Doing Quantum Physics with Split-Complex Numbers

Theory of Photons (Electromagnetism) and Wave-Particle Duality of Light

Earlier investigations on the nature of light show that, light must be described by electromagnetic waves or by particles (wave-particle duality). de Broglie hypothesized that what is true for photons should be valid for any particle. We may assign a particle with mass $m$, propagating uniformly with velocity $v$ through field-free space, an energy $E$ and momentum $\mathbf{p}$. In the wave picture, the same particle may be described by a frequency $\omega$ and a wave vector $\mathbf{k}$.  We require that these quantities satisfy the equations
$$E=\hbar\omega,\ \mathbf{p}=\hbar\mathbf{k}.$$
In fact, these equations are satisfied by the photon and that a photon is described by the plane wave
$$\psi(\mathbf{r},t)=A\exp[i(\mathbf{k}\cdot\mathbf{r}-\omega t)].$$
Following de Broglie, to every free particle, a plane wave shown above is assigned. So, naturally quantum mechanics was formulated in terms of complex numbers.

The Trouble with Quantum Mechanics

Quantum mechanics as it was formulated appears to be seamless and it has had decades of success with its great advancement as theory of matter and also with its so many wonderful applications in science and engineering. Nevertheless, there are still some fundamental issues with current quantum mechanics. One issue is about antiparticles. In non-relativistic quantum mechanics, there is no clear indication that antiparticles exist. In relativistic quantum mechanics, antiparticles are considered as ones that have negative energies (due to P.A.M. Dirac) or equivalently ones that travel back in time. (Since wave functions themselves are not observables, those particles travel back in time do not violate causality.) But then this may imply that antiparticles have negative rest masses. So to me, considering antiparticles as particles that have negative energies appears to be unnatural and unphysical. Another issue is about the path integral. In quantum mechanics, the amplitude of a particle to propagate from a point $q_I$ to a point $q_F$ in time $T$ is given by
\begin{equation}\label{eq:pathint}\langle q_F|e^{-\frac{i}{\hbar}\hat HT}|q_I\rangle=\int Dq(t)e^{\frac{i}{\hbar}\int_0^TdtL(\dot q,q)}\end{equation}
where $L(\dot q,q)$ is the Lagrangian
$$L(\dot q,q)=\frac{m}{2}\dot q^2-V(q)$$
and $Dq(t)$ is the Feynman measure given by
$$\int Dq(t):=\lim_{N\to\infty}\left(\frac{-im\hbar}{2\pi\delta t}\right)^{\frac{N}{2}}\left(\prod_{k=1}^{N-1}\int dq_k\right)$$
with $\delta t=\frac{T}{N}$. This path integral, while it makes perfect sense physically, does not converge due to the oscillatory factor appeared as the integrand. What Physicists do about this problem is to take the Wick rotation $t\mapsto it$ which turns Minkowski spacetime to Euclidean spacetime. Accordingly, the path integral turns into Euclidean path integral
$$\langle q_F|e^{-\frac{i}{\hbar}\hat HT}|q_I\rangle=\int Dq(t)e^{-\frac{1}{\hbar}\int_0^TdtL(\dot q,q)}.$$
The integrand becomes a decaying exponential whose maximum value occurs at the minimum of the Euclidean action. Most physicists appear to be satisfied with this resolution, however to me it is troublesome that the path integral \eqref{eq:pathint} cannot be calculated in actual spacetime and that it must be calculated in Euclidean spacetime which is not physical spacetime. Besides, most Euclidean solutions are approximations and there is no guarantee that these solutions will be stable when they are brought to Minkowski spacetime. Furthermore, analytic continuation via Wick rotation works when the spacetime is flat. So Euclideanization will have a problem when the spacetime is curved i.e. gravitation is considered.

$i=\sqrt{-1}$ is the Problem!

I believe that this issue with the path integral has its origin in the way our current quantum mechanics was built. It was built upon complex numbers! And the reason complex numbers entered in the formulation of quantum mechanics is that the founders used the theory of photons (electromagnetism) to build the theory of matter, i.e. theory that explains the physics of other particles, in particular of massive particles such as electrons. However, it may be that complex numbers are really for light (photons) and they are not meant for describing other particles, especially massive particles.

