nach oben

Soft Computing

Erschienen in:

02.12.2019 | Focus

Evolution of quantum observables: from non-commutativity to commutativity

verfasst von: S. Fortin, M. Gadella, F. Holik, M. Losada

Erschienen in: Soft Computing | Ausgabe 14/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe this transition is to consider the Gamow vectors, which introduce exponential decays in the evolution. In this paper, we give two mathematical models in which this transition happens in the infinite time limit. In the first one, we consider operators acting on the space of the Gamow vectors, which represent quantum resonances. In the second one, we use an algebraic formalism from scattering theory. We construct a non-commuting algebra which commutes in the infinite time limit.

Vorheriger Artikel An equational theory for -complete orthomodular lattices

Nächster Artikel How images combine meaning: quantum entanglement in visual perception

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

Antoine JP (1969) Dirac formalism and symmetry problems in quantum mechanics. I. General Dirac formalism. J Math Phys 10:53–69MathSciNetMATH

Antoniou I, Laura R, Suchanecki Z, Tasaki S (1997) Intrinsic irreversibility of quantum systems with diagonal singularity. Phys A 241:737–772

Antoniou IE, Gadella M, Pronko GP (1998) Gamow vectors for degenerate scattering resonances. J Math Phys 39:2459–2475MathSciNetMATH

Antoniou I, Gadella M, Suchanecki Z (1998) Some general properties of Liouville spaces. In: Bohm A, Doebner HD, Kielanowski P (eds) Irreversibility and causality. Lecture notes in physics, vol 504. Springer, Berlin, pp 38–56

Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–843MathSciNetMATH

Bleistein N, Handelsman R (1986) Asymptotic expansion of integrals. Dover Inc., New YorkMATH

Bohm A (1978) The rigged Hilbert space and quantum mechanics. Springer lecture notes in physics, vol 78. Springer, New York

Bohm A (1981) Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics. J Math Phys 22:2813–2823MathSciNet

Bohm A (1993) Quantum mechanics: foundations and applications. Springer, BerlinMATH

Bohm A, Gadella M (1989) Dirac kets, Gamow vectors and Gelfand triplets. Springer lecture notes in physics, vol 348. Springer, New York

Bohm A, Erman F, Uncu H (2011) Resonance phenomena and time asymmetric quantum mechanics. Turk J Phys 35:209–240

Castagnino M, Fortin S (2013) Formal features of a general theoretical framework for decoherence in open and closed systems. Int J Theor Phys 52:1379–1398MathSciNetMATH

Castagnino M, Gadella M (2006) The problem of the classical limit of quantum mechanics and the role of self-induced decoherence. Found Phys 36:920–952MathSciNetMATH

Castagnino M, Lombardi O (2005) Self-induced decoherence and the classical limit of quantum mechanics. Philos Sci 72:764–776MathSciNet

Castagnino M, Gadella M, Gaioli F, Laura R (1999) Gamow vectors and time asymmetry. Int J Theor Phys 38:2823–2865MathSciNetMATH

Castagnino M, Gadella M, Betán RI, Laura R (2001) Gamow functionals on operator algebras. J Phys A Math Gen 34:10067–10083MathSciNetMATH

Castagnino M, Fortin S, Lombardi O (2010) The effect of random coupling coefficients on decoherence. Mod Phys Lett A 25:611–617MATH

Celeghini E, Gadella M, del Olmo MA (2016) Applications of rigged Hilbert spaces in quantum mechanics and signal processing. J Math Phys 57:072105MathSciNetMATH

Celeghini E, Gadella M, del Olmo MA (2017) Lie algebra representations and rigged Hilbert spaces: the SO(2) case. Acta Polytech (Prag) 57:379–384

Celeghini E, Gadella M, del Olmo MA (2018) Spherical harmonics and rigged Hilbert spaces. J Math Phys 59:053502MathSciNetMATH

Civitarese O, Gadella M (2004) Physical and mathematical aspects of Gamow states. Phys Rep 396:41–113MathSciNet

Dalla Chiara ML, Giuntini R, Greechie R (2004) Reasoning in quantum theory. Kluwer Academic, DordrechtMATH

Exner P (1984) Open quantum systems and Feynman integrals. Reidel, DordrechtMATH

Fischer MC, Gutiérrez-Medina B, Reizen MG (2001) Observation of the quantum Zeno and anti-Zeno effects in an unstable system. Phys Rev Lett 87:40402

Fonda L, Ghirardi GC, Rimini A (1978) Decay theory of unstable quantum systems. Rep Prog Phys 41:587–631

Fortin S, Vanni L (2014) Quantum decoherence: a logical perspective. Found Phys 44:1258–1268MathSciNetMATH

Fortin S, Holik F, Vanni L (2016) Non-unitary evolution of quantum logics. In: Bagarello F, Passante R, Trapani C (eds) Non-hermitian Hamiltonians in quantum physics. Springer proceedings in physics, vol 184. Springer, ChamMATH

Friedrichs KO (1948) On the perturbation of continuous spectra. Commun Appl Math 1:361–406MathSciNetMATH

Gadella M (2014) Quantum resonances: theory and models. In: Kielanowski P, Bieliavsky P, Odesskii A, Odzijewicz A, Schlichenmaier M, Voronov T (eds) Geometric methods in physics, XXXII workshop Bialowieza. Springer Basel AG, Poland, pp 99–118

