We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus Ω1 depending on the Hamiltonian p2/2m + V(x), and a flat quantum connection ∇ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or “geodesics”) is exactly Schrödinger’s equation. The connection ∇ preserves a non-symmetric quantum metric given by the canonical symplectic structure lifted to a rank (0, 2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to obtain a novel flow generated by the Klein–Gordon operator on Minkowski spacetime with a background electromagnetic field, by formulating quantum “geodesics” on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include quantum geodesics that look like a relativistic free particle wave packet and a hydrogen-like atom.

The idea of geometry extended to a possibly noncommutative “coordinate algebra” A 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. There are various approaches and we will use particularly the constructive approach in Ref. 1 and references therein, based on a chosen differential graded algebra (Ω, d) of “differential forms,” a quantum metric g ∈ Ω1AΩ1 and a quantum Levi–Cività connection ∇: Ω1 → Ω1AΩ1 to build up the “quantum Riemannian geometry.” The ∇ here is a bimodule connection in the sense of Refs. 2 and 3. This approach is complementary to the well-known Connes’ approach where the noncommutative geometry is encoded in a spectral triple4 or abstract Dirac operator as starting point. The two approaches can be compatible and with interesting results where they meet.5,6 The bimodule approach has been used to construct toy models of quantum gravity7–9 and more recently to generate particle masses for scalar fields via a Kaluza-Klein mechanism with noncommutative “extra dimensions.”10 

In the present paper, we apply the 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 . 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 the latter in actual quantum systems has already been noted in Connes’ approach, for example to understand the quantum Hall effect.11 The role of the quantum Riemannian geometry formalism,1 however, has not been explored so far in this context but makes sense once we note that the “quantum metric” need not be symmetric, so can be equally applied to objects which classically would be antisymmetric. Indeed, we will be led to a “generalised quantum metric” on an extended phase space with time adjoined and which quantises an antisymmetric tensor related to the symplectic structure. This generalised quantum metric will also 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 the broadest 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 will make particular use of a notion of “quantum geodesics” with respect to any bimodule connection ∇, as recently introduced and studied in Refs. 12–14. The preliminary Sec. II A provides the algebraic definition particularly of quantum geodesic(s) via the notion of a “geodesic A-B bimodule” E where B=C(R) is the geodesic-time coordinate algebra. One choice of E recovers in the classical case a single classical geodesic in a manifold, while in Sec. II B another choice of E recovers classically 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 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 Sec. III 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 Sec. IV to A the Heisenberg algebra in the Schrödinger representation, allowing us to express the standard Schrödinger equation for a Hamiltonian h = p2/2m + 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, including a generalised quantum metric G, a compatible vector field X and a bimodule connection ∇. Moreover, while the Schrödinger representation necessarily entails the usual baggage of quantum mechanics, the noncommutative geometric structures themselves make sense at an algebraic level.

Specifically, the algebra A in Sec. IV will be the standard R2n Heisenberg algebra
[xi,pj]=iδij,[xi,xj]=[pi,pj]=0,1i,jn,
(1.1)
but equipped now with a certain differential calculus ΩA1 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 Ref. 15, where it was shown how Newtonian gravity can be encoded in the choice of differential structure on quantum spacetime, but now applied to quantum mechanics. The exterior algebra that we are led to in Proposition 4.2 is itself an interesting outcome of the paper and has the commutation relations
[dpi,pj]=i2Vxixjθ,[dpi,xj]=[dxi,pj]=0,[dxi,xj]=imδijθ
(1.2)
between functions and differentials, where θ′ is a graded-central extra direction initially with no classical analogue but dictated by the algebra. Here
θ=mi[xi,dxi]
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.16–19 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 à where we adjoin a central variable t. This fits in with the idea that highly noncommutative systems tend to generate their own evolution.16 (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
ωidpi+Vxiθ,ηidxipimθ
(1.3)
are covariantly constant under ∇ and killed by the geodesic velocity field so that X(ωi) = X(ηi) = 0. If we identify θ′ = dt for an external time variable t, then setting ωi = ηi = 0 exactly reproduces a quantum version
dpi=Vxidt,dxi=pimdt
(1.4)
of the Hamilton-Jacobi equations of motion in our approach. The generalised quantum metric in Proposition 4.4 is
G=ωiηiηiωi=dpidxidxidpi+O(θ)
where the second expression shows its origin in a lift of the classical symplectic 2-form ω = dpi ∧dxi and the first expression shows that G vanishes on solutions of the Hamilton-Jacobi equation of motion. The classical interior product iXh(ω)=dh saying that Xh 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 generalised quantum metric. Thus, there are some unusual aspects but broadly speaking the Heisenberg algebra of quantum mechanics admits a natural quantum Riemannian geometry coming out of the Schrödinger equation.
Section V does the same as Sec. IV but for A now the electromagnetic Heisenberg algebra with xa, pb Minkowski (e.g., 4-vectors and 4-covectors) and
[xa,pb]=iδab,[xa,xb]=0,[pa,pb]=iqFab,
(1.5)
as appropriate to an external U(1) gauge potential with curvature F. The exterior algebra
[dxa,xb]=imηabθ,[dxa,pc]=iqmηabFbcθ=[dpc,xa],
(1.6)
[dpc,pd]=iqFac,ddxaq2mηab(Fbc,ad+2iqFacFbd)θ
(1.7)
in Proposition 5.1 is itself an interesting outcome of the paper. Here t = x0/c is a spacetime coordinate variable (with metric −1 in this direction and c the speed of light) and we now use a different symbol s for the external geodesic time. The differential algebra is constructed so as to be compatible with the natural quantum geodesic flow equation
si2mηabDaDbϕ=0
defined by the minimally coupled Klein–Gordon (KG) operator. This flow is not something one usually considers in Physics, not least because the “wave functions” ϕ are now over spacetime. Moreover, the differential algebra has a central 1-form
ζ=dt+p0mcθ
and in the natural quotient of ΩA1 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 Sec. IV.

Note that this Sec. V is driven by the quantum algebra as a natural relativistic version of Sec. IV, and is very different from previous discussions of proper time in the Klein–Gordon context, such as Ref. 20 where the proper time and rest mass come from a canonically conjugate pair of observables. We illustrate our approach on 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 VI rounds off the paper with a self-contained Poisson-level extended phase space formalism as suggested by our results of Sec. IV 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 Sec. VII provide directions for further work.

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 Ref. 12 in terms of A-B bimodule connections.1 A possibly noncommutative unital algebra A equipped with an exterior algebra (ΩA, 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 (ΩB, 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 Ref. 12, Proposition 2.2 is essentially in Ref. 12 but reworked for right connections (which is needed to mesh later with conventions in quantum mechanics), while Corollary 2.3 is new.

We recall that geodesics on a smooth Riemannian or pseudo-Riemannian manifold M can be expressed as the autoparallel condition γ̇γ̇=0 for a curve γ in M, parametrised appropriately. Explicitly, this is
γ̈μ+Γμαβγ̇αγ̇β=0
(2.1)
and as such makes sense for any linear connection on a manifold (it does not have to be the Levi–Cività 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 γ̇ 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 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 ΩA1 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:AΩA1 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 ΩA1 to also have a *-operation for which (a.da′)* = (da′*).a* for all a, a′ ∈ A. One normally demands that ΩA1 is spanned by elements of the form ada′ for a, a′ ∈ A, otherwise one has a generalised differential calculus. Any ΩA1 then extends to an exterior algebra ΩA with product denoted ∧ and exterior derivative increasing degree by 1 and obeying a graded-Leibniz rule and d2 = 0. There is a canonical “maximal prolongation” of any ΩA1 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:ΩA1A (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-B-bimodule (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-connection1 on E as a map E:EEBΩB1 subject to two Leibniz rules. On the right,
E(e.b)=(Ee).b+edb,eE,bB
(2.2)
as usual for a right connection in noncommutative geometry. From the other side,
E(a.e)=a.Ee+σE(dae),eE,aA;σE:ΩA1AEEBΩB1
(2.3)
for a certain bimodule map σE 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-B-bimodule connection. If X is a vector field ΩB1B then we have an associated covariant derivative DX = (id ⊗ X)∇E: EE. The collection of categories EBA of such A-B-bimodule connections itself forms a coloured monoidal category, i.e., bicategory1 with a tensor product EBA×ECBECA defined by
EF=(idσF)(Eid)+idF
for all (E,E)EBA and (F,F)ECB. We defer discussion of the *-operation to where we need it in Sec. II C.

The above generalises the monoidal category EAA of right A-A-bimodule connections for any fixed differential algebra A (i.e., an algebra with differential structure). This diagonal case, in a left-handed version, is more familiar in noncommutative geometry.2,3,21 By a linear connection on A, we mean an A-A-bimodule connection ΩA1 with associated braiding σΩA1 (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 X:ΩA1ΩA1. We will also adopt an explicit notation ∇ξ = ξ(1)Aξ(2) (summation understood), so that ∇X(ξ) = ξ(1)X(ξ(2)) for all ξ ∈ Ω1.

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 ΩB1. We take ∇dt = 0 as defining a trivial classical linear connection acting on this. Here (bdt)=dbBdt=ḃdtdt 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 σE in (2.3). Each of the factors has a connection and hence we have two tensor product A-B-bimodule connections
ΩA1AE=(idσE)(id)+idE:ΩA1AEΩA1AEBΩB1
(2.4)
EBΩB1=(idσΩB1)(Eid)+idB:EBΩB1EBΩB1BΩB1
(2.5)
(albeit in our case σΩB1 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,12 proposed
(σE)EBΩB1σE(σEid)ΩA1AE=0
(2.6)
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: AB 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*:ΩA1ΩB1 for the pull-back action on ΩB1 by γ*(ada′) = γ(a)dγ(a′) for a, a′ ∈ A, see Ref. 1. In our case, since ΩB1 has basis dt, we can also write γ* explicitly as
γ*(ξ)γ*[ξ]dt,γ*[ada]=γ(a)γ̇(a),
(2.7)
where γ*[ξ] ∈ B. Also note that E = B is an A-B-bimodule by
ae=γ(a)e,e.b=eb,aA,eE,bB
and in this case ∇E, σE are maps
E:EEBΩB1=ΩB1,σE:ΩA1AEEBΩB1=ΩB1
with the proviso that A acts from the left on ΩB1 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 connectionξξ(1)Aξ(2) and let γ: AB be a differentiable algebra map and E = B an A-B-bimodule as above. Then the trivial connection Ee=de=ėdt is an A-B-bimodule connection with
σE(ξAe)=γ*(ξ)e
and ∇∇(σE) = 0 reduces to
ddtγ*[ξ]=γ*[ξ(1)]γ*[ξ(2)],ξΩA1
where γ*[ξ] is defined by (2.7).

Proof.
Here Ee=ėdt (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
E(a.e)=E(γ(a)e)=E(eγ(a))=E(e)γ(a)+eγ̇(a)dt=ėγ(a)dt+eγ̇(a)dt=γ(a)ėdt+γ̇(a)edt
and for the left Leibniz rule, this should equal
σE(dae)+a.E(e)=σE(dae)+γ(a)ėdt.
Comparing these and extending σE 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 ∇∇(σE) = 0. Here EBΩB1=ΩB1 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
ΩA1AE(ξe)=ξ(1)Aγ*(ξ(2))e+ξAdeΩA1AEBΩB1=ΩA1AΩB1
with the above proviso for the left action of A on ΩB1. It follows that ∇∇(σE) = 0 appears as
(γ*(ξ)e)=((γ*Bγ*)ξ)e+γ*(ξ)BdeΩB1BΩB1.
Expanding the linear connection on B on the left by the right Leibniz rule we cancel the de=ėdt term from both sides. This can be written more explicitly in terms of B as
ddtγ*[ξ]e=γ*[ξ(1)]γ*[ξ(2)]e
and requiring this for all eE is the condition stated.□

This equation makes sense for any differential algebra. We could also have defined Ee=(ė+eκt)dt slightly more generally with the same σE, albeit this generalisation is of no particular interest at this level. If A = C(M) and γ:RM is a smooth curve, then in local coordinates, γ*[ξidxi]=γ*(ξi)γ̇iB, where γ*(ξi)(t) = ξi(γ(t)) pulls back the coefficients of a 1-form ξ. If we also write dxi=Γijkdxjdxk for Christoffel symbols Γijk, then the algebraic geodesic equation reduces to (2.1), as analysed in Ref. 12. Thus, we have introduced ∇∇(σE) = 0 in generality as the notion of a “geodesic bimodule” and shown that the simplest choice E = B as a noncommutative bimodule, where the left action is defined by a curve γ, reduces to a single classical geodesic in the classical case.

