# Equivalence Principle for Quantum Mechanics in the Heisenberg Picture

Otto C. W. Kong  Department of Physics and Center for High Energy and High
Field Physics, National Central University, Chung-li, Taiwan 32054

###### Abstract

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 $p_{\mu}$, rather than
$p^{\mu}$. 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.
Keywords : Equivalence Principle, Quantum Geodesic Equation, Heisenberg
Picture, Quantum Gravity, Noncommutative Geometry

††preprint:  NCU-HEP-k101 Aug 2023

##  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 toward the theory of quantum gravity PPV . There has been a lot of
work on the subject matter, at least since the paper by Greenberger in 1968 G
. Summaries of many available results and good lists of references are
available in Refs.ah ; PPV . 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 kind of a
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. 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’ DM  or effective geodesic equation PPV .

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 motion of constant momentum as quantum observables, i.e.
$\frac{d\hat{p}^{i}}{dt}=0$. 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
$\frac{d\hat{p}^{i}}{dt}=0$ 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.

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 in the place of the
Newtonian time 096 . For all the analyses here, that can be identified with
the particle proper time. Such a description of quantum dynamics, while
available since at least around 1940 Fo ; S  (see for example the books of
Refs.H ; F  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 JohnsJ ) and hence not give a Lorentz covariant
formulation 096 . 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 096 ; 087 , 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 078  as a noncommutative geometry C ; Ma . 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 Ha .

##  II Geodesic and Hamiltonian Dynamics – Classical Case

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

$\displaystyle
S_{o}=\int\\!\\!dsL_{o}(s)=\int\\!\\!ds\sqrt{-g_{\mu\nu}(x)\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}}\;,$
(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 $p^{\mu}$. 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

$\displaystyle\frac{dF}{ds}=\\{F,H\\}=\frac{\partial H}{\partial
p_{\mu}}\frac{\partial F}{\partial x^{\mu}}-\frac{\partial F}{\partial
p_{\mu}}\frac{\partial H}{\partial x^{\mu}}\;,$ (2)

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

$\displaystyle\\{x^{\mu},x^{\nu}\\}=\\{p_{\mu},p_{\nu}\\}=0\;,\qquad\\{x^{\mu},p_{\nu}\\}=\delta^{\mu}_{\nu}\;.$
(3)

Note that in the presence of a nontrivial $g_{\mu\nu}$, $x^{\mu}$ are not
components of a four-vector, but $p_{\mu}$ are. We avoid writing $x_{\mu}$
here but $p^{\mu}=g^{\mu\nu}p_{\nu}$ are well-defined. Yet, even for canonical
$p_{\mu}$, while one has $\\{x^{\mu},p^{\nu}\\}=g^{\mu\nu}$,
$\\{p^{\mu},p^{\nu}\\}$ are generally nonzero. For a generic Riemannian
manifold with $x^{\mu}$ 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 variable, we have the Poisson bracket given by

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

with the canonical condition given by

$\displaystyle{\mathcal{P}}^{\mu\nu}={\mathcal{P}}_{\tilde{\mu}\tilde{\nu}}=0\;,\qquad{\mathcal{P}}^{\mu}_{\tilde{\nu}}=\delta^{\mu}_{\nu}\;,$
(5)

with indices $\mu$ and $\nu$ referring to the position coordinates and
$\tilde{\mu}$ and $\tilde{\nu}$ 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
$\rho(\tau)=0$). In terms of the classical metric

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

with

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

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

$\displaystyle{H}_{\\!a}(s)=\eta^{\mu\nu}\frac{p_{\mu}p_{\nu}}{2m}-\frac{ma^{2}}{2c^{2}}\eta_{\mu\nu}x^{\mu}x^{\nu}\;,$
(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 $\tau$ 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 $\tau=s$. 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,
of course, 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

$\displaystyle p^{ct}=\frac{\partial ct}{\partial\rho}p^{\rho}+\frac{\partial
ct}{\partial c\tau}p^{c\tau}\;,\qquad p^{x}=\frac{\partial
x}{\partial\rho}p^{\rho}+\frac{\partial x}{\partial c\tau}p^{c\tau}\;.$ (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 $p^{\prime}_{\mu}dx^{\prime\mu}=p_{\nu}dx^{\nu}$, 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 $\rho=0$ 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 Eq.(7).
We have $\tau=s$ and $\rho$ and $p^{\rho}$ maintaining their vanishing values,
while $p^{c\tau}=mc$, 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 $g_{\mu\nu}$
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.

