2014/07/10 by Matthew F. Brown, Brown, Matthew F.
Computer Science · Mathematics · Physics and Astronomy · #81S22 #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Information Theory (cs.IT) #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Statistical Mechanics and Entropy #advanced mathematical theories #cs.IT #math-ph #math.DS #math.IT #math.MP #msc:81S22 #quant-ph
paper · pdf · doi:10.48550/arxiv.1407.2875
openalex publication_date 2014/07/10 · arxiv created 2015/11/08 · arxiv updated 2015/11/10 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
In this paper we shall re-visit the well-known Schrödinger and Lindblad dynamics of quantum mechanics. However, these equations may be realized as the consequence of a more general, underlying dynamical process. In both cases we shall see that the evolution of a quantum state Pψ=\varrho(0) has the not so well-known pseudo-quadratic form ∂t\varrho(t)=V^⋆\varrho(t)V where V is a vector operator in a complex Minkowski space and the pseudo-adjoint V^⋆ is induced by the Minkowski metric \boldsymbolη. The interesting thing about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin Formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be `unraveled' in a second-quantized Minkowski space. Working in such a space provided the author with the means to construct a QS (quantum stochastic) Duhamel principle and known applications to a Schrödinger dynamics perturbed by a continual measurement process are considered. What is not known, but presented here, is the role of the Lorentz transform in quantum measurement, and the appearance of Riemannian geometry in quantum measurement is also discussed.