1 Introduction

The supposition that the wavefunction \(\psi \) provides the most complete description of the state of a physical system that is in principle possible has permeated interpretational discourse since quantum theory’s inception. Yet, it is an arbitrary and unproven conjecture adopted out of theoretical choice rather than empirical imperative. Indeed, the almost boundless paradoxical and fantastical narratives that flow from it dissolve when enhanced concepts of the quantum state are admitted. The most successful example of such a completion is the de Broglie–Bohm causal interpretation, or pilot-wave theory, where in addition to \(\psi \), conceived as a physically real-guiding field, the state comprises a material corpuscle traversing a well-defined spacetime trajectory. Rather than pertaining to mutually exclusive experiments, ‘wave–particle duality’ becomes an objective and permanent feature of matter, and all within ‘one world’.

Whilst the virtues of the de Broglie–Bohm theory cannot nowadays be gainsaid, its standard exposition is open to question. In the next section, we identify three areas where the theory would benefit from theoretical development, relating to the justification of its postulates and how the wave–particle interaction is modelled, and sketch how these problems may be addressed by founding the theory on analytical principles. The remainder of the paper works out this programme in detail and culminates in a proposed reformulation of the de Broglie–Bohm theory based on a unified field whose law of motion combines those of the field and the particle.

2 Critique of the theory of de Broglie and Bohm

2.1 The postulates

For a single-body system, the de Broglie–Bohm theory is usually presented in terms of the following three postulates governing the system’s wave and particle constituents (\(i,j,k,{\ldots } = 1,2,3\); \(\partial _i =\partial /{\partial x_i })\):

  1. 1.

    The wave is described mathematically by the wavefunction \(\psi (x,t)\) obeying the Schrödinger equation

    $$\begin{aligned} i\hbar \frac{\partial \psi }{\partial t}=-\frac{\hbar ^{2}}{2m}\partial _{ii} \psi +V\psi . \end{aligned}$$
    (2.1)
  2. 2.

    Writing \(\psi =\sqrt{\rho } \exp (iS/\hbar )\), the particle is a point obeying the guidance equation

    $$\begin{aligned} \left. {m{\dot{q}}_i =\partial _i S\left( {x,t} \right) } \right| _{x=q\left( {t,q_0 } \right) } \end{aligned}$$
    (2.2)

    whose solution \(q_i (t,q_0 )\) depends on the initial coordinates \(q_{0i} \). The latter coordinate any space point where the initial wavefunction \(\psi _0 (x)\ne 0\) and they identify the trajectory uniquely.

  3. 3.

    For an ensemble of wave–particle systems with a common \(\psi \) component, the particle spatial probability density at time t is \(\rho (x,t)\).

The theory is easily extended to an n-body system where the configuration space wavefunction is accompanied by n corpuscles moving in three-dimensional space [2].

These are the bare bones of the theory, although other concepts, such as energy and force, play a fundamental role [2]. Might the postulates be replaced by others that are perhaps better motivated? In examining this question, we start by exhibiting two simple ways in which the postulates may be usefully reformulated, relating to their interdependence and conceptual flexibility. These reformulations will be useful later. In this endeavour, it is helpful to write the Schrödinger Eq. (2.1) as two real equations, valid where \(\psi \ne 0\):

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\frac{1}{m}\partial _i \left( {\rho \partial _i S} \right) =0 \end{aligned}$$
(2.3)
$$\begin{aligned}&\frac{\partial S}{\partial t}+\frac{1}{2m}\partial _i S\partial _i S+Q+V=0, \end{aligned}$$
(2.4)

where \(Q(x,t)=-({\hbar ^{2}}/{2m\sqrt{\rho }})\partial _{ii} \sqrt{\rho }\) is the quantum potential.

In the first reformulation, we show that postulates 1 and 2 imply that postulate 3 need be asserted only at one time. This follows on using (2.2) to write (2.3) equivalently as the local law of conservation of probability:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left[ {\rho \left( {q\left( t \right) ,t} \right) d^{3}q\left( t \right) } \right] =0, \end{aligned}$$
(2.5)

where \(\mathrm{d}/{\mathrm{d}t}=\partial /{\partial t}+{\dot{q}}_i \partial _i \). Then

$$\begin{aligned} \rho \left( {q\left( t \right) ,t} \right) d^{3}q\left( t \right) =\rho _0 \left( {q_0 } \right) d^{3}q_0, \end{aligned}$$
(2.6)

so that, given the trajectory, \(\rho \) is implied by \(\rho _0 \): \(\rho (x,t)=\det (\partial q/\partial q_0 )\rho _0 (q_0 )\) with \(q_{0i} = q_{0i} (x,t)\). It follows that postulate 3 may be replaced without loss of generality by the postualteFootnote 1

3*:

For an ensemble of wave–particle systems with a common \(\psi \) component, the initial spatial particle probability density is \(\rho _0 (x)\).

In the second reformulation, we show how the first-order (‘Aristotelian’ [3]) law of postulate 2 may be expressed equivalently as a second-order (‘Newtonian’) equation, which brings out the fundamental role played by force in the causal explanation. In the context of the second-order equation, the status of the first-order law is that it constrains the initial velocity of the particle, the constraint being preserved by the second-order equation. This result may be formulated as follows [4]:

Proposition 1

(equivalence of first- and second-order laws) Let \(m{\dot{q}}_i =\partial _i S_0 (q)\) at \(t = 0\). Then, for all t, \(\left. {m{\dot{q}}_i =\partial _i S(x,t)} \right| _{x=q\left( t \right) } \) if and only if \(m\ddot{q}_i =\left. {-\partial _i (V(x,t)+Q(x,t))} \right| _{x=q\left( t \right) } \). (Proof: Appendix A)

It is easy to show that the second-order equation also preserves the single-valuedness condition \(\oint \partial _{i} S dx_{i}=nh\) and hence this need only be postulated at \(t = 0\) [5].

Evidently, the first-order version of the dynamics is no less contingent than the second-order one since both hinge on the initial constraint; if that is disobeyed, the first-order version fails completely and the second-order version admits ‘too many’ solutions, i.e., trajectories that in general do not preserve the probability (2.6) and hence may violate postulate 3, even if postulate 3* is obeyed (for a more general discussion of this point based on a quantum Liouville equation, see [6]). These remarks reinforce the falsity of regarding one of the equivalent formulations of the guidance law as more ‘fundamental’ than the other. Indeed, understanding how the ‘piloting’ or ‘guidance’ of the particle comes about, epithets that are commonly assigned to the first-order version, necessitates invoking the second-order force law. It is, of course, reasonable, in a theory whose remit is to counteract the vagueness of conventional interpretations, to utilize the full set of concepts that it is able to make precise. For the theory is not about the particle trajectory per se but about why it changes, so that the configuration of matter at one instant is causally and continuously connected to its later and preceding configurations. The limited explanatory power of the first-order law is apparent in many examples. For example, in stationary states where the particle or system of particles may be in uniform motion, it is the quantum force that explains phenomena such as the stability of the atom, the attraction of neutral atoms to form a molecule, the Casimir effect, and the pressure exerted by a gas [2, 6].

2.2 Limitations

A weakness of the stated postulates is that the choice of guidance law in postulate 2 is dictated, in the first instance, by compatibility with postulate 3, i.e., it is justified by the statistics it conserves. However, a wide class of nontrivially distinct guidance laws is compatible with a conserved \(\rho \hbox {-}\) distribution, if that is the only selection criterion [7]. This criterion is akin in classical mechanics to attempting to justify Newton’s law for the motion of an individual by inference from Liouville’s equation describing the evolution of an ensemble. Moreover, conclusions drawn from the particle law beyond its remit of conserving the quantum probability may be suspect. For example, if we wish to dispense with postulate 3 and attempt to derive it using an argument based on the guidance law, as first suggested by Bohm [3], we risk circularity if that postulate has already been invoked to justify the law. Another example is the use of the trajectory law to compute quantities that go beyond the standard quantum formalism, such as transit time [8].

Hence, it is desirable to find a justification for the guidance law by studying the dynamics of an individual wave–particle composite. But here we encounter two further shortcomings of the customary model, stemming from its somewhat primitive representation of wave–particle duality.

The first issue concerns what is arguably the model’s most striking aspect (although historically one of its least analyzed): the tacit assumption of postulates 1 and 2 that the particle responds passively to the wave without reciprocal action. The absence of particle reaction is not a logical problem; indeed, it is crucial if one aims to avoid disturbing the Schrödinger evolution and hence maintain the usual predictions of quantum mechanics (which of course are independent of the corpuscle). Nevertheless, it is a singular occurrence in physics that solicits scrutiny. To set it in context, and bearing in mind the hydrodynamic analogy in quantum mechanics [2], the no-reaction assumption is comparable to introducing a tracer into a classical fluid and assuming that it will follow a streamline, i.e., adopt the local fluid velocity along its route. In fact, such a hypothesis needs careful examination of the internal structures and mutual actions of the dirigible particle and the fluid, according to the dynamical principles expected to govern such interactions (Newton’s laws in that case, e.g., [9]). Likewise, one may seek a suitable dynamical framework for the quantum wave–particle interaction within which the absence of reaction is a natural attribute rather than an incidental oddity. Of course, the details of such a theory need not mirror classical treatments.

