Equivalence principle for quantum mechanics in the Heisenberg picture

We present an exact quantum observable analog of the weak equivalence principle for a ‘relativistic’ quantum particle. The quantum geodesic equations are obtained from Heisenberg equations of motion as an exact analog of a fully covariant classical Hamiltonian evolution picture, with the proper identification of the canonical momentum variables as p µ , rather than p µ . We discuss the meaning of the equations in relation to projective measurements as well as equations with solution curves as ones in the noncommutative geometric picture of spacetime, and a plausible approach to quantum gravity as a theory about quantum observables as physical quantities including the notion of quantum coordinate transformation.


I. INTRODUCTION
The question of if there is or what could be an equivalence principle (EP) for quantum mechanics is obviously an important one at the trailhead of our exploration of the theory of quantum gravity [1].There has been a lot of work on the subject matter, at least since the paper by Greenberger in 1968 [2].Summaries of many available results and good lists of references are available in Refs.[1,3].Here, we are focusing on the dynamics of a quantum particle, hence the weak version of the principle.Contrary to most, if not all, of the results in the literature, we are going to give an exact analog of the classical picture.That is to say, we have the exact weak equivalence principle (WEP), that a quantum particle moves along a quantum geodesic independent of its mass, there is local equivalence between gravity and acceleration for it, and its inertial mass agrees with its gravitational mass.That is to be obtained in the Heisenberg picture of quantum dynamics, in an exact 'relativistic' setting.
Heisenberg picture analysis of the subject matter has, apparently, hardly been performed.
For the Schrödinger picture analyses, basically, there is a kind of consent that the exact weak EP as we have in classical physics has to be compromised in some way.A naive reasoning is that a quantum particle cannot have a definite path of motion, in a classical geometric model of space(time)to be exact, hence cannot follow a geodesic in the latter.And of course, the Schrödinger picture and the Heisenberg picture are equivalent descriptions of the same dynamics.The answer to the apparent mystery is a fundamentally different perspective as in our term of a quantum geodesic, instead of 'quantum corrected geodesic' [4] or effective geodesic equation [1].
A geodesic equation is a differential equation of a distance or length parameter that has the shortest path as the solution.Physically, it is the equation of motion for a free particle.
The equation, of course, governs how the position observables change with the motion.Our quantum geodesic is exactly such a differential equation for the quantum position observable, and that is independent of the state.It is a Heisenberg equation of motion for a free quantum particle.To think about the simplest 'nonrelativistic' case, we certainly have a motion of constant momentum as quantum observables, i.e. dp i dt = 0. Note that a conservation law of this kind is actually exact and of no less importance than its classical analog.It is common to read statements that the uncertainty principle says that conservation laws are compromised in the quantum setting.Such statements are terribly misleading, if not completely wrong.A conservation statement such as dp i dt = 0 certainly cannot give you single constant eigenvalue answers in projective measurement for any particular pi , so long as you are not working on an eigenstate of the observable.Yet, the conservation statement says a lot about the time-independent properties of the momentum.Not only that the expectation values are time-independent, but every physical property or related mathematical result dependent only on pi are not changing with time.With projective measurements, all the statistical distributions of eigenvalue results obtained for any time instance (precisely any fixed time after the preparation of the states) for any observable as a function of only the three pi would not change with time.Each state, or ensemble of the same state, would give different constant distributions.But the constant behavior is a result as important and as exact as its classical counterpart.The time-independent nature of any such statistical distribution of results of projective measurements can be experimentally verified to any required precision in principle.Likewise, a Heisenberg equation of motion is a prediction from the quantum theory that can be verified precisely free of any concern about quantum uncertainty.Physics is about physical quantities, i.e. observables, and their behavior.Our analysis here hence offers an alternative approach to look at quantum physics in the presence of gravity that may open a new path towards quantum gravity.Within practical quantum physics, it could offer useful results complimentary to the Schrödinger picture ones.We used the 'nonrelativistic' setting to clarify the key background perspective above for a good reason.Our preferred 'relativistic' theory for dynamics is one with an invariant evolution parameter s in the place of Newtonian time [5].For all the classical analyses here, the solution gives the particle proper time as a linear function of the s parameter, hence essentially identifying the two physically.It is important to note that we are talking about a Hamiltonian dynamical theory with genuinely four degrees of freedom, instead of three as in a theory assuming the proper time to be the evolution parameter resulting in the velocity constraint.There is no a priori assumption about a relation between s and any dynamical variable.The geodesic problem in any manifold as a variational problem is, of course, one with as many degrees of freedom as the dimension of the manifold.In the quantum case, all coordinates and the particle proper time should be seen as quantum observables, or operators, while s stays as a real parameter characterizing the Hamiltonian evolution.Such a description of quantum dynamics, while available since at least around 1940 [6,7] (see for example the books of Refs.[8,9] and references therein), is not what is commonly presented in textbooks.Even the 'standard' presentation of classical Hamiltonian dynamics does not do that (see however Johns [10]) and hence does not give a Lorentz covariant formulation [5].
