Symplectic Quantization III: Non-relativistic Limit

Year: 2024

Authors: Gradenigo G., Livi R., Salasnich L.

Autors Affiliation: Gran Sasso Sci Inst, Viale F Crispi 7, I-67100 Laquila, Italy; INFN, LNGS, Via G Acitelli 22, I-67100 Laquila, Italy; Univ Firenze, Dipartimento Fis & Astron, Via G Sansone 1, I-50019 Sesto, Fiorentino, Italy; Univ Firenze, CSDC, Via G Sansone 1, I-50019 Sesto, Fiorentino, Italy; INFN, Sez Firenze, Via G Sansone 1, I-50019 Sesto, Fiorentino, Italy; CNR, ISC, Via Madonna Piano 10, I-50019 Sesto Fiorentino, Italy; Univ Padua, Dipartimento Fis & Astron Galileo Galilei, Via F Marzolo 8, I-35131 Padua, Italy; Univ Padua, Padua QTech Ctr, Via F Marzolo 8, I-35131 Padua, Italy; INFN, Sez Padova, Via F Marzolo 8, I-35131 Padua, Italy; CNR, INO, Via N Carrara 1, I-50019 Florence, Italy.

Abstract: First of all we shortly illustrate how the symplectic quantization scheme (Gradenigo and Livi, Found Phys 51(3):66, 2021) can be applied to a relativistic field theory with self-interaction. Taking inspiration from the stochastic quantization method by Parisi and Wu, this procedure is based on considering explicitly the role of an intrinsic time variable, associated with quantum fluctuations. The major part of this paper is devoted to showing how the symplectic quantization scheme can be extended to the non-relativistic limit for a Schr & ouml;dinger-like field. Then we also discuss how one can obtain from this non-relativistic theory a linear Schr & ouml;dinger equation for the single-particle wavefunction. This further passage is based on a suitable coarse-graining procedure, when self-interaction terms can be neglected, with respect to interactions with any external field. In the Appendix we complete our survey on symplectic quantization by discussing how this scheme applies to a non-relativistic particle under the action of a generic external potential.

Journal/Review: FOUNDATIONS OF PHYSICS

Volume: 54 (4)      Pages from: 50-1  to: 50-19

More Information: Open access funding provided by Gran Sasso Science Institute – GSSI within the CRUI-CARE Agreement.We warmly thank F. Bigazzi, P. Di Cintio, S. Franz, D. Seminara and S. Wimberger for useful discussions. G.G. is partially supported by the project MIUR-PRIN2022 Emergent Dynamical Patterns of Disordered Systems with Applications to Natural Communities code 2022WPHMXK and acknowledges the Physics Department of Sapienza, University of Rome, and the Physics and Astronomy Department Galileo Galilei, University of Padova, for kind hospitality during some stages along the preparation of this manuscript. R.L. acknowledges partial support from project MIUR-PRIN2017 Coarse-grained description for non-equilibrium systems and transport phenomena (CO-NEST) n. 201798CZL. L.S. is partially supported by the European Union-NextGenerationEU within the National Center for HPC, Big Data and Quantum Computing [Project No. CN00000013, CN1 Spoke 10: Quantum Computing], by the BIRD Project Ultracold atoms in curved geometries of the University of Padova, by Iniziativa Specifica Quantum of Istituto Nazionale di Fisica Nucleare, by the European Quantum Flagship Project PASQuanS 2, and by the PRIN 2022 Project Quantum Atomic Mixtures: Droplets, Topological Structures, and Vortices of the Italian Ministry for University and Research. L. S. also acknowledges the Project Frontiere Quantistiche (Dipartimenti di Eccellenza) of the Italian Ministry for Universities and Research.
KeyWords: Symplectic quantization; Non-relativistic limit; Non-relativistic quantum field theory; Quantum mechanics
DOI: 10.1007/s10701-024-00783-5

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