Contact Quantization: Quantum Mechanics = Parallel transport
CCONTACT QUANTIZATION:
QUANTUM MECHANICS = PARALLEL TRANSPORT
G. HERCZEG (cid:93) , E. LATINI (cid:91) & ANDREW WALDRON (cid:92)
Abstract.
Quantization together with quantum dynamics can be simultaneously formulatedas the problem of finding an appropriate flat connection on a Hilbert bundle over acontact manifold. Contact geometry treats time, generalized positions and momentaas points on an underlying phase-spacetime and reduces classical mechanics to contacttopology.
Contact quantization describes quantum dynamics in terms of parallel trans-port for a flat connection; the ultimate goal being to also handle quantum systems interms of contact topology. Our main result is a proof of local, formal gauge equiv-alence for a broad class of quantum dynamical systems—just as classical dynamicsdepends on choices of clocks, local quantum dynamics can be reduced to a problem ofstudying gauge transformations. We further show how to write quantum correlatorsin terms of parallel transport and in turn matrix elements for Hilbert bundle gaugetransformations, and give the path integral formulation of these results. Finally, weshow how to relate topology of the underlying contact manifold to boundary conditionsfor quantum wave functions.
Contents
1. Introduction 12. Strict Contact Structures and Quantization 22.1. Contact geometry 32.2. Constraint analysis 42.3. Flat connections 42.4. Contact deformation quantization 93. Flat Sections and Dynamics 113.1. Parallel transport 113.2. Path integrals 133.3. Topology 134. Discussion and Conclusions 14Acknowledgements 15References 151.
Introduction
To understand why a study of contact geometry is fundamental to quantum mechanics,it is useful to think about the standard Copenhagen intepretation in a novel way: Ac-cording to the Copenhagen interpretation, one prepares an initial quantum state, allowsit to evolve for some time, and then calculates the probability of observing some choiceof final state. The basic data here is a Hilbert space and a one parameter family of uni-tary operators that determine time evolution. This parameter typically corresponds to a r X i v : . [ h e p - t h ] M a y Herczeg, Latini & Waldron time intervals as measured in a classical laboratory. Two modifications of this standardparadigm will lead us to a—rather propitious—reformulation of quantum mechanics asa theory of flat connections on a Hilbert bundle over a contact manifold:(i) Because it ought be possible to describe quantum dynamics for any choice of labo-ratory time coordinate (for example one may conceive of notions of time that mixvarying combinations of classical-laboratory measurements), we replace the timeinterval with a classical “phase-spacetime” manifold Z , which can be thought ofas a classical phase space augmented by a timelike direction that enjoys generalcoordinate (diffeomorphism) invariance.(ii) Instead of viewing quantum dynamics as trajectories in a single given Hilbertspace H , we associate—in a manner reminiscent of gauge theories and generalrelativity—a copy of the Hilbert space to every point in the phase-spacetime Z .This structure is a Hilbert bundle Z (cid:110) H , viz. a vector bundle whose fibers areHilbert spaces [1]. We use the warped product notation Z (cid:110) H to indicate that,locally in Z , the Hilbert bundle is a direct product, although this need not globallybe the case.Given the geometric data of the vector bundle Z (cid:110) H , we wish to compare Hilbertspace states at distinct points in Z . For that we need a connection ∇ . Concretely ∇ = d + (cid:98) A , where d is the exterior derivative on Z and i (cid:98) A is a one-form taking values in the spaceof hermitean operators on H . In particular, if H is simply L ( R n ) , we may consider (cid:98) A to take values in the self-adjoint subspace of the corresponding Weyl algebra.To construct the connection ∇ , additional data is required. In Section 2, we will showthat giving the phase-spacetime manifold a strict contact stucture endows the Hilbertbundle Z (cid:110) H with a flat connection. Physically, this strict contact data correspondsto specifying classical dynamics on Z . The construction we give is partly motivated byearlier BRST studies of Fedosov quantization [2] for symplectic manifolds [3]. Solutionsto the quantum Schrödinger equation are then parallel sections of the Hilbert bundle—quantum dynamics amounts to parallel transport of states from one Hilbert space fiberto another. The main theorem of of Section 2 establishes that solutions for connectionsobeying the flatness condition are locally and formally gauge equivalent. The method ofproof is close to that employed in Fedosov’s original work on deformation quantizationof Poisson structures [2]. The key advantage is that our contact approach not only in-corporates dynamics, but also establishes a very general local gauge equivalence betweendynamical quantum systems.In Section 3, we focus on the description of dynamics in terms of parallel sectionsof the Hilbert bundle. In particular we show how to reduce the problem of computingquantum correlators to that of finding the matrix element of a gauge transformation. Wealso give a path integral description of correlators in terms of paths in a novel extendedphase-spacetime description of contact Reeb dynamics. We also show how topology of theunderlying contact manifold determines boundary conditions for quantum wavefunctions.Open problems and future prospects are discussed in Section 4.2. Strict Contact Structures and Quantization
