Equivalence principle for quantum mechanics in the Heisenberg picture

Otto C W Kong

doi:10.1088/1361-6382/ad359f

1. 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 [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, not been a focus of much attention. Most authors apply analyses based on a Schrödinger wavefunction formulation, within which there is, basically, 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. $\frac{d\hat{p}^i}{dt} = 0$ . A conservation law of this kind is 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 very misleading, if not completely wrong. A conservation statement such as $\frac{d\hat{p}^i}{dt} = 0$ certainly cannot give you single constant eigenvalue answers in projective measurement for any particular $\hat{p}^i$ , 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 $\hat{p}^i$ 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 $\hat{p}^i$ 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. As the Heisenberg picture is all about the observables, i.e. physical quantities, we see that as conceptually more fundamental. As a quantization approach, we see the Heisenberg-Dirac one through the identification of the canonical coordinate observables and the quantum Poisson bracket as the preferred one.

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 [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 3. 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, 16].

2. Geodesic and Hamiltonian dynamics—classical case

For the background analysis used in the section, we follow the presentation in the lecture note by Tong [17]. The latter gives a careful derivation of the geodesic equation through minimizing the action

$\begin{eqnarray} &&S_o = \int\!\!ds L_o\left(s\right) = \int\!\!ds \sqrt{- g_{\mu\nu}\left(x\right) \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} } \;, \end{eqnarray} \tag{ 1 }$

that we are not repeating here. Clearly, one would obtain the same equation taking the Lagrangian $L(s) = -\frac{m}{2} L_o^2$ . The latter, called a 'useful trick' by Tong, is the exact one that gives, through a Legendre transformation, a covariant Hamiltonian formulation of the free particle dynamics that is the exact 'relativistic' extension of the 'nonrelativistic' Hamiltonian formulation. The Lagrangian and the Hamiltonian, $H(s) = p_\mu \dot{x}^\mu - L(s)$ are both just the kinetic term $\frac{m}{2} g_{\mu\nu}(x) \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} = \frac{1}{2m} g^{\mu\nu}(x) p_\mu p_\nu$ , where the canonical momentum variables are $p_\mu = \frac{\partial L}{\partial \dot{x}^\mu}$ , $\dot{x}^\mu \equiv \frac{dx^\mu}{ds}$ , instead of p^µ. Hamilton's equations of motion, $\frac{dx^\mu}{ds} = \frac{\partial H}{\partial p_\mu}$ and $\frac{dp_\mu}{ds} = -\frac{\partial H}{\partial x^\mu}$ , are special examples of equation of motion for all observables $F(x^\mu,p_\nu)$ given by

$\begin{eqnarray} &&\frac{dF}{ds} = \left\{F,H\right\} = \frac{\partial H}{\partial p_\mu}\frac{\partial F}{\partial x^\mu}-\frac{\partial F}{\partial p_\mu}\frac{\partial H}{\partial x^\mu} \;, \end{eqnarray} \tag{ 2 }$

and the canonical condition for the coordinates of the covariant phase space is given through the Poisson brackets

$\begin{eqnarray} &&\left\{x^\mu, x^\nu \right\} = \left\{p_\mu, p_\nu\right\} = 0 \;, \qquad \left\{x^\mu, p_\nu \right\} = \delta^\mu_\nu \;. \end{eqnarray} \tag{ 3 }$

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 $p^\mu = g^{\mu\nu} p_\nu$ are well-defined. Yet, even for canonical p_µ, while one has $\{x^\mu, p^\nu \} = g^{\mu\nu}$ , $\{p^\mu, p^\nu \}$ 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

$\begin{eqnarray} &&\left\{F,G\right\} = {\mathcal P}^{\mu\nu} \frac{\partial F}{\partial x^\mu} \frac{\partial G}{\partial x^\nu} + {\mathcal P}^{{\mu}}_{\tilde{\nu}} \left( \frac{\partial F}{\partial x^\mu} \frac{\partial G}{\partial p_{\nu}} -\frac{\partial G}{\partial x^\mu} \frac{\partial F}{\partial p_\nu} \right) + {\mathcal P}_{\tilde{\mu}\tilde{\nu}} \frac{\partial F}{\partial p_{{\mu}} } \frac{\partial G}{\partial p_{{\nu}} } \;, \end{eqnarray} \tag{ 4 }$

with the canonical condition given by

$\begin{eqnarray} &&{\mathcal P}^{\mu\nu} = {\mathcal P}_{\tilde{\mu}\tilde{\nu}} = 0 \;, \qquad {\mathcal P}^{\mu}_{\tilde{\nu}} = \delta^{\mu}_{\nu} \;, \end{eqnarray} \tag{ 5 }$

with indices µ and ν referring to the position coordinates and $\tilde{\mu}$ and $\tilde{\nu}$ referring to the corresponding momentum coordinates. The mathematical validity of the above picture of the canonical coordinates for any Hamiltonian system with a nontrivial metric (on the configuration space) can be verified through the implementation of a coordinate transformation from the exact Euclidean one including taking that to the Minkowski case (say by adding a sign flip to one x_µ but not x^µ).