Gadella M (2015) A discussion on the properties of Gamow states. Found Phys 45:177–197MathSciNetMATH

Gadella M, Gómez F (2002) A unified mathematical formalism for the Dirac formulation of quantum mechanics. Found Phys 32:815–869MathSciNet

Gadella M, Gómez F (2003) On the mathematical basis of the Dirac formulation of quantum mechanics. Int J Theor Phys 42:2225–2254MathSciNetMATH

Gadella M, Laura R (2001) Gamow dyads and expectation values. Int J Quantum Chem 81:307–320

Gadella M, de la Madrid R (1999) Resonances and time reversal operator in rigged Hilbert spaces. Int J Theor Phys 38:93–113MathSciNetMATH

Gadella M, Pronko GP (2011) The Friedrichs model and its use in resonance phenomena. Fortschr Phys 59:795–859MathSciNetMATH

Gadella M, Kuru Ş, Negro J (2017) The hyperbolic step potential: antibound states, SUSY partners and Wigner time delays. Ann Phys 379:86–101MATH

Gelfand IM, Vilenkin NY (1964) Generalized functions: applications to harmonic analysis. Academic, New York

Gell-Mann M, Hartle JB (1990) Quantum mechanics in the light of quantum cosmology. In: Zurek WH (ed) Complexity, entropy and the physics of information. Addison-Wesley, Reading

Gell-Mann M, Hartle JB (1993) Classical equations for quantum systems. Phys Rev D 47:3345–3382MathSciNet

Griffiths RB (2002) Consistent quantum theory. Cambridge University Press, CambridgeMATH

Horvath J (1966) Topological vector spaces and distributions. Addison-Wesley, ReadingMATH

Khalfin LA (1972) CPT invariance of CP-noninvariant theory of K0 and Kbar0 Mesons and permissible mass distributions of the KS and KL Mesons. JETP Lett 15:388–392

Kiefer C, Polarski D (2009) Why do cosmological perturbations look classical to us? Adv Sci Lett 2:164–173

Losada M, Fortin S, Holik F (2018) Classical limit and quantum logic. Int J Theor Phys 57:465–475MathSciNetMATH

Losada M, Fortin S, Gadella M, Holik F (2018) Dynamics of algebras in quantum unstable systems. Int J Mod Phys A 33:1850109MathSciNetMATH

Melsheimer O (1974) Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory. J Math Phys 15:902–916MathSciNet

Misra B, Sudarshan ECG (1977) The Zeno’s paradox in quantum theory. J Math Phys 18:756–763MathSciNet

Mondragón A, Hernández E (1993) Degeneracy and crossing of resonance energy surfaces. J Phys A Math Gen 26:5595–5611MathSciNet

Nakanishi N (1958) A theory of clothed unstable particles. Progr Theor Phys 19:607–621MathSciNetMATH

Nussenzveig HM (1972) Causality and dispersion relations. Academic Press, New York

Omnès R (1999) Understanding quantum mechanics. Princeton University Press, PrincetonMATH

Ramírez R, Reboiro M (2019) Dynamics of finite dimensional non-hermitian systems with indefinite metric. J Math Phys 60:012106MathSciNetMATH

Ramírez R, Reboiro M (2019) Optimal spin squeezed steady state induced by the dynamics of non-hermitian Hamiltonians. Phys Scr 94:085220

Reed M, Simon B (1978) Analysis of operators. Academic Press, New YorkMATH

Reed M, Simon B (1981) Functional analysis. Academic Press, New York

Roberts JE (1966) Rigged Hilbert spaces in quantum mechanics. Commun Math Phys 3:98–119MathSciNetMATH

Rothe C, Hintschich SI, Monkman AP (2006) Violation of the exponential-decay law at long times. Phys Rev Lett 96:163601

Schlosshauer M (2007) Decoherence and the quantum-to-classical transition. Springer, Berlin

Urbanowski K (2009) General properties of the evolution of unstable states at long times. Eur Phys J D 54:25–29

Wigner EP (1967) Symmetries and reflections. Indiana University Press, Bloomington, pp 38–39

Wigner EP (1994) Group theoretical concepts and methods in elementary particle physics. Gordon and Breach, New York, pp 37–38

Zurek WH (2009) Quantum darwinism. Nat Phys 5:181–188

Titel: Evolution of quantum observables: from non-commutativity to commutativity
verfasst von: S. Fortin
M. Gadella
F. Holik
M. Losada
Publikationsdatum: 02.12.2019
Verlag: Springer Berlin Heidelberg
Erschienen in: Soft Computing / Ausgabe 14/2020
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI: https://doi.org/10.1007/s00500-019-04546-7

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 14/2020

An AdaBoost-modified classifier using stochastic diffusion search model for data optimization in Internet of Things

Explaining versus describing human decisions: Hilbert space structures in decision theory

Fuzzy representation of finite-valued quantum gates

Hierarchical detection of abnormal behaviors in video surveillance through modeling normal behaviors based on AUC maximization

Novel neutrality aggregation operator-based multiattribute group decision-making method for single-valued neutrosophic numbers

A new hybrid discriminative/generative model using the full-covariance multivariate generalized Gaussian mixture models

Premium Partner