We have specified a four-dimensional diffusion for the spacetime coordinates and now one needs to consider the respective Hamilton-Jacobi-Bellman equation. In principle it is straightforward, we follow11,14. In essence, we need to be careful with two things: first, we need to remember that the inner product is determined by the Minkowski metric so that covariant and contravariant objects are generally different, as the sign changes in the time-coordinate. Second, what used to be time, is now proper time. Additionally, we need to insert the imaginary unit properly into the HJB equation, due to the invariant volume form and thus due to the imaginary Hamiltonian and imaginary Lagrangian. Taking all this into account, the HJB equation is:

$$\frac{\partial J}{\partial \tau }-iV({\bf{x}})-\frac{1}{2im}{

abla }_{\mu }J{

abla }^{\mu }J+\frac{1}{2}{\sigma }^{2}{

abla }_{\mu }{

abla }^{\mu }J=0$$ (25)

Notice how in the equation we have both the contravariant and the covariant nabla operator and hence the Laplacian is the Laplace-Beltrami operator, although in this case as the metric tensor does not depend on the coordinates in the Minkowski spacetime, it is just the d’Alembertian. We have used Einstein summation convention and the index μ to represent the coordinates.

We can understand the equation better, when we notice that the nonlinear term is just representing the kinetic energy of the system via the four-momentum:

$$\frac{\partial J}{\partial \tau }-iV({\bf{x}})+iK+\frac{1}{2}{\sigma }^{2}{

abla }_{\mu }{

abla }^{\mu }J=0$$ (26)

or

$$\frac{\partial J}{\partial \tau }+i(K-V)+\frac{1}{2}{\sigma }^{2}{

abla }_{\mu }{

abla }^{\mu }J=0$$ (27)

where \(K=\frac{{P}_{\mu }{P}^{\mu }}{2m}\) and P μ = i▽ μ J is four-momentum. The momentum comes from the optimal feedback control policy: \({u}_{\mu }=-\,\frac{1}{im}{

abla }_{\mu }J\Leftarrow m{u}_{\mu }=i{

abla }_{\mu }J\)

Note how this definition for linear momentum explains the operator substitution postulate. The sign is reverted due to the fact that the HJB equation is solved backwards in time. Instead of postulating the operator substitution rules in a completely ad hoc manner, here we have derived them in a meaningful way.

We are looking for a linear PDE in general, as we are looking an equation describing ‘matter waves’, therefore we need to couple the scaling factor in such a way that the PDE

$$\frac{\partial J}{\partial \tau }-iV({\bf{x}})-\frac{1}{2im}{

abla }_{\mu }J{

abla }^{\mu }J+\frac{1}{2}{\sigma }^{2}{

abla }_{\mu }{

abla }^{\mu }J=0$$ (28)

becomes linear and we therefore choose:

$${\sigma }^{2}=\frac{i}{m}=-\frac{1}{im}$$ (29)

where the the variance term in the HJB equation is the following sum of the individual variances of the each diffusion component in the Minkowski spacetime (see the determination of σ2 in the book11):

$${\sigma }^{2}=-\,{\tilde{\sigma }}_{0}^{2}+{\tilde{\sigma }}_{1}^{2}+{\tilde{\sigma }}_{2}^{2}+{\tilde{\sigma }}_{3}^{2}$$ (30)

Note the minus sign stemming from the Minkowski metric. Then we must have

$${\sigma }^{2}=-\,{\tilde{\sigma }}_{0}^{2}+{\tilde{\sigma }}_{1}^{2}+{\tilde{\sigma }}_{2}^{2}+{\tilde{\sigma }}_{3}^{2}=\frac{i}{m}$$ (31)

Let

$${\tilde{\sigma }}_{1}^{2}+{\tilde{\sigma }}_{2}^{2}+{\tilde{\sigma }}_{3}^{2}=R\in {\mathbb{R}}$$ (32)

We can choose in particular \({\tilde{\sigma }}_{0}^{2}=R+\frac{1}{im}\) . Then this particular variance structure of the Minkowski spacetime implies that the variances are real in the spatial coordinates and that the diffusion scaling factor in the time coordinate is a proper complex number. This could be of further interest as such, because as \({\tilde{\sigma }}_{0}=\sqrt{R+\frac{1}{im}}\) is a proper complex number with real and imaginary parts, it implies that time can be understood as a two-dimensional object - it lives in the complex plane - and it has both a real component and a purely imaginary component. This can be seen from the temporal diffusion model:

$$d(c{X}_{0})={u}_{0}ds+{\tilde{\sigma }}_{0}d{W}_{0}={u}_{0}ds+\sqrt{R+\frac{1}{im}}d{W}_{0}$$ (33)

This ‘complex time’ is a mathematical consequence and requirement to linearise the HJB equation. The wave function is also complex-valued, but it is still a useful object in physics, whether or not it is ontologically ‘real’.

