Quantum mechanics in curved space(time) is a theory that sits between the well-established theory of basic quantum mechanics and the very challenging theory of quantum gravity. Classical gravity is encoded in the metric, which is firstly given geometrically as what defines an inner product among the (tangent) vectors, and then covectors, independent of their coordinate components. A fundamental notion of a physical vector quantity is that it has an invariant magnitude square, as the inner product with itself. It is firstly through the invariant magnitude square of the physical momentum vector , that gravity, or curvature of space(time) affects the dynamics of a particle. The free particle Hamiltonian is  which gives motion along geodesics as solutions. In our opinion, the role of the metric is the core of its physical meaning in the particle theory. Our formalism successfully maintains those features and accommodates an exact Weak Equivalence Principle we illustrated in a previous publication [1]. Apparently, the formalism is incompatible with the Schrödinger wavefunction representation in general. We contrast our formalism against the alternative available [2], based on the latter approach, and discuss why that is considered theoretically less desirable, especially as it loses the notion of physical vector quantities with invariant magnitude square.

We have presented the Weak Equivalence Principle for a quantum particle under a constant gravitational field as an exact quantum version of the classical results, in the language of covariant Hamiltonian dynamics in the Heisenberg picture [1]. There is no operator ordering ambiguity going from the classical to the quantum formulation for that problem even in terms of the position and momentum observables of the accelerated frame (with Kittel–Möller position coordinates). Our Hamiltonian formulation is a consistent mathematical description of the classical and quantum geodesic problem as a variational problem. The quantum geodesic equations as Heisenberg equations of motion for a free particle, in the accelerated frame, are exact operator analogs of the classical ones, with nontrivial operator ordering consistent with the following generic form of the vanishing of the expression  given, in the same form as the general classical case, by Note that the Christoffel symbols are functions of the commuting position observables, yet they generally fail to commute with the velocity observables of . Whether the velocity observables commute among themselves is another question. The quantum Poisson bracket is the commutator divided by . The canonical moment components are , and  can have nontrivial Poisson brackets among themselves. The present article presents the general picture of quantum mechanics in a curved space(time) with any expression for the metric tensor with components as arbitrary functions of the position observables. An expression of the quantum geodesic equation in terms of Christoffel symbols is, however, not feasible in the final results.

We first focus, however, on the ‘nonrelativistic’ case, i.e. for a metric with an Euclidean signature, in three dimensions. Basically, we are looking into the problem of a consistent formulation of quantum mechanics in a curved space. In our opinion, the problem is of fundamental importance and should be seen as a necessary foundation for any theory of quantum gravity, but has not received the attention it deserves. Apparently, only a 1952 paper from DeWitt [2] addresses it seriously, though it is only explicitly named as a presentation of the ‘nonrelativistic’ dynamics under a generic position coordinate picture. The work has not been discussed often enough in the literature of the subject matter, not to mention being critically reexamined. DeWitt’s starting point is the Schrödinger wavefunction representation, which is the opposite end from our perspectives. The formalism, in our opinion and as illustrated below, has a few less-than-desirable basic features and a quite undesirable result. Most importantly, one cannot have the velocity and momentum observables as vector quantities, having a notion of an invariant magnitude. In that case, we see the metric as losing most, if not all, of its physical meaning in the particle theory. Physics is, of course, about getting the ‘correct’ theories, ones that successfully describe Nature as probed through observations and experiments in the relevant domains. Yet, within the confine of the latter, most theorists would see it right to go after ‘simplicity and beauty’, so to speak. This is mostly about the conceptual picture and the mathematical form of the theories, probably also some of their key results. An unifying feature is an important example. It is more desirable to see unifying principles and themes among different theories. For the subject matter at hand, it is more desirable to find principles and features shared by our classical gravitational theory and quantum theory. Those should be seen as the more reliable starting point for pursuing the theory of quantum gravity. Most would also welcome a quantum theory the metric observable of which maintains more of its key features in the classical theory.

