( SO (3) , Z ) ∼ = Z /
2, can only be written as a globally-defined term in the lagrangian ifwe pass to a principal bundle over SO (3). There are two choices of such bundle, both of whichare isomorphic to central extensions of SO (3); the bundle is either U (2), or SO (3) × U (1). Weshow that the former choice corresponds to a term in the action phase that evaluates to − solving theSchr¨odinger equation for such quantum mechanical systems with magnetic backgrounds. Ourmethod exploits the group-theoretic structure of the problem, by decomposing the Hilbert spaceinto unitary irreducible representations of the central extension ˜ G . The method is thus verygeneral; indeed, we show that it is a suitable match for the generality of the problem which weare attempting to solve. Because the Hilbert space carries a bona fide representation of the group26 G (but not the group G , in which the Hilbert space carries only a projective representation), weexpect that such a decomposition should yield a solution for the spectrum of the correspondinghamiltonian. In the example of the fermionic rigid body mentioned above, we immediately seethe appearance of spin- representations in the spectrum by decomposing into representationsof ˜ G = U (2), thus exhibiting the non-trivial connection between topological terms in the actionand representation theory.We proceed to illustrate in all our examples how methods from harmonic analysis can beused to decompose the Hilbert space into representations of a central extension ˜ G , and in allcases this decomposition is found to be fruitful, typically reducing the SE to a family of ODEswhose solutions might be known. Our chosen examples range over some much-loved problems inquantum mechanics, including that of a particle moving on a plane in a uniform perpendicularmagnetic field, a charged particle moving in the field of a magnetic monopole, and a chargedparticle moving in the field of a dyon. This last example illustrates the virtues of our methodeven in cases where the group G acts non-transitively on M , in reducing the problem to one onthe space of orbits of G . We also study some new examples, including a particle moving on theHeisenberg group in the presence of a magnetic background, for which the Schr¨odinger equationis found to reduce, after decomposing into irreducible representations of a central extension ofthe Heisenberg group, to that of an anharmonic oscillator.We anticipate that there are many more quantum mechanics problems which can be de-scribed by dynamics on a manifold with invariance under a Lie group action, and a couplingto a magnetic field, because this setup is a very general one. For example, the cases where M = R n or SO ( n ) appear ubiquitously in physics and chemistry, and one might describe morerealistic molecular systems moving in magnetic fields, for example, by using a perturbative anal-ysis around these simple cases. Another possible source of examples, of interest to condensedmatter physicists and particle theorists, might be provided by quantum field theories admittinginstanton solutions, in which great insight can be gained by solving for quantum mechanics onthe instanton moduli space. Since such theories typically also contain topological terms in theaction, the method of solution we have outlined in this paper, in which we first construct the bona fide symmetry group using central extensions, and then bring to bear the heavy machineryof harmonic analysis, would be applicable.Finally, we observe that all the quantum mechanical problems studied in this paper have hadtopological terms that are linear in time derivatives. This is not, however, the only possibilityfor lagrangians which are quasi-invariant under the action of a symmetry Lie group G . Foran example where this is not the case, consider a free non-relativistic particle. This can bedescribed in terms of motion in space which has a transitive action by the Galileo group, but issuch that the lagrangian is not invariant, but shifts by a total derivative under a boost. It turnsout that the familiar kinetic term for such a non-relativistic free particle, viz. m ˙ x , which is quadratic in time derivatives rather than linear, is nonetheless the result of a topological termin the action. To formulate and solve this example using the methods employed here requiresthe use of so-called inverse Higgs constraints. These constraints are equivalent to the removalof Goldstone bosons by the equations of motion, and they add complications to the methodsintroduced in this paper; in particular, once the inverse Higgs constraint is applied we canno longer na¨ıvely rewrite the topological term as the holonomy of a connection on a principalbundle. This, and the other complications that arise in such cases, will be addressed in a futurework. Acknowledgments
We thank Nakarin Lohitsiri for helpful discussions. JD is supported by The Cambridge Trustand STFC consolidated grant ST/P000681/1. BG is partially supported by STFC consolidatedgrant ST/P000681/1 and King’s College, Cambridge. JTS is supported by STFC consolidated27rant ST/S505316/1.