Contact geometry may be viewed as a unification of Hamiltonian dynamics and sym-plectic geometry. Therefore, before discussing quantization, we introduce the salientfeatures of contact structures [4, 5]. ontact Quantization 3
Contact geometry. A strict contact structure is the data ( Z, α ) where Z is a n +1 dimensional manifold and α is a contact one-form , meaning that the volume form(2.1) Vol α := α ∧ ϕ ∧ n is nowhere vanishing , where the two form ϕ := dα , determines the Levi-form along the distribution; we therefore also term ϕ the Levi.The data ( Z, α ) allows us to formulate classical dynamics via the action principle(2.2) S = ˆ γ α , defined by integrating the contact one-form along unparameterized paths γ in Z . Requir-ing S to be extremal under compact variations of the embedding γ (cid:44) → Z yields equationsof motion(2.3) ϕ ( ˙ γ, · ) = 0 . Since the Levi-form necessarily has maximal rank, the above condition determines thetangent vector to γ up to an overall scale. The choice of solution ˙ γ = ρ to Equation (2.3)with normalization α ( ρ ) = 1 is called the Reeb vector . Classical evolution is thereforegoverned by flows of the Reeb vector; and in this context is dubbed
Reeb dynamics . It isnot difficult to verify that these obey a contact analog of the classical Liouville theorem,namely that the volume form is preserved by Reeb dynamics: L ρ Vol α = 0 , where L · denotes the Lie derivative.The contact Darboux theorem is particularly powerful; it ensures that locally thereexists a diffeomorphism on Z that brings any contact form to the normal form(2.4) α = π A d χ A − d ψ , where ( π A , χ A , ψ ) are n + 1 local coordinates for Z . On this coordinate patch the Reebvector ρ = − ∂∂ ψ so that dynamics are locally trivial. Observe that in the worldlinediffeomorphism gauge ψ = τ , where τ is a worldline parameter along γ , the action (2.2)becomes S = ˆ dτ (cid:2) π a ˙ χ a − (cid:3) . This is the Hamiltonian action principle for a system with Darboux symplectic form d π a ∧ d χ a and trivial Hamiltonian H = 1 . A contact structure is the data of a maximally non-integrable hyperplane distribution; the kernelof α (viewed as a map on tangent spaces T P Z → R ) determines precisely such a distribution (as doesany fα where < f ∈ C ∞ Z ). Note also, that it is interesting to consider models for which the Levi-form ϕ = dα has maximal rank, but Vol α may vanish (either locally or globally). The massless relativisticparticle falls into this class. Herczeg, Latini & Waldron
Constraint analysis.
Our quantum BRST treatment of Reeb dynamics requiresthat we examine the constraint structure of the model (2.2). Firstly observe that theaction principle (2.2) is worldline diffeomorphism invariant, and in a choice of coordi-nates z i for Z reads S = ´ α i ( z ) ˙ z i dτ . Therefore the canonical momenta p i for ˙ z i obey n + 1 constraints C i := p i − α i ( z ) = 0 , of which n are second class (because these constraints Poisson commute to give themaximal rank Levi-form: { C i , C j } PB = ϕ ij ) and one is first class (corresponding toworldline diffeomorphisms). By introducing n “fiber coordinates” s a (see [6]), localclassical dynamics can be described by an equivalent extended action principle for paths Γ in Z := Z × R n for which all constraints are first class :(2.5) S ext = ˆ Γ (cid:104) s a J ab ds b + A ( s ) (cid:105) . In the above J ab is a constant, maximal rank antisymmetric matrix (and therefore aninvariant tensor for the Lie algebra sp (2 n ) ). The one-form A is given by A ( s ) = α + e a J ab s b + ω ( s ) , where the soldering forms e a together with the contact one-form α are a basis for T ∗ Z such that the Levi-form decomposes as ϕ = 12 J ab e a ∧ e b , and e a ( ρ ) = 0 . The extended action (2.5) enjoys n + 1 gauge invariances (and hence n + 1 , abelian, first class constraints) when A obeys the zero curvature type condition dA + 12 { A ∧ A } PB = 0 . This condition can be used to determine the one-form ω ( s ) to any order in a formalpower series in s (and therefore exactly for contact forms expressible as polynomials insome coordinate system). The main ingredients for quantization are now ready.2.3. Flat connections.
Because the constraints are now abelian and first class, it isstraightforward to quantize the extended Reeb dynamics defined by the action (2.5) usingthe Hamiltonian BRST technology of [7]. The resultant nilpotent BRST charge may beinterpreted as a flat connection ∇ on the Hilbert bundle Z (cid:110) H . [An analogous connectionhas been constructed for symplectic manifolds in [8].] In detail, ∇ = d + (cid:98) A , where (cid:98) A is a one-form taking hermitean values in the enveloping algebra U ( heis ) of theHeisenberg algebra(2.6) heis = span { , ˆ s a } , [ˆ s a , ˆ s b ] = i (cid:126) J ab . In particular i (cid:98) A = α (cid:126) + e a J ab ˆ s b (cid:126) + i (cid:98) Ω , To analyze global dynamics one ought promote Z to a bundle Z (cid:110) R n . For a pair of one-forms A and B , we denote { A ( s ) ∧ B ( s ) } PB := J ab ∂A∂s a ∧ ∂B∂s b where the inversematrix J ab obeys J ab J bc = δ ca . ontact Quantization 5 where (cid:126) (cid:98) Ω is a hermitean operator, potentially involving higher powers of the genera-tors ˆ s a , that is expressible as a formal power series in (cid:126) . It is formally determined by thezero curvature condition(2.7) ∇ = 0 . Example 2.1 (Hamiltonian dynamics) . Let Z = R = { p, q, t } and α = pdq − H ( p, q, t ) dt , with Hamiltonian H given by a (possibly time-dependent) polynomial in p and q . Noticethat ϕ = e ∧ f where e := dp + ∂H∂q dt and f := dq − ∂H∂p dt , so we make a choice of soldering e a = ( f, e ) which we use to construct the flat connection:(2.8) ∇ = d + i (cid:126) (cid:104) dp S − dq (cid:16) p + (cid:126) i ∂∂S (cid:17)(cid:105) + i (cid:126) dt (cid:98) H , where the operator (cid:98) H := (cid:16) H (cid:0) q + S, p + (cid:126) i ∂∂S (cid:1)(cid:17) Weyl is given by Weyl ordering the operators ˆ s a := ( S, (cid:126) i ∂∂S ) (This ensures formal self-adjointness of the operator (cid:98) H .) The Schrödinger equation (2.9) may be solved by setting Ψ = exp( − i (cid:126) pS ) ψ ( q + S, t ) , where ψ ( Q, t ) obeys the standard time dependent Schrödingerequation i (cid:126) ∂ψ ( Q, t ) ∂t = (cid:16) H (cid:0) Q, (cid:126) i ∂∂Q (cid:1)(cid:17) Weyl ψ ( Q, t ) . This example therefore shows how contact quantization recovers standard quantum me-chanics.To better understand the space of flat connections ∇ , we further organize the expansionin powers of operators ˆ s by assigning a grading gr to the operators ˆ s and (cid:126) where gr ( (cid:126) ) = 2 , gr (ˆ s a ) = 1 . Thus, arranging the connection in terms of this grading we have ∇ = αi (cid:126) (cid:124)(cid:123)(cid:122)(cid:125) − + e a J ab ˆ s b i (cid:126) (cid:124) (cid:123)(cid:122) (cid:125) − + d w (cid:124)(cid:123)(cid:122)(cid:125) + (cid:98) ω (cid:124)(cid:123)(cid:122)(cid:125) (cid:62) , where d ω := d + 12 i (cid:126) ω ab ˆ s a ˆ s b . Here the symmetric part of ω ab gives an sp (2 n ) -valued one-form (or connection) whilethe antisymmetric part is necessarily pure imaginary in order that (cid:98) Ω is hermitean. Also,the terms with strictly positive grading are (cid:98) ω := (cid:98) Ω − i (cid:126) ω ab ˆ s a ˆ s b . Note that we have made the choice of Hilbert space H = L ( R ) here as well as a polarization forthe space of wavefunctions. Different choices of polarization differ only by gauge transformations—recallthat in its metaplectic representation, compact elements of sp (2 n ) act by Fourier transform on Schwartzfunctions. When applied to sums of terms inhomogeneous in the grading, we define gr by the grade of thelowest grade term. Herczeg, Latini & Waldron
Observe that this grading is invariant under rewritings of products of the operators ˆ s given by quantum reorderings, for example gr (ˆ s a ˆ s b ) = gr (cid:16) ˆ s b ˆ s a + i (cid:126) J ab (cid:17) . In other words, gr filters U ( heis ) . The projection of an element in U ( heis ) to the part ofgrade k is denoted by gr k ( · ) .In Theorem 2.2 we shall show that locally, every solution to the flatness condition 2.7is formally gauge equivalent to a connection where (cid:98) Ω = 0 . Moreover the latter suchsolutions always exist.Realizing ˆ s a by hermitean operators representing the Heisenberg algebra acting on H ,the (principal) connection ∇ gives a connection on the (associated) Hilbert bundle Z (cid:110) H .The Schrödinger equation is then simply the parallel transport condition(2.9) ∇ Ψ = 0 on Hilbert bundle sections Ψ ∈ Γ( Z (cid:110) H ) . Indeed, modulo (non-trivial) global issues,the problem of quantizing a given classical system now amounts to solving the above flatconnection problem (2.7), while quantum dynamics amounts to parallel transport. Theorem 2.2.
Any two flat connections ∇ = d + (cid:98) A and ∇ (cid:48) = d + (cid:98) A (cid:48) where gr − ( (cid:98) A ) = αi (cid:126) = gr − ( (cid:98) A (cid:48) ) , are locally, formally gauge equivalent.Proof. The contact Darboux theorem ensures that locally, there exists a set of closed one-forms dE a = 0 , such that ϕ = J ab E a ∧ E b and ι ρ E a = 0 . (In the normal form (2.4), E a = ( d χ A , d π A ) .) Hence the connection(2.10) ∇ D := αi (cid:126) + E a J ab ˆ s b i (cid:126) + d solves the flatness condition (2.7). Our strategy is to construct the gauge transformationbringing a general flat ∇ to this “Darboux form”.Firstly, the flatness condition of a general ∇ = d + (cid:98) A at grade − implies that dαi (cid:126) + (cid:16) gr − (cid:0) (cid:98) A (cid:1)(cid:17) = 0 . This is solved, as discussed earlier, by i (cid:126) gr − (cid:98) A = e a J ab ˆ s b , We also employ gr K ( · ) , where K ⊂ Z , to denote projection to subspaces with the correspondinggrades. For the exterior derivative, we define gr ( d ) = 0 . The terms formally equivalent here are defined to mean that gauge transformations exist givingconnections that are equal to any chosen order in the grading gr . To be sure, we are not claiming that this means all quantum dynamics on a given Hilbert space areequivalent, rather having identified the physical meaning of variables for a given connection ∇ , the “gaugeequivalent” (in the bundle sense) connection ∇ (cid:48) = (cid:98) U ∇ (cid:98) U † will in general describe different dynamics.This is much like the case of active diffeomorphisms for a theory in a fixed generally curved background.Moreover, it is a highly useful feature, because at least locally, it allows complicated dynamics to bedescribed in terms of simpler ones. ontact Quantization 7 where ϕ = J ab e a ∧ e b and ι ρ e a = 0 . Comparing the line above with the first display of this proof, we see there must (pointwisein some neighborhood in Z ) exist an invertible linear transformation U ∈ GL (2 n ) suchthat E a = U ab e b . Moreover, U must preserve J and hence is in fact Sp (2 n ) -valued with unit determinant.Thus, we may write U = exp( u ) . In turn it follows that gr {− , − } (cid:0) exp(ˆ u ) (cid:98) A exp( − ˆ u ) (cid:1) = αi (cid:126) + E a J ab ˆ s b i (cid:126) , where ˆ u = J ac u cb ˆ s a ˆ s b i (cid:126) . Essentially, we have just intertwined U in the fundamental representation of Sp (2 n ) toits metaplectic representation.We