Complex Numbers are Really for Light

Maxwell’s equations in vacuum are:
\begin{align*}
\nabla\cdot\mathbf{B}&=0,\ \nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}=0\\
\nabla\cdot\mathbf{E}&=0,\ \nabla\times\mathbf{B}-\frac{\partial\mathbf{E}}{\partial t}=0
\end{align*}
The transformation
$$\mathbf{B}\mapsto\mathbf{E},\ \mathbf{E}\mapsto -\mathbf{B}$$
takes the first pair of equations to the second and vice versa. This symmetry is called Electric-Magnetic Duality. The duality hints that the electric and magnetic fields are part of a unified whole, the electromagnetic field. Let us introduce a complex-valued vector field
$$\mathcal{E}=\mathbf{E}+i\mathbf{B}.$$
Then the duality amounts to the transformation
$$\mathcal{E}\mapsto -i\mathcal{E}$$
and the vacuum Maxwell’s equations boil down to two equations for $\mathcal{E}$:
$$\nabla\cdot\mathcal{E}=0,\ \nabla\times\mathcal{E}=i\frac{\partial\mathcal{E}}{\partial t}.$$
Let $\mathbf{k}$ be a vector in $\mathbb{R}^3$ and let $\omega=|\mathbf{k}|$. Fix $\mathbf{E}\in\mathbb{C}^3$ with $\mathbf{E}\cdot\mathbf{k}=0$ and $\mathbf{E}\times\mathbf{k}=i\omega\mathbf{E}$. Then the plane wave
$$\mathcal{E}(\mathbf{r},t)=\mathbf{E}\exp[i(\mathbf{k}\cdot\mathbf{r}-\omega t)]$$
satisfies the vacuum Maxwell’s equations. That is, complex-valued plane wave functions may describe light (electromagnetic waves).

Here is another example that shows complex numbers are for light. The light cone in Minkowski spacetime $\mathbb{R}^{3+1}$ is the hyperquadric
$$\mathbb{N}^3=\{(t,x,y,z)\in\mathbb{R}^{3+1}: t^2-x^2-y^2-z^2=0\}.$$
Let $\mathbb{N}^3_+$ and $\mathbb{N}^3_-$ denotes the future and the past light cones, respectively. The multiplicative group $\mathbb{R}^+$ acts on $\mathbb{N}^3_+$ and $\mathbb{N}^3_-$ respectively by scalar multiplication. The orbit spaces $\mathbb{N}^3_+/\mathbb{R}^+$ and $\mathbb{N}^3_-/\mathbb{R}^+$ are identified with the two-sphere $S^2$. Physically this means that for an observer at the origin (the event), light rays through his eye correspond to null lines through the origin and the past null directions constitute the field of vision of the observer which is the two-sphere $S^2$. This two-sphere is called the Celestial Sphere in astronomy. In mathematics, the two-sphere $S^2$ is the extended complex plane $\mathbb{C}\cup\{\infty\}$ called the Riemann sphere.

In the beginning, God might have said:

“Let there be complex numbers!”

Wave Functions are Real?

In current quantum physics, a wave functions itself is not considered as a physical reality but rather a manifestation of something that is both particle and wave. What if we assume that wave functions are real, say they represent actual waves in spacetime? If so, there can be wave functions that are split-complex-valued. I assert that split-complex numbers might have been the right choice for the theory of matter, theory of massless particles other than photons and particularly that of massive particles. It turns out that quantum mechanics can be completely rebuilt based upon split-complex numbers. This new quantum mechanics, split-Hermitian quantum mechanics, exhibits distinct features. In split-Hermitian quantum mechanics, antiparticles arise naturally. Remarkably, the path integral may be calculated in Minkowski spacetime without turning it into Euclidean path integral via Wick rotation. Furthermore, the path integral also can be transformed to the one looks just like Euclidean path integral, but it is still defined in Minkowski spacetime. Hence, what physicists have believe to be Euclidean path integral is a form of path integral defined in actual spacetime.

Baryon Asymmetry and Twin Universes

Split-Hermitian quantum mechanics may also offer an explanation on baryon asymmetry, i.e. an explanation as to why there aren’t as many antiparticles as particles in the universe (which is a fortunate thing for us). Although it is highly speculative at this juncture, split-Hermitian quantum mechanics appears to hint that Big Bang may have created twin (not exactly identical though) universes, one with signature $(- + + +)$ made mostly of matter (our universe) and the other with signature $(+ – ++)$ made mostly of antimatter.