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 have in mind not one geodesic but a distribution of them with every point moving on a geodesic. Their collective tangents define a time-dependent velocity field Xt (a path in the space of vector fields) subject to the velocity equation
Ẋt+XtXt=0.
(2.8)
The idea is to take Xt as the starting point and first solve this equation with some initial value X0 ∈ Vect(M). Any one geodesic is then recovered as a curve γ(t) such that
γ̇(t)=Xt(γ(t)),γ̇(0)=X0(γ(0)).
(2.9)
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 ∇∇(σE) = 0 equation for a different choice of bimodule, namely now E = AB, 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 AB.

Proposition 2.2.
cf Ref. 12 A right A-B-bimodule connectionE on E = AB has the form
σE(ξe)=Xt(ξ.e)dt,Ee=(ė+eκt+Xt(de))dt,
where Xt is a left vector field on A and κtA. Let :ΩA1ΩA1AΩA1 be a right bimodule connection. Then
  1. ∇∇(σE) is a bimodule map if and only if
    (σEid)(idσE)(σid)id=0,
    which is equivalent to
    Xt(idXt)(σid)=0;
  2. ∇∇(σE) = 0 if and only if in addition, for all ξΩA1,
    Ẋt(ξ)+[Xt,κt](ξ)+Xt(dXt(ξ))Xt(idXt)(ξ)=0.

Proof.

This is a right-handed version of a result for left connections in Ref. 12, but we include a brief proof for completeness. Since ΩA1AE=ΩA1B and EBΩB1=AΩB1 in the obvious way, and since dt is a basis of ΩB1, the content of the bimodule map σE is a bimodule map ΩA1BAB which when restricted to ΩA11 implies it is given by a time dependent left vector field Xt on A as stated.

Next, writing e = af(t) we have ∇E(e) = ∇E(a.1 ⊗ f) = a.∇E(1 ⊗ 1.f) + σE(daf) = a.∇E(1 ⊗ 1).f + a ⊗ df + σE(da ⊗ 1).f gives the formula for ∇E, for some undetermined ∇E(1 ⊗ 1) = κt ⊗ dt and some σE(da ⊗ 1) = Xt(da) ⊗ dt.

Next, by similar arguments to those at the start of the proof of [Ref. 1, Lemma 4.13], we see that ∇∇(σE) is a bimodule map if and only if
(σEid)σΩA1E=σEΩB1(idσE),
where σΩA1E=(idσE)(σid) and σEΩB1=σEid as σΩB1 is the identity, which is (1). Since σE is given by Xt, we obtain the second form as a map ΩA1AΩA1A.
For (2), as ∇∇(σE) is a right module map, we only have to calculate, for ξΩA1,
(σE)(ξ1)=idΩB1+(idσΩB1)(Eid)(Xt(ξ)dt)(σEid)idE+(idσE)(id)(ξ1)=E(Xt(ξ))dt(σEid)ξE(1)+(idσE)((ξ)1)=Ẋt(ξ)+Xt(ξ)κt+Xt(dXt(ξ))Xt(ξκt)Xt(idXt)(ξ)dtdt
and the vanishing of this is (2). As Xt is a left vector field, we wrote (Xt.κt)(ξ) = Xt(ξ)κt and (κt.Xt)(ξ) = Xt(ξκt).□

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, [Ref. 1, Lemma 3.70] tells us that ∇L = σ−1∇ is a left connection on ΩA1. If we further assume that ΩA1 is finitely generated projective as a left module, which classically reduces to saying that the cotangent space is locally trivial, then by [Ref. 1, Proposition 3.80] we can dualise a left bimodule connection ∇L on ΩA1 to a right one X on XhomA(ΩA1,A), the space of left-module map vector fields. This is a bimodule with (a.X.b)(ω) = (X(ωa))b for all a, bA and ωΩA1. In terms of the evaluation map ev:ΩA1AXA, we have
dev(ξX)=(idev)(Lid)+(evid)(idX)(ξX),(idev)(σLid)=(evid)(idσX):ΩA1AΩA1AXΩA1.
We can also define σXX:XAXXAX such that
(evid)(idσXX)=(idev)(σXid):ΩA1AXAXX.

Corollary 2.3.

In Proposition 2.2, lethave σ invertible, ΩA1 be finitely generated projective as a left module and X be the associated right connection on X. In these terms, the corresponding conditions are

  1. σXX(XtXt)=XtXt;

  2. Ẋt+[Xt,κt]+(idXt)X(Xt)=0.

Proof.
From Proposition 2.2 (1), we have Xt(id ⊗ Xt)σ = Xt(id ⊗ Xt), so (2) can be rewritten as
Ẋt(ξ)+[Xt,κt](ξ)+Xt(dXt(ξ))Xt(idXt)σ1(ξ)=0,
which by duality is the displayed Eq. (2). The other part is given by
ev(idevid)(σLXtXt)=ev(idevid)(ididσXX(XtXt))
as a function :ΩA1AΩA1A, where σL = σ−1 is inverse to σ in Proposition 2.2 (1).□

Next, returning to our classical model at the start of Sec. II B, 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 Xt, then conservation of mass (the continuity equation in fluid mechanics22) dictates
ρ̇+Xt(dρ)+ρdiv(Xt)=0.
In our algebraic formulation, we were led to eE=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
ė+Xt(de)+eκt=0,κt+κ̄t=div(Xt)
(2.10)
for the amplitude e(t), which is exactly ∇Ee = 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–Cività connection, we want to maintain
ϕ0(ρ(t))Mdx|g|ρ(t)=1
as the probability density ρ(t) evolves. Here |g| is the determinant of gμν. There is an associated inner product
f(t)|e(t)=ϕ0(f̄(t)e(t)),
where e(t), f(t) ∈ L2(M) with respect to the Riemannian measure as above, and we used the usual bra-ket notation. From this point of view, ∇Ee = 0 ensures that ⟨e(t)|e(t)⟩ = 1 as e(t) evolves. Thus, our approach to geodesics in Sec. II B leads us naturally into a framework of states and inner products in common with quantum mechanics, even though we are doing classical geodesics with A = C(M).
For the algebraic formalism, we take A and B to be *-algebras with *-differential structures as in Sec. II A. Any A-B-bimodule E has a conjugate Ē, which is a B-A-bimodule with elements ē for eE and vector space structure ē+f̄=e+f̄ and λē=λ̄ē for λC and e, fE, see Ref. 1. The algebra actions are ē.a=a*.ē and b.ē=e.b*̄ for aA and bB. Now suppose that E is equipped with a B-valued inner product ,:ĒAEB which is bilinear and hermitian in the sense ē,f*=f̄,e, where ē,a.f=ē.a,f for all aA. When B is a dense subalgebra of a C*-algebra, we call the inner product positive if ē,e>0 for all eE. In this context, a right A-B-bimodule connection ∇E is said to preserve the inner product if for all e, fE we have1 
dē,f=(id,)(Ē(ē)f)+(,id)(ēE(f)).
(2.11)
Here the left connection Ē:ĒΩB1BĒ is defined by Ē(ē)=ξ*p̄ if ∇E(e) = pξ (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” tR with t* = t. Then d on the left is derivative in the R coordinate. Hence, ifE preserves ⟨, ⟩ and e obeys ∇Ee = 0 as above for geodesic evolution then ddtē,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 = AB [or E=C(R,A)]. Since A could be noncommutative, instead of a measure we fix a positive linear functional ϕ0:AC or “vacuum state” and define f̄,e=ϕ0(f*e) as above, pointwise at each t so that the result is B-valued. Equivalently, we can suppose we are given ⟨, ⟩ and define ϕ0(a)=1̄,a, where a is viewed in E as constant in time.

Proposition 2.4.
Ref. 12 The connection on E = AB in Proposition 2.2 preserves the inner product on E if and only if for all aA and ξΩA1,
1̄,κt*a+aκt+Xt(da)=0,1̄,Xt(ξ*)Xt(ξ)*=0.

Proof.
This is again from12 but we provide a short explanation. The condition for preservation is, for a, cA,
0=cκt+Xt(dc)̄,a+c̄,aκt+Xt(da).dt=1̄,κt*c*a+Xt(dc)*a+c*aκt+c*Xt(da).dt=1̄,κt*c*a+Xt(a*dc)*+c*aκt+Xt(c*da).dt
and putting c = 1 gives the first displayed equation. Using this with c*a instead of c in the condition for preservation gives
0=1̄,Xt(a*dc)*Xt(d(c*a))+Xt(c*da)=1̄,Xt(a*dc)*Xt(dc*.a).
.□

We call the first of the displayed equations in Proposition 2.4 the divergence condition for κt and the second the reality condition for Xt. 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: AB or “state” given by
ϕ(a)=e|a|e=ē,a.e=a*.ē,e=ϕ0(e*ae)
(2.12)
in our two notations for the inner product. Here, positive means ϕ(a*a) ≥ 0 for all aA 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 notion23 of a Hilbert C* bimodule upon completion with respect to the induced norm |e|2=ē,eB. In our case of interest, B=C(R) and for every eE we have a possibly un-normalised state ϕt at each time defined by ϕt(a) = ⟨ e(t)|a|e(t)⟩.
We now go beyond the noncommutative differential geometric formulation of geodesics12 covered in Sec. II, extending this to a *-algebra A of observables represented on a Hilbert space H as in quantum mechanics. We still employ the “universal equation” ∇∇(σE) = 0 but for a new choice of bimodule E=HB with B=C(R), or more precisely its completion E=C(R,H) with its canonical A-B-bimodule structure
(a.ψ)(t)=ρt(a)ψ(t),(ψ.b)(t)=ψ(t)b(t).
Here ψE and ψ(t)H, while ρt 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 ρt 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 = ρt(a)◦T and T.a = Tρt(a) for all TL(H). Our possibly unbounded operators will be differential operators and hence applicable and composable on suitable domains. These are issues already in ordinary quantum mechanics and we proceed on the same basis, being concerned here only with the general structure rather than analytic aspects.

Lemma 3.1.
In this context, a bimodule map σE:ΩA1AEEC(R)Ω1(R) necessarily has the form
σE(ξψ)=X̃(ξ)(ψ)dt
for all ξΩA1, for some A-C(R) bimodule map X̃:ΩA1C(R,L(H)) and suitable ψE.

Proof.
As σE is a right C(R)-module map and Ω1(R) has basis dt, we can write σE(ξψ)=X̃(ξ)(ψ)dt for ψH and some linear map X̃(ξ)C(R,L(H)) which we apply pointwise to H. Now
X̃(ξa)(ψ)=σE(ξaψ)=σE(ξρ(a)ψ)=X̃(ξ)ρ(a)(ψ),X̃(aξ)(ψ)=σE(aξψ)=aσE(ξψ)=ρ(a)X̃(ξ)(ψ)
gives the bimodule map.□

Thus, the data here is an “operator-valued time-dependent vector field” which for each t is a bimodule map X̃t:ΩA1L(H) in the sense X̃t(a.ξ)=ρt(a)X̃t(ξ) and X̃t(ξ.a)=X̃t(ξ)ρt(a). We can also think of X̃(ξ) as an element of the dense subspace L(H)C(R).

Lemma 3.2.
For a time dependent operator htC(R,L(H)) we define a right C(R) connection on E=C(R,H) by
(Eψ)(t)=ψ̇(t)+ht(ψ(t))dt.
This is a A-C(R)-bimodule connection with σE as above if and only of
X̃t(da)=[ht,ρt(a)]+ρ̇t(a)
extends to a well defined bimodule map X̃t:ΩA1L(H) at each t, in which case σE(ξψ)(t)=X̃t(ξ)(ψ(t))dt.