now observe that(2.11) gr (cid:0) exp(ˆ u )( d + (cid:98) A ) exp( − ˆ u ) (cid:1) = d − iα + ω ab ˆ s a ˆ s b i (cid:126) , where α is some real-valued, (cid:126) -independent one-form and the one-form ω ab = ω ba (theHeisenberg algebra (2.6) may be used to absorb an antisymmetric part of ω ab in α ).We now again employ flatness of ∇ and closedness of the E a ’s to obtain gr − (cid:16)(cid:0) exp(ˆ u )( d + (cid:98) A ) exp( − ˆ u ) (cid:1) (cid:17) = ω ab ∧ E a ˆ s b i (cid:126) . We decompose the one-form ω ab with respect to the (local) basis ( α, e a ) for T ∗ Z as ω ab = W ab α + W abc E c . The above display then implies that the functions W ab mustvanish and W abc E a ∧ E c = 0 . Hence W abc is totally symmetric in the indices a, b, c .We now gauge away the term ω ab ˆ s a ˆ s b / (2 i (cid:126) ) = W abc ˆ s a ˆ s b E c / (2 i (cid:126) ) in Equation (2.11).Since we are working formally order by order in the grading, we may employ the Baker–Campbell–Hausforff formula exp(ˆ u ) (cid:99) W exp( − ˆ u ) = exp([ˆ u, · ])( (cid:99) W ) . In particular gr (cid:16) exp(ˆ u ) E a J ab ˆ s b i (cid:126) exp( − ˆ u ) (cid:17) = − W abc ˆ s a ˆ s b E c i (cid:126) , for the choice ˆ u = W abc ˆ s a ˆ s b ˆ s c / (3! i (cid:126) ) . Hence we have achieved gr {− , − , } (cid:16) exp(ˆ u ) exp(ˆ u )( d + (cid:98) A ) exp( − ˆ u ) exp( − ˆ u ) (cid:17) = αi (cid:126) + E a J ab ˆ s b i (cid:126) + d − iα . At this juncture, we have established the base case for an induction. Proceedingrecursively we now assume that the flat connection ∇ = d + (cid:98) A obeys gr {− ,...,k } ( (cid:98) A ) = α + (cid:126) α + · · · + (cid:126) [( k +1) / α [( k +1) / i (cid:126) + E a J ab ˆ s b i (cid:126) + d + ˆ ω k , where α i are (cid:126) -independent one-forms and, without loss of generality, take gr (ˆ ω k ) = k . Herczeg, Latini & Waldron
Employing the flatness condition for ∇ along the same lines explained above to ˆ ω k shows that i (cid:126) ˆ ω k = k +2)! W a ...a k +3 ˆ s a · · · ˆ s a k +2 E a k +3 + (cid:126) k ! W a ...a k +1 ˆ s a · · · ˆ s a k E a k +1 + · · · + (cid:126) ( k +1) / W a a ˆ s a E a , k odd , k +2)! W a ...a k +3 ˆ s a · · · ˆ s a k +2 E a k +3 + (cid:126) k ! W a ...a k +1 ˆ s a · · · ˆ s a k E a k +1 + · · · + (cid:126) k/ W a a a ˆ s a ˆ s a E a + (cid:126) ( k +2) / α ( k +2) / , k even , where the tensors W are totally symmetric and α ( k +2) / is some one-form. Both the W ’sand α ( k +2) / are (cid:126) -independent. Indeed, all terms save the one-form α ( k +2) / can— mutatis mutandis —be removed by higher order analogs of the gauge transformation exp(ˆ u ) employed in the base step above. Hence we have now proven that locally, gaugetransformations achieve the form (formally to any power in the grading) ∇ = ∇ D − i (cid:88) j> (cid:126) j − α j . It only remains to apply the flatness condition one more time to show that the one-form α (cid:126) := (cid:80) j> (cid:126) j − α j is closed and therefore locally α (cid:126) = dβ (cid:126) for some function β (cid:126) . Thus exp( iβ (cid:126) ) ∇ exp( − iβ (cid:126) ) = ∇ D . (cid:3) Example 2.3 (The harmonic oscillator) . Let Z = R = { p, q, t } and α = pdq −
12 ( p + q ) dt . The Levi form ϕ = d π ∧ d χ , where π = 12 ( p + q ) , χ = − t − arctan( p/q ) . Indeed, setting ψ = − pq , we have α = π d χ − d ψ , so ( π , χ , ψ ) are local Darboux coor-dinates and (denoting ˆ s a := ( ˆ S, ˆ P ) ) the Darboux normal form (2.10) for the connectionbecomes(2.12) ∇ D := π d χ − d ψ i (cid:126) + ˆ Sd π − ˆ P d χ i (cid:126) + d Let us now run the steps of the above proof in reverse to show how to find gaugetransformations bringing ∇ D to the Hamiltonian dynamics form of 2.8.The closed soldering forms E a = ( d χ , d π ) are related to those of the Hamiltoniandynamics Example 2.1 (given here by e a = ( dq − pdt, dp + qdt ) =: ( f, e ) ) according tothe Sp (2) transformation E a := (cid:18) d χ d π (cid:19) = (cid:18) p π − q π q p (cid:19) (cid:18) dq − pdtdp + qdt (cid:19) =: U ab e b . Writing U = exp( u ) and then intertwining to its metaplectic representation (cid:98) U :=exp (cid:0) J ac u cb ˆ s a ˆ s b i (cid:126) (cid:1) , we have (cid:98) U − (cid:0) αi (cid:126) + E a J ab ˆ s a i (cid:126) (cid:1) (cid:98) U = αi (cid:126) + e a J ab ˆ s a i (cid:126) , while a short computa-tion shows that the sp (2) -valued one-form U − dU is given explicitly by U − dU = (cid:32) − dtdt (cid:33) + − ( p − q )( pe + qf )4 π (3 p + q ) qe − ( p − q ) pf π ( p − q ) qe +( p +3 q ) pf π ( p − q )( pe + qf )4 π . ontact Quantization 9 It is not difficult to verify that the last term in the above display, which can be re-expressed as W abc e c where the tensor W abc (moving indices with the antisymmetricbilinear form J ) is totally symmetric . Moreover, interwining the first term to the meta-plectic representation gives the standard harmonic oscillator hamiltonian i (cid:126) dt ( ˆ P + ˆ S ) .Hence the difference between the gauge transformed Darboux connection and the Hamil-tonian dynamics connection of Equation 2.8 is (cid:98) U − ∇ D (cid:98) U − ∇ = ˆ s a ˆ s b W abc e c i (cid:126) . The above term is order in the grading gr and therefore seeds the recursion described inthe proof of Theorem 2.2. It is removed by a grade gauge transformation exp(ˆ u ) with ˆ u = ˆ s a ˆ s b ˆ s c W abc i (cid:126) . It would desirable to have an efficient recursion to compute all higherterms with respect to the grading gr for the gauge transformation between ∇ and ∇ D ,because in a general setting this would facilitate computation of quantum correlators.2.4. Contact deformation quantization.
