A Note on Schrödinger Operator Relations and Power-Law Energies

Abstract

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-momentum relation to determine new families of nonlinear partial differential relations. The Planck–de Broglie duality principle arises from Planck’s energy expression $e=h\nu$, de Broglie’s equation for momentum $p=h/\lambda$, and Einstein’s special relativity energy, where h is the Planck constant, $\nu$ and $\lambda$ are the frequency and wavelength, respectively, of an associated wave having a wave speed $w=\nu \lambda$. The author has extended these relations to a family that is characterised by a second fundamental constant $h^{\prime }$ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. In this note, we apply generalized Schrödinger operator relations and the power-law relations to generate a new family of nonlinear partial differential equations that are characterised by the constant $\kappa =h^{\prime }/h$ such that $\kappa =0$ corresponds to the Klein–Gordon equation. The resulting partial differential equation is unusual in the sense that it admits a stretching symmetry giving rise to both similarity solutions and simple harmonic travelling waves. Three simple solutions of the partial differential equation are examined including a separable solution, a travelling wave solution, and a similarity solution. A special case of the similarity solution admits zeroth-order Bessel functions as solutions while generally, it reduces to solving a nonlinear first-order ordinary differential equation.

1. Introduction

The classical distinction between waves and particles breaks down at the quantum level, and all matter may exhibit both particle-like and wave-like properties so that matter may behave either as a localised particle or as a spreading wave, depending on the circumstance and how it is observed. Particles occupy a specific location in space and are characterised by their energy e and momentum p and are adequately described by Einstein’s special relativity. On the other hand, waves spread out over space, are characterised by frequency, wavelength, and amplitude, and are described by quantum wave theory for which the energy and momentum variables become operators involving probability density wave functions. Schrödinger’s quantum wave theory is not compatible with special relativity, and the determination of a formal connection between special relativity and quantum mechanics attracted many researchers, including Einstein, resulting in Dirac’s theory and the Klein–Gordon equation (see [1,2,3,4]) and culminating in quantum field theory, which provides a fully relativistic description of quantum phenomena (see, for example, [5,6,7]).

The formal relations underpinning the particle-wave duality are the Planck–de Broglie relations for energy $e=m{c}^2$ and momentum $p=mu$ of an elementary particle, where m is the mass and u is the particle velocity. de Broglie [8,9,10] supplemented Planck’s particle energy expression $e=h\nu$ with the equation for particle momentum $p=h/\lambda$ to establish the dual particle-wave nature of matter, where h is the Planck constant, $\nu$ and $\lambda$ are the frequency and wavelength, respectively, of the associated wave having a wave speed $w=\nu \lambda$ (see [11,12,13,14] for further articles relating to de Broglie’s work). In [15], these basic relations have been generalised to involve a second fundamental constant $h^{\prime }$, and these new relations are shown to apply to the Lorentz invariant power-law energy-momentum relations proposed in [16] (see also [17] (p. 49)).

Einstein’s special relativistic mass-energy expression together with the Schrödinger operator relations yield the special relativistic Klein–Gordon partial differential Equation (21). In this note, we assume that the particle motion is described by the Lorentz invariant power-law energy-momenta relations derived in [16], and for generalised Schrödinger operator relations, we derive a family of nonlinear partial differential Equation (60) arising from one possible operator interpretation of the corresponding algebraic identity. This particular interpretation arises from following a similar development to that for the Klein–Gordon equation. The resulting partial differential Equation (60) is characterised by a stretching symmetry which, in particular, means that there is a similarity solution for which the partial differential equation may be reduced to a second-order ordinary differential equation and which may subsequently be reduced to one of first order. There are other possible operator interpretations which are most likely distinct from one another, one of which gives rise to fractional partial differential equations which we do not consider here.

In the following section, we present preliminary material in four sub-sections for the conventional Schrödinger operator relations including special relativity and Lorentz transformations; the Lorentz invariant extended Newton’s second law; the operator relations based on two Lorentz invariants, and the Klein–Gordon partial differential equation. In the subsequent section, we summarise the Lorentz invariant power-law energy-momenta relations from [16], which we suppose apply to an elementary particle. In the next section, we propose compatible operator relations which generalise Schrödinger’s operator relations, which we then use to generalise the Klein–Gordon equation, namely (59) or (60). In the next section, three simple solutions of the resulting nonlinear partial differential equation are examined, including a separable solution, a travelling wave solution, and a similarity solution. The final section of the paper contains a summary and some conclusions, and Appendix A contains some formal details involved in the reduction of the second-order nonlinear Equation (78) to one of first order, namely (93).

2. Schrödinger’s Operator Relations

In this section, we make some general comments relating to the Schrödinger operator relations. In the formulation of Schrödinger’s quantum wave theory, the variables become operators involving wave functions such that for a single spatial dimension x, the wave theory operator relations for momentum p and energy e become, respectively, $p⟶-i\hslash \partial /\partial x$ and $e⟶i\hslash \partial /\partial t$, namely

$$\begin{array}{c}\hfill p=-i\hslash {\displaystyle \frac{\partial \psi }{\partial x}},e=i\hslash {\displaystyle \frac{\partial \psi }{\partial t}},\end{array}$$

(1)

for some function $\psi (x,t)$. However, to the author’s knowledge, it appears that with the passing of time, the original motivation and formulation of these specific relations seems to have been lost. For some historical information relating to the formulation of wave mechanics, we refer the reader to [18,19]. Here, we simply note that with the usual definition of particle velocity $u=dx/dt=p{c}^2/e$, the relations (1) correspond to a certain orthogonality scalar product condition. Thus,

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \psi }{\partial x}}cdt+{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \psi }{\partial t}}dx=0.\end{array}$$

(2)

2.1. Special Relativity and Lorentz Transformations

With reference to Figure 1, capital letters refer to the fixed $(X,T)$ reference frame, and lower-case letters refer to the moving $(x,t)$ reference frame. If v denotes the constant relative frame velocity, then the Lorentz transformations $(0⩽v<c)$ are given by

$$\begin{array}{c}\hfill x={\displaystyle \frac{X-vT}{{[1-{(v/c)}^2]}^{1/2}}},t={\displaystyle \frac{T-vX/c^2}{{[1-{(v/c)}^2]}^{1/2}}},\end{array}$$

(3)

with the identity transformation $x=X$ and $t=T$ given by $v=0$. For $0⩽v<c$ the Lorentz invariant energy-momentum relations are given by

$$\begin{array}{c}\hfill p={\displaystyle \frac{P-Ev/c^2}{{[1-{(v/c)}^2]}^{1/2}}},e={\displaystyle \frac{E-Pv}{{[1-{(v/c)}^2]}^{1/2}}},\end{array}$$

(4)

and together Equation (4) are referred to as the Lorentz invariant energy-momentum relations corresponding to the Lorentz transformation (3). From each of these relations, the two Lorentz invariants ${\left(ct\right)}^2-x^2={\left(cT\right)}^2-X^2$ and $e^2-{\left(pc\right)}^2=E^2-{\left(Pc\right)}^2$ are apparent and the following basic identities may be verified

$$\begin{array}{cc}\hfill & ct+x={\left({\displaystyle \frac{1-v/c}{1+v/c}}\right)}^{1/2}(cT+X),ct-x={\left({\displaystyle \frac{1+v/c}{1-v/c}}\right)}^{1/2}(cT-X),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & e+pc={\left({\displaystyle \frac{1-v/c}{1+v/c}}\right)}^{1/2}(E+Pc),e-pc={\left({\displaystyle \frac{1+v/c}{1-v/c}}\right)}^{1/2}(E-Pc),\hfill \end{array}$$

(6)

and it is a simple matter to show that

$$\begin{array}{c}\hfill \left({\displaystyle \frac{ct+x}{ct-x}}\right)=\left({\displaystyle \frac{1-v/c}{1+v/c}}\right)\left({\displaystyle \frac{cT+X}{cT-X}}\right),\left({\displaystyle \frac{e+pc}{e-pc}}\right)=\left({\displaystyle \frac{1-v/c}{1+v/c}}\right)\left({\displaystyle \frac{E+Pc}{E-Pc}}\right),\end{array}$$

(7)

from which we might deduce the combined Lorentz invariant

$$\begin{array}{c}\hfill \left({\displaystyle \frac{e+pc}{e-pc}}\right)\left({\displaystyle \frac{ct-x}{ct+x}}\right)=\left({\displaystyle \frac{E+Pc}{E-Pc}}\right)\left({\displaystyle \frac{cT-X}{cT+X}}\right).\end{array}$$

(8)

From a special relativistic perspective, the conventional signatures of the operators (1) are entirely meaningful in the sense of being precisely what is required to produce the correct Lorentz invariances. Further, from Equation (3), we have the chain-rule differential formulae

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial x}}={\displaystyle \frac{1}{{(1-{(v/c)}^2)}^{1/2}}}\left\{{\displaystyle \frac{\partial }{\partial X}}+{\displaystyle \frac{v}{c^2}}{\displaystyle \frac{\partial }{\partial T}}\right\},{\displaystyle \frac{\partial }{\partial t}}={\displaystyle \frac{1}{{(1-{(v/c)}^2)}^{1/2}}}\left\{{\displaystyle \frac{\partial }{\partial T}}+v{\displaystyle \frac{\partial }{\partial X}}\right\},\end{array}$$

(9)

so that if we adopt the corresponding relations $P⟶-i\hslash \partial /\partial X$ and $E⟶i\hslash \partial /\partial T$, then along with $p⟶-i\hslash \partial /\partial x$ and $e⟶i\hslash \partial /\partial t$, the transformation differential Formulae (9) are precisely equivalent to the Lorentz invariant energy-momentum relations (4). Furthermore, as shown below, the given signatures of the operators are also precisely those required to ensure the Lorentz invariances of the operators arising from the Lorentz invariants $\xi =ex-c^{2}pt$ and $\eta =px-et$ (see also [17] (p. 40) for further discussion on the Lorentz invariants $\xi$ and $\eta$).