The second deficiency of the wave–particle model is that, as remarked by de Broglie (see Appendix B) and Bohm [1], the innate dissimilarity of the point particle and field enlisted as constituents of the composite is belied by the core assumption of the model that these entities are inseparable. There is nothing in their natures that would dictate that one cannot be present without the other. Can the two elements be connected at a more basic level so that their intimacy becomes an aspect of a unified description?

To summarize, we have identified three avenues of enquirty where improvements to the conventional presentation of the de Broglie–Bohm theory may be usefully sought, viz., detaching the dynamical law of the particle from probability, finding a theoretical environment to represent the absence of particle reaction, and developing a harmonious model where the wave and particle become aspects of a unified structure.

2.3 Alternative approach

We shall show that the clutch of issues just mentioned may be addressed, together, within an analytical theory of the wave–particle interaction. This approach develops previous work devoted to the first two problems, i.e., finding a nonstatistical justification for the particle law and a theoretical context to represent no reaction [6, 10]. In this work, it was assumed that the wave’s action on the particle is mediated by a scalar potential, and that the composite exhibits no observable differences with a quantum system. Thus, when the composite is the ‘source’ of another system—of an electromagnetic field through the current it generates, for instance, or of a gravitational field through its energy–momentum complex (regarded as the low-energy limit of a relativistic tensor)—that system should not ‘see’ more than a ‘quantum system’, i.e., the composite should behave in this regard like the bare Schrödinger field. Specifically, we required that the conserved matter, energy and momentum densities of the composite coincide with their Schrödinger counterparts. On the basis of these constraints (and another, time-reversal covariance, that was assumed tacitly), it was found that the particle variables obey the de Broglie–Bohm law, while the interaction with the wave is mediated by the quantum potential. To our knowledge, this was the first physical (as opposed to statistical or mathematical) justification for the de Broglie–Bohm law that does not involve invoking further processes, such as stochastic fluctuations.

The formulas for the number, energy, and momentum densities of the composite employed in this scheme are unknown a priori. A natural way to obtain them is as consequences of symmetries that the system may reasonably be expected to possess (specifically, global spacetime, and gauge translations), for then we may employ Noether’s theorem. This procedure also allows us to investigate the role of symmetries in establishing a consistent quantum particle theory, an issue that has been examined extensively in the case of Lorentz covariance but is otherwise underexplored.

The previous presentation was rather terse and no reference was made to the potential role of the analytical theory in addressing the third problem described above, that of finding a unified description. Our purpose here is to give a more thorough account of the analytical theory and bring out its implications for a unified theory.

To summarize, we develop the theory of a corpuscle interacting with a Schrödinger wave, conceived as a single physical system, using the following four assumptions:

  1. 1.

    The system may be treated using analytical methods.

  2. 2.

    The particle is acted upon by \(\psi \) via a scalar potential without reaction.

  3. 3.

    The theory admits global gauge, space, and time translations as symmetries.

  4. 4.

    The corresponding conserved matter, energy, and momentum densities implied by Noether’s theorem coincide with their Schrödinger values.

We shall see that these assumptions provide a consistent theory in which the particle law of motion and the interaction potential are fixed as functions of the wavefunction up to an undetermined constant. No further constraints are obtainable using the method of equal densities. The undetermined constant may be fixed using either of the following additional assumptions, each of which results in the de Broglie–Bohm law and the quantum potential as mediating potential:

  1. 5a.

    The interaction potential is a scalar under time reversal.

  2. 5b.

    For an ensemble of composite systems with a common \(\psi \) component, the particle spatial probability density at time t is \(\rho (x,t)\).

Case 5a implies the guidance formula for an individual system (our original goal) and 5b shows how an alternative statistical derivation works in this context.

This method clarifies other issues associated with the de Broglie–Bohm theory: the nature of the mass of the composite system; and how the energy and momentum of the composite are conserved under the usual conditions on the external potential required in quantum mechanics.

Our principal results pertain to the novel unified perspective that emerges as a concomitant of the analytical technique. The absence of reciprocal action is incorporated since \(\psi \) is represented as the homogeneous component of an inhomogeneous unified field and so is naturally source-free. Moreover, the unified field—to whose source the particle contributes—integrates into its structure both the \(\psi \) field and the particle (as a highly concentrated amplitude), and its inhomogeneous wave equation unites the Schrödinger and particle-guidance equations. These features survive extension of the theory to many-body systems.

It turns out that this work bears a close affinity with de Broglie’s putative ‘double solution’ theory, which we review in Appendix B. This programme was connected with de Broglie’s critical analysis of the pilot-wave theory, a chronicle that seems not to be widely known. In fact, the problem of finding a matter equation of the type desired (but never given) by de Broglie was in effect solved by the author in the earlier work treating the back-reaction problem [6, 10], but no reference was made to the double solution since the connection was not noticed.

3 Analytical derivation of the guidance equation for a wave–particle composite

3.1 Euler–Lagrange equations for a corpuscle interacting with a Schrödinger wave

We consider a system comprising a field whose state is described by the wavefunction \(\psi (x,t)\) and a particle whose state is given by the Cartesian coordinates \(q_i (t)\). This composite is our ‘quantum system’. The particle moves under the influence of the field via, we shall assume, a scalar potential which is to be determined. The field obeys an unmodified Schrödinger equation due to the requirement that the particle does not react upon it.

For variational purposes, the wavefunction \(\psi \) and its complex conjugate \(\psi ^{*}\) are regarded as independent coordinates (equivalent to two real fields) and are varied independently. The variational procedure may be formulated as a constraint problem: a particle moves under the influence of fields constrained to obey Schrödinger’s equation and its complex conjugate. A suitable Lagrangian is

$$\begin{aligned} L\left( {x,q} \right) =\frac{1}{2}m{\dot{q}}_i {\dot{q}}_i -V\left( {q\left( t \right) ,t} \right) -V_Q \left( {q\left( t \right) ,t} \right) \hbox {+ }\int {\left\{ {\frac{1}{2}u^{*}\left[ {i\hbar {\dot{\psi }}+\left( {{\hbar ^{2}}/{2m}} \right) \partial _{ii} \psi -V\psi } \right] +\hbox {cc}} \right\} \mathrm{d}^{3}x}. \end{aligned}$$
(3.1)

Here, ‘cc’ means ‘complex conjugate’, \({\dot{\psi }}=\partial \psi (x,t)/\partial t\), V is the classical external potential energy, and the potential energy \(V_Q (\psi ,\psi ^{*})\) represents the quantum effects on the particle. The complex functions u(x, t) and u*(x, t) are independent Lagrange multipliers. The significance of the function u will be explored below, but its notation is chosen deliberately as it will be found to possess many of the key properties de Broglie demanded of the u component of the ‘double solution’.

It will prove convenient sometimes to use the polar variables \(\rho \) and S in place of \(\psi \) and to mix these representations. It is assumed that \(\psi \rightarrow 0\) as \(x_i \rightarrow \pm \infty \). The action \(\smallint L\mathrm{d}t\) (and variants of it used below) is taken to be invariant with respect to global gauge, space, and time translations. To ensure this, we make the following assumptions about the functions \(V_Q \) and u:

  1. (a)

    \(V_Q \) depends on \(x_i ,t\) only implicitly through local functions of \(\psi \), \(\psi ^{*}\), and their derivatives.

  2. (b)

    \(V_Q \) is a scalar with respect to independent global gauge, space, and time translations (see Sect. 3.3), and so depends on \(\psi \) and \(\psi ^{*}\) via \(\rho \) and the space derivatives of \(\rho \) and S to any order (but not S itself due to the gauge symmetry).

  3. (c)

    \(V_Q \) is independent of the time derivatives of \(\psi \) and \(\psi ^{*}\). This is ensured if \(V_Q \) is a scalar under Galilean boosts (\({x}'_i =x_i -w_i t,\hbox { }{t}'=t)\), but we shall derive this property rather than posit it since it requires that \(V_Q \) is free of \(\partial _i S\) from the outset.

  4. (d)

    u transforms like \(\psi \) under global gauge, space, and time translations.

