Reasonable and viable mathematical models of relativistic and non-relativistic quantum theories require a huge variety of topological, algebraic, and topological-algebraic structures, mirroring their basic mechanisms. For instance, partial inner-product spaces and distribution spaces (with their normal subspaces of distributions and their Hilbert and pre-Hilbert subspaces) reveal much richer structures than Hilbert spaces and Gelfand's triples.

Beyond the known classic structures, that constitute the cornerstone of this Special Issue together with the relative evolution equations, are the Schwartz Distribution-based mathematical structures, which not only allow insights in many theoretical directions but also provide meaningful applications. Schwartz distributions are widely used in quantum theories, and some basic definitions of Quantum Mechanics even require them. However, their essential foundational role at a structural level is hardly recognized in the literature. Often the structures inside distribution spaces are obscured, despite the evidence that they reveal a better representation of quantum physics theoretical framework than the architectures of Hilbert spaces.

This Special Issue aims to provide a forum in which to investigate the foundational role of mathematical structures (including Hilbert subspaces of distribution spaces and manifolds modeled on distribution spaces), within the scope of relativistic and non-relativistic quantum physics, at the level of systematization, analysis, and development. We welcome original research and review articles regarding the mathematical architectures of quantum physics that emphasize the role of the various mathematical structures, beyond Hilbert spaces and groups.

Potential topics include but are not limited to the following:

• Mathematical structures and axiomatizations of Quantum Mechanics
• Schwartz distribution theory in quantum mechanics and quantum field theory
• Quantum probabilities and Gleason-type measures
• Quantum-Relativity foundations
• Haag-Kastler axiomatic framework
• C*-algebra theory in QM
• Reasonable proposals for the construction of an axiomatic Yang-Mills theory
• Schwartz kernels and nuclear spaces in quantum mechanics