Classical and quantum cosmology with York time

We consider a solution to the problem of time in quantum gravity by deparameterisation of the ADM action in terms of York time, a parameter proportional to the extrinsic curvature of a spatial hypersurface. We study a minisuperspace model together with a homogeneous scalar field, for which we can solve the Hamiltonian constraint exactly and arrive at an explicit expression for the physical (non-vanishing) Hamiltonian. The scale factor and associated momentum cease to be dynamical variables, leaving the scalar field as the only physical degree of freedom. We investigate the resulting classical theory, showing how the dynamics of the scale factor can be recovered via an appropriate interpretation of the Hamiltonian as a volume. We then quantise the system in the Schr\"odinger picture. In the quantum theory we recover the dynamics of the scale factor by interpreting the spectrum and expectation value of the Hamiltonian as being associated with volume rather than energy. If trajectories in the sense of de~Broglie-Bohm are introduced in the quantum theory, these are found to match those of the classical theory. We suggest that these trajectories may provide the basis for a perturbation theory in which both background and perturbations are quantised.Comment: 29 page

Gravitation and Cosmology with York Time

Despite decades of inquiry an adequate theory of \u27quantum gravity\u27 has remained elusive, in part due to the absence of data that would guide the search and in part due to technical difficulties, prominently among them the \u27problem of time\u27. The problem is a result of the attempt to quantise a classical theory with temporal reparameterisation and refoliation invariance such as general relativity. One way forward is therefore the breaking of this invariance via the identification of a preferred foliation of spacetime into parameterised spatial slices. In this thesis we argue that a foliation into slices of constant extrinsic curvature, parameterised by \u27York time\u27, is a viable contender. We argue that the role of York time in the initial-value problem of general relativity as well as a number of the parameter\u27s other properties make it the most promising candidate for a physically preferred notion of time. A Hamiltonian theory describing gravity in the York-time picture may be derived from general relativity by \u27Hamiltonian reduction\u27, a procedure that eliminates certain degrees of freedom --- specifically the local scale and its rate of change --- in favour of an explicit time parameter and a functional expression for the associated Hamiltonian. In full generality this procedure is impossible to carry out since the equation that determines the Hamiltonian cannot be solved using known methods. However, it is possible to derive explicit Hamiltonian functions for cosmological scenarios (where matter and geometry is treated as spatially homogeneous). Using a perturbative expansion of the unsolvable equation enables us to derive a quantisable Hamiltonian for cosmological perturbations on such a homogeneous background. We analyse the (classical) theories derived in this manner and look at the York-time description of a number of cosmological processes. We then proceed to apply the canonical quantisation procedure to these systems and analyse the resulting quantum theories. We discuss a number of conceptual and technical points, such as the notion of volume eigenfunctions and the absence of a momentum representation as a result of the non-canonical commutator structure. While not problematic in a technical sense, the conceptual problems with canonical quantisation are particularly apparent when the procedure is applied in cosmological contexts. In the final part of this thesis we develop a new quantisation method based on configuration-space trajectories and a dynamical configuration-space Weyl geometry. There is no wavefunction in this type of quantum theory and so many of the conceptual issues do not arise. We outline the application of this quantisation procedure to gravity and discuss some technical points. The actual technical developments are however left for future work. We conclude by reviewing how the York-time Hamiltonian-reduced theory deals with the problem of time. We place it in the wider context of a search for a theory of quantum gravity and briefly discuss the future of physics if and when such a theory is found

