The Quantum Darboux Theorem,

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a"phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a"quantum Darboux theorem"that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.


Introduction
A fundamental problem in quantum mechanics is to compute correlators where the states |i , |f are elements of some Hilbert space H, the operatorĤ is a quantum Hamiltonian, and t i , t f are classical times measured in some laboratory. It is useful to view the coordinates t i , t f as labels belonging to points in an odd-dimensional phasespacetime manifold Z, that corresponds to all possible classical laboratory measurements of generalized times, positions and momenta. Then the unitary time evolution operator exp − i Ĥ (t f − t i ) can be replaced by a Wilson line HereÂ is the connection form for a connection ∇ = d +Â acting on sections of a Hilbert bundle (see [1] and also [2]) HZ, over the base manifold Z with Hilbert space fibers H. Requiring that ∇ is flat (and if necessary, quotienting by non-trivial holonomies), the correlator f |P γ exp −´γÂ |i only depends on the endpoints of γ and points |i , |f in the fibers of HZ above these two endpoints: We attach the moniker quantum dynamical system to the data (HZ, ∇) of a Hilbert bundle equipped with a flat connection ∇. The goal of this article is to use the gauge covariance of Wilson line operators, to map complicated quantum dynamics to simpler ones.
The article is structured as follows: We first review how to formulate a classical dynamical system in terms of an odd-dimensional analog of a symplectic geometry. We then explain, following [3], how to obtain a quantum dynamical system from the BRST quantization of a classical dynamical system. Here the BRST charge corresponds to the flat connection ∇. We then treat Hamiltonian-Jacobi theory in terms of contact geometry. (Note that a modern treatment of Hamilton-Jacobi actions and their formal quantization is given in [4].) Thereafter we describe the quantum analog of the Darboux theorem for quantum dynamical systems [5]. After that we primarily focus on quantum mechanics of a particle on a line. The underlying classical dynamical system gives evolution on a three-dimensional phase-spacetime manifold Z, for which we give an explicit account of the (classical) Darboux theorem. The remainder of the article is devoted to developing a diagrammatic calculus for computing the asymptotics of the gauge transformations appearing in the quantum Darboux theorem of [5], and applying these to the computation of correlators.

Dynamical Systems
Quite generally, dynamics is a rule governing the time evolution of a system, so a dynamical system is a one-parameter family of maps Φ τ : Z → Z where points τ ∈ R are viewed as times and Z is the state space of the system. Fixing a point z 0 ∈ Z and allowing τ to vary gives a parameterized path γ : R → Z in Z with initial condition z 0 . Thus, dynamics amounts to a set of parameterized paths (locally) foliating the space Z. Often, Z is taken to be a symplectic manifold or phase space, in which case it is even-dimensional and points in that space label generalized positions and momenta. However, to maintain general covariance also with respect to choices of clocks, we shall demand that Z is an odd dimensional manifold which we term a phase-spacetime. We shall therefore view dynamics as a set of unparameterized paths (locally) foliating the odd-dimensional phase-spacetime Z 2n+1 .
In general it is unusual to know explicitly the map Φ τ determining a dynamical system, instead one has some kind of local rule generating dynamics. For example, on a phasespace or symplectic manifold one typically considers a Hamiltonian vector field X H determined by a Hamiltonian function H whose integral flow determines dynamics in terms of parameterized paths. However, for Z a phase-spacetime, we only need determine unparameterized geodesics. For that a local rule is generated by a maximally non-degenerate (so rank 2n) two-form ϕ ∈ Ω 2 Z, because an unparametrized path γ can be determined from the equation of motion (2.1) ϕ(γ, ·) = 0 .
In the aboveγ is the tangent vector to γ with respect to any choice of parameterization.
Example 2.1. Massless relativistic particle: Let Z = R 7 ( k, x 0 , x) and which is the tangent vector to the trajectory of a massless particle in Minkowski space moving in the directionk.
