Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Interpretations of QM with Applications for Formal Epistemology Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

An Introduction to Symmetric Inflated Probabilities

verfasst von : Mark Burgin

Erschienen in: Quantum Interaction

Verlag: Springer International Publishing

Einloggen

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

search-config
loading …

Abstract

Traditionally, probability is treated as a function that takes values in the interval [0, 1]. All conventional interpretations of probability support this assumption, while all popular formal descriptions, e.g., axioms for probability, such as Kolmogorov’s axioms, canonize this premise. However, researchers found that negative, as well as larger than 1 probabilities could be a useful tool in physics. Some even assert that probabilities that can be negative, larger than 1 or less than −1 are necessary for physics. Here we develop an axiomatic system for such probabilities, which are called symmetric inflated probabilities and reflect interaction of particles and antiparticles, and study their properties.

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 "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

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
Vorheriges Kapitel On the Interpretation of Probabilities in Generalized Probabilistic Models
Nächstes Kapitel A First Attempt at Ordinal Projective Measurement
Literatur
Abramsky, S., Brandenburger, A.: An operational interpretation of negative probabilities and no-signalling models. In: Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. A Tribute to Prakash Panangaden. LNCS, vol. 8464, pp. 59–75. Springer, Heidelberg (2014). doi:10.​1007/​978-3-319-06880-0_​3 CrossRef
Allen, E.H.: Negative probabilities and the uses of signed probability theory. Philos. Sci. 43(1), 53–70 (1976)MathSciNetCrossRef
Baker, G.A.: Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space. Phys. Rev. 109, 2198 (1958)MathSciNetCrossRef
de Barros, J.A.: Decision making for inconsistent expert judgments using negative probabilities. In: Atmanspacher, H., Haven, E., Kitto, K., Raine, D. (eds.) QI 2013. LNCS, vol. 8369, pp. 257–269. Springer, Heidelberg (2014). doi:10.​1007/​978-3-642-54943-4_​23 CrossRef
de Barros, J.A., Oas, G.: Negative probabilities and counter-factual reasoning in quantum cognition. Phys. Scr. T163, 014008 (2014)CrossRef
de Barros, J.A., Oas, G.: Quantum cognition, neural oscillators, and negative probabilities. In: Haven, E., Khrennikov, A. (eds.) The Palgrave Handbook of Quantum Models in Social Science: Applications and Grand Challenges. Palgrave MacMillan, Basingstoke (2015)
Bednorz, A., Belzig, W.: On the problem of negative probabilities in time-resolved full counting statistics, 25th international conference on low temperature physics (LT25). J. Phys. Conf. Ser. 150, 022005 (2009)
Belinskii, A.V.: How could you measure a negative probability? JETP Lett. 59, 301–311 (1994)MathSciNet
Billingsley, P.: Probability and Measure. Wiley-Interscience, Hoboken (1995)MATH
Burgin, M.: Interpretations of negative probabilities. Preprint in Quantum Physics, quant-ph/1008.1287, 17 p. (2010). http://​arXiv.​org
Burgin, M.: Negative probability in the framework of combined probability. Preprint in Probability (math.PR), 1306.1166 (2013). http://​arXiv.​org
Burgin, M., Krinik, A.C.: Probabilities and hyperprobabilities. In: 8th Annual International Conference on Statistics, Mathematics and Related Fields, Conference Proceedings, Honolulu, Hawaii, pp. 351–367 (2009)
Burgin, M., Meissner, G.: Negative probabilities in modeling random financial processes. Integr. Math. Theor. Appl. 2(3), 305–322 (2010)MATH
Burgin, M., Meissner, G.: Negative probabilities in financial modeling. Wilmott Magazine, pp. 60–65 (2012)
Burgin, M., Meissner, G.: Larger than one probabilities in mathematical and practical finance. Rev. Econ. Financ. 2(4), 1–13 (2012)
Dirac, P.A.M.: Note on exchange phenomena in the Thomas atom. Proc. Camb. Philos. Soc. 26, 376–395 (1930a)CrossRefMATH
Dirac, P.A.M.: A theory of electrons and protons. Proc. R. Soc. Lond. Ser. A 126, 360–365 (1930b)CrossRefMATH
Dirac, P.A.M.: Quantum electrodynamics. Commun. Dublin Inst. Adv. Stud. 1, 1 (1943)MATH
Duffie, D., Singleton, K.: Modeling term structures of defaultable bonds. Rev. Financ. Stud. 12, 687–720 (1999)CrossRef
Ferrie, C., Emerson, J.: Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations. J. Phys. A: Math. Theor. 41, 352001 (2008)MathSciNetCrossRefMATH
Feynman, R.P.: The concept of probability theory in quantum mechanics. In: The Second Berkeley Symposium on Mathematical Statistics and Probability Theory. University of California Press, Berkeley, California (1950)
Feynman, R.P.: Negative probability. In: Quantum Implications: Essays in Honour of David Bohm, pp. 235–248. Routledge & Kegan Paul Ltd., London & New York (1987)
Feynman, R.P.: The reason for antiparticles. In: Elementary Particles and the Laws of Physics, The 1986 Dirac memorial lectures, pp. 56–59. Cambridge University Press, Cambridge (1987a)
