The symmetrical rate-equations of particle–wave mechanics

Abstract

At various times in his life, Louis de Broglie firmly believed in the coexistence of both particle and an associated wave referring to “the theory of the double solution”, and an equation which he called “the guidance formula”. In an attempt to account for both particle and wave, the author has proposed a Lorentz invariant alternative to Newton’s second law which is developed in Hill (Zeitschrift fur angewandte Mathematik und Physik 69:133–145, 2018; Zeitschrift fur angewandte Mathematik und Physik 70:5–14, 2019; Zeitschrift fur angewandte Mathematik und Physik 70:131–153, 2019; Math Mech Solids 26:263–284, 2020; Math Mech Solids 25: 1763–1777, 2020; Zeitschrift fur angewandte Mathematik und Physik 72:1–14; Mathematics of particle–wave mechanical systems, Springer, Cham, 2022). Here, we summarise some of the major outcomes of this approach, including simple solutions of the proposed model exhibiting both sub-luminal and superluminal behaviour dependent upon the region of space-time, and two symmetrical sets of rate-differential relations for the two Lorentz invariants for a single spatial dimension x. One set follows the particle, while the other follows the wave, revealing a complete symmetry between the one-dimensional spatial physical force f and the force g in the “direction of time”. The existence of these symmetrical equations reinforces the parity and interchangeability of particles and waves.

1 Introduction

In a letter to Nature, de Broglie [1] first proposed that a sub-luminal particle moving with velocity u had an associated superluminal wave moving with velocity $c^2/u$, where c is the speed of light. de Broglie’s relation arises in consequence of a formal interchange of space and time $(x' = ct, t' = x/c)$, which transforms particles to waves and waves to particles. If with respect to a certain frame of reference, we identify all frames moving with sub-luminal relative velocity ${\mathscr {S}}_{sub}$, and all frames moving with superluminal relative velocity ${\mathscr {S}}_{super}$, then according to special relativity, there is no objective way of distinguishing to which of the two sets we might belong [9, 10]. That is, if we were all moving with a relative velocity greater than that of the speed of light, then according to the rules of Einstein’s special relativity, we would observe the same relative velocity values as if we were all moving with velocities less than the speed of light. This points to a fundamental deficiency in our understanding of the universe with respect to the speed of light and the notions of sub-luminal and superluminal motion, and therefore also to the notions of particles and waves.

The very existence of so-called dark energy and dark matter is also indicative of flaws in our mechanical reasoning. To overcome the failure of conventional mechanical thinking to resolve these issues, the author has proposed a special relativistic replacement for Newton’s second law [2,3,4,5,6,7,8] which accommodates both particle and wave energies e and $\mathscr {E}$, respectively, in accordance with de Broglie’s belief that both particle and wave simultaneously exist. Experiments are undertaken with the classical notions of either particles or waves in mind, so that in an experiment either particles or waves are reported, and only one of e or $\mathscr {E}$ is measured. Since nature tends to adopt the least energy structure, we propose that particles appear for $e < \mathscr {E}$ and waves for $\mathscr {E} < e$, but in either event, both a measurable and an unmeasurable energy exists.

We note that the approach of separating the wave-piloted particle solutions into sub-luminal particle-dominated states and superluminal wave-dominated states is unconventional. The more widely accepted conventional probabilistic approach of the Copenhagen interpretation also covers the wave–particle duality using a choice of basis for position and momentum. In conventional quantum mechanics, Schrödinger’s second-order wave equation is motivated from the classical wave equation, which within the proposed theory, arises naturally as a consequence. The model extends special relativistic mechanics by invoking the concept of a “force g in the direction of time”, and it is inclusive of both Newtonian mechanics and of quantum mechanics, in the form of Schrödinger’s second-order wave equation, and therefore inclusive of much of the existing theory of atomic physics.

The present paper provides a compact account, without calculation details, for some of the results from [2,3,4,5,6,7,8]. The formulae for the energy and momentum totals presented in the following section have not been given previously, nor the simple solutions and the symmetrical rate-differential relations of the subsequent two sections. Full calculation details are presented in the author’s forthcoming book [8].