Proof.
We only prove this for the stated form of E:EEBΩB1, but in fact this is a reasonably general case as follows. First ∇E: when restricted to time independent ψH should be of the form (Eψ)(t)=ht(ψ)dt for some time dependent linear operator ht. Then multiplying ψH by a function of time and using the Leibniz rule (2.2) gives the form for general ψE in the statement. Next, for a bimodule connection we need a map σE and by Lemma 3.1 we assume X̃t:ΩA1L(H) and σE(ξψ)(t)=X̃t(ξ)(ψ(t))dt. As in the lemma, each X̃t is left A-module map. Similarly at each t, σE is a well-defined map from ΩA1AH with X̃t(ξ)ρt(a)=X̃t(ξ.a), i.e., each X̃t is a right A-module map. Now we use (2.3) to write
σE(dae)=E(a.ψ)a.Eψ=E(ρt(a)(ψ))ρt(a)Eψ=ρ̇t(a)(ψ)+ρt(a)(ψ̇)+htρt(a)(ψ)ρt(a)ψ̇ρt(a)ht(ψ)dt=([ht,ρt(a)]+ρ̇t(a))(ψ(t))dt
which gives us the required form of X̃t.□

There are three important comments to make about this result. The first is that the well definedness of X̃t could be read, for a given ΩA1, as a condition on ht. However, in this paper we prefer to start with ht and read the condition as a constraint on the calculus ΩA1. The second comment is that in many applications the time dependent operator ht will be the action of a time dependent element of the algebra HtA via the representation, i.e., ht=ρt(Ht). 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 Xt:ΩA1A satisfying
ρt(Xt(da))=ρt([Ht,a])+ρ̇t(a).
(3.1)
This Xt would then be a geometric time dependent vector field on A and X̃t in the lemma is then its image. The third comment is that condition (3.1) does typically hold, e.g., if we suppose that H as an A-module has a cyclic vector x0H (typically a vacuum vector in physics), such that {ρt(a)(x0): aA} is dense in H at each t and that the operators contain x0 in their domain, then we have X̃t(ξ)(ρt(a)(x0))=X̃t(ξa)(x0)H at each t (smoothly in t). This means that we have a left A-module map X̃:ΩA1C(R,H) given at each t by X̃t(ξ)=X̃t(ξ)(x0) and recovering the bimodule map X̃ on the dense subset ρt(A)(x0) by the formula X̃t(ξ)(ρt(a)(x0))=X̃t(ξa). If we further suppose that ρt satisfies ρt(a)(x0) = ρt(a′)(x0) only when a = a′, and that every X̃t(ξ) maps the cyclic vector x0 into the dense subset ρt(A)(x0), then we obtain a left module map X:ΩA1C(R,A) characterised by
ρtXt(ξ)(x0)=X̃t(ξ).
As X̃ is also a bimodule map, it follows that ρtXt(ξ)=X̃t(ξ) on the dense subset ρt(A)(x0), and therefore that ρtXt(ξ)=X̃t(ξ) as desired. We can also work with the image of X in the dense subspace AC(R).

We now proceed as in Sec. II to take the trivial linear connection ∇dt = 0 acting on ΩB1 and an arbitrary linear right connection ∇ on ΩA1, and the tensor product connections (2.4) and (2.5) on the domain and codomain of σE.

Proposition 3.3.
For E=C(R,H) andE defined by ht, ∇∇(σE) = 0 is equivalent to an auxiliary condition of the same form as (1) in Proposition 2.2 and the further condition
[ht,X̃t(ξ)]+X̃̇t(ξ)=X̃t(idX̃t)ξ,ξΩA1.
This can be read as a time evolution equation for X̃t, and this is consistent with an initial A-bimodule map X̃0 giving a bimodule map X̃t 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,
[ht,[ht,ρt(a)]]+[ḣt,ρt(a)]+2[ht,ρ̇t(a)]+ρ̈t(a)=X̃t(idX̃t)da,aA.

Proof.
To calculate ∇∇(σE), we need
EΩR1σE(ξψ)=EΩR1(X̃t(ξ)(ψ(t))dt)=X̃t(ξ)(ψ̇(t))+X̃̇t(ξ)(ψ(t))+htX̃t(ξ)(ψ(t))dtdt
and also
(σEid)ΩA1E(ξψ)=(σEid)(idσE)(ξψ)+σE(ξ(ψ̇(t)+htψ(t)))dt=(X̃t(idX̃t)ξ)ψ+X̃t(ξ)(ψ̇(t)+htψ(t))dtdt.
So we get the equation
(X̃t(idX̃t)ξ)ψ+X̃t(ξ)(ψ̇(t)+htψ(t))=X̃t(ξ)(ψ̇(t))+X̃̇t(ξ)(ψ(t))+htX̃t(ξ)(ψ(t))
which gives the first displayed equation in the statement. Replacing ξ with ξ.a in the displayed equation and using X̃t being a right module map gives X̃̇t(ξ.a)=X̃̇t(ξ)ρt(a)+X̃t(ξ)ρ̇t(a), showing that the time evolution equation for X̃t is consistent with X̃t being a right A-module map. Similarly replacing ξ with a.ξ shows that the evolution equation for X̃t is consistent with X̃t being a left A-module map, but this requires using (1) of Proposition 2.2. Now just use the formula for X̃t(da) for the second displayed equation in the statement.□

The single geodesic case of Sec. II A is recovered by H=C and ρt(a) = a(γ(t)) i.e., the evaluation representation along the image of the curve in the classical case, or ρt(a) = γ(a)(t) in the algebraic version. In this case, ht is some function of t and does not enter, while
X̃:Ω1(M)C(R),X̃t(adxi)=a(γ(t))γ̇i(t)
in the classical case or X̃t(ξ)=γ*[ξ] for ξΩA1 in the algebraic version. Then Proposition 3.3 reduces to Proposition 2.1.

Similarly, the geodesic flow of Sec. II B is recovered with H=C(M) (or some completion thereof) in the classical case or H=A with the left regular representation (or more precisely a completion thereof) in a potentially noncommutative version. In the classical case, compatibility with the calculus Ω1(M) requires the Hamiltonian to have the form ht=X̃t+κt for some time dependent vector field X̃t and function κt acting by left multiplication on C(M) and the condition in Propostion 3.3 reduces to the velocity field Eq. (2.8). In the noncommutative case with ρ the left regular representation, we have discussed in (3.1) how X̃t can arise as the image of a bimodule map Xt. This is not quite as general as Proposition 2.2, where we only assumed a left vector field.

It remains to extend Sec. II C to our more general setting. We suppose that ρt is a *-homomorphism for each t. ϕt(a) = ⟨e(t)|a|e(t)⟩ = ⟨e(t)|ρte(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 ϕ0 or a preferred element 1 ∈ E as was possible before.

Corollary 3.4.

In the setting of Proposition 3.3, the inner product ,:ĒAEB is preserved byE if and only if ht is anti-hermitian.

Proof.
For ψ, ζE we have
(id,)(Ē(ζ̄)ψ)+(,id)(ζE(ψ))=((ζ̇+htζ)̄,ψ+ζ̄,ψ̇+htψ)dt
and this is required to equal
tζ̄,ψdt=(ζ̇̄,ψ+ζ̄,ψ̇)dt,
which is just the condition that ht is anti-hermitian.□

If we consider Proposition 2.2 with X̃t a bimodule map as a special case with ht as in (3.1), then ht 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 Ref. 12, if ψ obeys ∇Eψ = 0 then
ddtψ|a|ψ=ψ|X̃t(da)|ψ.
(3.2)
This comes from ∇E preserving the inner product and the definition of σ, so that dψ|aψ=(||id)(idσ)(ψ̄daψ)=ψ|X̃t(ξ)|ψdt.
Our second remark is that if we are in the setting of (3.1) where X̃t is the image of a bimodule map Xt(da) = [Ht, a] for all aA (with ρ time-independent and faithful) then we also have a solution of the geodesic velocity equation in Proposition 2.2 with κt = 0. This implies a quantum geodesic flow on E=AC(R) which, from ∇E in Proposition 2.2, comes out as
ȧt=X̃t(dat)=[Ht,at]
(3.3)
for at a time-dependent element of A (denoted eE = AB there). This is minus the usual Heisenberg evolution for an actual quantum system with Hamiltonian ht and Ht = iht/, so we call it the “anti-Heisenberg flow” underlying the “Schrödinger flow” studied above. We could, of course, redefine Ht to have the usual sign but we want to explain that the version with the minus sign is what emerges due to a different relationship with the Schrödinger flow compared to the usual context for the Heisenberg evolution. First of all, we can interpret this flow probabilistically as in Sec. II C if we fix a hermitian inner product on the *-algebra A by means of a positive linear functional ϕ0:AC. For the unitarity conditions in Proposition 2.4 to apply with κt = 0, we need
ϕ0(Xt(da))=ϕ0([Ht,a])=0
(3.4)
for all aA, which happens automatically if ϕ0 is a trace, and
ϕ0(Xt((da)*)(Xt(da))*)=ϕ0([Ht,a*][Ht,a]*)=ϕ0([Ht+Ht*,a*])=0
which is automatic as Ht = iht/ with ht Hermitian. This then applies to ξ = adb and hence to general ξΩA1 since Xt is a bimodule map. Note that ϕ0 does not have to be a trace, for example we can let ϕ0(a) = ⟨ψ|ρ(a)|ψ⟩ where |ψ⟩ is any eigenvector of the Hamiltonian (such as the ground state). For then, ϕ0([Ht,a])=ψ|[ht,ρ(a)]|ψ=ht*ψ|ρ(a)|ψψ|ρ(a)|htψ=0 if ψ is an eigenvector. Similarly for ϕ0 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 X̃t to a time dependent at 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 L2 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 ddtψ|a|ψ=ψ|[Ht,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 Xt, the geodesic velocity equation in Proposition 2.2 also holds with arbitrary κt since then [Xt, κt] = 0. This more general flow is then not connected with the Schrödinger flow above but we can still consider it. Moreover, choosing κt = Ht means that the unitarity conditions in Proposition 2.4 now hold automatically for any ϕ0 as then κt*a+aκt+Xt(da)=0 for the first condition, while the second condition holds automatically as already noted. For this second choice of κt, the quantum geodesic flow on A given by ∇E = 0 becomes
ȧt=atκtXt(dat)=Htat
(3.5)
for the evolution of atA. The significance of this second “non-standard flow” is unclear as it is not something we would normally consider in quantum mechanics.
In this section, we consider what the quantum geodesic formalism of Sec. III amounts to for A the standard Heisenberg algebra with generators xi and pi for i = 1, …, n and relations
[xi,pj]=iδij,[xi,xj]=[pi,pj]=0
for a suitable choice of differential calculus. We fix our Hamiltonian in the standard form
h=p12++pn22m+V(x1,,xn)
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 ΩB1dt=0 and σΩB1 the identity map.
Now we choose E=C(R,L2(Rn)) with A acting in the standard Schrödinger representation. Here ψE is a time dependent element ψ(t)L2(Rn), where Rn has standard basis x1, …, xn, and xiA act on ψ by multiplication and pi by ixi. The geodesic flow will be given by E:EEBΩB1=EBdt (we can also view the connection as an operator ∇E: EE if we leave ⊗Bdt understood). We will take this to be
Eψtψ1ihψdt
so that ∇Eψ = 0 lands on the standard Schrödinger equation. We will show:

Theorem 4.1.

Let A be the Heisenberg algebra and hA a Hamiltonian as above. There exists a canonical differential structure ΩA1, generalised quantum metric GΩA1AΩA1 and a metric compatible connectionon ΩA1 such that the standard Schrödinger equation is realised as a quantum geodesic flow equationEψ = 0 with respect to a geodesic velocity field XhomA(Ω1,A).

The proof will be a series of results starting with a class of “almost commutative” centrally extended differential structures17 where the classical commutation relations of differentials on phase space acquire a multiple of a central 1-form θ′.

Proposition 4.2.
There is a unique centrally extended differential calculus ΩA1 on the Heisenberg algebra such thatE is an A-B-bimodule connection with σE(θ′ ⊗ ψ) = ψ ⊗ dt, namely with the bimodule relations
[dpi,pj]=i2Vxixjθ,[dpi,xj]=[dxi,pj]=0,[dxi,xj]=imδijθ.
Moreover σE(ξψ) = X(ξ)ψdt for all ξΩA1, for a bimodule map
X:ΩA1A,X(θ)=1,X(dpi)=Vxi,X(dxi)=pim
acting on ψ in the Schrödinger representation.

