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Fefetltl

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- TL;DR Summary
- Solutions of "vanishing particles" may arise from the very basic 1D wave equation, what about their physical meaning ? Could we link them to vacuum fluctuation?

Hello everyone. The 1D wave equation is written:

$$ \left( \partial_t^2/c^2 - \partial_x^2 \right) \Psi = 0$$

An electromagnetic wave or matter wave, like free electron (unnormalized here), can be written with the following wave function ##\Psi_m## of energy ## \hbar k c ##:

$$ \Psi_m \propto e^{-i (kx + c t)}$$

is a solution of wave equation. Now if we take ##\Psi_v## (## x > 0 ##):

$$ \Psi_v \propto e^{-(kx+ c t)} $$

is also a solution of wave equation but it is not a wave (it does not oscillate): it is something that appears (or is already here, depends on initial condition) and disappears then. I heard about Feynman using of Klein-Gordon equation to describe vacuum fluctuations, with some unclear statements (but we are talking about QED...) like the mass problem, the energy-momentum which is not the same that for matter (negative kinetic energy and son on)...

So you guys have an idea if such a solution could exist in physics (we can extend it to 3D also) ? As an appearing pairs of particles ? Or it is something completely wrong and not relevant ?

$$ \left( \partial_t^2/c^2 - \partial_x^2 \right) \Psi = 0$$

An electromagnetic wave or matter wave, like free electron (unnormalized here), can be written with the following wave function ##\Psi_m## of energy ## \hbar k c ##:

$$ \Psi_m \propto e^{-i (kx + c t)}$$

is a solution of wave equation. Now if we take ##\Psi_v## (## x > 0 ##):

$$ \Psi_v \propto e^{-(kx+ c t)} $$

is also a solution of wave equation but it is not a wave (it does not oscillate): it is something that appears (or is already here, depends on initial condition) and disappears then. I heard about Feynman using of Klein-Gordon equation to describe vacuum fluctuations, with some unclear statements (but we are talking about QED...) like the mass problem, the energy-momentum which is not the same that for matter (negative kinetic energy and son on)...

So you guys have an idea if such a solution could exist in physics (we can extend it to 3D also) ? As an appearing pairs of particles ? Or it is something completely wrong and not relevant ?