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Classical-Level Background Independence and the Problem of Time. i. Time and Configuration

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The Problem of Time

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

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Abstract

This and the next chapter consider Background Independence features at the classical level; difficulties arising with formulating physics in this way constitute the Problem of Time. These are the main topics that the rest of this book expands upon, by which this chapter is essential preliminary reading for all subsequent chapters bar Chap. 11. The above-mentioned difficulties are indeed due to different physical paradigms having distinct conceptualizations of time, space, spacetime, frames, observers, clocks, and ‘rods’, which we laid out in Chaps. 1 to 8. The current chapter’s approach is from points of view in which space, configuration space or dynamics are primary. Isham and Kuchař moreover gave a conceptual classification of such differences as Problem of Time facets; the below enumeration follows an updated version of this emphasizing the Background Independence aspect underlying each facet.

  1. 1)

    The Temporal Relationalism aspect of Background Independence is that there is no time for the universe as a whole at the primary level. This is implemented by e.g. reparametrization-invariant actions along the lines of Jacobi’s principle. Constraints quadratic in the momenta ensue; for general relativity, this returns the Hamiltonian constraint. This starts to answer Wheeler’s question about first principles for the particular form taken by this important constraint, which, moreover, provides a timeless alias frozen wave equation at the quantum level: the Frozen Formalism facet of the Problem of Time. We also delineate strategies for this problem involving finding a hidden time, appending a matter time, or obtaining an emergent Machian time.

  2. 2)

    The Configurational Relationalism aspect of Background Independence incorporates a physically irrelevant group of transformations – spatial and/or internal—acting on the system’s configuration space. This generalizes Wheeler’s Thin Sandwich interpretation of general relativity, which encounters the Thin Sandwich Problem facet of the Problem of Time. We proceed via an intermediate generalization of Barbour’s—Best Matching—which produces constraints linear in the momenta. (The thin sandwich equation itself is a particular form for general relativity’s momentum constraint.) Minisuperspace—the spatially homogeneous simplification of general relativity often used as a model arena—cannot model this aspect, motivating our introduction of another model arena: relational mechanics.

  3. 3)

    With Temporal and Configurational Relationalism both producing constraints, it is crucial to test that these are consistent and whether they require further constraints to close as an algebraic structure. In this way, Constraint Closure enters as the third aspect of Background Independence; the tests in question are conducted with the Dirac algorithm; problems arising thus constitute the Constraint Closure Problem facet of the Problem of Time.

  4. 4)

    Furthermore, quantities which brackets-commute with the constraints play a distinguished role; these are observables (or beables). Assignment of Observables (or Beables) is the fourth aspect of Background Independence. Difficulties with this—known to occur especially in gravitational theory—comprise the Problem of Observables (or Beables) facet of the Problem of Time.

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Notes

  1. 1.

    See Sects. 11.1–11.2 and 11.6 for these other approaches toward Quantum Gravity’s own most significant precursor papers.

  2. 2.

    As regards other names, ‘Quantum GR’ will not do in this role due to implying the specific Einstein field equations. Contrast with how the Quantum Gestalt position remains open-minded as to which Relativistic Theory of Gravitation is involved. Quantum Gestalt is also in contradistinction to ‘Background Independent Quantum Gravity’. This is since the latter may carry connotations that the Background Independent and Gravitational inputs are separate rather than part of a coherent whole, whose classical counterpart—GR—already forms such a coherent whole.

  3. 3.

    Here indices \(I, J\) run over 1 to \(N\) (particle number), indices \(i\) run over 1 to \(d\) (spatial dimension) and \(m_{I}\) are the particles’ masses.

  4. 4.

    Chapter 12 outlines various other answers, which constitute distinct strategies for addressing the Frozen Formalism Problem.

  5. 5.

    In this book, given a time variable \(t\), its calendar year zero adjusted version \(t - t(0)\) is denoted by the corresponding oversized \(t\).

  6. 6.

    The name and concept of Gauge Theory is used here in a somewhat broader manner than that of Particle Physics, covering also e.g. Molecular Physics [624] and cosmological perturbations [110, 671].

  7. 7.

    The overline denotes densitization, i.e. inclusion of a factor of \(\sqrt{ \mbox{h}}\), which in the current case resides in \(\mathbf{M}\).

  8. 8.

    In RPMs, linear constraints and inhomogeneities are logically-independent features. This is in contrast with the Minisuperspace version, in which both are concurrently trivialized by homogeneity rendering the spatial derivative operator \(\mathcal{D}_{i}\) meaningless.

  9. 9.

    Many of these are quantum-level approaches, for which the motivation is stronger, as further detailed in Chaps. 12, 26 and 51).

  10. 10.

    Field Theory constraint algebraic structures are most usefully presented in terms of smearing functions, as explained in footnote 1; the undefined symbols in this Sec’s presentations of Electromagnetism, Yang–Mills Theory and GR are just such smearings.

  11. 11.

    See Appendix V.6 and [154] if interested in algebroids in general or the Dirac algebroid in particular. Hitherto this has usually been called ‘Dirac algebra’, though ‘Dirac algebroid’ is both more mathematically correct and not open to confusion with fermionic theory’s Dirac algebra (6.9). Moreover, the Dirac algebroid is manifested even in Minkowski spacetime \(\mathbb{M}^{n}\), upon considering arbitrary spatial hypersurfaces therein [250]; these correspond to fleets of arbitrarily accelerating observers.

  12. 12.

    Moreover, this generalization does not concern a change of definition, but is rather a more inclusive context in which the entities are interpreted. The term ‘beables’ was originally coined by physicist John Bell [126, 127]. Both his and the Author’s use of this word extend to include realist interpretations of QM. However, our uses differ as to what ‘realist interpretations’ we wish to include; most of those from Bell’s day have ceased to be tenable positions, with the ones I discuss specifically in Part III being largely conceptually and technically unrelated to these earlier uses (Sect. 50.4). To a lesser extent, classical whole-universe modelling also motivates use of the beables concept. Finally note that these two motivations combine further in the arena of Quantum Cosmology.

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Anderson, E. (2017). Classical-Level Background Independence and the Problem of Time. i. Time and Configuration. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_9

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