2003/12/09 by A. Mostafazadeh, Mostafazadeh, A., F. Zamani +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nuclear Experiment (nucl-ex) #Quantum Physics (quant-ph) #gr-qc #hep-th #math-ph #math.MP #nucl-ex #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0312078
36 pages
arxiv created 2003/12/09 · arxiv updated 2009/12/01
For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities Jaμ, with a∈(-1,1), and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density \cal Jaμ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both Jaμ and \cal Jaμ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike Jaμ the current density \cal Jaμ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of Jaμ. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter a is rational. It is the multiplicative group of positive real numbers if a is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.