Quantum geodesics in quantum mechanics

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or `geodesics') is exactly Schr\"odinger's equation. The connection $\nabla$ preserves a generalised `skew metric' given by the canonical symplectic structure lifted to a certain rank (0,2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to the Klein Gordon equation on Minkowski spacetime with a background electromagnetic field, formulating quantum `geodesics' on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include a relativistic free particle wave packet and a hydrogen-like atom.


Introduction
Quantum Riemannian geometry, in the sense of quantum metrics and connections on possibly noncommutative 'coordinate algebras', has been extensively developed since the 1980s and now has an accepted role as a plausibly better description of spacetime (i.e.'quantum spacetime') that includes Planck scale effects.Many authors have written on this topic and we refer to our book [11] for a bibliography as well as an introduction to a particular constructive approach developed in [7,8,9], among other works, which we will adopt.This approach uses bimodule connections [14,27] and quantum metrics to build up the quantum Riemannian geometry on a chosen differential algebra, making it complementary to the well-known Connes' approach where the noncommutative geometry is encoded in a spectral triple [13] or abstract Dirac operator as starting point.The two approaches can be compatible and with interesting results where they meet [9,20].Recent applications to models of quantum gravity itself are in [26,1,21].
In the present paper, we apply this powerful machinery of quantum Riemannian geometry to the more obvious context of ordinary quantum mechanics and quantum theory.Here the noncommutativity parameter will not be the Planck scale but just the usual ̵ h.The noncommutativity inherent in quantum theory has long been one of the motivations for results in operator algebras in general and noncommutative geometry in particular, and the role of latter in actual quantum systems has already been noted in Connes' approach, for example to understand the quantum Hall effect [3].The role of the quantum Riemannian geometry formalism [11], however, has not been explored so far in this context but makes sense once we note that the 'quantum metric' need not be symmetric.Rather, we will be led to an antisymmetric 'generalised quantum metric' and on an extended phase space where we adjoin time.The quantum metric will be degenerate and ∇, although compatible with it, will have a small amount of torsion, both features relating to the extra time direction.Thus, there are some differences but in broad terms we will effectively formulate ordinary quantum mechanics somewhat more in the spirit of gravity, rather than the more well-studied idea of formulating gravity in a quantum manner.
We make use of the notion of 'quantum geodesics' with respect to any ∇, as recently introduced in [4].The preliminary Section 2.1 provides the algebraic definition of a 'geodesic bimodule' E and its application to one classical geodesic in a manifold, and Section 2.2 its further application for another choice of E to a dust of particles with density ρ, where each particle moves along a classical geodesic.The tangent vector to all these particles will be a vector field X obeying an autoparallel 'geodesic velocity equation'.The actual particle flows are then given classically by exponentiating the vector field X to a diffeomorphism of the manifold, while the natural way to do this in our algebraic formulation turns out to be a corresponding flow equation not for ρ but for an amplitude ψ, where ρ = ψ 2 .This algebraic formalism then makes sense when the coordinate algebra of the manifold is replaced by a noncommutative algebra A, i.e. in noncommutative geometry.For our purposes now, we need to go further and Section 3 introduces a new choice of E in which the 'geodesic flow' takes place more generally on a representation space of an algebra A rather than on A itself.We can then apply this in Section 4 to A the Heisenberg algebra in the Schrödinger representation, allowing us to express the standard Schrödinger equation for a Hamiltonian h = p 2 2m + V (x) as a quantum geodesic flow.The new result here is not the flow equation, which is just Schrödinger's equation, but the noncommutative geometric structures that we find behind it: quantum differential forms, the quantum vector field X, the antisymmetric quantum metric and the metric compatible connection ∇.
Specifically, the algebra A in Section 4 will be the standard R 2n Heisenberg algebra (1.1) [x i , p j ] = i ̵ hδ i j , [x i , x j ] = [p i , p j ] = 0, 1 ≤ i, j ≤ n, but equipped now with a certain differential calculus Ω 1  A defined by the Hamiltonian h.This idea to use the freedom of the noncommutative differential structure to encode the physical dynamics is in the spirit of [25] where one of the authors showed how Newtonian gravity can be encoded in the choice of differential structure on quantum spacetime, but now applied to quantum mechanics.Specifically, the exterior algebra in Proposition 4.1 has the commutation relations between functions and differentials, where θ ′ is a graded central extra direction initially with no classical analogue but is dictated by the algebra.Here for any fixed i makes clear that this has its origins in the noncommutativity of the quantum geometry.The need for an extra direction θ ′ in the cotangent bundle has emerged in recent years as a somewhat common phenomenon in noncommutative model building [31,22,23,24].Its associated partial derivative in the expansion of the exterior derivative d is typically a second order 'Laplacian' of some kind and that will be our case as well.Mathematically, it means that the calculus we use is a central extension of a commutative differential calculus on the Heisenberg algebra, which is recovered by projection via θ ′ = 0.All of our results become empty if we set θ ′ = 0, which means that our entire point of view is purely quantum and not visible at the classical level.Rather, we find that the natural interpretation of this emergent 1-form is θ ′ = dt on an extended heisenberg algebra Ã where we adjoin a central variable t.This fits in with the idea that highly noncommutative systems tend to generate their own evolution [22].(This is different from but reminiscent of the observation that von Neumann algebras have an associated modular automorphism group.)In our specific case, and using this extra cotangent dimension, we will arrive at a rather unusual geometric picture in which the central 1-forms are covariantly constant under ∇ and killed by the geodesic velocity field so that X(ω i ) = X(η i ) = 0.If we identify θ ′ = dt for an external time variable t, then setting ω i = η i = 0 exactly reproduces a quantum version dp i = − ∂V ∂x i dt, dx i = pi m dt of the Hamilton-Jacobi equations of motion in our approach.The antisymmetric quantum metric in Proposition 4.3 is where the second expression shows its origin in a lift of the classical symplectic 2form ω = dp i ∧dx i and the first expression shows that G vanishes on solutions of the Hamilton-Jacobi equation of motion.The classical interior product i X h (ω) = dh saying that X h is the Hamiltonian vector field for the Hamiltonian function appears differently now in our extended phase space geometry as (X⊗id)(G) = (id⊗X)(G) = 0, i.e. as the kernel of the antisymmetric quantum metric.
Section 5 does the same as Section 4 but for A now the Heisenberg algebra with x a , p b Minkowski (eg 4-vectors and covectors) and with quantum differential calculus defined now by an external U (1) gauge potential.In this section, t = x 0 c is a coordinate variable (with metric -1 in this direction and c the speed of light) and the quantum geodesic time parameter is instead denoted s.Moreover, [p a , p b ] are no longer zero when there is electromagnetic curvature.This time, we propose a novel quantum 'geodesic' evolution based on the Klein-Gordon operator, which was not previously studied but is suggested by our formalism as a relativistic version of Section 4. The geodesic parameter s now plays a role more like proper time.Moreover, the differential algebra has a central 1-form and in the natural quotient of Ω 1 A where this is set to zero, θ ′ has the same role as the relativistic proper time interval in relation to the Minkowski coordinate time interval dt.The geodesic time element ds plays a similar role to θ ′ but as before it is external to the calculus on A; identifying the two now imposes the time-dilation relation in a similar spirit to the way that we imposed the Hamilton-Jacobi equations in Section 4. Our approach here is very different from previous discussions of proper time in the Klein Gordon context, such as [28] where the proper time and rest mass come from a canonically conjugate pair of observables.We illustrate our proposal with the easy case of a free particle in 1+1 Minkowski space, where we analyse a proper time wave packet centred around an on-shell Klein Gordon field (Example 5.5), and we also outline a proper time atomic model similar to a hydrogen atom.Section 6 ends the paper with a self-contained Poisson-level extended phase space formalism as suggested by our results of Section 4 at the semiclassical level.This helps to clarify the geometric content of our constructions and also provides the physical meaning of ∇ as infinitesimal data for the quantisation of the differential structure in the same way as a Poisson bracket is usually regarded as the data for the quantisation of the algebra.Some concluding remarks in Section 7 provide directions for further work.