2.2. Lorentz Invariant Extended Newton’s Second Law

The mechanical model proposed in [17,20,21] formally includes both Newton’s second law and Schrödinger’s quantum wave theory and remains invariant under the combined transformations (3) and (4) (see [17] (p. 185)). Thus, for a single spatial dimension x, the proposed equations become simply

$$\begin{array}{c}\hfill f={\displaystyle \frac{\partial p}{\partial t}}+{\displaystyle \frac{\partial e}{\partial x}},g={\displaystyle \frac{1}{c^2}}{\displaystyle \frac{\partial e}{\partial t}}+{\displaystyle \frac{\partial p}{\partial x}},\end{array}$$

(10)

where f and g denote certain applied external forces. It can be shown that these equations remain invariant under the Lorentz transformations (3) and the Lorentz invariant energy-momentum relations (4). In other words, the following relations hold:

$$\begin{array}{c}\hfill f={\displaystyle \frac{\partial p}{\partial t}}+{\displaystyle \frac{\partial e}{\partial x}}={\displaystyle \frac{\partial P}{\partial T}}+{\displaystyle \frac{\partial E}{\partial X}},g={\displaystyle \frac{1}{c^2}}{\displaystyle \frac{\partial e}{\partial t}}+{\displaystyle \frac{\partial p}{\partial x}}={\displaystyle \frac{1}{c^2}}{\displaystyle \frac{\partial E}{\partial T}}+{\displaystyle \frac{\partial P}{\partial X}},\end{array}$$

(11)

and these details can be found in [17] (p. 185). Importantly, we observe from (1) that the operator relations (10) are such that $f=0$ and $g={\displaystyle \frac{i\hslash }{c^2}}\left({\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}\right)$, and on setting g as the source of an applied potential, together with the wave equation operator, provides the usual basis for the derivation of Schrödinger’s quantum wave equation [17] (p. 282).

2.3. Invariants Invariants $\xi =ex-c^{2}pt$ and $\eta =px-et$

Following [17] (p. 40), we adopt the convention that the operator corresponding to a given variable is subscripted, so that the operators for momentum p and energy e become, respectively,

$$\begin{array}{c}\hfill L_p=-i\hslash {\displaystyle \frac{\partial }{\partial x}},L_e=i\hslash {\displaystyle \frac{\partial }{\partial t}},\end{array}$$

(12)

and from $\xi =ex-c^{2}pt$ and $\eta =px-et$, we have $L_{\xi }$ and $L_{\eta }$ defined by

$$\begin{array}{c}\hfill L_{\xi }=i\hslash \left(x{\displaystyle \frac{\partial }{\partial t}}+c^{2}t{\displaystyle \frac{\partial }{\partial x}}\right),L_{\eta }=-i\hslash \left(x{\displaystyle \frac{\partial }{\partial x}}+t{\displaystyle \frac{\partial }{\partial t}}\right).\end{array}$$

(13)

These are Lorentz invariant operators, since from (3) and (9), we may eventually deduce

$$\begin{array}{c}\hfill L_{\xi }=i\hslash \left(X{\displaystyle \frac{\partial }{\partial T}}+c^{2}T{\displaystyle \frac{\partial }{\partial X}}\right),L_{\eta }=-i\hslash \left(X{\displaystyle \frac{\partial }{\partial X}}+T{\displaystyle \frac{\partial }{\partial T}}\right),\end{array}$$

(14)

and consequently, $L_{\xi }$ and $L_{\eta }$ are Lorentz invariant. The two operators can be shown to commute, namely $L_{\xi }{L}_{\eta }=L_{\eta }{L}_{\xi }$, which means that the corresponding observables $\xi$ and $\eta$ are simultaneously measurable and that they share the same eigenfunctions (see for example [22] (p. 101)). Also, the operator $L_{\xi }$ coincides with the Lorentz operator $L_v$ arising from the one-parameter group (3), which is given by

$$\begin{array}{c}\hfill L_v=-\left(t{\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{x}{c^2}}{\displaystyle \frac{\partial }{\partial t}}\right).\end{array}$$

(15)

2.4. Klein–Gordon Partial Differential Equation

In this subsection, assuming Einstein’s energy-momentum relation $e^2-{\left(pc\right)}^2=e_0^2$, we show that the Lorentz invariant operators (13) give rise to the Klein–Gordon partial differential Equation (21) arising as the operator equivalent of the algebraic identity ${\xi }^2-{\left(c\eta \right)}^2=e_0^2(x^2-{\left(ct\right)}^2)$. The various properties of the two operators $L_{\xi }$ and $L_{\eta }$ are most apparent in terms of the characteristic coordinates $\alpha =ct+x$ and $\beta =ct-x$, and we comment that in terms of $\alpha$ and $\beta$, the constraint (2) becomes simply ${\displaystyle \frac{d\alpha }{d\beta }}={\displaystyle \frac{\partial \psi }{\partial \beta }}/{\displaystyle \frac{\partial \psi }{\partial \alpha }}$.

From the differential formulae

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial x}}={\displaystyle \frac{\partial }{\partial \alpha }}-{\displaystyle \frac{\partial }{\partial \beta }},{\displaystyle \frac{\partial }{\partial t}}=c\left({\displaystyle \frac{\partial }{\partial \alpha }}+{\displaystyle \frac{\partial }{\partial \beta }}\right),\end{array}$$

(16)

we may deduce

$$\begin{array}{c}\hfill L_{\xi }=i\hslash c\left(\alpha {\displaystyle \frac{\partial }{\partial \alpha }}-\beta {\displaystyle \frac{\partial }{\partial \beta }}\right),L_{\eta }=-i\hslash \left(\alpha {\displaystyle \frac{\partial }{\partial \alpha }}+\beta {\displaystyle \frac{\partial }{\partial \beta }}\right),\end{array}$$

(17)

and from which there arises the simple formulae

$$\begin{array}{c}\hfill L_{\xi -c\eta }=2i\hslash c\alpha {\displaystyle \frac{\partial }{\partial \alpha }},L_{\xi +c\eta }=-2i\hslash c\beta {\displaystyle \frac{\partial }{\partial \beta }}.\end{array}$$

(18)

Thus, the formal operator equation corresponding to the identity ${\xi }^2-{\left(c\eta \right)}^2=e_0^2(x^2-{\left(ct\right)}^2)$ or $(\xi -c\eta )(\xi +c\eta )=-e_0^2\alpha \beta$ gives rise to the Klein–Gordon Equation (21) ([22] (p. 312) or [17] (p. 289))

$$\begin{array}{c}\hfill L_{\xi -c\eta }{L}_{\xi +c\eta }\psi =4{(\hslash c)}^2\alpha \beta {\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}=-e_0^2\alpha \beta \psi ,\end{array}$$

(19)

for some function $\psi (x,t)$ and from which we have

$$\begin{array}{c}\hfill 4{\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}=-{\left({\displaystyle \frac{e_0}{\hslash c}}\right)}^2\psi .\end{array}$$

(20)

Alternatively, in terms of the conventional $(x,t)$ wave equation operator, we have the more usual form of the Klein–Gordon equation [22] (p. 313):

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}=-{\left({\displaystyle \frac{e_0}{\hslash }}\right)}^2\psi .\end{array}$$

(21)

For the simple harmonic travelling wave, thus

$$\begin{array}{c}\hfill \psi (x,t)={\psi }_{0}exp\left(i(kx-\omega t)\right),\end{array}$$

(22)

where ${\psi }_0$ is a constant, $k=2\pi /\lambda$ and $\omega =2\pi \nu$, the dispersion relation ${\omega }^2-{\left(kc\right)}^2={(e_0/\hslash )}^2$ arises from the Klein–Gordon Equation (21) since substitution of (22) into (21) yields the dispersion relation ${\omega }^2-{\left(kc\right)}^2={(e_0/\hslash )}^2$, which, when using the Planck–de Broglie relations, $e=h\nu$ and $p=h/\lambda$, where $\nu$ is the frequency and $\lambda$ is the wavelength, so that $\omega =e/\hslash$ and $k=p/\hslash$, is equivalent to the Einstein energy relation $e^2-{\left(pc\right)}^2=e_0^2$.

3. Elementary Particle Power-Law Energy-Momenta Relations

In this paper, for a single space dimension x, we suppose $(x,t)$ characterises the location x of an elementary particle at time t which is moving with velocity $u(x,t)=dx/dt$, with energy $e=m{c}^2$ and momentum $p=mu$, and given by the Lorentz invariant power-law relations

$$\begin{array}{cc}\hfill & e\left(u\right)={\displaystyle \frac{e_0}{{(1-{(u/c)}^2)}^{1/2}}}{\left({\displaystyle \frac{1+(u/c)}{1-(u/c)}}\right)}^{\kappa /2},\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill & cp\left(u\right)={\displaystyle \frac{e_0(u/c)}{{(1-{(u/c)}^2)}^{1/2}}}{\left({\displaystyle \frac{1+(u/c)}{1-(u/c)}}\right)}^{\kappa /2},\hfill \end{array}$$

(23b)

where $\kappa$ denotes an arbitrary constant and $e_0$ denotes the particle rest energy. These relations, first obtained in [16] (see also [17] (p. 49)), extend Einstein’s energy-momentum relations which are included by the case $\kappa =0$. We comment that the Lorentz invariance of these relations is revealed through the arbitrary constant $\kappa$, which we may give meaning both through the generalised Planck-de Broglie relations (24) for which $\kappa =h^{\prime }/h$, as well as through the generalised Schrödinger operator relations (38) for which $\kappa =\mu /\tau$. As described below (see Equation (35)), this is a less restrictive Lorentz invariance than conventional Lorentz invariance in the sense that it allows for the power-law factors in (23) for $\kappa \ne 0$ and includes conventional special relativity invariance through the special case $\kappa =0$.

In [15], the conventional Planck–de Broglie relations $e=h\nu$ and $p=h/\lambda$ are generalised by the expressions

$$\begin{array}{c}\hfill e=h\nu +h^{\prime }c/\lambda ,p=h/\lambda +h^{\prime }\nu /c,\end{array}$$

(24)

where $h^{\prime }$ denotes a new fundamental constant and the family of energy expressions (23) bears exactly the same relationship with this generalisation as does Einstein’s energy expression with the Planck–de Broglie formulae where $\kappa =h^{\prime }/h$. The dimensions of $h^{\prime }$ are those of h, and physically, the relations (24) reflect the parity of space and time in special relativity. Planck’s constant is a space characteristic so that $h^{\prime }$ could be referred as “Planck’s constant in time”.