Our results here also illustrate an advantage of that formulation of 'relativistic' dynamics, classical and quantum.
Our analysis starts in the next section with a presentation of the classical picture, especially focusing on a formulation in terms of Hamiltonian dynamics with the phase space seen as a cotangent bundle of the spacetime as the configuration space of the particle.The quantum dynamics in the Heisenberg picture and the quantum geodesic equation are then straightforward to obtain along the line.That is presented in section III.The last section gives careful discussions of various related aspects of the theory of quantum mechanics.
While the Heisenberg picture analysis is not dependent on the explicit theory for the corresponding, even abstract vector space, Schrödinger picture, our new theory of 'relativistic' quantum mechanics with a notion of Minkowski metric operator for the vector space of states [5,11], defining the inner product, is conceptually deeply connected to our Heisenberg picture results.In particular, we emphasize a perspective that takes seriously the quantum observables as physical quantities to be understood beyond the framework of classical physical concepts.In particular, the position and momentum observables may be seen as coordinate observables of the quantum phase space [12] as a noncommutative geometry [13,14].Some projections on taking an approach to quantum gravity focusing on the dynamical behavior of the physical quantities as quantum observables/operators are also discussed, including the important notion of quantum coordinate transformations in relation [15].

II. GEODESIC AND HAMILTONIAN DYNAMICS -CLASSICAL CASE
For the background analysis used in the section, we follow the presentation in the lecture note by Tong [16].The latter gives a careful derivation of the geodesic equation through minimizing the action and the canonical condition for the coordinates of the covariant phase space is given through the Poisson brackets Note that in the presence of a nontrivial g µν , x µ are not components of a four-vector, but p µ are.We avoid writing x µ here but p µ = g µν p ν are well-defined.Yet, even for canonical p µ , while one has {x µ , p ν } = g µν , {p µ , p ν } are generally nonzero.For a generic Riemannian manifold with x µ as local coordinates, the canonical momentum is still a cotangent vector and the phase space as the cotangent bundle is a symplectic manifold by construction.
The geometry of the latter dictates the structure of the local Hamiltonian dynamics.s as the parameter of the Hamiltonian flows has its mathematical nature fixed by the choice of Hamiltonian.For the case at hand, one can easily see that it is indeed essentially a geodesic length parameter and the physical proper time of the particle.For general use of configuration and momentum variables, we have the Poisson bracket given by with the canonical condition given by with indices µ and ν referring to the position coordinates and μ and ν referring to the corresponding momentum coordinates.
Let us go on adopting Tong's illustration of the WEP with the supplement of the Hamiltonian picture.We consider particle dynamics under (constant) gravity in a Minkowski spacetime and its description in the instantaneous frame of a free-falling observer (with the Kottler-Möller coordinates, at ρ(τ ) = 0).In terms of the classical metric with , is the constant acceleration, as the value of d 2 x dt 2 at the observer's rest frame.A Hamiltonian for the classical dynamics of the constant acceleration, under the original frame of reference, can be given by with η µν = diag{−1, 1, 1, 1} as the Minkowski metric.Note that the simple kinetic term guarantees p µ = m dx µ ds , which says s is the particle's proper time.The variable τ above is taken as the time coordinate of the instantaneous free-falling frame, hence differs from s in general.That is to say, other than the special solution of Eq.( 7), we do not have τ = s.