Preliminaries: Algebraic formulation of geodesics
Here, we give a minimal but self-contained account of the algebraic set up of differentials and connections and the formulation of quantum geodesics introduced in [4] in terms of A-B bimodule connections [11].A possibly noncommutative unital algebra A equipped with an exterior algebra (Ω A , d) will play the role of a manifold, and B = C ∞ (R) expresses a geodesic time parameter t with its classical differential dt.The formalism also allows for more general and possibly noncommutative B and (Ω B , d), but we will only need the classical choice in the present paper.Proposition 2.1 is a general version of the classical case treated in [4], Proposition 2.2 is essentially in [4] but reworked for right connections (which is needed to mesh later with conventions in quantum mechanics), while Corollary 2.3 is new.
2.1.Algebraic set up and the case of a single geodesic.We recall that geodesics on a smooth Riemannian or pseudo-Riemannian manifold M can be expressed as the autoparallel condition ∇ γ γ = 0 for a curve γ in M , parametrised appropriately.Explicitly, this is and as such makes sense for any linear connection on a manifold (it does not have to be the Levi-Civita connection for a metric if we are not seeking to obey a variational principle).In quantum geometry, there is not yet a convincing calculus of variations and instead, by 'geodesic', we mean this autoparallel sense with respect to any linear connection (albeit one of geometric interest).Also note that γ is not actually a vector field, being defined only along a particular curve.Fortunately ∇ is only being taken along the same curve, but this does suggest that there there is a more geometric point of view.To explain it, we will need a fair bit of algebra.
Our first task for an algebraic version is the differential structure.If A is any unital algebra, we define a 'differential structure' formally by fixing a bimodule Ω 1 A over A of 1-forms.This means a vector space where we can associatively multiply by elements of A from either side and a map d ∶ A → Ω 1 A sending a 'function' to a 'differential form' obeying the Leibniz rule d(aa ′ ) = da.a′ + a.da ′ ).In the * -algebra case over C, we require Ω 1 A to also have a * -operation for which (a.da ′ ) * = (da ′ * ).a * for all a, a ′ ∈ A. One normally demands that Ω 1  A is spanned by elements of the form ada ′ for a, a ′ ∈ A, otherwise one has a generalised differential calculus.Any Ω 1 A then extends to an exterior algebra Ω A with exterior derivative increasing degree by 1 and obeying a graded-Leibniz rule and d 2 = 0.There is a canonical 'maximal prolongation' of any Ω 1  A which will actually be sufficient in our examples, but one can also consider quotients of it for (Ω, d).In fact the choice of higher degrees does not directly impact the geodesic theory but is relevant to the torsion and curvature of a connection.We also define left and right vector fields as respectively left and right module maps X ∶ Ω 1 A → A (i.e.maps which are tensorial in the sense of commuting with the left and right multiplication by A).
Next, let A, B be unital algebras with differential structure and E an A-Bbimodule (so we can associatively multiply elements of E by elements of A from the left and of B from the right).We define a right A-B-connection [11] on E as a map as usual for a right connection in noncommutative geometry.From the other side, for a certain bimodule map σ E as shown, called the 'generalised braiding'.Being a bimodule map means that it is fully tensorial in the sense of commuting with the algebra actions from either side.This map, if it exists, is uniquely determined by ∇ E and the bimodule structure and we say in this case that ∇ E is a (right) A-Bbimodule connection.If X is a vector field Ω 1 B → B then we have an associated covariant derivative D X = (id ⊗ X)∇ E ∶ E → E. The collection of categories A E B of such A-B-bimodule connections itself forms a coloured bicategory [11] with a tensor product We defer discussion of the * -operation to where we need it Section 2.3.
The above generalises the monoidal category A E A of right A-A-bimodule connections for any fixed differential algebra A. This diagonal case, in a left-handed version, is more familiar in noncommutative geometry [14,15,27].By a linear connection on A, we mean an A-A-bimodule connection ∇ Ω 1 A with associated braiding σ Ω 1 A (or just ∇ with associated braiding σ when the context is clear).In this case, the covariant derivative associated to a left vector field X will be denoted A .We will also adopt an explicit notation ∇ξ = ξ (1) ⊗ A ξ (2) (summation understood), so that ∇ X (ξ) = ξ (1) X(ξ (2) ) for all ξ ∈ Ω 1 .
We now express dependence on a time variable t by values in an algebra B = C ∞ (R) with its usual dt and the usual (commutative) bimodule structure on Ω 1 B .We take ∇dt = 0 as defining a trivial classical linear connection acting on this.Here ∇(bdt) = db⊗ B dt = ḃdt⊗dt and σ(dt⊗dt) = dt⊗dt.Now consider a linear connection ∇ on A and an A-B-bimodule connection E with the domain and codomian of σ E in (2.3).Each of the factors has a connection and hence we have two tensor product A-B-bimodule connections (albeit in our case σ Ω 1 B acts as the identity map so one does not need to include it).Next, any bimodule map between A-B-bimodules with connection has covariant derivative ∇ ∇ which measures the extent to which the map fails to intertwine the connections (classically, in the familiar diagonal case, this would be the induced covariant derivative of the map viewed as a tensor).With this machinery, [4] proposed as a kind of universal 'geodesic equation' including and generalising (2.1), depending on the choice of E.
To describe a single geodesic, note that a smooth curve in a manifold M defines an algebra map γ ∶ A → B compatible with the differential structures.Here A = C ∞ (M ) for the classical setting, but we can proceed at the algebraic level more generally.We say that γ is 'differentiable' if it extends to an A-bimodule map γ * ∶ Ω 1 A → Ω 1 B for the pull-back action on Ω 1 B by γ * (ada ′ ) = γ(a)dγ(a ′ ) for a, a ′ ∈ A, see [11].In our case, since Ω 1 B has basis dt, we can also write γ * explicitly as (2. and in this case ∇ E , σ E are maps with the proviso that A acts from the left on Ω 1 B via γ.The trivial choice is ∇ E = d.As above, we also fix the trivial linear connection with ∇dt = 0 on B. Proposition 2.1.Let A be a differential algebra with linear connection ∇ξ ∶= ξ (1) ⊗ A ξ (2) and let γ ∶ A → B a differentiable algebra map and E = B an A-Bbimodule as above.Then the trivial connection ∇ E e = de = ė⊗dt is an A-B-bimodule connection with Proof.
Here ∇ E e = ė ⊗ dt (the same as ∇ on B).We have a right connection as this is the same as classically.Using this, the left action and that B is commutative, we have and for the left Leibniz rule, this should equal Comparing these and extending σ E as a left module map gives the formula stated, which is well-defined by the assumption that γ is differentiable.Hence we have a bimodule connection.There is therefore a well-defined equation ∇ ∇(σ E ) = 0.Here B has the trivial linear connection ∇dt = 0 but just viewed as an A-B-bimodule connection with A acting by pull back along γ.Then with the above proviso for the left action of A on Ω 1 B .It follows that ∇ ∇(σ E ) = 0 appears as Expanding the linear connection on B on the left by the right Leibniz rule we cancel the de = ė ⊗ dt term from both sides.This can be written more explicitly in terms of B as 1) ]γ * [ξ (2) ]e and requiring this for all e ∈ E is the condition stated.
This equation makes sense for any differential algebra.We could also have defined ∇ E e = ( ė + eκ t ) ⊗ dt slightly more generally with the same σ E , albeit this generalisation is of no particular interest at this level.
) pulls back the coefficients of a 1-form ξ.If we also write ∇dx i = −Γ i jk dx j ⊗ dx k for Christoffel symbols Γ i jk , then the algebraic geodesic equation reduces to (2.1), as analysed in [4].Thus, we have introduced ∇ ∇(σ E ) = 0 in generality as the notion of a 'geodesic bimodule' and shown that the simplest choice E = B as noncommutative bimodule where the left action is defined by a curve γ reduces to a single classical geodesic in the classical case.

