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Follow the Math!: The Mathematics of Quantum Mechanics as the Mathematics of Set Partitions Linearized to (Hilbert) Vector Spaces

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Abstract

The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac’s Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being “definite all the way down”.

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Notes

  1. The slogan “Follow the money” means that finding the source of an organization’s or person’s money may reveal their true nature. In a similar sense, we use the slogan “Follow the math!” to mean that finding “where the mathematics of QM comes from” reveals a good deal about the key concepts and machinery of the theory.

  2. Since our purpose is conceptual clarity, not mathematical generality, we will stick to the finite sets and dimensions throughout.

  3. Many of the older texts [5] presented the “lattice of partitions” upside down, i.e., with the opposite partial order, so the join and meet as well as the top and bottom were interchanged.

  4. In his sympathetic interpretation of Aristotle’s treatment of substance and form, Heisenberg refers to the substance as: “a kind of indefinite corporeal substratum, embodying the possibility of passing over into actuality by means of the form” [26, p. 148]. Heisenberg’s “potentiality” “passing over into actuality by means of the form” should be seen as the actual indefinite “passing over into” the actual definite by being objectively in-formed through the making of distinctions.

  5. The indefiniteness interpretation of qubit \(a\left| 0\right\rangle +b\left| 1\right\rangle\) is more routine in quantum information and computation theory [33] as opposed to the being simultaneously \(\left| 0\right\rangle\) and \(\left| 1\right\rangle\) version of superposition.

  6. There is at least an analogy between superposition in QM and abstraction in mathematics. In Frege’s example of a set of parallel directed line segments oriented in the same way, the abstraction “direction” is definite on what is common between the lines and indefinite on how they differ [18]. In QM, the emphasis is on the indefiniteness between the definite eigenstates (glass half-empty) while in abstraction, the emphasis is on the common definiteness between the instances (glass half-full).

  7. There is the linearization map (functor) which takes a set U to the vector space \(\mathbb {C} ^{U}\) where u lifts to the basis vector \(\delta _{u}=\chi _{\left\{ u\right\} }:U\rightarrow \mathbb {C}\), but we apply the Yoga to many different concepts.

  8. In view of the duality between subsets and set partitions, there is the induced duality between subspaces and direct-sum decompositions. Hence the Birkhoff-von-Neumann quantum logic of subspaces [6] has a dual quantum logic of DSDs [17].

  9. Heisenberg’s German word was “Unbestimmtheit” which could well be translated as “indefiniteness” or “indeterminacy” rather than ”uncertainty”.

  10. This is particularly clear in the pedagogical model of the double-slit experiment in QM over \(\mathbb {Z} _{2}\) [16] where the interference in the evolving 0, 1-vectors has no resemblence to waves.

  11. In the general vector space case, for two commuting (diagonalizable) operators, \(FG=GF\), if both are represented in a basis of F-eigenvectors and \(\left( u_{i},u_{k}\right)\) is a qudit of F, then \(G_{ik}=0\) [44, p. 4].

  12. The Yoga is used to generate vector space concepts corresponding to set concepts. There is no implication that every instance of a vector space concept, e.g., a commuting F, must come from an instance of the set concept, e.g., a commuting f..

References

  1. Ainsworth, T.: Form vs. Matter. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Spring 2016 Ed.) (2016) https://plato.stanford.edu/archives/spr2016/entries/form-matter/

  2. Auletta, G., Fortunato, M., Parisi, G.: Quantum Mechanics. Cambridge University Press, Cambridge UK (2009)

    Book  MATH  Google Scholar 

  3. Bell, J.S.: Against “Measurement". In: Miller, A.I. (ed.) Sixty-Two Years of Uncertainty, pp. 17–31. Plenum Press, New York (1990)

    Chapter  Google Scholar 

  4. Bennett, C.H.: Quantum information: Qubits and quantum error correction. Int. J. Theor. Phys. 42, 153–176 (2003). https://doi.org/10.1023/A:1024439131297