Variation of the action with respect to u* and u yields the Schrödinger equation

$$\begin{aligned} i\hbar \frac{\partial \psi \left( x \right) }{\partial t}=-\frac{\hbar ^{2}}{2m}\partial _{ii} \psi \left( x \right) +V\left( x \right) \psi \left( x \right) \end{aligned}$$
(3.2)

and its complex conjugate, respectively. Next, varying with respect to \(\psi ^{*}\) and \(\psi \) gives the equation obeyed by the Lagrange multiplier

$$\begin{aligned} i\hbar \frac{\partial u\left( {x,t} \right) }{\partial t}=-\frac{\hbar ^{2}}{2m}\partial _{ii} u\left( {x,t} \right) +V\left( x \right) u\left( {x,t} \right) +2\left. {\frac{\delta V_Q \left( {\psi \left( q \right) ,\psi ^{*}\left( q \right) } \right) }{\delta \psi ^{*}\left( x \right) }} \right| _{q=q\left( t \right) } \end{aligned}$$
(3.3)

and the complex conjugate equation, respectively. Finally, varying the variables \(q_i \) generates the particle equation

$$\begin{aligned} m\ddot{q}_i =\left. {-\frac{\partial }{\partial q_i }\left( {V+V_Q } \right) } \right| _{q=q\left( t \right) } . \end{aligned}$$
(3.4)

The functional derivative in (3.3) is defined in Appendix C. The equations are completed by specifying the potential \(V_Q \) and the initial conditions \({\dot{q}}_{0i} \), \(q_{0i} \), \(\psi _0 \), and \(u_0 \). Expressions for \({\dot{q}}_{0i}, u_{0}\) and \(V_Q \) will be obtained below from the consistency constraints imposed on the system. The only free variables in the system will then be \(q_{0i} \) and \(\psi _0 \), on which u will depend.

As required, the Schrödinger Eq. (3.2) is unmodified by the particle variables, while the wavefunction appears in the particle law (3.4). To bring out the significance of the Lagrange multipliers u and u*, we note that they are the canonical momenta corresponding to \(\psi \)* and \(\psi \), respectively (up to multiplicative constants):

$$\begin{aligned} p_{\psi ^{*}} =\frac{\delta L}{\delta {\dot{\psi }}^{*}\left( x \right) }=-\frac{i\hbar }{2}u\left( x \right) ,\quad p_\psi =\frac{\delta L}{\delta {\dot{\psi }}\left( x \right) }=\frac{i\hbar }{2}u^{*}\left( x \right) . \end{aligned}$$
(3.5)

The more significant property of the function u here, however, is that its dynamical equation [Eq. (3.3)] is an inhomogeneous Schrödinger equation that exhibits on the right-hand side a (singular) source term that depends on \(\psi \) and on \(q_i (t)\) via \(\delta (x-q(t))\) and its derivatives, which we shall examine below.

3.2 Fully field-theoretic formulation

To obtain local conserved quantities associated with the composite system using Noether’s theorem, it will be helpful to first write the particle and field variables appearing in the Lagrangian (3.1) in a unified field-theoretic language. Consider the following Lagrangian density:

$$\begin{aligned} {\mathfrak {L}}'=\left[ {\frac{1}{2}mv_i v_i -V\left( x \right) -V_Q \left( {\psi \left( {x,t} \right) } \right) } \right] \delta \left( {x-q\left( t \right) } \right) +\left\{ {\frac{1}{2}u^{*}\left[ {i\hbar {\dot{\psi }}+\left( {{\hbar ^{2}}/{2m}} \right) \partial _{ii} \psi -V\left( x \right) \psi } \right] +\hbox {cc}} \right\} \hbox { } \end{aligned}$$
(3.6)

obtained by a change of coordinates \(q_i (q_0 ,t)\rightarrow x_i \) where the two sets of coordinates are connected by the relation

$$\begin{aligned} v_i \left( {x,t} \right) =\left. {{\dot{q}}_i \left( {q_0 ,t} \right) } \right| _{q_0 \left( {x,t} \right) } . \end{aligned}$$
(3.7)

The integral of the density gives the Lagrangian (3.1): \(L=\smallint {\mathfrak {L}}'d^{3}x\). In this formulation, the particle component of the dynamics is recast as a ‘single-particle fluid dynamics’ whose Eulerian picture is characterized by the density field \(\rho _p (x,t)=\delta (x-q(q_0 ,t))\) and velocity field \(v_i (x,t)\). L now has a field-theoretic form, but this is not suitable to obtain the particle law of motion (3.4) through a variational procedure, because the time derivatives of the particle variables are missing, a well-known issue when passing from discrete to field descriptions. This problem may be tackled by introducing a Clebsch parameterization for the velocity [11] (the constant m is introduced for later convenience):

$$\begin{aligned} v_i =m^{-1}\left( {\partial _i \theta +\alpha \partial _i \beta } \right) . \end{aligned}$$
(3.8)

Employing the new fields \(\alpha \left( {x,t} \right) ,\beta \left( {x,t} \right) \) and \(\theta \left( {x,t} \right) \), we shall use the following redefined Lagrangian density:

$$\begin{aligned} \mathfrak {L}= & {} -\rho _p \left( {x,t} \right) \left[ {{\dot{\theta }}+\alpha {\dot{\beta }}+\left( {2m} \right) ^{-1}\left( {\partial _i \theta +\alpha \partial _i \beta } \right) ^{2}+V\left( x \right) +V_Q \left( {\psi \left( x \right) } \right) } \right] \nonumber \\&+\left\{ {\frac{1}{2}u^{*}\left[ {i\hbar {\dot{\psi }}+\left( {{\hbar ^{2}}/{2m}} \right) \partial _i \partial _i \psi -V\left( x \right) \psi } \right] +\hbox {cc}} \right\} , \end{aligned}$$
(3.9)

where \({\dot{\theta }}=\partial \theta (x,t)/\partial t\), etc. As desired, this gives a fully field-theoretic formulation of the interacting systems, with the particle variables represented by the fields \(\rho _p \), \(\alpha ,\beta ,\theta \), and the associated requisite time derivatives. The associated Lagrangian \({L}'=\smallint \mathfrak {L} d^{3}x\) differs from L in (3.1) by more than an added total time derivative, so the associated action functionals are not the same: \(\smallint {L}'\mathrm{d}t\ne \smallint L\mathrm{d}t\). Nevertheless, the revised set of Euler–Lagrange equations still give (3.2–3.4). The latter assertion is obvious for the variations of \(\psi \), \(\psi \)*, u and u*. We shall check explicitly that the particle equation [Eq. (3.4)] is obtained via the variations of \(\alpha ,\beta ,\theta \) and \(\rho _p \). We have

$$\begin{aligned}&\delta \rho _p :\quad {\dot{\theta }}+\alpha {\dot{\beta }}+\frac{1}{2m}\left( {\partial _i \theta +\alpha \partial _i \beta } \right) ^{2}+V\left( x \right) +V_Q \left( x \right) =0 \end{aligned}$$
(3.10)
$$\begin{aligned}&\delta \theta :\quad \frac{\partial \delta \left( {x-q\left( t \right) } \right) }{\partial t}+\partial _i \left[ {\delta \left( {x-q\left( t \right) } \right) v_i } \right] =0 \end{aligned}$$
(3.11)
$$\begin{aligned}&\delta \beta ,\;\delta \alpha :\quad {\dot{\alpha }}+v_i \partial _i \alpha =0,\quad {\dot{\beta }}+v_i \partial _i \beta =0. \end{aligned}$$
(3.12)

The last two equations follow upon using (3.11) which implies (3.7) using the identity (cf. Appendix G)

$$\begin{aligned} \frac{\partial \delta \left( {x-q\left( t \right) } \right) }{\partial t}+{\dot{q}}_i \partial _i \left[ {\delta \left( {x-q\left( t \right) } \right) } \right] =0. \end{aligned}$$
(3.13)

To deduce the particle law, we take the gradient of (3.10) and use the relations (3.8) and (3.12) to obtain an Euler-type force law:

$$\begin{aligned} m\left( {\frac{\partial v_i }{\partial t}+v_j \partial _j v_i } \right) =-\partial _i \left( {V+V_Q } \right) . \end{aligned}$$
(3.14)

In the field-theoretic picture, the motion is characterized by the velocity field rather than the trajectory (as in the Eulerian picture in hydrodynamics). To obtain the trajectory version, we transform the independent variables \(x_i \rightarrow q_i (q_0 ,t)\) via the relation (3.7) and the Euler law (3.14) becomes the particle Eq. (3.4) with initial condition \(v_i (x,0)={\dot{q}}_{0i} \). The ‘equation of motion’ (3.7) is essentially a conversion formula between the field and trajectory views of the particle motion. The Eqs. (3.10) and (3.12) hold along each of the tracks \(q_i (q_0 ,t)\) potentially occupied by the particle that are obtained by varying \(q_{0i} \).