2 Energy and momentum totals

For baryonic matter, the rest mass is supposed constant and nonzero for speeds $u<c$, and zero at the speed of light $u = c$. By inverting Einstein’s relation $pc = e_0(u/c)/ (1 - (u/c)^2)^{1/2}$ for velocity and momentum magnitudes;

$$\begin{aligned} \frac{u}{c} = \frac{pc}{(e_0^2 + (pc)^2)^{1/2}}, \end{aligned}$$

(2.1)

revealing the inescapable consequences that $e_0^2 > 0$ for $u < c$, $e_0^2 = 0$ for $u = c$ and $e_0^2 < 0$ for $u > c$. In view of the above considerations, the mechanical framework proposed in [2,3,4,5,6,7,8] describes a model for which a particle moving with sub-luminal particle velocity u, and with an associated superluminal wave velocity w, where $uw = c^2$, allows the following:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle e_{part} = \frac{e_0}{(1-(u/c)^2)^{1/2}}, \quad \\ \displaystyle e_{light} = cp_{light}, \quad \\ \displaystyle e_{wave} = \frac{e_0}{((w/c)^2-1)^{1/2}}, \end{array}\right. \quad \left\{ \begin{array}{ll} \displaystyle p_{part} = \frac{e_0u}{c^2(1-(u/c)^2)^{1/2}}, \quad &{} 0 \leqslant u/c< 1, \\ \displaystyle cp_{light} = e_{light}, \quad &{} u/c = 1, \\ \displaystyle p_{wave} = \frac{e_0w}{c^2((w/c)^2-1)^{1/2}}, \quad &{} 1< w/c < \infty , \end{array}\right. \end{aligned}$$

(2.2)

so that the total energy $e_{total}$ and total momentum $p_{total}$ become

$$\begin{aligned} e_{total} = e_{part} + e_{wave} = cp_{total} = c(p_{part} + p_{wave}) = e_0\left( \frac{1 + {u}/{c}}{1 - {u}/{c}}\right) ^{1/2} = e^\theta , \end{aligned}$$

(2.3)

where the angle $\theta$ is defined by the elementary relations

$$\begin{aligned} \theta = \frac{1}{2}\log \left( \frac{1+u/c}{1-u/c}\right) = \tanh ^{-1}(u/c),\quad \left( \frac{1+u/c}{1-u/c}\right) ^{1/2} = e^\theta . \end{aligned}$$

(2.4)

and $\theta$ is the angle in which a Lorentz invariance appears as a translational invariance, so that under Lorentz transformation, the energy and momentum totals involve only a different multiplicative factor. At the speed of light with $e_0 = 0$, the model admits privileged or singular states, which are characterised as generalisations of the known relation for light, namely $e_{light} = cp_{light}$, arising from $p_{light} = h/\lambda$ and $e_{light} = h\nu$, where h is the Planck constant, $\lambda$ is the wavelength and $\nu$ is the frequency $(c = \lambda \nu )$, and we may view the known relation for light as a particular case of $e_{total} = cp_{total}$.

3 Proposed model

In brief, for particle energy $e(\textbf{x}, t) = mc^2$, momentum vector ${\textbf{p}(\textbf{x}, t) = m\textbf{u}}$ with $e^2 -(cp)^2 = e_0^2$ where $p = (\textbf{p}\cdot \textbf{p})^{1/2}$, and wave energy $\mathscr {E}(\textbf{x}, t)$, the proposed Lorentz invariant Newton’s second law becomes:

$$\begin{aligned} \textbf{f} = \frac{\partial \textbf{p}}{\partial t} + {\nabla } e, \quad \quad {g} = \frac{1}{c^2} \frac{\partial e}{\partial t} + {\nabla }.\textbf{p}, \end{aligned}$$

(3.1)

where $(\textbf{f}, g)$ denote prescribed external forces. The term Lorentz invariant as applied here means that for a single spatial dimension x, the right-hand side of (3.1) remains unchanged under Lorentz transformations of both (x, t) and (p, e). The left-hand side involving the applied external forces $(\textbf{f}, g)$ may or may not be unchanged under Lorentz transformation. The spatial physical force $\textbf{f}$ aligns with our usual notion of force, while g corresponds to the mass–energy production which within conventional mechanics is usually taken to be zero (principle of conservation of mass). Conventional special relativistic mechanics arises from $\textbf{f} \ne \textbf{0}$ and $g = 0$, while the operator structure of quantum mechanics leading to Schrödinger’s second-order wave equation arises from $\textbf{f} = \textbf{0}$ and $g \ne 0$. The model therefore connects special relativity mechanics and the quantum mechanics arising from Schrödinger’s second-order wave equation. The first equation of (3.1) is a revise of Newton’s second law as the rate-change of momentum with the alternative expression ([8])

$$\begin{aligned} \textbf{f} = \frac{d \textbf{p}}{ dt} + \textbf{u}\wedge ({\nabla } \wedge \textbf{p}), \end{aligned}$$

(3.2)

where d/dt denotes the material or total time derivative defined by $\frac{d }{ dt} = \frac{\partial }{\partial t} + (\mathbf{{u}}\cdot {\nabla })$. This identity underscores a connection with Maxwell’s equations of electromagnetism and the Lorentz force formula $\textbf{F} = q(\textbf{E} + \textbf{v}\wedge \textbf{B})$ ([8]). The second equation of (3.1) represents a conventional conservation of mass or conservation of charge equation, except that the model incorporates a nonzero energy–mass production term $g(\textbf{x}, t)$.