A key question is if the transformation should be seen as a canonical one.We are taking the configuration space coordinate transformation x → x ′ onto the phase space by enforcing the momentum four-vector to transform as dictated by that, which preserves the metric independent result p ′µ = m dx ′µ ds .Explicitly, we have That is to say, one simply takes the coordinate transformation on the configuration manifold to its cotangent bundle.Conceptually, the p ′ µ , from p ′ µ dx ′µ = p ν dx ν , are still the canonical components of the cotangent vector.One can confirm that analytically by checking the canonical condition through evaluating the Poisson brackets among the new canonical position and momentum variables (cτ, ρ, y, z) and (p cτ , p ρ , p y , p z ).The kinetic term of H a (s) maintains the quadratic form, as , while the potential term, neglecting the y and z dependent part, is −m(c 2 +aρ) 2 2c 2 , giving free particle motion instantaneously at for ρ = 0 at s = 0. Explicitly, with the initial values at s = 0 for all original phase space coordinate variables being zero except for x = c 2 a and p ct = mc, one obtains the solution of Eq.( 7).We have τ = s and ρ and p ρ maintaining their vanishing values, while p cτ = mc, and H a has value of mc 2 .
Complete results of the Hamiltonian dynamics above in the new and old coordinates, as well as the exact geodesic equations under the g µν metric can easily be worked out.We refrain from giving them explicitly here but their exact quantum analogs are to be given below.We want to note that in the Hamiltonian H a , the mass parameter m in the kinetic term is really the inertial mass, as in p µ = m dx µ ds , while the one in the potential term is a gravitational one.Without the equality of the two masses, there is no mass-independent free fall as obtained.

III. GEODESIC AND HAMILTONIAN DYNAMICS -QUANTUM CASE
Now in this section, we are going to illustrate the exact quantum analog of the classical analysis above with the classical observables replaced by quantum observables.The Hamiltonian approach can be used as a useful guideline, and results can most easily be seen from the Schrödinger representation with x′µ = x ′µ and p′ µ = −i ∂ ∂x ′µ , taking = 1, but is only a consequence of the commutation relation among the operators, or rather just as abstract quantum observables.The commutation relations are essentially Poisson bracket relations.
In fact, we will use the terms operators below, without really committing to any concretely given vector space they have to act on.If one prefers to think about that, it is certainly fine, so long as it gives a consistent representation of the algebra involved.We are more interested in working in the free-falling frame, with the nontrivial g µν .The exact quantum which is in fact trivially satisfied under Schrödinger representation for any choice of the set of position observables.Otherwise, so long as one adopts the commutation relations among the xµ and pµ , for the case of the Minkowski metric, they can be directly derived by promoting the classical coordinate transformation to one among the quantum observables (which can be seen as noncommutative coordinates of the quantum phase space [12]).Explicitly, that means taking Eq.( 7) and Eq.( 9) as for the operators.
For the Hamiltonian dynamics, we promote H a of Eq.( 8) directly to the Hamiltonian operator and write, in the free-falling frame, Note that we have dropped the part for the ŷ and ẑ degrees of freedom, for simplicity, and have put in an extra parameter ǫ for the easy tracing of the exact free case with the geodesic in the equations of motion.That is, ǫ = 1 gives the exact quantum dynamics of what is sketched in the last section, while ǫ = 0 gives the geodesic motion.The Heisenberg equations of motion we are interested in are given by and pcτ = m dcτ ds and pρ = m dρ ds .They have exactly the same form as the classical equations, with the classical observables replaced by quantum ones.We have, however, as the quantum version of the classical geodesic one.Note the sum of the two terms on the right-hand side for the classical limit with commutating variables.The operator ordering ambiguity from the classical to the quantum case is resolved through the Hamiltonian formulation.As in the classical case, the second-order differential equations for the position observables are mass-independent.Explicitly, they are Again, the ǫ = 0 case is the free motion, i.e. quantum geodesic equations.For the latter, as well as the ǫ = 1 case, the equations of motion are mass-independent, so long as the inertial mass and the gravitational mass as the m is the kinetic and potential terms of Eq.( 11), respectively, are taken the same.All are exact analogs of the classical case.

IV. DISCUSSIONS
A. More about the quantum theory We have obtained exact quantum analogs of the WEP for the classical case as treated in Tong's book supplemented by a fully covariant Hamiltonian dynamical picture based on an invariant evolution parameter that is essentially the classical particle proper time.The Hamiltonian evolution equations, with the clear identification of canonical variables, are of great help here for getting around otherwise nontrivial operator ordering issues.We have emphasized the cotangent bundle structure of the manifold as a phase space that gives the right picture of the coordinate transformation for the spacetime, the configuration space of the particle, as a canonical coordinate transformation for the Hamiltonian dynamics.