When the two-form ϕ is closed we may write down an action principle for local dynamics, because locally ϕ = dα for some α ∈ Ω 1 Z. Then the action is extremized under compactly supported variations of the path γ precisely when the Equation of Motion (2.1) holds. Changing α by an exact term dβ does not change the equation of motion. Because we wish to focus on systems with an action principle, from now on, we shall assume that ϕ is closed. When ϕ ∈ Ω 2 Z is closed and maximally non-degenerate, we call (Z, ϕ) a dynamical phase-spacetime. The closed, maximally nondegenerate two form ϕ is termed an odd symplectic form. (Sometimes the additional data of a volume form is added to the definition of an odd symplectic manifold, see [6].) The data of α modulo gauge transformations α → α + dβ, whose curvature ϕ = dα is maximally non-degenerate, also determines the dynamics, and we shall call this a dynamical connection. Given a dynamical system (Z, ϕ), a function Q ∈ C ∞ Z is said to be conserved when for any vector field ρ obeying ϕ(ρ, ·) = 0. Note that L f ρ Q = f L ρ Q for any f ∈ C ∞ Z. A vector field u is said to generate a symmetry when Conserved quantities and symmetries obey a Noether theorem: Given Q conserved, any solution u to is a symmetry because L u ϕ = d ι u ϕ = d 2 Q = 0. Vector fields f ρ correspond to trivial (constant) conserved charges. The equation displayed above is always solvable for u because ι ρ dQ = 0, and the solution for u is unique modulo a term f ρ. Conversely, given a symmetry u, then the one-form ϕ(u, ·) is exact because dι u ϕ = L u ϕ = 0. Hence, locally, there always exists a smooth conserved Q solving the above display.
Experimentally, the initial conditions for a dynamical system cannot be determined with infinite precision. Instead, one associates an experimental uncertainty to an open ball in Z about about some initial point z 0 ∈ Z. Therefore, a notion of volume is needed to ascertain the accuracy of such an initial measurement. Because ϕ is maximally nondegenerate, the form ϕ ∧n ∈ Ω 2n Z is non-vanishing. To make a top form, we need to exterior multiply this by some one-form. When ϕ = dα-which is certainly true locally and for many physical systems and most experimental apparatus can anyway only handle a measurements in a finite range-a natural choice is the one-form α, in which case we define A priori, the "volume form" Vol α need not be non-degenerate, and moreover it depends, up to an exact term, on the choice of α; Vol α+dβ = Vol α +d(βϕ ∧n ). For some systems, it may suffice only to have a non-degenerate volume form defined on suitable hypersurfaces of Z by ϕ ∧n , but generally we focus on systems where Vol α is a volume form and view the choice of α as part of the data of the system required to give a good measurement theory. The case where ϕ is globally the d of a given α and Vol α is non-degenerate is in some sense optimal. In this case α is called a contact one-form. The data (Z, α) is a strict contact structure and the dynamics on Z determined by the uniquely defined vector field ρ ∈ Γ(T Z) such that are called Reeb dynamics and ρ is the Reeb vector field. The canonical parameterization of path γ in Z such thatγ = ρ is akin to that of geodesics in a Riemannian manifold by their proper length. A contact structure (Z, ξ) is the data of a hyperplane distribution ξ in T Z determined by the kernel of a contact one-form. A useful starting reference describing contact geometry is [7].
Example 2.2. Massive relativistic particle: Let Z = R 3 (p, t, x) and Then so α is a contact one-form. It not difficult to verify that the Reeb trajectories correspond to those of a mass m, relativistic particle in two dimensional Minkowski space.

Quantization
Our main focus is quantization of dynamical phase-spacetimes. We follow the treatment in [3]. The framework is quite close to that developed by Fedosov for deformation quantization of symplectic manifolds [8] (see also [9]). To quantize a dynamical phasespacetime (Z, ϕ) we begin with an action on paths γ in Z where α is any primitive of ϕ, ϕ = dα .
For locally supported variations of γ, the potential failure of α to be unique or to exist globally, is irrelevant. The above action can be reformulated as a Hamiltonian system on a coisotropic submanifold C of a larger symplectic manifold Z, where In the above, ξ * is the bundle of hyperplanes in T * Z defined by the kernel of the ray defined by the kernel of ϕ. So, for example, if ϕ(r, ·) = 0 for the ray r, then ξ * = ker r. In the case that α is a global contact form, this is the dual of the maximally non-integrable distribution ξ defining a contact structure. Motivated by this correspondence, we call ξ * a codistribution. The direct sum above is the Whitney vector bundle sum, so Z has base Z z i and fiber R 2n+1 ⊕ R 2n (p i , s a ). We call Z an extended phasespace, it has a symplectic current (or Liouville form) Λ obtained from the sum of the tautological one-form on T * Z and the standard Liouville form λ s on the R 2n fibers of ξ * . In local coordinates (z i , p i , s a ) for Z, This makes Z into a symplectic manifold. Here j ab = −j ba is an odd bilinear form defining the invariant tensor of Sp(2n). Also, we employ the convention X a = j ab X b . A coisotropic submanifold C of this, is determined by the 2n + 1 first class constraints where the one-form A = α + κ(z, s) + a(z, s) obeys the Cartan Maurer equation Also, the leading fiberwise jet of the one-form κ(z, s) must obey the maximal rank condition κ(z, s) = e a (z)s a , In the above, the 2n one-forms e a are a basis, or adapted coframe, for the codistribution ξ * , such that e a (ρ) = 0. We shall often refer to these as soldering forms. The extended action for paths Γ in ξ, obtained by integrating out the momenta p i by solving the constraints C i , and given by then gives a (gauge) equivalent description of the dynamics given by the original action S[γ] above. Moreover, the constraints C i are abelian, so the classical BFV-BRST charge is simply Q BRST = c i (p i − A i (s, z)) where c i are ghosts. This charge is "nilpotent" (strictly its BFV Poisson bracket with itself vanishes) by virtue of the Cartan-Maurer equation. In the above discussion we have effectively converted a system with a mixture of first and second class constraints to one with only first class constraints, this is an example of a general technique due to [10]. The extended action easily quantized by viewing the ghosts c i as one-forms on the base manifold Z (see for example [11]), so that i c i p i becomes the exterior derivative d acting on forms on Z and the fiber coordinates s a become operatorsŝ a ∈ End H acting on some choice of Hilbert space H and subject to In fact, since states are defined by complex rays, we only need to consider the projective Hilbert space P(H) [|ψ ] = [[e iθ |ψ ] (for any real θ). We will abbreviate the notation P(H) by H in what follows.
