The Madelung Picture as a Foundation of Geometric Quantum Theory

The Madelung Picture as a Foundation of Geometric Quantum Theory

Abstract

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem.
In this article we argue that a quantum theory recursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we show that the Schrödinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by E. Madelung, naturally ground the Schrödinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modeling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.

\dedication

Keywords: Geometric Quantization   -   Interpretation of Quantum Mechanics   -   Geometric Quantum Theory   -   Madelung Equations   -   Classical Limit

1 Introductory Discussion

1.1 Critique of Quantization and a new Methodology

The idea of quantization was first put forward by Dirac [Dirac] in 1925 in an attempt to extend Heisenberg’s theory of matrix mechanics [Heisenberg]. He based the concept on a formal analogy between the Hamilton and the Heisenberg equation and on the principle of correspondence, namely that a quantum theoretical model should yield a “classical” one in some limit. This analogy motivated Dirac to develop a scheme that constructs one or more quantum analogues of a given “classical system” formulated in the language of Hamiltonian mechanics.1 When it was discovered that Dirac’s scheme, nowadays known as canonical quantization, was ill-defined (see [Groenewold, Hove] for the original works by Groenewold and van Hove, also [Abraham]*§5.4, in particular [Abraham]*Thm. 5.4.9), physicists and mathematicians attempted to develop a more sophisticated machinery rather than questioning the ansatz. The result has been a variety of quantization algorithms, one of which is particularly noteworthy: Geometric quantization (cf. [Woodhouse, Enriquez] for an introduction).

In his seminal paper, Segal [Segal] expressed the need to employ the language of differential geometry in quantum theory. He understood that determining the relevant differential-geometric structures, spaces and their relation to the fundamental equations of quantum theory creates the mathematical coherence necessary to adequately address foundational issues in the subject. By merging this ansatz with Kirillov’s work in representation theory [Kirillov], Segal, Kostant [Kostant] and Souriau [Souriau] were able to construct the algorithm of geometric quantization. However, rather than elaborating on the relation between quantum and classical mechanics, geometric quantization unearthed a large amount of geometric structures [Hall]*§23.2, introduced in an ad hoc manner.

It is tempting to blame this state of affairs on the inadequacy of the geometric ansatz or the theory, but instead we invite the reader to take a step back. What is the reason for the construction of a quantization algorithm? Why do we quantize? Certainly, quantum mechanics should agree with Newtonian mechanics in some approximation, where the latter is known to accord with experiment, but is it reasonable to assume the existence of an algorithm that constructs the new theory out of the old one?

These questions are of philosophical nature and it is useful to address them within the historical context. Clearly, the step from Newtonian mechanics to quantum mechanics was a scientific revolution, which is why we find the work of the philosopher and physicist Thomas Kuhn [Kuhn] of relevance to our discussion. Kuhn is known for his book “The Structure of Scientific Revolutions” [Kuhn], in which he analyzed the steps of scientific progress in the natural sciences. For a summary see [Pajares].

Kuhn argues that, as a field of science develops, a paradigm is eventually formed through which all empirical data is interpreted. As, however, the empirical evidence becomes increasingly incompatible with the paradigm, it is modified in an ad hoc manner in order to allow for progress in the field. Ultimately, this creates a crisis, as attempts to account for the evidence become increasingly ad hoc, unmanageably elaborate and ultimately contradictory. Unless a new paradigm is presented and withstands experimental and theoretical scrutiny, the crisis persists and deepens, because of the internal and external inconsistencies of the current paradigm.

This process can be directly observed in the history of quantum theory. When Newtonian mechanics was faced with the problem of describing the atomic spectra and the stability of the atom in the beginning of the twentieth century [Bohr1], it was ad hoc modified by adding the Bohr-Sommerfeld quantization condition [Bohr1, Sommerfeld], despite its known inconsistency with then accepted principles of physics [Bohr2, Heisenberg1]. This ad hoc modification of Newtonian mechanics continued with Werner Heisenberg’s [Heisenberg] and Erwin Schrödinger’s [Schroedinger1] postulation of their fundamental equations of quantum mechanics, two descriptions later shown to be formally equivalent by von Neumann in his constitutive work [Neumann]. Schrödinger’s and Heisenberg’s description can be viewed as an ad hoc modification, because their equations are formulated on a Newtonian spacetime and intended to replace Newton’s second law without being based on postulated principles of nature. With his quantization algorithm [Dirac], Dirac supplied a convenient way to pass from the mathematical description of a physical system in Newtonian mechanics to the then incomplete, new theory. In accordance with Kuhn’s description, it was a pragmatic, ad hoc step, not one rooted in deep philosophical reflection. Nonetheless, the concept of quantization is ingrained in quantum theory as of today [Weinberg0], while the as of now futile search for unity in physics has become increasingly ad hoc and elaborate [Woit]*§19.