The quantum part of the story is then based on the adoption of position and momentum observables satisfying the canonical condition, Eq.( 10).The condition is independent of the (configuration space) metric.After all, symplectic geometric structure, and hence the Hamiltonian formulation, is independent of even the very existence of a Riemannian metric, for the configuration space or otherwise.The quantum geodesic equation governs the quantum evolution of the position observables, which we have argued are exact equations of motion that can be verified.Our result may well complement those available in the literature and provide a peek into an alternative approach to quantum gravity based on the quantum observables.Note that only they are the representations of the physical quantities in the theory.Spacetime as a physical object should be described with physical quantities, hence quantum, instead of classical, observables in a quantum theory.In a theory of particle dynamics, the only physical notion about spacetime is the observable position coordinates of a particle.The thinking about the validity of the classical geometric models of Newtonian space or Minkowski spacetime for the quantum theory hence lacks justification.In relation to that, it is interesting to note that the particle proper time as the time coordinate in the particle rest frame is really an observable ( g µν (x)x µ xν ), while the parameter s characterizing the Hamiltonian evolution stays as a real parameter, as the analog of Newtonian time in the 'nonrelativistic' theory.The possibility of going beyond that is a fundamental question about the concept of symmetry of quantum systems as exemplified by the notion of quantum reference frame transformations [17].This has very important implications for the EP and quantum gravity [15] beyond the classical coordinate transformation analyzed here, to be discussed more below.
Our Heisenberg picture analysis relies only on the Poisson bracket relations among the observables and the Hamiltonian formulation of the dynamics among them.It does not depend on any particular representation picture having the observables as explicit operators on a vector space of states, hence not even the notion of Hermiticity.The latter is really to be defined based on the chosen inner product on the vector space of states.Even the imaginary number i and suppressed in the commutation relations may be unnecessary.

One can replaced the
where the exact parallel of the classical and the quantum case would be plainly obvious.
The triviality in the case is because the Hamiltonian analysis for it involves no ordering ambiguity going from classical to quantum.
It is well known that Schrödinger quantum mechanics is Hamiltonian dynamics.Hence, the same must be true for the Heisenberg picture description as well.In fact, one can illustrate clearly that the Heisenberg equation of motion is exactly a Hamiltonian equation of motion for observables with the Poisson bracket as identified, essentially already be Dirac.
One can take the (projective) Hilbert space as the symplectic manifold.Each operator β on the Hilbert space can be matched to its expectation value function f β = φ|β|φ φ|φ as a 'classical' observable for which the two pictures of the Hamiltonian dynamics can be matched perfectly with the introduction of a noncommutative (Kähler) product among such function satisfying [12,18] The a Casimir invariant for the irreducible representation to be interpreted as Newtonian mass [5].An abstract Fock state basis as well as coherent state wavefunction description of that has been essentially presented [5,11].What is particularly interesting to note, in relation to the present analysis, is that the theory has a representation space that is Krein, instead of Hilbert.That is, the inner product is not positive definite [20].In fact, it is defined in terms of Pauli's metric operator [22,23,32] that is for the case Minkowski, denoted by η.
The position and momentum operators are exactly η-Hermitian, i.e. satisfying One can see that as a special case, the Minkowski case, of a quantum theory with an inner product defined in terms of a metric operator denoted by ĝ, as The proper definition of the adjoint or Hermitian conjugate of an operator β is then given by β † g satisfying g-Hermitian operators then generate one-parameter groups of (pseudo)-unitary transformations that preserve the inner product.The usual Hilbert space theory is exactly the case for the metric operator being the identity, giving an Euclidean metric on it.The Minkowski nature of such four-vector coordinate observables beyond the naive notion of switching between lower and upper indices is what is needed to justify seeing the position and momentum operators as noncommutative coordinates of the quantum phase space (the vector space of states or rather its projective space) with a Minkowski metric.