The above proof of gauge equivalence of flatconnections is very close in spirit to Fedosov’s formal quantization for symplectic andPoisson structures . That work is concerned with constructing a quantum deformation ofthe Moyal star product, while here we wish to describe both dynamics and quantization.Nonetheless, we can employ’s Fedosov’s method to our quantized contact connection ∇ ,to find a quantum deformation of the commutative algebra of classical solutions.To study the algebra of operators, instead of the Hilbert bundle over Z , we consider a Heisenberg bundle Z (cid:110) U ( heis ) , defined in the same way as the Weyl bundle, except thatinstead of working with fibers given by functions of R n with a non-commutative Moyalstar product, we work directly with operators . For our purposes, the key point is thatlocal sections ˆ a of the Heisenberg bundle are functions of Z taking values in U ( heis ) ,which can be expressed with respect to the grading gr as ˆ a = a ( − i (cid:126) (cid:124) (cid:123)(cid:122) (cid:125) − + a ( − a ˆ s a i (cid:126) (cid:124) (cid:123)(cid:122) (cid:125) − + a (0) ab ˆ s a ˆ s b i (cid:126) − ia (0) (cid:124) (cid:123)(cid:122) (cid:125) + · · · Importantly, a ( k ) are (cid:126) independent, and we do not allow negative powers of (cid:126) greaterthan one.Requiring total symmetry of the tensors a ( k ) a ...a j (cid:54) k appearing in the above expansionuniquely determines a function of (cid:126) which—following Fedosov—we call the abelian part of ˆ a and denote by σ (ˆ a ) := a ( − + (cid:126) a (0) + (cid:126) a (2) + · · · . We call ˆ a − i (cid:126) σ (ˆ a ) the non-abelian part of ˆ a .The flat connection ∇ acts on sections of the Heisenberg bundle by the adjoint action ∇ ˆ a := d ˆ a + [ (cid:98) A, ˆ a ] . Note that W = (3 p + q ) p π , W = − ( p − q ) q π , W = − ( p − q ) p π , W = (3 p + q ) q π . Deformation quantization dates back to the seminal work of Bayen et al [9], see also [10] for areview of symplectic connections. Recall that the Moyal star product amounts simply to coordinatizing the space of operators U ( heis ) in terms of functions of R n by employing a Weyl-ordered operator basis, and then encoding their algebrausing a non-commutative (cid:63) -multiplication of functions. The following lemma locally characterizes parallel sections.
Lemma 2.4.
Let f (cid:126) ∈ C ∞ Z [[ (cid:126) ]] obey L ρ f (cid:126) = 0 . Then locally, there is a unique section ˆ a ∈ Γ( Z (cid:110) U ( heis )) such that ∇ ˆ a = 0 and σ (ˆ a ) = f (cid:126) . Proof.
