We study the mathematical structure of the notion of measurement space, which extends aspects of noncommutative topology that are based on quantale theory. This yields a geometric model of physical measurements that provides a realist picture, yet also operational, such that measurements and classical information arise interdependently as primitive concepts. A derived notion of classical observer caters for a mathematical formulation of Bohr's classical/quantum divide. Two important classes of measurement spaces are obtained, respectively from C*-algebras and from second-countable locally compact open sober topological groupoids. The latter yield measurements of classical type and relate to Schwinger's notion of selective measurement. We show that the measurement space associated to the reduced C*-algebra of any second-countable locally compact Hausdorff étale groupoid is canonically equipped with a classical observer, and we establish a correspondence between properties of the observer and properties of the groupoid.
We study the mathematical structure of the notion of measurement space, which extends aspects of noncommutative topology that are based on quantale theory. This yields a geometric model of physical measurements that provides a realist picture, yet also operational, such that measurements and classical information arise interdependently as primitive concepts. A derived notion of classical observer caters for a mathematical formulation of Bohr's classical/quantum divide. Two important classes of measurement spaces are obtained, respectively from C*-algebras and from second-countable locally compact open sober topological groupoids. The latter yield measurements of classical type and relate to Schwinger's notion of selective measurement. We show that the measurement space associated to the reduced C*-algebra of any second-countable locally compact Hausdorff étale groupoid is canonically equipped with a classical observer, and we establish a correspondence between properties of the observer and properties of the groupoid.
A discussion is given of the structure of a physical theory and an ``ideal form'' for such a theory is proposed. The essential feature is that all concepts should be defined in operational terms. Quantum (and classical) mechanics is then formulated in this way (the formulation being, however, restricted to the kinematical theory). This requires the introduction of the concept of a mixed test, related to a pure test (or ``question'') just as a mixed state is related to a pure state. In the new formulation, the primitive concepts are not states and observables but certain operationally accessible mixed states and tests called physical. The notion of a C*-system is introduced; each such system is characterized by a certain C*-algebra. The structure of a general C*-system is then studied, all concepts being defined in terms of physical states and tests. It is shown first how pure states and tests can be so defined. The quantum analog of the phase space of classical mechanics is then constructed and on it is built a mathematical structure, called a q-topology, which is a quantum analog of the topology of classical phase space. Mathematically, a q-topology is related to a noncommutative C*-algebra as an ordinary topology is related to a commutative C*-algebra. Some properties of the q-topology of a C*-system are given. An appendix contains some physically motivated examples illustrating the theory.
In this paper, we will present the main features of what can be called Schwinger’s foundational approach to Quantum Mechanics. The basic ingredients of this formulation are the selective measurements, whose algebraic composition rules define a mathematical structure called groupoid, which is associated with any physical system. After the introduction of the basic axioms of a groupoid, the concepts of observables and states, statistical interpretation and evolution are derived. An example is finally introduced to support the theoretical description of this approach.
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be jointly measurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital C*- algebra A we associate a geometric object—a diagram of topological spaces collating quotient spaces of the noncommutative space underlying A —that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category C can be applied directly to these geometric objects to automatically yield an extension F~ acting on all unital C*-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a finitary variant K~f of the extension K~ of the topological K-functor. We then delineate a C*-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C*-algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.
The kinematical foundations of Schwinger's algebra of selective measurements were discussed in a previous paper (arXiv:1905.12274) and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analysed. For that, the algebra generated by the observables, as well as the notion of state, are dicussed, and the structure of the transition functions, that plays an instrumental role in Schwinger's picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac-Schr\"odinger and Born-Jordan-Heisenberg pictures is discussed.
Bell’s theorem is a foundational result in quantum theory, demonstrating that no local hidden-variable model satisfying certain assumptions can reproduce quantum correlations. While mathematically correct within its formal framework, the theorem implicitly relies on a specific representational choice: measurement outcomes are modeled as scalar, commutative variables (±1). This assumption, though natural for flat and static configurations, is not mandated by physical necessity. In this work, we examine the consequences of relaxing this representational constraint when measurement outcomes arise as projections of states with nontrivial topology. We show that when physical correlations are naturally organized as elements of an S³-valued structure — such as those associated with spin and rotational degrees of freedom — the induced measurement algebra becomes noncommutative. Under these conditions, the key algebraic factorizations employed in Bell-type derivations no longer apply. We construct an explicit local, realistic model in which observable correlations reproduce the quantum expectation E(a, b) = − cos θ and violate the CHSH bound S ≤ 2, without invoking non-local interactions or superluminal influences. The effective S³×I structure arises not as a fundamental substrate, but as the natural geometric form taken by the actualization of physical processes within an underlying relational framework. Importantly, this structure was not introduced to address quantum entanglement specifically; it emerged independently in the study of cosmological expansion, emergent gravity, and mass generation. Its relevance to Bell-type correlations follows as a consequence of geometry, not as an ad hoc modification. These results suggest that quantum entanglement may reflect topological features of physical realization rather than intrinsic non-locality. Possible experimental implications and connections to geometric and substrate-based approaches to fundamental physics
The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appear in IJGMMP (2019)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures is discussed.