Nicolai's theorem suggests a simple stochastic interpetation for supersymmetric Euclidean qua... more Nicolai's theorem suggests a simple stochastic interpetation for supersymmetric Euclidean quantum theories, without requiring any inner product to be defined on the space of states. In order to apply this idea to supergravity, we first reduce to a one-dimensional theory with local supersymmetry by the imposition of homogeneity conditions. We then make the supersymmetry rigid by imposing gauge conditions, and quantise to obtain the evolution equation for a time-dependent wave function. Owing to the inclusion of a certain boundary term in the classical action, and a careful treatment of the initial conditions, the evolution equation has the form of a Fokker-Planck equation. Of particular interest is the static solution, as this satisfies all the standard quantum constraints. This is naturally interpreted as a cosmological probability density function, and is found to coincide with the square of the magnitude of the conventional wave function for the wormhole state.
Several new results regarding the quantum cosmology of higher-derivative gravity theories derived... more Several new results regarding the quantum cosmology of higher-derivative gravity theories derived from superstring effective actions are presented. After describing techniques for solving the Wheeler-DeWitt equation with appropriate boundary conditions, results are compared with semiclassical theories of inflationary cosmology and implications for various different string cosmology models are outlined. 1. Classical Cosmological Space-times of Higher-Derivative Theories Since solutions to the general relativistic field equations contain initial curvature singularities whenever the dominant energy condition is satisfied, one of the motivations for developing quantum cosmology has been the theoretical justification of the elimination of the singularities. Curvature singularities predicted by general relativity can be avoided by introducing boundary conditions in the path integral defining the quantum theory which restrict the integral over all four-geometries and matter fields to Riema...
Stochastic quantisation normally involves the introduction of a fictitious extra time parameter, ... more Stochastic quantisation normally involves the introduction of a fictitious extra time parameter, which is taken to infinity so that the system evolves to an equilibrium state.In the case of a locally supersymmetric theory, an interesting new possibility arises due to the existence of a Nicolai map. In this case it turns out that no additional time parameter is required, as the existence of the Nicolai map ensures that the same job can be done by the existing time parameter after Euclideanisation. This provides the quantum theory with a natural probabilistic interpretation, without any reference to the concept of an inner product or a Hilbert space structure. 1 1
In the canonical approach to general relativity it is customary to parametrize the phase space by... more In the canonical approach to general relativity it is customary to parametrize the phase space by initial data on spacelike hypersurfaces. However, if one seeks a theory dealing with observations that can be made by a single localized observer, it is natural to use a different description of the phase space. This results in a different set of Dirac observables from that appearing in the conventional formulation. It also suggests a possible solution to the problem of time, which has been one of the obstacles to the development of a satisfactory quantum theory of gravity. 1 1
Several new results regarding the quantum cosmology of higher-derivative gravity theories derived... more Several new results regarding the quantum cosmology of higher-derivative gravity theories derived from superstring effective actions are presented. After describing techniques for solving the Wheeler-DeWitt equation with appropriate boundary conditions, results are compared with semiclassical theories of inflationary cosmology and implications for various different string cosmology models are outlined. AMS Classification: 83C45, 83E30, 83F051. Classical Cosmological Space-times of Higher-Derivative Theories Since solutions to the general relativistic field equations contain initial curvature singularities whenever the dominant energy condition is satisfied, one of the motivations for developing quantum cosmology has been the theoretical justification of the elimination of the singularities. Curvature singularities predicted by general relativity can be avoided by introducing boundary conditions in the path integral defining the quantum theory which restrict the integral over all fou...
Continuous-time random walks can be used as phenomenological models of high-frequency time dynami... more Continuous-time random walks can be used as phenomenological models of high-frequency time dynamics in financial markets. Empirical analyses show that the intertrade durations (or waiting-times) are non-exponentially distributed. This fact imposes constraints on agent-based models of financial markets based on continuousdouble auctions.
Nicolai’s theorem suggests a simple stochastic interpetation for supersymmetric Euclidean quantum... more Nicolai’s theorem suggests a simple stochastic interpetation for supersymmetric Euclidean quantum theories, without requiring any inner product to be defined on the space of states. In order to apply this idea to supergravity, we first reduce to a one-dimensional theory with local supersymmetry by the imposition of homogeneity conditions. We then make the supersymmetry rigid by imposing gauge conditions, and quantise to obtain the evolution equation for a time-dependent wave function. Owing to the inclusion of a certain boundary term in the classical action, and a careful treatment of the initial conditions, the evolution equation has the form of a Fokker-Planck equation. Of particular interest is the static solution, as this satisfies all the standard quantum constraints. This is naturally interpreted as a cosmological probability density function, and is found to coincide with the square of the magnitude of the conventional wave function for the wormhole state. Supersymmetric theo...