The BRST Hilbert space H BRST is then differential forms on Z taking values in H. At ghost number zero, these are sections of a Hilbert bundle HZ which is a vector bundle with fibers given by H and the base manifold is Z. In general BRST wavefunc- The ghost number grading for space is given by form degree. Note that the frame bundle of the dual ξ := (ξ * ) * of the codistribution ξ * canonically defines a principal Sp(2n) bundle over Z, because ϕ endows ξ with non-degenerate, skew symmetric bilinear form. This canonically and globally defines the associated vector bundle with fibers given by H. In turn the principal bundle of orthonormal frames with respect to the Hilbert space inner product has structure group U (H). Let us denote by HZ the associated vector bundle with fibers H transforming under the fundamental representation of U (H). This is the bundle of wavefunctions defined up to unitary equivalence where the base manifold Z plays the rôle of a generalized time coordinate. . The quantum BRST charge is a flat connection form on HZ; this extends by linearity to higher forms in H BRST . It is given by whereÂ is a one-form taking values in Hermitean operators on H is the quantization of the one-form A, and similarly to the classical solution A(z, s) to the Cartan-Maurer, decomposes as We require thatÂ ∈ End(Γ(HZ)) ⊗ Ω 1 Z gives a solution of the flatness condition Given a connection form d +Â, the map is called the quantum calibration map. We require that it obeys the Heisenberg algebra, in the sense that for any u, v ∈ Γ(T Z) This can be solved locally using the coframe by writingκ = e aŝ a . Therefore this map calibrates quantum operatorsŝ to the underlying classical phase-spacetime manifold Z. Indeed the above map always exists, because H comes equipped with a representation of the Heisenberg algebra; this is uniquely defined up unitary equivalence by the Stone von Neumann theorem. The mapκ is a symplectic analog of Clifford multiplication for spinor bundles, see [12]. For any quantum system with a classical limit, or that arises via quantization, there must be a flow with respect to some parameter encoding how quantum quantities respond to changes in . Therefore, we introduce a grading operator where the operator E : Γ(HZ) → Γ(HZ) obeys This grading operator is also needed to state the quantum analog of the classical higher jet condition in Equation (3.1). For example, in any choice of polarizationŝ Let us say that an operatorÔ ∈ End(Γ(HZ) has grade k when We require that the operatorâ in Equation (3.2) has grade greater than or equal to zero. Then, given the data of a dynamical connection α and a quantization mapκ, a solution to the flatness condition ∇ 2 = 0 will be called a quantum connection. When only a formal power series solution for ∇ is given, we call this a formal quantum connection. Not that the first and third term of the above connection are the starting point for the geometric quantization of contact manifolds developed in [13]. Local existence of formal quantum connections given the data (Z, α,κ, gr) is not difficult to establish [3] (alternately, see the Darboux construction of a flat quantum connection given in Section 3.1). Indeed, when the calibration map is given by coframes, the operatorâ can be expanded order by order in the gradingâ In the above, the one-forms ω a 1 ...a j are totally symmetric in the labels a 1 , . . . , a j , and are determined by solving a system of algebraic, zero curvature, equations.
3.1. Quantum Darboux Theorem. Locally, the contact Darboux theorem states that there exist coordinates ( π, χ, ψ) such that any contact form α can be written as Here the Reeb vector is ρ = − ∂ ∂ψ so that evolution is along straight lines of constant π and χ. In the case where only a dynamical phase-spacetime (Z, ϕ) is given, because ϕ is closed we may always locally write ϕ = dα . Moreover, in the case that α is not contact, because ϕ is non degenerate, we may add to α an exact term dβ such that α = α + dβ is a contact form, at least locally. Therefore, the contact Darboux theorem applies to dynamical phase-spacetimes as well.
