2000/02/29 by A. P. Balachandran
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Axiom #Commutative property #Geometry #Hilbert space #Mathematics #Noncommutative and Quantum Gravity Theories #Observable #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum Mechanics and Applications #Quantum mechanics #Theoretical physics #Topology (electrical circuits) #gr-qc #hep-th #quant-ph
paper · pdf · doi:10.1007/s12043-001-0120-y
published as Pramana 56:223-237,2001 · 19 pages, Latex. Talk presented at the Winter Institute on Foundations of Quantum Theory and Quantum Optics,S.N.Bose National Centre for Basic Sciences,Calcutta,January 1-13,2000. A hep-th number in a reference has been corrected
arxiv created 2000/03/03 · openalex publication_date 2001/02/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac's `Quantum Mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its CK- and C^∞- structures) can be reconstructed using Gel'fand - Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.