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 $\rho(\tau) = 0$ ). In terms of the classical metric

$\begin{eqnarray} &&ds^2 = -c^2dt^2+dx^2+dy^2+dz^2 = -\left(1+\frac{a\rho}{c^2} \right)^2 c^2 d\tau^2 +d\rho^2 +dy^2+dz^2\;, \end{eqnarray} \tag{ 6 }$

with

$\begin{eqnarray} &&ct = \left(\rho +\frac{c^2}{a} \right) \sinh\!\left(\frac{a\tau}{c} \right)\;, \qquad x = \left(\rho +\frac{c^2}{a} \right) \cosh\!\left(\frac{a\tau}{c} \right) \;. \end{eqnarray} \tag{ 7 }$

$a = \sqrt{-(a^t)^2+ (a^x)^2}$ , $a^x = \frac{d^2x}{d\tau^2} = a \cosh\!\left(\frac{a\tau}{c} \right)$ and $a^t = \frac{d^2ct}{d\tau^2} = a \sinh\!\left(\frac{a\tau}{c} \right)$ , is the constant acceleration, as the value of $\frac{d^2x}{dt^2}$ 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

$\begin{eqnarray} &&{H}_{\!a}\left(s\right) = \eta^{\mu\nu} \frac{p_\mu p_\nu}{2m} - \frac{ma^2}{2c^2}\eta_{\mu\nu}x^\mu x^\nu \;, \end{eqnarray} \tag{ 8 }$

with $\eta_{\mu\nu} = \mbox{diag}\{-1,1,1,1\}$ as the Minkowski metric. Note that the simple kinetic term guarantees $p^\mu = m \frac{dx^\mu}{ds}$ , 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 equation (7), we do not have $\tau = s$ . A key question is if the transformation should be seen as a canonical one. We are taking the configuration space coordinate transformation $x \to x^{^{\prime}}$ onto the phase space by enforcing the momentum four-vector to transform as dictated by that, which preserves the metric independent result $p^{^{\prime}\mu} = m \frac{dx^{^{\prime}\mu}}{ds}$ . Explicitly, we have

$\begin{eqnarray} &&p^{ct} = p^\rho \frac{\partial ct}{\partial \rho} + p^{c\tau} \frac{\partial ct}{\partial c\tau}\;, \qquad p^x = p^\rho \frac{\partial x}{\partial \rho} + p^{c\tau} \frac{\partial x}{\partial c\tau} \;. \end{eqnarray} \tag{ 9 }$

That is to say, one simply takes the coordinate transformation on the configuration manifold to its cotangent bundle. Conceptually, the $p^{^{\prime}}_\mu$ , from $dx^{^{\prime}\mu} p^{^{\prime}}_\mu = dx^\nu p_\nu$ , 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 $(c\tau,\rho,y,z)$ and $(p_{c\tau}, p_\rho ,p_y,p_z)$ . The kinetic term of ${H}_{\!a}(s)$ maintains the quadratic form, as $g^{\mu\nu}(x^{^{\prime}})\frac{p^{^{\prime}}_\mu p^{^{\prime}}_\nu}{2m}$ , while the potential term, neglecting the y and z dependent part, is $\frac{-m(c^2+a\rho)^2}{2c^2}$ , 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 $x = \frac{c^2}{a}$ and $p^{ct} = mc$ , one obtains the solution of equation (7). We have $\tau = s$ and ρ and p^ρ maintaining their vanishing values, while $p^{c\tau} = mc$ , and $H_{\!a}$ has value of mc².

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 $p^\mu = m\frac{dx^\mu}{ds}$ , 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.