Algebraic setting of geodesic velocity fields.
In noncommutative geometry there can often not be enough algebra maps and one indeed needs a more general concept such as a correspondence or, in our case, an A-B-bimodule E. The root of this is that if there are not enough points, one should not expect enough curves either if these are defined pointwise.The natural thing to do here from a physical point of view is to consider not one geodesic but a density distribution of them with every point moving on a geodesic.Their collective tangents define a time-dependent velocity field X t (a path in the space of vector fields) subject to the velocity equation The idea is to take X t as the starting point and first solve this equation with some initial value X 0 ∈ Vect(M ).Any one geodesic is then recovered as a curve γ(t) such that This is different from the notion of 'geodesic spray', being more directly tied to the manifold itself.
In algebraic terms, this is just our universal ∇ ∇(σ E ) = 0 equation for a different choice of bimodule, namely now E = A ⊗ B, where A is a differential algebra in the role of the manifold equipped with a linear connection and B = C ∞ (R) as above with its classical calculus and trivial linear connection ∇dt = 0.More precisely, for a topological algebra, we can take E = C ∞ (R, A) with the left action by A and right action by B when viewed as subalgebras in the obvious way, but to keep things simple we will give formulae for A ⊗ B.
where X t is a left vector field on A and κ t ∈ A.
A be a right bimodule connection.Then (1) ∇ ∇(σ E ) is a bimodule map if and only if Proof.This is a right-handed version of a result for left connections in [4], but we include a brief proof for completeness.Since Ω in the obvious way, and since dt is a basis of Ω 1 B , the content of the bimodule map σ E is a bimodule map Ω 1 A ⊗ B → A ⊗ B which when restricted to Ω 1 A ⊗ 1 implies it is given by a time dependent left vector field X t on A as stated.
Next, by similar arguments to those at the start of the proof of [11,Lemma 4.13], we see that ∇ ∇(σ E ) is a bimodule map if and only if B is the identity, which is (1).Since σ E is given by X t , we obtain the second form as a map Ω 1 A ⊗ A Ω 1 A → A. For (2), as ∇ ∇(σ E ) is a right module map, we only have to calculate, for ξ ∈ and the vanishing of this is (2).As X t is a left vector field, we wrote It is also possible to restate the conditions in Proposition 2.2 in terms of a connection on the vector fields, bypassing the 1-forms entirely, but we have to be careful of the sides of the connections.To do this, we first assume that ∇ above has σ invertible.In this case, [11,Lemma 3.70] A .If we further assume that Ω 1 A is left finitely generated projective as a left module, which classically reduces to saying that the cotangent space is locally trivial, then by [11,Prop. 3.80] we can dualise a left bimodule connection∇ L on Ω 1  A to a right one ∇ X on X.In terms of the evaluation map ev Corollary 2.3.In Proposition 2.2, let ∇ have σ invertible, Ω 1 A be left finitely generated projective as a left module and ∇ X be the associated right connection on X.In these terms, the corresponding conditions are Proof.From Proposition 2.2 (1), we have X t (id ⊗ X t )σ = X t (id ⊗ X t ), so (2) can be rewritten as which by duality is the displayed equation (2).The other part is given by ev(id 2.3.Probabilistic geodesic flow and the ∇ E e = 0 equation.Next, returning to our classical model at the start of Section 2.2, if we have a perfect fluid with an evolving density ρ(t) on a manifold M , where each particle moves according to a velocity field X t , then conservation of mass (the continuity equation in fluid mechanics [30]) dictates ρ + X t (dρ) + ρ div(X t ) = 0.
In our algebraic formulation, we were led to e ∈ E = C ∞ (R, C ∞ (M )) or e(t) ∈ C ∞ (M ) at each t, with complex values, and we now identify ρ(t) = e(t)e(t) as playing the role of the probability density.Its evolution then corresponds to (2.10) ė + X t (de) + eκ t = 0, κ t + κ t = div(X t ) for the amplitude e(t), which is exactly ∇ E e = 0 in the classical limit of Proposition 2.2.For an actual probabilistic interpretation, we need a measure and to maintain the total probability with respect to it.For example, in the Riemannian case with the Levi-Civita connection, we want to maintain as the probability density ρ(t) evolves.Here g is the determinant of g µν .There is an associated inner product where e(t), f (t) ∈ L 2 (M ) with respect to the Riemannian measure as above, and we used the usual bra-ket notation.From this point of view, ∇ E e = 0 ensures that ⟨e(t) e(t)⟩ = 1 as e(t) evolves.Thus, our approach to geodesics in Section 2.2 leads us naturally into a quantum mechanics-like interpretation, even though we are doing classical geodesics with A = C ∞ (M ).
For the algebraic formalism, we take A and B * -algebras with * -differential structures as in Section 2.1.Any A-B-bimodule E has a conjugate E, which is a B-A-bimodule with elements e for e ∈ E and vector space structure e + f = e + f and λ e = λ e for λ ∈ C and e, f ∈ E, see [11].The algebra actions are e.a = a * .e and b.e = e.b * for a ∈ A and b ∈ B. Now suppose that E is equipped with a B valued inner product ⟨ , ⟩ ∶ E ⊗ A E → B which is bilinear and hermitian in the sense ⟨e, f ⟩ * = ⟨f , e⟩, where ⟨e, a.f ⟩ = ⟨e.a,f ⟩ for all a ∈ A. When B is a dense subalgebra of a C * -algebra, we call the inner product positive if ⟨e, e⟩ > 0 for all e ∈ E. In this context, a right A-B-bimodule connection ∇ E is said to preserve the inner product if for all e, f ∈ E we have [11] (2.11) d⟨e, Here the left connection sum of such terms implicit).Both A and B could be noncommutative.
In our case of interest, B = C ∞ (R) and we consider the B-valued output to define a function of 'time' t ∈ R with t * = t.Then d on the left is derivative in the R coordinate.Hence, if ∇ E preserves ⟨ , ⟩ and e obeys ∇ E e = 0 as above for geodesic evolution then d dt ⟨e, e⟩ = 0. We adopted a more mathematical notation but this is equivalent to the usual bra-ket notion other than the values being in B.
It remains to analyse the content of inner product preservation for our specific E where E = A ⊗ B (or E = C ∞ (R, A)).Since A could be noncommutative, instead of a measure we fix a positive linear functional φ 0 ∶ A → C or 'vacuum state' and define ⟨ f , e⟩ = φ 0 (f * e) as above, pointwise at each t so that the result is B-valued.Equivalently, we can suppose we are given ⟨ , ⟩ and define φ 0 (a) = ⟨ 1, a⟩, where a is viewed in E as constant in time.
Proposition 2.4.[4] The connection on E = A ⊗ B in Proposition 2.2 preserves the inner product on E if and only if for all a ∈ A and ξ ∈ Ω 1 A , Proof.This is again from [4] but we provide a short explanation.The condition for preservation is, for a, c ∈ A, and putting c = 1 gives the first displayed equation.Using this with c * a instead of c in the condition for preservation gives We call the first of the displayed equations in Proposition 2.4 the divergence condition for κ t and the second the reality condition for X t .The first generalises the second half of (2.10) to potentially noncommutative A and the second would be automatic on a real manifold.When A is a noncommutative, one cannot think of ρ(t) = e(t) * e(t) as a time-dependent probability density, but rather we adopt the usual formalism of quantum theory where any e implies an associated positive linear functional φ ∶ A → B or 'state' given by (2.12) φ(a) = ⟨e a e⟩ = ⟨ē, a.e⟩ = ⟨a * .e,e⟩ = φ 0 (e * ae) in our two notations for the inner product.Here, positive means φ(a * a) ≥ 0 for all a ∈ A and usually we normalise it so that φ(1) = ⟨ē, e⟩ = ⟨e e⟩ = 1 as we have assumed above.If A and B were C * -algebras then we would have the standard notion [19] of a Hilbert C * bimodule upon completion with respect to the induced norm e 2 = ⟨e, e⟩ B .In our case of interest, B = C ∞ (R) and for every e ∈ E we have a possibly un-normalised state φ t at each time defined by φ t (a) = ⟨e(t) a e(t)⟩.

Quantum mechanical geodesics on Hilbert spaces
We now go beyond the noncommutative differential geometric formulation of geodesics [4] covered in the preceding section, extending this to a * -algebra A of observables represented on a Hilbert space H as in quantum mechanics.We still employ the 'universal equation' Here ψ ∈ E and ψ(t) ∈ H, while ρ t at each t is a representation of A on a vector space H (we will only use the constant case where ρ is fixed but the more general case costs little to include and will be needed to recover the case of a single geodesic).Here ρ t should not be confused with probability densities ψ 2 which we no longer consider separately.We let L(H) be the (possibly unbounded) linear operators from H to itself, and make this into an A-bimodule in the obvious way by a.T = ρ t (a) ○ T and gives the bimodule map.
If we think of being valued in C ∞ (R) as dependence on t, the required data will be an 'operator-valued time-dependent vector field' which for each t is a bimodule map