At this point, there are no known experimental values for the constant $h^{\prime }$. However, there is evidence (such as [23,24] and the numerous references therein) that the fine structure constant, which involves the Planck constant h and the velocity of light c, may not be an absolute constant and may be connected to the expansion of space. This gives rise to the possibility that Planck’s constant may not be strictly constant (see [25,26]). We further note that the expressions (24) are only novel when the wave is not travelling at the speed of light, since in that case, $\lambda \nu =c$ and (24) become $e=(h+h^{\prime })\nu$ and $p=(h+h^{\prime })/\lambda$. This means that experimentally, in order to identify $h^{\prime }$, data from either sub-luminal or superluminal waves must be examined.

In terms of the wave number $k=2\pi /\lambda$ and angular frequency $\omega =2\pi \nu$, these new relations become $e=\hslash \omega +{\hslash }^{\prime }ck$ and $pc=\hslash ck+{\hslash }^{\prime }\omega$, from which we may deduce the relation

$$\begin{array}{c}\hfill {\displaystyle \frac{e+cp}{e-cp}}={\displaystyle \frac{(\hslash +{\hslash }^{\prime })(\omega +ck)}{(\hslash -{\hslash }^{\prime })(\omega -ck)}}=\left({\displaystyle \frac{\hslash +{\hslash }^{\prime }}{\hslash -{\hslash }^{\prime }}}\right)\left({\displaystyle \frac{\omega +ck}{\omega -ck}}\right)=\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)\left({\displaystyle \frac{\omega +ck}{\omega -ck}}\right),\end{array}$$

(25)

where $\hslash =h/2\pi$, ${\hslash }^{\prime }=h^{\prime }/2\pi$ and $\kappa =h^{\prime }/h={\hslash }^{\prime }/\hslash$.

The relations (23) arise from the following Lorentz invariant relations

$$\begin{array}{c}\hfill {\displaystyle \frac{cp\left(u\right)}{e\left(u\right)}}={\displaystyle \frac{u}{c}},{\displaystyle \frac{de}{dp}}=c\left({\displaystyle \frac{\kappa +u/c}{1+\kappa u/c}}\right),\end{array}$$

(26)

and also satisfy

$$\begin{array}{c}\hfill e{\left(u\right)}^2-{\left(cp\left(u\right)\right)}^2=e_0^2{\left({\displaystyle \frac{1+(u/c)}{1-(u/c)}}\right)}^{\kappa }.\end{array}$$

(27)

On using $u/c=cp/e$, this equation can be alternatively expressed as

$$\begin{array}{c}\hfill e^2-{\left(cp\right)}^2=e_0^2{\left({\displaystyle \frac{e+cp}{e-cp}}\right)}^{\kappa },\end{array}$$

(28)

which in turn becomes

$$\begin{array}{c}\hfill {(e-cp)}^{\kappa +1}=e_0^2{(e+cp)}^{\kappa -1}.\end{array}$$

(29)

The invariance of (26) under the Lorentz transformations (3) and (4) may be established either by using Einstein’s addition of velocities law (see [16] or [17] (p. 49)) or by using the variables $\rho =e+pc$ and $\sigma =e-pc$ and the Lorentz transformations (4) and (7)₂. In terms of these new variables, Equation (26)₂ becomes simply

$$\begin{array}{c}\hfill {\displaystyle \frac{d\rho }{d\sigma }}=-{\displaystyle \frac{(1+\kappa )}{(1-\kappa )}}{\displaystyle \frac{\rho }{\sigma }},\end{array}$$

(30)

and from (7)₂, it is clear that

$$\begin{array}{c}\hfill {\displaystyle \frac{\rho }{\sigma }}={\displaystyle \frac{e+pc}{e-pc}}=\left({\displaystyle \frac{1-v/c}{1+v/c}}\right)\left({\displaystyle \frac{E+Pc}{E-Pc}}\right),\end{array}$$

(31)

and that (30) is unchanged by (31) and integrates immediately to give ${\rho }^{\lambda }\sigma =constant$ or ${\rho }^{1-\kappa }{\sigma }^{1+\kappa }=constant$, consistent with (29).

The important issue is that for $\kappa \ne 0$, while the two relations (26) are fully Lorentz invariant, it nevertheless constitutes a less restricted Lorentz invariance in the sense that the resulting integral (either (28) or (29)) are not fully Lorentz invariant without the additional assumption that the constant rest energies $e_0$ and $E_0$ vary with the relative frame velocity v. Thus,

$$\begin{array}{c}\hfill E_0=e_0{\left({\displaystyle \frac{1-(v/c)}{1+(v/c)}}\right)}^{\kappa /2}.\end{array}$$

(32)

This assumption, together with (7)₂, means that the actual invariant is

$$\begin{array}{c}\hfill e_0^2{\left({\displaystyle \frac{e+pc}{e-pc}}\right)}^{\kappa }=E_0^2{\left({\displaystyle \frac{E+Pc}{E-Pc}}\right)}^{\kappa },\end{array}$$

(33)

and therefore both the left and right-hand sides of (28) are Lorentz invariants. For conventional special relativity ($\kappa =0$), the rest energies are such that $e_0=E_0$ so that this is not an issue, and the equation $e^2-{\left(cp\right)}^2=e_0^2$ is fully Lorentz invariant and provides the formal basis giving rise to the linear Klein–Gordon equation of quantum wave theory. However, for $\kappa \ne 0$, the correct operator interpretation of either (28) or (29) hinges on the determination of the correct Lorentz invariant formulation.

For $\kappa \ne 0$, the machinery of Schrödinger’s quantum wave theory is not as meaningful as for $\kappa =0$ in the sense that for $\kappa \ne 0$, there is no unique operator interpretation of either (28) or (29). Here, because of (33) and both the left and right-hand sides of (28) are Lorentz invariants, we adopt (28) in preference to (29). We exploit (28) and (33) in the form

$$\begin{array}{c}\hfill e_0^2{\left({\displaystyle \frac{e+pc}{e-pc}}\right)}^{\kappa }=e_0^2{\left({\displaystyle \frac{d\alpha }{d\beta }}\right)}^{\kappa }=e_0^2{\left(\left({\displaystyle \frac{\tau +\mu }{\tau -\mu }}\right){\displaystyle \frac{\partial \psi }{\partial \beta }}/{\displaystyle \frac{\partial \psi }{\partial \alpha }}\right)}^{\kappa },\end{array}$$

(34)

and on making use of (38) and (41) below, we then follow a similar development to that for $\kappa =0$ to deduce the highly nonlinear second order partial differential Equation (59) or (60). We do not attempt the alternative development based upon (29) because firstly, this version of (28) does not involve natural Lorentz invariants, and secondly, it necessarily gives rise to fractional nonlinear partial differential equations, which are notoriously difficult to solve.

With reference to the power-law expressions (23) and the assumption (32), we comment that from a special relativistic perspective, (32) is an unusual assumption. Physically, it singles out the rest energy as a characteristic that is relative velocity dependent and provides a link to the constant $\kappa$. We further comment that as discussed in [16] and [17] (p. 52) since $e(-u)\ne e\left(u\right)$, for $\kappa \ne 0$, there is a directional dependence, which is also inherent in the combined Planck–de Broglie relations $e=\pm pc$. Both [23,24] suggest that the universe may have a directional dependence so that the properties of the universe may not be isotropic and a preferred direction may exist. Further, in terms of the actual particle energy e, the assumption (32) implies that

$$\begin{array}{c}\hfill E{\left(1-{\left({\displaystyle \frac{U}{c}}\right)}^2\right)}^{1/2}=e{\left(1-{\left({\displaystyle \frac{u}{c}}\right)}^2\right)}^{1/2},\end{array}$$

(35)

which is certainly the case in conventional special relativity and follows immediately from (32) when using the Einstein addition of velocities law in the following form (see, for example, [17] (p. 18)):

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1+U/c}{1-U/c}}\right)=\left({\displaystyle \frac{1+u/c}{1-u/c}}\right)\left({\displaystyle \frac{1+v/c}{1-v/c}}\right).\end{array}$$

(36)

4. Generalised Schrödinger Operator Relations

In this section, based upon the Lorentz invariant power-law energy-momentum relations (23), we generalise the one-dimensional Schrödinger quantum wave theory operator relations $p⟶-i\hslash \partial /\partial x$ and $e⟶i\hslash \partial /\partial t$. We adopt the basic idea suggested in [15] that both sub-luminal (particle) and superluminal (wave) characteristics might contribute to any fundamental physical relations. As previously noted, it is only for $\kappa =0$ that the machinery of Schrödinger’s quantum wave theory is particularly meaningful, in the sense that for $\kappa \ne 0$ nonlinear partial differential equations are obtained from (28) rather than the special relativistic linear Klein–Gordon Equation (21).

4.1. Generalised Operator Relations

For a single spatial dimension, on noting the superluminal space-time transformation $x^{\prime }=ct$ and $t^{\prime }=x/c$, and using

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial x^{\prime }}}={\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t^{\prime }}}=c{\displaystyle \frac{\partial }{\partial x}},\end{array}$$

(37)

the quantum wave theory operator relations, $p⟶-i\hslash \partial /\partial x$ and $e⟶i\hslash \partial /\partial t$, might be extended as follows:

$$\begin{array}{cc}\hfill & p=-\tau {\displaystyle \frac{\partial \psi }{\partial x}}+\mu {\displaystyle \frac{\partial \psi }{\partial x^{\prime }}}=-\tau {\displaystyle \frac{\partial \psi }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial \psi }{\partial t}},\hfill \end{array}$$

(38a)

$$\begin{array}{cc}\hfill & e=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-\mu {\displaystyle \frac{\partial \psi }{\partial t^{\prime }}}=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \psi }{\partial x}},\hfill \end{array}$$

(38b)

for some function $\psi (x,t)$, where $\tau$ and $\mu$ denote arbitrary constants such that conventionally $\mu =0$ and the constant $\tau$ is pure imaginary, namely $\tau =i\hslash$. From the relations in (38), we may deduce

$$\begin{array}{c}\hfill {\displaystyle \frac{e+cp}{e-cp}}=\left({\displaystyle \frac{\tau +\mu }{\tau -\mu }}\right)\left({\displaystyle \frac{\frac{\partial \psi }{\partial t}-c\frac{\partial \psi }{\partial x}}{\frac{\partial \psi }{\partial t}+c\frac{\partial \psi }{\partial x}}}\right)=\left({\displaystyle \frac{\tau +\mu }{\tau -\mu }}\right){\displaystyle \frac{\partial \psi }{\partial \beta }}/{\displaystyle \frac{\partial \psi }{\partial \alpha }},\end{array}$$

(39)

where $\alpha =ct+x$ and $\beta =ct-x$.

