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It is known that the Klein-Gordon equation does not admit a positive definite conserved probability density. Nonetheless, in Wikipedia (for example), you can read that with the $\textit{appropriate}$ interpretation it can be used to describe a spinless particle (such as the pion I think it says).

My question is the following, which is this appropriate interpretration? and how can we get the wave-function of a particle described with the Klein Gordon equation?

Qmechanic
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Yossarian
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    Quantum Field Theory. – user26143 Feb 23 '14 at 22:40
  • Yes one could also give a wavefunction intepretation, with positive probabilities of finding the particle localized in regions of the 3-space, but it needs great care and produces a non local structures, see my answer here http://physics.stackexchange.com/q/98711/ – Valter Moretti Feb 24 '14 at 07:52
  • the problem of negative probabilities of the Klein-Gordon field gets naturally resolved in the framework of pseudo-Hermitian QM; see http://arxiv.org/abs/quant-ph/0307059 and http://arxiv.org/abs/0810.5643 for more background information – Christoph Feb 24 '14 at 16:11

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Solutions to the Klein-Gordon equation can be interpreted as wave functions or operator fields.

Interpreting solutions as wave functions leads to relativistic quantum mechanics (RQM). This has all the nasty negative probabilities you've heard of. RQM is rarely taught in classes; people who need it learn it.

Interpreting solutions as an operator field, however, leads to one of the useful quantum field theories (QFT), namely spin-zero field theory or scalar field theory, or whatever else it's called. This theory has positive probabilities & energies. Yay.

A particularly straightforward way to derive spin-zero QFT is to start with the RQM solutions to the Klein-Gordon equation. Then, you just simply re-interpret the solutions as an operator-valued field, and impose particular commutation relations between the operators. Bam, QFT.

Source: Chapter 3 of Student Friendly Quantum Field Theory by R. Klauber, some of which is available free online here.

BMS
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