This is a bimodule connection if and only of Xt
Proof.We prove the result in the given direction but, because of the complications of unbounded operators and topologies, we just give a motivation as to why this is a reasonably general case.First the linear map B when restricted to time independent ψ ∈ H should be of the form (∇ E ψ)(t) = h t (ψ(t)) ⊗ dt for some time dependent linear operator h t .Then multiplying ψ ∈ H by a function of time and using the Leibniz rule (2.2) gives the form for general ψ ∈ E in the statement.Now we assume that this assumption is true in what follows.
For a bimodule connection we can similarly justify a time dependent linear map as required.We require this to be a well-defined bimodule map as analysed in Lemma 3.1.
There are two important comments to make about this result.The first is that the well definedness of Xt could be read, starting with the calculus Ω 1 A as a condition on h t .However, in this paper we prefer to start with h t and read the condition as a constraint on the calculus Ω 1 A .The second is that in many applications the time dependent operator h t will be the action of a time dependent element of the algebra H t ∈ A via the representation, i.e. h t = ρ t (H t ).On the assumption that the representation is faithful, the condition in Lemma 3.2 for a bimodule connection is then equivalent to whether there is a well defined bimodule map This X t would then be a geometric time dependent vector field on A and Xt in the lemma is then its image.This is typically the case as follows.Assuming that the left A-module H has a cyclic vector (often a vacuum vector in physics), i.e. x 0 ∈ H so that {ρ t (a)(x 0 ) ∶ a ∈ A} is dense in H at each t and that the operators contain x 0 in their domain, we have X(ξ)(ρ t (a)(x 0 )) = X(ξa)(x 0 ) ∈ H.This means that the left module map X ∶ Ω 1 A → H⊗C ∞ (R) given by X(ξ) = X(ξ)(x 0 ) recovers the bimodule map X on the dense subset ρ t (A)(x 0 ) by the formula X(ξ)(ρ t (a)(x 0 )) = X(ξa).If we further suppose that ρ t satisfies ρ t (a)(x 0 ) = ρ t (a ′ )(x 0 ) only when a = a ′ , and that every Xt (ξ) maps the cyclic vector x 0 into the dense subset ρ t (A)(x 0 ), then we obtain a left module map As X is also a bimodule map, it follows that ρ t X t (ξ) = Xt (ξ) on the dense subset ρ t (A)(x 0 ), and therefore that ρ t X t (ξ) = Xt (ξ) as desired.
We now proceed as in Section 2 to take the trivial linear connection ∇dt = 0 acting on on Ω 1 B and an arbitrary linear right connection ∇ on Ω 1 A , and the tensor product connections (2.4)-(2.5)on the domain and codomain of σ E .Proposition 3.3.For E = C ∞ (R, H) and ∇ E defined by h t , ∇ ∇(σ E ) = 0 is equivalent to an auxiliary condition of the same form as (1) in Proposition 2.2 and the further condition A .This can be read as a time evolution equation for Xt , and this is consistent with an initial A-bimodule map X0 giving a bimodule map Xt for t ≥ 0. (In the absence of a uniqueness result for the differential equation we cannot make a stronger statement.)For such a bimodule map the evolution equation is determined by its value on ξ = da, which gives the first displayed equation in the statement.Replacing ξ with ξ.a in the displayed equation and using Xt being a right module map gives Ẋt (ξ.a) = Ẋt (ξ)ρ t (a) + Xt (ξ) ρt (a), showing that the time evolution equation for Xt is consistent with Xt being a right A-module map.Similarly replacing ξ with a.ξ shows that the evolution equation for Xt is consistent with Xt being a left A-module map, but this requires using (1) of Proposition 2.2.Now just use the formula for Xt (da) for the second displayed equation in the statement.
The single geodesic case of Section 2.1 is recovered by H = C and ρ t (a) = a(γ(t)) i.e. the evaluation representation along the image of the curve in the classical case, or ρ t (a) = γ(a)(t) in the algebraic version.In this case, h t is some function of t and does not enter, while A in the algebraic version.Then Proposition 3.3 reduces to Proposition 2.1.
Similarly, the geodesic flow of Section 2.2 is recovered with H = C ∞ (M ) (or some completion thereof) in the classical case or H = A with the left regular representation (or a completion thereof) in a potentially noncommutative version.In the classical case, compatibility with the calculus Ω 1 (M ) requires the Hamiltonian to have the form h t = Xt + κ t for some time dependent vector field Xt and function κ t acting by left multiplication on M and the condition in Propostion 3.3 reduces to the velocity field equation (2.8).In the noncommutative case with ρ the left regular representation, we have discussed in (3.1) how Xt can arise as the image of a bimodule map X t .This is not quite as general as Proposition 2.2, where we only assumed a left vector field.
It remains to extend Section 2.3 to our more general setting.We suppose that ρ t is unitary for each t.φ t (a) = ⟨e(t) a e(t)⟩ = ⟨e(t) ρ t e(t)⟩ or φ(a) = ⟨ē, a.e⟩ when viewed as a B-valued inner product.The main difference is that now the inner product is assumed in the Hilbert space and not given by a vacuum state φ 0 or a preferred element 1 ∈ E as was possible before.
Corollary 3.4.In the setting of Proposition 3.3, the inner product which is just the condition that h t is antihermitian.
If we consider Proposition 2.2 with Xt a bimodule map as a special case with h t as in (3.1), then h t anti-hermtian essentially reduces to the two conditions in Proposition 2.4.
We close with a couple of remarks in the case where the representation ρ does not depend on t.The first is, by the same reasoning as in [4], if ψ obeys ∇ E ψ = 0 then This comes from ∇ E preserving the inner product and the definition of σ, so that In terms of cochain complexes, this says that the expectation map Our second remark is that if we are in the setting of (3.1) where Xt is the image of a bimodule map X t (da) = [H t , a] for all a ∈ A (with ρ time-independent and faithful) then we also have a solution of the geodesic velocity equation in Proposition 2.2 with κ t = 0.This implies a quantum geodesic flow on E = A⊗C ∞ (R) which, from ∇ E in Proposition 2.2, comes out as for a t a time-dependent element of A (denoted e ∈ E = A ⊗ B there).This is minus the usual Heisenberg evolution for an actual quantum system with Hamiltonian h t and H t = ih t ̵ h, so we call it the 'anti-Heisenberg flow' underlying the 'Schrödinger flow' studied above.We can also interpret this flow probabilistically as in Section 2.3 if we fix a hermitian inner product on the * -algebra A by means of a positive linear functional φ 0 ∶ A → C. For the unitarity conditions in Proposition 2.4 to apply with κ t = 0, we need for all a ∈ A, which happens automatically if φ 0 is a trace, and which is automatic as H t = ih t ̵ h with h t Hermitian.This then applies to ξ = adb and hence to general ξ ∈ Ω 1 A since X t is a bimodule map.Note that φ 0 does not have to be a trace, for example we can let φ 0 (a) = ⟨ψ ρ(a) ψ⟩ where ψ⟩ is any eigenvector of the Hamiltonian (such as the ground state).For then, φ 0 ([H t , a]) = ⟨ψ [h t , ρ(a)] ψ⟩ = ⟨h * t ψ ρ(a) ψ⟩ − ⟨ψ ρ(a) h t ψ⟩ = 0 if ψ is an eigenvector.Similarly for φ 0 any convex linear combination of pure states given by eigenvectors of the Hamiltonian.The reason for the opposite sign in (3.3) is that applying the flow for the same Xt to a time dependent a t has the same general flavour (but in a noncommutative algebra) as applying the flow to ψ(t) ∈ H or to the density ψ(t) 2 if H is L 2 of a configuration space as in usual quantum mechanics.It is also comparable to the von Neumann evolution for density operators which has opposite sign to the Heisenberg evolution.By contrast, the usual equivalence between the Schrödinger and Heisenberg evolution equations is based on equating d dt ⟨ψ a ψ⟩ = ⟨ψ [H t , a] ψ⟩ if ψ obeys the Schrödinger equation and a is constant, to ⟨ψ ȧ ψ⟩ for a time-dependent and ψ constant from the Heisenberg point of view.This is a contravariant relationship in that ⟨ψ a ψ⟩ is being interpreted from dual points of view, evolving due to ψ or evolving due to a.This is very different from our more direct point of view.
We also note in passing that given our assumption of a bimodule map X t , the geodesic velocity equation in Proposition 2.2 also holds with arbitrary κ t since then [X t , κ t ] = 0.This more general flow is then not connected with the Schrödinger flow above but we can still consider it.Moreover, choosing κ t = H t means that the unitarity conditions in Proposition 2.4 now hold automatically for any φ 0 as then κ * t a + aκ t + X t (da) = 0 for the first condition, while the second condition holds automatically as already noted.For this second choice of κ t , the quantum geodesic flow on A given by ∇ E = 0 becomes for the evolution of a t ∈ A. The significance of this second 'non-standard flow' is unclear as it is not something we would normally consider in quantum mechanics.

Quantum geodesics for the Heisenberg algebra
In this section, we consider what the quantum geodesic formalism of Section 3 amounts to for A the standard Heisenberg algebra with generators x i and p i for i = 1, . . ., n and relations = 0 for a suitable choice of differential calculus.We fix our Hamiltonian in the standard form for some real potential V .We avoid any normal ordering problems due to the decoupled form.The algebra B = C ∞ (R) as usual, with its classical calculus and trivial linear connection with where R n has standard basis x 1 , . . ., x n , and x i ∈ A act on ψ by multiplication and p i by −i ̵ h ∂ ∂x i .The geodesic flow will be given by we can also view the connection as an operator ∇ E ∶ E → E if we leave ⊗ B dt understood).We will take this to be so that ∇ E ψ = 0 lands on the standard Schrödinger equation.Our new result will be all the quantum geometry for this to work, i.e. for Schrödinger's equation to appear as a quantum geodesic flow: the choice of calculus Ω 1 A , a linear connection ∇ on Ω 1  A , the geodesic velocity field X, and an associated generalised metric that ∇ is metric compatible with.To achieve this we consider a class of 'almost commutative' centrally extended differential structures [23] where the classical commutation relations of differentials on phase space acquire a multiple of a central 1-form θ ′ .Proposition 4.1.There is a unique centrally extended differential calculus Ω 1  A on the Heisenberg algebra such that ∇ E is an A-B-bimodule connection with σ E (θ ′ ⊗ ψ) = ψ ⊗ dt, namely with the bimodule relations p i m acting on ψ in the Schrödinger representation.
Proof.Recall that a bimodule connection involves the existence of a bimodule map , and that by definition of σ E we have For the central element we assume that σ E (θ ′ ⊗ ψ) = ψ ⊗ dt.Now we find the commutation relations in the calculus as follows: From these we are led to the commutation relations as stated, which can be shown give a calculus.
For the last part, the value of X(θ ′ ) is a definition and for the other values of X, we use the formula In the lemma, X is an operator-valued map but we see that this operator factors through a map X ∶ Ω 1 A → A and the Schrödinger representation of A (the p i , x i in the formulae for σ E act on ψ).We then check directly that this X respects the commutation relations of the calculus so as to give a bimodule map, i.e. a left and right vector field.