Adopting the definition of particle velocity $u={\displaystyle \frac{dx}{dt}}={\displaystyle \frac{p{c}^2}{e}}$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{c}}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{-\tau c\frac{\partial \psi }{\partial x}+\mu \frac{\partial \psi }{\partial t}}{\tau \frac{\partial \psi }{\partial t}-c\mu \frac{\partial \psi }{\partial x}}},\end{array}$$

(40)

and the intermediate relation $\tau \left({\displaystyle \frac{\partial \psi }{\partial t}}dx+c^2{\displaystyle \frac{\partial \psi }{\partial x}}dt\right)=c\mu d\psi$ which in terms of $\alpha =ct+x$ and $\beta =ct-x$, becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{e+cp}{e-cp}}={\displaystyle \frac{d\alpha }{d\beta }}=\left({\displaystyle \frac{\tau +\mu }{\tau -\mu }}\right){\displaystyle \frac{\partial \psi }{\partial \beta }}/{\displaystyle \frac{\partial \psi }{\partial \alpha }},\end{array}$$

(41)

consistent with (39). Further, on assuming the extended Newton’s second law (10) we obtain $f={\displaystyle \frac{\mu }{c}}\left({\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}\right)$ and $g={\displaystyle \frac{\tau }{c^2}}\left({\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}\right)$, and using the relation $\kappa =\mu /\tau$ implies that the assumption (38) is equivalent to the assumption that the applied forces f and g in (10) satisfy $f=\kappa cg$.

In terms of the requirement to determine the appropriate operator interpretation of the energy-momentum identities, the proposal (42) below does not make a great deal of sense since the major objective is the determination of a single partial differential equation for a single function $\psi (x,t)$. Despite this and as an aside, it may be worth noting that we may use the relations in (38) as an opportunity to introduce a second function $\varphi (x,t)$ such that

$$\begin{array}{c}\hfill p=-\tau {\displaystyle \frac{\partial \psi }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial \varphi }{\partial t}},e=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \varphi }{\partial x}},\end{array}$$

(42)

and the immediately above relations are not greatly changed. Thus, from

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{c}}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{-\tau c\frac{\partial \psi }{\partial x}+\mu \frac{\partial \varphi }{\partial t}}{\tau \frac{\partial \psi }{\partial t}-c\mu \frac{\partial \varphi }{\partial x}}},\end{array}$$

(43)

we have the intermediate relation $\tau \left({\displaystyle \frac{\partial \psi }{\partial t}}dx+c^2{\displaystyle \frac{\partial \psi }{\partial x}}dt\right)=c\mu d\varphi$, and in terms of $\alpha =ct+x$ and $\beta =ct-x$, this equation becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{d\alpha }{d\beta }}=\left(\tau {\displaystyle \frac{\partial \psi }{\partial \beta }}+\mu {\displaystyle \frac{\partial \varphi }{\partial \beta }}\right)/\left(\tau {\displaystyle \frac{\partial \psi }{\partial \alpha }}-\mu {\displaystyle \frac{\partial \varphi }{\partial \alpha }}\right).\end{array}$$

(44)

The extended Newton’s second law (10) yields $f={\displaystyle \frac{\mu }{c}}\left({\displaystyle \frac{{\partial }^2\varphi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}\right)$ and $g={\displaystyle \frac{\tau }{c^2}}\left({\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}\right)$ so that in the absence of further assumptions, there is no implied relation connecting the external applied forces f and g. There is, however, the definite implication that the function $\varphi (x,t)$ is closely associated with the sub-luminal or spatial world, while the function $\psi (x,t)$ is associated with the superluminal or wave world, which we conventionally understand as the quantum world. In other words, $\psi (x,t)$ is the wave function arising in Schrödinger’s quantum wave theory.

We further comment that the appropriate extension of the relations (38) to three dimensions is not altogether obvious but might be expected to have the following general structure:

$$\begin{array}{c}\hfill \mathbf{p}=-\tau \nabla \psi +{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial \mathbf{A}}{\partial t}},e=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu \nabla ·\mathbf{A},\end{array}$$

(45)

for a certain scalar $\psi =\psi (x,y,z,t)$ and vector $\mathbf{A}=\mathbf{A}(x,y,z,t)$ and where ∇ denotes the usual del-operator, and noting that (38) is recovered from (45) in the case when the vector $\mathbf{A}$ has the single component $\psi$. In such a general model, any anisotropy of space might be incorporated with three distinct values corresponding to the three directions of space, namely $({\mu }_1,{\mu }_2,{\mu }_3)$.

4.2. Some Standard Operator Relations

Following [17] (p. 288) and the convention introduced there that the operator corresponding to a given variable is subscripted, the above two operators arising from momentum p and energy e become

$$\begin{array}{c}\hfill L_p=-\tau {\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial }{\partial t}},L_e=\tau {\displaystyle \frac{\partial }{\partial t}}-c\mu {\displaystyle \frac{\partial }{\partial x}}.\end{array}$$

(46)

We observe that the conventional commutation relations remain the same with $i\hslash$ replaced by $\tau$. Thus, for example, we have,

$$\begin{array}{c}(L_p{L}_x-L_x{L}_p)\psi =\left(-\tau {\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial }{\partial t}}\right)\left(x\psi \right)-\left(-\tau {\displaystyle \frac{\partial \psi }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial \psi }{\partial t}}\right)=-\tau \psi ,\hspace{1em}\hfill \end{array}$$

(47a)

$$\begin{array}{c}(L_e{L}_t-L_t{L}_e)\psi =\left(\tau {\displaystyle \frac{\partial }{\partial t}}-c\mu {\displaystyle \frac{\partial }{\partial x}}\right)\left(t\psi \right)-t\left(\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \psi }{\partial x}}\right)=\tau \psi ,\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(47b)

$$\begin{array}{c}(L_p{L}_e-L_e{L}_p)\psi \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(47c)

$$\begin{array}{c}=\left(-\tau {\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial }{\partial t}}\right)\left(\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \psi }{\partial x}}\right)-\left(\tau {\displaystyle \frac{\partial }{\partial t}}-c\mu {\displaystyle \frac{\partial }{\partial x}}\right)\left(-\tau {\displaystyle \frac{\partial \psi }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial \psi }{\partial t}}\right)\hfill \end{array}$$

(47d)

$$\begin{array}{c}=0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(47e)

and the remainder follow similarly.

4.3. Invariant Operators $L_{\xi -c\eta }$ and $L_{\xi +c\eta }$

Now, based upon the Lorentz transformations (3) and (4), the quantities $\xi =ex-c^{2}pt$ and $\eta =px-et$ (see [17] (p. 40) for details) are Lorentz invariant. Therefore, we might introduce corresponding Lorentz invariant operators $L_{\xi }$ and $L_{\eta }$ that are simplest in terms of the characteristic coordinates $\alpha =ct+x$ and $\beta =ct-x$. Using the differential formulae

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial x}}={\displaystyle \frac{\partial }{\partial \alpha }}-{\displaystyle \frac{\partial }{\partial \beta }},{\displaystyle \frac{\partial }{\partial t}}=c\left({\displaystyle \frac{\partial }{\partial \alpha }}+{\displaystyle \frac{\partial }{\partial \beta }}\right),\end{array}$$

(48)

we might deduce the Lorentz invariant operators

$$\begin{array}{c}\hfill L_{\xi }=(\tau -\mu )c\alpha {\displaystyle \frac{\partial }{\partial \alpha }}-(\tau +\mu )c\beta {\displaystyle \frac{\partial }{\partial \beta }},L_{\eta }=-(\tau -\mu )\alpha {\displaystyle \frac{\partial }{\partial \alpha }}-(\tau +\mu )\beta {\displaystyle \frac{\partial }{\partial \beta }},\end{array}$$

(49)

and from which we may obtain the commuting operators

$$\begin{array}{c}\hfill L_{\xi -c\eta }=2(\tau -\mu )c\alpha {\displaystyle \frac{\partial }{\partial \alpha }},L_{\xi +c\eta }=-2(\tau +\mu )c\beta {\displaystyle \frac{\partial }{\partial \beta }}.\end{array}$$

(50)

The two cases $\kappa =0$ and $\kappa \ne 0$ are necessarily separate, and for $\kappa \ne 0$, there may be no unique operator interpretation of (27)–(29). Here, we follow the development for $\kappa =0$ using (28) and (34) to obtain (59) and (60). The alternative development based upon (29) results in fractional partial differential equations which we do not attempt here.

4.4. Operator Interpretation for $\kappa =0$

In this subsection, we show that the Klein–Gordon partial differential Equation (21) formally arises from the generalised Schrödinger operator relations (38) when coupled with Einstein’s energy-momentum relation. If for the time being we formally ignore the condition $\kappa =\mu /\tau$, then for $\kappa =0$, Equation (29) becomes $e^2-{\left(cp\right)}^2=e_0^2$, which together with the relations $\xi +c\eta =-\beta (e+cp)$ and $\xi -c\eta =\alpha (e-cp)$ gives the identity ${\xi }^2-{\left(c\eta \right)}^2=e_0^2(x^2-{\left(ct\right)}^2)$ or $(\xi -c\eta )(\xi +c\eta )=-\alpha \beta e_0^2$. This latter identity and the operator relations (50) formally give

$$\begin{array}{c}\hfill L_{\xi -c\eta }{L}_{\xi +c\eta }\psi =-4({\tau }^2-{\mu }^2)c^2\alpha \beta {\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}=-e_0^2\alpha \beta \psi ,\end{array}$$

(51)

for some function $\psi (x,t)$ and from which we obtain

$$\begin{array}{c}\hfill 4({\tau }^2-{\mu }^2)c^2{\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}=e_0^2\psi ,\end{array}$$

(52)

which is again the Klein–Gordon partial differential Equation (20) or (21), and it is clear that the constants $\tau$ and $\mu$ satisfy the condition ${\mu }^2-{\tau }^2={\hslash }^2$ and that $\tau$ is pure imaginary when $\mu$ is assumed to be zero.

