Noah Linden

Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory

The recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell‐Mann and Hartle for a generalized form of quantum theory based on the ideas of histories and decoherence functionals is analyzed and developed herein. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory.

Continuous histories and the history group in generalized quantum theory

We treat continuous histories within the histories approach to generalized quantum mechanics. The essential tool is the ‘‘history group:’’ the analog, within the generalized history scheme, of the canonical group of single‐time quantum mechanics.

The classification of decoherence functionals: An analog of Gleason’s theorem

Gell‐Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are ‘‘histories’’ and decoherence functionals. Within this scheme it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where UP is the lattice of projectors P(V) in some Hilbert space V; in effect, one seeks the history analog of Gleason’s famous theorem in standard quantum theory. In the present article the solution to this problem for the case where V is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈P(V) can be written in the form d(α,β)=trV⊗V(α⊗βX) for some operator X on the tensor‐product space V⊗V.

Continuous time and consistent histories

We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.

The geometry of inequivalent quantizations

We describe how inequivalent quantizations (superselection sectors) arise within two related algebraic approaches to quantum mechanics (viz. quantization by canonical groups and by C∗-algebras). By construction of the quantum hamiltonian and the path integral of a particle moving on a coset space, we show that the inequivalent quantizations manifest themselves as the particle coupling to a certain fictitious external gauge field, in a representation depending on the superselection sector; various well-known topologically non-trivial Yang-Mills field configurations emerge in this way. The general theory is illustrated by taking the coset space to be a circle and a sphere, which puts θ-angles (hence the Aharonov-Bohm effect) and the Dirac charge quantization condition, respectively, in a new light.

Superselection rules from Dirac and BRST quantisation of constrained systems

Abstract Adopting the point of view that inequivalent quantisations correspond to superselection sectors, i.e. unitarily inequivalent representations of the algebra of observables, we show how the superselection sectors of a particle moving on a coset space G/H follow from its quantisation as a constrained system (the unconstrained system being the phase space T ∗ G). Both the Dirac and BRST method are examined; the former works well for compact H, whereas the latter runs into several difficulties. Accordingly, a possible improvement to the BRST procedure is suggested.

Predicting British financial indices: An approach based on chaos theory

Abstract Non-linear deterministic systems are capable of generating chaotic output that mimics the output of stochastic systems. We test British financial indices to see if they are chaotic. The results of our tests of chaotic dynamics are not conclusive. We found some difficulties in calculating the correlation dimension due to insufficient economic data. Our main contribution is to generate short-term prediction for our series which have better predictive properties than traditional linear auto-regressive procedures. We use a non-parametric procedure advocated by biologists in predicting time series in a chaos framework.

Path integrals and unitarity in quantum cosmology

Abstract We explore a simple cosmological model in the minisuperspace approach. The global structure of the model leads to subtleties in its quantization which are treated using the affine, rather than the Heisenberg, algebra. We develop a path-integral formulation for this algebra, and apply it to our cosmological model. We investigate the Friedmann-Robertson-Walker universe with a scalar field as source and discuss in detail the constraint that the universe have positive radius.

Inequivalent quantisations of the Neumann model

Abstract Using the algebraic approach to quantisation, it is shown that there are many inequivalent quantum versions of the Neumann model, each of which is integrable.

New designs on spacetime foam

What is spacetime like at the smallest scale? Despite decades of work, the answer to this basic question is still unknown. Two recent theoretical research projects, the first involving Abhay Ashtekar of Syracuse University, USA, and collaborators and the second, Chris Isham of Imperial College, London, may, however, provide new insights into this fundamental area. Interestingly, the two programmes approach the relevant subject – quantum gravity – with almost diametrically opposed perspectives.