2024/12/09 by Christoph Chiaffrino, Chiaffrino, Christoph, Olaf Hohm +2 · 1 citation
Mathematics · Computer Science · #Advanced Topics in Algebra #Rings, Modules, and Algebras #Quantum Computing Algorithms and Architecture
paper · pdf · doi:10.48550/arxiv.2412.06912
We present, for the harmonic oscillator and the spin-(1)/(2) system, an alternative formulation of quantum mechanics that is `off-shell': it is based on classical off-shell configurations and thus similar to the path integral. The core elements are Batalin-Vilkovisky (BV) algebras and factorization algebras, following a program by Costello and Gwilliam. The BV algebras are the spaces of quantum observables \rm Obsq(I) given by the symmetric algebra of polynomials in compactly supported functions on some interval I⊂ℝ, which can be viewed as functionals on the dynamical variables. Generalizing associative algebras, factorization algebras include in their data a topological space, which here is ℝ, and an assignment of a vector space to each open set, which here is the assignment of \rm Obsq(I) to each open interval I. The central structure maps are bilinear \rm Obsq(I1)⊗ \rm Obsq(I2)→ \rm Obsq(J) for disjoint intervals I1 and I2 contained in an interval J, which here is the wedge product of the symmetric algebra. We prove, as the central result of this paper, that this factorization algebra is quasi-isomorphic to the factorization algebra of `on-shell' quantum mechanics. In this we extend previous work by including half-open and closed intervals, and by generalizing to the spin-(1)/(2) system.