2018/01/01 by Pierre-Cyril Aubin-Frankowski, Aubin-Frankowski, Pierre-Cyril
Mathematics · Physics and Astronomy · #46E22 #81S40 #Algebraic and Geometric Analysis #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.1801.00488
openalex publication_date 2018/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This study shows how Aronszajn's theory of reproducing kernels can be of use for the construction the Hilbert spaces of quantum theory. We show that the Feynman propagator is an example of a reproducing kernel under a boundedness condition. To every Lagrangian thus corresponds a Hilbert space that does not need to be postulated a priori. For the free non-relativistic particle, we justify mathematically the concept of space-time granularity. Reproducing kernels allow for a functional, rather than distributional, description of the Hilbert spaces of quantum theory, including the Fock space.