Proof.
Recall that a bimodule connection involves the existence of a bimodule map σE:ΩA1AEEBΩB1=EBdt, and that by definition of σE we have
σE(dpiψ)=E(piψ)piE(ψ)=Vxiψdt,σE(dxiψ)=E(xiψ)xiE(ψ)=1mpiψdt.
For the central element we assume that σE(θ′ ⊗ ψ) = ψ ⊗ dt. Now we find the commutation relations in the calculus as follows:
σE([dpi,pj]ψ)=σE(dpipjψ)pjσE(dpiψ)=i2Vxixjψdt,σE([dpi,xj]ψ)=σE(dpixjψ)xjσE(dpiψ)=0,σE([dxi,pj]ψ)=σE(dxipjψ)pjσE(dxiψ)=0,σE([dxi,xj]ψ)=σE(dxixjψ)xjσE(dxiψ)=imδijψdt.
From these we are led to the commutation relations as stated, which can be shown to 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 X(db)=[h,ρ(b)] from Lemma 3.2 in the time-independent case, with
h=1iρ(h).
(4.1)
In the lemma, X is an operator-valued map but we see that this operator factors through a map X:ΩA1A and the Schrödinger representation of A (the pi, xi in the formulae for σE 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 ΩA1 and σE, which is uniquely determined from the pre-chosen ∇E once ΩA1 is fixed. The form of σE 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 xi only, then
df(x)=fxidxii2m(2f)θ,
(4.2)
where 2f=i2fxixi is the Rn Laplacian. The structure of the calculus here is that of a central extension17,18 by θ′ of the more trivial 2n-dimensional calculus on A where we set θ′ = 0.
Next, we turn to the geodesic velocity equation ∇∇(σE) = 0 which depends on the choice of linear connection ∇ acting on ΩA1. By Proposition 3.3, this is equivalent to the auxiliary Eq. (1) in Proposition 2.2 and an autoparallel equation,
(XX)(σid)=0,(XX)da=1i[X(da),h],aA,
(4.3)
where we prefer to write X(id ⊗ X) as XX with the product of the result in A understood, given that X is a bimodule map.

Proposition 4.3.
On the above ΩA1, we have a natural right bimodule connection obeying ∇∇(σE) = 0, namely ∇(θ′) = 0 and
(dxi)=1mθdpi,(dpi)=2Vxixjθdxj+i2m2Vxiθθ,
σ(dxidxj)=dxjdxi,σ(dpidpj)=dpjdpi,
σ(dxidpj)=dpjdxi+im2Vxjxiθθ,σ(dpidxj)=dxjdpiim2Vxixjθθ
and σ = flip when one factor is θ′.

Proof.
The second half of (4.3), explicitly, is
(XX)(θ)=0,(XX)(dpi)=1mVxixjpj+i2m2Vxi,(XX)(dxi)=1mVxi.
The stated ∇ is then easily seen to obey these. The calculation of σ is then routine. Thus,
σ(dxidpj)=dpjdxi+[(dpj),xi]=dpjdxi+2Vxjxkθdxk+i2m3Vxjxkxkθθ,xi=dpjdxi+im2Vxjxiθθ,
σ(dpidxj)=(pi.dxj)pi.(dxj)=([pi,dxj])+(dxj.pi)pi.(dxj)=dxjdpi+[(dxj),pi]=dxjdpi+1mθdpj,pi=dxjdpiim2Vxixjθθ,
σ(dpidpj)=(pi.dpj)pi.(dpj)=([pi,dpj])+(dpj.pi)pi.(dpj)=dpjdpi+i2Vxixjθ+[(dpj),pi]=dpjdpi+iθd2Vxixj+2Vxjxkθdxk+i2m3Vxjxkxkθθ,pi=dpjdpi+iθd2Vxixj2Vxjxk,piθdxk+i2m3Vxjxkxk,piθθ=dpjdpi+iθd2Vxixji3Vxixjxkθdxk+(i)22m4Vxixjxkxkθθ.
Substituting (4.2) 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
(XX)(σid)(dxidxj)=[X(dxj),X(dxi)]=0(XX)(σid)(dpidpj)=[X(dpj),X(dpi)]=0(XX)(σid)(dxidpj)=[X(dpj),X(dxi)]+im2Vxjxi=0(XX)(σid)(dpidxj)=[X(dxj),X(dpi)]im2Vxjxi=0,
which can be checked to hold for the form of ∇. For example, for the last equation
[X(dxj),X(dpi)]=pjm,Vxi=im2Vxjxi
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 ΩA1 and uniquely characterised by this as follows.

Proposition 4.4.

  1. ΩA1 has 2n central 1-forms
    ωi=dpi+iVθ,ηi=dxipimθ
    such that X(ωi) = X(ηi) = 0 for all i. Moreover,is the unique right connection such that
    θ=ωi=ηi=0,i.
  2. ΩA1AΩA1 has a central element
    G=dpidxidxidpi+θdhdhθ+im2Vθθ;dh=(dxi)iV+pimdpi
    such that
    (Xid)G=(idX)G=0,G=0,σ(G)=G.

Proof.

For (1), the commutation relations of Proposition 4.2 immediately give that ωi, ηi are central, clearly annihilated by X as stated there. That these are covariantly constant requires (dxi)=(θpim)=θdpim+(θ)pim and ∇(dpi) = −∇(θiV) = −θ′ ⊗ diV − (∇θ′)iV if ∇ is a right connection. We then assume ∇θ′ = 0 and use (4.2).

Part (2) is immediate from part (1) once we compute that
ωiηiηiωi=(dpi+iVθ)(dxipimθ)(dxipimθ)(dpi+iVθ)=dpidxidpipimθ+iVθdxiiVpimθθdxidpidxiiVθ+pimθdpi+pimiVθθ=dpidxidxidpi+θ(iVdxi+pimdpi)(iVdxi+pimdpi)θ+im2Vθθ=G
where we used the commutation relations of the calculus to reorder (dpi)pi and (dxi)iV (the two corrections cancel) and the usual Heisenberg relations for the θ′ ⊗ θ′ term for the third equality. We then recognise the answer in terms of dh (where a reorder of iVdxi to match dh cancels between the two terms). The formula for dh follows from (4.2) and the commutation relations of the calculus. We then apply σ from Proposition 4.3.□

To discuss the quantum geometry further, we now need to specify ΩA2. For every ΩA1 there is a canonical “maximal prolongation” obtained by applying d to the degree 1 relations, and other choices are a quotient.

Definition 4.5.

Let ΩA2 be the quotient of the maximal prolongation of ΩA1 by the additional relations dθ′ = 0 and θ2 = 0.

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. Explicitly, the rest of the relations (by applying d to the degree 1 relations) are
{dxi,dxj}={dxi,dpj}={θ,dxi}={θ,dpi}=0,{dpi,dpj}=iV,ijkdxkθ.
Also, since θ2 = 0, corrections from the commutation relations vanish and one similarly has
{ηi,ηj}={ηi,ωj}=0,{ωi,ωj}=iV,ijkηkθ.

Corollary 4.6.
For ΩA2 as in Definition 4.5, the linear connectionin Proposition 4.3 is flat and has torsion
T(θ)=0,T(dxi)=1mdpiθ,T(dpi)=2Vxjxidxjθ.
Moreover, the 2-form
ω̃=(G)=2(dxidpi+dhθ)
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 4.4 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 R = (id ⊗ d + ∇ ∧id)∇ = 0 as clearly R(ωi) = R(ηi) = R(θ′) = 0, since ∇ itself vanishes on these.□

We see that G is not quantum-symmetric in the sense of ∧(G) = 0 as needed for a strict quantum metric;1 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–Cività connection” in the sense of quantum Riemannian geometry either. Rather, G is if anything antisymmetric with respect to σ (but this depends on ∇) and moreover, one can quotient out θ′ = 0 to work in the unextended calculus on A, in which case ω̃ 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 Sec. VI.

It should also be clear that the remarks at the end of Sec. III apply since the quantum vector field used above is explicitly constructed in Proposition 4.2 as the action of a bimodule map X:ΩA1A. 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
X(dX(dxi))=X(dpim)=1miV,X(dX(dpi))=X(diV)=ijVpjm+i2m2iV
are the same expressions as computed for (XX)∇dxi and (XX)∇dpi in the Proof of Proposition 4.3. 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 ϕ0 such that (3.4) holds and, as explained there, the natural way to do this is to take a convex linear combination ϕ0(a) = ∑iρiψi|a|ψi⟩ of pure states associated to normalised energy eigenvectors |ψi⟩ in the Schrödinger representation. Here ρi ∈ [0, 1], ∑iρi = 1. We can, for example, just take the ground state, ϕ0(a) = ⟨0|a|0⟩. We also noted a non-standard quantum geodesic flow with κ = ih/.
We look briefly at the case of V=12mν2ixi2. For the calculus, geodesic velocity field and connection, we have
[xi,pj]=iδij,[dxi,xj]=imδijθ,[dpi,pj]=imν2δijθ,[dxi,pj]=[dpj,xi]=0,
X(dxi)=pim,X(dpi)=mν2xi,dxi=θdpim,dpi=mν2θdxi,
σ(dxidpj)=dpjdxi+iν2δijθθ,σ(dpidxj)=dxjdpiiν2δijθθ
and from Proposition 4.4 we have the central element and 1-forms
G=dpidxidxidpi+θdhdhθ+iν2θθ
dh=mν2(dxi)xi+pimdpi,ωi=dpi+mν2xiθ,ηi=dxipimθ
(sum over repeated indices) showing the expected symmetry between xi, pi. This quantum geometry quantises an extended phase space geometry as we discuss further in Sec. VI, and underlies our interpretation of the usual Schrödinger evolution as a quantum geodesic flow. It means, for example, that
ddtψpi2ψ=mν2ψpixi+xipiψ
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:ΩA1A and the machinery of Sec. II B with κ = 0. This is ȧt=[Ht,at] which in our case is
ȧt=[hi,at];at=ethia0ethi
as usual but with a reversed sign. We can make this concrete by restricting to the form
at=χi(t)xi+ψi(t)pi
(4.4)
for some evolving complex coefficients χi(t), ψi(t). Then
χ̇i=ψimν2,ψ̇i=χim
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
χ̈i=(κ2+ν2)χi2κχ̇i.
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 ϕ0 or (ii) κ = Ht and any positive linear functional ϕ0 if we want a unitary flow according to Proposition 2.4.
For case (i), the natural way to construct ϕ0 is as a convex linear combination of pure states obtained from energy eigenvectors. The latter are of course |n⟩ labeled by n = 0, 1, 2, … and given by Hermite functions with eigenvalue En=(n+12)ν of h. For example, the vacuum expectation value ϕ0(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
ȧt=hiat;at=ethia0,
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.,
n|at|n=ei(n+12)νtn|a0|n.
The evolution is nevertheless *-preserving in the sense of Proposition 2.4 and is best understood in terms of the associated positive linear functionals ϕt defined by at according to (2.12), namely
ϕt(b)=ϕ0(at*bat)=ϕ0(a0*ethibethia0)=ϕ0(btheis)
where btheis=ethibethi is the usual Heisenberg flow of an initial bA and ϕ0=ϕ0(a0*()a0) is the initial value of ϕt. If ϕ0 happens to be a convex linear combination of pure states given by energy eigenvectors then ϕt comes out the same as the positive linear functional corresponding to the anti-Heisenberg flow of Case (i). In this case, if a0 commutes with h then ϕt = ϕ0 does not evolve. These remarks are not specific to the harmonic oscillator but features of our formalism.

Our goal in this section is to extend the quantum geodesic flow of Sec. IV to a relativistic setting with flat spacetime metric η = diag(−1, 1, 1, 1) and an electromagnetic background with gauge potential Aa in place of the potential in the Hamiltonian in Sec. IV. This is done in Sec. V A using the Klein–Gordon operator but it is important to note that we are not proposing this as a way of solving the Klein–Gordon equation itself nor as an alternative to its established role in quantum field theory. Rather, this is something new which, unlike the nonrelativistic version, does not land on an established equation, not least due to the external geodesic time parameter in addition to the spacetime time. As a first look at what we have, Sec. V B discusses how it could nevertheless be visualised in a quantum-mechanics like manner in a laboratory frame and shows what we get in some examples.

Our flow is loosely motivated by an analogy with GR(General Relativity), where a geodesic extremises the proper distance between two points but where in practice it is equivalent and more natural to omit the square root and extremise the integral of the square of the proper velocity. Similarly in a field theory context, let Da=xaiqAa, 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 . We introduce an external time parameter s for the geodesic flow so now B=C(R) for this parameter and we set x0 = ct in terms of the usual time coordinate t. Our first guess for the natural flow is to consider
sϕicηabDaDbϕ=0
motivated by the formula for proper time in GR, but it is unpleasant to work with square roots and we will instead consider the flow
ϕsi2mηabDaDbϕ=0
(5.1)
In effect, we write ηabDaDb=ηabDaDb/ηcdDcDd and replace the denominator by its on-shell value mc where m is the particle rest mass. The half is to allow for the idea that any kind of variation of ηabDaDb brings down a 1/2 in comparison to that of ηabDaDb.

