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Time and the Algebraic Theory of Moments

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Re-Thinking Time at the Interface of Physics and Philosophy

Part of the book series: On Thinking ((ONTHINKING,volume 4))

Abstract

We introduce the notion of an extended moment in time, the duron. This is a region of temporal ambiguity which arises naturally in the nature of process which we take to be basic. We introduce an algebra of process and show how it is related to, but different from, the monoidal category introduced by Abramsky and Coecke. By considering the limit as the duration of the moment approaches the infinitesimal, we obtain a pair of dynamical equations, one expressed in terms of a commutator and the other which is expressed in terms of an anti-commutator. These two coupled real equations are equivalent to the Schrödinger equation and its dual.

We then construct a bi-algebra, which allows us to make contact with the thermal quantum field theory introduced by Umezawa. This allows us to link quantum mechanics with thermodynamics. This approach leads to two types of time, one is Schrödinger time, the other is an irreversible time that can be associated with a movement between inequivalent vacuum states. Finally we discuss the relation between our process algebra and the thermodynamic origin of time.

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Notes

  1. 1.

    This example will be appreciated by cricketers everywhere.

  2. 2.

    The label a was suppressed by Dirac leaving it understood provided no ambiguity arose.

  3. 3.

    Symbolically we write \(\rangle \langle \;\times \;\rangle \langle \; =\langle \rangle \;\rangle \langle \;\) with \(\langle \rangle \;= 1\).

  4. 4.

    This is essentially the same idea that led to the notion of the anti-particle “going backwards in time,” but here we are not considering “exotic” anti-matter.

  5. 5.

    In modern parlance these functions are the generating functions of the symplectomorphisms in classical mechanics (see de Gosson [32]).

  6. 6.

    If R is a noncommutative ring, a left ideal is a subset I L such that if a ∈ I L then ra ∈ I L for all r ∈ R.

  7. 7.

    The inner automorphism is a way of expressing the enfolding and unfolding movement.

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Acknowledgements

I would like to thank the members of the TPRU for their invaluable help in trying to straighten out the ideas expressed in this paper. I would like to thank Albrecht von Müller and Thomas Filk for inviting me to participate in their stimulating meeting at the Parmenides Foundation in Munich.

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Authors and Affiliations

  1. TPRU, Birkbeck, University of London, Malet Street, London, WC1E 7HX, UK

    B. J. Hiley

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  1. B. J. Hiley
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Correspondence to B. J. Hiley .

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Editors and Affiliations

  1. Parmenides Stiftung, Pullach, Bayern, Germany

    Albrecht von Müller

  2. Institute of Physics, University of Freiburg, Freiburg, Germany

    Thomas Filk

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Hiley, B.J. (2015). Time and the Algebraic Theory of Moments. In: von Müller, A., Filk, T. (eds) Re-Thinking Time at the Interface of Physics and Philosophy. On Thinking, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-10446-1_7

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