There is evidently some redundancy in the field-theoretic formalism for the particle component of the system since the variables \(\alpha ,\beta \) play no essential role. We shall remove these superfluous quantities by choosing the solutions \(\alpha =\beta =0\) to (3.12). Then the velocity field reduces to \(v_i =(1/m)\partial _i \theta \), the usual relation obtained in Hamilton–Jacobi theory for a structureless particle in an external scalar potential if the function \(\theta \) is identified with the Hamilton–Jacobi function. That this is a correct identification follows since (3.10) reduces to the Hamilton–Jacobi equation. The field-theoretic version of the particle law of motion is, therefore, expressed by the equations

$$\begin{aligned} v_i \left( {x,t} \right) =m^{-1}\partial _i \theta ,\quad {\dot{\theta }} + \left( {2m} \right) ^{-1}\partial _i \theta \partial _i \theta +V+V_Q =0. \end{aligned}$$
(3.15)

As shown above, these equations imply the particle law in the form (3.4) with initial conditions \({\dot{q}}_{0i} =m^{-1}\partial _i \theta _0 (q_0 )\) and \(q_{0i} \) chosen freely.

Henceforth, we take (3.2), (3.3), and (3.15) as the dynamical equations for the composite system. The Lagrangian density is

$$\begin{aligned} \mathfrak {L}= & {} -\delta \left( {x-q\left( t \right) } \right) \left[ {{\dot{\theta }}+\left( {2m} \right) ^{-1}\partial _i \theta \partial _i \theta } \right. \left. +V\left( {x,t} \right) +V_Q \left( {\psi \left( x \right) } \right) \right] \nonumber \\&+\left\{ {\frac{1}{2}u^{*}\left[ {i\hbar {\dot{\psi }}+\left( {{\hbar ^{2}}/{2m}} \right) \partial _{ii} \psi -V\left( x \right) \psi } \right] +\hbox {cc}} \right\} . \end{aligned}$$
(3.16)

3.3 Conserved densities obtained from Noether’s theorem

We now apply Noether’s theorem (see Appendix C) which asserts: to each continuous transformation that leaves the action \(\smallint \mathfrak {L} d^{3}x\,\mathrm{d}t\) invariant, there corresponds a density \(\mathrm{P}\) and current density \(\mathrm{P}_i \) that together obey the continuity equation (C.7).

Using formula (C.5), the conserved number density corresponding to invariance of the action under an infinitesimal gauge transformation \({\theta }'=\theta +\varepsilon \eta \), \({\psi }'=\psi +i\varepsilon \eta \psi /\hbar \) (so that \({S}'=S+\varepsilon \eta )\), \({u}'=u+i\varepsilon \eta u/\hbar \) with \(\eta \) = constant and all other variable invariant is

$$\begin{aligned} J_0 =\mathrm{P}/{\left( {-\eta } \right) }=\delta \left( {x-q\left( t \right) } \right) +\frac{1}{2}\left( {u^{*}\psi +u\psi ^{*}} \right) . \end{aligned}$$
(3.17)

Although it does not assist in proving the guidance theorem, it is instructive to give the corresponding current density for the gauge symmetry obtained from (C.6):

$$\begin{aligned} J_i ={\mathrm{P}_i }/{\left( {-\eta } \right) }=m^{-1}\partial _i \theta \delta \left( {x-q\left( t \right) } \right) +\left( {{i\hbar }/{4m}} \right) \left( {\psi \partial _i u^{*}-u^{*}\partial _i \psi +u\partial _i \psi ^{*}-\psi ^{*}\partial _i u} \right) -X_i \end{aligned}$$
(3.18)

with

$$\begin{aligned} X_i =-\delta \left( {x-q} \right) \frac{\partial V_Q }{\partial \left( {\partial _i S} \right) }+\partial _j \left[ {\delta \left( {x-q} \right) \frac{\partial V_Q }{\partial \left( {\partial _{ij} S} \right) }} \right] -\partial _{jk} \left[ {\delta \left( {x-q} \right) \frac{\partial V_Q }{\partial \left( {\partial _{ijk} S} \right) }} \right] +...\;. \end{aligned}$$
(3.19)

Using (3.2), (3.3), and the easily proved result

$$\begin{aligned} \partial _i X_i =\frac{\delta V_Q }{\delta S}, \end{aligned}$$
(3.20)

it is readily confirmed that the number density and number current density obey the continuity equation (C.7):

$$\begin{aligned} \frac{\partial J_0 }{\partial t}+\partial _i J_i =0. \end{aligned}$$
(3.21)

In fact, the point particle (delta function) and field contributions to the density and current density obey separate continuity equations, which reflects the invariance of the action under independent shifts in \(\theta \) and S. The continuity equation obtained in this way for the particle alone, corresponding to a displacement in \(\theta \), is the identity (3.11).

Next, the momentum density follows from invariance of the action under a space translation \({x}'_i =x_i +\varepsilon \xi _i \), \({q}'_i =q_i +\varepsilon \xi _i \) with \(\xi _i \) = constant and all other variables invariant:

$$\begin{aligned} \mathfrak {P}_i =\mathrm{P}/{\left( {\xi _i } \right) }=\partial _i \theta \delta \left( {x-q\left( t \right) } \right) +\left( {{i\hbar }/2} \right) \left( {-u^{*}\partial _i \psi +u\partial _i \psi ^{*}} \right) . \end{aligned}$$
(3.22)

Finally, the energy density follows from invariance under a time translation \({t}'=t+\varepsilon \xi _0 \) with \(\xi _0 \) = constant and all other variables invariant:

$$\begin{aligned} \mathfrak {H}= & {} \mathrm{P}/{\left( {-\xi _0 } \right) } \nonumber \\= & {} \delta \left( {x-q\left( t \right) } \right) \left( {\frac{1}{2m}\left( {\partial _i \theta } \right) ^{2}+V+V_Q } \right) -\left( {{\hbar ^{2}}/{4m}} \right) \left( {u^{*}\partial _{ii} \psi +u\partial _{ii} \psi ^{*}} \right) +\frac{1}{2}\left( {u^{*}\psi +u\psi ^{*}} \right) V. \end{aligned}$$
(3.23)

As noted, we interpret \(J_0 (x,q_0 ,t)\) as the number, or matter, density, so that \(\smallint J_0 d^{3}x\) is the number of particles in the system. Two points are noteworthy about the expression (3.17): (a) we expect that \(J_0 \) will be nonnegative and that in ensuring this the term in brackets may become negative; (b) although \(J_0 \) exhibits in the first delta function term the number density of the particle, the right-hand side is not a decomposition into independent particle and field contributions, because the function u may depend on the particle variables. The latter remark applies also to the momentum and energy densities: the right-hand sides of (3.22) and (3.23) are not decompositions into independent particle and field components.

To obtain the corresponding densities for the Schrödinger field alone, we take its Lagrangian density to be

$$\begin{aligned} \mathfrak {L}_S \left( x \right)= & {} \frac{1}{2}\psi ^{*}\left[ {i\hbar {\dot{\psi }}+\left( {{\hbar ^{2}}/{2m}} \right) \partial _{ii} \psi -V\psi } \right] +\hbox {cc} \nonumber \\= & {} -\rho \left[ {{\dot{S}}+\left( {1/{2m}} \right) \left( {\partial _i S} \right) ^{2}+Q+V} \right] . \end{aligned}$$
(3.24)

The function \(\psi ^{*}\) (\(\psi )\) acts like a Lagrange multiplier whose variation implies that the Schrödinger equation for \(\psi \) (\(\psi ^{*})\) is obtained as a constraint. Formulas (C.5) and (C.6) give the following expressions for the number, number current, momentum, and energy densities:

$$\begin{aligned}&j_0 =\rho ,\quad j_i =\left( {1/m} \right) \rho \partial _i S \end{aligned}$$
(3.25)
$$\begin{aligned}&\mathfrak {P}_{Si} =\mathrm{P}/{\left( {-\xi _i } \right) }=\rho \partial _i S \end{aligned}$$
(3.26)
$$\begin{aligned}&\mathfrak {H}_S =\mathrm{P}/{\left( {-\xi _0 } \right) }=\rho \left[ {\left( {1/{2m}} \right) \left( {\partial _i S} \right) ^{2}+Q+V} \right] . \end{aligned}$$
(3.27)

Note that \(\mathfrak {P}_{Si} =mj_i \) for the pure Schrödinger field, whereas \(\mathfrak {P}\ne m J_i \) for the wave–particle composite.