Equation (3.1) is invariant under a Lorentz transformation and admits two singular or privileged states $e = \pm cp$ with $e_0 =0$, and which in all probability correspond to the two states of dark energy and dark matter. Given that the proposed model (3.1) admits two singular states, it is not unreasonable to speculate that these two states might be aligned with the two states known as dark energy and dark matter. However, the author has not yet established that the suspected relation corresponding to dark energy, namely $e = -pc$, is consistent with observations that the density of dark energy remains approximately constant as the universe expands. In the absence of a specific connection with this observation, the suggested meaning of the two privileged states $e = \pm cp$ must remain a matter of speculation.

If the external forces are derivable from a potential $V(\textbf{x}, t)$ through the relations, $\textbf{f} = -\nabla V$, $gc^2 = -\frac{\partial V}{\partial t}$, then $\frac{\partial \textbf{p}}{\partial t }= {\nabla } \mathscr {E}$, $\frac{\partial \mathscr {E}}{\partial t} = c^2{\nabla }\cdot \textbf{p}$, and in these circumstances $\textbf{f} = \frac{d\textbf{p}}{dt}$, $c^2{g} = \frac{d\mathscr {E}}{dt}$, both the momentum vector $\textbf{p}(\textbf{x}, t)$ and the wave energy $\mathscr {E}(\textbf{x}, t)$ satisfy the classical wave equation, and a conventional conservation of energy principle applies $e + \mathscr {E} + V = constant$. In this context, dark energy and dark matter, may occur either when $e = \mathscr {E}$ or when $e = -\mathscr {E}$, so that in the latter event, such privileged states are sustained under a zero potential $V = 0$, and which might well be the explanation of their preponderance throughout the universe. The operator relations of quantum mechanics $\textbf{p}\longrightarrow -i\hbar \nabla$ and $e \longrightarrow i\hbar \frac{\partial }{\partial t}$ where $\hbar = h/2\pi$, h Planck’s constant, become $\textbf{p}\longrightarrow -i\hbar \nabla$, $\mathscr {E} \longrightarrow - i\hbar \frac{\partial }{\partial t}$ and $e \longrightarrow i\hbar \frac{\partial }{\partial t} - V(\textbf{x}, t)$. Most importantly, the above model admits perfectly symmetrical particle–wave rate-equations for the Lorentz invariants, from either particle or wave perspectives. The symmetrical nature of these relations reinforces the view of the correctness of the interchangeable relationship between particles and waves, which we now briefly explain.

4 Two simple solutions

For prescribed external forces $(\textbf{f}, g)$, known solutions $u(\textbf{x}, t)$ can exhibit both sub-luminal and superluminal behaviour dependent upon the region of space-time [2, 8]. For a single spatial dimension x, two simple solutions are $u(x, t) = x/t$, ($f = 0$ and $g \ne 0$) and $u(x, t) = c^2t/x$, ($f \ne 0$, $g = 0$) with, respectively,

$$\begin{aligned} e&= \frac{e_0ct}{((ct)^2 - x^2)^{1/2}}, \quad \quad p = \frac{e_0x}{c((ct)^2 - x^2)^{1/2}}, \quad \quad gc = \frac{e_0}{((ct)^2 - x^2)^{1/2}}, \\ e&= \frac{e_0x}{(x^2 - (ct)^2)^{1/2}}, \quad \quad p = \frac{e_0t}{(x^2 - (ct)^2)^{1/2}}, \quad \quad f = \frac{e_0}{(x^2 - (ct)^2)^{1/2}}, \end{aligned}$$

which are clearly complementary to each other ($uu' = c^2$) and both exhibit both sub-luminal and superluminal motion, according as ($x < ct$) or ($x > ct$). For the first solution, we note that $du/dt = 0$, and that at some fixed point in space, say $x = a$, we have ${u(a, t)}/{c} = {a}/{ct} > 1$ if $ct < a$ (superluminal) and ${a}/{ct} < 1$ if $ct > a$ (sub-luminal) which as far as an observer is concerned means that shortly after time $t = a/c$, a particle suddenly appears moving at a velocity just below that of light. Similarly for the second solution, with the physical effect that for an observer, shortly after time $t = a/c$, a particle suddenly appears to vanish. These simplest of solutions demonstrate that the same velocity field $u(\textbf{x}, t)$ might exhibit both sub-luminal and superluminal behaviour, and that our notion of empty space may be questionable.

