On the Deformed Algebras Applications to Quantum Physics
Julio Cesar Jaramillo Quiceno, Plamen Neytchev Nechev
TL;DR
The paper surveys $q$-deformations of algebraic structures in quantum physics and develops a rigorous link to equipped Hilbert spaces via Jackson calculus. It defines the Jackson derivative $D^{(q)}_x$ and Jackson integrals, builds a $[n]_q$-based Fock-like basis with operators $\hat{x}$ and $\hat{p}$, and analyzes the corresponding deformed Heisenberg algebra and oscillator representations. It examines existence and uniqueness for deformed integral equations, exposes convergence and monotonicity subtleties, and discusses the practical viability and current limitations of applying these deformed algebras to Quantum Mechanics and Field Theory, including potential lattice formulations. The work emphasizes correcting prior inaccuracies and outlines future directions for mathematically rigorous applications in lattice field contexts while highlighting fundamental issues with operator commutativity and the interpretation of $\hat{N}$.
Abstract
Some possible applications of deformed algebras to Quantum Physics are considered based on a rigorous approach. Jackson integrals are expressed in the context of the equipped separable Hilbert space. Jackson integrals are expressed in the context of the equipped separable Hilbert spaces using point measures where possible. Along the way, certain errors and/or inaccuracies made by different authors of the cited references have been corrected. A brief analysis at the end of the article indicates that there are still problems in applying deformed algebras to Quantum Mechanics and Field Theory.