We are thus reminded of our historical position and the original intention behind quantization: We would like to be able to mathematically describe microscopic phenomena, having at hand neither the fundamental equations describing those phenomena nor a proper understanding of the physical principles involved allowing us to derive such equations. That is, what we lack with respect to our knowledge of microscopic phenomena is, in Kuhn’s words, a paradigm. Rather than having a set of principles of nature, which we use to intuitively understand and derive the fundamental laws of quantum theory, we physicists assume the validity of the old theory, namely Newtonian mechanics or special relativity in its Hamiltonian formulation, only to apply an ad hoc algorithm to obtain laws we have inadequately understood. This is why the concept of quantization itself is objectionable.

Indeed, even if a mathematically well defined quantization scheme existed, it would remain an ad hoc procedure and one would still need additional knowledge which quantized systems are physical (cf. [Waldmann]*§5.1.2 for a discussion of this in German). From a theory builder’s perspective, it would then be more favorable to simply use the quantized, physically correct models as a theoretical basis and deduce the classical models out of these, rather than formulating the theory in the reverse way. Hence quantization can be viewed as a procedure invented to systematically guess quantum-theoretical models. This is done with the implicit expectation of shedding some light on the conceptual and mathematical problems of quantum theory, so that one day a theory can be deduced from first principles. Thus a quantum theory, which is constructed from a quantization scheme, must necessarily be incomplete. More precisely, it has not been formulated as a closed entity, since for its formulation it requires the theory it attempts to replace and which it potentially contradicts.

As a result of this development, quantum mechanics and thus quantum theory as a whole has not been able to pass beyond its status as an ad hoc modification of Newtonian mechanics and relativity to date. For a recapitulation of the history of quantum theory illustrating this point, see e.g. the article by Heisenberg [Heisenberg1].

Fortunately, our criticism does not apply to the theory of relativity, which to our knowledge provides an accurate description of phenomena [Will], at least in the macroscopic realm. As the principles of relativity theory are known (cf. [Kriele]*p. XVII), the ridiculousness of “relativizing” Newtonian mechanics is obvious. Indeed, in the theory of relativity physics still finds a working paradigm.

Rejecting quantization neither leads to a rejection of quantum theory itself, nor does it imply that previous attempts to put quantum theory into a geometric language were futile. If we reject quantization, we are forced to view quantum theory as incomplete and phenomenological, which raises the question of what the underlying physical principles and observables are. Considering that the theory of relativity is mainly a theory of spacetime geometry, asking, as Segal did, for the primary geometric and physical quantities in quantum theory offers a promising and natural approach to this question.

Therefore, we reason that we theorists should look at the equations of quantum theory with strong empirical support and use these to construct a mathematically consistent, probabilistic, geometric theory, tied to fundamental physical principles as closely as possible. But how is this to be approached?

1.2 The Madelung Equations as a Geometric Ansatz

In the year 1926, the same year Schrödinger published his famous articles [Schroedinger1, Schroedinger2, Schroedinger3], the German physicist Erwin Madelung reformulated the Schrödinger equation into a set of real, non-linear partial differential equations [Madelung] with strong resemblance to the Euler equations [Chorin]*§1.1 found in hydrodynamics. The so-called Madelung equations are2

(1.1)
(1.2)
(1.3)

where is the mass of the particle, is a real vector field, called the drift (velocity) field, is the probability density (by an abuse of terminology), the external force and denotes the so-called material derivative (cf. [Chorin]*p. 4) of along itself. Madelung already believed3 that these equations could serve as a foundation of quantum theory. He reached this conclusion, because the equations exhibit a strong link between quantum mechanics and Newtonian continuum mechanics [Madelung]. Thus Madelung used these equations to interpret quantum behavior by exploiting the analogy to the Euler equations. At this point in history, it was not clear how to interpret the wave function as the Born rule and the ensemble interpretation had just recently emerged [Born]. Madelung’s misinterpretation of quantum mechanics may perhaps be the reason why it took almost 25 years for his approach to become popular again, when Bohm employed the Madelung equations to develop what is now known as Bohmian mechanics [Bohm1, Bohm2]. Nonetheless a clear distinction should be drawn [Tsekov2] between the Madelung equations and the Bohmian theory [Bohm1, Bohm2]. Despite the popularity of Bohm’s approach, a discussion of the Madelung equations on their own [Holland1, Janossy1, Janossy2, Janossy3, Takabayasi, Wallstrom] seems less common.