Gravitation and Cosmology with York Time

Despite decades of inquiry an adequate theory of \u27quantum gravity\u27 has remained elusive, in part due to the absence of data that would guide the search and in part due to technical difficulties, prominently among them the \u27problem of time\u27. The problem is a result of the attempt to quantise a classical theory with temporal reparameterisation and refoliation invariance such as general relativity. One way forward is therefore the breaking of this invariance via the identification of a preferred foliation of spacetime into parameterised spatial slices. In this thesis we argue that a foliation into slices of constant extrinsic curvature, parameterised by \u27York time\u27, is a viable contender. We argue that the role of York time in the initial-value problem of general relativity as well as a number of the parameter\u27s other properties make it the most promising candidate for a physically preferred notion of time. A Hamiltonian theory describing gravity in the York-time picture may be derived from general relativity by \u27Hamiltonian reduction\u27, a procedure that eliminates certain degrees of freedom --- specifically the local scale and its rate of change --- in favour of an explicit time parameter and a functional expression for the associated Hamiltonian. In full generality this procedure is impossible to carry out since the equation that determines the Hamiltonian cannot be solved using known methods. However, it is possible to derive explicit Hamiltonian functions for cosmological scenarios (where matter and geometry is treated as spatially homogeneous). Using a perturbative expansion of the unsolvable equation enables us to derive a quantisable Hamiltonian for cosmological perturbations on such a homogeneous background. We analyse the (classical) theories derived in this manner and look at the York-time description of a number of cosmological processes. We then proceed to apply the canonical quantisation procedure to these systems and analyse the resulting quantum theories. We discuss a number of conceptual and technical points, such as the notion of volume eigenfunctions and the absence of a momentum representation as a result of the non-canonical commutator structure. While not problematic in a technical sense, the conceptual problems with canonical quantisation are particularly apparent when the procedure is applied in cosmological contexts. In the final part of this thesis we develop a new quantisation method based on configuration-space trajectories and a dynamical configuration-space Weyl geometry. There is no wavefunction in this type of quantum theory and so many of the conceptual issues do not arise. We outline the application of this quantisation procedure to gravity and discuss some technical points. The actual technical developments are however left for future work. We conclude by reviewing how the York-time Hamiltonian-reduced theory deals with the problem of time. We place it in the wider context of a search for a theory of quantum gravity and briefly discuss the future of physics if and when such a theory is found

Non-Quantum Behaviors of Configuration-Space Density Formulations of quantum mechanics

The trajectories of the pilot-wave formulation of quantum mechanics and hence its empirical predictions may be recovered via the dynamics of a density function on the configuration space of a system, without reference to a physical wave function. We label such formulations `CSD frameworks.' But this result only holds if a particular, apparently ad hoc condition, broadly speaking equivalent to the single-valuedness of the wave function in standard quantum mechanics, is imposed. Here we relax this condition. We describe the types of scenarios in which this would lead to deviations from quantum mechanics. Using computational models we ask how the degree of `non-quantumness' of a state, suitably defined, changes with time. We find that it remains constant in time even under non-trivial dynamics, and argue that this implies that a dynamical justification of the Wallstrom condition is unlikely to be successful. However, we also make certain observations about stationary states in CSD frameworks, which may offer a way forward in justifying the Wallstrom condition.Comment: 8 pages, forthcoming in "Advances in Pilot Wave Theory - From Experiments to Foundations", fixed references in this versio

Tenfold your photons -- a physically-sound approach to filtering-based variance reduction of Monte-Carlo-simulated dose distributions

X-ray dose constantly gains interest in the interventional suite. With dose being generally difficult to monitor reliably, fast computational methods are desirable. A major drawback of the gold standard based on Monte Carlo (MC) methods is its computational complexity. Besides common variance reduction techniques, filter approaches are often applied to achieve conclusive results within a fraction of time. Inspired by these methods, we propose a novel approach. We down-sample the target volume based on the fraction of mass, simulate the imaging situation, and then revert the down-sampling. To this end, the dose is weighted by the mass energy absorption, up-sampled, and distributed using a guided filter. Eventually, the weighting is inverted resulting in accurate high resolution dose distributions. The approach has the potential to considerably speed-up MC simulations since less photons and boundary checks are necessary. First experiments substantiate these assumptions. We achieve a median accuracy of 96.7 % to 97.4 % of the dose estimation with the proposed method and a down-sampling factor of 8 and 4, respectively. While maintaining a high accuracy, the proposed method provides for a tenfold speed-up. The overall findings suggest the conclusion that the proposed method has the potential to allow for further efficiency.Comment: 6 pages, 3 figures, Bildverarbeitung f\"ur die Medizin 202