##  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 obtained from the Schrödinger
representation with $\hat{x}^{\prime\mu}={x}^{\prime\mu}$ and
$\hat{p}^{\prime}_{\mu}=-i\partial_{\mu}$, taking $\hbar=1$, but is only a
consequence of the commutation relation among the operators, or rather just as
abstract quantum observables. 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
$g_{\mu\nu}$. The exact quantum Poisson bracket is $-i[\cdot,\cdot]$, giving
the canonical condition as

$\displaystyle-i[\hat{x}^{\prime\mu},\hat{x}^{\prime\nu}]=-i[\hat{p}^{\prime}_{\mu},\hat{p}^{\prime}_{\nu}]=0\;,\qquad-i[\hat{x}^{\prime\mu},\hat{p}^{\prime}_{\nu}]=\delta^{\mu}_{\nu}\;,$
(10)

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 $\hat{x}^{\mu}$ and $\hat{p}_{\mu}$, for
the case of the Minkowski metric, they can be directly derived but promoting
the classical coordinate transformation to one among the quantum observables
(which can be seen as noncommutative coordinates of the quantum phase space
078 ). 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,

$\displaystyle\hat{H}_{\\!a}=-\frac{c^{4}}{(c^{2}+a\hat{\rho})^{2}}\frac{(\hat{p}_{c\tau})^{2}}{2m}+\frac{(\hat{p}_{\rho})^{2}}{2m}-\epsilon\frac{m(c^{2}+a\hat{\rho})^{2}}{2c^{2}}\;.$
(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 $\epsilon$ for the
easy tracing of the exact free case with the geodesic in the equations of
motion. That is, $\epsilon=1$ gives the exact quantum dynamics of what is
sketched in the last section, while $\epsilon=0$ gives the geodesic motion.
The Heisenberg equations of motion we are interested in are given by

$\displaystyle\frac{d\hat{p}_{c\tau}}{ds}=0\;,\qquad\frac{d\hat{p}^{\rho}}{ds}=\frac{d\hat{p}_{\rho}}{ds}=-\frac{a(c^{2}+a\hat{\rho})}{mc^{4}}(\hat{p}^{c\tau})^{2}+\epsilon\frac{ma(c^{2}+a\hat{\rho})}{c^{2}}\;,$
(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,

$\displaystyle\frac{d\hat{p}^{c\tau}}{ds}=-i[\hat{p}^{c\tau},\hat{H}_{\\!a}]=-\frac{a}{c^{2}+a\hat{\rho}}\frac{\hat{p}^{c\tau}\hat{p}^{\rho}}{m}-\frac{\hat{p}^{\rho}\hat{p}^{c\tau}}{m}\frac{a}{c^{2}+a\hat{\rho}}$
(13)

as the quantum version of the classical geodesic one. Note 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

$\displaystyle\frac{d^{2}{c\hat{\tau}}}{ds^{2}}+\frac{a}{c^{2}+a\hat{\rho}}\frac{dc\hat{\tau}}{ds}\frac{d\hat{\rho}}{ds}+\frac{d\hat{\rho}}{ds}\frac{dc\hat{\tau}}{ds}\frac{a}{c^{2}+a\hat{\rho}}=0\;,$
$\displaystyle\frac{d^{2}\hat{\rho}}{ds^{2}}+\frac{a(c^{2}+a\hat{\rho})}{c^{4}}\frac{dc\hat{\tau}}{ds}\frac{dc\hat{\tau}}{ds}-\epsilon\frac{a(c^{2}+a\hat{\rho})}{c^{2}}=0\;.$
(14)

Again, the $\epsilon=0$ case is the free motion, i.e. quantum geodesic
equations.

##  IV Discussions

###  IV.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, 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
$\hat{\tau}$), 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 qrf .
This has very important implications for the EP and quantum gravity Ha  beyond
the classical coordinate transformation analyzed here, to be discussed more
below.

Our Heisenberg picture analysis relies only on the commutation 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 not 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 $\beta$
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 cmp ; 078

$\displaystyle
f_{\\!\scriptscriptstyle\beta}\star_{\\!\scriptscriptstyle\kappa}f_{\\!\scriptscriptstyle\gamma}=f_{\\!\scriptscriptstyle\beta\gamma}\;.$
(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 078 ; 081
. 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 $Y_{\mu}$ and $P_{\mu}$
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 096 . An abstract Fock
state basis as well as coherent state wavefunction description of that has
been essentially presented 096 ; 087 . 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 Bo . In fact, it is defined in terms of
Pauli’s metric operator P ; D ; M  that is for the case Minkowski, denoted by
$\hat{\eta}$. The position and momentum operators are exactly
$\eta$-Hermitian, i.e. satisfying

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

$\displaystyle\prescript{}{{}_{g}}{\\!\left\langle\cdot|\beta^{{\dagger}^{g}}\cdot\right\rangle}=\prescript{}{g}{\\!\left\langle\beta\cdot|\cdot\right\rangle}\;.$
(17)

$g$-Hermitian operators then generate one-parameter groups of (pseudo)-unitary
transformations that preserve the inner product. For our ‘relativistic’
quantum mechanics, in particular, we have
$\hat{\eta}\beta^{{\dagger}^{\eta}}_{\mu}\hat{\eta}^{-1}=\eta_{\mu\nu}\beta^{\nu}$
for any four-vector observable which reduces to
$\hat{\eta}\beta_{\mu}\hat{\eta}^{-1}=\eta_{\mu\nu}\beta^{\nu}$ for the
$\eta$-Hermitian physical observables such as the position and momentum
observables. 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.