3.4 The guidance theorem (single system version)

To ensure that the composite system qualifies for the appellation ‘quantum system’, our aim is to find the relations linking the functions \(\psi \), \(q_i \), \(V_Q \), u and \(\theta \), so that the composite densities (3.17), (3.22), and (3.23) equal their pure Schrödinger counterparts (3.25–3.27). These relations must be consistent with the dynamical equations (3.2), (3.3), and (3.15). It is assumed that there are no constraints involving \(\psi \) alone since the theory applies to arbitrary wavefunctions. We find that the condition of equal densities determines \(V_Q \) and u as functions of \(\psi \) up to an arbitrary constant, and \(\theta \) up to two arbitrary constants:

Proposition 2

(equality of Noetherian densities) Suppose that the variables \(\psi \), u, and \(\theta \) solve the Eqs. (3.2), (3.3), and (3.15). Then, the conserved number, momentum, and energy densities coincide with their pure Schrödinger counterparts if and only if the solutions and \(V_Q \) are connected by the relations

$$\begin{aligned}&\theta =S-\kappa \log \rho +\mathrm{constant} \end{aligned}$$
(3.28)
$$\begin{aligned}&V_Q =Q-\left( {{\kappa ^{2}}/{2m}} \right) \partial _i \log \rho \partial _i \log \rho -\left( {\kappa /m} \right) \partial _{ii} S \end{aligned}$$
(3.29)
$$\begin{aligned}&u=\psi -\left( {1+\frac{2i\kappa }{\hbar }} \right) \frac{1}{\psi ^{*}}\delta \left( {x-q\left( t \right) } \right) , \end{aligned}$$
(3.30)

where \(\kappa \) is an undetermined constant. (Proof: Appendix D)

As mentioned previously, the functional relations that we have found exhaust the constraints obtainable by the method of equating Noetherian densities derived from variational symmetries. Substituting relations (3.28–3.30), the composite Lagrangian density (3.16) reduces to the Schrödinger form (3.24), and the number current density (3.18) and the momentum and energy current densities derived from (C.6) all return their Schrödinger values. Conserved densities derived from the number, momentum, and energy densities and known functions of t and \(x_i \) likewise automatically reduce to their Schrödinger counterparts when the constraints are applied. Examples include the angular momentum density \(\varepsilon _{ijk} x_j \mathfrak { P}_k =\varepsilon _{ijk} x_j \mathfrak {P}_{Sk} \) (corresponding to invariance of the action under an infinitesimal rotation \({x}'_i =x_i -\varepsilon _{ijk} \xi _j x_k )\) and the Galilean density \(t\mathfrak {P}_i -x_i J_0 =t\mathfrak {P}_{Si} -x_i j_0 \) (corresponding to invariance under an infinitesimal boost \({x}'_i =x_i -\xi _i t)\).

To determine the constant \(\kappa \), we examine the impact of a time-reversal transformation

$$\begin{aligned} {x}'_i =x_i ,\quad {t}'=-t,\quad {\psi }'\left( {{x}',{t}'} \right) =\psi ^{*}\left( {x,t} \right) \end{aligned}$$
(3.31)

for which

$$\begin{aligned} {q}'_i \left( {{t}'} \right) =q_i \left( t \right) ,\quad {\rho }'\left( {{x}',{t}'} \right) =\rho \left( {x,t} \right) ,\quad {S}'\left( {{x}',{t}'} \right) =-S\left( {x,t} \right) . \end{aligned}$$
(3.32)

The corresponding transformations of the fields (3.28–3.30) are:

$$\begin{aligned} {\theta }'=-\theta -2\kappa \log \rho ,\quad {V}'_Q =V_Q +2\kappa \partial _{ii} S/m,\quad {u}'=u^{*}-\left( {{4i\kappa }/{\hbar \psi }} \right) \delta \left( {x-q\left( t \right) } \right) . \end{aligned}$$
(3.33)

Assuming that the external potential V is a scalar, time reversal is a symmetry of the Schrödinger equation (3.2), but the theory of the composite as a whole is not covariant. The symmetry is broken by the first equation in (3.15), which links the field \(\theta \) with the particle velocity (\(v_i =m^{-1}\partial _i \theta )\). According to the relations (3.31) and (3.32), the velocity of the particle reverses:

$$\begin{aligned} {{\dot{q}}}'_i \left( {{t}'} \right) =\frac{d{q}'_i \left( {{t}'} \right) }{d{t}'}=\frac{dq_i \left( t \right) }{d\left( {-t} \right) }=-{\dot{q}}_i \left( t \right) \quad \hbox {or}\quad {v}'_i \left( {{x}',{t}'} \right) =-v_i \left( {x,t} \right) . \end{aligned}$$
(3.34)

However, using the first relation in (3.33), we find for the transformed velocity field

$$\begin{aligned} {v}'_i \left( {{x}',{t}'} \right) =m^{-1}{\partial }'_i {\theta }'=-v_i \left( {x,t} \right) -2\kappa m^{-1}\partial _i \log \rho \end{aligned}$$
(3.35)

which contradicts (3.34) when \(\kappa \ne 0\). We conclude that fixing \(\kappa \) (= 0) is tied to requiring time-reversal covariance. Consulting (3.33), this may be accomplished by assuming that \(V_Q \) is a scalar under the transformation (but evidently alternative assumptions may be invoked, e.g., that the velocity reverses sign):

Proposition 3

(time reversal) Under the conditions of Proposition 2, \(V_Q \) is a scalar under time reversal if and only if \(\kappa =0\).

Combining Propositions 2 and 3, we have:

Proposition 4

(full set of constraints) The assumptions of Propositions 2 and 3 are obeyed if and only if the functions \(\theta \), u, and \(V_Q \) are fixed as functions of \(\psi \) as follows:

$$\begin{aligned}&\theta \left( {x,t} \right) =S\left( {x,t} \right) +\hbox {constant} \end{aligned}$$
(3.36)
$$\begin{aligned}&V_Q =Q \end{aligned}$$
(3.37)
$$\begin{aligned}&u=\psi -\frac{1}{\psi ^{*}}\delta \left( {x-q\left( t \right) } \right) . \end{aligned}$$
(3.38)

We note that, given \(V_Q =Q\), the condition (3.36) is equivalent to the condition \(\partial _i \theta =\partial _i S\), as is easily seen by subtracting (2.4) from (3.15) (and, using Proposition 1 (Sect. 1), (3.36) need only be stated at \(t = 0\)). Gathering the above results, we obtain the following theorem which states the conditions under which the variables \({\dot{q}}_i \), u, and \(V_Q \) are fixed as functions of \(\psi \):

Guidance Theorem 1

(single system version) Suppose that the variables \(\psi \), u, and \(q_i \) solve the Eqs. (3.2–3.4) where the interaction potential \(V_Q \) is a function of \(\psi \). Let the action with Lagrangian density (3.16) and the function \(V_Q \) be invariant, and u transform like \(\psi \), with respect to global phase, space, and time translations. Then, the conserved number, momentum, and energy densities (3.17), (3.22), and (3.23) coincide with their quantum counterparts (3.25), (3.26), and (3.27), respectively, and \(V_Q \) is a time-reversal scalar, if and only if the functions \(\psi \), \(q_i \), \(V_Q \), and u are connected by the relations

$$\begin{aligned}&{\dot{q}}_i \left( t \right) =m^{-1}\left. {\partial _i S\left( {x,t} \right) } \right| _{x=q\left( t \right) } \end{aligned}$$
(3.39)
$$\begin{aligned}&V_Q =Q \end{aligned}$$
(3.40)
$$\begin{aligned}&u\left( {x,t,q_0 } \right) =\psi \left( {x,t} \right) -\frac{1}{\psi ^{*}\left( {x,t} \right) }\delta \left( {x-q\left( {q_0 ,t} \right) } \right) . \end{aligned}$$
(3.41)

Thus, the interaction potential is the quantum potential and the Lagrange multiplier u is determined by the wavefunction and the particle coordinates. As anticipated, the only free variables are \(q_{0i} \) and \(\psi _{0}\). The dynamical equations governing the wave–particle system become, therefore, the Schrödinger equation (3.2) and the de Broglie–Bohm guidance equation (3.39) or, following Proposition 1 (Sect. 1)

$$\begin{aligned} m\ddot{q}_i =-\frac{\partial }{\partial q_i }\left( {V\left( q \right) +Q\left( q \right) } \right) ,\quad {\dot{q}}_i \left( 0 \right) =m^{-1}\partial _i S_0 \left( {q_0 } \right) . \end{aligned}$$
(3.42)

The Lagrange multiplier obeys the inhomogeneous Schrödinger equation

$$\begin{aligned} i\hbar \frac{\partial u}{\partial t}=-\frac{\hbar ^{2}}{2m}\partial _{ii} u+V\left( x \right) u+2\frac{\delta Q\left[ {\rho \left( {q\left( t \right) } \right) } \right] }{\delta \psi ^{*}\left( {x,t} \right) } \end{aligned}$$
(3.43)

with solution (3.41). The Hamilton–Jacobi equation (3.15) coincides with (2.4). These equations form a mutually consistent set. Equation (3.43) and its solution (3.41) were first given in [8].

