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Time and Ordinary Quantum Mechanics (QM)

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Book cover The Problem of Time

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 190))

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Abstract

This chapter recollects the postulates and laws of (non-relativistic) quantum mechanics, now emphasizing in particular the forms that time, space, clocks, ‘rods’ and observers take in quantum theory. We contrast these with their classical Newtonian counterparts. In particular, quantum mechanics has distinct external, dynamical and observable notions of time. We also comment on the time–energy uncertainty principle’s unusual features. Temporal notions additionally enter the postulates of quantum mechanics, in roles such as equal-time commutation relations, measurements at a given time, and evolution. We touch upon atomic clocks, and how these and some inputs into measuring lengths, are quantum-mechanical. We finally indicate how clocks are inherently imprecise at the quantum level, and have a finite probability for occasionally running backwards.

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Notes

  1. 1.

    If unfamiliar with any of the non-temporal material in this Chapter, see e.g. [487, 599]; some of this Chapter’s material on time in QM may however be new. See the start of Part III if interested in a more general and detailed treatment of Quantization.

  2. 2.

    This is a complete inner product space (see Appendices C.2 and H.2 for a bit more mathematical context for this). This term is often used to mean the infinite version, but the current use covers the finite version as well. The lack of physical distinctions between \(| \Psi \rangle \) and the phase-shifted \(\mbox{exp}(i \phi )| \Psi \rangle \) results in the Projective Geometry notion of a ray in Hilbert space is a further suitable means of modelling quantum wavefunctions.

  3. 3.

    For unconstrained theories, the corresponding notion of (for now unconstrained) classical observable is entirely trivial: any function of the classical variables \(A = A(\boldsymbol{Q} , \boldsymbol{P})\) is a classical observable; see Chaps. 9, 25 and 50 for the constrained case.

  4. 4.

    See Sect. 5.2 for why we use \(\widehat{J} _{i}\) in place of \(\widehat{L}_{i}\) here.

  5. 5.

    The guarantee here is actually for an operator whose eigenvalues form a discrete and non-degenerate set, but one can more laboriously generalize one’s way round these limitations [487].

  6. 6.

    Any symmetry transformation in QM can be represented by an operator on the Hilbert space of states \(\boldsymbol{\mathfrak{H}}\mathrm{ilb}\) that is either i) linear and unitary or ii) antilinear and antiunitary. See [885] for an accessible proof, and both that and [269, 401] for commentary.

  7. 7.

    These statistics are named after physicists Enrico Fermi, Paul Dirac, Einstein, and Satyendra Bose.

  8. 8.

    See (J.27) for the classical precursor of \(\mathcal{T}\). In the case of a conservative system, this is just \(t - t(0)\), which can be interpreted as \(t^{\mbox{\tiny N}\mbox{\tiny e}\mbox{\tiny w}\mbox{\tiny t}\mbox{\tiny o}\mbox{\tiny n}}\) up to choice of calendar year zero, by which \(\mathcal{T} = t ^{\mbox{\tiny N}\mbox{\tiny e}\mbox{\tiny w}\mbox{\tiny t}\mbox{\tiny o}\mbox{\tiny n}}\) and \(\Delta {\mathcal{T}} = \Delta t^{\mbox{\tiny N}\mbox{\tiny e}\mbox{\tiny w}\mbox{\tiny t}\mbox{\tiny o}\mbox{\tiny n}}\) in (5.17).

  9. 9.

    This is a Uniqueness Theorem, named after mathematician Marshall Stone and noted mathematician and polymath John von Neumann. It is phrased for the exponentiated (alias Weyl, after mathematician Hermann Weyl) commutation relations; these have a generic unitary representation; see e.g. [407] for a commented proof. Finite Theories lie within the remit of this result, whereas Field Theories do not.

  10. 10.

    This is valid for clocks that are dynamical systems that keep track of their own state, e.g. a pendulum with hands rather than just a pendulum.

  11. 11.

    This is the Laser Interferometer Gravitational Wave Observatory; [839] see Chap. 7 for further discussion.

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  1. DAMTP, Centre for Mathematical Sciences, Cambridge, UK

    Edward Anderson

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Anderson, E. (2017). Time and Ordinary Quantum Mechanics (QM). In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_5

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