This dictates both the differential calculus Ω 1
A and σ E , which is uniquely determined from the pre-chosen ∇ E once Ω 1  A is fixed.The form of σ E then determined X uniquely.Note that the exterior derivative on general elements is determined from the Leibniz rule and the stated commutation relations.For example, if f (x) is a function of the x i only, then where The structure of the calculus here is that of a central extension [23,24] by θ ′ of the more trivial 2n-dimensional calculus on A where we set θ ′ = 0.
Next, we turn to the geodesic velocity equation ∇ ∇(σ E ) = 0 which depends on the choice linear connection ∇ acting on Ω 1 A .By Proposition 3.3, this is equivalent to the auxiliary equation ( 1) in Proposition 2.2 and an autoparallel equation, where we prefer to write X(id ⊗ X) as X ⊗ X with the product of the result in A understood, given that X is a bimodule map.
Proposition 4.2.On the above Ω 1 A , we have a natural right bimodule connection obeying ∇ ∇(σ E ) = 0, namely ∇(θ ′ ) = 0 and Proof.The second half of (4.3), explicitly, is ∂V ∂x i .The stated ∇ is then easily seen to obey these.The calculation of σ is then routine.Thus 2) gives the result stated that σ on the generators is just the flip.
Finally, we have to check the first of (4.3).We have which can be checked to hold for the form of ∇.For example, for the last equation and similarly for the others.
We are not asserting that ∇ is unique, although we are not aware of any other solutions at least for generic V (x).It is natural in the sense of playing well with central 1-forms in Ω 1  A and uniquely characterised by this as follows.Proposition 4.
Part ( 2) is immediate from part (1) once we compute that where we used the commutation relations of the calculus to reorder (dp i )p i and (dx i )∂ i V (the two corrections cancel) and the usual Heisenberg relations for the θ ′ ⊗θ ′ term for the 3rd equality.We then recognise the answer in terms of dh (where a reorder of ∂ i V dx i to match dh cancels between the two terms).The formula for dh follows from (4.2) and the commutation relations of the calculus.
To discuss the quantum geometry further, we now need to specify Ω 2 A .For every Ω 1  A there is a canonical 'maximal prolongation' obtained by applying d to the degree 1 relations, and other choices are a quotient.In our case, we impose dθ ′ = 0 and θ ′2 = 0 and then the rest of the exterior algebra relations, obtained by applying d to the degree 1 relations, are This has the expected dimension in degree 2 as for a 2n+1-dimensional manifold and is therefore a reasonable quotient of the maximal prolongation, where we imposed the θ ′ behaviour.Since θ ′2 = 0, corrections from the commutation relations vanish and one similarly has Corollary 4.4.The linear connection ∇ in Proposition 4.2 is flat and has torsion is closed and covariantly constant under ∇.
Proof.The torsion of a right connection is T ∇ = ∧∇ + d and comes out as shown.The formula for ∧(G) is immediate from the form of G stated in Proposition 4.3 given that θ ′ anticommutes with 1-forms.Note that the torsion and curvature for a right connection are right module maps but not necessarily bimodule ones, which is indeed not the case for T ∇ here.It follows in our case that We see that G is not quantum-symmetric in the sense of ∧(G) = 0 as needed for a strict quantum metric [11]; it is 'generalised quantum metric' in the notation there (and is, moreover, degenerate).Likewise, the torsion tensor does not vanish, so ∇ is not a 'quantum Levi-Civita connection' in the sense of quantum Riemannian geometry either.Certainly, one can quotient out θ ′ = 0 to work in the unextended calculus on A, in which case ω has the same form as the canonical symplectic 2-form in the classical case, and G becomes its lift.The geodesic velocity field X, however, does not descend to this quotient while ∇ = 0 at this quotient level, at least in the Heisenberg case studied here.Thus, the geometric picture is not exactly a quantum version of symplectic geometry either.We return to this in Section 6.
It should also be clear that the remarks at the end of Section 3 apply since the quantum vector field used above is explicitly constructed in Proposition 4.1 as the action of a bimodule map X ∶ Ω 1 A → A. In our case it is time independent and the Schrödinger representation used above is also time independent and faithful.It follows that X obeys the geodesic velocity equations in Proposition 2.2, but we can also verify this directly since are the same expressions as computed for (X ⊗ X)∇dx i and (X ⊗ X)∇dp i in the proof of Proposition 4.2.Since X is a bimodule map, [X, κ] = 0 for any κ.For the probabilistic interpretation in Proposition 2.4 to apply with κ = 0, we need a positive linear functional φ 0 such that (3.4) holds and, as explained there, the natural way to do this is to take a convex linear combination φ 0 (a) = ∑ i ρ i ⟨ψ i a ψ i ⟩ of pure states associated to normalised energy eigenvectors ψ i ⟩ in the Schrödinger representation.Here ρ i ∈ [0, 1], ∑ i ρ i = 1.We can, for example, just take the ground state, φ 0 (a) = ⟨0 a 0⟩.We also noted a non-standard quantum geodesic flow with κ = ih ̵ h.

4.1.
Example of the harmonic oscillator.We look briefly at the case of V = For the calculus, geodesic velocity field and connection, we have (sum over repeated indices) showing the expected symmetry between x i , p i .This quantum geometry quantises an extended phase space geometry as we discuss further in Section 6, and underlies our interpretation of the usual Schrödinger evolution as a quantum geodesic flow.It means, for example, that d dt ⟨ψ p i 2 ψ⟩ = −mν 2 ⟨ψ p i x i + x i p i ψ⟩ .
according to (3.2) whenever ψ⟩ is a solution of the time-dependent Schrödinger equation for a quantum harmonic oscillator (this is also easy enough to see directly).
We also have an underlying 'anti-Heisenberg' flow (3.3) using X ∶ Ω 1 A → A and the machinery of Section 2.2 with κ = 0.This is ȧt = −[H t , a t ] which in our case is as usual but with a reversed sign.We can make this concrete by restricting to the form (4.4) for some evolving complex coefficients χ i (t), ψ i (t).Then χi = ψ i mν 2 , ψi = − χ i m which is the expected simple harmonic motion among the coefficients.We can also introduce any κ as far as Proposition 2.2 is concerned, e.g. a constant κ modifies the above with a damping term such that χi = −(κ 2 + ν 2 )χ i − 2κ χi .
On the other hand, such a flow is no longer unitary and indeed the natural choices in our analysis are either (i) κ = 0 and a suitable φ 0 or (ii) κ = H t and any positive linear functional φ 0 if we want a unitary flow according to Proposition 2.4.
For case (i), the natural way to construct φ 0 is as a convex linear combination of pure states obtained from energy eigenvectors.The latter are of course n⟩ labelled by n = 0, 1, 2, ... and given by Hermite functions with eigenvalue E n = (n + 1  2 ) ̵ hν of h.For example, the vacuum expectation value φ 0 (a) = ⟨0 a 0⟩ provides a natural Hermitian inner product on A with respect to which the anti-Heisenberg evolution is unitary.In the classical limit, the elements of A become functions on phase space and the evolution becomes that of the classical harmonic oscillator.Case (ii), by contrast, is rather non-standard with which is no longer closed for the linear ansatz (4.4).Unlike Case (i), the expectation values in an energy eigenstate now evolve by phases, e.g.
The evolution is nevertheless * -preserving in the sense of Proposition 2.4 and is best understood in terms of the associated positive linear functionals φ t defined by a t according to (2.12), namely h is the usual Heisenberg flow of an initial b ∈ A and φ 0 = φ 0 (a * 0 ( )a 0 ) is the initial value of φ t .If φ 0 happens to be a convex linear combination of pure states given by energy eigenvectors then φ t comes out the same as the positive linear functional corresponding to the anti-Heisenberg flow of Case (i).In this case, if a 0 commutes with h then φ t = φ 0 does not evolve.These remarks are not specific to the harmonic oscillator but features of our formalism.

Electromagnetic Klein Gordon equation as geodesic flow
Our goal in this section is to extend the geodesic picture of quantum mechanical evolution to a relativistic setting with flat spacetime metric η = diag(−1, 1, 1, 1) and an electromagnetic background with gauge potential A a in place of a Hamiltonian potential.Motion in such a background is not geodesic motion in a usual sense, so this will be a novel application of our formalism.Let D a = ∂ ∂x a − i q ̵ h A a , where q is the particle charge and we use a physical normalisation so that background electromagnetic fields will appear in the classical limit without extraneous factors of ̵ h.We introduce an external time parameter s for the geodesic flow so now B = C ∞ (R) for this parameter and we set x 0 = ct in terms of the usual time coordinate t.Our first guess might be motivated by the formula for proper time in GR, but it is unpleasant to work with square roots and, as when working out geodesics in GR (where it is easier to extremise the integral of the proper velocity squared), we prefer to consider In effect, we write η ab D a D b = η ab D a D b η cd D c D d and replace the denominator by its on-shell value mc ̵ h where m is the particle rest mass.The half is to allow for the idea that any kind of variation of η ab D a D b brings down a 1 2 in comparison to that of η ab D a D b .Although somewhat novel, we will see that this ansatz lends itself to a reasonable quantum geodesic formulation.That this works out will appear as a minor miracle in terms of the amount of algebra, which in itself lends credibility to our hypothesis.

Electromagnetic Heisenberg differential calculus. Motivated as above,
we consider H = L 2 (R 1,3 ) with its 4D Schrödinger representation of the electromagnetic Heisenberg algebra A with commutation relations given that [D a , D b ] = −i q ̵ h F ab and p a is represented by −i ̵ hD a .Here F ab = A b,a −A a,b and the algebra is associative due to dF = 0.
We set E = H ⊗ C ∞ (R), or more precisely E = C ∞ (R, H), and We can also write σ E (da ⊗ φ) = X(da).φ⊗ ds, where (5.1) maps to the algebra and its output then acts on H in the 4D Schrödinger representation.For j = 1, 2, 3, we have X(dx i ) = pi m so in some sense dx i ds is being identified with the value p i m which is consistent with Special Relativity if s were to be proper time.We also define the Hamiltonian as the operator in ∇ E relevant to our formalism.We now provide a suitable calculus for the above.
Proposition 5.1.The spacetime Heisenberg algebra A has a first order differential calculus with an extra central direction θ ′ , given by Proof.We give the two most difficult checks, first d applied to the commutator of two ps: and which agree when we remember to use the centrally extended formula for d.Now we check the three ps Jacobi identity: and so and as we see that the three ps Jacobi identity is satisfied.
We next want to choose ∇ on Ω 1 A such that the conditions (4.3) for ∇ ∇(σ E ) in Proposition 3.3 hold.