By virtue of Theorem 2.2 we know that locally ∇ = exp(ˆ u ) ◦ ∇ D ◦ exp( − ˆ u ) , for some ˆ u ∈ Γ( Z (cid:110) U ( heis )) and ∇ D is given by Equation (2.10). Therefore we begin byestablishing that the equation(2.13) ∇ D ˆ b = 0 has a solution such that(2.14) σ (exp(ˆ u ) ˆ b exp( − ˆ u )) = f (cid:126) , because ˆ a = exp(ˆ u ) ˆ b exp( − ˆ u ) will then solve ∇ ˆ a = 0 with the correct boundary condition σ (ˆ a ) = f (cid:126) . (We deal with uniqueness at the end of this proof.)We now work order by order in the grading gr . Firstly, we must solve gr − ( ∇ D ˆ b ) = db ( − + b ( − a E a i (cid:126) . From Equation (2.14) we have b ( − = a ( − = gr − f (cid:126) , but by assumption L ρ f (cid:126) = 0 soCartan’s magic lemma gives ι ρ db ( − = 0 , whence db ( − ∈ span { E a } . Hence we can solvethe equation in the above display (uniquely) for b ( − a .At the next order in the grading we must now solve gr − ( ∇ D ˆ b ) = db ( − a ˆ s a + b (0) ab E a ˆ s b i (cid:126) . By virtue of the Darboux coordinate system, b ( − a cannot depend on ψ so ι ρ db ( − a =0 . Hence the above display (uniquely) determines b (0) ab (and once again ι ρ db (0) ab = 0 ).The abelian term − ib (0) is at this point not determined. However for that we imposeEquation (2.14) to the order in the grading, which now determines b (0) in terms of f (cid:126) andother ψ -independent quantities. This establishes the pattern for an obvious recursion,which completes the existence part of this proof.To show uniqueness, suppose ˆ a (cid:48) also obeys ∇ ˆ a (cid:48) = 0 such that σ (ˆ a (cid:48) − ˆ a ) = 0 . Now, let ∇ = αi (cid:126) + e a J ab ˆ s b i (cid:126) + · · · . Then gr − (cid:0) ∇ (ˆ a (cid:48) − ˆ a ) (cid:1) = ( a (cid:48) ( − a − a ( − a ) e a i (cid:126) ⇔ a (cid:48) ( − a = a ( − a . Indeed, the same pattern holds at all higher orders in the grading gr , so that ˆ a (cid:48) = ˆ a , asrequired. (cid:3) Remark . Calling ξ a = ( χ i , π i ) , the Darboux connection (2.10) obeys [ ∇ D , ˆ s a − ξ a ] = 0 . ontact Quantization 11 So taking ˆ b equal to any polynomial P (ˆ s a − ξ a ) solves the parallel section condition (2.13).This in turn immediately solves the parallel section problem for f (cid:126) expressible as poly-nomial in Darboux coordinates. Note however, that in general, replacing P by a formalpower series in ˆ s a − ξ a , may not give a well defined formal power series in Weyl orderedsymbols of ˆ s a . (Quantum reordering terms potentially involve infinite, non-convergent,sums of the coefficients of the original power series.)Let us denote by σ − the map C ∞ Z [[ (cid:126) ]] ∩ ker( L ρ ) (cid:51) f (cid:126) (cid:55)→ ˆ a as defined by the abovelemma. Now consider a pair of solutions f (cid:126) , g (cid:126) ∈ C ∞ Z [[ (cid:126) ]] to the classical equations ofmotion: L ρ f (cid:126) = 0 = L ρ g (cid:126) . Then we have a pair of parallel sections σ − ( f (cid:126) ) and σ − ( g (cid:126) ) of Z (cid:110) U ( heis ) . These maybe multiplied pointwise along Z using the operator product on fibers. Therefore, a lá Fedosov [2], we may define a (cid:63) -multiplication of functions f (cid:126) and g (cid:126) by f (cid:126) (cid:63) g (cid:126) = σ (cid:0) σ − ( f (cid:126) ) σ − ( g (cid:126) ) (cid:1) . This gives a contact analog of deformation quantization. Observe that it reduces thedeformation problem to a gauge transformation. However, unlike Fedosov’s work, thismeans that the above uniqueness proof for flat sections is local. It ought however bepossible to improve this to a global statement and preliminary results indicate that thisis the case; we reserve those results for a later publication, where we also plan to detail theprecise map between the above display and Fedosov’s deformation formula for symplecticstructures. 3.
Flat Sections and Dynamics
As discussed in the previous section, solving for a flat connection ∇ on the Hilbertbundle Z (cid:110) H is analogous to finding an operator quantizing a classical Hamiltonian,while the parallel transport equation (2.9) is the analog of the Schrödinger equationwhich controls quantum dynamics. We now turn our attention to solving the latter andcomputing correlators.3.1. Parallel transport.
Let us suppose we have prepared a state |E i (cid:105) ∈ H z i where H z i is the Hilbert space associated with a point z i ∈ Z (one may think of z ∈ Z as ageneralized laboratory time coordinate). We would like to compute the probability ofmeasuring a state |E f (cid:105) ∈ H z f at some other point z f ∈ Z . For that, observe that we canparallel transport the “initial” state |E i (cid:105) from the Hilbert space H z i to any other Hilbertspace H z using a line operator(3.1) |E ( z ) (cid:105) = (cid:16) P γ exp (cid:0) − ˆ zz i (cid:98) A (cid:1)(cid:17) |E i (cid:105) ∈ H z , where P γ denotes path ordering and γ is any path in Z joining z i and z . Since ∇ = d + ˆ A ,it follows that the section Ψ( z ) = |E ( z ) (cid:105) of Z (cid:110) H solves the Schrödinger equation (2.9).Since the connection ∇ is flat, if the fundamental group π ( Z ) is trivial, this solution isindependent of the choice of path γ between z i and z . When this is not the case, we mustbe more careful with the choice of Hilbert space fibers. We discuss this further below. Fedosov constructs a deformation of the Moyal star product for Weyl ordered operators in the Weylalgebra given the data of a symplectic manifold. Here we skip the Moyal star and work directly withoperators in the Weyl algebra.