Let us recall the basic perspectives of our approach to the problem and quantum theories in general, emphasizing how that may share common themes and principles with the theory of General Relativity (GR). We emphasize a point of view based firstly on the observables. They are the physical quantities, period. It is not difficult to see mathematical relations among the quantum observables would directly give the corresponding classical results when the observables are taken as classical ones. Recovery of the equation of motion for a classical state from the Schrödinger equation, is a less trivial business. The two certainly do not share the same form. Our second key theme is dynamics as given under a Hamiltonian formulation. From the formulation, apart from the identification of the physical Hamiltonian among generic Hamiltonian functions, the mathematical structure of the system is fully dictated by the symplectic geometry of the phase space. So, we have dynamics from geometry, as in GR. The phase space is mostly conveniently described in terms of canonical coordinates. For a (spin zero) particle, they are the canonical pairs of position and momentum variables. For the real variables in classical mechanics, we have the basic Poisson bracket relations The expressions are independent of even the existence, not to say the form, of a metric tensor. Note that the generic Poisson bracket relations can be verified through an implementation of a transformation of the position coordinates from the simple exact Euclidean ones. The particle phase space can be seen as the cotangent bundle of the configuration/position space, hence having  as components of the momentum vector as firstly a covector, though  as generic position coordinates in a curved space cannot be taken as components of any vector. The cotangent bundle of a manifold, with or without a Riemannian structure, has a natural symplectic geometry. The symplectic structure gives curves of Hamiltonian flow for a generic Hamiltonian function  as integral curves of the Hamiltonian vector field  where the mathematical nature of the flow parameter  is dictated by . For classical mechanics within the formulation, the only proper way to look at the Newtonian time is that it is such a parameter for physical Hamiltonian. The Hamiltonian flow is then to be interpreted as the physical time evolution. While the position variables as part of the phase space coordinates are part of the kinematic set-up of the theory, with their validity subjected to the validity of the theory, time is completely about the dynamics of specific systems. The picture of Newtonian space–time as a coset space of the Galileo group, and the exact idea of the latter as the relativity symmetry behind Newtonian mechanics is not justified and indeed incorrect [3]. Though the formulation as given through Newton’s Laws requires a model for space and time as the starting point, Galilean time translation can only give the correct picture for a particle when it is free, but not when it has nontrivial dynamics. From the point of view of Hamiltonian mechanics, relativity symmetry should be about reference frame transformations for the phase space.

Here, we are talking about the metric tensor in the usual sense, i.e. as one firstly for the configuration space. In the case of a classical particle, the configuration space is our physical space or rather our classical model of it. We want to emphasize the logic that it is from our successful theory of particle dynamics that we can retrieve from it a successful model of our physical space. That is to be seen as the totality of all possible positions of a particle. Any otherwise notion of (empty) space in itself is not physical. For quantum mechanics, we see no justification to maintain the idea that the classical model of our physical space is the proper model to look at Nature as described by the theory. The phase space for a quantum particle certainly does not have the three-dimensional real manifold as a configuration subspace. We have presented a picture of the quantum phase space as a noncommutative symplectic geometry with the quantum position and momentum observables as canonical coordinates [4]. The Poisson bracket is the standard one, the commutator divided by . From the discussion above, one would conclude that the picture should remain valid with the introduction of a nontrivial metric with components as functions of the position observables. Hence, we should have quantum position and momentum observables satisfying the canonical Poisson bracket conditions of Eq. (2). In the general mathematical theory of noncommutative geometry [5], [6], [7], [8], Riemannian structure has been analyzed [7]. While the language in the latter literature hardly fits our purpose, and it is not trivial to see if there could be such a picture (of the so-called spectral geometry) for the position subspace of our quantum phase space, some common features to the theory of quantum mechanics we are seeking here is worth attention. Connes [7] has geodesic flows as a one-parameter group of automorphism on the (observable) algebra, with a generator as essentially the magnitude square of a cotangent vector. Such a group of automorphism is exactly one of unitary transformation in the Schrödinger picture. They are Hamiltonian, with a generator as the free particle Hamiltonian proportional to the magnitude of the canonical momentum vector, an exact analog of the classical, commutative, case of . As the readers will see, all the features discussed are what we want, and have successfully retained in our formulation presented below.