4.5. Operator Interpretation for $\kappa \ne 0$

For $\kappa \ne 0$, in place of the algebraic identity $(\xi -c\eta )(\xi +c\eta )=-\alpha \beta e_0^2$, from (28) and (41), we have instead

$$\begin{array}{c}\hfill (\xi -c\eta )(\xi +c\eta )=-\alpha \beta (e^2-{\left(cp\right)}^2)=-\alpha \beta e_0^2{\left({\displaystyle \frac{e+cp}{e-cp}}\right)}^{\kappa }==-\alpha \beta e_0^2{\left({\displaystyle \frac{d\alpha }{d\beta }}\right)}^{\kappa },\end{array}$$

(53)

so that in place of (51), we have

$$\begin{array}{c}\hfill L_{\xi -c\eta }{L}_{\xi +c\eta }\psi =-4({\tau }^2-{\mu }^2)c^2\alpha \beta {\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}=-e_0^2\alpha \beta {\left({\displaystyle \frac{d\alpha }{d\beta }}\right)}^{\kappa }\psi ,\end{array}$$

(54)

and on using (41), this equation becomes

$$\begin{array}{c}\hfill 4({\tau }^2-{\mu }^2)c^2{\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}=e_0^2{\left({\displaystyle \frac{\tau +\mu }{\tau -\mu }}\right)}^{\kappa }{\left({\displaystyle \frac{\partial \psi }{\partial \beta }}/{\displaystyle \frac{\partial \psi }{\partial \alpha }}\right)}^{\kappa }\psi .\end{array}$$

(55)

On making a term-by-term identification of the generalised operator relations (38) with the generalised Planck–de Broglie formulae given in [15],

$$\begin{array}{c}\hfill e=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \psi }{\partial x}},p=-\tau {\displaystyle \frac{\partial \psi }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial \psi }{\partial t}},\end{array}$$

(56)

$$\begin{array}{c}\hfill e=h\nu +{\displaystyle \frac{h^{\prime }c}{\lambda }},p={\displaystyle \frac{h}{\lambda }}+{\displaystyle \frac{h^{\prime }\nu }{c}},\end{array}$$

(57)

then it is clear that $\kappa =h^{\prime }/h=\mu /\tau$. If we preserve the relationship ${\mu }^2-{\tau }^2={\hslash }^2$, then combining this with $\kappa =\mu /\tau$ gives

$$\begin{array}{c}\hfill \tau ={\displaystyle \frac{i{\hslash }^2}{{({\hslash }^2-{\hslash }^{\prime 2})}^{1/2}}},\mu ={\displaystyle \frac{i\hslash {\hslash }^{\prime }}{{({\hslash }^2-{\hslash }^{\prime 2})}^{1/2}}},\end{array}$$

(58)

which means that we may represent the constants in (55) with a variety of forms. Here, it is simpler to assume only the relation $\kappa =\mu /\tau$, noting that the key issue for the representations of (55) is the magnitude of $\kappa$. Given the possibilities for $\mu$ and $\tau$ including pure imaginary, the sign of the constant in (59) and (60) may only be determined on a case-by-case basis.

Although not essential, here for definiteness, we assume that $\kappa <1$ and Equation (55) becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}={\left({\displaystyle \frac{e_0}{2c\tau {(1-{\kappa }^2)}^{1/2}}}\right)}^2{\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)}^{\kappa }{\left({\displaystyle \frac{\partial \psi }{\partial \beta }}/{\displaystyle \frac{\partial \psi }{\partial \alpha }}\right)}^{\kappa }\psi ,\end{array}$$

(59)

which is a highly nonlinear partial differential equation that for $\kappa =0$ formally includes the linear Klein–Gordon partial differential Equation (20) or (21). In terms of $(x,t)$ variables, this equation becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}={\left({\displaystyle \frac{e_0}{\tau {(1-{\kappa }^2)}^{1/2}}}\right)}^2{\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)}^{\kappa }{\left({\displaystyle \frac{\frac{\partial \psi }{\partial t}-c\frac{\partial \psi }{\partial x}}{\frac{\partial \psi }{\partial t}+c\frac{\partial \psi }{\partial x}}}\right)}^{\kappa }\psi .\end{array}$$

(60)

Equation (60) is a nonlinear wave equation which overall will exhibit similar physical characteristics to those of other well known wave equations such as the Klein–Gordon equation and the sine–Gordon equation, except that its behaviour is modulated by both the sign and magnitude of the constant K, which is defined by

$$\begin{array}{c}\hfill K={\left({\displaystyle \frac{e_0}{\tau {(1-{\kappa }^2)}^{1/2}}}\right)}^2{\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)}^{\kappa },\end{array}$$

(61)

and by both the sign and magnitude of the factor ${\left({\displaystyle \frac{\frac{\partial \psi }{\partial t}-c\frac{\partial \psi }{\partial x}}{\frac{\partial \psi }{\partial t}+c\frac{\partial \psi }{\partial x}}}\right)}^{\kappa }$. The constant $\kappa$ is a non-dimensional constant that reflects the relative space-time contributions, in the sense that for small $\kappa$, space or particle characteristics are the dominant contributor, while if $\kappa$ is large then time or wave characteristics dominate, and clearly, the case $\kappa =\pm 1$, when there are perfectly balanced contributions, and the prospect of singular behaviour at both $\kappa =\pm 1$ arises for the constant K. However, the constant K, which is a dimensional constant with the dimensions of $T^{-2}$, may assume a wide variety of characteristics depending upon the nature of the constant $\tau$. For example, if we adopt the values in (58), then K is negative and given by

$$\begin{array}{c}\hfill K=-{\left({\displaystyle \frac{2\pi e_0}{h}}\right)}^2{\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)}^{\kappa },\end{array}$$

(62)

which has singular behaviour at both $\kappa =\pm 1$. We note that if $\kappa >1$, then there are a number of subtle changes throughout the subsequent analysis which need to be accommodated. However, the most significant change arises from a sign change in the constant ${\left({\displaystyle \frac{e_0}{2c\tau {(1-{\kappa }^2)}^{1/2}}}\right)}^2$, since we have instead $-{\left({\displaystyle \frac{e_0}{2c\tau {({\kappa }^2-1)}^{1/2}}}\right)}^2$.

5. Some Solutions for $\mathbf{\kappa }\ne \mathbf{0}$

Although (59) is nonlinear, the independent variables $\alpha$ and $\beta$ only appear through the partial derivatives; therefore, there are many ad hoc solution techniques that are available (see [27,28,29]). The majority of the techniques described in [27] assume a certain functional form involving arbitrary functions and parameters, which are then determined by substitution into the given nonlinear partial differential equation, and the assumed form of the solution may or may not be compatible with the differential equation. Both [28,29] describe the use of one-parameter transformation groups for the determination of solutions of nonlinear partial differential equations, which, although complicated, is a more systematic solution procedure. In the following three sub-sections, we examine three such solutions. The first two involve ad hoc assumptions, while the third is based upon invariance under a one-parameter group of transformations.

5.1. Separable Solution for $\kappa \ne 0$

If we assume a separable solution of the form $\psi (\alpha ,\beta )=A\left(\alpha \right)B\left(\beta \right)$, then on substituting the partial derivatives into (59), we find that $A\left(\alpha \right)$ and $B\left(\beta \right)$ are necessarily determined from the differential relations

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{A^{\prime }\left(\alpha \right)}{A\left(\alpha \right)}}\right)}^{\kappa +1}=C_1,{\left({\displaystyle \frac{B^{\prime }\left(\beta \right)}{B\left(\beta \right)}}\right)}^{\kappa -1}=C_2,\end{array}$$

(63)

where $C_1$ and $C_2$ denote constants such that

$$\begin{array}{c}\hfill {\displaystyle \frac{C_1}{C_2}}={\left({\displaystyle \frac{e_0}{2c\tau {(1-{\kappa }^2)}^{1/2}}}\right)}^2{\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)}^{\kappa }.\end{array}$$

(64)

On integration of (63), we obtain

$$\begin{array}{c}\hfill A\left(\alpha \right)=exp\left(C_1^{1/(\kappa +1)}\left(\alpha +{\alpha }_0\right)\right),B\left(\beta \right)=exp\left(C_2^{1/(\kappa -1)}\left(\beta +{\beta }_0\right)\right),\end{array}$$

(65)

where ${\alpha }_0$ and ${\beta }_0$ denote arbitrary constants. Thus, altogether, we find that $\psi (\alpha ,\beta )=A\left(\alpha \right)B\left(\beta \right)$ is given by

$$\begin{array}{c}\hfill \psi (\alpha ,\beta )=exp\left(C_1^{1/(\kappa +1)}\left(\alpha +{\alpha }_0\right)+C_2^{1/(\kappa -1)}\left(\beta +{\beta }_0\right)\right),\end{array}$$

(66)

where the constants $C_1$ and $C_2$ satisfy the constraint (64). This solution may be confirmed by direct substitution of (66) into (59) and has the alternative form

$$\begin{array}{c}\hfill \psi (\alpha ,\beta )=C_{3}exp\left(C_1^{1/(\kappa +1)}\alpha +C_2^{1/(\kappa -1)}\beta +C_4\right),\end{array}$$

(67)

where $C_3$ and $C_4$ denote further arbitrary constants, noting that depending upon the constants $C_1$ and $C_2$, this solution may become unbounded. If either $C_1^{1/(\kappa +1)}$ or $C_2^{1/(\kappa -1)}$ is real, then (67) may become unbounded in either space or time, while if both $C_1^{1/(\kappa +1)}$ or $C_2^{1/(\kappa -1)}$ are pure imaginary, then (67) is essentially the travelling wave solution examined in the following sub-section.