B. On quantum mechanics in nontrivial gravitational background, and beyond
The discussion in the last paragraph, apart from giving a solid vector space formulation of the 'relativistic' quantum dynamics behind the otherwise abstract Heisenberg picture dynamics analyzed above, along but different from the line of fully Lorentz covariant formulation [8,9], also brings up an interesting perspective for a plausible picture of a theory of quantum mechanics in a generic curved spacetime and its interpretation as particle dynamics on a curved noncommutative geometry.For example, one can think about a Schrödinger wavefunction representation of the Minkowski picture with the integral inner product between φ(x) and ψ(x) giving by where ηS is the explicit representation of η as operator acting on the wavefunction.When we apply the configuration space coordinate transforming of Eq.( 7) and Eq.( 9), the x → x ′ transformation taking the state wavefunctions to φ ′ (x ′ ) and ψ ′ (x ′ ) would take the above integral to One can reasonably expect ηS to depends on the position operators xµ only, then the inner product integral can be as That is to say, the new description of the quantum theory with states described by Schrödinger wavefunction φ(cτ, ρ, y, z) would have a inner product defined in terms of the new form of Pauli's metric operator ĝS .In any case, as the metric (tensor) changes, the inner product changes accordingly, as what is to be expected from the consistency of the notion of the quantum metric realized in the form of the inner product on the vector space of states.
In fact, in our main analysis in section III above, we have implicitly taken components of the metric tensor g µν , or its inverse, as functions of the position observables x′ .In the Schrödinger representation, x′ of course are just x ′ .That is to say, we have already taken the classical g µν (x) to a quantum ĝµν = g µν (x).Our quantum geodesic equations further illustrate an explicit notion of quantum Christoffel symbols.Up to operator ordering issues, we have them simply as direct operator versions of the classical ones.They can be seen as among the class of special elements of the quantum observable algebra that characterize the geometry of the quantum picture of the spacetime.While Ref. [1] has the kind of operators written down, they are not used much in the actual analysis, and certainly not from our perspective.The success of the approach here may be seen, naively, as suggesting an approach to quantum gravity as simple as recasting Einstein's theory in operator form.However, other than issues about the proper operator ordering, there are still important questions to address.We name two here.One is the notion of a quantum coordinate transformation we have touched on, which is to be discussed below.The other is related to Pauli's metric operator.The latter gives a metric really on the vector space of states, which is generally a Kähler manifold, hence the symplectic structure is tied to the metric structure.
The corresponding metric tensor is one for the infinite -dimensional real/complex manifold.
The metric tensor operator ĝµν is, however, a metric for the configuration space with the position observables, x′ , or coordinate observables.It is hard to think about that space as a spacetime any different from the classical one.The answer to the puzzle lies in the fact that the quantum phase space, unlike its classical analog, is an irreducible representation of the background (relativity) symmetry.In the classical theory as an approximation, in which the commutation relations are trivialized, the representation is reducible to that of the configuration and momentum space.That is very much like the relation of the classical Minkowski spacetime to its 'nonrelativistic' approximation as Newtonian space and time.
At the 'relativistic' level, there is no separate notion of space and time.At the quantum level, there is no separate notion of configuration and momentum space.The noncommutative geometric picture for the quantum phase space is the quantum model of the physical spacetime.It is not clear then if one should be thinking about only a metric tensor operator as for the x′ part only, though that is naturally doubled with a copy for the p′ part.
For example, a generic coordinate system could have mixed the position and momentum coordinate observables.The question is really if quantum gravity is about the metric of the quantum phase space as a noncommutative geometry.
Lämmerzahl stated that 'Quantum mechanics is a non-local description of matter' [24].
That statement is probably the first idea many physicists have in mind when approaching the problems related to the EP.Obviously, if a quantum particle generally cannot go along an exact path, in a classical model of space(time), it cannot follow such a geodesic.We have already discussed much about the idea of our quantum geodesic.With the idea of the quantum model of spacetime being a noncommutative geometry, motion along a definite path therein is completely feasible.In the Schrödinger picture, the state of the particle certainly evolves along a definitive path.Apart from the spacetime geometry picture, quantum nonlocality as in entanglement between parts of a composite system may be seen as the deeper meaning of Lämmerzahl's statement.Yet, even that notion of nonlocality has been challenged, interestingly enough, in the Heisenberg picture [25].Ways to describe the complete quantum information for a composite system as information about a set of local basic observables, such as the position and momentum of the individual particles have been presented [26].For our result of the quantum geodesic, as equations of motion for the observables, they are state-independent.One can think about two quantum particles in simultaneous free fall.To the extent that we can neglect the gravitational pull between them, the free motion Hamiltonian for the system would only be a sum of the individual kinetic terms.Each particle then has the same quantum geodesic equation governing its motion, irrespective of the actual composite state of the two particles and to what extent they are entangled.Of course, the story may be very different in a full treatment from a theory of quantum gravity where one cannot simply take the metric as a fixed background.

