Abstract
Quantum probability is used to provide a new organization of basic quantum theory in a logical, axiomatic way. The principal thesis is that there is one fundamental time evolution equation in quantum theory, and this is given by a new version of Born’s Rule, which now includes both consecutive and conditional probability as it must, since science is based on correlations. A major modification of one of the standard axioms of quantum theory allows the implementation of various mathematically distinct models (commonly called ‘pictures’) of them. A natural definition of isomorphism of models is presented next. For a given Hamiltonian the standard ‘pictures’ of quantum theory (e.g., Schrödinger, Heisenberg, interaction) are isomorphic models, whose probabilities are nonetheless model independent. The Schrödinger equation remains valid in one model of the axioms, even though it is no longer the fundamental equation of time evolution. All the usual calculations of standard, textbook quantum theory remain valid, but they are put in a new light. For example, the ‘collapse’ of the state is now viewed in the Schrödinger model as one step in a two step algorithm for calculating one conditional probability. In the isomorphic Heisenberg model, where states are constant in time, one has the same conditional probability given by the same formula. Also, entanglement is defined in a new, more general model independent way as an aspect of quantum probability without using ‘collapse’ or tensor products.
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Notes
The error that only Type I von Neumann algebras are relevant in quantum theory.
References
Sontz, S.B.: An Introductory Path to Quantum Theory. Springer, Berlin (2020)
Dirac, P.A.M.: Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford (1958)
von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press (1955). (English translation of: Mathematische Grundlagen der Quantenmechanik. Springer (1932).)
Sontz, S.B.: A New Approach to Quantum Theory, treatise to be submitted
Russo, L.: The Forgotten Revolution. Springer, Berlin (2004)
Isham, C.: Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press, London (1995)
Weidmann, J.: Linear Operators in Hilbert Spaces. Springer (1980). (English translation of: Lineare Operatoren in Hilberträumen, B.G. Teubner) (1976)
Kolmogorov, A.N.: Foundations of the Theory of Probability, 2nd edn. Chelsea Pub. Co., (1956). (English translation of: Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, 1933.)
Gudder, S.P.: Quantum Probability. Academic Press, Cambridge (1988)
Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Birkhäuser, Basel (1992)
Beltrametti, E.G., Cassinelli, G.: The Logic of Quantum Mechanics. Addison-Wesley, Boston (1981)
Mackey, G.: The mathematical foundations of quantum mechanics. Synthese 42, 1–70 (1963)
Born, M.: Zur Quantemmechanik der Stossvorgänge. Z. Phys. 37, 863–867 (1926)
Iwai, T.: Geometry, Mechanics, and Control in Action for the Falling Cat. Springer, Berlin (2021)
Kadison, R.V.: Fundamentals of the Theory of Operator Algebras, vol. I. Academic Press, Cambridge (1983)
Cassinelli, G., Zanghi, N.: Conditional probabilities in quantum mechanics. Nuovo Cimento 73(B), 237–245 (1983)
Busch, P., et al.: Quantum Measurement. Springer, Berlin (2016)
Schmidt, E.: Zur Theorie der linearen und nichtlinearen Integralgleichungen. Math. Ann. 63, 433 (1907)
Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)
Lüders, G.: Über die Zustandsänderung durch den Messenprozess, Ann. Phys. (Liepzig) 8, 322–328 (1951). (English translation: Concerning the state-change due to the measurement process, Ann. Phys. (Liepzig) 15, 663–670 (2006).)
Griffiths, R.B.: Consistent Quantum Theory. Cambridge University Press, Cambridge (2002)
Bohm, D.: A suggested interpretation of quantum theory in terms of ‘Hidden’ Variables, I and II. Phys. Rev. 85, 166–193 (1952)
Frohlich, J., Pizzo, A.: The time-evolution of states in quantum mechanics according to the ETH-approach. Commun. Math. Phys. 389, 1673–1715 (2022)
Everett, H.: Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957)
Mermin, N.D.: The Ithaca interpretation of quantum mechanics. Pramana 51, 549–565 (1998)
Mermin, N.D.: Copenhagen computation: how i learned to stop worrying and love Bohr. IBM Res. J. Res. Dev. 48, 53 (2004)
Deutsch, D., Marletto, C.: Constructor theory of information. Proc. R. Soc. A 471, 20140540 (2015)
Moyal, J.E.: Quantum mechanics as a statistical theory. Math. Proc. Camb. Philos. Soc. 45, 99–124 (1949)
Acknowledgements
I thank Micho Ɖurđevich for his valuable comments and encouraging interest. I am also grateful to an anonymous reviewer whose comments motivated me to clarify several matters.
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Sontz, S.B. A New Organization of Quantum Theory Based on Quantum Probability. Found Phys 53, 49 (2023). https://doi.org/10.1007/s10701-023-00691-0
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DOI: https://doi.org/10.1007/s10701-023-00691-0