3. 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 $\hat{x}^{^{\prime}\mu} = {x}^{^{\prime}\mu}$ and $\hat{p}^{^{\prime}}_\mu = -i\frac{\partial ~}{\partial x^{^{\prime}\mu}}$ , taking $\hbar = 1$ , 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 Poisson bracket is $-i[\cdot,\cdot]$ , giving the canonical condition as

$\begin{eqnarray} &&-i\left[\hat{x}^{^{\prime}\mu}, \hat{x}^{^{\prime}\nu}\right] = -i \left[\hat{p}^{^{\prime}}_\mu, \hat{p}^{^{\prime}}_\nu\right] = 0 \;, \qquad -i\left[ \hat{x}^{^{\prime}\mu}, \hat{p}^{^{\prime}}_\nu\right] = \delta^\mu_\nu \;. \end{eqnarray} \tag{ 10 }$

So long as one adopts the commutation relations among the $\hat{x}^\mu$ and $\hat{p}_\mu$ , 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 equations (7) and (9) as for the operators.

For the Hamiltonian dynamics, we promote $H_{\!a}$ of equation (8) directly to the Hamiltonian operator and write, in the free-falling frame,

$\begin{eqnarray} &&\hat{H}_{\!a} = - \frac{c^4}{\left(c^2+a\hat\rho\right)^2} \frac{\left(\hat{p}_{c\tau} \right)^2}{2m} + \frac{\left(\hat{p}_\rho\right)^2}{2m} - \epsilon \frac{m\left(c^2+a\hat\rho\right)^2}{2c^2} \;. \end{eqnarray} \tag{ 11 }$

Note that we have dropped the part for the $\hat{y}$ and $\hat{z}$ degrees of freedom, for simplicity, and have put in an extra parameter ε for the easy tracing of the exact free case with the geodesic in the equations of motion. That is, ε = 1 gives the exact quantum dynamics of what is sketched in the last section, while ε = 0 gives the geodesic motion. The Heisenberg equations of motion we are interested in are given by

$\begin{eqnarray} &&\frac{d\hat{p}_{c\tau}}{ds} = 0 \;, \qquad \frac{d\hat{p}^\rho}{ds} = \frac{d\hat{p}_\rho}{ds} = -\frac{a\left(c^2+a\hat\rho\right)}{mc^4} \left(\hat{p}^{c\tau} \right)^2 + \epsilon \frac{ma\left(c^2+a\hat\rho\right)}{c^2} \;, \end{eqnarray} \tag{ 12 }$

and $\hat{p}^{c\tau} = m \frac{d{c\hat\tau}}{ds}$ and $\hat{p}^\rho = m \frac{d\hat\rho}{ds}$ . They have exactly the same form as the classical equations, with the classical observables replaced by quantum ones. We have, however,

$\begin{eqnarray} &&\frac{d\hat{p}^{c\tau}}{ds} = -i \left[\hat{p}^{c\tau}, \hat{H}_{\!a}\right] = - \frac{a}{c^2+a\hat\rho} \frac{\hat{p}^{c\tau} \hat{p}^\rho}{m} - \frac{\hat{p}^\rho \hat{p}^{c\tau} }{m} \frac{a}{c^2+a\hat\rho} \end{eqnarray} \tag{ 13 }$

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

$\begin{align} &\frac{d^2{c\hat\tau}}{ds^2} + \frac{d c\hat\tau} {ds} \frac{a}{c^2+a\hat\rho} \frac{d \hat{\rho}}{ds} + \frac{d \hat{\rho}}{ds} \frac{a}{c^2+a\hat\rho} \frac{d c\hat\tau} {ds}= 0 \;, \nonumber \\ &\frac{d^2\hat\rho}{ds^2} + \frac{d c\hat\tau} {ds} \frac{a\left(c^2+a\hat\rho\right)}{c^4} \frac{d c\hat\tau} {ds} -\epsilon \frac{a\left(c^2+a\hat\rho\right)}{c^2} = 0 \;. \end{align} \tag{ 14 }$

Again, the ε = 0 case is the free motion, i.e. quantum geodesic equations. For the latter, as well as the ε = 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 equation (11), respectively, are taken the same. All are exact analogs of the classical case.

