Quantum mechanics of Klein Gordon fields I: Hilbert Space, localized states, and chiral symmetry
Abstract
We derive an explicit manifestly covariant expression for the most general positivedefinite and Lorentzinvariant inner product on the space of solutions of the KleinGordon equation. This expression involves a oneparameter family of conserved current densities Jaμ, with a ∈ (1, 1), that are analogous to the chiral current density for spin half fields. The conservation of Jaμ is related to a global gauge symmetry of the KleinGordon fields whose gauge group is U (1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positivedefinite inner product on the space of real KleinGordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.
 Publication:

Annals of Physics
 Pub Date:
 September 2006
 DOI:
 10.1016/j.aop.2006.02.007
 arXiv:
 arXiv:quantph/0602151
 Bibcode:
 2006AnPhy.321.2183M
 Keywords:

 Quantum Physics;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 31 pages, part one of a series of two papers, to appear in Ann. of Phys. (NY)