There is a quantum analog of the Darboux theorem for formal quantum connections [5]. Before discussing this we need to talk about gauge transformations for quantum connections. In the previous section we stipulated that the quantum connection ∇ was a flat connection form on the bundle HZ. This is an associated vector bundle to a principal U (H) bundle over Z. The connection form obeys a self-adjoint condition , for any u ∈ Γ(T Z), and the adjoint is defined fiberwise using the adjoint of H.
The quantum Darboux theorem states that locally any pair of formal quantum connections for a given choice of dynamical phase-spacetime (Z, ϕ) are formally gauge equivalent [5]. The result is established inductively in the grading gr by showing that there exists a formal U (H) gauge transformationÛ such that Here ∇ D is a quantum connection whose quantum calibration mapκ is closed so that To see that such a connection always exists locally, one can use a Darboux coordinate ball, for which ϕ = dα and where α is contact with enjoys coordinates ( π, χ, ψ) such that Then the quantum calibration mapκ = e aŝ a can be built from closed coframes e a = (d π, d χ) .
The quantum calibration map for a general quantum connection ∇ will not be given in terms of closed frames, but an Sp(2n) gauge transformation in the frame bundle of ξ * can be employed to achieve this. Locally, as discussed earlier, this lifts to an M p(2n) gauge transformationÛ 0 . The strategy to findÛ is first to find the metaplectic transformationÛ 0 . Then, , for someâ 0 of grade 0 or greater. Thereafter one solves for a form gauge transfor-mationÛ 1 of grade one higher, given as a formal series in the grading, such that This means we must solve the equation is the adjoint action of ∇. Note that acting with ∇ again on the above, just returns ∇ 2 0 = 0 as an integrability condition. The above display has a formal (and possibly only local) solution forÛ −1 1 [5]. The proof is by induction in the grading.
A main aim of this article, is to study explicit, global (but possibly formal), solutions to the above equation for quantum mechanical systems describing dynamics on a line. Knowledge of the gauge transformationÛ 1 is powerful, because it relates non-trivial interacting systems to their trivial Darboux counterparts. The first step is to study the classical Darboux theorem for these models.

Contact Hamilton-Jacobi Theory
Let us consider a one-dimensional system with time-dependent Hamiltonian H(p, q, t). The standard Hamilton-Jacobi theory for this system (see [4] for a modern treatment) can be recovered by studying diffeomorphisms on a three-dimensional dynamical phasespacetime manifold Z with local coordinates (p, q, t) and odd symplectic form This gives dynamicsγ where where ∝ denotes equality up to multiplication by some non-vanishing function on Z. Calling the worldline parameter τ , and choosing this function to be unity, the equations of motion are In the gauge t(τ ) = τ + c these are the standard Hamilton's equations.
The odd symplectic form ϕ can be written as the exterior derivative of the one-form Away from the zero locus of p ∂H ∂p − H, the above form is in fact contact. The contact Darboux theorem ensures that locally we can find a new local coordinate system (π, χ, ψ) such that Our aim in this section is to give an (as) explicit (as possible) formula for the diffeomorphism bringing α, at least on some open set U , to its Darboux form displayed above.
The Reeb vector for α as in Equation (4.1), is given in these Darboux coordinates by ρ = − ∂ ∂ψ , while in the original coordinate system To proceed we need to know one conserved quantity K ∈ C ∞ U ,  H(p, q) is itself a solution. In what follows we assume that a solution for K is known (possibly approximately or even numerically). Then we make an ansatz for the sought after diffeomorphism: π = K(p, q, t) , Since L ρ π = 0 = L ρ χ and L ρ ψ = −1, using Equation (4.3) we must have Comparing the right hand sides of Equations (4.1) and (4.2) and using the Ansatz (4.5) gives a triplet of PDEs which we wish to use to determine φ and λ: Differentiating the second equation with respect to p and the first with respect to q and then taking the difference yields Assuming that the equation ε = K(p, q, t) can be solved for p = p(ε, q, t) we can solve the PDE given by the above display by first writing φ(p, q, t) =: φ(K(p, q, t), q, t) .
Similarly, multiplying the second equation in Display (4.6) by ∂K/∂p and the first by ∂K/∂q, the difference yields This can be solved for λ(p, q, t) =: λ(K(p, q, t), q, t) using the same method employed for φ: .
In summary, the diffeomorphism to the Darboux coordinate system is given by π = K(p, q, t) , .