5.2. Travelling Wave Solution for $\kappa \ne 0$

Generally, for nonlinear partial differential equations, there are no super-postion principles available, so it is perhaps surprising that the travelling wave solution (22) also applies to (59), which is a consequence of the simple form of (22). The travelling wave solution (22) is a solution of the linear wave equation and generally it would be inapplicable to most nonlinear partial differential equations so that Equation (59) might be regarded as unusual in this respect. However, this does not mean that linear super-position applies to (59). The simple harmonic travelling wave $\psi (x,t)={\psi }_{0}exp\left(i(kx-\omega t)\right)$, where ${\psi }_0$ is a constant, $k=2\pi /\lambda$ and $\omega =2\pi \nu$, has wave velocity $w=\lambda \nu =\omega /k$. On using $\mu =\kappa \tau$, we obtain from the generalised expressions for energy and momentum

$$\begin{array}{cc}\hfill & e=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \psi }{\partial x}}=-i\tau (\omega +\kappa kc){\psi }_{0}exp\left(i(kx-\omega t)\right),\hfill \end{array}$$

(68a)

$$\begin{array}{cc}\hfill & pc=-\tau c{\displaystyle \frac{\partial \psi }{\partial x}}+\mu {\displaystyle \frac{\partial \psi }{\partial t}}=-i\tau (kc+\kappa \omega ){\psi }_{0}exp\left(i(kx-\omega t)\right),\hfill \end{array}$$

(68b)

so that the particle velocity u as defined by $u/c=pc/e$ becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{u}{c}}={\displaystyle \frac{kc+\kappa \omega }{\omega +\kappa kc}},\end{array}$$

(69)

and we observe that for $\kappa =0$, the de Broglie relation $uw=c^2$ emerges, while for large $\kappa$ values, the particle velocity coincides with the wave velocity; thus, $u=w$.

In terms of $\alpha =ct+x$ and $\beta =ct-x$, (22) becomes

$$\begin{array}{c}\hfill \psi (x,t)={\psi }_{0}exp\left(i(kx-\omega t)\right)={\psi }_{0}exp\left\{{\displaystyle \frac{i}{2c}}[(kc-\omega )\alpha -(kc+\omega )\beta ]\right\},\end{array}$$

(70)

with partial derivatives

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \psi }{\partial \alpha }}={\displaystyle \frac{i}{2c}}(kc-\omega )\psi ,{\displaystyle \frac{\partial \psi }{\partial \beta }}=-{\displaystyle \frac{i}{2c}}(kc+\omega )\psi ,{\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}={\displaystyle \frac{1}{4c^2}}[{\left(kc\right)}^2-{\omega }^2]\psi .\end{array}$$

(71)

From these expressions and (59), we obtain the following dispersion relation:

$$\begin{array}{c}\hfill {\left(kc\right)}^2-{\omega }^2={\displaystyle \frac{e_0^2}{{\tau }^2(1-{\kappa }^2)}}{\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)}^{\kappa }{\left({\displaystyle \frac{\omega +kc}{\omega -kc}}\right)}^{\kappa }.\end{array}$$

(72)

It is easy to see why the simple harmonic travelling wave is also a solution, since if we look for general solutions of (60) of the form $\psi (x,t)=F(x-wt)$, where w is the constant wave velocity, we obtain

$$\begin{array}{c}\hfill F^{″}={\displaystyle \frac{K^2}{c^2({(w/c)}^2-1)}}{\left({\displaystyle \frac{w/c+1}{w/c-1}}\right)}^{\kappa }F,\end{array}$$

(73)

where K is the constant given by (61), primes here denote differentiation with respect to the argument $x-wt$, and apart from different constants, Equation (73) is essentially that arising from the Klein–Gordon equation. It is also interesting to observe that (72) may also be deduced from the relations $e+cp=(\hslash +{\hslash }^{\prime })(\omega +ck)$ and $e-cp=(\hslash -{\hslash }^{\prime })(\omega -ck)$, which are used in the derivation of (25), since from (28), we have

$$\begin{array}{c}\hfill ({\hslash }^2-{\hslash }^{\prime 2})({\omega }^2-{\left(ck\right)}^2)=e_0^2{\left({\displaystyle \frac{\hslash +{\hslash }^{\prime }}{\hslash -{\hslash }^{\prime }}}\right)}^{\kappa }{\left({\displaystyle \frac{\omega +ck}{\omega -ck}}\right)}^{\kappa },\end{array}$$

(74)

and (72) now follows on noting the relations $\tau =i\hslash$ and $\kappa =h^{\prime }/h={\hslash }^{\prime }/\hslash$.

Many fundamental physical problems are self-similar, and in the following sub-section, we examine a particular similarity solution of (59) arising from a stretching group of invariant transformations.

5.3. Self-Similar Solution for $\kappa \ne 0$

If a partial differential equation and any associated boundary conditions remain invariant under a one-parameter group of transformations, then generally, there is a major simplification of the problem (see for example [29]). Here, without reference to possible boundary data, it is easy to see that (59) remains invariant under the following simple one-parameter group of stretching transformations (75):

$$\begin{array}{c}\hfill {\alpha }^{*}={\lambda }^{\kappa -1}\alpha ,{\beta }^{*}={\lambda }^{\kappa +1}\beta ,{\psi }^{*}={\lambda }^m\psi ,\end{array}$$

(75)

where $\lambda$ is the one-parameter and m denotes any arbitrary constant. Now, since $\zeta ={\alpha }^{*\kappa +1}/{\beta }^{*\kappa -1}={\alpha }^{\kappa +1}/{\beta }^{\kappa -1}$ is one invariant of (75), and since m is completely arbitrary, essentially, we may adopt ${\psi }^{*}/{\alpha }^{*m}=\psi /{\alpha }^m$ as a second independent invariant so that the similarity solution of (59) takes the form

$$\begin{array}{c}\hfill \psi (\alpha ,\beta )={\alpha }^m\mathrm{\Psi }\left(\zeta \right),\zeta ={\displaystyle \frac{{\alpha }^{\kappa +1}}{{\beta }^{\kappa -1}}},\end{array}$$

(76)

where $\mathrm{\Psi }\left(\zeta \right)$ denotes a function of $\zeta$ to be determined. The functional form of the solution (76) arises by taking one invariant to be a function of the other. Of course, for nonlinear partial differential equations, the major significance of (76) is that if correct, then upon substitution into (59), the partial differential equation will reduce to an ordinary differential equation that is considerably simpler to solve. The arbitrary nature of the constant m relates to the invariance of (60) under arbitrary scaling, or in other words, the same Equation (60) is obtained whatever units are adopted.

On evaluating the partial derivatives for (76), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \psi }{\partial \alpha }}={\alpha }^{m-1}[(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }+m\mathrm{\Psi }],{\displaystyle \frac{\partial \psi }{\partial \beta }}={\displaystyle \frac{{\alpha }^m}{\beta }}(1-\kappa )\zeta {\mathrm{\Psi }}^{\prime },\hfill \end{array}$$

(77a)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^2\psi }{\partial \alpha \partial \beta }}={\displaystyle \frac{{\alpha }^{m-1}}{\beta }}(1-\kappa )\zeta {[(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }+m\mathrm{\Psi }]}^{\prime },\hfill \end{array}$$

(77b)

where primes denote derivatives with respect to $\zeta$. On substitution of these expressions into (59), we might eventually deduce the following nonlinear ordinary differential equation:

$$\begin{array}{c}\hfill {\left(1+{\displaystyle \frac{m\mathrm{\Psi }}{(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }}}\right)}^{\kappa }{\left(\zeta {\mathrm{\Psi }}^{\prime }+{\displaystyle \frac{m}{(1+\kappa )}}\mathrm{\Psi }\right)}^{\prime }={\mathcal{K}}^2\mathrm{\Psi },\end{array}$$

(78)

where $\mathcal{K}$ is the constant defined by

$$\begin{array}{c}\hfill \mathcal{K}={\displaystyle \frac{e_0}{2c\tau (1-{\kappa }^2)}}.\end{array}$$

(79)

As we have previously noted, the constant $\kappa$ is a non-dimensional constant that reflects the relative space-time contributions, such that for small $\kappa$, space dominates, while for large $\kappa$, time is the major contributor, and the two cases $\kappa =\pm 1$ arise when the contributions balance. In these cases, the constant $\mathcal{K}$ becomes singular. The constant $\mathcal{K}$, which is a dimensional constant with dimensions of $L^{-1}$, may assume a variety of characteristics depending on the nature of the constant $\tau$. For example, if we adopt the values in (58), then $\mathcal{K}$ itself is pure imaginary and given by $\mathcal{K}=-i\pi e_0/hc{(1-{\kappa }^2)}^{1/2}$ so that ${\mathcal{K}}^2$, which is the constant appearing in (78), is negative.

We observe that for the special case when $m=0$, (78) becomes simply $\zeta {\mathrm{\Psi }}^{″}+{\mathrm{\Psi }}^{\prime }-{\mathcal{K}}^2\mathrm{\Psi }=0$, which for ${\mathcal{K}}^2>0$ has the modified Bessel function solutions $\mathrm{\Psi }\left(\zeta \right)=C_1{I}_0\left(2\mathcal{K}{\zeta }^{1/2}\right)+C_2{K}_0\left(2\mathcal{K}{\zeta }^{1/2}\right)$, where here, $C_1$ and $C_2$ denote arbitrary constants, noting that since $K_0\left(z\right)$ and its derivative $K_0^{\prime }\left(z\right)=-K_1\left(z\right)$ have logarithmic singularities, the constant $C_2$ may be zero. We further note that for $\kappa >1$ and with a slightly redefined constant ${\mathcal{K}}^{*}$, we would obtain the ordinary Bessel function solutions $\mathrm{\Psi }\left(\zeta \right)=C_1{J}_0\left(2{\mathcal{K}}^{*}{\zeta }^{1/2}\right)+C_2{Y}_0\left(2{\mathcal{K}}^{*}{\zeta }^{1/2}\right)$. We further note that for $\kappa =0$, (78) becomes $\zeta {\mathrm{\Psi }}^{″}+(m+1){\mathrm{\Psi }}^{\prime }-{K_0}^2\mathrm{\Psi }=0$, which for ${K_0}^2>0$ has the modified Bessel function solutions $\mathrm{\Psi }\left(\zeta \right)={\zeta }^{-m/2}\left\{C_1{I}_m\left(2K_0{\zeta }^{1/2}\right)+C_2{K}_m\left(2K_0{\zeta }^{1/2}\right)\right\}$, where $K_0=e_0/2\tau c$. We comment that Bessel functions appear throughout physics and applied mathematics as a known special function, and their appearance in the present context may or may not be physically significant in terms of establishing a correspondence between widely differing physical phenomena.