A Mathematical prerequisites
In this Appendix we will present, through an example, a brief summary of some of the math-ematical concepts used in this paper. A more detailed discussion is given in e.g. [29, 30, 31],which are the main references for this Appendix.We start by defining a fibre bundle, using as our prototype the (principal) fibre bundleintroduced in § P the total space and M the base space , and a surjective map π : P → M between them called the projection . In our example the total space is P = S , which can be embedded in C using the parametrization ( z = cos( θ/ e i ( χ + φ ) / , z =sin( θ/ e i ( χ − φ ) / ) ∈ C , where θ ∈ [0 , π ], φ ∈ [0 , π ), and χ ∈ [0 , π ). The base space ishere M = S , which we embed in R , with the projection map π : S → S defined by π ( z , z ) = (sin( θ ) cos( φ ) , sin( θ ) sin( φ ) , cos( θ )). The pre-image π − ( m ), for any point m ∈ M , is diffeomorphic to the same differential manifold, F , known as the typical fibre of thefibre bundle. For the bundle π : S → S the typical fibre is F = S , as can be seen from π − ((0 , , { ( e i χ , e i χ ) | χ ∈ [0 , π ) } , for example.The base space M is equipped with an open covering { U i } and a collection, { φ i } , of localtrivialisations . Local trivialisations are diffeomorphisms of the form φ i : π − ( U i ) → U i × F with π ◦ φ i ( m, f ) = m for all m ∈ U i . On double intersections of open sets, there are transitionfunctions , t ij : U i ∩ U j → G , from M to some group G , known as the structure group . Thereis a left-action of the group G on the fibre F defined such that φ − j ( m, f ) = φ − i ( m, t ij ( m ) f ).In the context of our example π : S → S , an open covering of S is given by the charts { U + , U − } defined in § φ + ( z , z ) = (cid:0) ( θ, φ ) , e i ( χ + φ ) / (cid:1) and φ − ( z , z ) = (cid:0) ( θ, φ ) , e i ( χ − φ ) / (cid:1) . The structure group is U (1)with a single transition function given by t + − ( θ, φ ) = e − iφ . The inverse of these trivialisa-tions are given by φ − (( θ, φ ) , e is + ) = (cid:0) cos( θ/ e is + , sin( θ/ e i ( s + − φ ) (cid:1) and φ − − (( θ, φ ) , e is − ) = (cid:0) cos( θ/ e i ( s − + φ ) , sin( θ/ e is − (cid:1) .In this work we make frequent use of a specific type of fibre bundle, known as a principal(fibre) bundle. In a principal bundle, the structure group G is a Lie group which, as a manifold,is diffeomorphic to the typical fibre F . In addition, the Lie group G has a right action, denoteit R g , on P such that π ◦ R g = π , and that acts both freely and transitively on each fibre. Forour example, G = U (1) which is diffeomorphic to S as a manifold, and we can define a suitableright action R g , for g = e iδ ∈ U (1), by R e iδ φ − ± (( θ, φ ) , e is ) = φ − ± (( θ, φ ) , e is + iδ ), which is bothfree and transitive.Next, we define the concept of a local section , σ i , which is a smooth map σ i : U i → P such that π ◦ σ i = id M . In this paper we have at times described wavefunctions as sections onthe hermitian line bundle associated with the U (1)-principal bundle P . This refers to a set offunctions, s i : U i → C , defined for each open set U i in our cover, which on double intersectionsare related by s j = t ij s i , where t ij are the U (1)-valued transition functions