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The Emergent Nature of Time and the Complex Numbers in Quantum Cosmology

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The Arrows of Time

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 172))

Abstract

The nature of time in quantum mechanics is closely related to the use of a complex, rather than say real, Hilbert space. This becomes particularly clear when considering quantum field theory in time dependent backgrounds, such as in cosmology, when the notion of positive frequency ceases to be well defined. In spacetimes lacking time orientation, i.e without the possibility of defining an arrow of time, one is forced to abandon complex quantum mechanics. One also has to face this problem in quantum cosmology. I use this to argue that this suggests that, at a fundamental level, quantum mechanics may be really real with not one, but a multitude of complex structures. I relate these ideas to other suggestions that in quantum gravity time evolution may not be unitary, possibly implemented by a super-scattering matrix, and the status of CPT.

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Notes

  1. 1.

    See Pullin’s contribution in this volume.

  2. 2.

    Conversely, as shown by Dyson [3] in his three-fold way, if His time-reversal invariant one may pass to a real (boson) or quaternionic (fermion) basis.

  3. 3.

    Of course in splitting the discussion into two parts, in Part Iwe take the view that Quantum Logic like its classical Aristotelian special case is timeless. This avoids appealing to Temporal Logic to resolving such paradoxes as that of “the sea fight tomorrow” [72–74] and puts the burden of its resolution firmly where it belongs, in Part II.

  4. 4.

    By the principle of binary coding, Classical Boolean Logic may, for finite sets at least, be thought of as projective geometry over the Galois field of two elements. We shall also ignore the exceptional case of the octonions.

  5. 5.

    Strictly speaking the inverse.

  6. 6.

    Or recall what we might know about Kähler manifolds; Quantum Mechanics makes use of a Kählerian vector space.

  7. 7.

    cf Hyper-Kähler manifolds such as K3.

  8. 8.

    Vis not \({\mathcal{H}}_{\mathrm{qm}}\)thought of as real! A good reference for the properties of Clifford algebras used here is [112], see also [111].

  9. 9.

    A similar point has been made recently by Schucking [62] but he plumps for the quaternions.

  10. 10.

    We use real (Majorana) commuting spinors for convenience: there use is not essential.

  11. 11.

    Bernard Kay has implemented this argument more rigourously within an algebraic framework [120].

  12. 12.

    Amusingly CTC’s seem to be quiet innocuous from this point of view. It seems that they can be compatible with quantum mechanics.

  13. 13.

    See Hartle and Witt [114].

  14. 14.

    This is clear from the periodicity modulo eight of Clifford algebras \(\mathrm{Cliff}(s + 8,t) \equiv \mathrm{Cliff}(s,t) \otimes{M}_{16}(\mathbb{R})\)and the easily verified fact that the that \(\mathrm{Cliff}(2, 0; \mathbb{R}) \equiv {M}_{2}(\mathbb{R})\).

  15. 15.

    The matrices \({\Gamma }_{a}, {\Gamma }_{11}\)generate the M-theory Clifford algebra \(\mathrm{Cliff}(10, 1; \mathbb{R}) \equiv {M}_{32}(\mathbb{R}) \oplus{M}_{32}(\mathbb{R})\).

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Acknowledgements

I would like to thank, Thibault Damour, Stanley Deser, Marc Henneaux and John Taylor for helpful discussions and suggestions about the material in Section 11.

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    Rudy Vaas

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Gibbons, G.W. (2012). The Emergent Nature of Time and the Complex Numbers in Quantum Cosmology. In: Mersini-Houghton, L., Vaas, R. (eds) The Arrows of Time. Fundamental Theories of Physics, vol 172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23259-6_6

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