We mention briefly and in general terms only one possible class of fundamental problems for which the similarity solution (75) might apply. This is the class of moving boundary problems, which involve two regions with distinct physical characteristics that are separated by a boundary that is in motion. Such problems frequently occur, and more often than not, a constant similarity variable defines the moving boundary. The case $m=0$ is potentially important since this allows the moving boundary $x=x_0\left(t\right)$ to move into a region of constant $\psi (x,t)$, say $\psi ={\psi }_0$. These problems are difficult to solve analytically, and we mention below some of the issues that may be relevant. For heat-related moving boundary problems, which are referred to as Stefan moving boundary problems involving the determination of moving melting or freezing fronts, we refer the reader to [30].

For a specific example, we know that $x_0\left(t\right)=\pm ct$ defines the demarcation boundary between the particle and wave worlds, and we might pose the question as to whether or not other such particle-wave boundaries might exist. We have in mind the separate regions, say $0\le x\le x_0\left(t\right)$ and $x_0\left(t\right)\le x\le \infty$, with the common boundary $x=x_0\left(t\right)$. We suppose that a power-law energy-momentum relation applies in the region $0\le x\le x_0\left(t\right)$, and that the similarity solution (76)

$$\begin{array}{c}\hfill \psi (\alpha ,\beta )={(ct+x)}^m\mathrm{\Psi }\left(\zeta \right),\zeta ={\displaystyle \frac{{(ct+x)}^{\kappa +1}}{{(ct-x)}^{\kappa -1}}},\end{array}$$

(80)

applies in the region $x_0\left(t\right)\le x\le \infty$. We further suppose that the similarity variable $\zeta$ defines the moving boundary through the relation

$$\begin{array}{c}\hfill \zeta ={\displaystyle \frac{{\alpha }^{\kappa +1}}{{\beta }^{\kappa -1}}}=\alpha \beta {\left({\displaystyle \frac{\alpha }{\beta }}\right)}^{\kappa }=({\left(ct\right)}^2-x^2){\left({\displaystyle \frac{ct+x}{ct-x}}\right)}^{\kappa }={\zeta }_0,\end{array}$$

(81)

where ${\zeta }_0$ is a constant, perhaps yet to be determined, and we use the subscript zero to designate the value along the boundary. In general, this is a complicated implicit equation that defines the moving boundary $x=x_0\left(t\right)$ as a function of time. For $\kappa =0$, $x_0\left(t\right)=\pm \sqrt{{\left(ct\right)}^2-{\zeta }_0}$ and $u_0\left(t\right)=c^{2}t/x_0\left(t\right)$, while for $\kappa =\pm 1$, we have $x_0\left(t\right)=\pm \sqrt{{\zeta }_0}\pm ct$ and $u_0\left(t\right)=\pm c$.

For any $\kappa$, on taking the total logarithmic derivative with respect to time of the expression in (81), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{c\zeta }}{\displaystyle \frac{d\zeta }{dt}}={\displaystyle \frac{(\kappa +1)(1+u/c)}{(ct+x)}}-{\displaystyle \frac{(\kappa -1)(1-u/c)}{(ct-x)}},\end{array}$$

(82)

and from this basic relation equated to zero, we may deduce that the velocity of the moving boundary $u_0=d{x}_0/dt$ is given by the simple expression

$$\begin{array}{c}\hfill {\displaystyle \frac{u_0}{c}}={\displaystyle \frac{1}{c}}{\displaystyle \frac{d{x}_0}{dt}}={\displaystyle \frac{\kappa x_0-ct}{\kappa ct-x_0}},\end{array}$$

(83)

which we observe is entirely consistent with the above special cases arising from $\kappa =0$ and $\kappa =\pm 1$. In deriving (83), we are assuming that the velocity of the boundary may be determined by simply assuming the similarity variable to be a constant and then differentiating. The question is whether the particle velocities as defined by $u=p{c}^2/e$ and predicted by both the power-law energy momentum relations (23) and from the broader wave theory (in other words from the general expressions (38) or (56) and not just the constant similarity variable) are consistent with the boundary velocity as given by (83). In order for the moving boundary $x=x_0\left(t\right)$ to be well defined from both sides, the particle velocity must be continuous across the boundary, although we may allow jumps in both energy and momentum.

The power-law energy-momentum relations (23) are functions of particle velocity, so that on assuming (83) for the velocity on the boundary, and in conjunction with (81) in the form

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{ct+x_0}{ct-x_0}}\right)}^{\kappa /2}={\displaystyle \frac{{\zeta }_0^{1/2}}{{({\left(ct\right)}^2-x_0^2)}^{1/2}}},\end{array}$$

(84)

we might establish the following boundary values for energy and momentum:

$$\begin{array}{cc}\hfill & e={\displaystyle \frac{e_0{\zeta }_0^{1/2}(\kappa ct-x_0)}{{({\kappa }^2-1)}^{1/2}({\left(ct\right)}^2-x_0^2)}}{\left({\displaystyle \frac{\kappa -1}{\kappa +1}}\right)}^{\kappa /2},\hfill \end{array}$$

(85a)

$$\begin{array}{cc}\hfill & pc={\displaystyle \frac{e_0{\zeta }_0^{1/2}(\kappa x_0-ct)}{{({\kappa }^2-1)}^{1/2}({\left(ct\right)}^2-x_0^2)}}{\left({\displaystyle \frac{\kappa -1}{\kappa +1}}\right)}^{\kappa /2},\hfill \end{array}$$

(85b)

which are evidently consistent with (83) through the definition $u/c=pc/e$, as indeed they must.

The question is whether the boundary values for the similarity solution (80) can be arranged so that the predicted velocity on the boundary coincides with (83) for any boundaries other than $x_0\left(t\right)=\pm ct$. On partially differentiating (81), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\zeta }}{\displaystyle \frac{\partial \zeta }{\partial x}}={\displaystyle \frac{2(\kappa ct-x)}{{\left(ct\right)}^2-x^2}},{\displaystyle \frac{1}{c\zeta }}{\displaystyle \frac{\partial \zeta }{\partial t}}=-{\displaystyle \frac{2(\kappa x-ct)}{{\left(ct\right)}^2-x^2}},\end{array}$$

(86)

so that from the expressions

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \psi }{\partial x}}=m{(ct+x)}^{m-1}\mathrm{\Psi }\left(\zeta \right)+{(ct+x)}^m{\mathrm{\Psi }}^{\prime }\left(\zeta \right){\displaystyle \frac{\partial \zeta }{\partial x}},\hfill \end{array}$$

(87a)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \psi }{\partial t}}=mc{(ct+x)}^{m-1}\mathrm{\Psi }\left(\zeta \right)+{(ct+x)}^m{\mathrm{\Psi }}^{\prime }\left(\zeta \right){\displaystyle \frac{\partial \zeta }{\partial t}},\hfill \end{array}$$

(87b)

and the new relations (56), we might eventually deduce

$$\begin{array}{cc}\hfill & e=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \psi }{\partial x}}=(1-\kappa )\tau c{\alpha }^{m-1}\left\{2(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }\left(\zeta \right){\displaystyle \frac{ct}{\beta }}+m\mathrm{\Psi }\left(\zeta \right)\right\},\hfill \end{array}$$

(88a)

$$\begin{array}{cc}\hfill & pc=-\tau c{\displaystyle \frac{\partial \psi }{\partial x}}+\mu {\displaystyle \frac{\partial \psi }{\partial t}}=(1-\kappa )\tau c{\alpha }^{m-1}\left\{2(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }\left(\zeta \right){\displaystyle \frac{x}{\beta }}-m\mathrm{\Psi }\left(\zeta \right)\right\},\hfill \end{array}$$

(88b)

and from these expressions, we may deduce that the velocity u is given by

$$\begin{array}{c}\hfill {\displaystyle \frac{u}{c}}={\displaystyle \frac{pc}{e}}={\displaystyle \frac{\left(1+\frac{2(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }\left(\zeta \right)}{m\mathrm{\Psi }}\right)x-ct}{\left(1+\frac{2(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }\left(\zeta \right)}{m\mathrm{\Psi }}\right)ct-x}},\end{array}$$

(89)

which is consistent with (83) provided that on the moving boundary $\zeta ={\zeta }_0$, the function $\mathrm{\Psi }\left(\zeta \right)$ satisfies the condition

$$\begin{array}{c}\hfill {\zeta }_0{\mathrm{\Psi }}^{\prime }\left({\zeta }_0\right)+{\displaystyle \frac{m}{2}}\left({\displaystyle \frac{1-\kappa }{1+\kappa }}\right)\mathrm{\Psi }\left({\zeta }_0\right)=0.\end{array}$$

(90)

This is the critical condition that the function $\mathrm{\Psi }\left(\zeta \right)$ must satisfy in order for the particle and wave theories to be meaningful at the common boundary $x=x_0\left(t\right)$. In principle, since m is arbitrary, it might always be possible to satisfy this condition.

Typically, the two constants $C_1$ and $C_2$, say in the Bessel function solution for $m=0$, are determined from a boundary constraint such as $C_1{I}_0\left(2\mathcal{K}{\zeta }_0^{1/2}\right)+C_2{K}_0\left(2\mathcal{K}{\zeta }_0^{1/2}\right)={\psi }_0$ along with a jump condition which connects the boundary velocity with jumps in energy and momentum across the boundary. The jump or shock relations are referred to as the Rankine–Hugoniot conditions, and a concise account with examples may be found in [31] (page 89). If the constant $C_2$ is set to zero, then generally, this leads to a transcendental equation for the determination of the constant ${\zeta }_0$, which is usually solved numerically. However, we do not pursue such matters further here.