Modulo this issue, the probability P f , i of observing |E f (cid:105) ∈ H z f having prepared |E i (cid:105) ∈ H z i is P f , i = (cid:12)(cid:12)(cid:12) (cid:104)E f | (cid:16) P γ exp (cid:0) − ´ z f z i (cid:98) A (cid:1)(cid:17) |E i (cid:105) (cid:12)(cid:12)(cid:12) (cid:104)E f |E f (cid:105) (cid:104)E i |E i (cid:105) . In [11] we showed how to extract quantum mechanical Wigner functions from correlators(3.2) W E f , E i ( z f , z i ) := (cid:104)E f | (cid:16) P γ exp (cid:0) − ˆ z f z i (cid:98) A (cid:1)(cid:17) |E i (cid:105) . This correlator is gauge covariant. In particular, in a contractible local patch aroundthe path γ , by virtue of Theorem 2.2, we can find a gauge transformation (cid:98) U such that (cid:98) U ∇ (cid:98) U − = ∇ D , where the Darboux normal form is given in Equation (2.4). Hence theline operators for these two connections are related by(3.3) (cid:16) P γ exp (cid:0) − ˆ z f z i (cid:98) A (cid:1)(cid:17) = (cid:98) U ( z f ) − ◦ (cid:16) P γ exp (cid:0) − ˆ zz i (cid:98) A D (cid:1)(cid:17) ◦ (cid:98) U ( z i ) . Inserting resolutions of unity ´ dS | S (cid:105)(cid:104) S | = 1 = ´ dP | P (cid:105)(cid:104) P | for H (where ˆ s a = ( ˆ S A , ˆ P A ) and ˆ S A | S (cid:105) = S A | S (cid:105) , ˆ P A | P (cid:105) = P A | P (cid:105) ) in the above identity, and putting this in thecorrelator (3.2) gives (3.4) W E f , E i ( z f , z i ) := ˆ dSdP (cid:104)E f | (cid:98) U ( z f ) − | P (cid:105) (cid:104) P | (cid:16) P γ exp (cid:0) − ˆ z f z i (cid:98) A D (cid:1)(cid:17) | S (cid:105) (cid:104) S | (cid:98) U ( z i ) |E i (cid:105) . Since the line operator for the connection (cid:98) A D in the Darboux frame is essentially trivial(see directly below), knowledge of the gauge transformations (cid:98) U determines the correlator. Example 3.1 (The Darboux correlator) . Consider a pair of points z i = ( π i , χ i , ψ i ) and z f = ( π f , χ f , ψ f ) in the contact three-manifold Z = ( R , π d χ − d ψ ) . Since here we wantto study a line operator for a flat connection ∇ D on a trivial manifold, we may chooseany path between these two points, so take γ = γ ψ ∪ γ π ∪ γ χ where γ π := { (1 − t ) π i + t π f , χ i , ψ i ) } ,γ χ := { ( π f , (1 − t ) χ i + t χ f , ψ i ) } ,γ ψ := { ( π f , χ f , (1 − t ) ψ i + t ψ f ) } , where t ∈ [0 , . Then, along these three paths the potential (cid:98) A for the Darboux connection(see Equation 2.12) takes the form (cid:98) A γ π = 1 i (cid:126) dt ( π f − π i ) ˆ S , (cid:98) A γ χ = 1 i (cid:126) dt ( χ f − χ i )( π f − ˆ P ) , (cid:98) A γ ψ = − i (cid:126) dt ( ψ f − ψ i ) . Hence the correlator in Darboux frame is simply (cid:104) P | (cid:16) P γ exp (cid:0) − ˆ z f z i (cid:98) A D (cid:1)(cid:17) | S (cid:105) = exp (cid:16) − ( χ f − χ i )( π f − P ) + ( π f − π i ) S − ψ f + ψ i i (cid:126) (cid:17) . The above result combined with Equation 3.4 indeed shows that knowledge of the gaugetransformation (cid:98) U bringing a connection to its Darboux form determines correlators. Of course, one could equally well insert other resolutions of unity, for example, replacing ´ dP | P (cid:105)(cid:104) P | with ´ dS (cid:48) | S (cid:48) (cid:105)(cid:104) S (cid:48) | is a propitious choice used in the next example. ontact Quantization 13 Path integrals.
In general, one does not have access to the explicit diffeomorphismbringing the contact form to its Darboux normal form (let alone the gauge transforma-tion (cid:98) U ). Instead correlators can be computed in terms of path integrals. For that, perits definition, we split the path ordered exponential of the integrated potential (cid:98) A intoinfinitesimal segments dz i along the path γ , and insert successive resolutions of unity. Inparticular, using that, for dz i small, (cid:104) P | exp( − (cid:98) A i ( ˆ S, ˆ P ) dz i ) | S (cid:105) ≈ exp (cid:0) i (cid:126) P A S A − A N ( S, P ) (cid:1) , where A N ( S, P ) is the normal ordered symbol of the operator ˆ A , we have the operatorrelation exp( − (cid:98) A i dz i ) ≈ ˆ dSdP | P (cid:105) exp (cid:0) i (cid:126) P A S A − A N ( S, P ) (cid:1) (cid:104) S | . Concatentating this expression along the path γ gives the path integral formula for thecorrelator between states | S i (cid:105) and (cid:104) P f |W P f ,S i ( z f , z i ) = ˆ P ( z f )= S f S ( z i )= S i [ dP dS ] exp (cid:16) − i (cid:126) ˆ γ (cid:0) P A dS A + A N ( S, P ) (cid:1)(cid:17) . In the above γ is any path in Z connecting z i and z f . When ∇ has trivial holonomy(otherwise see below), neither the correlator nor its path integral representation dependson this choice. Notice that the path integration in the above formula is only performedfiberwise. We do not integrate over paths γ in Z , but rather paths in the total space Z = Z (cid:110)R n above the path γ in Z . Indeed, calling s a := ( S A , P A ) and writing P A dS A = s a J ab ds b we see that the action appearing in the exponent of the above path integral isthe quantum corrected analog of the extended action of Equation (2.5)(computing theoperator (cid:98) A and its normal ordered symbol A N will in general produce terms proportionalto powers of (cid:126) ).3.3. Topology.