Theorem 5.2. There is a right bimodule connection ∇ on Ω 1
A given by ∇θ ′ = 0 and Here σ is the flip map when one factor is θ ′ and Proof.First we have (on using the commutation relations for the p a s), 2m 2 η ar η eb F ec F rb,a and we can check that this obeys the autoparallel equation in (4.3) From the value of ∇(dx d ), we calculate and we check that this is a bimodule map, the difficult case being as required.From our value of ∇(dx d ), we also calculate as given.We check that this is commutes with commutators, as required, and as required.Later, it will be convenient to set (defining M a c ) where we can calculate and using and also giving the stated value.The proof that these formulae for σ are consistent with this extending as a bimodule map is extremely tedious and relegated to the appendix.Finally, we have to check the condition that (X ⊗ X)(σ − id) = 0. From the form of σ in the statement, this means checking the following equations: 2m 2 η rp η ab (F pd F be,ar − F pd,ar F be ) .For example, we check the last and hardest case, computing using the fact that a commutator with a fixed element is a derivation.Expanding the commutators in the three parts of the RHS gives the expression claimed.
Proposition 5.3.The above calculus on the Heisenberg algebra has a quotient Ω 1 red with relations whereby the commutation relations of dx i , dp i imply those required for dx 0 , dp 0 .Moreover, X and ∇ descend to this quotient.
Proof.For the calculus we just check the hardest case which agrees with [dp 0 , p a ].That X descends is immediate and for ∇ descending, the hardest case is showing that ∇(dp 0 ) is the same as where we have used d for functions f of the x a in the analogous form to (6.10), with ∆(f ) = η ab f ,ab .Ordering functions to the left and amalgamating terms gives the above as The dx 0 relation says that in this theory it is natural to identify θ ′ with the proper time interval dτ given that in Special Relativity dx 0 dτ = − p0 m for our metric.With this in mind, the other relation roughly speaking can be interpreted as the quantum analogue of dp depending on which side we place the F 0i before making our interpretation (with the 2nd term vanishing if we average the two).Here F 0i = − Ei c so the first term here is the expected rate of change of energy −cp 0 due to the work done by the electric field E i , while the 'quantum correction' term is the divergence F 0i,i = − ∂⋅E c proportional to the charge density of the external source.
In our formalism, we can also see this more precisely in terms of expectation values in an evolving s-dependent vector φ(s) = φ⟩ ∈ H, where we have noted in general that (3.2) holds, except that the geodesic time t there is now being denoted by s, i.e. which says that the expected proper acceleration is governed by the Lorentz force law again as classically.The c = 0 instance of this is the relation discussed at the operator level in the quotient, now at the level of expectation values.If the F ab are not constant then we will have order ̵ h corrections due to the form of (5.1).We next turn to the static case where A a are time independent.Lemma 5.4.If A a is time independent then u ∶= −p 0 −qA 0 is central in the Heisenberg algebra and [u, x 0 ] = i ̵ h.Moreover, there is a subalgebra A with subcalculus Ω 1

A
of Ω 1 red generated by x i , p i , dx i , dp i , θ ′ , u where u is central in Ω 1 A and du = 0.Moreover, ∇ restricts on the generators to a connection on Ω 1 A and h ∈ A.
Proof.Clearly, u is always canonically conjugate to x 0 as p 0 was.Also [u, x i ] = 0 and when A a are time independent then −[u, p i ] = [p 0 , p i ] + [qA 0 , p i ] = i ̵ hqF 0i + i ̵ hqA 0,i = 0.For the differentials working in Ω 1 red in the time independent case, We also have, using x 0 invariance of the A a and (5.4) with ∆ defined by η ab , as A 0,i = −F 0i and ∆A 0 = A 0,ii = −F 0i,i by the relations in Ω 1 red .Next, we omit x 0 from our algebra as under our assumptions it does not appear in F ab or on the right hand side of any of the commutation relations other than as dx 0 = −p 0 θ ′ m.The remaining generators and relations are (5.6)-(5.8)as listed below albeit u a closed central generator.Further, ∇ restricts to this subcalculus as any dx 0 terms given by ∇ can be rewritten in terms of u by the relations.
We can clearly restrict X as well, and obtain the equations for a quantum geodesic flow on A with this calculus.Moreover as u is central and closed, we can consider it instead as a fixed real parameter.We denote this quotient by A u , with calculus Ω 1 Au given by (5.8) This then deforms the Heisenberg algebra on spatial R 3 in Section 4 by the background electrostatic and magnetostatic potentials A i , A 0 with u regarded as a real parameter.This suggests to decompose our representation H into fields where u has constant value and this is what we will do in the next section.Thus we take and ∇ E given by the same Hamiltonian as above, now viewed as the representation of an element of A u on functions on R 3 with fixed value of u.In this case, we have quantum geodesic motion for the reduced Heisenberg algebra A u with calculus Ω 1 Au as in Section 4 but now with Hamiltonian that contains magnetic potentials in the p i and an electric potential in the form of V .5.2.Relativistic amplitudes and hydrogen atom.We now consider the probabilistic interpretation of the quantum geodesic evolution constructed above.We recall that x 0 = ct where c is the speed of light and time t is in usual units in an inertial frame.So far, we considered the relativistic Heisenberg algebra acting by multiplication and D a on φ ∈ C ∞ (R 1,3 ) at each s ∈ R.However, φφ on spacetime is not suitable for a probabilistic interpretation in any laboratory as it involves probabilities spread over past and future in the laboratory frame time.To address this, we work with fields ψ(u, x i ) Fourier transformed from t to a Fourier conjugate variable u say, so In physical terms, we can think of amplitudes ψ with a probability distribution of energies and spatial positions.The Heisenberg algebra (as well as the Lorentz group) acts unitarily on this new space of fields completed to L 2 (R 4 ) in these variables, just because it did before and Fourier transform in one variable can be viewed as an isometry (if also completed to L 2 (R 4 ) on the spacetime side.) In this form, we can fix one well-known problem with the Klein Gordon equations, namely we are only interested in fields with positive energy, i.e. we restrict support to u ≥ 0. Another way of looking at this is that the Klein-Gordon equation is second order in time, so to solve it we need additional information, e.g. the time derivative of φ at a fixed t.So far, u stands for the classical Fourier conjugate variable to t but we also would like to identify it with the eigenvalues of an operator in the Heisenberg algebra.We chose this to be cu defined by −p 0 = u + qA 0 which then acts by multiplication by u on our fields.This choice of cu is adapted to the time-independent case but we can use it more generally also.The c is needed since u is conjugate to x 0 = ct.The minus sign is needed due to the − + ++ signature as classically p 0 = −p 0 is positive for a future pointing time-like geodesic.The action of the electromagnetic Heisenberg algebra on ψ(u, x i ) is by x i and p i = −i ̵ hD i as before, and Moreover, ∇ E as before now appears as (5.9) This is clear from the Fourier transform, but if one wants to check it directly, p i , x i are already represented as before and as A 0 (−i ̵ h ∂ ∂u , x i ) does not depend on u, we still have [p i , x 0 ] = 0.Meanwhile, Now suppose that A a are indeed independent of t.Then by Lemma 5.4 we can write ψ(u, x i ) = Ψ(x i ) and regard u as a fixed parameter energy since it is central in the Heisenberg algebra.In this case the ∇ E on Ψ is governed by a similar operator as in Section 4 but with p i = −i ̵ hD i for a particle minimally coupled to the A i as a magnetostatic gauge potential and with potential energy where the upper index potential connects to usual conventions.Thus, ∇ E Ψ = 0 looks very much like Schrödinger's equation except that the geodesic time parameter is not t but proper time s.Moreover, we have maintained Lorenz invariance (we could change our laboratory frame) in (5.9) before we fixed the energy u in our laboratory frame.Spacetime is still present and a mode concentrated at a specific u appears in our original KG field φ(t, x i ) as e − i ̵ h ut Ψ(x i ) with Ψ(x i ) the amplitude distribution for such modes at different positions in space.Such a mode will not appear as concentrated at a fixed energy in another frame; this is our choice in the laboratory frame but the geodesic evolution is not dependent on this.We have suppressed that both φ and Ψ are evolving and depend on the geodesic time parameter s.
Example 5.5.(Free particle proper time relativistic wave packet.)We consider the simplest case of a scalar field for mass m in 1+1 Minkowski spacetime with zero electromagnetic potential.Then h .The on-shell fields (i.e.solving the KG equation) just evolve by an unobservable phase e i smc 2 2 ̵ h but, as in GR, we need to also look at nearby off-shell ones (in the case of GR to see that we are at maximum proper time).More precisely, we look at a wave packet which in spatial momentum space is centred on the positive on-shell value corresponding to energy u but includes a Gaussian spread about this.We evolve this from s = 0 to general s: as plotted in Figure 1.It is easy enough to check the expectation values as expected.We also find using x = −i ̵ h ∂ ∂k on ψ k inside the upper of the ratio of integrals that which verifies our identity (5.5) and shows that our quantum geodesic evolves with proper velocity given by the average spatial momentum/ m.We can also compute t = −i ̵ h ∂ ∂u applied in the upper of the ratio of integrals to find as expected respectively for the proper velocity in the time direction in Special Relativity and the lab velocity v in our case.Note that the Gaussian parameter β > 0 does not enter into these expectation values but is visible in Ψ as it sets the initial spread (which then increases during the motion).
Although our quantum geodesic flow equation ∇ E Ψ = 0 is not Schrödingers equation, its close similarity means that we can use all the tools and methods of quantum mechanics with s in place of time there and u as a parameter in the Hamiltonian, as in the preceding example.This approach is somewhat different from the usual derivation of Schrödingers equation as a limit of the KG equation, which involves writing φ(x 0 , x i ) = e −i mc 2 ̵ h t Ψ KG (t, x i ) where ct = x 0 and Ψ KG (t, x i ) is slowly varying to recover Schrödinger's equation for Ψ KG with corrections.The minus sign is due to the − + ++ conventions.We do not need to make such slow variation assumptions and in fact we proceed relativistically.Our approach means that our differences from Schrödingers equation are now of a different nature from the usual ones coming from the KG equation, although they share some terms in common.
Example 5.6.(Hydrogen atom revisited.)We consider a hydrogen-like atom or more precisely an electron of charge q = −e in around a point source nucleus of atomic number Z or charge Ze, so Φ(r) = Ze 4π 0 r .
Then the geodesic flow equation since there is no magnetic potential in the D i (we write ∂ 2 = ∑ ∂ 2 i for the spatial Laplacian).We are effectively in the Schrödinger equation setting of Section 4 with h = p 2 2m + V except that the geodesic parameter is now proper time while u corresponds to a plane wave in laboratory time direction as explained above.We can still use the methods of ordinary quantum mechanics, with where qΦ = − Ze 2 4π 0 r = − ̵ hc Zα r in terms of the fine struture constant α ∼ 1 137.We solve this for eigenmodes where E is positive due to a large negative offset in h u further minus a binding energy.This is solved by the same methods as the usual hydrogen atom by separation of variables, namely set Ψ = R(r)χ l where χ l has only angular dependence and is given by an integer l and a quantum number m = −l, ⋯, l which does not change the energy.The remaining radial equation is then which has the same form as for a usual atom but with shifted angular momentum l ′ = l − δ l defined as in [18, Chapter 2.3] by For every n such that n − (l + 1) = d is a positive integer i.e. l = 0, ..., n − 1, one has mc 2 in terms of a generalised Laguerre polynomial of degree d.This gives eigenvalues (5.11) for our Schrödinger-like geodesic flow equation.From our point of view, we first consider when modes are on-shell, meaning the associated KG field obeys the Klein-Gordon equation.Given the way ∇ E was defined, this means to find the spectrum of u such that the eigenvalue E as above is mc 2 2 .From (5.11), these are 1 + ( Zα n−δ l ) 2 in agreement with the allowed 'Schrödinger-like' energy spectrum coming from directly solving the KG equation [18,Chapter 2.3].More generally, we are not obliged to stick to on-shell states and indeed we should not as we saw in the preceding example.For example, we can solve for each fixed u as above and then a general evolution would be n,l,m at fixed u as sketched above, and initial values set by coefficients c n,l,m (u).One could then compute expectation values along the quantum geodesic flow in a similar manner to Example 5.5.
Finally, although not our main purpose, it is tempting to actually think of quantum geodesic flow as a modification of an atomic system and see what the differences are.For this we set u = mc 2 so that we have the correct 1 r term for an atom at least if we ignore that one should use the reduced mass and that s is proper time.In this case where the first term should be ignored and the second would be minus the Rydberg binding energy for atomic number Z if δ l had been zero.In terms of potentials, at u = mc 2 we have of which we discard the constant term so that the effective potential is For a hydrogen atom, the two terms are of equal size at , where α is the fine structure constant, λ c is the Compton wavelength and a 0 is the Bohr radius.For one electron, this critical radius is about 1.4 × 10 −15 metres compared to 0.85 × 10 −15 metres for the size of a proton.But for a large atomic number Z the critical radius would be Z times this, so well outside the nucleus itself.However, in the more careful analysis above, we still need Z < 1 2α to have solutions for l ′ as known in the context of solving the KG equation for this background [18,Chapter 2.3].We see it directly from the potential and without the complications from double time derivatives.This correction is also different from the usual 1 r 3 spin-orbit correction from allowing for the spin of the electron.Since s is more like proper time, there would also be a relativistic correction compared to coordinate time much as in the usual relativistic correction to the p 2 component of the Hamiltonian.

