A quantum route to the classical Lagrangian formalism
| Autor: | Giuseppe Marmo, Fabio Di Cosmo, Alberto Ibort, Florio M. Ciaglia, Alessandro Zampini, Luca Schiavone |
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| Přispěvatelé: | Ciaglia, F. M., Di Cosmo, F., Ibort, A., Marmo, G., Schiavone, L., Zampini, A. |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Lie algebroid
Nuclear and High Energy Physics Dirac (software) FOS: Physical sciences General Physics and Astronomy 01 natural sciences symbols.namesake Mathematics::Category Theory 0103 physical sciences Quantum system 0101 mathematics Quantum Mathematics::Symplectic Geometry Mathematical Physics Mathematical physics Physics 010308 nuclear & particles physics Mathematics::Operator Algebras 010102 general mathematics Astronomy and Astrophysics Mathematical Physics (math-ph) Function (mathematics) State (functional analysis) Lagrangian in quantum mechanic Manifold Lie groupoids and Lie algebroids Von Neumann algebra symbols classical limit of quantum mechanic |
| Popis: | Using the recently developed groupoidal description of Schwinger's picture of Quantum Mechanics, a new approach to Dirac's fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function $\ell$ on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call {\itshape q-Lagrangian}, can be described in terms of a new function $\mathcal{L}$ on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold $M$, the quadratic expansion of $\mathcal{L}$ will reproduce the standard Lagrangians on $TM$ used to describe the classical dynamics of particles. 12 pages |
| Databáze: | OpenAIRE |
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