So far, the role of the field u has been to assist in building an analytical theory. Its further significance will be examined below. It is easy to evaluate the source term in (3.43) explicitly: using the second of the conversion formulas

$$\begin{aligned} \frac{\delta }{\delta \psi }=\psi ^{*}\frac{\delta }{\delta \rho }-\frac{i\hbar }{2\psi }\frac{\delta }{\delta S},\quad \frac{\delta }{\delta \psi ^{*}}=\psi \frac{\delta }{\delta \rho }+\frac{i\hbar }{2\psi ^{*}}\frac{\delta }{\delta S}, \end{aligned}$$
(3.44)

and the following expression for the quantum potential

$$\begin{aligned} Q\left( q \right) =\frac{\hbar ^{2}}{4m\rho \left( q \right) }\left[ {\left( {\frac{1}{2\rho }\frac{\partial \rho \left( q \right) }{\partial q_i }\frac{\partial \rho \left( q \right) }{\partial q_i }-\frac{\partial ^{2}\rho \left( q \right) }{\partial q_i \partial q_i }} \right) } \right] , \end{aligned}$$
(3.45)

the source term is

$$\begin{aligned} 2\frac{\delta Q\left( q \right) }{\delta \psi ^{*}\left( x \right) }=\frac{\hbar ^{2}}{2m\psi ^{*}\left( x \right) }\left[ {\delta \left( {x-q} \right) \partial _{ii} \log \rho \left( x \right) +\partial _i \delta \left( {x-q} \right) \partial _i \log \rho \left( x \right) -\partial _{ii} \delta \left( {x-q} \right) } \right] . \end{aligned}$$
(3.46)

In addition to a term proportional to \(\delta (x-q)\) expected in classical particle-source theory, this expression also contains first- and second-order derivatives of the delta function. Note that the factor 2 on the left-hand side of (3.46) is an artefact of the definition of u and can be removed by replacing \(u\rightarrow 2u\) in (3.16) and (3.41).

4 Properties of the wave–particle composite

4.1 Number of particles and mass

We have interpreted \(J_0 =\delta (x-q(t))+u\psi ^{*}\) in (3.17) as the number density of particles in the system. We expect then that, for each \(q_{0i} \), the number of particles \(\smallint J_0 d^{3}x\) will be unity. This is confirmed by inserting \(u\psi ^{*}=\rho -\delta (x-q(t))\) in \(J_0 \). Note that the unity result is due not to the appearance of the particle density (delta function) in \(J_0 \), which by construction cancels out, but to the normalized field density \(\rho \). In the n-body case (see Sect. 8), \(J_0 \) is the number density of configuration space ‘particles’ and is again normalized to unity since the n particles make up a single system point.

Hitherto we have not attributed mass to the corpuscle, although it is tacit in the Lagrangian (3.1) that the particle has mass and that this is m. If we make the latter assumption and bear in mind that the composite system comprises what is usually termed ‘a quantum system of mass m’ (i.e., the \(\psi \) field) in addition to the corpuscle, what is the mass of the composite - 2m? Defining the mass density to be \(mJ_0 \), an argument similar to that just given for the number density shows that the mass of the composite is, in fact, m; by construction, the mass density is \(m\rho \).

4.2 Conservation of energy and momentum

The self-contained nature of the wave–particle composite may be illustrated by considering the conditions under which its energy and momentum are conserved. Using (3.22) and (3.23), these quantities are given by

$$\begin{aligned} \int {\mathfrak {H} d^{3}x}= & {} \frac{1}{2}m{\dot{q}}_i {\dot{q}}_i +V\left( {q,t} \right) +V_Q \left( {q\left( t \right) ,t} \right) \nonumber \\&+ \int {\left[ {-\left( {{\hbar ^{2}}/{4m}} \right) \left( {u^{*}\partial _{ii} \psi +u\partial _{ii} \psi ^{*}} \right) +\frac{1}{2}\left( {u^{*}\psi +u\psi ^{*}} \right) V} \right] d^{3}x} \end{aligned}$$
(4.1)
$$\begin{aligned} \int {\mathfrak {P}_i d^{3}x}= & {} m{\dot{q}}_i +\left( {{i\hbar }/2} \right) \int {\left( {-u^{*}\partial _i \psi +u\partial _i \psi ^{*}} \right) d^{3}x} . \end{aligned}$$
(4.2)

Differentiating each of these expressions with respect to time and using the dynamical equations (3.2–3.4) gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\int {\mathfrak {H} d^{3}x} =\frac{\partial V}{\partial t}+\int {u\psi ^{*}\frac{\partial V}{\partial t}d^{3}x} \end{aligned}$$
(4.3)
$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\int {\mathfrak {P}_i d^{3}x} =-\frac{\partial V}{\partial q_i }-\int {u\psi ^{*}\partial _i Vd^{3}x}. \end{aligned}$$
(4.4)

It follows that the energy is conserved when there is no external source of power (\({\partial V}/{\partial t}=0)\) and the momentum is conserved when there is no external force (\(\partial _i V=0)\). The wave–particle composite as we have defined it is, therefore, an isolated system when the ‘external’ agents of change are absent, as expected from the latter’s nomenclature. These conditions on the external potential coincide with those under which energy and momentum are conserved according to the usual quantum formalism. Indeed, as we expect, when the solution (3.41) is inserted, (4.3) and (4.4) reduce to the usual quantal expressions for the mean values of the external power and force, respectively:

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\int {\mathfrak {H}_S d^{3}x} =\int {\rho \frac{\partial V}{\partial t}d^{3}x} \end{aligned}$$
(4.5)
$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\int {\mathfrak {P}_{Si} d^{3}x} =-\int {\rho \partial _i Vd^{3}x} . \end{aligned}$$
(4.6)

The energy and momentum of the particle in isolation, whose equations of change are \(\mathrm{d}E/\mathrm{d}t=\partial (V+Q)/\partial t\) and (3.42), respectively, are not conserved under these conditions due to the presence of the quantum potential.

5 Probability interpretation: alternative derivation of the guidance equation

The derivation of the de Broglie–Bohm law enshrined in Guidance Theorem 1 (Sect. 3) applies to a single composite system. We thus achieve our aim set out in Sect. 2 of decoupling the law of the individual from a statistical assumption about \(\rho \), the latter being regarded in our derivation as a physical field extending throughout space. As remarked in Sect. 2, the statistical assumption is insufficiently stringent in its usual implementation to entail a specific form for the guidance law (we assume that \(\rho \) determines the particle probability as a secondary feature).

We now show that the statistical assumption is sufficient to obtain the guidance law if implemented in the context of the assumptions made in Proposition 2 (Sect. 3). This provides an alternative (probabilistic) completion of the derivation of the law of motion in place of the assumption of time-reversal covariance of an individual system used in Proposition 3. There is no reason to doubt the validity of time-reversal covariance in this context, but, proceeding on this alternative basis, this covariance becomes a derived property. In place of Proposition 3, we have:

Proposition 5

(ensemble) Under the conditions of Proposition 2, consider an ensemble of wave–particle composite systems for which the wave aspect \(\psi \) is identical and the particle position is distributed as \(\mathrm{P}(x,t)=\rho (x,t)\). Then \(\kappa =0\). (Proof: Appendix E)

In place of Guidance Theorem 1, we may now assert:

Guidance Theorem 2

(ensemble version) In Guidance Theorem 1, replace the condition that \(V_Q \) is a time-reversal scalar by the condition that an ensemble of particles is distributed as \(\rho (x,t)\) for all t. Then, the functions \(\psi \), \(q_i \), \(V_Q \), and u are connected by the relations (3.39–3.41).

Note that the converse is not true: as shown in Sect. 2, the de Broglie–Bohm law does not imply \(\mathrm{P}(x,t)=\rho (x,t)\) for all t unless this condition holds at one instant.

6 Unification of the Schrödinger and guidance equations

We have seen that, for an interacting wave–particle system, the quantum potential and the guidance condition emerge from the requirements that there is no reciprocal action of the particle on the wave and no empirical discrepancy between the behaviour of the composite and a ‘quantum system’. To this end, an auxiliary field was introduced, the Lagrange multiplier u, which, as noted above, is determined completely by the variables \(q_i \) and \(\psi \). In fact, the relations (3.39–3.43) that define the theory exhibit a general mutual dependence that suggests the possibility of reversing fundamental roles. To develop this idea, we take \(V_Q =Q\) and focus on the close connection between the guidance equation (3.39) and the inhomogeneous equation (3.43) by deriving the latter’s general solution:

Proposition 6

(general solution of the inhomogeneous equation) The general solution of (3.43) is

$$\begin{aligned} u\left( {x,q_0 ,t} \right)= & {} \phi \left( {x,t} \right) -\frac{1}{\psi ^{*}\left( {x,t} \right) }\delta \left( {x-q\left( {t,q_0 } \right) } \right) \nonumber \\&+\int {\left[ {{\dot{q}}_i \left( {{t}'} \right) -\frac{1}{m}\frac{\partial S\left( {q\left( {{t}'} \right) ,{t}'} \right) }{\partial q_i \left( {{t}'} \right) }} \right] \frac{\partial }{\partial q_i \left( {{t}'} \right) }\left[ {\frac{G\left( {x-q\left( {{t}'} \right) ,t-{t}'} \right) }{\psi ^{*}\left( {q\left( {{t}'} \right) ,{t}'} \right) }} \right] } d{t}', \end{aligned}$$
(6.1)

where \(\phi \) and \(\psi \) obey the homogeneous (Schrödinger) equation and \(G(x-{x}',t-{t}')\) is the latter’s retarded Green function. (Proof: Appendix F)