C. Against Poincaré symmetry and on-shell mass condition
So far as obtaining the geodesic equation is concerned, the Hamiltonian approach if the same as the Lagrangian one which is just about minimizing the action dsL, or ds L.
One may think about it completely independent of any dynamics.However, physics is dynamics.We have emphasized that from the physical point of view, the proper model of spacetime should be taken from the successful theory of particle dynamics rather than assumed as given.As a parallel, the notion of a geodesic equation should only be taken as an equation of motion for a free particle.One may want to replace the latter with a photon.But that is really taking it beyond a theory of particle dynamics.In the other part of the analysis above for the WEP we have the Hamiltonian H a with a potential from the gravitational pull as seen from a classical Minkowski spacetime to start with.The success of the Hamiltonian picture is clear.However, vigilant readers may have noticed that the on-shell mass condition as −p µ p µ = m 2 E c 2 is not necessarily respected; it generally allows −p µ p µ to have nontrivial s-evolution.Admitting a potential in violation of the on-shell mass condition is a general feature of the kind of fully Lorentz covariant Hamiltonian formulation [8,9].Note that we used a new notation m E , instead of m here.After all, the concept of Einstein rest mass is not the same as the Newtonian inertial and gravitational mass.
We have a quite elaborated discussion on the related issues which is closely connected to the Poincaré symmetry that we do not see as the right symmetry to formulate a theory of 'relativistic' or Lorentz covariant quantum dynamics in Ref. [5].Note that any quantum theory with wavefunctions as functions fo otherwise free Minkowski four-vector variables x µ and pµ as −i( )∂ µ as in Klein-Gordon or Dirac equations are really not obtainable from the Poincaré symmetry.And when we have the version of the equations with the presence of an electromagnetic interaction (through the covairant derivative), the canonical momentum pµ for the charged particle is no longer conserved.We do not even have pµ = m dx µ dτ .The same holds in the classical case.Einstein was well aware that the generally important quantity as the conserved momentum in any closed system may not be the quantity of mass times velocity and does not necessarily obey any on-shell mass condition [27].Note that the latter is non-negotiable for a particle corresponding to an irreducible representation of the Poincaré symmetry.Our H a with the gravitational acceleration a in a potential term is legitimate and its success in term speaks for the strength of the covariant Hamiltonian formulation.

D. Quantum coordinate transformations and quantum gravity
Even when we take the operator version of Eq.( 7) and Eq.( 9) for the coordinate transformation, it is really only a classical one.We talk about going to the free-falling frame.But that is only the free-falling frame of a classical particle.The reference frame transformation sees no quantum properties of the particle, and the key parameter describing the transformation, the gravitational acceleration a, is only taken as a classical quantity.To stick fully to the idea of the free-falling frame of the physical particle which is a quantum object, one should have a version of quantum reference frame transformation [17].One should take the gravitational acceleration of the quantum particle seen in the Minkowski frame as what it should be -a quantum observable â.Yet, how to do it properly is a very challenging question.To start with, going for a version of the operator coordinate transformation with an â has nontrivial operator ordering issues between â and the position observables ρ and τ has to be resolved.
