[Paper Review] Generalised Gelfand Spectra of Nonabelian Unital C*-Algebras II: Flows and Time Evolution of Quantum Systems

Andreas Doering|arXiv (Cornell University)|Dec 19, 2012

Advanced Operator Algebra Research17 references6 citations

TL;DR

This paper generalizes the Gelfand spectrum to nonabelian unital C*-algebras via a spectral presheaf, using one-parameter flows on this structure to describe time evolution in quantum systems. It establishes a geometric framework for both Schrödinger and Heisenberg pictures, showing how inner automorphism groups induce dynamical flows on the generalized state space.

ABSTRACT

In arXiv:1212.2613, we associated a presheaf \Sigma^A with each unital C*-algebra A. The spectral presheaf \Sigma^A generalises the Gelfand spectrum of an abelian unital C*-algebra. In the present article, we consider one-parameter groups of automorphisms of the spectral presheaf, in particular those arising from one-parameter groups of inner automorphisms of the algebra. We interpret the spectral presheaf as a (generalised) state space for a quantum system and show how one can use flows on the spectral presheaf and on associated structures to describe the time evolution of non-relativistic quantum systems, both in the Schrodinger picture and the Heisenberg picture.

Motivation & Objective

• To extend the Gelfand duality concept from abelian to nonabelian unital C*-algebras by introducing a generalized state space via a spectral presheaf.
• To model time evolution in non-relativistic quantum systems using one-parameter groups of automorphisms on the spectral presheaf.
• To unify the Schrödinger and Heisenberg pictures of quantum dynamics within a single geometric framework based on the spectral presheaf.
• To interpret flows on the spectral presheaf as physical time evolution, particularly those induced by inner automorphisms of the C*-algebra.

Proposed method

• Constructs a spectral presheaf Σ^A from a unital C*-algebra A, generalizing the Gelfand spectrum for abelian algebras.
• Defines one-parameter groups of automorphisms on Σ^A, particularly those induced by inner automorphisms of A.
• Uses the action of these automorphism groups to define continuous flows on the spectral presheaf and associated structures.
• Applies the flow structure to represent time evolution in the Schrödinger picture as evolution of states on Σ^A.
• Reconstructs the Heisenberg picture as the dual action on observables via pullbacks along the flow morphisms.
• Establishes a correspondence between the dynamics of the C*-algebra and geometric flows on the generalized state space.

Experimental results

Research questions

• RQ1 How can the Gelfand spectrum be generalized to nonabelian C*-algebras to serve as a state space for quantum systems?
• RQ2 What is the role of one-parameter groups of automorphisms on the spectral presheaf in describing time evolution?
• RQ3 How do flows on the spectral presheaf correspond to the Schrödinger and Heisenberg pictures of quantum dynamics?
• RQ4 Can inner automorphisms of a C*-algebra generate physically meaningful dynamical flows on the generalized state space?
• RQ5 What is the geometric and algebraic structure underlying time evolution in non-relativistic quantum systems via spectral presheaves?

Key findings

• The spectral presheaf Σ^A provides a generalized state space for nonabelian unital C*-algebras, extending the Gelfand spectrum beyond the abelian case.
• One-parameter groups of automorphisms on Σ^A, especially those induced by inner automorphisms, define continuous flows that model time evolution.
• The Schrödinger picture is realized as the flow of states on Σ^A, with states evolving along the flow trajectories.
• The Heisenberg picture emerges as the dual action on the algebra of observables, with operators evolving via pullbacks along the flow morphisms.
• The framework unifies both pictures of quantum dynamics within a single geometric structure based on the spectral presheaf.
• The construction establishes a natural correspondence between the dynamical structure of the C*-algebra and the geometric evolution on the generalized state space.