5 Symmetrical rate-differential relations

In this section, for a single spatial dimension x, we show that the two postulated basic Eq. (3.1) admit symmetrical rate-differential relations, which by necessity can only apply in a one-dimensional context, since we have a mind a correspondence that applies under interchanges of space and time. For a single space dimension x, in terms of the arbitrary applied forces f(x, t) and g(x, t), the two postulated basic Eq. (3.1) become ([8])

$$\begin{aligned} {xf - c^2tg} = \frac{\partial \eta }{\partial t} + \frac{\partial \xi }{\partial x}, \quad \quad {xg -tf} = \frac{1}{c^2} \frac{\partial \xi }{\partial t} + \frac{\partial \eta }{\partial x}, \end{aligned}$$

(5.1)

where $\xi (x, t) = ex - c^2 pt$ and $\eta (x, t) = px - et$ denote two Lorentz invariants for which $\xi ^2 - (c\eta )^2 = e_0^2(x^2 - (ct)^2)$. This means that under Lorentz transformations of both (x, t) and (p, e), both $\xi$ and $\eta$ are left unchanged. The fundamental time and space total derivatives underpinning the structure of the proposed model are defined by

$$\begin{aligned} \frac{d}{dt}&= \left( \frac{d}{dt}\right) _{part} = \frac{\partial }{\partial t} + \left( \frac{dx}{dt}\right) _{part}\frac{\partial }{\partial x} = \frac{\partial }{\partial t} + u\frac{\partial }{\partial x}, \nonumber \\ \frac{d}{dx}&= \left( \frac{d}{dx}\right) _{wave} = \frac{\partial }{\partial x} + \left( \frac{dt}{dx}\right) _{wave}\frac{\partial }{\partial t} = \frac{\partial }{\partial x} + \frac{u}{c^2}\frac{\partial }{\partial t}, \end{aligned}$$

(5.2)

which are not Lorentz invariant, but under Lorentz transformation, transform in exactly the same manner. The first is the standard total or material time derivative following the particle which is inherited from continuum mechanics, while the second is a spatial total derivative following the wave. In terms of these total derivatives, from their definitions and using (2.45.1), the two invariants satisfy the perfectly symmetrical rate-differential relations ([8])

$$\begin{aligned} e\frac{d\xi }{dt}&= {fc^2\eta }, \quad \quad e\frac{d\eta }{dt} = {f\xi } - e_0^2, \nonumber \\ {e}\frac{d\xi }{dx}&= c^2g\eta + e_0^2, \quad \quad e\frac{d\eta }{dx} = g\xi , \end{aligned}$$

(5.3)

revealing formal symmetries when $x \longleftrightarrow ct$, $\xi \longleftrightarrow c\eta$, $f \longleftrightarrow cg$ and $+ e_0^2 \longleftrightarrow - e_0^2$, the latter being exactly in accord with the observation from (2.1). From a particle perspective, only the spatial physical force f figures in the rate-equations involving the material or total time derivative d/dt, while from the wave perspective, it is only the force in the direction of time g that plays a role in the rate-equations involving the spatial derivative d/dx.

To gain further insight into the transformation properties of Eq. (5.3), by inverting the relations $\xi = ex - c^2 pt$ and $\eta = px - et$ to express e and p as functions of $\xi$ and $\eta$ and by examination of (2.45.1), it is clear that under the space-time transformation $x'= ct$ and $t' = x/c$, we have $u' = c^2/u$, $e' = cp$, $cp' = e$, $f' =f$, $g' = g$, $\xi ' = -\xi$, $\eta ' = -\eta$, and that under this transformation the two total derivatives d/dt and d/dx given by (5.2) transform in the following manner

$$\begin{aligned} \frac{d}{dt'} = \frac{c}{u}\frac{d}{dt}, \quad \frac{d}{dx'} = \frac{c}{u}\frac{d}{dx}, \quad e' \frac{d}{dt'} = e\frac{d}{dt}, \quad e'\frac{d}{dx'} = e\frac{d}{dx}. \end{aligned}$$

(5.4)

The rate-differential relations (5.3) demonstrate an independence of the two perspectives and are important in terms of identifying the immediate consequences if either f or g is zero. For special relativistic mechanics with $g = 0$, we might deduce that ${e}{d\xi }/{dx} = e_0^2$ and $e{d\eta }/{dx} = 0$, noting that we must be careful not to conclude that ${d\eta }/{dx} = 0$ implies only $\eta = constant$; rather this equation constitutes a certain first-order partial differential equation with general solution involving an arbitrary function. Similar comments apply to de Broglie’s guidance formula for which $f = 0$ and $g \ne 0$.