Extended phase space Poisson geometry
Traditionally in physics, one starts at the Poisson level and then 'quantises'.In our case the situation was reversed with the quantum geometry of the Heisenberg algebra in Section 4 dictated by the algebraic set up.We now semiclassicalise this and similar models to a Poisson level version and present that independently.The first thing we notice is that there is an extra dimension θ ′ in the calculus, which is not a problem when ̵ h ≠ 0 but which means that we do not have an actual differential calculus when ̵ h = 0 as θ ′ is still present and not generated by functions and differentials of them.This suggests that to have an honest geometric picture, we should work on M = M × R where (M, ω, ∇) is a symplectic manifold with symplectic connection ∇ and symplectic form ω µν in local coordnates (we denote its inverse by ω µν with upper indices for the associated Poisson bivector inverse to it) and R corresponds to an external time variable t with θ ′ = dt.The latter recognises that noncommutative systems can generate their own time in a way that is not explicable in the classical limit.By a symplectic connection ∇ we mean torsion free and preserving the symplectic form.(Such connections always exist but are not unique.) In Section 6.1, we obtain a self-contained formulation where we fix a Hamiltonian function h with X the associated Hamiltonian vector field, which we extend to X = ∂ ∂t + X.We likewise extend ω µν to a (0,2) tensor G on the extended phase space with G µν = ω µν , G 0µ = −G µ0 = ∂ µ h and we also extend ω µν to a Poisson bivector on the extended phase space with ω 0µ = −ω 0ν = τ µ for a suitable vector field τ = τ µ ∂ µ on M .Our convention is that Greek indices exclude zero.Both extensions are degenerate and no longer mutually inverse.For simplicity, both h and τ are taken as time independent, i.e. defined by data on M .The main result will be to extend ∇ to a linear connection ∇ on the extended phase space compatible with G such that autoparallel curves are solutions of the original Hamiltonian-Jacobi equations with velocity vector field X.
If ω = dθ then ∧(G) = dΘ where Θ = θ + 2hθ ′ for the usual contact form Θ on extended phase space as in [32].On the other hand, our specific results in this section are not related as far as we can tell to metrics on phase space such as the Jacobi metric in [16,29].Nevertheless, we do make use of a natural (possibly degenerate) classical metric g µν on M induced by the Hamiltonian and we do not exclude the possibility that different approaches to geometry on phase space could be linked in future work.
Rather, the bigger picture from our point of view is that ω µν , τ µ = ω 0µ provide the Poisson bracket and hence quantisation data for the extended phase space: and the extended linear connection ∇ similarly provides semiclassical data for the quantisation of the differential structure cf [17,5,11] , which for the natural ∇ found in (6.4) is We require the connection to be Poisson compatible in the sense of [5] and, for an associative algebra and calculus at order ̵ h 2 , we require ω µν , τ µ to obey the Jacobi identity and ∇ to be flat.Neither of the last two conditions is needed at the semiclassical level, while the Poisson compatibility holds if ∇ is symplectic and τ obeys some conditions deferred to the end of Section 6.1.Section 6.2 checks the semiclassical limit of Section 4 extended by central t with θ ′ = dt, and shows that we obtain an example with ∇ = τ = 0. 6.1.Hamiltonian vector fields as autoparallel on extended phase space.Let M be a symplectic manifold with coordinates x µ , µ = 1, . . ., 2n, symplectic form ω µν with inverse ω µν , and let ∇ be a symplectic connection with Christoffel symbols Γµ νρ defined by ∇µ dx ν = − Γν µρ dx ρ .We fix a function h ∈ C ∞ (M ) with Hamiltonian vector field Xµ = ω µν h ,ν (we use h ,ν for the partial derivative of h with respect to x ν ).It is easy to see that in general X is not autoparallel as where we define the possibly degenerate metric inner product (dx µ , dx ν ) = g µν = ω µγ ω νρ (h ,ρ ) ;γ (with semicolon the ∇ connection).This is symmetric as ∇ is torsion free.
To resolve this obstruction to X being autoparallel, we work on the extend phase space with x 0 = t and X = ∂ ∂t + X, and write down an extension of the symplectic connection on forms 0 for some additional Christoffel symbol data as shown.Lemma 6.1.For generic h, X is autoparallel with respect to ∇ if and only if Proof.We require ∂ ∂x µ = 0 which gives the result stated.
We now turn to the classical symplectic form ω µν dx µ ∧ dx ν and its torsion free symplectic connection ∇.In our extended calculus, it would be reasonable to find a related 2-form which is preserved by the extended covariant derivative ∇.To the symplectic form, it is reasonable to add something wedged with dt in order to be closed.Lemma 6.2.The extended covariant derivative ∇ preserves a 2-form of the form ω αβ dx α ∧ dx β + df ∧ dt for generic f (derivative not vanishing identically on any open region) and has X autoparallel if and only if ∇µ (d(f +2h)) = 0, Γ µ α0 = g µβ ω βα and Γ µ 0α = 0.
In a local patch with coordinates such that ω µν are constant and if Γµ νρ = 0, then these 1-forms are also killed by ∇.One can view them along with dt as a local parallelisation of M .The quotient of the cotangent bundle where we set η µ = 0, is dual to a sub-bundle the tangent bundle of M spanned by X.This, at any point of M , is the tangent to the Hamilton-Jacobi equations of motion regarded as a flow in M through that point.
On the other hand, this extended connection from Lemma 6.2 necessarily has torsion in this extended direction, T µ ν0 = −T µ 0ν = g µβ ω βν .We recall that adding torsion to a connection does not change the autoparallel curves but causes the directions normal to the curves to rotate about them.If we had taken the symmetric form of extension where Γ µ α0 = Γ µ 0α = 1 2 g µβ ω βα , we would have had zero torsion but not compatibility with ω.Proposition 6.3.The extended ∇ in (6.4) with torsion has curvature and zero for R with index 0 in all other positions, where Rα βγδ is the curvature of ∇.
Proof.The curvature can be computed from the usual Christoffel symbol formula with Roman indices including the index 0, and the derivative in the 0 direction vanishing, thus Recalling that Γ 0 ab = Γ a 0b = 0 we observe that R a bcd is zero if any of a, c, d are zero.Using Greek indices which cannot be zero, we observe from the formula that the only possible nonzero values apart from R If we use semicolon for covariant differentiation with respect to ∇, then Γ α δ0 = g αβ ω βδ = ω βδ ω αγ ω βρ (h ,ρ ) ;γ = −ω αγ (h ,δ ) ;γ = −ω αγ (h ,γ ) ;δ as ∇ has zero torsion.Now, as ω has zero covariant derivative with respect to ∇, and substituting this into the equation for the Riemann tensor gives R α 0γδ = ω ατ (h ,τ ) ;γ;δ − ω ατ (h ,τ ) ;δ;γ = ω ατ h ,κ Rκ τ γδ , where we have used ∇ torsion free again.
Note that the connection (6.4), its torsion and curvature can be written in our more algebraic way, as a right connection, Rµ ναβ η ν ⊗ dx β ∧ dx α where we used symmetry in the first two indices of the curvature of ∇ when raised by ω, see [12].
From the point of view of this extended phase space geometry, we can now write (6.1)-(6.2) compactly as , where a, b, c.d include 0 with x 0 = t and ω 0µ = −ω µ0 = τ µ .The condition for this to be Poisson compatible follows easily from the characterisation of Poisson compatibility in [11,Lem. 9.21] and comes out as (6.7) ∇τ = 0, T µ ν0 τ ν = 0, given our assumptions on ω and the form of the extensions above.
6.2.Quantum geometry of the extended Heisenberg algebra.Here we extend the Heisenberg algebra A to Ã = A ⊗ B where B = C ∞ (R), in the form of an additional time variable t, but with calculus which makes sense as θ ′ and dt are both graded central in the tensor product and killed by d.We then show that its semiclassical limit is an example of Section 6.1.
As t is central, we have τ µ = 0.
The first step is to identify the underlying classical symplectic structure and Poisson tensor The induced (possibly degenerate) metric g µν = (dx µ , dx ν ) = ω µα ω νβ ∇ α h ,β in Section 6.1 is given by (6.8) and other entries zero.Here dx i and dx i+n = dp i define dx µ .Writing as characterised by Leibnizator L ∆ (f, g) = 2(df, dg) for all f, g on phase space (this makes sense without assuming g µν invertible, but in the invertible case it would be the usual Laplacian-Beltrami operator ∆f = g µν ∇ µ ∂ ν f ).
The extended classical Hamiltonian vector field for our chosen Hamiltonian as defined in Section 6.1 is We have already used these values in the quantum case without change.We also combine the covariant 1-forms η i and ω i = η i+n as 1-forms η µ .Corollary 6.4.The relations of the differential calculus, the quantum linear connection and the invariant 1-forms η µ appear in Section 4 appear in terms of the above phase space structures as along with ∇θ ′ = 0 and σ = flip when one factor is θ ′ .These expressions also apply to Ã with θ ′ = dt.
Proof.Note that the last of the calculus relations is given by applying d to the middle relations (these relations do not need the metric to be invertible).The middle form of ∇(dx µ ) requires some explanation.In fact, the quantum differential calculus in Proposition 4.1 has the structure of a general 'central extension' [23,24] by a 1-form θ ′ of the extended calculus on the Heisenberg algebra where we set θ ′ = 0.In this way, with θ ′ = dt one has for f (x, p) normal ordered so that x is to the left of p and ∆ a certain 2nd order operator on the Heisenberg algebra which reduces to (6.9) when f (x, p) is normal ordered and at most quadratic in p. Otherwise ∆ is more complicated with O( ̵ h) terms arising from (6.8) not being a bimodule map for general V (x) on the unextended calculus.We used this ∆ for ∇(dx µ ) and recognised the result in terms of d(X(dx µ )).
Comparing with (6.2), we see that the calculus corresponds to ∇ at the semiclassical level with ∇ = 0 (as well as τ µ = 0).The first term of the first form of ∇(dx µ ) also then agrees with ∇ in (6.4), with a further quantum correction.In this way, the formulae in Section 4 can be written more geometrically on the extended phase space and the meaning of the connection ∇ with respect to which Schrödinger's equation is 'quantum geodesic flow' emerges as the semiclassical data for the quantum differential calculus.This also suggests how Section 4 could potentially be extended to other quantisations of symplectic manifolds, though this remains to be done.We have only considered the time-independent theory and it seems likely that the above will extend also to the time-dependent case.Again, this remains to be done.

