2006/06/15 by Frieder Kleefeld, F. Kleefeld · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Non-Hermitian Physics #hep-ph #hep-th #quant-ph
paper · pdf · doi:10.1007/s10582-006-0395-9
published as Czech.J.Phys.56:999-1006,2006 · 8 pages, no figures; correction of obvious typos and marginal improvement of formulations
arxiv created 2006/06/15 · openalex publication_date 2006/09/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29
A simple derivation of a meaningful, manifestly covariant inner product for real Klein-Gordon (KG) fields with positive semi-definite norm is provided which turns out - assuming a symmetric bilinear form - to be the real-KG-field limit of the inner product for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December, 2003, and February, 2006 (quant-ph/0312078, quant-ph/0602151, quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for real KG fields measures the number of active positive and negative energy Fourier modes of the real KG field on the relativistic mass shell. The very existence of an inner product with positive semi-definite norm for the considered real, i.e. neutral, KG fields shows that the metric operator entering the inner product does not contain the charge-conjugation operator. This observation sheds some additional light on the meaning of the C operator in the CPT inner product of PT-symmetric Quantum Mechanics defined by C.M. Bender, D.C. Brody and H.F. Jones.