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Symmetry, Ontology and the Problem of Time on the Interpretation and Quantisation of Canonical Gravity

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Symmetry, Ontology and the Problem of Time on the Interpretation and Quantisation of Canonical Gravity UNIVERSITY OF SYDNEY DOCTORAL THESIS Symmetry, Ontology and the Problem of Time On the Interpretation and Quantisation of Canonical Gravity Karim P.Y. THEBAULT´ Declaration and Copyright This thesis is an account of research undertaken initially undertaken between August 2008 and August 2011 at the Centre for Time, Department of Philosophy, School of Philosoph- ical and Historical Inquiry, University of Sydney, Australia. Subsequent changes have been in this the final text in line with recommendations of the examiners. Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university. Sections of this thesis have been reproduced in papers for The British Journal for the Philosophy of Science (Thebault´ (2012)), Symmetry (Thebault´ (2011)), and Foundations of Physics (Gryb and Thebault´ (2012)). Where relevant, the copyright rests with the journal publishers. Otherwise it is retained by the author. Karim P. Y. Thebault´ July, 2012 i Preface The true path is along a rope, not a rope suspended way up in the air, but rather only just over the ground. It seems more like a tripwire than a tightrope. Franz Kafka The cause of our confusion is often that the solution to the problem at hand is so near to us that we cannot help but trip over it, and then assume it to be part of the problem itself. So, it will be argued, is the case of the problem of time in canonical gravity. Moreover, as we shall see, the realisation essential to understanding the true role of time in canonical gravity is that we have failed to notice that the solutions – in the sense of the true dynamical paths – lie in precisely the direction that is perpendicular to where we are accustomed to looking. The following work is constituted by the confluence of a variety of different ideas, is- sues and questions that grow out of the interconnected tasks of interpreting and quantising the general theory of relativity. Essentially its focus is on both the classical and quantum facets of the problem of time in canonical gravity and, as such, much attention is given to ideas – both technical and conceptual – targeted directly or indirectly at unpicking the Gordian knot of this multifaceted problem. For all that, as intimated above, in essence our analysis relies on a single, fundamental realisation which we can state in concise but technical terms: the null directions associated with the Hamiltonian constraints of canon- ical general relativity must be understood as dynamical not unphysical directions. The formal and conceptual basis behind this statement, as well as the relevant qualifications and implications, will be detailed at length during the body of our discussion. Before then it will be useful to give some historical background to our problem. To understand the history of the problem of time in canonical gravity we must first understand the history of the canonical formulation of gravity itself. Thus, in placing our iii iv project in its proper context it is necessary to go back to the physics of the late nineteen fifties and principally the work of Paul Dirac and Peter Bergmann.1 Dirac’s contribution is particularly significant since both the general constrained Hamiltonian formalism (Dirac (1958a)) and the first application of this formalism to general relativity (Dirac(1958b)) can be traced back to him. A few years later Arnowitt, Deser and Misner (Arnowitt et al. (1960, 1962)) were able to simplify the formalism and it is their ‘ADM’ action that we shall study below. The motivation for much of this early work was to find a canonical path towards a quantum theory of gravity. It was expected that if we could gain a good understanding of: i) how to quantise constrained Hamiltonian theories in general terms; and ii) how to formulate general relativity as a constrained Hamiltonian theory, then we would have gone a long way towards the holy grail of a quantum theory of gravity. Unfortunately, things turned out not to be so simple and, despite some admirable progress, half a century later the canonical quantisation program is still beset by severe technical and conceptual problems. The particular problem that we will be considering in great detail here was identified early on in its classical manifestation and is connected to the fact that the local Hamilto- nian functions responsible for generating evolution within the canonical theory are first class constraints. According to the the Dirac prescription (Dirac(1964)), all the first class constraints that occur within constrained Hamiltonian theory should be understood as generating infinitesimal transformations that do not change the physical state; thus we have that time is in some sense unphysical! A directly analogous problem can be found within the definition of observables (see Bergmann(1961)). According to Bergmann’s definition (which follows from Dirac’s