Today, the importance of the Madelung equations lies in the fact that they naturally generalize the Schrödinger equation and in doing so expose the sought-after geometric structures of quantum theory and its classical limit. As a byproduct, one obtains a natural answer to the question why complex numbers arise in quantum mechanics. The Madelung equations, by their virtue of being formulated in the language of Newtonian mechanics, make it possible to construct a wide class of quantum theories by making the same coordinate-independent modifications found in Newtonian mechanics, without any need to construct a quantization algorithm as, for example, in geometric [Woodhouse, Enriquez] and deformation quantization [Waldmann]. This greatly simplifies the construction of new quantum theories and therefore makes the Madelung equations the natural foundation of quantum mechanics and the natural ansatz for any attempts of interpreting quantum mechanics.

For some of these modifications it is not possible to construct a Schrödinger equation and for others the Schrödinger equation becomes non-linear, which suggests that there exist quantum-mechanical models that cannot be formulated in the language of linear operators acting on a vector space of functions. From a conceptual point of view, this might prove to be a necessity to remove the mathematical and conceptual problems that plague relativistic quantum theory today or at least expose the origins of these problems. In fact, the Madelung equations admit a straight-forward (general-)relativistic generalization leading to the Klein-Gordon equation, which is, however, not discussed here and arguably unphysical.4 The Madelung equations and their modifications are henceforth particularly suited for studying quantum theory from the differential-geometric perspective. We thus believe that they will take a central role both in the future construction of an internally consistent, geometric quantum theory as well as the realist understanding of microscopic phenomena.

1.3 Outline and Conventions

In this article we formalize the Madelung picture of quantum mechanics and thus provide a rigorous framework for further development. A first step is made by postulating a modification intended to model particle creation and annihilation. In addition, we give a possible interpretation of quantum mechanics that is an extension of the stochastic interpretation developed by Tsekov [Tsekov1], which in turn originated in ideas from Bohm and Vigier [Bohm3] in the 1950s.

Our article is organized as follows: We first construct a spacetime model on which to formulate the Madelung equations using relativistic considerations. In section 3 on page 3, we further motivate the need for the Madelung equations in the formulation of quantum mechanics and then give a theorem stating the equivalence of the Madelung equations and the Schrödinger equation, if the force is irrotational and a certain topological condition is satisfied. We also address concerns raised in the literature \citelist[Takabayasi] [Wallstrom0] [Wallstrom] [Holland]*§3.2.2 regarding this point. We introduce some terminology and proceed with a basic, mathematical discussion. In section 4 on page 4, we discuss the operator formalism in the Schrödinger picture and its relation to the Madelung equations. We proceed by giving a formal interpretation of the Madelung equations in section 5.1 on page 5.1 and then speculate in section 5.2 on page 5.2 that quantum mechanical behavior originates in noise created by random irregularities in spacetime curvature, that is random, small-amplitude gravitational waves. How the violation of Bell’s inequality can be achieved in this stochastic interpretation is also discussed. In section 6 on page 6, we propose a modification of the Madelung equations, intended to model particle creation and annihilation, and show how this in general leads to a non-linearity in the Schrödinger equation. We conclude this article on page 7 with a brief review of our results including a table and an overview of some open problems.

Some prior remarks: To fully understand this article, an elementary knowledge of Riemannian geometry, relativity and quantum mechanics is required. We refer to [Rudolph]*Chap. 1-4, [Carroll, Wald] and [Ballentine], respectively. The mathematical formalism of the article is, however, not intended to deter anyone from reading it and should not be a hindrance to understanding the physics we discuss, which is not merely of relevance to mathematical physicists. For the sake of clarification, we have attempted to provide some intuitive insight along the lines of the argument. Less mathematically versed readers should skip the proofs and the more technical arguments while being aware that precise mathematical arguments are required, as intuition fails easily in a subject this far away from everyday experience. Moreover, we stress that section 5.2 should be considered fully separate from the rest of the article. At this point the stochastic interpretation, however well motivated, is speculation, but this does not invalidate the rest of the argument.