Galvao, E.F.: Discrete Wigner functions and quantum computational speedup. Phys. Rev. A 71, 04230 (2005)CrossRef
Gell-Mann, M., Hartle, J.B.: Decoherent Histories Quantum Mechanics with One ‘Real’ Fine-Grained History. Preprint in Quantum Physics, quant-ph/1106.0767 (2011). http://​arXiv.​org
Halliwell, J., Yearsley, J.: Negative probabilities, Fine’s theorem, and linear positivity. Phys. Rev. A 87(2), 022114 (2013)CrossRef
Han, Y.D., Hwang, W.Y., Koh, I.G.: Explicit solutions for negative-probability measures for all entangled states. Phys. Lett. A 221(5), 283–286 (1996)MathSciNetCrossRefMATH
Haug, E.G.: Why so negative to negative probabilities. Wilmott Magazine, pp 34–38 (2004)
Heisenberg, W.: Über die inkohärente Streuung von Röntgenstrahlen. Physik. Zeitschr. 32, 737–740 (1931)MATH
Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Distribution functions in physics: fundamentals. Phys. Rep. 106(3), 121–167 (1984)MathSciNetCrossRef
Hofmann, H.F.: How to simulate a universal quantum computer using negative probabilities. J. Phys. A Math. Theor. 42, 275304 (2009). (9 pp)
Holland, R.E., Lynch, F.J., Perlow, G.J., Hanna, S.S.: Time spectra of filtered resonance radiation of Fe. Phys. Rev. Lett. 4, 181–182 (1960)CrossRef
Howard, J., Schnitzer, J., Sudarshan, E.C.: Quantum mechanical systems with indefinite metric II. Phys. Rev. 123, 2193–2201 (1961)MathSciNetCrossRefMATH
Khrennikov, A.: Interpretations of Probability. Walter de Gruyter, Berlin/New York (2009)CrossRefMATH
Kline, M.: Mathematics: The Loss of Certainty. Oxford University Press, New York (1980)MATH
Knuth, D.: The Art of Computer Programming. Seminumerical Algorithms, vol. 2. Addison-Wesley, Reading, Mass, Boston (1997)MATH
Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitrechnung. Ergebnisse Der Mathematik (English translation: Foundations of the Theory of Probability, Chelsea P.C. 1950) (1933)
Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Dover Publications, New York (1999)MATH
Koenig, A.: Patterns and antipatterns. J. Object Oriented Program. 8(1), 46–48 (1995)
Kronz, F.: Actual and virtual events in the quantum domain. Ontol. Stud. 9, 209–220 (2009)
Laplante, P. A., Neill, C. J.: Antipatterns: Identification, Refactoring and Management. Auerbach Publications, Boca Raton, FL (2005) CrossRef
Le Couteur, K.J.: The indefinite metric in relativistic quantum mechanics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 196(1045), 251–272 (1949)MathSciNetCrossRefMATH
Lynch, F.J., Holland, R.E., Hamermesh, M.: Time dependence of resonantly filtered gamma rays from Fe57. Phys. Rev. 120, 513–520 (1960)CrossRef
Mack, T.: Schadenversicherungsmathematik, 2nd edn. Verlag Versicherungswirtschaft (2002)
Martinez, A.A.: Negative Math: How Mathematical Rules Can Be Positively Bent. Princeton University Press, New Jersey (2006)MATH
Mattessich, R.: From accounting to negative numbers: a signal contribution of medieval India to mathematics. Acc. Historians J. 25(2), 129–145 (1998)
Mückenheim, W.: An extended-probability response to the Einstein-Podolsky-Rosen argument. Quantum Mechanics versus Local Realism: The Einstein-Podolsky-Rosen Paradox, pp. 345–364. Plenum Press, New York (1988)CrossRef
Nyambuya, G.: Deciphering and Fathoming Negative Probabilities in Quantum Mechanics (2011). (viXra.org > Quantum Physics > viXra:1102.0031)
Oas, G., de Barros, J.A., Carvalhaes, C.: Exploring non-signalling polytopes with negative probability. Phys. Scr. T163, 014034 (2014)CrossRef
Scully, M.O., Walther, H., Schleich, W.: Feynman’s approach to negative probability in quantum mechanics. Phys. Rev. A 49, 1562 (1994)MathSciNetCrossRef
Sjöstrand, T.: Monte Carlo generators. In: Fleischer, R (ed.) 2006 European School of High-Energy Physics, CERN-2007-005, pp. 51–73 (2007)
Sokolovski, D.: Weak values, “negative probability”, and the uncertainty principle. Phys. Rev. A 76, 042125 (2007)MathSciNetCrossRef
Sokolovski, D., Connor, J.N.L.: Negative probability and the distributions of dwell, transmission, and reflection times for quantum tunneling. Phys. Rev. A 44, 1500–1504 (1991)CrossRef
Székely, G.J.: Half of a coin: negative probabilities. Wilmott Magazine, 66–68, July 2005
Venter, G.: Generalized linear models beyond the exponential family with loss reserve applications. Astin Bull. 37(2), 345–364 (2007)MathSciNetCrossRefMATH
Weisskopf, V., Wigner, E.: Berechnung der nat urlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z. Phys. 63, 54–73 (1930)CrossRefMATH
Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)CrossRefMATH
Wu, C.S., Lee, Y.K., Benczer-Koller, N., Simms, P.: Frequency distribution of resonance line versus delay time. Phys. Rev. Lett. 5(9), 432–435 (1960)CrossRef
Metadaten
Titel
An Introduction to Symmetric Inflated Probabilities
verfasst von
Mark Burgin
Copyright-Jahr
2017
Buch
Quantum Interaction

Print ISBN: 978-3-319-52288-3

Electronic ISBN: 978-3-319-52289-0

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-52289-0_17