Ian C. Percival

A variational principle for invariant tori of fixed frequency

A fixed frequency Lagrangian variational principle is formulated for the invariant tori of conservative dynamical systems. It avoids the singularities due to small frequency divisors, and for pure rotation provides a strict bound which can be used as a basis for an effective variational method.

DECOHERENT HISTORIES AND QUANTUM STATE DIFFUSION

We demonstrate a close connection between the decoherent histories (DH) approach to quantum mechanics and the quantum state diffusion (QSD) picture, for open quantum systems described by a master equation of Lindblad form. The (physically unique) set of variables that localize in the QSD picture also define an approximately decoherent set of histories in the DH approach. The degree of localization is related to the degree of decoherence, and the probabilities for histories prescribed by each approach are essentially the same.

PRIMARY STATE DIFFUSION

Primary quantum state diffusion (PSD) theory is an alternative quantum theory from which classical dynamics, quantum dynamics and localization dynamics are derived. It is based on four principles, that a system is represented by an operator, its state by a normalized state vector, the state vector satisfies a Langevin-Itô state diffusion equation, and the resultant density operator for an ensemble must satisfy an equation of elementary Lindblad form. There are three conditions. The ז0 first determines the operator, to within an undetermined universal time constant ז0. The second and third conditions put opposing bounds on ז0. Dissipation of coherence is distinguished from destruction of coherence. The state diffusion destroys coherence and produces the localization or reduction that makes classical dynamics possible. PSD theory is a development of the environmental quantum state diffusion theory of Gisin and Percival and particularly resembles earlier proposals by Gisin and by Milburn. It is also related to the spontaneous localization theories of Ghirardi, Rimini and Weber, of Diósi and of Pearle. The non-relativistic PSD theory is of value only for systems which occupy small regions of space. Special relativity is needed for more extended systems even when they contain only slowly moving massive particles. Experiments on coherence lifetimes and matter interferometry are proposed which either measure ז0 or put bounds on it, and which might distinguish between PSD and ordinary quantum mechanics.

Quantum spacetime fluctuations and primary state diffusion

Non-differentiable fluctuations in spacetime on a Planck scale introduce stochastic terms into the equations for quantum states, resulting in a proposed new foundation for an existing alternative quantum theory: primary state diffusion (PSD). Planck-scale stochastic spacetime structure results in quantum fluctuations, whilst larger-scale curvature is responsible for gravitational forces. The gravitational field and the quantum fluctuation field are the same, differing only in scale. The quantum mechanics of small systems, classical mechanics of large systems and the physics of quantum experiments are all derived dynamically, without any prior division into classical and quantum domains, and without any measurement hypothesis. Unlike the earlier derivation of PSD, the new derivation, based on a stochastic spacetime differential geometry, has essentially no free parameters. However, many features of this structure remain to be determined. The theory is falsifiable in the laboratory, and critical matter interferometry experiments, to distinguish it from ordinary quantum mechanics, might be feasible within the next decade.

Quantum transfer functions, weak nonlocality and relativity

Abstract The method of transfer functions is developed as a tool for studying Bell inequalities, alternative quantum theories and the associated physical properties of quantum systems. Non-negative probabilities for transfer functions result in Bell-type inequalities. The method is used to show that all realistic Lorentz-invariant quantum theories, which give unique results and have no preferred frame, can be ruled out on the grounds that they lead to weak backward causality.

Thermal equilibrium in the quantum state diffusion picture

Abstract We use the theory of quantum state diffusion to examine the evolution of an oscillator coupled to a finite temperature heat bath.

Non-integrable systems with algebraic singularities in complex time

Dynamical arguments are presented which suggest that there are non-integrable systems without clustering of singularities, without infinite singularities, or singularities with an infinite number of branches in the complex t-plane. Several examples with only algebraic singularities are studied, for which strong numerical evidence is presented for non-integrability and infinitely sheeted solutions. Weak-Painleve potentials are also analysed from this point of view, and all integrable cases are found to possess only finitely sheeted solutions.

BellTest and CHSH experiments with more than two settings

Strong nonlocality allows signals faster than light. Weak nonlocality is a statistical property of classical events for which there is no realistic local theory. This requires the violation of at least one general Bell inequality. The theory of ideal quantum measurements predicts weak nonlocality but not strong nonlocality. Bell experiment here refers to any experiment designed to demonstrate weak nonlocality. BellTest is a computer program generally available on the Web to help planning and analysis of such Bell experiments. In Mode 1 it obtains general Bell inequalities. In Mode 2 it tests for their violation. We describe its use, with some new results for illustration.

Atom interferometry, spacetime and reality

Einsteins general theory of relativity dominates the science of the very large, including gravity and cosmology. Quantum theory reigns supreme on the scale of atoms and smaller. What is more natural than to try to unify these two theories into a single quantum theory of gravity? Modern theories of gravity, including superstring theory, give quantum properties to space and time. These theories have been inspired by Diracs principles of beauty and elegance. But Dirac had the advantage of experimental evidence, and modern theories of quantum gravity do not. Moreover, beauty and elegance are subjective – they are a guide to the correct theory, not a guarantee. For any scientific theory, experiment is the final test.

Stochastic wave equations versus parallel world components

In response to a recent Letter of Zeh, we present the case for adding stochastic terms to the Schrijdinger equation, rather than introducing different observers in different world-components. The similarities and the contrasts between these alternative approaches are presented, with emphasis on recent results obtained with the first approach. We agree with many of the points put by Zeh [ 1 ] in response to our Letter [ 21, in particular the importance of the environment and decoherence for reduction, position localisation and measurement, and the proven advantages of continuous physical theories over discrete ones. We agree that for some practical purposes the “decoherence can be interpreted as the occurrence of dynamical discontinuities”. We also agree that “density matrices are effectively equivalent to ensembles of narrow wave packets”. But unlike Zeh we have an equation for the wave packets, and it is this to which he appears to object, since the equation is different from the Schrodinger