Although the flow (5.1) is not something usually considered in physics, we will see that it lends itself to a quantum geodesic formulation. Indeed, that this works out will be a minor miracle in terms of the amount of algebra, which in itself suggests that this is a natural relativistic generalisation of the quantum geodesic flow in Sec. IV.

Motivated as above, we consider H=L2(R1,3) with its 4D Schrödinger representation of the electromagnetic Heisenberg algebra A with commutation relations
[xa,pb]=iδab,[xa,xb]=0,[pa,pb]=iqFab,
(5.2)
given that [Da,Db]=iqFab and pa is represented by −iℏDa. Here Fab = Ab,aAa,b and the algebra is associative due to dF = 0.
We set E=HC(R), or more precisely E=C(R,H), and
Eϕ=ϕsi2mηabDaDbϕds,σE(dxaϕ)=imηabDbϕds.
We can also write σE(daϕ) = X(da).ϕ ⊗ ds, where
X(dxa)=1mηabpb,X(dpc)=q2mηab(2FcapbiFcb,a)
(5.3)
maps to the algebra and its output then acts on H in the 4D Schrödinger representation. For j = 1, 2, 3, we have X(dxi)=pim so in some sense dxids is being identified with the value pi/m which is consistent with Special Relativity if s were to be proper time. We also define the Hamiltonian
h=ηab2mpapb
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
[dxa,xb]=imηabθ,[dxa,pc]=iqmηabFbcθ=[dpc,xa],
[dpc,pd]=iqFac,ddxaq2mηab(Fbc,ad+2iqFacFbd)θ
such that X extended by X(θ′) = 1 is a bimodule map ΩA1A.

Proof.
We give the two most difficult checks, first d applied to the commutator of two ps:
[dpc,pd]+[pc,dpd]=[dpc,pd][dpd,pc]=iq(Fac,dFad,c)dxaq2mηab(Fbc,ad+2iqFacFbdFbd,ac2iqFadFbc)θ=iqFdc,adxa2q2mηabFdc,abθ=iqFdc,adxai2mηabFdc,abθ
which we compare to
d([pc,pd])=(i)2iqdFcd=iqdFcd=iqdFdc
and which agree when we remember to use the centrally extended formula for d. Now we check the three ps Jacobi identity:
[[dpc,pd],pe]=iqFac,ddxaq2mηab(Fbc,ad+2iqFacFbd)θ,pe=2qFac,dedxaiqFac,d[dxa,pe]q2mηab(Fbc,ad+2iqFacFbd),peθ=2qFac,dedxaiqFac,diqmηabFbeθi2q2mηab(Fbc,ade+2iqFac,eFbd+2iqFacFbd,e)θ=2qFac,dedxai2q2mηab(Fbc,ade+2iqFac,eFbd+2iqFad,eFbc+2iqFac,dFbe)θ
and so
[[dpc,pd],pe][[dpc,pe],pd]=i2q2mηab(2iqFae,dFbc+2iqFad,eFbc)θ=2q2mηabFde,aFbcθ.
As
[[pd,pe],dpc]=(i)2iq[dpc,Fde]=iqiqmηabFbcθFde,a,
we see that the three ps Jacobi identity is satisfied.□

We next want to choose ∇ on ΩA1 such that the conditions (4.3) for ∇∇(σE) in Proposition 3.3 hold.

Theorem 5.2.
There is a right bimodule connectionon ΩA1 given byθ′ = 0 and
(dxd)=qmηcdFacθdxa+θiq2m2ηabηcdFbc,aθ,(dpc)=qFdc,edxddxeξcθθηc+Ncθθ
such that ∇∇(σE) = 0, where
Nc=iq22m2ηnmηab2FanFmc,b+Fbn,aFmc+2q4m2ηnmηabFbc,anm,ξc=iq2mηnmFac,nmdxa,ηc=iq2mηnmFnc,madxaq2mηebFecFabdxa.
Here σ is the flip map when one factor is θand
σ(dxedxd)=dxddxe+iqm2ηcdFacηaeθθ,σ(dpedxd)=dxddpe+iqmηdcθFae,cdxaqmFacηabFbeθ+i2mηabFae,cbθ,σ(dxadpc)=dpcdxa+iqmηeaFdc,edxdθiq2m2ηbnηraFncFrb2q2m2ηarηnbFbc,nrθθ,σ(dpedpd)=dpddpe+iq2mηrpFrdFae,pθdxaFreFad,pdxaθ+iq3m2ηrpηbaFpeFadFrbθθ+2q22m2ηrpηab(FpdFbe,arFpd,arFbe)θθ.

Proof.
First we have (on using the commutation relations for the pas),
[hi,X(dxd)]=q2m2ηabηcd(2FacpbiFbc,a),[hi,X(dpc)]=qm2ηabηdeFac,epbpdiq2m2ηabηdeFac,eFdb+iq2mηde(Fac,ed+Fdc,ea)X(dxa)+2q4m2ηabηdeFbc,aed+q2mηebFecFabX(dxa)iq22m2ηarηebFecFrb,a
and we can check that this obeys the autoparallel equation in (4.3), in particular that X(ηc)X(ξc)+Nc=[hi,X(dpc)].
From the value of ∇(dxd), we calculate
σ(dxedxd)=dxddxe([dxd,xe])+[(dxd),xe]=dxddxeθqmηcdFac[dxa,xe]=dxddxe+iqm2ηcdFacηaeθθ
(5.4)
and we check that this is a bimodule map, the difficult case being
[dxedxd,pc]=iqmηebFbcθdxd+dxeiqmηdbFbcθ=iqmηebFbcθdxd+ηdbFbcdxeθimηeaηdbFbc,aθθσ([dxedxd,pc])=iqmηebFbcdxdθ+ηdbFbcθdxeimηeaηdbFbc,aθθ[σ(dxedxd),pc]=[dxddxe,pc]+iqm2ηbd[Fab,pc]ηaeθθ=iqmηdbFbcθdxe+dxdiqmηebFbcθ+i22qm2ηbdFab,cηaeθθ=iqmηdbFbcθdxe+ηebFbcdxdθimηdaηebFbc,aθθ+imηbdFab,cηaeθθ
as required. From our value of ∇(dxd), we also calculate
σ(dpedxd)=dxddpe([dxd,pe])+[(dxd),pe]=dxddpeiqmηdbFbeθθqmηcd[Facdxa,pe]+θiq2m2ηabηcd[Fbc,a,pe]θ=dxddpeiqmηdcθdFceθiqmηcdFac,edxaθqmηcdFaciqmηabFbeθθ2q2m2ηabηcdFbc,aeθ=dxddpe+iqmηdcθdFceFac,edxaqmFacηabFbeθ+i2mηabFbc,aeθ=dxddpe+iqmηdcθ(Fce,a+Fac,e)dxaqmFacηabFbeθ+i2mηab(Fce,ab+Fac,be)θ=dxddpe+iqmηdcθFae,cdxaqmFacηabFbeθ+i2mηabFae,cbθ
as given. We check that this commutes with commutators,
σ([dpedxd,xc])=σiqmηcbFbeθdxddpeimηdcθ=iqmηcbFbedxdθimηdcθdpe[σ(dpedxd),xc]=[dxddpe,xc]iqmηdcθFae,c[dxa,xc]=imηdcθdpe+dxdiqmηcaFaeθ+iqmηdcθFae,cimηacθ
as required, and
[dpedxd,pc]=iqFae,cdxaq2mηab(Fbe,ac+2iqFaeFbc)θdxd+dpeiqmηdbFbcθ=iqFae,cdxadxdq2mηab(Fbe,ac+2iqFaeFbc)θdxd+iqmηdbFbcdpeθ+iqmηdaFac,riqmηrbFbeθθσ([dpedxd,pc])=iqFae,cdxddxa+iqm2ηbdFrbηraθθq2mηab(Fbe,ac+2iqFaeFbc)dxdθ+iqmηdbFbcθdpe+iqmηdaFac,riqmηrbFbeθθ
versus
[σ(dpedxd),pc]=[dxddpe,pc]+iqmηdrθFae,rdxaqmFarηabFbeθ+i2mηabFae,rbθ,pc=iqmηdbFbcθdpe+dxdiqFae,cdxaq2mηab(Fbe,ac+2iqFaeFbc)θ2qmηdrθFae,rcdxaqmFar,cηabFbeθqmFarηabFbe,cθ+i2mηabFae,rbcθiqmηdrθFae,riqmηabFbcθ=iqmηdbFbcθdpeiqdxdFae,cdxaq2mηab(Fbe,ac+2iqFaeFbc)dxdθ+i2q2m2ηab(Fbe,acr+2iqFae,rFbc+2iqFaeFbc,r)ηdrθθ+2qmηdrFae,rcθdxa+2q2m2ηdrFae,rηabFbc+Far,cηabFbe+FarηabFbe,cθθi3q2m2ηdrηabFae,rbcθθ=iqmηdbFbcθdpeiqFae,cdxddxaq2mηab(Fbe,ac+2iqFaeFbc)dxdθ+2q2m2ηdrηabFar,cFbe+FarFbe,cFaeFbc,rθθ
as required. Now we consider those values of σ depending on ∇(dpc). In particular, remembering that d involves θ
σ(dxadpc)=dpcdxaiqmηab(Fbcθ)qFdc,e[dxddxe,xa][ξc,xa]θθ[ηc,xa]+[Nc,xa]θθ=dpcdxa+iqmηeaFdc,edxdθ2q2m2ηabηdeFbc,deθθ[ξc,xa]θθ[ηc,xa]+[Nc,xa]θθ
as required. Later, it will be convenient to set (defining Mac)
σ(dxadpc)=dpcdxa+iqmηeaFdc,edxdθ+Macθθ
(5.5)
where we can calculate
Mde=iq2m2ηbaηrdFaeFrb+2q2m2ηdrηabFbe,ar.
To calculate σ(dpd ⊗ dpc), we use
σ(dpddpc)=dpcdpd([dpc,pd])q[Fdc,edxddxe,pd][ξc,pd]θθ[ηc,pd]+[Nc,pd]θθ=dpcdpdiqFac,ddxaq2mηab(Fbc,ad+2iqFacFbd)θiqFac,eddxadxeiq2mFac,e(ηabFbdθdxe+ηebdxaFbdθ)[ξc,pd]θθ[ηc,pd]+[Nc,pd]θθ
and using
(Fac,ddxa)=dxaFac,d+imηabθFac,db=(dxa)Fac,d+dxadFac,d+imηabθdFac,db=qmηraFerθdxeFac,d+iq2m2ηebηraFbr,eFac,dθθ+dxadFac,d+imηabθdFac,db=qmηraFerFac,dθdxe+iqm2ηraηebFerFac,dbθθ+iq2m2ηebηraFbr,eFac,dθθ+dxaFac,dedxei2mηnmdxaFac,dnmθ+imηabFac,dbeθdxeimi2mηnmηabFac,dbnmθθ=qmηraFerFac,dθdxe+iqm2ηraηebFerFac,dbθθ+iq2m2ηebηraFbr,eFac,dθθ+Fac,dedxadxei2mηnmFac,dnmdxaθ
and also
ηab(Fbc,ad+2iqFacFbd)θ=ηabθd(Fbc,ad+2iqFacFbd)=ηab(Fbc,ade+2iqFac,eFbd+2iqFacFbd,e)θdxei2mηnmηab(Fbc,adnm+2iqFac,nmFbd+4iqFac,nFbd,m+2iqFacFbd,nm)θθ
we have
σ(dpddpc)=dpcdpd+iqqmηraFerFac,dθdxe+iqm2ηraηebFerFac,dbθθ+iq2m2ηebηraFbr,eFac,dθθi2mηnmFac,dnmdxaθ+q2mηab(Fbc,ade+2iqFac,eFbd+2iqFacFbd,e)θdxeq2mi2mηnmηab(Fbc,adnm+2iqFac,nmFbd+4iqFac,nFbd,m+2iqFacFbd,nm)θθiq2mFac,e(ηabFbdθdxe+ηebdxaFbdθ)[ξc,pd]θθ[ηc,pd]+[Nc,pd]θθ=dpcdpd2q22m2ηraηeb2FerFac,db+Fbr,eFac,dθθ+2q2mηnmFac,dnmdxaθ+q2mηab(Fbc,ade+2iqFacFbd,e2iqFebFac,d)θdxeq2mi2mηnmηab(Fbc,adnm+2iqFac,nmFbd+4iqFac,nFbd,m+2iqFacFbd,nm)θθiq2mηebFac,eFbddxaθ2q2m2ηebηarFac,eFbd,rθθ[ξc,pd]θθ[ηc,pd]+[Nc,pd]θθ,
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 (XX)(σ − id) = 0. From the form of σ in the statement, this means checking the following equations:
[X(dxe),X(dxd)]=iqm2ηcdFacηae[X(dpe),X(dxd)]=iqmηdcFae,cX(dxa)qmFacηabFbe+i2mηabFae,cb[X(dxa),X(dpc)]=iqmηeaFdc,eX(dxd)iq2m2ηbnηraFncFrb+2q2m2ηarηnbFbc,nr[X(dpe),X(dpd)]=iq2mηrpFrdFae,pFreFad,pX(dxa)+iq3m2ηrpηbaFpeFadFrb+2q22m2ηrpηab(FpdFbe,arFpd,arFbe).
For example, we check the last and hardest case, computing
[X(dpe),X(dpd)]=q24m2ηabηrp[(2FaepbiFbe,a),(2FrdppiFpd,r)]=q24m2ηabηrp2Fae[pb,2Frd]pp2Frd[pp,2Fae]pb+2Fae2Frd[pb,pp]i2Frd[Fbe,a,pp]+2Fae[pb,Fpd,r]
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.□

The constructions so far are manifestly Lorentz invariant as long as θ′ is taken to transform trivially. We will also have recourse to the following quotient which is adapted to the observer in the chosen inertial frame but which is covariant in that one can make this in any inertial frame.