###  IV.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 H ; F , 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

$\displaystyle\int\\!d^{4}x\psi_{\eta}^{*}(x)\phi(x)=\int\\!d^{4}x[\hat{\eta}_{\scriptscriptstyle
S}\psi(x)]^{*}\phi(x)\;,$ (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 Eq.(7) and Eq.(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

$\displaystyle\int\\!d^{4}x^{\prime}\left|\frac{\partial x}{\partial
x^{\prime}}\right|[\hat{\eta}_{\scriptscriptstyle
S}\psi^{\prime}(x^{\prime})]^{*}\phi^{\prime}(x^{\prime})=\int\\!d^{4}x^{\prime}\left(1+\frac{a\rho}{c^{2}}\right)[\hat{\eta}_{\scriptscriptstyle
S}\psi^{\prime}(x^{\prime})]^{*}\phi^{\prime}(x^{\prime})\;.$ (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

$\displaystyle\int\\!d^{4}x^{\prime}\left(1+\frac{a\rho}{c^{2}}\right)\hat{\eta}_{\scriptscriptstyle
S}(x)\psi^{\prime}(x^{\prime})^{*}\phi^{\prime}(x^{\prime})=\int\\!d^{4}x^{\prime}[\hat{g}_{\scriptscriptstyle
S}(x^{\prime})\psi^{\prime}(x^{\prime})]^{*}\phi^{\prime}(x^{\prime})\;.$ (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 III above, we have implicitly taken
components of the metric tensor $g_{\mu\nu}$, or its inverse, as functions of
the position observables $\hat{x}^{\prime}$. In the Schrödinger
representation, $\hat{x}^{\prime}$ of course are just $x^{\prime}$. 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 Ref.PPV  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. 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 $\hat{x}^{\prime}$
part only, though that is naturally doubled with a copy for the
$\hat{p}^{\prime}$ 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’ L . 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 DH . 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 097 . 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.

###  IV.3 Against Poincaré symmetry and on-shell mass condition

So long as obtaining the geodesic equation is concerned, the Hamiltonian
approach if the same as the Lagrangian one which is just about minimizing the
action $\int\\!\\!dsL$, or $\int\\!\\!ds\hat{L}$. 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 $-p_{\mu}p^{\mu}=m_{\\!\scriptscriptstyle
E}^{2}c^{2}$ is not necessarily respected; it generally allows
$-p_{\mu}p^{\mu}$ 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 H ; F . 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 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.096 . Note that any quantum theory with
wavefunctions as functions fo otherwise free Minkowski four-vector variables
$x^{\mu}$ and $p_{\mu}$ as $-i(\hbar)\partial_{\mu}$ 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, the canonical momentum $p^{\mu}$ for the charged particle is no
longer conserved. We do not even have $p^{\mu}=m\frac{dx^{\mu}}{d\tau}$.
Einstein was well aware of that only the canonical momentum is the generally
important quantity and it does not necessarily obey any on-shell mass
condition E . 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.

###  IV.4 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 fullly 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 qrf . One should take the gravitational acceleration of
the quantum particle seen in the Minkowski frame as what it should be – a
quantum observable $\hat{a}$. 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 $\hat{a}$ has nontrivial operator ordering issues
between $\hat{a}$ and the position observables $\hat{\rho}$ and $\hat{\tau}$
to be resolved.

While the idea of quantum frames of reference is not new qf1 ; qf2 , a more
more recent study qrf  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
$e^{i\hat{x}_{\\!\scriptscriptstyle A}\hat{p}_{\scriptscriptstyle B}}$. The
latter can be read naively as $\hat{p}_{\scriptscriptstyle B}$ generating a
translation of $\hat{x}_{\scriptscriptstyle B}$ by ‘an amount’
$\hat{x}_{\\!\scriptscriptstyle A}$ giving the position observable for
particle $B$ in the reference frame of particle $A$ as
$\hat{x}_{\\!\scriptscriptstyle B}-\hat{x}_{\\!\scriptscriptstyle A}$. 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
$\hat{x}_{\scriptscriptstyle A}$ and $\hat{x}_{\scriptscriptstyle B}$. 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 $\hat{x}_{\scriptscriptstyle
B}-\hat{x}_{\scriptscriptstyle A}$, seen in the original frame. One can also
startswith 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
$e^{i\hat{x}_{\scriptscriptstyle A}\hat{p}_{\scriptscriptstyle B}}$ is a
canonical transformation, both seen from the Hilbert space point of view or
that of the noncommutative symplectic geometry of $\hat{x}$-$\hat{p}$. The
transformation as given also serves as a momentum translation of
$\hat{p}_{\scriptscriptstyle A}$ generated by the operator
$\hat{x}_{\scriptscriptstyle A}$ by ‘an amount’ $-\hat{p}_{\scriptscriptstyle
B}$, hence maintaining the canonical condition among all pairs of position and
momentum observables under the different frames.

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 HardyHa . 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 081 ; 079 , 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 093 . 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
097 . 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 $\hat{x}_{\scriptscriptstyle A}$ that the quantum
spatial transformation discussed above translates an
$\hat{x}_{\scriptscriptstyle B}$ with all notions about changes in Heisenberg
uncertainty and even entanglement successfully described 093 . 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.

## Acknowledgement

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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