As an aside and an interesting related issue, Equation (83) may be re-arranged to give

$$\begin{array}{c}\hfill {\displaystyle \frac{x_0}{ct}}={\displaystyle \frac{1+\kappa \frac{u_0}{c}}{\kappa +\frac{u_0}{c}}},\end{array}$$

(91)

and we observe that the factor involved in this equation also arises in (26)₂, leading to the condition $x_{0}d{e}_0-c^{2}td{p}_0=0$, which together with (26)₁ gives the interesting implication that along the moving boundary, the Lorentz invariant $\xi =ex-c^{2}pt$ is a constant, namely $d{\xi }_0=0$. This interesting outcome leads to us asking whether the second Lorentz invariant $\eta =px-et$ is also constant along the moving boundary, which is indeed the case. Again, using both (26)₁ and (23a) in conjunction with (83) and (81) in the form (84), we may eventually show that the constant values of $\xi =ex-c^{2}pt$ and $\eta =px-et$ along the moving boundary are given by

$$\begin{array}{c}\hfill {\xi }_0={\displaystyle \frac{e_0}{{({\kappa }^2-1)}^{1/2}}}{\left({\displaystyle \frac{\kappa -1}{\kappa +1}}\right)}^{\kappa /2}{\zeta }_0^{1/2},{\eta }_0=-{\displaystyle \frac{\kappa e_0}{c{({\kappa }^2-1)}^{1/2}}}{\left({\displaystyle \frac{\kappa -1}{\kappa +1}}\right)}^{\kappa /2}{\zeta }_0^{1/2},\end{array}$$

(92)

giving rise to the condition $\kappa {\xi }_0+c{\eta }_0=0$ along the boundary, where $e_0$ as usual refers to the constant rest energy.

Generally, (78) is a highly nonlinear second-order ordinary differential equation. However, it is apparent from (78) that the equation remains invariant under the one-parameter group of stretching transformations ${\mathrm{\Psi }}^{*}=\lambda \mathrm{\Psi }$, where $\lambda$ is the one-parameter. This means that the second-order differential equation may be reduced to one of first order. The formal calculation details are presented in Appendix A.

In brief, if we adopt $\mathrm{\Phi }=1+{\displaystyle \frac{m\mathrm{\Psi }}{(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }}}$ as the new dependent variable, which we note is an invariant of the stretching group ${\mathrm{\Psi }}^{*}=\lambda \mathrm{\Psi }$. Then, the second-order differential Equation (78) may be shown to reduce to the following first-order differential equation:

$$\begin{array}{c}\hfill {\displaystyle \frac{d\mathrm{\Phi }}{d\zeta }}-{\displaystyle \frac{m\mathrm{\Phi }}{(1+\kappa )\zeta }}=-{\displaystyle \frac{{\mathcal{K}}^2(1+\kappa )}{m}}{\displaystyle \frac{{(\mathrm{\Phi }-1)}^2}{{\mathrm{\Phi }}^{\kappa }}}.\end{array}$$

(93)

Further details are given in Appendix A. We note that although (93) is certainly of first order, it is not one of the readily solvable first-order differential equations. However, it is conceivable that particular values of the two constants m and $\kappa$ may give rise to equations that are analytically solvable.

Some details are given in Appendix A for an approximate asymptotic solution to (78) of the form

$$\begin{array}{c}\hfill \mathrm{\Phi }\left(\zeta \right)=A{\zeta }^n+B{\zeta }^{2n}+C{\zeta }^{3n}+\mathcal{O}\left({\zeta }^{4n}\right),\end{array}$$

(94)

which applies for small values of $\zeta$ ($\zeta ⋘1$), and where A, B, C, and n denote certain constants which are determined by substitution of (94) into (93) and equating the leading powers. We further comment that the boundary condition (90) for the function $\mathrm{\Psi }\left(\zeta \right)$, for the function $\mathrm{\Phi }=1+{\displaystyle \frac{m\mathrm{\Psi }}{(1+\kappa )\zeta {\mathrm{\Psi }}^{\prime }}}$, becomes simply $\mathrm{\Phi }\left({\zeta }_0\right)=-\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)$, and assuming that we may use this constant value in the differential Equation (93), we may deduce the following estimate of ${\zeta }_0$. Thus, altogether, we have

$$\begin{array}{c}\hfill \mathrm{\Phi }\left({\zeta }_0\right)=\left({\displaystyle \frac{\kappa +1}{\kappa -1}}\right),\sqrt{{\zeta }_0}\approx {\displaystyle \frac{m}{2\mathcal{K}}}{\left({\displaystyle \frac{\kappa +1}{\kappa -1}}\right)}^{(\kappa -1)/2},\end{array}$$

(95)

where $\mathcal{K}$ is the constant defined by (79).

6. Summary and Conclusions

In this paper, we use the Lorentz invariant power-law relations

$$\begin{array}{c}\hfill e{\left(u\right)}^2-{\left(cp\left(u\right)\right)}^2=e_0^2{\left({\displaystyle \frac{1+(u/c)}{1-(u/c)}}\right)}^{\kappa },\end{array}$$

(96)

where $\kappa$ denotes an arbitrary parameter, in conjunction with both the generalisation of the Planck–de Broglie relations $e=h\nu$ and $p=h/\lambda$, namely

$$\begin{array}{c}\hfill e=h\nu +h^{\prime }c/\lambda ,p=h/\lambda +h^{\prime }\nu /c,\end{array}$$

(97)

where $h^{\prime }$ denotes a new fundamental constant such that $\kappa =h^{\prime }/h$, and the generalisation of the Schrödinger operator relations, namely

$$\begin{array}{c}\hfill e=\tau {\displaystyle \frac{\partial \psi }{\partial t}}-c\mu {\displaystyle \frac{\partial \psi }{\partial x}},p=-\tau {\displaystyle \frac{\partial \psi }{\partial x}}+{\displaystyle \frac{\mu }{c}}{\displaystyle \frac{\partial \psi }{\partial t}},\end{array}$$

(98)

where in this context, $\kappa =\mu /\tau$, to derive the following nonlinear generalisation of the Klein–Gordon equation

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-c^2{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}}={\left({\displaystyle \frac{e_0}{\tau {(1-{\kappa }^2)}^{1/2}}}\right)}^2{\left({\displaystyle \frac{1+\kappa }{1-\kappa }}\right)}^{\kappa }{\left({\displaystyle \frac{\frac{\partial \psi }{\partial t}-c\frac{\partial \psi }{\partial x}}{\frac{\partial \psi }{\partial t}+c\frac{\partial \psi }{\partial x}}}\right)}^{\kappa }\psi .\end{array}$$

(99)

We have proposed that the nonlinear partial differential Equation (99) (also (59) and (60)), involving the arbitrary power-law parameter $\kappa$, as a generalisation of the Klein–Gordon Equation (20) or (21), arises from (99) in the special case $\kappa =0$. The partial differential Equation (99) is based upon the generalisation (98) of the Schrödinger operator relations, which are motivated from the basic idea proposed in [15] that sub-luminal (particle) and superluminal (wave) characteristics both contribute to fundamental physical relations. Further, these operator relations are coupled with the Lorentz invariant energy-momentum power-law relations (96) proposed in [16] and with the generalised Planck–de Broglie formulae (97) derived in [15], where $h^{\prime }$ is a new fundamental constant. The generalised Planck–de Broglie formulae are supported by the family of Lorentz invariant energy relations (96) characterised by the arbitrary constant $\kappa$, and in this context, $\kappa =h^{\prime }/h$.

For nonlinear partial differential equations, there are no super-position principles available for generating general solutions, and apart from numerical solutions, often, the best that can be achieved analytically are particular solutions satisfying artificially constructed boundary data. The major characteristic of the resulting nonlinear partial differential Equation (99) (or (59) or (60)) is its invariance under a one-parameter group of stretching transformations. This stretching symmetry gives rise to a family of similarity solutions and means that the partial differential equation may be reduced to a nonlinear second-order ordinary differential equation, which may be then reduced to a highly nonlinear first-order ordinary differential equation.

Many of the nonlinear solution techniques proposed in [27,28,29] are available to determine analytical solutions of (99). Here, we have examined three such solutions for $\kappa \ne 0$, including (67), which arises from assuming a separable solution of the form $\psi (\alpha ,\beta )=A\left(\alpha \right)B\left(\beta \right)$, where $\alpha =ct+x$, $\beta =ct-x$, $A\left(\alpha \right)$, $B\left(\beta \right)$ are determined from the partial differential Equation (59). We have also examined the simple travelling wave (70) with dispersion relation (72) or (74), noting that the two derivations of this relation are distinct. We have also noted that the structure of the partial differential Equation (59) must be regarded as unusual in the sense that the travelling wave (22) is the solution of a linear equation and generally would not apply to most nonlinear partial differential equations. Finally, we have examined the similarity solution (76), involving the arbitrary parameter m, which permits the reduction of the partial differential Equation (59) to the second-order ordinary differential Equation (78). The arbitrariness in the parameter m is a consequence of the invariance of (99) under the one-parameter group of transformations (75). We have shown that the special case $m=0$ has zeroth-order Bessel function solutions, while generally for $m\ne 0$, (78) may be reduced to the highly nonlinear first order ordinary differential Equation (93) for which we have provided in Appendix A some details for an approximate asymptotic solution of the form (94).

In terms of future work and interesting topics that have not been addressed, to date, we have not determined any numerical values from experimental results for the new constant $h^{\prime }$ and its relative magnitude with respect to Planck’s constant h. In addition, we have not determined any numerical solutions of either the nonlinear partial differential Equation (99) or the first-order nonlinear ordinary differential Equation (93), nor have we examined in any detail the major physical implications of the derived solutions of (99), which are given in Section 5. The larger questions of deriving the operator relations (38) from a more fundamental approach such as a variational framework or the relationship of (99), if any, with other nonlinear relativistic wave equations, such as the sine–Gordon equation, have also not so far been addressed.