    Article  MathSciNet  MATH  Google Scholar 

  5. Birkhoff, G.: Lattice Theory. American Mathematical Society, New York (1948)

    MATH  Google Scholar 

  6. Birkhoff, G., Von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–43 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  7. Castellani, E.: Symmetry and equivalence. In: Brading, K., Castellani, E. (eds.) Symmetries in Physics: Philosophical Reflections, pp. 425–436. Cambridge University Press, Cambridge (2003)

    Chapter  Google Scholar 

  8. Chen, J.-Q., Gao, M.-J., Ma, G.-Q.: The representation group and its application to space groups. Rev. Mod. Phys. 57, 211–278 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  9. Chen, J.-Q., Ping, J., Wang, F.: Group Representation Theory for Physicists, 2nd edn. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  10. Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics: Volumes 1 and 2. Wiley, New York (2005)

    MATH  Google Scholar 

  11. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Clarendon Press, Oxford (1958)

    MATH  Google Scholar 

  12. Dubreil, P., Dubreil-Jacotin, M.-L.: Théorie algébrique des relations d’équivalence. J. de Mathématique. 18, 63–95 (1939)

    MATH  Google Scholar 

  13. Eddington, A.S.: New Pathways in Science (Messenger Lectures 1934). Cambridge University Press, Cambridge UK (1947)

    Google Scholar 

  14. Ellerman, D.: Counting distinctions: on the conceptual foundations of Shannon’s information theory. Synthese 168, 119–149 (2009). https://doi.org/10.1007/s11229-008-9333-7

    Article  MathSciNet  MATH  Google Scholar 

  15. Ellerman, D.: The logic of partitions: introduction to the dual of the logic of subsets. Rev Symb Logic. 3, 287–350 (2010). https://doi.org/10.1017/S1755020310000018

    Article  MathSciNet  MATH  Google Scholar 

  16. Ellerman, D.: Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus. Synthese 194, 4863–4896 (2017). https://doi.org/10.1007/s11229-016-1175-0

    Article  MathSciNet  MATH  Google Scholar 

  17. Ellerman, D.: The quantum logic of direct-sum decompositions: the dual to the quantum logic of subspaces. Logic J IGPL. 26, 1–13 (2018). https://doi.org/10.1093/jigpal/jzx026

    Article  MathSciNet  MATH  Google Scholar 

  18. Ellerman, D.: On abstraction in mathematics and indefiniteness in quantum mechanics. J. Philos. Log. 50, 813–835 (2021). https://doi.org/10.1007/s10992-020-09586-1

    Article  MathSciNet  MATH  Google Scholar 

  19. Ellerman, D.: New Foundations for Information Theory: Logical Entropy and Shannon Entropy. SpringerNature, Cham (2021)

    Book  MATH  Google Scholar 

  20. Ellerman, D.: Introduction to logical entropy and its relationship to Shannon entropy. 4Open Special Issue: Logical Entropy 5, 1–33 (2022). https://doi.org/10.1051/fopen/2021004

  21. Fano, U.: Description of states in quantum mechanics by density matrix and operator techniques. Rev. Mod. Phys. 29, 74–93 (1957)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1968)

    MATH  Google Scholar 

  23. Feyerabend, P.: Problems of microphysics. In: Colodny, R.G. (ed.) Frontiers of Science and Philosophy, pp. 189–283. University Press of America, Lanham MD (1983)

    Google Scholar 

  24. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics: Quantum Mechanics, vol. III. Addison-Wesley, Reading (1965)