On a technical note, we usually assume that all mappings and manifolds are smooth. This assumption can be considerably relaxed in most cases, but this would lead to additional, currently unnecessary technicalities. Our notation mostly originates from [Rudolph], but is quite standard in physics or differential geometry. For example, is the pushforward and the pullback of the smooth map , is tensor contraction of adjacent entries or the Euclidean inner product, d the Cartan derivative, the Levi-Civita symbol, the Lie bracket (of vector fields), denotes the space of smooth vector fields and the space of smooth -forms on the smooth manifold , respectively. We use the Einstein summation convention and, where relativistic arguments are used, the metric signature is , which gives tangent vectors of observers positive “norm”. Definitions are indicated by italics.

2 Construction of Newtonian Spacetime

In order to be able to construct a rigorous proof of the equivalence of the Schrödinger and Madelung equations, we first construct a spacetime model suitable for our purposes. For a discussion on prerelativistic spacetimes see e.g. [Kriele]*§1.1 to §1.3 and [ArnoldV]*Chap. 1.

To describe the motion of a point mass of mass in Newtonian physics, we consider an open subset of , which has a canonical topology and smooth structure. The need to restrict oneself to open subsets of arises, for instance, from the fact that it is common for forces in Newtonian physics to diverge at the point where the source is located. We exclude such points from the manifold. For similar reasons we also allow non-connected subsets.

To be able to measure spatial distances within the Newtonian ontology, one intuitively needs a degenerate, Euclidean metric. However, this construction should obey the principle of Galilean relativity (cf. [Kriele]*Postulate 1.3.1).

Principle 1 (Galilean relativity)

For any two non-accelerating observers that move relative to each other with constant velocity all mechanical processes are the same.

Therefore, if we formulate physical laws coordinate-independently with some (degenerate) metric and attribute to it a physical reality, then all observers should measure the same distances. However, in physical terms, whether one travels some distance at constant velocity or is standing still, fully depends on the observer, hence the coordinate system chosen to describe the system. This is a deep problem within the conceptual framework of Newtonian mechanics. One way to circumvent this, is to prevent the measurement of distances for different times. For a mathematical treatment of such Neo-Newtonian or, better to say, Galilean spacetimes see [ArnoldV]*Chap. 1. A less complicated and physically more satisfying approach is to consider a Newtonian spacetime as a limiting case of a special-relativistic one. More precisely, a Newtonian spacetime is an approximative spacetime model appropriate for mechanical systems involving only small velocities relative to an inertial frame of reference and relative to the speed of light, not involving the modeling of light itself and with negligible spacetime curvature. In this relativistic ontology, the above conceptual problem does not occur, as the notion of spatial and temporal distance is made observer-dependent, which is necessary due to the phenomenon of time dilation and length contraction.

As quantum mechanics is formulated in a Newtonian/Galilean spacetime, it is consequently necessary to view it as a theory in the so-called Newtonian limit. This limit is naively defined by neglecting terms of the order in equations involving only physically measurable quantities, where is the speed corresponding to the velocity of any mass point relative to the inertial frame and is the speed of light (in vacuum). Obviously, this is not a rigorous definition, but this naive approach suffices for our purposes here. We will give a more thorough discussion of the Newtonian limit in a future work [ReddigerB0]. Also note that is dimensionless and hence the Newtonian limit is independent of the chosen system of units.

Our reasoning directly leads us to the definition of Newtonian spacetime.

Definition 2.1 (Newtonian spacetime)
A Newtonian spacetime is a tuple , where
  1. is an open subset of equipped with the standard topology and smooth structure,

  2. the time form is an exact, non-vanishing -form and the spatial metric is a symmetric, non-vanishing, covariant -tensor field, such that there exist coordinates on with

    (2.1a)

    for .

  3. the Newtonian orientation is a (smooth) -reduction of the frame bundle defined as follows (see e.g. \citelist[Rudolph]*§6.1 [Poor]*Def. 9.6 for definitions). Consider the Lie group

    (2.1b)

    the vector field , defined by

    (2.1c)
    (2.1d)

    and the -right action

    (2.1e)

    for and . Then is a -reduction of the frame bundle with the property that there exists a global frame field satisfying and for all , such that

    (2.1f)
  4. The tangent bundle is equipped with a covariant derivative, called the Newtonian derivative , which is

    1. compatible with the temporal metric :

      (2.1g)
    2. compatible with the spatial metric:

      (2.1h)
    3. torsion-free, i.e. :

      (2.1i)
The vector field is called the intrinsic observer (vector) field. An (ordered) triple of tangent vectors at some is called right-handed, if . Analogously we define right-handedness of a triple of vector fields. Coordinates satisfying (2.1a) are called Eulerian coordinates, if in addition is right-handed.