4. Discussions

4.1. 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, equation (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 ( $\sqrt{g_{\mu\nu}(\hat{x})\hat{x}^\mu \hat{x}^\nu}$ ), 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 [18]. 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 $\hbar$ in the commutation relations may be unnecessary. One can replaced the $-i [\cdot,\cdot]$ by $\{\cdot,\cdot\}_{\!\scriptscriptstyle Q}$ or even just $\{\cdot,\cdot\}$ as the (quantum) Poisson bracket., 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 $f_{\!\scriptscriptstyle\beta} = \frac{\left\langle\phi\right| \beta\left|\phi\right\rangle}{\left\langle\phi |\phi\right\rangle}$ 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, 19]

$\begin{eqnarray} &&f_{\!\scriptscriptstyle\beta} \star_{\!\scriptscriptstyle\kappa} f_{\!\scriptscriptstyle\gamma} = f_{\!\scriptscriptstyle\beta\gamma} \;. \end{eqnarray} \tag{ 15 }$

The 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 $f_{\!\scriptscriptstyle\beta}$ 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, 20]. 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 $H_{\!\scriptscriptstyle R}(1,3)$ 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 $[m\hat{x}_\mu, \hat{p}_\nu] = [\hat{y}_\mu, \hat{p}_\nu] = i(\hbar) \eta_{\mu\nu} \hat{m} = i(\hbar) \eta_{\mu\nu} m\hat{I}$ , where m is effectively 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 [21]. In fact, it is defined in terms of Pauli's metric operator [22–24] that is for the case Minkowski, denoted by $\hat{\eta}$ . The position and momentum operators are exactly η-Hermitian, i.e. satisfying

$\begin{eqnarray} &&{}{_\eta}{\!\left\langle \cdot | \beta \cdot \right\rangle} = {}{\eta}{\!\left\langle \beta \cdot | \cdot \right\rangle} \;. \end{eqnarray} \tag{ 16 }$

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 $\hat{g}$ , as ${}{_g}{\!\left\langle \cdot | \cdot \right\rangle } = \left\langle \cdot | \hat{g} |\cdot \right\rangle$ , or equivalently ${}{_g}{\!\left\langle \cdot \right|} = \left\langle \cdot \right| \hat{g}$ . The proper definition of the adjoint or Hermitian conjugate of an operator β is then given by $\beta^{\dagger^g}$ satisfying

$\begin{eqnarray} &&{}{_g}{\!\left\langle \cdot | \beta^{\dagger^g} \cdot \right\rangle} = {}{g}{\!\left\langle \beta\cdot | \cdot \right\rangle} \;. \end{eqnarray} \tag{ 17 }$

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.

4.2. 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 $\phi(x)$ and $\psi(x)$ giving by

$\begin{eqnarray} &&\int\!d^4x \psi_\eta^* \left(x\right) \phi\left(x\right) = \int\!d^4x \left[\hat\eta_{\scriptscriptstyle S} \psi \left(x\right)\right]^* \phi\left(x\right) \;, \end{eqnarray} \tag{ 18 }$

where $\hat\eta_{\scriptscriptstyle S}$ is the explicit representation of $\hat\eta$ as operator acting on the wavefunction. When we apply the configuration space coordinate transforming of equations (7) and (9), the $x \to x^{^{\prime}}$ transformation taking the state wavefunctions to $\phi^{^{\prime}}(x^{^{\prime}})$ and $\psi^{^{\prime}}(x^{^{\prime}})$ would take the above integral to

$\begin{eqnarray} &&\int\!d^4x^{^{\prime}} \left|\frac{\partial x}{\partial x^{^{\prime}}}\right| \left[\hat\eta_{\scriptscriptstyle S} \psi^{^{\prime}}\left(x^{^{\prime}}\right)\right]^* \phi^{^{\prime}}\left(x^{^{\prime}}\right) = \int\!d^4x^{^{\prime}} \left( 1+ \frac{a\rho}{c^2}\right) \left[\hat\eta_{\scriptscriptstyle S} \psi^{^{\prime}}\left(x^{^{\prime}}\right)\right]^* \phi^{^{\prime}}\left(x^{^{\prime}}\right) \;. \end{eqnarray} \tag{ 19 }$

One can reasonably expect $\hat\eta_{\scriptscriptstyle S}$ to depends on the position operators $\hat{x}^\mu$ only, then the inner product integral can be as

$\begin{eqnarray} &&\int\!d^4x^{^{\prime}} \left( 1+ \frac{a\rho}{c^2}\right) \hat\eta_{\scriptscriptstyle S}\left(x\right) \psi^{^{\prime}}\left(x^{^{\prime}}\right)^* \phi^{^{\prime}}\left(x^{^{\prime}}\right) = \int\!d^4x^{^{\prime}} \left[\hat{g}_{\scriptscriptstyle S}\left(x^{^{\prime}}\right) \psi^{^{\prime}}\left(x^{^{\prime}}\right)\right]^* \phi^{^{\prime}}\left(x^{^{\prime}}\right) \;. \end{eqnarray} \tag{ 20 }$