While the idea of quantum frames of reference is not new [28,29], a more more recent study [17] has brought it back to the attention of many authors.In the example of a quantum (spatial) translation, as an explicit notion of the position of particle B relative to particle A, the paper gives it as a unitary transformation on the quantum phase space of the form e ix We have only given a brief sketch of the generic notion of quantum coordinate transformation above.Its important relevancy to a theory of quantum gravity has been noted by Hardy [15].He adopts an approach to the problem quite different from us though.Quantum gravity has to be about the quantum geometry of the spacetime.The picture of a simple quantum spatial translation clearly illustrates the necessity to beyond any classical geometric picture.Hardy looked at that quantum geometry as a superposition of classical ones.We want to look at it as a single noncommutative geometry, a geometry with the observables as coordinates.A more solid picture of the latter is given by a notion of noncommutative values for the observables [19,30], as a representation of the full quantum information a state bears for the observable.Instead of identifying a state as defining the evaluational functional as given by the expectation values of observables, it can be promoted to an evaluational homomorphism that maps the observables to local representations of its expectation value functions as a noncommutative algebra of the state-specific quantum values based on the Kähler product of Eq.( 15).Each such noncommutative value can be represented by a complex number sequence with terms essentially given by the coefficients of the Taylor series expansion around the state.In the wavefunction picture, we have essentially the expectation values themselves and their functional derivatives with respect to the wavefunction [31].Details we refer the readers to the references.From the theoretical point of view, the notion of the noncommutative value for an observable is an element of a state-specific noncommutative algebra that encodes the complete mathematical information the theory of quantum mechanics contains when matching the observable to the state [26].The algebraic relations among the observables as variables are exactly preserved among their values.The noncommutative value for the position observable, for example, offers exactly an answer to what is the actual 'amount', that value of xA that the quantum spatial transformation discussed above translates any xB with all notions about changes in Heisenberg uncertainty and even entanglement successfully described [31].The noncommutative value, on the one hand, provides the quantum generalization of the classical real number parameter of a coordinate transformation to make the latter quantum.On the other hand, it gives a picture of that noncommutative/quantum geometry as the space of all possible (noncommutative) values of the set of position and momentum observables beyond the picture of classical geometries.
Definitive points, and hence a definite geodesic path of the quantum particle within that quantum model of spacetime are to be specified by fixed values of the coordinate observables.
Hardy's picture of the quantum spacetime geometry is then the Schrödinger picture of our sketched Heisenberg picture here.
It is interesting to note, in relation to our discussion, a comment from Penrose n the problem of compatibility of quantum mechanics and the principle of relativity [32].Basically, Penrose was pointing to the fact that no reference frame transformation can take a position eigenstate to another state that is not a position eigenstate.But that is true only when the classical picture of all possible eigenvalues of the position observable is taken to give the model of that space(time).With the notion of quantum reference frame transformation, one can apply a translation by exactly the (noncommutative value) amount of difference in the definite noncommutative values of the two positions to take one to the other.A point in the noncommutative space(time) can be described by the different noncommutative coordinate values under different choices of reference frames.There is no intrinsic difference between a point with a noncommutative position coordinate value that corresponds to an eigenstate and one that has a more nontrivial noncommutative position coordinate value.This illustrates well then quantum general relativity is about quantum reference frame transformations that certainly cannot be formulated in terms of a classical space(time) manifold.
expression of the Heisenberg equation of motion in terms of the such functions for all the observables is the exact Hamilton equation of motion for any f β in terms of the exact Poisson bracket for the (projective) Hilbert space.The set of the corresponding equations of motion for the real or complex number coordinates characterizing the state is the exact content of the Schrödinger equation.We have suggested the interpretation of the Heisenberg picture as the noncommutative coordinate picture of exactly the same symplectic geometry and established a consistent differential geometric picture of that as a noncommutative geometry [12, 19].An exact Lorentz covariant version of that can be obtained, from a symmetry theoretical formulation of the 'relativistic' quantum picture based on a Lie group/algebra we called H R (1, 3) with Lorentz symmetry plus Minkowski four-vector generators Y µ and P µ together with a central charge M giving the above canonical condition/commutation relation in the form [mx µ , pν ] = [ŷ µ , pν ] = i( )η µν m = i( )η µν m Î, where m is effectively A pB .The latter can be read naively as pB generating a translation of xB by 'an amount' xA giving the position observable for particle B in the reference frame of particle A as xB − xA .Apply to explicit states of particles A and B, interesting quantum features can be retrieved.For example, if we can start with such a composite state that has the two positions completely entangled yet far from eigenstates of xA and xB .The position of particle B as observed from particle A, exactly as obtained from the transformation, gives B as in a position eigenstate.What we have is really an eigenstate of xB − xA , seen in the original frame.One can also starts with a product state between A and B. From the frame of A, particle B would be seen as fully entangled to the particle C that represents the original frame of reference.Quantum properties, as such Heisenberg uncertainty and entanglement, generally change under a quantum coordinate transformation.Note the unitary transformation e ix A pB is a canonical transformation, both seen from the Hilbert space point of view or that of the noncommutative symplectic geometry of x-p.The transformation as given also serves as a momentum translation of pA generated by the operator xA by 'an amount' −p B , hence maintaining the canonical condition among all pairs of position and momentum observables under the different frames.