We introduce next the notion of Pauli metric operator (PMO), denoted by . In the classical geometric picture of the quantum phase space as the (projective) Hilbert space, it is a Kähler manifold with the metric structure tied to the symplectic structure. The feature cannot be maintained in a noncommutative geometric picture, as the ‘coordinate transformation’ between the two pictures does not respect the complex structure [4]. The two pictures are essentially the Schrödinger and the Heisenberg pictures. Following a study by Dirac [9], basically trying to put a notion of Minkowski metric onto a vector space of quantum states, Pauli introduced the metric operator that defines the inner product as  [10], that is also the key feature behind the more recent notion of pseudo-Hermitian quantum mechanics [11]. When the PMO is trivial, we have the usual Hilbert space for quantum mechanics, in flat space. What is important to note is that the proper notion of operator Hermiticity for an inner product space is to be defined relative to the inner product. Explicitly, with the notation , the Hermitian conjugate of operator  relative to the inner product is given by the operator  satisfying the equation. In terms of the naive Hermitian conjugate for the case with a trivial metric, which we denote by the usual , we have Note that while we can have many different inner products introduced for a vector space, only one can be considered the physical inner product, as there is one physical metric. In a theory of ‘relativistic’ quantum mechanics [12], we have illustrated an explicit notion of a Minkowski metric of the vector space of state having a Minkowski PMO  that is directly connected to a naive notion of a Minkowski metric tensor in the noncommutative geometric picture. We have position and momentum observables  and  being -Hermitian satisfying  and , with . In the naive Schrödinger representation which Pauli mostly worked with, for the ‘nonrelativistic’ case, we have  as , from the invariant volume element [13]. The key point is that the notion of Hermiticity may generally be metric-dependent, hence dependent on the choice of coordinates. In the spirit of GR, the choice of frame of reference with the coordinates as part of it is like a gauge choice. The coordinate observables are then gauge-dependent quantities. We are using the term observables in the more general sense as in the literature of mathematical physics, without requiring them to be Hermitian in any sense. We use the term physical observables for observables as described in basic quantum mechanics textbooks, hence Hermitian. There is nothing so unphysical about a non-Hermitian element of the observable algebra though. A simple product of two Hermitian observables which do not commute would not be Hermitian. As to be presented below, we have various notions of Hermiticity on the same quantum observable algebra, each defined by a PMO, referred to as -Hermiticity, relevant in a different coordinate picture of the related noncommutative geometry. Now, if what can be considered a physical (gauge-dependent) observable depends on the gauge choice, it may not be considered unreasonable.

Another important feature of the noncommutative geometric picture we want to mention here is the notion of quantum reference frame transformations [14], [15]. We are concerned here with how to think about the dynamical theory from different systems of coordinates or frames of reference. The focus is on a generic transformation of the quantum position coordinates as , as the exact parallel of what we have in classical physics. The position observables are taken as functions of the old ones in the usual classical sense. Then all the position observables old and new commute with one another. When we talked about going to an accelerated frame in Ref. [1], for example, the relative acceleration between the old and new frames is essentially taken as a classical quantity. It was assumed at least to commute with the position observables. Such coordinate transformations are classical in nature, though they have quite nontrivial implications on momentum observables as quantum quantities. A full consideration of the free-falling frame of a quantum particle needs to consider the gravitational acceleration as a quantum quantity. Solving the problem may be still quite a challenge to be taken. In a quantum spatial translation position eigenstates are not necessarily taken to position eigenstates, as well illustrated in Refs. [14], [15]. In particular, we have presented a way to look at a simple quantum spatial translation, as needed for the description of the position of a quantum particle relative to another quantum particle in a generic state. In terms of the full information about the quantum position of the particles seen in the original frame, an explicit description of the quantum quantities about the exact ‘amounts’ the position observables of the other particle differs between the old and new frames have been presented [15]. Heisenberg uncertainties in the position observables and related features of entanglement changes are fully encoded in those exact ‘amounts’. The important relevancy of quantum reference frame transformations to a theory of quantum gravity has been addressed by Hardy [16]. After all, the Relativity Principle has a fundamental role in GR. Nature does not use any frame of reference and is not expected to prefer anyone over another. Any practically defined frame of reference has a physical nature and hence relies on the physical properties of matter that are quantum in nature. Penrose has emphasized a supposed incompatibility between quantum mechanics and the Relativity Principle [17]. Essentially, it is the observation that no reference frame transformation can take a position eigenstate to one that is a nontrivial linear combination of such eigenstates. However, that is true only when we limit ourselves to transformations that are classical in nature. Otherwise, it can easily be done with a simple quantum translation. A particle in a position eigenstate would not be in an eigenstate of its position as observed from another, observing, particle that in itself is not in an eigenstate. A quantum translation from the original frame to the frame of the observing particle would change the nature of the first, observed, particle as a position eigenstate or not in a way dependent on the corresponding properties of the second particle. Such reference frame transformations cannot be pictured in any classical/commutative geometry. But they are natural in our noncommutative geometric picture [4].

In the next section, we discuss the various key issues from the perspective of noncommutative symplectic geometry for the phase space corresponding to the quantum observable algebra. DeWitt’s formalism is also sketched to illustrate what we see as limitations and unpleasant features of it. Analyzing all that sets a clear platform for the appreciation of the key features of our formalism, which is presented in Section 3, focusing on the case of Euclidean metric signature. The corresponding picture for the case with Minkowski metric signature incorporating our results of Ref. [1] is addressed in the section after. Concluding remarks are presented in the last section.