of the principalbundle P .On a principal bundle, π : P → M , we can define a principal-connection -form (or simplya connection for short). This is a 1-form on P with value in the Lie algebra, g , of the Lie group G . A connection must also satisfy the following conditions A ( X ) = X,R ∗ g A = Ad g − A, (A.1)where X is in the Lie algebra g , and the vector field X on P is defined by X f ( p ) = ddt f ( R e itX · p ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 (A.2)28or p ∈ P and f : P → R .On the principal bundle π : S → S , the 1-form A = dχ/ θ dφ/ R -valued which is as required since the Lie algebraof U (1) is R . Secondly, from the right action of e itδ ∈ U (1) on P we can deduce that thevector field X = 2 δ ∂∂χ , which implies A ( X ) = δ = X ∈ g as required. Lastly, it can be seenthat both terms of A are invariant under R ∗ g , meaning the second condition is satisfied sinceAd g − A = A for U (1).Throughout this paper we will often resort to using local expressions for the connection,which can be obtained using a corresponding pair of sections and trivialisations. Notably, givenlocal sections σ i , the corresponding trivialisation, known as the canonical local trivialisation , isdefined by φ i ( p ) = ( π ( p ) , g i ) , (A.3)where p ∈ π − ( U i ) and g i are related by p = R g i σ i ( π ( p )). Given this, and letting A i = σ ∗ i A ,locally A | U i = g − i π ∗ A i g i − ig − i dg i , (A.4)where d is the exterior derivative on P . Equivalently, and going the other way, sections may bedefined from a given choice of local trivialisation.It turns out that the trivialisation defined above for our example is the canonical localtrivialisation that corresponds to the pair of sections σ + ( θ, φ ) = (cid:0) cos( θ/ , sin( θ/ e − iφ (cid:1) and σ − ( θ, φ ) = (cid:0) cos( θ/ e iφ , sin( θ/ (cid:1) , which can be seen by simply setting s + and s − to zero inthe formulae for φ − ± . Then A + = ( − θ )) dφ and A − = (1 + cos( θ )) dφ . Furthermorewe have that g + = e is + and g − = e is − , which gives us the local expressions for the connection, A | U + = ds + + ( − θ )) dφ and A | U − = ds − + (1 + cos( θ )) dφ .Finally, we must introduce the concepts of holonomy and horizontal lift. Given a connectionwe can define the horizontal lift of a curve γ ( t ) in M as a curve γ hl ( t ) in P such that γ ( t ) = π ( γ hl ( t )), and such that the tangent vector at each point, call it Y γ hl ( t ) , satisfies A ( Y γ hl ( t ) ) = 0, i.e. is horizontal with respect to the connection. The horizontal lift of a curve is unique, upto specifying the start point in the fibre above, say, γ (0). As an example, given our aboveconnection A = dχ/ θ dφ/
2, the horizontal lift of the curve γ ( t ) = (cos t, sin t,
0) in S ,starting at ( z = 1 , z = 0) ∈ S , is given simply by γ hl ( t ) = (1 , Y γ hl ( t ) = ∂∂φ − ∂∂χ .Using a horizontal lift we can define the holonomy. The holonomy of a loop γ ( t ) in M for t ∈ [0 , π ] is defined as the element g ∈ G such that γ hl (2 π ) = R g γ hl (0) . (A.5)For the specific γ hl in our example the holonomy is trivially 1, because γ hl (2 π ) = γ hl (0). We canalso derive an equivalent (and perhaps more familiar) formula for the holonomy which involvesintegrating the connection A . To wit, let ˜ γ ( t ) be a loop in P which projects down to γ ( t ) under π . For any such loop ˜ γ ( t ), the horizontal lift is related to ˜ γ ( t ) by γ hl ( t ) = R (cid:16) e − i R t γ ∗ A (cid:17) ˜ γ ( t ) . (A.6)Using (A.5) and (A.6), one finds that the holonomy of γ ( t ) (with respect to the connection A )is equal to e − i R π ˜ γ ∗ A . In our example, γ hl ( t ) is a already a loop and thus, again, it is obviousthat the holonomy is 1. B Rudiments of harmonic analysis with constraints
