Fwd: 129(6): Negative Probability Problem of the Klein Gordon Equation



Subject: 129(6): Negative Probability Problem of the Klein Gordon Equation
Date: Sat, 11 Apr 2009 06:04:03 EDT

The first attempt to quantize special relativity resulted in the Klein Gordon equation. This equation does not indicate that a particle may have spin. The Dirac equation was the first to do that. In the Klein Gordon equation the probability density may become negative as in the attached. Quantum field theory then abandons it as a single particle equation and uses second quantization to develop it as a field equation. This all takes place in the limit of Minkowski spacetime. I have never been persuaded by the standard argument, it seems to me that negative probability density, like negative energy, is meaningless, so the wavefunction of the Klein Gordon equation should be chosen to be a solution of the equation and also to give a positive probability and current density. However, I just give the standard argument here and in the next note will show how the Weyl equation produces a positive definite probability. I show that for a particle at rest the Klein Gordon equation gives a probability density of unity. In this case there is no negative probability problem. Dirac also discarded negative energy as meaningless, but his solution to the problem (the Dirac sea) is not particularly convincing, there are much simpler solutions.

Attachment: a129thpapernotes6.pdf


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