Proposition 5.3.
The above calculus on the Heisenberg algebra has a quotient Ωred1 with relations
dx0=p0mθ,dp0=qF0idxiiq2mF0i,iθ
whereby the commutation relations of dxi, dpi imply those required for dx0, dp0. Moreover, X anddescend to this quotient.

Proof.
For the calculus we just check the hardest case
[qF0idxiiq2mF0i,iθ,pa]=q[F0i,pa]dxi+qF0i[dxi,pa]iq2m[F0i,i,pa]θ=iqF0i,adxi+iq2mF0iFiaθ+2q2mF0i,iaθ,
which agrees with [dp0, pa]. That X descends is immediate and for ∇ descending, the hardest case is showing that ∇(dp0) is the same as
(qF0idxiiq2mF0i,iθ)=(qdxiF0i+iq2mθF0i,i)=q(dxi)F0i+qdxidF0i+iq2mθdF0i,i=q2mηciFacθdxaF0i+iq22m2ηabηciθFbc,aθF0i+qdxiF0i,adxaiq2mdxiΔ(F0i)θ+iq2mθF0i,iadxaiq2mθi2mΔ(F0i,i)θ,
where we have used d for functions f of the xa in the analogous form to (6.12),
df=f,adxai2mΔ(f)θ
(5.6)
with Δ(f) = ηabf,ab. Ordering functions to the left and amalgamating terms gives the above as
=q2mηciFacF0iθdxa+iq2m2ηciFacηabF0i,bθθ+iq22m2ηabηciFbc,aF0iθθ+qF0i,adxidxaiq2mΔ(F0i)dxiθiq2mF0i,iaθdxa2q4m2Δ(F0i,i)θθ
which is ∇(dp0) as required.□

The dx0 relation says that in this theory it is natural to identify θ′ with the proper time interval dτ given that in Special Relativity dx0dτ=p0m for our metric convention. With this in mind, the other relation roughly speaking can be interpreted as the quantum analogue of
dp0dτ=qF0idxidτiq2mF0i,i
depending on which side we place the F0i before making our interpretation (with the second term vanishing if we average the two). Here F0i=Eic so the first term here is the expected rate of change of energy −cp0 due to the work done by the electric field Ei, while the “quantum correction” term is the divergence F0i,i=Ec 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)=|ϕH, 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.,
ddsϕ|a|ϕ=ϕ|X(da)|ϕ
(5.7)
for all aA. Working in the full algebra (we do not need the above quotient), we have from (5.3) that
dϕ|xa|ϕds=ηabmϕ|pb|ϕ,
which says that in any state the expected momentum is m times the proper velocity as classically, and if the Fab are constant,
dϕ|pc|ϕds=qmηabFcaϕ|pb|ϕ=qFcadϕ|xa|ϕds,
which says that the expected proper acceleration is governed by the Lorentz force law again as classically. The index 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 Fab are not constant then we will have order corrections due to the form of (5.3). We next turn to the static case where Aa are time independent.

Lemma 5.4.

If Aa is time independent then up0qA0 is central in the Heisenberg algebra and [u,x0]=i. Moreover, there is a subalgebra A with subcalculus ΩA1 of Ωred1 generated by xi,pi,dxi,dpi,θ,u where u is central in ΩA1 and du=0. Moreover,restricts on the generators to a connection on ΩA1 and hA.

Proof.
Clearly, u is always canonically conjugate to x0 as p0 was. Also [u,xi]=0 and when Aa are time independent then [u,pi]=[p0,pi]+[qA0,pi]=iqF0i+iqA0,i=0. For the differentials, working in Ωred1 in the time independent case,
[u,dxi]=[p0+qA0,dxi]=iqm(Fi0A0,i)θ=0,
[dpi,u]=[dpi,p0+qA0]=iq2mFjiFj0θqiqmFijA0,jθ=0.
We also have, using x0 invariance of the Aa and (5.6) with Δ defined by ηab,
du=dp0+qdA0=dp0+qA0,idxiiq2mΔ(A0)θ=0
as A0,i = −F0i and ΔA0 = A0,ii = −F0i,i by the relations in Ωred1.

Next, we omit x0 from our algebra as under our assumptions it does not appear in Fab or on the right hand side of any of the commutation relations other than as dx0 = −p0θ′/m. The remaining generators and relations are (5.8)(5.10) as listed below albeit u a closed central generator. Further, ∇ restricts to this subcalculus as any dx0 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 Au, with calculus ΩAu1 given by
[xi,xj]=0,[xi,pj]=iδij,[pi,pj]=iqFij,
(5.8)
[dxi,xj]=imδijθ,[dxi,pj]=iqmFijθ=[dpj,xi],
(5.9)
[dpi,pj]=iqFki,jdxk2q2mFki,kjθ+iq2m(F0iF0jFkiFkj)θiqmF0i,j(u+qA0)θ.
(5.10)
This then deforms the Heisenberg algebra on spatial R3 in Sec. IV by the background electrostatic and magnetostatic potentials Ai, A0 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 Sec. V B. Thus we take E=L2(R3)C(R) and ∇E given by the same Hamiltonian as above, now viewed as the representation of an element of Au on functions on R3 with fixed value of u. In this case, we have quantum geodesic motion for the reduced Heisenberg algebra Au with calculus ΩAu1 as in Sec. IV but now with Hamiltonian that contains magnetic potentials in the pi and an electric potential in the form of V.

Here we consider a possible interpretation or way to visualise the quantum geodesic evolution constructed in Sec. V A in a manner that is a little analogous to a modification of quantum mechanics. This is for comparison purposes to start to get a feel for the content of this flow, given that it is not something usually considered.

We recall that x0 = 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 Da on ϕC(R1,3) at each sR. However, ϕ̄ϕ 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. With this in mind, we work with fields ψ(u, xi) Fourier transformed from t to a Fourier conjugate variable u say, so
ψ(u,xi)=dteiutϕ(t,xi),ϕ(t,xi)=dueituψ(u,xi).
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 L2(R4) in these variables, just because it did before and Fourier transform in one variable can be viewed as an isometry [if also completed to L2(R4) on the spacetime side.]
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 p0=u+qA0 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 x0 = ct. The minus sign is needed due to the −+ + + signature as classically p0 = −p0 is positive for a future pointing time-like geodesic. The action of the electromagnetic Heisenberg algebra on ψ(u, xi) is by xi and pi = −iℏDi as before, and
t=iu,u=uc,p0=ucqA0(t,xi).
Moreover, ∇E as before now appears as
Eψ=ψsi2mDiDiψi2m(uc+qA0)2ψds.
(5.11)
This is clear from the Fourier transform, but if one wants to check it directly, pi, xi are already represented as before and as A0(iu,xi) does not depend on u, we still have [pi, x0] = 0. Meanwhile,
[p0,pi]=iuc+qA0,Di=quc,Ai+iqA0,xi=iqF0i,[p0,x0]=uc+qA0,ciu=i.
Now suppose that Aa are indeed independent of t. Then by Lemma 5.4 we can write ψ(u, xi) = Ψ(xi) and regard u as a fixed energy parameter since it is central in the Heisenberg algebra. In this case the ∇E on Ψ is governed by a similar operator as in Sec. IV but with pi = −iℏDi for a particle minimally coupled to the Ai as a magnetostatic gauge potential and with potential energy
V(x)=12muc+qA0(x)2=12mc2(uqΦ(x))2,Φ=cA0=cA0,
where the upper index potential connects to usual conventions. Thus, ∇EΨ = 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 (5.11) 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, xi) as eiutΨ(xi) with Ψ(xi) 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 with mass m in 1 + 1 Minkowski spacetime with zero electromagnetic potential. Then ∇EΨ = 0 is
isΨ=huΨ,hu=(i)22mx2u22mc2
with huΨ=EkΨ with Ek=(k2u2c2)/2m for ψk(x)=eikx. The on-shell fields (i.e., solving the KG equation) just evolve by an unobservable phase eismc22 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 extremal 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:
Ψ(0,x)=dke(cku2m2c4)2βeikx,Ψ(s,x)=dkeisu2k2c22mc2e(cku2m2c4)2βeikx
as plotted in Fig. 1. One can see, and check, that dx|Ψ(s, x)|2 is constant in s as per our interpretation as an evolving amplitude. It is easy enough to check the expectation values
Ψ|p|ΨΨ|Ψ=dke2(cku2m2c4)2βkdke2(cku2m2c4)2β=1cu2m2c4,Ψ|p0|ΨΨ|Ψ=uc
as expected. We also find using x=ik on ψk inside the upper of the ratio of integrals that
Ψ|x|ΨΨ|Ψ=smcu2m2c4=smΨ|p|ΨΨ|Ψ,
which verifies our identity (5.7) and shows that our quantum geodesic evolves with proper velocity given by the average spatial momentum/m. We can also compute t=iu applied in the upper of the ratio of integrals to find
Ψ|t|ΨΨ|Ψ=sumc2=s1v2c2,vc1mc2u2=Ψ|x|ΨΨ|t|Ψ
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).

FIG. 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 = = m = 1 and u = 1.1. Images produced by Mathematica.

FIG. 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 = = m = 1 and u = 1.1. Images produced by Mathematica.

Close modal

Although our quantum geodesic flow equation ∇EΨ = 0 is not Schrödinger’s equation, its similarity at fixed u 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 is also somewhat different from the usual derivation of Schrödinger’s equation as a limit of the KG equation, which involves writing ϕ(x0,xi)=eimc2tΨKG(t,xi) where ct = x0 and ΨKG(t, xi) is slowly varying to recover Schrödinger’s equation for ΨKG 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 as far as the flow is concerned and in a choice of laboratory frame as far as the interpretation is concerned. This means that our differences from Schrödinger’s 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.
We consider a hydrogen-like atom or more precisely an electron of charge q = −e around a point source nucleus of atomic number Z or charge Ze, so
Φ(r)=Ze4πϵ0r.
Then the geodesic flow equation ∇EΨ = 0 at fixed u is
iΨs=huΨ,hu=(i)22m212mc2(uqΦ)2
(5.12)
since there is no magnetic potential in the Di (we write 2=i2 for the spatial Laplacian). We are effectively in the Schrödinger equation setting of Sec. IV with h=p22m+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
V(r)=12mc2(uqΦ)2=u22mc2umcZαr22mZ2α2r2,
where
qΦ=Ze24πϵ0r=cZαr
in terms of the fine struture constant α ∼ 1/137. We solve this for eigenmodes
huΨ=EΨ,
where E is positive due to a large negative offset in hu 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)χl where χl 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
rr2Rrl(l+1)R+mr222E+1mc2u+cZαr2R=0,
which has the same form as for a usual atom but with shifted angular momentum l′ = lδl defined as in [Ref. 24, Chap. 2.3] by
l(l+1)=l(l+1)Z2α2;δl=l+12(l+12)2Z2α2.
For every n such that n − (l + 1) = d is a positive integer i.e., l = 0, …, n − 1, one has
R(r)=rlekrLd1+2l(2kr);k2=m22Eu2mc2
in terms of a generalised Laguerre polynomial of degree d. This gives eigenvalues
En,l=u22mc21+Zαnδl2
(5.13)
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 mc22. From (5.13), these are
un,l=mc211+(Zαnδl)2
in agreement with the allowed “Schrödinger-like” energy spectrum coming from directly solving the KG equation [Ref. 24, Chap. 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
Ψ(s,xi)=n,l,meiEn,l(u)scn,l,m(u)ψn,l,m(u)(xi)
with eigenvectors ψn,l,m(u) at fixed u as sketched above, and initial values set by coefficients cn,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 in order to see what the differences are. For this we set u = mc2 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
En,l=mc221+Zαnδl2=mc22+mc2Z2α22(nδl)2,
where the first term should be ignored and the second would be minus the Rydberg binding energy for atomic number Z if δl had been zero. In terms of potentials, at u = mc2 we have
V=12mc2(uqΦ)2=mc22+qΦq22mc2Φ2
of which we discard the constant term so that the effective potential is
V(r)=cZαr22mZ2α2r2.
For a hydrogen atom, the two terms are of equal size at rcrit=e28πϵ0mc2=α4πλc=12a0α2, where α is the fine structure constant, λc is the Compton wavelength and a0 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<12α to have solutions for l′, as known in the context of solving the KG equation for this background [Ref. 24, Chap. 2.3]. We see this directly from the potential and without the complications from double time derivatives. This correction is also different from the usual 1/r3 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 p2 component of the Hamiltonian.