Example 4.1. For a time independent Hamiltonian
The stated approximations are accurate in the limit when the deformation of the harmonic oscillator v(q) < < q 2 . In the harmonic oscillator limit when v(q) = 0, it is easily checked that indeed Now that we can explicitly locally map a wide class of classical dynamical systems to one another using the Darboux theorem, we proceed to study the quantum analog that was discussed in Section 3.1. We shall focus on the quantum anharmonic oscillator.

The Quantum Anharmonic Oscillator
Let us consider the model with Hamiltonian with a single-well potential V that is smooth, concave up and obeys V (0) = V (0) = 0.
The classical phase-space curves for this model are concentric closed orbits. The phasespacetime is R 3 (p, q, t) and ϕ = dp + V (q)dt ∧ dq − pdt .
In the Darboux coordinates (χ, π, ψ), the physical trajectories are straight lines. Clearly for this model, these can be globally mapped to the trajectories in (p, q, t) space. These are helix-like and foliate the phase-spacetime. Standard quantization of the Hamiltonian (5.1) replaces the c-numbers p and q by quantum operators acting on the Hilbert space H = L 2 (R) given by square integrable, complex-valued functions of S. A common choice of quantum Hamiltonian is then . In principle, the space of all quantizations of the classical Hamiltonian in Equation (5.1) ought be encoded in the space of flat quantum connections on the phase-spacetime. This issue is of independent interest, but we avoid studying it for now, and instead focus on the simple quantization in the above display.
Our aim is to compute the evolution operator exp − i(t f −t i ) H or its matrix elements This the usual propagator problem in non-relativistic quantum mechanics, which can be handled perturbatively by various quantum mechanical techniques. Here we want to demonstrate a rather different approach based on the quantum Darboux theorem.
First we need to rewrite the operator exp − i H(t f − t i ) as a path ordered line operator P γ exp −´γ A for a quantum connection ∇ = d + A acting on a Hilbert bundle over phase-spacetime. For that, we need a solution for ∇ corresponding to the quantum Hamiltonian in Equation (5.2). A solution is given by [3] Observe that along the path the connection potential It is tempting to take a path γ along which both p and q vanish. However we avoid this choice because the Jacobian for a change of variables (p, q) → (π, φ) vanishes along such a path. Because ∂/∂S = ∂/∂(S + q), there is no difficulty working along a path with constant q = 0, because this just shifts the variable S in wavefunctions.
The soldering forms for the above connection are given by e a = dp + V (q)dt, dq − pdt =: (f, e) , so that ϕ = 1 2 j ab e a ∧ e b = e ∧ f and j 12 = 1 = −j 21 = j 12 = −j 21 . These forms vanish along γ. Also note thatŝ a = − S, i ∂ ∂S and [ŝ 1 ,ŝ 2 ] = −i . It now follows that Our next goal is to find a gauge transformation mapping (at least formally) the connection ∇ to a far simpler one. Before doing that, it useful to discuss quantum symmetries of this system.

Quantum Noether Theorem.
Since quantum dynamics is given by parallel transport with respect to a quantum connection ∇ = d + A, quantum symmetries ought be given by operators O on the Hilbert bundle that obey for any operator O , since if Ψ ∈ Γ(HZ) solves ∇Ψ = 0, then so too does OΨ.
Such operators O are easy to construct: Let u ∈ Γ(T Z) be a vector field. Then, because ∇ is nilpotent, the operator {ι u , ∇} commutes with ∇. Acting on sections of the Hilbert bundle, this gives the operator Note, that the Lie derivative is defined acting on sections of ∧ • Z ⊗ HZ is defined by the anticommutator {ι u , d}. Symmetries of the above type that hold for arbitrary vector fields u are tautological, in the sense that the first operator acts along the base Z, while the second acts on the Hilbert space fibers in a way that exactly compensates the former transformation. In BRST terms, these symmetries are BRST exact. However, specializing to vector fields u that solve (5.5) [L u , ∇] = dβ i for some β ∈ C ∞ Z, we can define a conserved quantum charge It is easy to verify that [∇, Q u ] = 0 .
Note that the leading term of Q u in the grading is α(u) − β which gives a classical conserved charge Q that solves Equation (2.2) because Here we used that Equation (5.5) implies L u α = dβ and ϕ(ρ, ·) = 0. Let us call vector fields u obeying Equation (5.5) quantum symmetries of ∇.
The simple model in the next example will be important.
