Abstract
This paper outlines the common ground between the algebraic approach to quantum phenomena proposed by Hans Primas and the ideas lying behind David Bohm’s notion of the implicate and explicate order. The latter emerged from what he called “an algebraic description of structure- process” which, in terms of formal logic, was a way to study the relation between a non-Boolean (implicate) quantum logic and its Boolean (explicate) projections. We show that in the implicate order, we have two time-evolution equations, one involving a commutator, which is essentially Heisenberg’s equation of motion, and the other involving an anti-commutator or Jordan product. Explicate orders emerge from projections into, or shadows on, Boolean sub-structures, a process that Primas has likened to “pattern recognition”. These projections produce equations that form the basis of what has been called the de Broglie–Bohm interpretation of quantum mechanics. By exploiting the properties of the orthogonal Clifford algebras, this model has been generalized to include relativistic systems with spin, giving a novel insight into the whole approach.
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Notes
- 1.
Note that what they called a B*-algebra in 1978 is nowadays usually referred to as a C*-algebra.
References
Abramsky, S., and Coecke, B. (2004): A categorical semantics of quantum protocols. In Logic in Computer Science, IEEE Computer Society, Washington DC, pp. 415–425.
Bohm, D. (1965): Space, time, and the quantum theory understood in terms of discrete structural process. Proceedings of the International Conference on Elementary Particles, Kyoto, pp. 252–287.
Bohm, D. (1971): Space-time geometry as an abstraction from spinor ordering. In Perspectives in Quantum Theory: Essays in Honour of Alfred Landé, ed. by W. Yourgrau, MIT Press, Cambridge, pp. 78–90.
Bohm, D. (1980): Wholeness and the Implicate Order, Routledge, London.
Bohm, D.J., Hiley, B.J., and Stuart, A.E.G. (1970): On a new mode of description in physics. International Journal of Theoretical Physics 3, 171–183.
Born M., and Jordan P. (1925): Zur Quantenmechanik. Zeitschrift für Physik 34, 858–888.
Brown, M.R., and Hiley, B.J. (2000): Schrödinger revisited: the role of Dirac’s “standard” ket in the algebraic approach. Preprint accessible at at http://arxiv.org/abs/quant-ph/0005026.
Clifford W.K. (1882): Further note on biquaternions. In Mathematical Papers XLII, ed. by R. Tucker, Macmillan, London, pp. 385–394.
Coecke, B. (2005): Kindergarten quantum mechanics. In Quantum Theory: Reconsiderations of Foundations III, ed. by A. Khrennikov, AIP Press, New York, pp. 81–98.
Crumeyrolle A. (1990): Orthogonal and Symplectic Clifford Algebras: Spinor Structures, Kluwer, Dordrecht.
Domb, C., and Hiley, B.J. (1962): On the method of Yvon in crystal statistics. Proceedings of the Royal Society A268, 506–526.
Eddington, A.S. (1936): Relativity Theory of Protons and Electrons, Cambridge University Press, Cambridge.
Eddington, A.S. (1958): The Philosophy of Physical Science, University of Michigan Press, Ann Arbor.
Finkelstein, D. (1968): Matter, space and logic. In Boston Studies in the Philosophy of Science V, ed. by R.S. Cohen and M.W. Wartowsky, Reidel, Dordrecht, pp. 199–215.
d’Espagnat, B. (2003): Veiled Reality: An Analysis of Present-Day Quantum Mechanical Concepts, Westview Press, Boulder.
Finkelstein, D. (1969): Matter, space, and logic. In Boston Studies in the Philosophy of Science V, ed. by R.S. Cohen and M.W. Wartowsky, Reidel, Dordrecht, pp. 199–215.
Finkelstein, D. (1987): All is flux. In Quantum Implications: Essays in Honour of David Bohm, ed. by B.J. Hiley and D. Peat, D., Routledge and Kegan Paul, London, pp. 289–294.
Finkelstein, D.R. (1996): Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg, Springer, Berlin.
Grassmann, H.G. (1894): Gesammelte mathematische und physikalische Werke, Teubner, Leipzig.
Grassmann, H.G. (1995): A New Branch of Mathematics: the Ausdehnungslehre of 1844 and Other Works, translated by L.C. Kannenberg, Open Court, Chicago.
Hamilton, W. R. (1967): Mathematical Papers, Vol. 3: Algebra, Cambridge University Press, Cambridge.
Heisenberg, W. (1958): Physics and Philosophy: The Revolution in Modern Science, George Allen and Unwin, London.
Hiley, B.J. (2001): A note on the role of idempotents in the extended Heisenberg Algebra. In Implications (ANPA 22), Alternative Natural Philosophy Association, Cambridge, pp. 107–121.
Hiley, B.J. (2015): On the relationship between the Moyal algebra and the quantum operator algebra of von Neumann. Journal of Computational Electronics 14, 869–878.
Hiley, B.J., Burke, T., and Finney, J. (1977): Self-avoiding walks on irregular structures. Journal of Physics A10, 197–204.
Hiley, B.J., and Callaghan, R.E. (2010): The Clifford algebra approach to quantum mechanics A: The Schrödinger and Pauli particles. Preprint accessible at arXiv:1011.4031.
Hiley, B.J., and Callaghan, R.E. (2012): Clifford algebras and the Dirac-Bohm quantum Hamilton-Jacobi equation. Foundations of Physics 42, 192–208.
Hiley, B.J., and Frescura, F.A.M. (1980): The implicate order, algebras and the spinor. Foundations of Physics 10, 7–31.
Jones, V.F.R. (1986): A new knot polynomial and von Neumann algebras. Notices of the American Mathematical Society 33, 219–225.
Jones, V.F.R. (2003): Von Neumann algebras. Lecture notes accessible at http://www.math.berkeley.edu/vfr/MATH20909/VonNeumann2009.pdf.
Kauffman, L.H. (2001): Knots and Physics, World Scientific, Singapore.
Khalkhali, M. (2009): Basic Non-Commutative Geometry, EMS Publishing, Zurich.
Murray, F.J., and von Neumann, J. (1936): On rings of operators. Annals of Mathematics 37, 116–229.
Onsager, L. (1944): Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review 65, 117–149.
Penrose, R. (1967): Twistor algebra. Journal of Mathematical Physics 8, 345–366.
Penrose, R. (1971): Angular momentum: A combinatorial approach to space-time. In Quantum Theory and Beyond, ed. by T. Bastin, Cambridge University Press, Cambridge, pp. 151–180.
Philippidis, C., Dewdney, C., and Hiley, B.J. (1979): Quantum interference and the quantum potential. Nuovo Cimento 52B, 15–28.
Primas, H.(1977): Theory reduction and non-Boolean theories. Journal of Mathematical Biology 4, 281–301.
Primas, H., and Müller-Herold, U. (1978): Quantum mechanical system theory: A unifying framework for observations and stochastic processes in quantum mechanics. Advances in Chemical Physics 38, 1–107.
Schönberg, M. (1957): Quantum mechanics and geometry. Anais da Academia Brasileira de Ciencias 29, 473–485.
Weyl, H. (1931): The Theory of Groups and Quantum Mechanics, Dover, London.
Wheeler, J.A. (1991): At Home in the Universe, AIP Press, New York.
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I should like to thank Glen Dennis for his suggestions and helpful comments.
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Hiley, B.J. (2016). Aspects of Algebraic Quantum Theory: A Tribute to Hans Primas. In: Atmanspacher, H., Müller-Herold, U. (eds) From Chemistry to Consciousness. Springer, Cham. https://doi.org/10.1007/978-3-319-43573-2_7
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