In this Appendix we will review, by way of an example, the form of harmonic analysis usedthroughout this paper. The example we will use is that of planar motion in a magnetic field,as discussed in § G , whichrecall is a central extension by U (1) of the original group G (constructed in § G . In our prototypical example, we have G = M = R and ˜ G = Hb, and the left-regularrepresentation of Hb is defined by ρ (( x ′ , y ′ , s ′ )) · Ψ( x, y, s ) = Ψ( x − x ′ , y − y ′ , s − s ′ − Bx ′ y ′ + By ′ x ) . (B.1)for Ψ( x, y, s ) ∈ H , where the Hilbert space H was defined in (2.5).In this example we first decompose a general ˜Ψ( x, y, s ) ∈ L (Hb) into unirreps of Hb,following [8]: ˜Ψ( x, y, s ) = X k Z drdt | k | π D k ( r, t ; x, y, s ) g k ( r, t ) ∈ L (Hb) , (B.2)where recall the unirreps D k are D k ( r, t ; x, y, s ) = e ik ( xr − s/B ) δ ( r + y − t ) , k/B ∈ Z , (B.3)which transform under the left-regular representation as ρ (( x ′ , y ′ , s ′ )) · D B ( q, t ; x, y, s ) = Z D − B ( q, r ; x ′ , y ′ , s ′ ) D B ( q, t ; x, y, s ) dq, (B.4) i.e. in the unirrep D − B . inverse transform is g k ( r, t ) = Z dxdyds (cid:16) D k ( r, t ; x, y, s ) (cid:17) ∗ Ψ( x, y, s ) . (B.5)These unirreps satisfy the Schur orthogonality relation Z dxdyds (cid:16) D k ( r, t ; x, y, s ) (cid:17) ∗ D k ′ ( r ′ , t ′ ; x, y, s ) = 4 π | k | δ kB , k ′ B δ ( r − r ′ ) δ ( t − t ′ ) . (B.6)Enforcing the constraint ( − i∂ s + 1) ˜Ψ = 0, and using the orthogonality relation (B.6), immedi-ately implies g k ( r, t ) = 0 , ∀ k = B . We can then writeΨ( x, y, s ) = Z drdt | B | π D B ( r, t ; x, y, s ) f ( r, t ) ∈ H , (B.7)thus recovering the decomposition in (2.6), where g k ( r, t ) = 2 πδ kB , f ( r, t ), and the inverse ofthis decomposition is given by f ( r, t ) = Z dx ′ dy ′ (cid:0) D B ( r, t ; x ′ , y ′ , s ′ ) (cid:1) ∗ Ψ( x ′ , y ′ , s ′ ) . (B.8)In other words, we may restrict our decomposition to those unirreps which satisfy the con-straint. This restricted subspace of unirreps (which satisfy the constraint) inherits the followingcompleteness relation Z drdt | B | π (cid:0) D B ( r, t ; x ′ , y ′ , s ′ ) (cid:1) ∗ D B ( r, t ; x, y, s ) = e − i ( s − s ′ ) δ ( x − x ′ ) δ ( y − y ′ ) . (B.9)It seems plausible that, under suitably general assumptions, one may decompose a general stateΨ ∈ H into a basis of unirreps of ˜ G which satisfy the constraint, following a similar procedure tothat used in this example. We have indeed found this to be the case in all examples considered,as can be verified on a case-by-case basis by obtaining a completeness relation on the Hilbertspace H , analogous to (B.9). 30 eferences [1] P. A. M. Dirac, Quantised singularities in the electromagnetic field , Proc. R. Soc. London,Ser. A (1931) 60.[2] I. Tamm,
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