For convenience, we identify the points with their Eulerian coordinate values, s.t. . Condition ii) can be read as an integrability condition, i.e. the coordinates are chosen in accordance with the geometric structures and not vice versa. Thus the definition is coordinate-independent. The Newtonian orientation iii) is necessary in the definition to be able to mathematically distinguish a physical system modeled on a Newtonian spacetime from its mirror image. It is easy to check that (2.1c) and (2.1d) uniquely determine to be

(2.2)

so our definition of is sensible. As it is the case for ordinary orientations on manifolds, there are precisely two possible Newtonian orientations on .

Clearly, the intrinsic observer field plays a special role. Condition (2.1d) means that the time form determines the parametrization of the integral curves of the intrinsic observer field, including its “time orientation”, and condition (2.1c) means that the integral curves of the observer field have no spatial length, or, equivalently, they describe mass points at rest. Therefore, due to the existence of a “preferred rest frame”, Principle 1 is actually violated in Definition 2.1, if one does not consider a Newtonian spacetime as the limiting case of a special relativistic model for a particular observer. Mathematically this is captured by the fact that Galilei boosts are not spatial isometries of a Newtonian spacetime, i.e. isometries with respect to the degenerate spatial metric. Within the special relativistic ontology, however, the Lorentz boosts are isometries of the physical spacetime and we can find a Newtonian spacetime corresponding to the boost by taking the Newtonian limit. This procedure yields two different spatial metrics, one for each observer. Therefore, Principle 1 is indeed satisfied on an ontological level.

Excluding point iv), Newtonian spacetimes trivially exist. The following lemma shows that the Newtonian connection is also well-defined.

Lemma 2.2 (Existence & Uniqueness of the Newtonian connection)
Let be a Newtonian spacetime. Then the Newtonian connection is unique and trivial in Eulerian coordinates, i.e. all the connection coefficients vanish.

Proof:

Consider . (2.1g) and (2.1h) in Definition 2.1 imply

(2.4)

Now is just the Levi-Civita connection with respect to the standard Riemannian metric in the global chart and the result follows.

We conclude that our construction is both physically and mathematically consistent. Yet before we can set up physical models on a Newtonian spacetime , we need to consider the relevant dynamical quantities as obtained from the theory of relativity. These considerations will yield two subclasses of tangent vectors.

Recall that in relativity theory the spacetime model is a time-oriented5 Lorentzian -manifold , equipped with the Levi-Civita connection . If a curve

(2.5)

defined on an open interval is assumed to describe physical motion, we require its tangent vector field to be timelike, future directed and to be parametrized with respect to proper time . For the latter

(2.6)

is a necessary and sufficient condition. Such curves are known as observers and if for a tangent vector an observer exists with for some , then is called an observer vector. Vector fields whose values are observer vectors at every are accordingly called observer (vector) fields. Since a region of physical spacetime with negligible curvature can be approximately described by special relativity, we may restrict ourselves to the case where is open and is the Minkowski metric. In standard coordinates on , we write the tangent vector of an observer as

(2.7)

where the dot denotes differentiation with respect to . On the other hand, condition (2.6) requires

(2.8)

where we used the notation

(2.9)

A first order Taylor expansion of (2.8) in

(2.10)

around yields

(2.11)

which is the expression for in the Newtonian limit. This implies

(2.12)

Plugging (2.11) and (2.12) back into (2.7) we get

(2.13)

If we carry this reasoning over to observer vectors at any , then we get in the Newtonian limit

(2.14)

with . This is the reason for naming in Definition 2.1 the ‘intrinsic observer vector field’.

To obtain the other important class of tangent vectors, we have a look at the dynamics. Hence we consider a test particle6, which is described by an observer and has mass . The force on the particle is defined by

(2.15)

which is just the generalization of Newton’s second law to general relativity. Note that gravity is not a force, but a pseudo-force. Due to metricity of the connection and condition (2.6) we obtain

(2.16)

which roughly means that the (relativistic) velocity is orthogonal to the (relativistic) acceleration. Applying this on (2.15), we get

(2.17)

hence is spacelike [O'Neill]*Chap. 5, 26. Lemma. In the Newtonian limit, the force field must stay “spacelike”. This is indeed the case, which we see by using the definition (2.15) of together with the approximation (2.13) for :

(2.18)

This directly shows that has to vanish in the Newtonian limit.