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 Sec. IV 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 ≠ 0 but which means that we do not have an actual differential calculus when = 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 M̃=M×R where (M,ω,̄) is a symplectic manifold with symplectic connection ̄ and symplectic form ωμν in local coordnates (we denote its inverse by ωμν 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 Sec. VI A, we obtain a self-contained formulation where we fix a Hamiltonian function h with X̄ the associated Hamiltonian vector field, which we extend to X=t+X̄. We likewise extend ωμν to a (0, 2) tensor G on the extended phase space with
Gμν=ωμν,G0μ=Gμ0=μh
(6.1)
and we also extend ωμν to a Poisson bivector on the extended phase space with
ω0μ=ω0μ=τμ
(6.2)
for a suitable vector field τ = τμμ 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 + 2′ for the usual contact form Θ on extended phase space as in Ref. 25. 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 Refs. 26 and 27. Nevertheless, we do make use of a natural (possibly degenerate) classical metric gμν 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 ωμν, τμ = ω0 μ provide the Poisson bracket and hence quantisation data for the extended phase space:
[xμ,xν]=iωμν,[t,xμ]=iτμ
(6.3)
and the extended linear connection ∇ similarly provides semiclassical data for the quantisation of the differential structure cf1,28,29
[xν,dxμ]=iωνββdxμ,[t,dxμ]=iτββdxμ,[xν,dt]=0,[t,dt]=0
to errors O(2), which for the natural ∇ found in (6.6) is
[xν,dxμ]=iωνβ̄βdxμ+igμνdt,[t,dxμ]=iτβ̄βdxμigμγωγβτβdt.
(6.4)
We require the connection to be Poisson compatible in the sense of Ref. 28 and, for an associative algebra and calculus at order 2, we require ωμν, τμ 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 Sec. VI A. Section VI B checks the semiclassical limit of Sec. IV extended by central t with θ′ = dt, and shows that we obtain an example with ̄=τ=0.
Let M be a symplectic manifold with coordinates xμ, μ = 1, …, 2n, symplectic form ωμν with inverse ωμν, and let ̄ be a symplectic connection with Christoffel symbols Γ̄μνρ defined by ̄μdxν=Γ̄νμρdxρ. We fix a function hC(M) with Hamiltonian vector field X̄μ=ωμνh,ν (we use h,ν for the partial derivative of h with respect to xν). It is easy to see that in general X̄ is not autoparallel as
X̄X̄μ=ωαβh,βωμν̄αh,ν=ωαβh,βωμν(h,ναΓ̄γανh,γ)=gμνh,ν
where we define the possibly degenerate metric inner product
(dxμ,dxν)=gμν=ωμγωνρ(h,ρ);γ
(with semicolon the ̄ connection). This is symmetric as ̄ is torsion free.
To resolve this obstruction to X̄ being autoparallel, we work on the extend phase space with x0 = t and X=t+X̄, and write down an extension of the symplectic connection on forms
μdxν=̄μdxνΓνμ0dt,0dxν=Γν0μdxμ,μdt=0dt=0
for some additional Christoffel symbol data as shown.

Lemma 6.1.

For generic h, X is autoparallel with respect toif and only if Γμα0+Γμ0α=gμβωβα.

Proof.
We require
XX=X̄X̄+0X̄+X̄(t)=(gμνh,ν+Γμ0αX̄α+Γμα0X̄α)xμ=(gμν+(Γμα0+Γμ0α)ωαν)h,νxμ=0
which gives the result stated.□

We now turn to the classical symplectic form ωμνdxμ ∧dxν and its torsion free symplectic connection ̄. In our extended calculus, we would like to find a related 2-form which is preserved by the extended covariant derivative ∇. Given that we have just added a variable t, it is reasonable to do this by extending the symplectic form by something wedged with dt.

Lemma 6.2.

The extended covariant derivativepreserves a 2-form of the form ωαβdxα ∧dxβ + 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, Γμα0=gμβωβα and Γμ0α=0.

Proof.
Begin by calculating
0(ωαβdxαdxβ)=2ωαβΓβ0μdxμdxαμ(ωαβdxαdxβ)=2ωαβΓβμ0dxαdt0(f,αdxαdt)=f,αΓα0βdxβdtμ(f,αdxαdt)=(f,αμf,βΓβμα)dxαdt.
By comparing these, we see that preserving the given 2-form requires that f,αΓα0β=0, which for generic f requires Γα0β=0. Now, from the autoparallel condition we have Γμα0=gμβωβα, and
̃μ(ωαβdxαdxβ+dfdt)=(f,αμf,βΓβμα2ωαβΓβμ0)dxαdt=(f,αμf,βΓβμα2ωαβgβγωγμ)dxαdt=(f,αμf,βΓβμα2ωαβωβπωγρ(h,ρ);πωγμ)dxαdt=(f,αμf,βΓβμα+2(h,μ);α)dxαdt
so we require ∇μ(d(f + 2h)) = 0.□

The obvious choice in this lemma is f = −2h and we henceforth make this choice. This means that the classical symplectic geometry has a natural extension
ω̃=ω2dhdt,
(6.5)
αdxμ=̄αdxμgμβωβαdt,0dxμ=0,dt=0
(6.6)
arranged so that μω̃=0ω̃=0 and X is autoparallel. Using interior product iX̄ with a vector field X̄ (defined as a graded derivation extending the degree 1 pairing), we obtain iX̄(ω)=2dh, where ω = ωαβdxα ∧dxβ, which now appears in the extended terms as a kernel condition
iXω̃=0.
Equivalently, one can check that the antisymmetric rank (0, 2) tensor
G=ωμνdxμdxν+dtdhdhdt=ωμνημην
is covariantly constant under ∇, where
ημdxμX̄μdt.
(6.7)
In a local patch with coordinates such that ωμν are constant and if Γ̄μνρ=0, then these 1-forms are also killed by ∇. One can view them along with dt as a local parallelisation of M̃. The quotient of the cotangent bundle where we set ημ = 0, is dual to a sub-bundle of the tangent bundle of M̃ spanned by X. This, at any point of M̃, is the tangent to the Hamilton-Jacobi equations of motion regarded as a flow in M̃ through that point.
On the other hand, this extended connection from Lemma 6.2 necessarily has torsion in this extended direction,
Tμν0=Tμ0ν=gμβωβν.
(6.8)
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 Γμα0=Γμ0α=12gμβωβα, we would have had zero torsion but not compatibility with ω̃.

Proposition 6.3.
The extendedin (6.6) with torsion has curvature
Rαβγδ=R̄αβγδ,Rα0γδ=ωατh,κR̄κτγδ
and zero for R with index 0 in all other positions, where R̄αβγδ 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
Rabcd=Γadb,cΓacb,d+ΓacsΓsdbΓadsΓscb
Recalling that Γ0ab=Γa0b=0 we observe that Rabcd 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 Rαβγδ=R̄αβγδ are
Rα0γδ=Γαδ0,γΓαγ0,δ+Γ̄αγπΓπδ0Γ̄αδπΓπγ0.
If we use semicolon for covariant differentiation with respect to ̄, then
Γαδ0=gαβωβδ=ωβδωαγωβρ(h,ρ);γ=ωαγ(h,δ);γ=ωαγ(h,γ);δ
as ̄ has zero torsion. Now, as ω has zero covariant derivative with respect to ̄,
Γαδ0,γ=(ωατ(h,τ);δ),γ=(ωατ(h,τ);δ);γ+Γ̄αγπωπτ(h,τ);δΓ̄ργδωατ(h,τ);ρ=ωατ(h,τ);δ;γΓ̄αγπΓπδ0+Γ̄ργδΓαρ0
and substituting this into the equation for the Riemann tensor gives
Rα0γδ=ωατ(h,τ);γ;δωατ(h,τ);δ;γ=ωατh,κR̄κτγδ,
where we have used ̄ torsion free again.□

Note that the connection (6.6), its torsion and curvature can be written in our more algebraic way, as a right connection,
dxμ=̄dxμgμνωναdtdxα,
T(dxμ)=gμνωναdxαdt,R(dxμ)=12R̄μναβηνdxβdxα
where we used symmetry in the first two indices of the curvature of ̄ when raised by ω, see Ref. 30.
From the point of view of this extended phase space geometry, we can now write (6.3) and (6.4) compactly as
[xa,xb]=iωab,[xa,dxb]=iωacΓbcdxd,
(6.9)
where a, b, c,d include 0 with x0 = t and ω0 μ = −ωμ0 = τμ. The condition for this to be Poisson compatible follows easily from the characterisation of Poisson compatibility in [Ref. 1, Lemma 9.21] and comes out as
̄τ=0,Tμν0τν=0,
(6.10)
given our assumptions on ω̄ and the form of the extensions above.
Here we extend the Heisenberg algebra A to Ã=AB where B=C(R), in the form of an additional time variable t, but with calculus
ΩÃ=(ΩAΩB)/θdt
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 Sec. VI A. As t is central, we have τμ = 0. We start with this semiclassical data.

Proposition 6.4.

  1. Let M=R2n with coordinates xμ = xi for μ = i and xμ = pi for μ = i + n, where ∈ {1, …, n}. We take symplectic structure ω with tensor ωμν, associated Poisson bivector ωμν, and symplectic connection
    12ω=dxidpi;ωi,j+n=ωi+n,j=δij,ωi,j+n=ωi+n,j=δij,Γ̄μνρ=0.
    The Hamiltonian vector field X̄μ associated to h=12mipi2+V(x) and the possibly degenerate inner product gμν are
    X̄i=1mpi,X̄i+n=V,i,gij=1mδij,gi+n,j+n=V,ij
    and zero otherwise.
  2. We extend to M̃=R2n+1 with an additional coordinate x0 = t. The extended connection and its torsion are
    Γij+n,0=Tij+n,0=Tij+n,0=δijm,Γi+nj0=Ti+nj0=Ti+n0j=V,ij
    and otherwise zero, with autoparallel vector field X extended by X0 = 1 (i.e., we add t). The preserved antisymmetric tensor and the 1-form η are
    Gμν=ωμν,G0i=Gi0=V,i,G0,i+n=Gi+n,0=pim,
    ηi=dxi1mpidt,ηi+n=dpi+V,idt.

Proof.

(1) We work through X̄μ=ωμνh,ν and gμν = (dxμ, dxν) = ωμαωνβαh,β in Sec. VI A. (2) For the extended connection and that the extended X is autoparallel, we use Lemma 6.2. The torsion is from (6.8), the curvature from Proposition 6.3, the invariant 2-form written down from (6.1) and ημ is from (6.7).□

At the semiclassical level the G tensor is antisymmetric and hence equivalent to a 2-form (but not symplectic as it is degenerate). The generalised quantum metric is a quantisation of this. Meanwhile, ηi+n quantises to what we called ωi, but is now combined with ηi to define ημ. We also note that, writing 2 = δijij, the associated classical second order Laplace-Beltrami operator is
Δ=1m2+V,ij2pipj
(6.11)
as characterised by Leibnizator LΔ(f, g) = 2(df, dg) for all f, g on phase space (this makes sense without assuming gμν invertible, but in the invertible case it would be the usual Laplacian-Beltrami operator Δf = gμνμνf).