Finally, we discuss the case when the fundamental group π ( Z ) is nontrivial . The holonomy of the connection ∇ may then be non-trivial, and the paralleltransport solution (3.1) to the Schrödinger equation can depend on the homotopy classof the path γ . A priori this seems to be a bug leading to loss of predictivity, howeverremembering that the topology of system can influence its quantum spectrum (consider afree particle in a box, for example), we have in fact hit upon a feature . Our quantizationprocedure is not complete until we impose that the holonomy of the connection ∇ actstrivially on the Hilbert space fibers. To explain this point better, as a running exampleconsider the contact form α = π d θ − d ψ , on the manifold Z = C × R where C is a cylinder with periodic coordinate θ ∼ θ + 2 π .Now let us study the quantizaton determined by the flat connection ∇ = d + (cid:98) A where (cid:98) A = αi (cid:126) + d π Si (cid:126) + d θ ∂∂S . Here we have picked some polarization for the Hilbert space fibers such that elementsare given by wavefunctions ψ ( S ) . To be precise, (cid:98) A is recovered by writing A ( S, P ) as a power series in P and S and then replacingmonomials P k S l by the operator ˆ P k ˆ S l . We owe the key idea of this section of modding out the Hilbert space fibers by the holonomy of ∇ to Tudor Dimofte. Along the path γ = { θ = θ o + θ, π = π o , ψ = ψ o : θ ∈ [0 , π ) } , we have (cid:98) A γ = i (cid:126) dθ (cid:0) π o − (cid:126) i ∂∂S (cid:1) . Hence the holonomy of ∇ at basepoint z o = ( θ o , π o , ψ o ) is hol z o ( (cid:98) A γ ) = exp (cid:16) − πi (cid:126) (cid:0) π o − (cid:126) i ∂∂S (cid:1)(cid:17) . Requiring that this holonomy acts trivially on the Hilbert space H over the base point z o ∈ Z , we impose that elements ψ z o ( S ) of that space obey exp (cid:16) − πi (cid:126) (cid:0) π o − (cid:126) i ∂∂S (cid:1)(cid:17) ψ z o ( S ) = ψ z o ( S ) . Hence ψ z o ( S + 2 π ) = e πi π o (cid:126) ψ z o ( S ) . So, up to a basepoint dependent phase, wavefunctions are periodic. In effect, the classicaltopology of the contact base manifold Z has enforced the desired boundary conditionson quantum wavefunctions.4. Discussion and Conclusions
Just as contact geometry reduces classical mechanics to a problem of contact topology(all dynamics is locally trivial by virtue of the contact Darboux theorem), the contactquantization we have presented does the same for quantum dynamics. Moreover, sinceour approach is completely generally covariant, even seemingly disparate systems canbe related by appropriate choices of clocks. This gives a concrete setting for quantumcosmology-motivated studies of the “clock ambiguity” of quantum dynamics [12, 13].Beyond providing a solid mathematical framework for philosophical questions of timeand measurement in quantum mechanics, it is very interesting to probe to which extentthe gauge freedom characterized in Theorem 2.2 can be used to solve or further thestudy of concrete quantum mechanical systems. As discussed in Section 3, knowledgeof the gauge transformation bringing the connection ∇ to its Darboux form can beused to compute correlators, which begs the question whether methods—perturbative,exact when symmetries are present, or numerical—can be developed to calculate thesetransformations.Along similar lines to the above remark, symmetries and integrability play a central rôle in the analysis of quantum systems. Again contact geometry and its quantizationought be an ideal setting for analyzing quantum symmetries and relating them to contacttopology. Preliminary results show that this is case, and we plan to report on suchquestions elsewhere.Lattice spin models and models with Fermi statistics are crucial for the descriptionof physical systems. Here one needs to study supercontact structures (see [14, 15, 16]);it is indeed not difficult to verify that our flat connection/quantizaton and parallel sec-tion/dynamics methodology can be applied directly in the supercontact setting; againwe plan to report on this interesting direction in the near future.In Section 2.4 we showed how to relate contact quantization to Fedosov’s deformationquantization. It would also be interesting to relate our approach to other quantizationmethods. In particular, it would be interesting to study the relation to Kontsevich’sexplicit deformation quantization formula for Poisson structures [17] and its Cattaneo–Felder sigma model derivation [18]. In addition, it would be interesting to study when wecan go beyond formal deformation quantization, perhaps along the lines of the A -modelapproach of Gukov–Witten to quantization [19], or geometric quantization in general. ontact Quantization 15 Indeed, Fitzpatrick has made a rigorous geometric quantization study of contact struc-tures [20] based on the proposal by Rajeev [5] to quantize Lagrange brackets (these arethe contact analog of the Poisson bracket). Note also that earlier work by Kashiwara [21]studies sheaves of pseudodifferential operators over contact manifolds, and Yoshioka hasperformed a contact analog of Fedosov quantization where the base manifold is a sym-plectic manifold and the fibers carry a contact structure [22].Finally, we mention that our construction of the connection ∇ is in spirit rather closeto the Cartan normal connection in parabolic geometries, see [23] for the general theoryand [24] for its application to contact structures compatible with a projective structure.These geometric methods may also end up being directly relevant to quantum mechanics. Acknowledgements
This work was presented in part in lectures and further developed at the 38th Ge-ometry and Physics Winter School in Srní. A.W. and E.L. thank the organizers forthis wonderful forum for interactions between geometry and physicists. We also thankAndy Albrecht, Roberto Bonezzi, Steve Carlip, James Conway, Olindo Corradini, Tu-dor Dimofte, Mike Eastwood, Rod Gover, Maxim Grigoriev, Jerry Kaminker, BrunoNachtergaele and Andrea Santi for discussions. A.W. was supported in part by a SimonsFoundation Collaboration Grant for Mathematicians ID 317562.
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E-mail address : [email protected] (cid:91)
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, andINFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy
E-mail address : [email protected] (cid:92) Center for Quantum Mathematics and Physics (QMAP), Department of Mathematics,University of California, Davis, CA95616, USA
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