Chapter 6 - Wave-Particle Duality

The following Wave-Particle PDE derivation is ridiculously simple (Occam's razor?) that does not require any specialized knowledge but Newtonian physics and a little bit craziness. Let's start the journey:

Newtonian physics $dK=-\lambda Kdt$ with chain rules shows the following:

$$v^2\frac{d^{2}K}{d{x}^2}=\frac{d^{2}K}{d{t}^2}+\frac{1}{2}\mathrm{\lambda <!--\; \lambda \; -->}\frac{dK}{dt}$$
Interestingly, it tells us two things:

(1) The spatial convexity on the left should be adjusted by local kinetic energy instead of a constant as in prevailing wave & diffusion PDEs.

(2) The 2nd order on the right is Wave term, and 1st order term is Diffusion term, with $\mathrm{\lambda <!--\; \lambda \; -->}\mathrm{/2}$ factor coming from dv/dx term.

If we assume space-time separation K(t,x,y,z)=T(t)S(x,y,z), which is commonly used when solving PDEs, we can replace ODE operands to PDE operands forming the wave-particle PDEs.

$$v^2\frac{{\partial }^{2}K}{\partial x^2}=\frac{{\partial }^{2}K}{\partial t^2}+\frac{1}{2}\mathrm{\lambda <!--\; \lambda \; -->}\frac{\partial K}{\partial t}$$
The $v^2$ term is basically local kinetic energy. For example, if the maximum kinetic energy $c^2$ is observed at the origin, then local kinetic energy should be $(c^2-\mathrm{\lambda <!--\; \lambda \; -->}{x}^2)$ With some boundary condition, it produced results just like wave propagating through the particle system. Furthermore, it can be linked to Klein-Gordon Equation that's is commonly used in atomic physics. Please see the paper for more details.

If you crazy enough to believe what's proposed so far, here's the key takeaway. Time, is not a true dimension in this construct. It is merely a measurements between energy and space -- as assumed by space-time separation. This is really not a surprising result actually, as Erwin Schrödinger has shown that time can be eliminated his his version of PDE, also commonly referenced in atomic physics.

Recall space-time separation K(t,x,y,z)=T(t)S(x,y,z). One formula has 3 variables. One of them is deemed redundant? It is just like defining a linear equation y = ax, where a is a variable. We can observe space S(x,y,z) and energy K(t,x,y,z), only time is not observable, hence a parameter. In other words, it tells us if K and S are true quantities, then T is merely measurement either based of K or based of S as below:
-λ Δ t = 2Δv v = ΔKK
If time measurement differs by a factor of 2, depending upon "changes in space" or "changes in energy", doesn't it tell us that for a $m$ particle system each with velocity c, the system energy would be $E=m{c}^2$, while observed kinetic energy is $E=1/2m{c}^2$ expressed in spatial flux? Or rather time index differs when looking energy vs spatial changes. After all, who would be the best candidate to explain why $E=m{c}^2$ other than Sir Issac Newton?

The paper also proposed that moving particle balls have smaller "diffusable" portion of energy, which would shrink the size of particle ball, pretty much the same as the Proper Length. And the derivation is Relativity-compatible. In essence, Lorentz factor can be viewed as local time measured with potential-energy change -- $\gamma =1/\sqrt{e^{\mathrm{-\lambda t}}}$, meaning integrating time $\mathrm{\lambda dt}$ measured from max velocity $c$ to local time $v$ in the particle ball system where $\lambda$ is constant.

Karl Popper believes that science is Falsifiability - It doesn't mean things are false, but merely means that if things are false, they can be proven so. The existence of time is one of those that can not be falsified. We only provide a "timeless" meeting between Newton and Einstein at the corner of Wall Street. But one can not really prove time existence or not, like we can not prove whether we live in space or in the Matrix?

Prevailing mainstream theorists believe that time-space are warped, or there are multiple versions of space coexists labeled with time, or even multiple version of reality each with a different outcome, etc. It is such an awful waste to store that much of information really for not much of use. The author, for one, believes that time does not exist. Time is only a ruler to measure changes when it occurs. What is the point of life without doing something? Of course, the reader is more than welcomed to challenge the validity of proposed framework. Changes makes our life meaningful?


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Disclaimer: All contents on this website have not been peer-reviewed, and it may contain errors. It is strictly the author’s personal ideas that do not reflect any organization’s view. Furthermore, it is in disagreement with generally accepted practice of Brownian process. The proposal is original to the best of author’s knowledge. The author also reserves all the rights to the contents, including but not limited to, any copy rights, rights to modify and remove any statements, and the right to distribution. Any application of the model is used at the readers own risks.