Corollary 6.5.
The relations of the differential calculus, the quantum linear connection and the invariant 1-forms ημ in Sec. IV appear in terms of the above phase space structures as
[xμ,xν]=iωμν,[xμ,dxν]=igμνθ,{dxμ,dxν}=igμν,ρdxρθ,(dxμ)=θgμνωναdxαi2Δ(X(dxμ))θθ=θd(X(dxμ)),σ(dxμdxν)=dxνdxμigμαgνβωαβθθ,ημ=dxμX(dxμ)θ
along withθ′ = 0 and σ = flip when one factor is θ. These expressions also apply to à with θ′ = dt.

Proof.
We write the quantum geometry in Sec. IV in terms of gμν, ωμν, ωμν as identified in Proposition 6.4. 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 ∇(dxμ) requires some explanation. In fact, the quantum differential calculus in Proposition 4.2 has the structure of a general “central extension”17,18 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
df(x,p)=fxidxi+fpidpii2(Δf)θ,
(6.12)
for f(x, p) normal ordered so that x is to the left of p and Δ a certain second order operator on the Heisenberg algebra which reduces to (6.11) when f(x, p) is normal ordered and at most quadratic in p. Otherwise Δ is more complicated with O() terms arising from (dxμ, dxν) not being a bimodule map for general V(x) on the unextended calculus. We used this Δ for ∇(dxμ) and recognised the result in terms of d(X(dxμ)).□

Comparing with (6.4), we see that the calculus corresponds to ∇ at the semiclassical level with ̄=0 (as well as τμ = 0). The first term of the first form of ∇(dxμ) also then agrees with ∇ in (6.6) with a further quantum correction. In this way, the formulae in Sec. IV 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 Sec. IV 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.

In Sec. III, we extended the formalism of “quantum geodesics” in noncommutative geometry as introduced in Ref. 12 using A-B-bimodule connections from Ref. 1 to geodesics in representation spaces. We then applied this to ordinary quantum mechanics and showed in Sec. IV 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.31 Such a freedom was already exploited to encode Newtonian gravity by putting the gravitational potential into the spacetime differential structure;15 our now results in Sec. IV 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 Sec. V to a novel relativistic version of Sec. IV 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 geodesic wave packet spreads off-shell on either side of a Klein–Gordon solution but on average evolves as expected. We also showed how our quantum geodesic flow in the case of a time-independent background field nevertheless amounts to some kind of 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 VI 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 Sec. IV we focused on time-independent 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 Sec. IV, the quantum connection on phase space was flat and preserved the extended quantum symplectic structure (rather than being a quantum Levi–Cività 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 Sec. V from an electromagnetic background on the representation space to a curved Riemannian background on the latter, i.e., to gravitational backgrounds such as the wave-operator black-hole models in Ref. 17. It could also be of interest to consider quantum geodesic flows using a Dirac operator or spectral triple4 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,28,32 which could also be of interest here.

The authors have no conflicts to disclose.

Edwin Beggs: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Shahn Majid: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created nor analyzed in this study.

Here we complete the Proof of Theorem 5.2 by checking the remaining cases that σ is a bimodule map. Begin with
[dxadpc,xe]=imηaeθdpc+iqmηebdxaFbcθ=imηaeθdpc+iqmηebFbcdxaθ+2qm2ηebηarFbc,rθθ[dpcdxa,xe]=imηaedpcθ+iqmηebFbcθdxaσ([dxadpc,xe])[dpcdxa,xe]=2qm2ηebηarFbc,rθθ.
and from (5.5),
σ([dxadpc,xe])[σ(dxadpc),xe]=2qm2ηebηarFbc,rθθiqmηbaFdc,b[dxd,xe]θ=2qm2ηebηarFbc,rθθ+iqmηbaFdc,bimηdeθθ=0.
Also
[dpedxd,pc]=iqFae,cdxa+q2mηab(Fbe,ac+2iqFaeFbc)θdxd+dpeiqmηdbFbcθ=iqFae,cdxadxdq2mηab(Fbe,ac+2iqFaeFbc)θdxd+iqmηdbFbcdpeθ+iqmηdaFac,riqmηrbFbeθθ,[dxddpe,pc]=dxdiqFae,cdxa+q2mηab(Fbe,ac+2iqFaeFbc)θ+iqmηdbFbcθdpe=iqFae,cdxddxaq2mηab(Fbe,ac+2iqFaeFbc)dxdθ+iqmηdbFbcθdpe2qmηdrFae,crθdxa+i2q2m2ηdrηab(Fbe,acr+2iqFae,rFbc+2iqFaeFbc,r)θθ
and as a result, using the formula for σ(dxd ⊗ dxa),
σ([dxddpe,pc])[dpedxd,pc]=iqFae,c(σ(dxddxa)dxadxd)2qmηdrFae,crdxaθ+i2q2m2ηdrηab(Fbe,acr+2iqFae,rFbc+2iqFaeFbc,r)θθiqmηdaFac,riqmηrbFbeθθ=2q2m2ηraηbdFae,cFbrθθ+2q2m2ηdaηrbFac,rFbeθθ2qmηdrFae,crdxaθ+2q2m2ηdrηab(iFbe,acr2qFae,rFbc2qFaeFbc,r)θθ=2q2m2ηbaηrd(Fae,cFrb+Frb,cFaeFae,rFbc)θθ2qmηdrFae,crdxaθ+i3q2m2ηdrηabFbe,acrθθ
and using (5.5),
σ([dxddpe,pc])[σ(dxddpe),pc]=2q2m2ηbaηrd(Fae,cFrb+Frb,cFaeFae,rFbc)θθ2qmηdrFae,crdxaθ+i3q2m2ηdrηabFbe,acrθθiqmηrd[Fae,rdxa,pc]θ[Mde,pc]θθ=2q2m2ηbaηrd(Fae,cFrb+Frb,cFaeFae,rFbc)θθ+i3q2m2ηdrηabFbe,acrθθiqmηrdFae,riqmηabFbcθθiMde,cθθ
so we deduce that σ([dxd ⊗ dpe, pc]) = [σ(dxd ⊗ dpe), pc]. Also
[dpadpc,xe]=iqmηeb(Fbaθdpc+dpaFbcθ)=iqmηebFbaθdpc+Fbcdpaθ+iqmηrdFdaFbc,rθθ
and as a result,
σ([dpadpc,xe])[dpcdpa,xe]=2q2m2ηebηrd(FdaFbc,rFdcFba,r)θθ
which, with a little work, implies σ([dpa ⊗ dpc, xe]) = [σ(dpa ⊗ dpc), xe]. Finally, we look at the condition σ([dpe ⊗ dpd, pc]) = [σ(dpe ⊗ dpd), pc], beginning with
[dpedpd,pc]=iqFae,cdxadpdq2mηab(Fbe,ac+2iqFaeFbc)θdpdiqdpeFad,cdxaq2mηabdpe(Fbd,ac+2iqFadFbc)θ=iqFae,cdxadpdq2mηab(Fbe,ac+2iqFaeFbc)θdpdiqFad,cdpedxaq2mηab(Fbd,ac+2iqFadFbc)dpeθ+2q2mFad,crηrpFpeθdxai2q22m2ηab(Fbd,acr+2iqFad,rFbc+2iqFadFbc,r)ηrpFpeθθ
we get
σ([dpedpd,pc])[dpddpe,pc]=iqFae,cσ(dxadpd)q2mηab(Fbe,ac+2iqFaeFbc)dpdθiqFad,cσ(dpedxa)q2mηab(Fbd,ac+2iqFadFbc)θdpe+2q2mFad,crηrpFpedxaθi2q22m2ηab(Fbd,acr+2iqFad,rFbc+2iqFadFbc,r)ηrpFpeθθ+iqFad,cdxadpe+q2mηab(Fbd,ac+2iqFadFbc)θdpe+iqFae,cdpddxa+q2mηab(Fbe,ac+2iqFaeFbc)dpdθ2q2mFae,crηrpFpdθdxa+i2q22m2ηab(Fbe,acr+2iqFae,rFbc+2iqFaeFbc,r)ηrpFpdθθ=iqFae,cσ(dxadpd)dpddxaiqFad,cσ(dpedxa)dxadpe+2q2mFad,crηrpFpedxaθi2q22m2ηab(Fbd,acr+2iqFad,rFbc+2iqFadFbc,r)ηrpFpeθθ2q2mFae,crηrpFpdθdxa+i2q22m2ηab(Fbe,acr+2iqFae,rFbc+2iqFaeFbc,r)ηrpFpdθθ.
So using (5.5),
σ([dpedpd,pc])[dpddpe,pc]=2q2mFae,cηpaFsd,pdxsθqmηbpηraFpdFrbi2mηarηpbFbd,prθθ+2q2mFad,cηapθFre,pdxrqmFrpηrbFbeθ+i2mηrbFre,pbθ+2q2mFad,crηrpFpedxaθi2q22m2ηab(Fbd,acr+2iqFad,rFbc+2iqFadFbc,r)ηrpFpeθθ2q2mFae,crηrpFpdθdxa+i2q22m2ηab(Fbe,acr+2iqFae,rFbc+2iqFaeFbc,r)ηrpFpdθθ
and here the dx containing terms are
2q2mηrp(Frd,cFae,p+Fae,crFpd)θdxa+(Fre,cFad,p+Fad,crFpe)dxaθ=2q2mηrp(FrdFae,p),cθdxa+(FreFad,p),cdxaθ
Next,
iq2mηrpFrdFae,pθdxaFreFad,pdxaθ,pc=2q2mηrp(FrdFae,p),cθdxa+(FreFad,p),cdxaθ2q3m2ηrpηabFrdFae,pFbcFreFad,pFbcθθ
so that
σ([dpedpd,pc])[dpddpe,pc]iq2mηrpFrdFae,pθdxaFreFad,pdxaθ,pc=2q2mFae,cqmηbpηraFpdFrbi2mηarηpbFbd,prθθ+2q2mFad,cηapqmFrpηrbFbe+i2mηrbFre,pbθθi2q22m2ηab(Fbd,acr+2iqFad,rFbc+2iqFadFbc,r)ηrpFpeθθ+i2q22m2ηab(Fbe,acr+2iqFae,rFbc+2iqFaeFbc,r)ηrpFpdθθ+2q3m2ηrpηabFpdFae,rFbcFpeFad,rFbcθθ=2q22m2Fae,cηbpηra(2qFpdFrbiFbd,pr)θθ+2q22m2Fad,cηrbηap2qFrpFbe+iFre,pbθθ2q22m2ηab(iFbd,acr2qFadFbc,r)ηrpFpeθθ+2q22m2ηab(iFbe,acr2qFaeFbc,r)ηrpFpdθθ=2q3m2ηrpηbaFae,cFpdFrb+Fad,cFrbFpeFadFbc,rFpe+FaeFbc,rFpdθθ+i3q22m2Fae,cηbpηraFbd,pr+Fad,cηrbηapFre,pbηabFbd,acrηrpFpe+ηabFbe,acrηrpFpdθθ=2q3m2ηrpηbaFpe,cFadFbr+Fad,cFrbFpeFadFbc,rFpe+FpeFrc,bFadθθ+i3q22m2ηrpηabFbe,cFpd,ar+Fpd,cFbe,arFpd,acrFbe+Fbe,acrFpdθθ=2q3m2ηrpηba(FpeFadFrb),cθθ+i3q22m2ηrpηab(FpdFbe,arFpd,arFbe),cθθ=iq3m2ηrpηbaFpeFadFrbθθ+2q22m2ηrpηab(FpdFbe,arFpd,arFbe)θθ,pc
and this gives a value for σ(dpe ⊗ dpd) which would imply the bimodule map condition. Subtracting the value from the last long calculation from the value calculated from ∇, we get the condition
0=2q22m2ηraηeb2FerFac,db+Fbr,eFac,dθθ+2q2mηnmFac,dnmdxaθ+q2mηab(Fbc,ade+2iqFacFbd,e2iqFebFac,d2iqFacFed,b)θdxei2q4m2ηnmηab(Fbc,adnm+2iqFac,nmFbd+4iqFac,nFbd,m+2iqFacFbd,nm)θθ2q2m2ηebηarFac,eFbd,rθθ+[ξc,pd]θ+θ[ηc,pd]+[Nc,pd]θθiq3m2ηrpηbaFpdFacFrbθθ2q22m2ηrpηab(FpcFbd,arFpc,arFbd)θθ
and substituting the values for ξc, ηc and Nc from the statement one can see that this holds.
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