One of the aims of this blog is to raise questions that, in my view, have never been raised before. So the reader can disagree with what I am discussing here but the main purpose is to make people think from different angles.
When I started rethinking about the foundational ideas in physics one of the things that bothered me was the way we take the non-relativistic limit of the relativistic dispersion relation to prove that Schrodinger's equation (SE) is the non-relativistic limit of the Dirac and Klein Gordon (KG) equation. Since all these equations are partial differential equations (PDEs) I started learning how PDEs are studied by mathematicians. I tried to find whether this transition from Klein Gordon to Schrodinger equation can be performed with partial differential equations and it turned out that the answer was no (as far as I went in studying PDEs). I spent a long time trying to understand how mathematicians dissect PDEs to study them and see if this can give us more information about these fundamental equations in physics. This whole effort led to a paper in which I tried to make a connection of physics with numerical analysis [1] (This is one of my most viewed papers on researchgate and has been cited several times).
An important concept that I learned from this was regarding the domain of dependence of a differential equation. It turns out that the Schrodinger equation is a parabolic PDE and these equations allow for physical information to travel with infinite speed [2]. I discussed in the paper on how to implement causality in the Schrodinger equation using the explicit method. But this insight appears to indicate that the Schrodinger equation might not be an exclusively non-relativistic equation. It does allow for solutions that are non-relativistic but this may not be true in general. This might be the reason that entanglement is possible which allows for instant information transfer between entangled quantum mechanical states. Later, I came to the same conclusion from a completely different angle, i.e., Lorentz violation, which I will explain below.
I wrote the numerical analysis paper in 2013 and then continued on my journey in trying to tackle some fundamental issues in physics. In 2015, I proposed an equation that was the equivalent of the Dirac equation, i.e., the linear form of Schrodinger equation. In the follow up paper I showed that the equation I proposed can be obtained from the non-relativistic limit of the Dirac equation [4]. Later, however, I realized that there is completely different way of obtaining the linear form of the Schrodinger equation from the Dirac equation, which is by adding enhanced Lorentz violating terms to it [5]. In other words, Schrodinger equation does not obey Lorentz symmetry and apparently violates it maximally. This may be the reason that its domain of dependence allows for information to travel with infinite speed.
The above discussion is to encourage readers (especially physicists) to think more about this. I am not saying that what I have concluded is the right answer but it does appear to indicate that this might be a missing link in our understanding of fundamental physics.
The above discussion is to encourage readers (especially physicists) to think more about this. I am not saying that what I have concluded is the right answer but it does appear to indicate that this might be a missing link in our understanding of fundamental physics.
[1] https://arxiv.org/pdf/1302.5601.pdf
[2] A good reference is Numerical Methods for Engineers and Scientists by Joe D. Hoffman, Ch 10.
[3] https://link.springer.com/article/10.1007%2Fs10701-015-9944-z
[4] https://arxiv.org/pdf/1511.07901.pdf
[5] https://arxiv.org/pdf/1403.7622.pdf
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