We have thus obtained the two types of tangent vectors (and hence curves and vector fields) of relevance in any physical model set in a Newtonian spacetime, i.e. tangent vectors with either or . Our discussion motivates the following definition.

Definition 2.3 (Newtonian vectors)
Let be a Newtonian spacetime. A tangent vector at is called Newtonian spacelike, if or, equivalently, in Eulerian coordinates
(2.19a)
is called a Newtonian observer vector, if or, equivalently,
(2.19b)
A tangent vector is called Newtonian, if is either a Newtonian observer vector or Newtonian spacelike. For a Newtonian vector , we call the spacelike component of .

It follows that a tangent vector , describing the velocity vector of a point mass in the Newtonian limit at some instant, is a Newtonian observer vector, and a vector , giving the force acting on such a particle according to (2.15) at that instant, is Newtonian spacelike (i.e. ).

Remark 2.4
The above terminology carries over to vector fields, e.g. a Newtonian observer (vector) field is one whose values are Newtonian observer vectors for every . We denote the space of (smooth) Newtonian vector fields by , the space of (smooth) Newtonian spacelike vector fields by and the space of (smooth) Newtonian observer vector fields by .
Note that there are not any “Newtonian lightlike” vectors. Indeed, for physical consistency we require .
The space of Newtonian spacelike vector fields forms a real vector space, the space of Newtonian observer vector fields does not. However, if we add a Newtonian spacelike vector field to a Newtonian observer vector field, we still have a Newtonian observer vector field. The intrinsic observer field is then the trivial Newtonian observer field, its integral curves physically correspond to observers at rest with respect to some inertial observer in Minkowski spacetime .

Instead of considering a single observer in Minkowski spacetime, let us now assume that it is the integral curve of an observer field . If each integral curve of describes the trajectory of a test particle of equal mass , then (2.15) adapted to this case yields

(2.21)

In the Newtonian limit, we obtain the Newtonian spacelike vector field and the Newtonian observer vector field , hence (2.21) is approximated by

(2.22)

We thus see that the Newtonian limit naturally gives rise to what is known as the material derivative in the fluid mechanics literature [Chorin]*p. 4. Intuitively, the material derivative of a Newtonian observer vector field along itself gives the acceleration of a point in space moving along the flow lines of at some time [Acheson]*§1.2. However, as we have obtained this from the Levi-Civita connection in the Newtonian limit and not in the context of fluids, we do not use this terminology here. Nonetheless we shall adapt our notation. So if is a Newtonian observer field and a Newtonian vector field then, according to Lemma 2.2, the Newtonian derivative of along can be written as

(2.23)

in full compliance with (2.22). If is Newtonian spacelike instead, then

(2.24)

This also shows that for Newtonian vector fields , the expression is always Newtonian spacelike.

For the special case of a Newtonian observer field , we use the notation

(2.25)

which has the natural interpretation of acceleration.

We still have to mathematically construct the relevant vector calculus operators on Newtonian spacetimes without the need to refer to the Newtonian limit.

So let be a Newtonian spacetime, define

(2.26)

and let be the natural inclusion. By the regular value theorem, there is a unique topology and smooth structure on such that it becomes an embedded, smooth submanifold of and it can then be naturally equipped with the flat Riemannian metric . It also inherits a natural orientation from the Newtonian orientation on . Thus is an oriented Riemannian -manifold and hence the vector calculus operators , and are well defined (cf. [Rudolph]*Ex. 4.5.8 for definitions). These can be naturally extended to operators on by considering as a linear subspace of for each .

Definition 2.5 (Vector Calculus on Newtonian spacetimes)
Let be a Newtonian spacetime, be a smooth Newtonian vector field, and let be defined as above for each such that . We then define for every
  1. the gradient of , denoted by , via

    (2.27a)
  2. the divergence of , denoted by , via

    (2.27b)
  3. the curl of , denoted by , via

    (2.27c)
  4. and the Laplacian of as

    (2.27d)

Note that this definition just yields the ordinary vector calculus operators on , naturally adapted to the setting of Newtonian spacetimes. Similarly, the cross product can be extended from to . Moreover, the definitions naturally extend to complex valued functions and vector fields.