That is to say, the new description of the quantum theory with states described by Schrödinger wavefunction $\phi(c\tau,\rho,y,z)$ would have a inner product defined in terms of the new form of Pauli's metric operator $\hat{g}_{\scriptscriptstyle S}$ . 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 3 above, we have implicitly taken components of the metric tensor g_µν, or its inverse, as functions of the position observables $\hat{x}^{^{\prime}}$ . In the Schrödinger representation, $\hat{x}^{^{\prime}}$ of course are just x'. That is to say, we have already taken the classical $g_{\mu\nu}(x)$ to a quantum $\hat{g}_{\mu\nu} = g_{\mu\nu}(\hat{x})$ . 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 [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 $\hat{g}_{\mu\nu}$ is, however, a metric for the configuration space with the position observables, $\hat{x}^{^{\prime}}$ , or coordinate observables. At a more fundamental level from the noncommutative geometric perspective, the relation between the metric and symplectic structure is worth more serious consideration.

Lämmerzahl stated that 'Quantum mechanics is a non-local description of matter' [25]. 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 too. 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 [26]. 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 [27]. 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.

It may be of interest to note that theories on noncommutative geometric models of the physical space generally have violations of the EP at least for composite systems. The simple example in [28] for the so-called canonical case illustrates that well, as the resulting noncommutative parameter for the center-of-mass position coordinates of a system of particles would be dependent on the particle masses. However, we are only treating standard quantum mechanics in the Heisenberg picture, supplemented by a perspective of seeing the quantum phase space as a noncommutative geometry, and the configuration space as a part of that. Hence, we have no noncommutativity among any of the position observables, for individual particles or the center-of-mass of a composite system. The latter still behaves in the same way as that of a particle.

4.3. Against Poincaré symmetry and on-shell mass condition

Our analysis starts with the variational formulation of the mathematical geometric geodesic problem, followed by taking the Lagrangian formulation to the matching Hamiltonian formulation. The mathematics of the problem is not dependent on the dimension of geometric space and its metric signature. The four-dimensional Minkowski case gives the mathematical Hamiltonian formulation with the Poisson brackets of equation (2), and through a spacetime coordinate transformation equation (4). For the corresponding mathematical formulation of the WEP, we take that to the quantum analog, for which we illustrate that there is no logical inconsistency and ambiguity, and solved for the quantum geodesic equation. All that can be taken completely independent of any dynamics. From the physics point of view, there is no good reason to doubt that a geodesic is about a path of free particle motion. That is indeed a basic theme in Einstein General Relativity. Our solution to the mathematical problem can only go along with the alternative formulations of 'relativistic' mechanics [8, 9] without the on-shell mass condition as a defining property for a particle, though one can retrieve it for a free particle as an initial condition under a class of reference frames [5] in an exact parallel to the 'nonrelativistic's case. The background relativity symmetry is a larger one with a Poincare subgroup, without $P_\mu P^\mu$ as a Casimir invariant, hence no on-shell mass condition [5]. We have constructed the corresponding theory of quantum mechanics with four-vector position and momentum operators and a Minkowski metric operator defining a Krein vector space of states [11]. The latter is necessary to have a noncommutative geometric picture of the quantum phase space with a quantum version of the metric tensor and a notion of Minkowski signature for it. That is what we see as the solid physics background for our current effort. It is this new line of thinking that allows us to get to the exact quantum analog of the WEP presented above. We believe the result is a desirable one, especially when compared to other results with complicated quantum corrections violating the WEP in terms of technical expressions no one has proposed any conceptual explanation of. The result in turn speaks for the merit of the approach.