  25. Fine, A.: The Shaky Game: Einstein, Realism, and the Quantum Theory. University of Chicago Press, Chicago (1986)

    Google Scholar 

  26. Heisenberg, W.: Physics & Philosophy: The Revolution in Modern Science. Harper Torchbooks, New York (1962)

    Google Scholar 

  27. Hoffman, K., Kunze, R.: Linear Algebra. Prentice-Hall, Englewood Cliffs NJ (1961)

    MATH  Google Scholar 

  28. Kolmogorov, A.N.: Combinatorial foundations of information theory and the calculus of probabilities. Russ. Math. Surv. 38, 29–40 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kung, J.P.S., Rota, G.-C., Yan, C.H.: Combinatorics: The Rota Way. Cambridge University Press, New York (2009)

    Book  MATH  Google Scholar 

  30. Lawvere, F.W., Rosebrugh, R.: Sets for Mathematics. Cambridge University Press, Cambridge MA (2003)

    Book  MATH  Google Scholar 

  31. Leibniz, G.W.: New Essays on Human Understanding. Cambridge University Press, Cambridge UK (1996)

    Book  Google Scholar 

  32. Lüders, G.: Über die Zustandsänderung durch Meßprozeß. Ann. Phys. 8, 322–328 (1951)

    MATH  Google Scholar 

  33. Nielsen, M., Chuang, I.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  34. Norsen, T.: Foundations of Quantum Mechanics. Springer International, Cham, Switzerland (2017)

    Book  MATH  Google Scholar 

  35. Pagels, H.: Perfect Symmetry: The Search for the Beginning of Time. Simon and Schuster, New York (1985)

    Google Scholar 

  36. Rota, G.-C.: Indiscrete Thoughts. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  37. Rota, G.-C.: Twelve problems in probability no one likes to bring up. In: Crapo, H., Senato, D. (eds.) Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota, pp. 57–93. Springer, Milano (2001)

    Chapter  Google Scholar 

  38. Sakurai, J.J., Napolitano, J.: Modern Quantum Mechanics. Addison-Wesley, Boston (2011)

    MATH  Google Scholar 

  39. Shimony, A.: The reality of the quantum world. Sci. Am. 258, 46–53 (1988)

    Article  ADS  Google Scholar 

  40. Shimony, A.: Philosophical and experimental perspectives on quantum physics. In: Philosophical and Experimental Perspectives on Quantum Physics: Vienna Circle Institute Yearbook 7, pp. 1–18. Springer, Dordrecht (1999)

  41. Sternberg, S.: Group Theory and Physics. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  42. Stone, M.H.: On one-parameter unitary groups in Hilbert Space. Ann. Math. 33, 643–648 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tamir, B., Piava, I.L., Schwartzman-Nowik, Z., Cohen, E.: Quantum logical entropy: fundamentals and general properties. 4Open Special Issue: Logical Entropy 5, 1–14 (2022). https://doi.org/10.1051/fopen/2021005

  44. Tinkham, M.: Group Theory and Quantum Mechanics. Dover, Mineola NY (1964)

    MATH  Google Scholar 

  45. Von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)

    MATH  Google Scholar 

  46. Wang, F.: A conceptual review of the new approach to group representation theory. In: Feng, D.H., Iachello, F., Ping, J.L., and Wang, F. (eds.) The Beauty of Mathematics in Science: The Intellectual Path of J. Q. Chen. pp. 1–8. World Scientific, Singapore (2004)

  47. Weinberg, S.: Dreams of a Final Theory. Vintage Books, New York (1994)

    Book  Google Scholar 

  48. Weyl, H.: Philosophy of Mathematics and Natural Science. Princeton University Press, Princeton (1949)

    MATH  Google Scholar 

  49. Wigner, E.P.: Invariance in physical theory. In: Symmetries and Reflections: Scientific Essays of Eugene P. Wigner, pp. 3–13. Ox Bow Press, Woodbridge CN (1967)

  50. Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Ellerman, D. Follow the Math!: The Mathematics of Quantum Mechanics as the Mathematics of Set Partitions Linearized to (Hilbert) Vector Spaces. Found Phys 52, 100 (2022). https://doi.org/10.1007/s10701-022-00608-3

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