Stochastic quantisation normally involves the introduction of a fictitious extra time parameter, ... more Stochastic quantisation normally involves the introduction of a fictitious extra time parameter, which is taken to infinity so that the system evolves to an equilibrium state. In the case of a locally supersymmetric theory, an interesting new possibility arises due to the existence of a Nicolai map. In this case it turns out that no additional time parameter is required, as the existence of the Nicolai map ensures that the same job can be done by the existing time parameter after Euclideanisation. This provides the quantum theory with a natural probabilistic interpretation, without any reference to the concept of an inner product or a Hilbert space structure.
Continuous-time random walks can be used as phenomenological models of high-frequency time dynami... more Continuous-time random walks can be used as phenomenological models of high-frequency time dynamics in financial markets. Empirical analyses show that the intertrade durations (or waiting-times) are non-exponentially distributed. This fact imposes constraints on agent-based models of financial markets based on continuousdouble auctions.
The quantum theory of a quadratic gravity theory derived from the heterotic string effective action
The renormalizability of a quadratic gravity theory with exponential coupling between dilaton fie... more The renormalizability of a quadratic gravity theory with exponential coupling between dilaton field and curvature terms is verified. The wave function for this model is then deduced to first order in e −Φ g 2 4 by solving a perturbed second-order Wheeler-DeWitt equation, assuming that the potential is slowly varying with respect to Φ. It is shown how formal expressions for the average paths in minisuperspace {a(t), Φ(t)} for this theory can be used to determine the shifts from the classical solutions a cl (t) and Φ cl (t), which occur only at third order in the expansion of the functional integrals representing the expectation values.
arXiv: General Relativity and Quantum Cosmology, 2004
Stochastic quantisation normally involves the introduction of a fictitious extra time parameter, ... more Stochastic quantisation normally involves the introduction of a fictitious extra time parameter, which is taken to infinity so that the system evolves to an equilibrium state.In the case of a locally supersymmetric theory, an interesting new possibility arises due to the existence of a Nicolai map. In this case it turns out that no additional time parameter is required, as the existence of the Nicolai map ensures that the same job can be done by the existing time parameter after Euclideanisation. This provides the quantum theory with a natural probabilistic interpretation, without any reference to the concept of an inner product or a Hilbert space structure.
The version of supergravity formulated by Ogievetsky and Sokatchev is almost identical to the con... more The version of supergravity formulated by Ogievetsky and Sokatchev is almost identical to the conventional N = 1 theory, except that the cosmological constant Λ appears as a dynamical variable which is constant only by virtue of the field equations. We consider the canonical quantisation of this theory, and show that the wave function evolves with respect to a dynamical variable which can be interpreted as a cosmological time parameter. The square of the modulus of the wave function obeys a set of simple conservation equations and can be interpreted as a probability density functional. The usual problems associated with time in quantum gravity are avoided. PACS numbers: 98.80.Hw, 04.65.+e e-mail address: graham@uni-essen.de e-mail address: luckock h@maths.su.oz.au
The wave function for the quadratic gravity theory derived from the heterotic string effective ac... more The wave function for the quadratic gravity theory derived from the heterotic string effective action is deduced to first order in ${{e^{-\Phi}}\over {g_4^2}}$ by solving a perturbed second-order Wheeler-DeWitt equation, assuming that the potential is slowly varying with respect to $\Phi$. Predictions for inflation based on the solution to the second-order Wheeler-DeWitt equation continue to hold for this theory. It is shown how formal expressions for the average paths in minisuperspace $\{ , \}$ determine the shifts from the classical solutions to $a_{cl}(t)$ and $\Phi_{cl}(t)$, which occur only at third order in the expansion of the integrals representing the expectation values.
The Hartle-Hawking State for the Bianchi IX Model in Supergravity
We solve the quantum constraints for homogeneous N=1 supergravity on 3- geometries with a Bianchi... more We solve the quantum constraints for homogeneous N=1 supergravity on 3- geometries with a Bianchi IX metric. Because these geometries admit Killing vectors with the same commutation relations as the angular momentum generators, there are two distinct definitions of homogeneity. The first of these is well-known and has been shown by D'Eath to give the wormhole state. We show that
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Papers by Hugh Luckock