With this, we have finished our construction of a spacetime model, the associated (differential) operators and the elementary concepts needed for any physical model constructed upon it.

3 Local Equivalence of the Schrödinger and Madelung Equations

We now employ the construction of the previous section to set up a model of a non-relativistic quantum system with one Schrödinger particle.

In the Schrödinger picture of quantum mechanics [Ballentine]*§4.1 to §4.3 such a system under the influence of an external force

(3.1)

with potential is described by a so called wave function , satisfying the Schrödinger equation [Schroedinger1, Schroedinger2, Schroedinger3]

(3.2)

together with the rule that gives the probability density for the particle’s position at fixed time.7 This description has a number of disadvantages:

  1. The function is complex and it is not apparent how and why this is the case. This in turn prevents a direct physical interpretation.

  2. The equation is already integrated, in the sense that it is formulated in terms of the potential and that the phase of is only specified up to an arbitrary real summand. This in turn suggests that the equation is not fundamental, i.e. it is not formulated in terms of directly measurable physical quantities.

  3. It is not apparent how to generalize the Schrödinger equation to the case where no potential exists for a given force .

  4. It is not entirely apparent how to generalize the Schrödinger equation to more general geometries, i.e. what happens in the presence of constraints, and what the underlying topological assumptions are.

  5. Related to this is the fact that, due to the term, there is no obvious relativistic generalization. This in turn reintroduces the conceptual problems with Principle 1 on page 1.

  6. Let and let be the canonical volume form on (cf. (2.26)) with respect to the metric , i.e. . The statement that for any Borel measurable the expression

    (3.3)

    gives the probability for the particle to be found within the region at time is inherently non-relativistic. Again this leads to problems with Principle 1.

In this section we will observe that these problems are strongly related to each other and find their natural resolution in the Madelung picture.

Before we state and prove the main theorem of this section, that is Theorem 3.7, we would like to remind the reader of the Weber identity [Weber] known from fluid dynamics, since it is essential for passing between the Newtonian and the Hamiltonian description.

Lemma 3.6 (Weber Identity)

Let be a Newtonian spacetime and let be a smooth Newtonian spacelike vector field.
Then

(3.4)

Proof:

Let as defined in (2.26). For the vector fields and the induced (standard) connection on , we have, using standard notation, as a standard result in vector calculus in (cf. [Chorin]*p. 165, Eq. 7) that

To obtain (3.4), we set and let vary.

We named Theorem 3.7 in the honor of Erwin Madelung, as it is mainly based on his article [Madelung] and we merely formalized it to meet the standards of mathematical physics. Note that the choice of sign of is pure convention. We choose it such that for future directed in Minkowski spacetime (cf. [Kriele]*Def. 3.1.3) and , the vector field

(3.5)

is future directed.

Theorem 3.7 (Madelung’s Theorem)
Let be a Newtonian spacetime, and let , be defined as in (2.26).
If is a Newtonian observer vector field, a Newtonian spacelike vector field, a strictly positive, real function and the first Betti number of vanishes for all , then
(3.6a)
(3.6b)
(3.6c)
(3.6d)
imply that there exist such that
(3.6e)
(3.6f)
(3.6g)
Moreover, if one defines
(3.6h)
then it satisfies
(3.6i)
Conversely, if and satisfy (3.6i), define , via (3.6f) and
(3.6j)
such that is a Newtonian observer vector field. Then (3.6a), (3.6b), (3.6c) and (3.6d) hold.