These considerations turn the HJB equation into

$$\frac{\partial J}{\partial \tau }-iV({\bf{x}})-\frac{1}{2im}({

abla }_{\mu }J{

abla }^{\mu }J+{

abla }_{\mu }{

abla }^{\mu }J)=0$$ (34)

Let us invoke a (Hopf-Cole) logarithmic transformation, so that \(J=\,\log \,\varphi \) then the HJB equation becomes linear.

$$\frac{\frac{\partial \varphi }{\partial \tau }}{\varphi }-iV({\bf{x}})-\frac{1}{2im}(-\frac{1}{{c}^{2}}\frac{{\varphi }_{tt}}{\varphi }+\frac{{\varphi }_{xx}}{\varphi }+\frac{{\varphi }_{yy}}{\varphi }+\frac{{\varphi }_{zz}}{\varphi })=0$$ (35)

Finally, multiplying through with \(i\varphi \), we obtain the following PDE:

$$i\frac{\partial \varphi ({\bf{x}})}{\partial \tau }=\frac{1}{2m}\square \varphi ({\bf{x}})-V({\bf{x}})\varphi ({\bf{x}})$$ (36)

where ☐ is the d’Alembertian partial differential operator. From this we can see that we have actually obtained the (time-reversed) Stueckelberg wave equation, which was invented already in 1941, see15. It can be understood as the Schrödinger equation in four dimensional Minkowski spacetime. Stueckelberg did not unfortunately explain either the imaginary structure of his generalised relativistic wave equation, he just postulated it. Stueckelberg’s wave equation is the foundation for what is called the approach of ‘Parameterized Relativistic Dynamics (PRD)’, see e.g.16.

The spacetime diffusion approach seems to be therefore connected also to considerations of antiparticles and particles15, where charge-reversal is related to time reversal. Nonlocality and the possible link with gravitation is considered for example in papers17,18 and references therein. In none of these papers, however, the current approach of coordinate invariant stochastic optimization is utilised as a teleological explanation for the resulting (complex) Stueckelberg field equations.

The missing link between the Stueckelberg equation and the Dirac equation: the Telegrapher’s equation

In this section we derive the Telegrapher’s equation from the Stueckelberg equation above. Telegrapher’s equation is very important as it is a hyperbolic PDE from which Klein-Gordon and Dirac equations can be derived from, see19. We recall that in Special Relativity the proper time τ is defined as:

$$d\tau =\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}dt$$ (37)

Therefore the Stueckelberg equation becomes

$$\frac{i}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}\frac{\partial \varphi ({\bf{x}})}{\partial t}=\frac{1}{2m}\square \varphi ({\bf{x}})-V({\bf{x}})\varphi ({\bf{x}})$$ (38)

Or in a more convenient form

$$\frac{1}{2m}\square \varphi ({\bf{x}})-\frac{i}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}\frac{\partial \varphi ({\bf{x}})}{\partial t}-V({\bf{x}})\varphi ({\bf{x}})=0$$ (39)

This shows that the HJB equation reduces to the Telegrapher’s equation in the present relativistic setting of optimal control. The Telegrapher’s equation is indeed appropriate, due to its property of finite speed of propagation, see e.g.20. This is required for a causal theory and therefore the present model of spacetime diffusion is superior to canonical stochastic control models in R3, as the respective HJB equation is in those contexts parabolic, thus leading to infinite speed of propagation. Moving into a relativistic optimal control setting hence represents somewhat a similar procedure as has been done in other physical contexts, such as in relativistic thermodynamics, see e.g.21.

The Klein-Gordon equation and the Dirac equation are closely related and even obtained from the Telegrapher’s equation, see the profound paper19. In this profound paper it is interesting that the authors also seem to struggle with the problem of the analytic continuation when they derive the Telegrapher’s equation using Poisson processes, see other approaches by22,23 and24. In the present paper no ad hoc analytic continuation is needed, as the imaginary unit comes naturally from the invariant volume form. It should be also noted that the Klein-Gordon equation is the stationary equation when one sets the partial derivative with respect to proper time to zero:

$$\square \varphi ({\bf{x}})-2mV({\bf{x}})\varphi ({\bf{x}})=0$$ (40)

Which is a hyperbolic PDE and manifestly Lorentz-covariant.

Obtaining the Schrödinger equation in the nonrelativistic limit and the relationship between probability and energy

Consider the Telegrapher’s equation above:

$$\frac{1}{2m}\square \varphi ({\bf{x}})-\frac{i}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}\frac{\partial \varphi ({\bf{x}})}{\partial t}-V({\bf{x}})\varphi ({\bf{x}})=0$$ (41)

Passing to the nonrelativistic limit, c→∞, the proper time is just the ordinary time and the first term of the d’Alembertian goes to zero. This in turn gives us

$$i\frac{\partial \varphi ({\bf{x}})}{\partial t}=\frac{1}{2m}(\frac{{\partial }^{2}}{\partial {x}^{2}}+\frac{{\partial }^{2}}{\partial {y}^{2}}+\frac{{\partial }^{2}}{\partial {z}^{2}})\varphi ({\bf{x}})-V({\bf{x}})\varphi ({\bf{x}})$$ (42)

Which is the Schrödinger equation with time reversed, but from the time-symmetry properties of the Schrödinger equation we know that if \(\varphi \) satisfies the above equation then its complex conjugate satisfies the canonical Schrödinger equation

$$i\frac{\partial {\varphi }^{\ast }({\bf{x}})}{\partial t}=-\frac{1}{2m}\Delta {\varphi }^{\ast }({\bf{x}})+V({\bf{x}}){\varphi }^{\ast }$$ (43)

Finally, it is worth considering that there is an interesting natural link between the Born rule and the minimal expected action, because we have the complex algebraic identity \({J}^{\ast }=\,\log \,{\varphi }^{\ast }\), from which it immediately follows that the Born rule gives \(p=\varphi {\varphi }^{\ast }={e}^{2a}\) where we assume that the value function is of the form J = a(x, y, z, t) + ib(x, y, z, t).