We see from (6.1) that the legitimacy of the guidance equation (3.39) is correlated with the solution to the inhomogeneous equation (3.43); a deviation in the former is represented in the latter. This result prompts a change of perspective. For it is evident that, while our previous results show that the auxiliary field u is derived from \(\psi \) and \(q_i \) (Guidance Theorem 1 implies that \(\phi =\psi )\), it may equally be regarded as the theory’s primary descriptive element. Treating \(\psi \) and \(q_i \) as u’s constituents, its equation unifies their (one-way) interaction: the homogeneous part is the Schrödinger equation obeyed by the unmodified \(\psi \) and, as we show next using the solution (3.41), the inhomogeneous equation is equivalent to the particle guidance equation:

Guidance Theorem 3

(unified version). Let a material system of mass m be associated with a complex field \(u(x,t,q_0 )\) that obeys the inhomogeneous Schrödinger equation

$$\begin{aligned} i\hbar \frac{\partial u\left( {x,t} \right) }{\partial t}=-\frac{\hbar ^{2}}{2m}\partial _{ii} u\left( {x,t} \right) +V\left( x \right) u\left( {x,t} \right) +2\left. {\frac{\delta Q\left( {\rho \left( q \right) } \right) }{\delta \psi ^{*}\left( x \right) }} \right| _{q=q\left( {t,q_0 } \right) }, \end{aligned}$$
(6.2)

where \(\psi \) satisfies the homogeneous (Schrödinger) equation, Q is the quantum potential constructed from \(\psi \), \(q_i (t,q_0 )\) are the coordinates of a mobile singularity with initial position \(q_{0i} \), and the source term is given in (3.46). Then, the function

$$\begin{aligned} u\left( {x,t,q_0 } \right) =\psi \left( {x,t} \right) -\frac{1}{\psi ^{*}\left( {x,t} \right) }\delta \left( {x-q\left( {t,q_0 } \right) } \right) \end{aligned}$$
(6.3)

satisfies (6.2) if and only if the singularity-particle coordinates obey the guidance formula

$$\begin{aligned} {\dot{q}}_i =m^{-1}\left. {\partial _i S\left( x \right) } \right| _{x=q\left( {t,q_0 } \right) } .\hbox { (Proof: Appendix G)} \end{aligned}$$
(6.4)

These results validate the following model: a physical system comprises wave (\(\psi )\) and particle (\(q_i )\) aspects, which together produce, via a current generated by the functional gradient of the quantum potential concentrated around the particle trajectory, a ‘matter field’ described by the amplitude (6.3). The latter integrates the characteristics of both the wave (through the linear component \(\psi )\) and the particle (through the delta function) into a single spacetime field. The function u may, therefore, be regarded as an alternative field-theoretic description of the state of the composite system. Employing the solution (6.3), Eq. (6.2) governing the state melds the Schrödinger equation for the wave (its homogeneous part) with the guidance Eq. (6.4) (the inhomogeneous equation), the latter being the dynamical law of the delta function singularity representing the particle. That \(\psi \) suffers no reaction from the particle finds an explanation, for it is the field u of which the particle (along with \(\psi )\) is a source, whereas \(\psi \) is its (sourceless) homogeneous part. In short: the wave–particle composite both contributes to the source of u and is represented in its structure.

7 Properties of the unified model

(i)Comparison with the double solution. In our model, the source (3.46), built from the wave \(\psi \) and particle \(q_i \), produces a field \(-(1/\psi ^{*})\delta (x-q)\) that itself has the character of a point particle, namely, a delta function peaked around the particle trajectory (modulated by \(1/\psi ^{*}\)). The particle is, therefore, modelled as a mobile singularity moving in accordance with the guidance formula (6.4). Outside the singularity, u coincides with the relatively weak background field \(\psi \). Referring to Appendix B, we have, therefore, derived a formula of the type (B.1), with \(C = 1\). The spreading of \(\psi \) implies the spreading of u, but this does not undermine the integrity of the singularity whose solitary character is a permanent feature of u’s structure. The particle may be identified with the singular component of the field u.

Because of the appearance of the inverse complex conjugate wavefunction in the solution u, the model also implies the equality of the phases of the u and \(\psi \) waves in all space, including the region occupied by the particle. Using the polar representation, we have

$$\begin{aligned} u\left( {x,t} \right) =\left[ {\sqrt{\rho \left( {x,t} \right) }-\frac{1}{\sqrt{\rho \left( {x,t} \right) }}\delta \left( {x-q\left( {t,q_0 } \right) } \right) } \right] e^{{iS\left( {x,t} \right) }/\hbar }. \end{aligned}$$
(7.1)

The particle and the \(\psi \) wave thus ‘beat in phase’ (and hence do not ‘interfere’). This provides support for de Broglie’s contention that the corpuscle comprises a periodic process locked into the surrounding \(\psi \) wave.

We have, therefore, shown that the ideas informing the pilot-wave and double-solution theories are compatible rather than antagonistic; in our approach, the latter is a reformulation of the former where, in particular, the \(\psi \) wave retains its dual characteristics of physical field and probability amplitude.

We mention also some points of disparity with de Broglie’s version. De Broglie’s assumption that the linear component \(\psi \) may be multiplied by a constant C as in (B.1) is not borne out, since, although the resulting function obeys the inhomogeneous equation, the conditions of Guidance Theorem 1 require C = 1. A further difference with de Broglie’s approach appears in the extension of our theory to an n-body system (Sect. 8) where, in general, a single many-body field u is associated with the system rather than a collection of n ‘one-body’ fields.

(ii)Superposition. Given a set of solutions \(\psi _\mu \) to the homogeneous equation, the linear superposition \(\sum _\mu c_\mu \psi _\mu \), where \(c_\mu \) is constant, is also a solution. To each wave \(\psi _\mu \) and associated trajectory \(q_{\mu i}(t)\), there corresponds a solution \(u_\mu \) of the sort (6.3) connected with a quantum potential \(Q_{\mu }(\psi _{\mu })\). The solution of the inhomogeneous equation corresponding to the superposition is

$$\begin{aligned} u\left( {x,q_0 ,t} \right) =\sum \nolimits _\mu {c_\mu \psi _\mu } -\frac{1}{\sum _\mu {\left( {c_\mu \psi _\mu } \right) ^{*}} }\delta \left( {x-q\left( {t,q_0 } \right) } \right) , \end{aligned}$$
(7.2)

where \(q_i (t)\) is computed using the total wavefunction, as is Q. There is no simple relation between u and the \(u_\mu \hbox {s}\), or Q and the \(Q_\mu \hbox {s}\). In particular, the combination \({u}'=\sum _\mu c_\mu u_\mu \) does not generally represent a solution to the inhomogeneous equation. Rather, it obeys an equation involving a sum of source terms, each depending on one of the \(\psi _\mu \hbox {s}\), and this sum cannot generally be expressed as a single source depending just on the combination \(\sum _\mu c_\mu \psi _\mu \).

(iii)Covariance group of the inhomogeneous wave equation. Eq. (6.2) is covariant under the continuous transformations of the Galileo group,

$$\begin{aligned} {t}'=t+d,\quad {x}'_i =A_{ij} x_j -w_i t+c_i ,\quad {q}'_i =A_{ij} q_j -w_i t+c_i ,\quad \det A=1, \end{aligned}$$
(7.3)

where \(d,c_i ,A_{ij} ,w_i \) are constants. This is readily checked on noting that the field (7.1) transforms like \(\psi \), that is,

$$\begin{aligned} {u}'\left( {{x}',{t}'} \right) =u\left( {x,t} \right) e^{{i\left( {\frac{1}{2}mw_i w_i t-mA_{ij} w_i x_j } \right) }/\hbar }, \end{aligned}$$
(7.4)

and that the source term (3.46) is \(1/\psi ^{*}\) times a scalar. The theory is also covariant with respect to time reversal (for scalar V):

$$\begin{aligned} {x}'_i =x_i ,\quad {t}'=-t,\quad {\psi }'\left( {{x}',{t}'} \right) =\psi ^{*}\left( {x,t} \right) ,\quad {q}'_i \left( {{t}'} \right) =q_i \left( t \right) ,\quad {u}'\left( {{x}',{t}'} \right) =u^{*}\left( {x,t} \right) . \end{aligned}$$
(7.5)

(iv)Analogy with field–particle interaction in electromagnetism. The theory based on the inhomogeneous equation (6.2), where the particle density appears in the source and a general solution u is the superposition of a free solution and a source solution, is analogous, in broad terms, to a typical classical theory of particle–field interactions. However, the details of the quantum version differ significantly, because the field \(\psi \) and particle enjoy a more intimate relationship than in the classical template, which tends to emphasize their separateness. For definiteness, and with due recognition that we have studied a nonrelativistic system, we shall compare with the electromagnetic case where the general solution of the inhomogeneous Maxwell equations with a point-charge source, \(\Box A^{\mu }(x)=e \int \dot{q}^{\mu } \delta (x-q(\tau ))d\tau \), is the superposition of a free field (obeying the homogeneous wave equation) and the Liénard–Wiechert potentials generated by the particle, and the particle is subject to the Lorentz force law (the ‘guidance equation’). Some key differences between the quantum and classical-electromagnetic cases are: (a) the solution \(\psi \) to the homogeneous equation is involved in the source (3.46) of the field u, which is evident also in the appearance of \(\psi \) as a factor of the delta function in (6.3); (b) the quantum source (3.46) is not localized just on the particle track, but receives contributions from its neighbourhood through derivatives of the delta function; (c) issues surrounding radiative energy loss, radiation reaction, and mass renormalization that are central to the electromagnetic theory [12] are absent here. In particular, by construction, the energy of the quantum composite is that of the homogeneous field; (d) the quantum wave and particle equations are incorporated in the equation for the unified field u.