Example 5.1. Let Z = R 3 (π, χ, ψ) and This equals dα where α = πdχ−dψ is contact and the coordinates (π, χ, ψ) are Darboux. Then a quantum connection is The vector fields ∂ ∂π , ∂ ∂χ , ∂ ∂ψ , are quantum symmetries with β equaling χ, 0, 0, respectively. Their conserved quantum charges are Acting on the Hilbert bundle, these charges obey the Heisenberg Lie algebra which is also the algebra of the three contact Hamiltonian vector fields ∂ ∂π + χ ∂ ∂ψ , ∂ ∂χ and ∂ ∂ψ . Returning to the anharmonic oscillator, we note that it has two non-trivial, independent, globally defined, conserved charges. One of these is the Hamiltonian H = 1 2 p 2 + V (q), or simply π in Darboux coordinates. The other is t − φ(p, q), or equivalently −χ. This says that the angle variable (see Equation (4.8)) obeys φ(p, q) = t + constant. Or in other words, the initial value of the angle variable is preserved along classical paths. It is interesting to study the quantization of the conserved quantities H and t − φ. The former is simple because for the quantum connection form given in Equation (5.3). Hence we can construct the quantum charge corresponding to the strict contactomorphism generated by ∂ ∂t . This gives It is easy to verify that the above operator commutes with ∇. Moreover, at p = 0 = q, this recovers the standard quantum Hamiltonian H. The quantum charge corresponding to t − φ(p, q) is more involved. The quantization given by the quantum connection form ∇ in Equation (5.3) does not obey Condition (5.5) for the vector field u = ∂ ∂π + χ ∂ ∂ψ . This does not mean that there is no corresponding charge, but rather the simple Formula (5.6) cannot be used. Instead, given a quantum gauge transformationÛ relating ∇ to ∇ D as in Equation (3.3), then the quantum chargeÛ −1 Q ∂ ∂πÛ commutes with ∇. Unfortunately we do not yet know the gauge trans-formationÛ . The computation of this operator is the subject of the next section.

Quantum Gauge Transformation.
We want to compute the quantum gauge transformationÛ given by a unitary endomorphism of the section space of the Hilbert bundle Γ(HZ) such that where ∇ is the connection form corresponding to the anharmonic oscillator in Equation (5.3) while ∇ D is the Darboux connection in Equation (5.7). For this we will work iteratively order by order in the grading. The first step is to find a gauge transformation relating the calibration maps of the two connection forms.

Metaplectic transformation.
To begin with, we need the relation between the soldering forms (dπ, dχ) of the Darboux connection ∇ D to those-given by (dp + H q dt, dq − H p )-of ∇. (For brevity, from now on we often denote partial derivatives by subscripts). This is given by The above display was computed using Equation (4.5) specialized to the case K = H(p, q) for which the functions φ and λ are t-independent. In the above is the Jacobian of the change of variables (p, q) → (π, φ). Notice also, that the last equality was achieved using that the Poisson bracket of {π, H} PB = 0 and {φ, H} PB = 1 (see Equation (4.7)). This implies that Hence the matrix U 0 is Sp(2)-valued with respect to the antisymmetric bilinear form We want to intertwine the Sp(2) group element U 0 expressed in the fundamental representation in Equation (5.10) to an operatorÛ 0 acting on sections of the Hilbert bundle. This operator must obeŷ I.e., the operatorÛ 0 transforms the Darboux solderings to those of ∇. The map from the matrix U 0 to the operatorÛ 0 is the intertwiner from the fundamental representation of Sp(2) to its metaplectic representation on the projective Hilbert space P(H). Note that strictly only the double cover M p(2) of Sp(2) has a metaplectic representation on the Hilbert space, but upon projectivizing, this gives a unitary Sp(2) representation. Formally, the projective metaplectic action of any Sp(2) matrix V is determined by the formulae The action of these operators on wavefunctions ψ(S) can be computed using suitable Fourier transforms. In particular (up to irrelevant normalizations) (5.11) Also, again up to an irrelevant normalization, For future use, note that at the level of the Lie algebra sp(2)-recycling the hat notation for this-one has In the above M a b := −a b c a and M a b =: j bc M ac . Also note that if AD − BC = 1, then The above formula is the intertwiner between the fundamental representation of Sp(2) and its projective metaplectic representation. By now we have achieved that The difference between ∇ 0 and ∇ in Equation (3.4) is given by (see also Equation (5.3)) In the above v 2 := V (q + S) − V (q) − V (q)S, and the above-displayed operator only has terms of grade zero and higher.