Proof:
” By the definition of curl (2.27c), we have for any fixed
(3.7a)
Since , all closed -forms are exact and hence :
(3.7b)
If we now let vary and observe that , the left hand sides yield smooth -forms on and so do the right hand sides. In other words, the function
(3.7c)
has smooth partial derivatives on for , but need not exist. However, if we integrate with respect to , we obtain a smooth function on , i.e. by choosing the integration constants appropriately we may assume . We then repeat this argument to obtain .
Choosing and , we get via (3.7b) and (2.27a), that (3.6e) and (3.6f) hold.
Define now
(3.7d)
and . Using the Weber identity (Lemma 3.6) together with (3.6c), equation (3.6a) reads
(3.7e)
Due to smoothness of and the Schwarz’ theorem, we have
(3.7f)
and hence
(3.7g)
Thus , as defined by the left side of (3.6g), depends only on . If , we can redefine via as then remains true. Hence (3.6g) follows.
We now define via (3.6h), and calculate in accordance with (2.27d):
(3.7h)
Plugging and (3.6e) into (3.6b) yields
(3.7i)
Since vanishes nowhere, we can multiply with , compare with (3.7h) and arrive at
(3.7j)
On the other hand, (3.6g) can also be reformulated in terms of and to yield
(3.7k)
By comparing this with (3.7h), we see that we can construct a by adding times the imaginary part of for which we have the expression (3.7j). This gives
(3.7l)
To take care of the right hand side, we notice
(3.7m)
Thus, by multiplying (3.7l) by , we finally arrive at the Schrödinger equation (3.6i).
” The reverse construction amounts to Madelung’s discovery [Madelung]. We may define the real function , yet, unfortunately, we cannot write as in (3.6h), since the complex exponential is not (globally) invertible. Instead we define and observe that by (3.6j)
(3.7n)
We now do the calculation backwards with instead of . So in analogy to (3.7h) we consider
(3.7o)
and in analogy to (3.7m) we obtain
(3.7p)
Dividing the Schrödinger equation (3.6i) by and inserting (3.7o) as well as (3.7p), we can take the imaginary part as well as the real part . This is done by employing the facts that both commute with derivatives, derivatives of divided by are purely imaginary and that for any complex number , we have and . Then after some further algebraic manipulation and using (3.7n), the imaginary part yields the continuity equation (3.6b) and the real part gives (3.6g) with instead of . For the latter, we again use the Weber identity from Lemma 3.6 and note
(3.7q)
Recalling the definition (3.6f) of we indeed obtain (3.6a). Finally, (3.6d) and (3.6c) are obtained by seeing that is a gradient vector field and by calculating
(3.7r)
This completes the proof.

Since for every the open ball centered at the point is canonically a Newtonian spacetime as well, the theorem shows that the Madelung equations for irrotational force fields and the Schrödinger equation are locally equivalent.

Remark 3.8 (On the ‘Quantization Condition’)
In the literature one finds the claim that a quantization condition needs to be added for the Schrödinger and the Madelung equations to be equivalent \citelist[Wallstrom0] [Wallstrom] [Takabayasi]*§6 [Holland]*§3.2.2, namely
(3.8a)
for all and all smooth loops . Note that, as observed by Holland [Holland]*§3.2.2, equation (3.8a) is astonishingly similar, yet inequivalent to the Bohr-Sommerfeld quantization condition in the old quantum theory [Sommerfeld]. Recalling Stoke’s theorem [Rudolph]*Thm. 4.2.14 and that the irrotationality (3.6c) of is equivalent to closedness of for all , we see that expression (3.8a) vanishes for and all if and only if . (3.8a) can therefore only be relevant for the case for some . Condition (3.8a) originates from the simplest quantum mechanical model of the Hydrogen atom and indeed excludes apparently unphysical bound states, but, as we will show in detail, is itself of topological origin.
We consider the Madelung equations for a particle with charge being attracted via the Coulomb force by a particle with charge fixed at position . The maximal domain where is smooth is , and we have . Together with irrotationality (3.6d) of , we can thus find a potential . Moreover, in spherical coordinates
(3.8b)
we can write the values of as , since is time-independent.
If we now look for stationary (i.e. -independent) solutions of the Madelung equations, we find the natural domains to be of the form with open , but in general we cannot assume . That is, to be able to write down the Schrödinger equation by application of Theorem 3.7, we have to formally restrict ourselves to a (maximal, non-unique) subset with . The set is the natural domain of and , but first we have to find the solution and then we may fix . Due to the rotational symmetry of the problem, we may already assume
(3.8c)
such that . Note that the assumption of stationarity implies , but may be time dependent. If we now proceed, as usual, by separation of variables in spherical coordinates, we obtain a splitting with and , a radial equation and a spherical one. The latter leads to with and the associated Legendre equation for given by
(3.8d)
Now one usually asks for the condition
(3.8e)
to be satisfied for some and for all , which constrains (and ultimately the other quantum numbers and ) to be integer. If were a global function, (3.8e) would follow from the continuity of on and the property of , that there exists an such that the curve , given by
(3.8f)
lies entirely in . However, assumption (3.8e) cannot be made if we only ask for to be continuous on . As equation (3.8d) also admits solutions for not integer \citelist[MacRobert]*p. 288ff [Hobson]*p. 180f, we may continue to solve the other equation8 and ultimately find that there are solutions with and , given by
(3.8g)
in spherical coordinates. Please keep in mind that also depends on