8 Many-body systems

For a composite system of n bodies with masses \(m_r \), \(r = 1,{\ldots }, n\), and wavefunction \(\psi (x_{1}, \ldots , x_{n},t)\), the function u obeys the inhomogeneous equation

$$\begin{aligned} i\hbar \frac{\partial u}{\partial t}=-\sum _{r=1}^n {\frac{\hbar ^{2}}{2m_r }\partial _{ri} \partial _{ri} u} +V\left( {x_1 , \ldots , x_n } \right) u+2\left. {\frac{\delta Q\left( {\rho \left( {q_1 , \ldots , q_n } \right) } \right) }{\delta \psi ^{*}\left( {x_1 , \ldots , x_n } \right) }} \right| _{q_r =q_r \left( t \right) }, \end{aligned}$$
(8.1)

where \(Q=-\sum _{r}(\hbar ^{2}/2m_{r}\sqrt{\rho })\partial _{r i}\partial _{r i}\sqrt{\rho }\). The n-body solution corresponding to the single-body formula (6.3) may be written down immediately by extending the index range:

$$\begin{aligned} u\left( {x_1 , \ldots , x_n ,q_{10} , \ldots , q_{n0} ,t} \right) =\psi \left( {x_1 , \ldots , x_n ,t} \right) -\frac{1}{\psi ^{*}\left( {x_1 , \ldots , x_n ,t} \right) }\prod _{r=1}^n {\delta \left( {x_r -q_r \left( {q_{10} , \ldots , q_{n0} ,t} \right) } \right) } . \end{aligned}$$
(8.2)

Following Guidance Theorem 3, this solution renders the inhomogeneous equation (8.1) equivalent to the many-body guidance equation

$$\begin{aligned} {\dot{q}}_{ri} =\frac{1}{m_r }\left. {\partial _{ri} S\left( {x_1 , \ldots , x_n } \right) } \right| _{x_r =q_r \left( {q_{r0} ,t} \right) } ,\quad r=1, \ldots , n. \end{aligned}$$
(8.3)

The function u is, therefore, defined irreducibly in the configuration spaces spanned by \(x_{1i}, \ldots , x_{ni} \) and \(q_{10i}, \ldots , q_{n0i} \). On the other hand, the single configuration space trajectory may be regarded as composed of n three-dimensional trajectories \(q_{ri} (q_{10}, \ldots , q_{n0},t)\), where each triplet of coordinates (\(i = 1,2,3\)) for given \(r = 1,{\ldots }\), n corresponds to one of the n corpuscles making up the particle component of the system. Each particle’s coordinates generally depend on the labels of the other n-1 particles, which expresses the nonlocal connection of the set.

When the wavefunction factorizes into n one-body factors, \(\psi (x_1, \ldots , x_n )=\prod _{r=1}^n \psi _r (x_r )\), the particle delta function does likewise, and the system decomposes into a set of n independent ‘single-body’ composite systems. The unified field u can be expressed in terms of the single-body fields \(u_r \) as a sort of factorization:

$$\begin{aligned} u-\psi =\prod \nolimits _{r=1}^n [u_r (x_r ,q_{r0} )-\psi _r (x_r )]. \end{aligned}$$
(8.4)

Using the function u in (8.4), (8.1) becomes equivalent to a set of n copies of the one-body guidance equation.

9 Conclusion

9.1 Revised postulates of the causal interpretation

Our initial aim was to find a nonstatistical vindication of the de Broglie–Bohm law that incorporates the nonreactive character of the particle on the \(\psi \) wave. To this end, we developed an analytical approach to the dynamics of a single system whose interacting wave and particle components, inseparable yet independent entities according to the usual de Broglie–Bohm theory, are treated as a unit insofar as key properties of the composite—conserved densities implied by assumed variational symmetries—coincide with those of a usual ‘quantum system’. In conjunction with the condition of time-reversal covariance, the particle equation and interaction potential become the de Broglie–Bohm law and the quantum potential, respectively. We also showed how the time-reversal assumption may be replaced by a statistical condition.

In the process, an alternative perspective for the causal theory emerged based on a function u, introduced initially as a Lagrange multiplier, for which the particle (together with \(\psi )\) is a source and whose governing inhomogeneous equation embraces both the de Broglie–Bohm law for the particle and the Schrödinger equation for the wave. The disparate elements of the de Broglie–Bohm theory now become aspects of the single field u, with the linear wave being its sourceless homogeneous part and the particle being represented by a highly concentrated amplitude that moves in accordance with the guidance law. The ‘true’ matter field of quantum theory is then to be identified not with \(\psi \) but with u, and we have argued that the model provides a realization of de Broglie’s hitherto unfulfilled double-solution programme. Since the representation of the particle by the u field is a (peaked) point following the track implied by the guidance law, its spatial probability distribution is identical to that of the particle postulated in the usual de Broglie–Bohm theory.

We summarize these findings in the following set of revised postulates for the causal theory:

\(1^{'}\):

A quantum system comprises a wave–particle composite whose state is described by a field u that obeys the inhomogeneous Eq. (6.2).

\(2^{'}\):

The field solution is \(u(x,t,q_0 )=\psi (x,t)-(1/\psi ^{*}(x,t))\delta (x-q(t,q_0 ))\) where the delta function represents the particle aspect.

\(3^{'}\):

For an ensemble of u fields with a common \(\psi \) component and an initial particle amplitude \(\delta (x-q_0 )\)whose location varies with \(q_{0i} \), the initial probability density is \(\rho _0 (q_0 )\).

The justification for postulate \(2^{'}\) is that of postulate 2 (see Guidance Theorem 1, Sect. 3). We have shown in Guidance Theorem 3 (Sect. 6) that the combination of postulates 1\('\) and 2\('\) furnishes the guidance law for the particle amplitude and this law is selected (within the analytical scheme we have used) as that which secures empirical equivalence with quantum theory. The extension of the postulates to many-body systems is straightforward, using the results of Sect. 8. The unified theory may be applied in an obvious way simply by replacing the point particle of the usual de Broglie–Bohm theory by the microscopic particle amplitude. It provides, therefore, a satisfactory alternative causal underpinning for quantum mechanics.

9.2 Open questions

In our analytical scheme, we demonstrated the constructive role played by spacetime (translation) and internal (gauge) symmetries in selecting dynamical equations. This was achieved through conditions imposed on the conserved densities generated by the symmetries via Noether’s theorem. The latter represents a potential limitation of the method; the correlation between a symmetry and a density established thereby is not unique since it depends intimately on the Lagrangian used in deriving the Euler–Lagrange equations. The latitude in the Lagrangian density goes considerably beyond the addition of a total divergence, and quite diverse expressions for a conserved density may accompany a given symmetry. Thus, while the analytical method detaches the guidance formula from the statistical postulate, the result may not be unique. Likewise, the inhomogeneous equation may be open to modification, although it should be noted that its property of unifying the Schrödinger and guidance equations is independent of the analytical procedure that produced it. Further justification of this equation may also focus on the physical basis of the absence of particle back reaction.

In examining alternative techniques to justify the revised postulates, we may consider utilizing a relativistic treatment since Lorentz symmetry is known to enforce a unique expression for the particle law in certain contexts [7]. We might also base the theory directly on an analysis of the forces acting within the system, and entertain less restrictive assumptions, such as allowing a more general interaction than a scalar potential or imbuing the particle with structure. In relation to the latter, the hydrodynamic analogy prompts the notion that the inertia of the particle may be acquired through a ‘virtual mass’ effect stemming from the displacement of the enveloping fluid [13]. We may also relax the requirement that the model reproduces exactly the current empirical content of quantum mechanics, although there are as yet no clues as to where deviations might occur. A further issue is the impact on the unified theory of employing the trajectory conception of the quantum state in place of the wavefunction [4, 5, 14]. This is a kind of ‘prequel’ to the de Broglie–Bohm theory in that the entire fleet of potential trajectories is utilized to define the quantum state (from which the time dependence of the wavefunction may be derived), but shorn of the additional corpuscle. This approach provides a congenial setting to introduce the latter.