2020/07/24 by Francisco F. López-Ruiz, Julio Guerrero, V. Aldaya +1
Mathematics · Physics and Astronomy · #Algebra over a field #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Fubini's theorem #Geometry #Inner product space #Lie group #Mathematical analysis #Mathematics #Multivector #Product (mathematics) #Pure mathematics #Quantum Electrodynamics and Casimir Effect #Scalar (mathematics) #Subspace topology #Unitary representation #Unitary state #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1103/physrevd.102.125010
2 figures
arxiv created 2020/07/24 · openalex publication_date 2020/12/01 · arxiv updated 2020/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We propose a new inner product for scalar fields that are solutions of the Klein-Gordon equation with m2<0. This inner product is nonlocal, bearing an integral kernel including Bessel functions of the second kind. The associated norm proves to be positive definite in the subspace of oscillatory solutions, as opposed to the conventional one, although some regularization procedure is required. Poincar'e transformations are unitarily implemented on this subspace, which is the support of a unitary and irreducible representation of the proper orthochronous Poincar'e group. We also provide a new Fourier transform between configuration and momentum spaces, which is unitary, and recover the projection onto the representation space. This new scenario suggests a revision of the corresponding quantum field theory.