To complete the computation ofâ 0 , we must calculateÛ −1 0 dÛ 0 . Because this oneform is Lie algebra-valued, we instead compute U −1 0 dU 0 . Note that the matrix U 0 in Equation (5.10) only depends on the variables p and q, or equivalently only on the pair (π, φ). Moreover, χ = −t + φ, so observe that-acting on functions that depend only on (p, q)-the exterior derivative can be written Then, using {φ, π} PB = 1, after some computation, it follows that (5.15) where the Hessian matrix Hess(f ) := f pp f pq f pq f qq . Because the Hessian is symmetric, it follows that J Hess(f ) ∈ sp(2).
Using Equation (5.13), orchestrating the above computations giveŝ where v 3 := V (q + S) − V (q) − V (q)S − 1 2! V (q)S 2 and the angle variable is given explicitly in Equation (5.9). Happily-and necessarily on general grounds-the first two terms in the above display have grade zero, and lie in the codistribution. Assuming real analyticity of V (S), the last term has grades one and higher. Next we need to compute the higher order gauge transformationÛ 1 subject to Equation (3.5).

Higher order gauge transformations.
To computeÛ 1 we work in a formal power series in the grading. Examining (3.5), we see that it is simpler to computeÛ −1 1 , which we expand asÛ Here W abc is some totally symmetric tensor to be determined. Then the lowest order contribution to Equation (3.5) implies that Using Equations (5.13,5.15) and calling Π a = (π, φ) and ∂ a = (∂ p , ∂ q ) we have that Moreover, using that ∂ [a Π e ∂ b] Π f is proportional to the Poisson bracket of Π e and Π f , and that {φ, π} PB = 1, it follows that W abc is totally symmetric. Now let us examine higher order terms inÛ −1 1 . Let us call the grade 1 object Along, similar lines, the grade zero part of ∇ in Equation (5.3) is Then, at grade 1, Equation (3.5) demands that To solve the above equation it is important to remember thatÛ −1 1 must be a unitary operator. Thus we must require so the hermitean part ofŴ (2) is given by The anti-Hermitean part aHe(W (2) ) is still undetermined. However Equation ( To proceed we need to compute the right hand side of the above expression. This is tractable computation, but clearly as we move to even higher orders the complexity of such computations will grow dramatically. Hence, we digress to develop a diagrammatic representation of the operator differential forms appearing in the above discussion.
Here the solid lines correspond to the indices a 1 , . . . , a n and these may be permuted at no cost because partial derivatives commute. The tensors j ab and its inverse j ab are denoted by directed line segments j ab := and j ab := .
Concatenated lines indicated index contractions so, for example, and (5.21) = j ab j cb = δ c a . The identity above allows the depicted concatenation to be removed from a diagram. Also, note that reversing the direction of any arrow multiplies the tensor depicted by a minus sign.
More complicated diagrams obtained by such concatenations have the drawback that symmetry of external legs may be broken, and thus it may no longer be possible to uniquely associate a tensor to such a picture. Moreover, we are typically interested in Hilbert space operators taking values in differential forms built from tensors made from derivatives and products of Π a 's. Hence we adopt a "heaven and earth notation" in which a line attached to the earth (a horizontal green line) denotes contraction with the operatorŝ a . Similarly, a line attached to heaven denotes contraction with a differential form, the choice of which will be labeled when this is not clear. For examplê Now the Poisson bracket {φ, π} = 1 implies that ∂ a Π c j cd ∂ b Π d = j ba so we have the diagrammatic identity (5.22) = .
These relations imply two identities for heaven and earth diagrams The last equalities used that [ŝ a ,ŝ b ] = i j ba and j ab dΠ a ∧ dΠ b = 2ϕ.
Since a solid line attached to a solid dot denotes a partial derivative of Π, there are further identities following from the product rule of, for example, the schematic type Hence, in a heaven and earth diagram, we can remove a leg from a solid dot and produce (minus) a sum of diagrams with that leg attached to all other solid dots plus a term where this derivative acts on the whole diagram. The latter is denoted by a solid line that ends midair between heaven and earth and which denotes a partial derivative acting to the right. An example is the following identity The last equality used thatŝ a ∂ a i = 0. The above is a diagrammatic demonstration that the tensor W abc = j f e ∂ a Π e ∂ b ∂ c Π f is totally symmetric. To emphasize this we diagrammatically denote := .
