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I was reading in Wikipedia about how the Dirac Equation and the Klein-Gordon Equation where built to introduce in the Schrödinger equation the relativistic description of the Energy–momentum relation: $$E = m^2c^4+m^2p^2$$

Where:

  1. Klein-Gordon works directly with the squared version: $\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} -\nabla^2 -\frac{m^2c^2}{\hbar^2}\right)\psi(t,\vec{x})=0$
  2. And Dirac equation made a clever matrix decomposition, as stated in Wikipedia, to avoid using $E=c\sqrt{p^2+m^2c^2}$ and "expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations." Finally this lead him find his equation $(i{\partial\!\!\!\big /} - m) \psi = 0$

But recently, I learned through the answers to this question that a differential equation $\ddot{x}=F(\dot{x},x)$ could have solutions that vanishes at a finite ending time only if the equation $F(\cdot,\cdot)$ have a point where is Non-Lipschitz.

As example, the equation $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$ could admit the solution $x(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$.

And since I believe that both Klein-Gordon and Dirac Equations are locally Lipschitz so their solutions at best vanishes at infinity, I am wonder if, similar to the example equation where the square root introduce a non-Lipschitz point, If there exist solutions with finite extinction times to the problem of pairing the Schrödinger equation directly with the term $E=c\sqrt{p^2+m^2c^2}$ (without using expansions), that are not considered by Kleing-Gordon or Dirac models.

As example of a modified version of the Schrödinger equation where a Non-Lipschitz component is introduced to achieve solutions with finite extinction times, I found the following paper:

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  • $\begingroup$ What, exactly, constitutes a "pairing" of the SE with relativistic energy, i.e. what are the criteria for any DE with finite extinction time to be an answer to this question? What is the physical motivation to want finite extinction times? $\endgroup$
    – ACuriousMind
    Jun 10 at 0:49
  • $\begingroup$ @ACuriousMind My main motivation is curiosity, I could give many ideas of Why but they will be a bit speculative for QM. But the main reason is in daily life phenomena we experience things that stops moving, but QM cannot accurately model them at least in their classic equations, since are never-ending and non-local due their analytic construction by power series. Also a never-ending solution could mess things with causality, and differently, an true-ending equation will broke time-symmetry, so by itself the existence of this solutions could be interesting. (continues...) $\endgroup$
    – Joako
    Jun 10 at 4:12
  • $\begingroup$ @ACuriousMind As example, on this video is explained that the power series construction in order to vanish at infinity "creates the quantization of energy in QM", and an analytic non-piecewise solution will never achieve a finite extinction time due the Identity Theorem. Or this other video where absorption is explained, is used a Gaussian envelope (let say $\exp(-\pi(x-t)^2)$), and is show that if the Kramers-Kronig relations aren't hold the signal becomes non-causal...(continues) $\endgroup$
    – Joako
    Jun 10 at 4:20
  • $\begingroup$ @ACuriousMind ... but the Gaussian envelope was already never-ending for the explanation purpose, so instead something of the form $(1-(x-t)^2+|1-(x-t)^2|)^\pi$ could be used (which approximate quite good the previous Gaussian and also solves the electromagnetic wave equation). Thinking in this last envelope as a pulse in a Schrödinger Equation of optical fiber, it will never be a solution in classic formulations since has finite extension in time, differently for a soliton as example, which is also never-ending: If I shoot 2 pulses the 2nd one have parts that precede parts of the 1st one. $\endgroup$
    – Joako
    Jun 10 at 4:26
  • $\begingroup$ @ACuriousMind but I don't really know what to expect "specifically", if I go into speculating, it will be awesome if a finite duration solution could describe what is happening during the jump of en electron between energy levels (which is now instantly in theory, I read news months ago of a paper explaining somehow the process in time), or maybe have localized solutions like smooth bump functions, among others... but at the end, is just curiosity, due things with a finite end looks intuitive for me (not meaning many things in QM surely are not). $\endgroup$
    – Joako
    Jun 10 at 4:35

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