Concluding remarks
In Section 3, we extended the formalism of 'quantum geodesics' in noncommutative geometry as introduced in [4] using A-B-bimodule connections from [11] to geodesics in representation spaces.We then applied this to ordinary quantum mechanics and showed in Section 4 that the usual Schrödinger equation can be viewed as a quantum geodesic flow for a certain quantum differential calculus on the quantum algebra of observables (the Heisenberg algebra) acting on wave functions in the Schrödinger representation.The quantum differential calculus here encodes the Hamiltonian much as in GR the Riemannian manifold determines the geodesic flow.This idea that physics has new degrees of freedom in the choice of quantum differential structure has been around for a while now and is particularly evident at the Poisson level [10].Such a freedom was already exploited to encode Newtonian gravity by putting the gravitational potential into the spacetime differential structure [25]; our now results in Section 4 are in the same spirit but now on phase space in ordinary quantum mechanics and not as part of Planck scale physics.
We then proceeded in Section 5 to a relativistic treatment based on the Klein-Gordon operator minimally coupled to an external field.Even the simplest 1+1 dimensional case without external field in Example 5.5 proved interesting, with relativistic proper time wave packets Ψ quantum geodesically flowing with constant velocity v = ⟨Ψ x Ψ⟩ ⟨Ψ t Ψ⟩ in the laboratory frame.The example illustrates well that quantum geodesic flow looks beyond the Klein-Gordon equation itself.Just as an ant moving on an apple has feet on either side of the geodesic which keeps it on the geodesic path, the quantum wave packet spreads off-shell on either side of a Klein-Gordon solution but on average evolves as expected.In that respect our relativistic approach to quantum mechanics is very different from previous ones and more resembles elements of scalar quantum field theory (where one goes off-shell in the functional integral).We showed how time-independent background fields nevertheless amount to a proper time Schrödinger-like equation if we analyse the geodesic flow at fixed energy u, allowing the usual tools of quantum mechanics to be adapted to our case.We illustrated this with a hydrogen-like atom of atomic number Z. Section 6 concluded with a look at the extended phase space geometry that emerges from our constructions at the semiclassical level.
Clearly, many more examples could be computed and studied using the formalism in this paper, including general (non-static) electromagnetic backgrounds to which the theory already applies.Also, in Section 4 we focussed on timeindependent Hamiltonians, but the general theory in Proposition 3.3 does not require this.It would be interesting to look at the time dependent case and the construction of conserved currents.The present formalism also allows the possibility of more general algebras B in place of C ∞ (R) for the geodesic time variable.
On the theoretical side, the formalism can be extended to study quantum geodesic deviation, where classically one can see the role of Ricci curvature entering.This is not relevant to the immediate setting of the present paper since, at least in Section 4, the quantum connection on phase space was flat and preserved the extended quantum symplectic structure (rather than being a quantum Levi-Civita connection).It will be looked at elsewhere as more relevant to quantum spacetime and quantum gravity applications, but we don't exclude the possibility of quantum mechanical systems where curvature is needed, e.g. with a more general form of Hamiltonian.Another immediate direction for further work would be to extend Section 5 from an electromagnetic background on the representation space to a curved Riemannian background on the latter, i.e. to gravitational backgrounds such as the quantum-geometric black-hole models in [2].It could also be of interest to consider quantum geodesic flows using a Dirac operator or spectral triple [13] as in Connes' approach instead of the Klein Gordon operator.
Finally, on the technical side, the role of θ ′ needs to be more fully understood from the point of view of the quantum extended phase space and its reductions.In our case, it arises as an obstruction to the Heisenberg algebra differential calculus, which forces an extra dimension, but we ultimately identified it with the geodesic time interval.However, a very different approach to handle this obstruction is to drop the bimodule associativity condition in the differential structure [5,6], which could also be of interest here.
7) γ * (ξ) ∶= γ * [ξ]dt, γ * [ada ′ ] = γ(a) γ(a ′ ) where γ * [ξ] ∈ B. Also note that E = B is an an A-B-bimodule by a ⋅ e = γ(a)e, e.b = eb, ∀a ∈ A, e ∈ E, b ∈ B and from Proposition 4.3 we have the central element and 1-forms = ⟨φ X(da) φ⟩ for all a ∈ A. Working in the full algebra (we do not need the above quotient), we have from (5.1) that d⟨φ x a φ⟩ ds = η ab m ⟨φ p b φ⟩, which says that in any state the expected momentum is m times the proper velocity as classically, and if the F ab are constant, d⟨φ p c φ⟩ ds = q m η ab F ca ⟨φ p b φ⟩ = qF ca d⟨φ x a φ⟩ ds ,

Figure 1 .
Figure 1.Proper time s relativistic wave packet dispersing as it moves down and to the right.Shown are the real and absolute values at c = ̵ h = m = 1 and u = 1.1.Images produced by Mathematica.