Foundations of Physics Letters

, Volume 11, Issue 4, pp 371–377 | Cite as

A Unified Algebraic Approach to Quantum Theory

  • N. A. M. Monk
  • B. J. Hiley


Conventional approaches to quantum mechanics are essentially dualistic. This is reflected in the fact that their mathematical formulation is based on two distinct mathematical structures: the algebra of dynamical variables (observables) and the vector space of state vectors. In contrast, coherent interpretations of quantum mechanics highlight the fact that quantum phenomena must be considered as undivided wholes. Here, we discuss a purely algebraic formulation of quantum mechanics. This formulation does not require the specification of a space of state vectors; rather, the required vector spaces can be identified as substructures in the algebra of dynamical variables (suitably extended for bosonic systems). This formulation of quantum mechanics captures the undivided wholeness characteristic of quantum phenomena, and provides insight into their characteristic nonseparability and nonlocality. The interpretation of the algebraic formulation in terms of quantum process is discussed.

quantum algebra quantum pre-space discrete Weyl algebra Clifford algebra implicate order 


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  1. Barnes, B. A. (1980). Proc. Edinburgh Math. Soc. 23, 229–238.MathSciNetCrossRefGoogle Scholar
  2. Bohm D. J. (1957). Causality and Chance in Modern Physics (Routledge & Kegan Paul, London).CrossRefGoogle Scholar
  3. Bohm D. J. (1980). Wholeness and the Implicate Order (Routledge & Kegan Paul, London).Google Scholar
  4. Bohm D. J. and Peat, F. D. (1988). Science, Order and Creativity (Routledge, London).Google Scholar
  5. Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford).zbMATHGoogle Scholar
  6. Dirac, P. A. M. (1965). Phys. Rev. 1393B, 684–690.ADSMathSciNetCrossRefGoogle Scholar
  7. Dirac, P. A. M. (1966). Lectures on Quantum Field Theory (Yeshiva University Belfer Graduate School of Science, New York).Google Scholar
  8. Fernandes, M. (1995). “Geometric Algebras and the Foundations of Quantum Theory,” Ph.D. thesis, University of London.Google Scholar
  9. Fernandes, M., and Hiley, B. J. (1996). “The metaplectic group, the symplectic spinor and the Güoy phase,” submitted.Google Scholar
  10. Frescura, F. A. M., and Hiley, B. J. (1980a). Found. Phys. 10, 7–31.ADSMathSciNetCrossRefGoogle Scholar
  11. Frescura, F. A. M., and Hiley, B. J. (1980b). Found. Phys. 10, 705–722.ADSMathSciNetCrossRefGoogle Scholar
  12. Frescura, F. A. M., and Hiley, B. J. (1984). Rev. Brasil. Phys., vol. especial 70 anos de Mario Schönberg, 49–86.Google Scholar
  13. Guillemin, V., and Sternberg, S. (1984). Symplectic Techniques in Physics (Cambridge University Press, Cambridge).zbMATHGoogle Scholar
  14. Hiley, B. J. (1980). Ann. Fond. Louis de Broglie 5, 75–103.Google Scholar
  15. Hiley, B. J., and Fernandes, M. (1997). In Time, Temporality and Now, E. Ruhnau, H. Atmanspacher, and W. Beiglboeck, eds. (Springer, Berlin).Google Scholar
  16. Hiley, B., and Monk, N. (1993). Mod. Phys. Lett. A 8, 3625–3633.ADSMathSciNetCrossRefGoogle Scholar
  17. Monk, N. A. M. (1994). “Algebraic Structures in the Light of the Implicate Order,” PhD thesis, University of London.Google Scholar
  18. Monk, N. A. M. (1997). Stud. Hist. Phil. Mod. Phys. 28, 1–34.MathSciNetCrossRefGoogle Scholar
  19. von Neumann, J. (1931). Math. Ann. 104, 570–578.MathSciNetCrossRefGoogle Scholar
  20. Weyl, H. (1950). The Theory of Groups and Quantum Mechanics (Dover, New York).zbMATHGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • N. A. M. Monk
    • 1
    • 2
  • B. J. Hiley
    • 1
  1. 1.Theoretical Physics Research UnitBirkbeck CollegeLondonUK
  2. 2.Developmental Genetics ProgrammeUniversity of Sheffield, Firth Court, Western BankSheffieldUK

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