We need one further ingredient to attack Equation (5.20); the exterior derivative. For that we note that acting on functions that are independent of t, we have Thus we have the diagrammatic notation for the exterior derivative Before proceeding, we need one further diagrammatic identity expressing that [ŝ a ,ŝ b ] = i j ba , namely where the shaded blobs denote the (unspecified) remainder of the diagram. Now we are ready to diagrammatically compute the right hand side of Equation (5.20). We begin with [d,Ŵ (1) ]: Now let Ω a be any pair of differential forms. Then we have the following identities Here we used that = and = 0. For the remaining commutator terms in Equation (5.20) we need the following computation

Now note that
where dT a := (0, dt) =: dΠ a − dX a . Here X a = (π, χ). Thus A check of this result is that it is antihermitean. This amounts to reversing the order of all the legs attached to the earth and then using the identity in Equation (5.24) to restore these to the original pictures. Hence we compute Here we used Equations (5.21) and (5.23) as well as the identity Thus we learn aHe(Ŵ (2) ) = − 1 24i + 1 16 Note that it is not difficult to see that the right hand side above is antihermitean using the identity Let us define where both sides of this diagrammatic picture define a totally symmetric rank four tensor. Then is not difficult to check that It is interesting to study even higher orders. For example, to computeŴ (3) , one can first consder the next order in Equation (5.19) which determines The remaining difficulty now is to calculate the heaven and earth diagrams for aHe(Ŵ (3) ).
Clearly this is possible but tedious. We have verified that in the model harmonic oscillator case where V (q) = 1 2 q 2 , the obvious conjecture that symmetrizing the diagram computes aHe(Ŵ (3) ), in fact fails. The problem of computing an all order expression forÛ is related to the problem of finding deformation quantizations for Poisson and symplectic structures. This problem is solved by Kontsevich's formality theorem [14], which is neatly explained by the perturbative expansion of a Poisson sigma model performed by Cattaneo and Felder [15]. It is possible that those ideas could be applied to the computation of a formal asymptotic series result for the gauge transformationÛ .

Correlators.
We are now ready compute the evolution operator, which using Equation (5.4) is given by Here γ is any path betwen z i = (0, q, t i ) and z f = (0, q, t f ) (in the (p, q, t) coordinate system). For the moment, let us focus on the quantity on the right hand side in the more general case that z i and z f are any two points in the manifold Z. In the previous section we computed the gauge transformationÛ such that where A D is the quantum connection potential in Darboux form as in Equation (5.7). The above operator is easy to compute (see for example [5]): Consider γ to be a path beginning at z i = (π i , χ i , ψ i ) and ending at z f = (π f , χ f , ψ f ). Because the connection ∇ is flat, we may take γ to particularly simple, for example one along which first only ψ changes by amount ψ f − ψ i , then π by π f − π i and finally χ by χ f − χ i . This gives Because γ is a path between z i = (0, q, t i ) and z f = (0, q, t f ), taking H = 1 2 p 2 + V (q) with V (0) = 0, the above becomes (see Equation (4.10)) simply a translation operator We also need to computeÛ |S i . Viewed as a wavefunction, the state |S i is represented by δ(S − S i ). Now, remember that whereÛ 0 is the metaplectic representation of the matrix U 0 given in Equation (5.10). Because p and q do not change along the path γ, at both the start and endpoint we have Then using Equation (5.14), we havê Next we imagine rewriting the remainder of the gauge transformation in normal order,i.e. The above Fourier convolution result for quantum mechanical propagators is a very strong result, but it comes with a caveat which we now describe.
The equality in the above convolution result assumes that we can find an exact expression for the operatorÛ 1 . However, in general, only an asymptotic series expression forÛ 1 will exist. To see why this is, it is useful to study the harmonic oscillator example V (q) = 1 2 q 2 . In that case the operatorÛ 1 must solve the condition Because the spectrum of the harmonic oscillator Hamiltonian is discrete, while that of the operator qS is continuous, no unitary operatorÛ 1 solving the above operator equation can exist. However, there is a asymptotic series solution. Let us complete our study of the quantum Darboux theorem by demonstratng how these asymptotics work for the harmonic oscillator. The first three asymptotic orders of the operatorÛ were computed for general models using heaven and earth diagrams in Section 5.2.3. To explicate this series it is useful to introduce the new variable in terms of which the display before last becomes For the harmonic oscillator, the pair Π = (π, φ) = ( 1 2 p 2 + 1 2 q 2 , − arctan p q ). Hence, at the start point of a path γ with p = 0, the operatorŴ (1) given in Equation (5.17) becomeŝ while the anti-Hermitean part ofŴ (2) given in Equation (5.25) is aHe(Ŵ (2) ) = ε 2 4! 2σ ∂ 3 ∂σ 3 + 6σ 3 ∂ ∂σ + 3 ∂ 2 ∂σ 2 + 9σ 2 . Thus (5.29)Û 1 = 1 +Ŵ (1) + aHe(Ŵ (2) ) + 1 2Ŵ .
It is not difficult to check thatÛ † 1Û 1 = 1 + O(ε 3 ) , and that . Note that as mentioned in the previous section, it can be easily checked that the anti-Hermitean part ofŴ (3) is not given solely by the five point diagram displayed there.