Non-Commutative Geometry, Categories and Quantum Physics

Communications in Mathematical Physics, 2019

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 K 0 -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.

Journal of Geometry and Physics, 1989

The structure of amanifold can be encoded in the commutative algebra of functions on the manifold it sell-this is usual-. In the case of a non com.mut.ative algebra thereis no underlying manifold and the usual concepts and tools of diffe.rential geometry (differentialforms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory.. .) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The presentpaper is an introduction for the non specialist and a review oftheprincipal results on the field.

Journal of Geometry and Physics, 1993

This is an introduction to the old and new concepts of non-commutative (N.C.) geometry. We review the ideas underlying N.C. measure and topology, N.C. differential calculus, N.C. connections on N.C. vector bundles, and N.C. Riemannian geometry by following A. Connes' point of view.

Particles and Fields: Proceedings of the X Jorge André Swieca Summer School, 1999

We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, Γ, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over Γ and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two "spacetime points"), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operatorD on H = H ⊗ K. The triple (A, H,D) can be viewed as the quantization of the family of the triples (A, H, D).

Journal of Mathematical Physics, 2000

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of spacetime, in particular, amounts to a postulated new force or physical effect called cogravity.

Clifford Algebras: Applications to Mathematics, Physics, and Engineering, 2004

A physical interpretation of the two-sheeted space, the most fundamental ingredient of noncommutative spectral geometry proposed by Connes as an approach to unification, is presented. It is shown that the doubling of the algebra is related to dissipation and to the gauge structure of the theory, the gauge field acting as a reservoir for the matter field. In a regime of completely deterministic dynamics, dissipation appears to play a key role in the quantization of the theory, according to the ’t Hooft’s conjecture. It is thus argued that the noncommutative spectral geometry classical construction carries the seeds of quantization, implicit in its feature of the doubling of the algebra.

2006

International Journal of Theoretical Physics, 2007

We propose a general scheme for the "logic" of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non Commutative Geometry. It involves Baire*-algebras, the non-commutative version of measurable functions, arising as envelope of the C*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of the state of "initial conditions" in the standard (W * -)algebraic approach to classical systems.