prescription) these are represented by phase space functions which have a (weakly) vanishing Poisson bracket with the first class constraints. Since the Hamiltonian is a constraint this means that the observables cannot have any dy- namical evolution. Thus, we have two aspects to the classical problem of time; the prob- lem of change and the problem of observables. Below we will consider two strategies which have been developed over the last twenty years and which address these prob- lems by either modifying the Dirac-Bergmann prescriptions or rejecting them outright. Understanding the ontology implied by either of these strategies involves tackling much conceptually challenging terrain and one of the key tasks of this project will be to provide 1This is not to discount the notable contributions of others, in particular James Anderson and Arthur Komar. For an early paper which deals with aspects of the problem of time from a non-canonical perspective see Misner(1957). For a fascinating account of the little known 1930s work on constraint theory by L eon´ Rosenfeld see Salisbury(2007, 2010). v clear guidance as to how the relevant interpretational frameworks should be thought to sit together. Perhaps more starkly problematic is the quantum mechanical manifestation of the problem of time. In his work on the quantisation of constrained Hamiltonian systems Dirac constructed a technique for canonical quantisation that can be successfully applied to theories such as electromagnetism. Since in that case this Dirac quantisation is found to lead to the hugely successful theory that is quantum electrodynamics one would expect that the technique itself has solid mathematical foundations. However, when applied to canonical general relativity Dirac quantisation, which involves the promotion of all first class constraint functions to operators annihilating the wavefunction, leads directly to the Wheeler-de Witt equation (DeWitt(1967)) – ‘ H^ = 0’2. This ‘wavefunction of the j i universe’ gives at best probability amplitudes on three dimensional spatial configurations. Thus, the application of standard quantisation techniques to canonical gravity leads to a quantum formalism that is in a fundamental sense without time. The question is then: how can we reconcile ourselves to a formalism that represents reality as frozen in an energy eigenstate when our phenomenology abounds with change? Much work over the last few decades has focused upon recovering the impression of dynamics from within the frozen formalism (see Anderson(2010) for a recent review). Here we will develop an entirely different approach whereby it is a problem with the Dirac quantisation technique itself that has led to an unphysical timelessness within the conventional canonical quantum gravity formalism. For a class of non-relativistic models we will offer an alternative methodology which involves modifying the Dirac technique such that we are able to retain dynamics. For the full case of general relativity, however, it is not entirely clear how one should proceed towards quantisation whilst retaining dynam- ics. This is in part due to the various subtleties involved in the two different formulations of canonical general relativity according to the two strategies for solving the classical problem of time mentioned above. It is in the context of attempting to get a better understanding of the relationship be- tween the problem of quantising gravity and the problem of reconciling two very different formulations of the classical theory, that the lengthly digression into the philosophy of sci- ence in the final quarter of this thesis should be seen. There we will examine the extent to which a generic problem of metaphysical underdetermination in science might be seen 2The inverted commas are because, as we shall see, this expression is far from well defined. vi to motivate a structuralist view as to the ontology of physical theory. The particular case that is most relevant to canonical gravity is when this underdetermination is driven by the existence of multiple formulations of a theory, and we shall find suggestive evidence that – given we are dealing with a classical theory – quantisation may be able to give us key insights into what structures are most significant. The final key idea that will be introduced below is that we may be able to invert this idea of quantisation as a guide to structure and use the isolation of common structure between different formulations of a theory as a heuristic towards finding the correct quantisation methodology. Thus, by re- solving the underdetermination in the case of our two formulations of classical canonical general relativity, we may be able to gain an invaluable insight into the correct way to proceed towards the quantum theory. The genesis of this thesis can be traced back to a suggestion made to me by Oliver Pooley during my undergraduate studies at Oxford that it would be interesting to con- sider the comparison between Julian Barbour’s and Carlo Rovelli’s timeless approaches to quantum gravity as a topic for my fourth year thesis.
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