We examine a series of developments in mathematics used in quantum physics. These include Hilbert space. We then examine how several developments in mathematics can be used in application to Quantum physics.
In this chapter, we examine a different case study, where the aim was to unify apparently unrelated domains, such as quantum states, probability assignments, and logical inference. This is John von Neumann’s development of an alternative framework to the Hilbert space formalism he pioneered: one articulated in terms of his theory of operators and what we now call von Neumann algebras. This allowed him to accommodate probabilities in the context of systems with infinite degrees of freedom. Here we find, in addition to ‘top-down’ moves from the mathematics to the physics, ‘bottom-up’ developments from empirical features, to a particular logic and thence to mathematical structures. Through a combination of such moves, crucially involving exploration of the structural relations that hold between the mathematical and the physical domains, von Neumann articulated the kind of unification across such domains that represents a further important aspect of the application of mathematics.
Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio
The work is based on two premises: local availability of mathematics to an observer at any space time location, and the observation that number systems, as structures satisfying axioms for the number type being considered, can be scaled by arbitrary, positive real numbers. Local availability leads to the assignment of mathematical universes, Vx, to each point, x, of space time. Vx contains all the mathematics that an observer, Ox, at x, can know. Each Vx contains many types of mathematical systems. These include the different types of numbers (natural numbers, integers, rationals, and real and complex numbers), Hilbert spaces, algebras, and many other types of systems. Space time dependent scaling of number systems is used to define representations, in Vx, of real and complex number systems in Vy. The representations are scaled by a factor ry,x relative to the systems in Vx. For y a neighbor point of x,ry,x is the exponential of the scalar product of a gauge field, A(x), and the vector from x to y. For y distant from x,ry,x is a path integral from x to y. Some consequences of the two premises will be examined. Number scaling has no effect on general comparisons of numbers obtained as computations or as experimental outputs. The effect is limited to mathematical expressions that include space or space time integrals or derivatives. The effect of A on wave packets and canonical momenta in quantum theory, and some properties of A in gauge theories, is described.