Admitting a potential in violation of any notion of 'on-shell mass condition' $-p_\mu p^\mu = m_{\!\scriptscriptstyle E}^2 (c^2)$ is a general feature of the kind of fully Lorentz covariant Hamiltonian formulation, as it generally allows $-p_\mu p^\mu$ to have nontrivial s-evolution. Note that we used a new notation $m_{\!\scriptscriptstyle E}$ , 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 presented a quite elaborate 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 [5]. Note that any quantum theory with wavefunctions as functions of otherwise free Minkowski four-vector variables x^µ and $\hat{p}_\mu$ as $-i(\hbar)\partial _\mu$ , essentially four independent coordinate derivatives, as in Klein–Gordon or Dirac equations are not truly obtainable from the Poincaré symmetry. The theories have all observables (independent of spin) as functions of x^µ and $-i(\hbar)\partial _\mu$ , yet no notion of the basic operator $x^{\scriptscriptstyle 0}$ can be retrieved out of the symmetry algebra which brings into question the nature of $-i(\hbar)\partial _{\scriptscriptstyle 0}$ . From the direct Poincaré symmetry construction, the Hilbert space should only be the span of all eigenstates of the independent three-momentum, hence the same as the one for the 'nonrelativistic' theory. When we have the version of the equations with the presence of an electromagnetic interaction (through the covariant derivative), we no longer have $\hat{p}^\mu = m \frac{d\hat{x}^\mu}{d\tau}$ and the magnitude square of the canonical momentum $\hat{p}^\mu$ for the charged particle is no longer constant. 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 [29]. Note that the latter is non-negotiable for a particle corresponding to an irreducible representation of the Poincaré symmetry. It would, in that case, be a defining property of the particle that holds even for the particle as a part of a composite system with interaction. In the quantum case, the operator condition could not be compromised for any 'virtual state' either. Physicists never stick to that. Our $H_{\!a}$ with the gravitational acceleration a in a potential term is legitimate and its success in turn speaks for the strength of the covariant Hamiltonian formulation.

The bottom line is that we have obtained, mathematically, the exact quantum analog of the WEP with a quantum geodesic equation, and presented a consistent physical picture of that. The picture is not compatible with the common believe in Poincaré symmetry and the straight on-shell mass condition, which we have criticized. For those whose believe in the latter being unquestionable, and would rather question the idea that geodesic is about free particle motion, we can only leave to them to decide what might be the physical meaning of our results.

4.4. Quantum coordinate transformations and quantum gravity

The important relevancy of quantum coordinate transformation (a notion that has been brought back to much attention in recent years [18]) to a theory of quantum gravity has been noted by Hardy [15]. The latter shares with us the perspective that quantum gravity has to be about the quantum geometry of spacetime. Hardy looks at quantum geometry as a superposition of classical ones. That may be considered the matching Schrödinger picture of our Heisenberg picture perspective looking at spacetime as a noncommutative geometry, a geometry with the quantum observables as coordinates. A more solid picture of the latter is given by a notion of noncommutative values for the observables [20, 30], as a representation of the full quantum information a state bears for the observable, that we have used to describe explicitly the results of a quantum spatial translation [16]. It is interesting to note a comment from Penrose on the compatibility of quantum mechanics and the principle of relativity [31]. Penrose was pointing, essentially, to the apparent 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 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.

For the problem analyzed in this paper, when we take the operator version of equations (7) and (9) for the coordinate transformation, it is only a classical transformation applied to the quantum system particle. The transformation to the free-falling frame sees no quantum properties of the particle. The gravitational acceleration a, as the key parameter describing the transformation, 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. One should take the gravitational acceleration of the quantum particle seen in the Minkowski frame as what it should be, namely a quantum observable $\hat{a}$ . The latter cannot be assumed to commute with the position observables. How to do that properly is a very challenging question, on which we hope to be able to address in the near future based on the mentioned background framework.

Acknowledgments

The author thanks P M Ho and J T Hsiang for discussions. Thanks also go to B L Hu for reading the manuscript, and comments. The work is partially supported by research Grant Number 112-2112-M-008-019 of the NSTC of Taiwan.

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No new data were created or analysed in this study.

Equivalence principle for quantum mechanics in the Heisenberg picture

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Abstract

1. Introduction

2. Geodesic and Hamiltonian dynamics—classical case

3. Geodesic and Hamiltonian dynamics—quantum case

4. Discussions

4.1. More about the quantum theory

4.2. On quantum mechanics in nontrivial gravitational background, and beyond

4.3. Against Poincaré symmetry and on-shell mass condition

4.4. Quantum coordinate transformations and quantum gravity

Acknowledgments

Data availability statement

Equivalence principle for quantum mechanics in the Heisenberg picture

Article metrics

Submit

Share this article

Author e-mails

Author affiliations

ORCID iDs

Dates

Peer review information

Abstract

1. Introduction

2. Geodesic and Hamiltonian dynamics—classical case

3. Geodesic and Hamiltonian dynamics—quantum case

4. Discussions

4.1. More about the quantum theory

4.2. On quantum mechanics in nontrivial gravitational background, and beyond

4.3. Against Poincaré symmetry and on-shell mass condition

4.4. Quantum coordinate transformations and quantum gravity

Acknowledgments

Data availability statement