# Abstracts

Basil Hiley

Concepts of Process as a basis for understanding non-commutative geometry

I will explain how David Bohm‚s notion of the implicate order can provide an overarching process based view of some of the ideas being used in non-commutative geometry. I will show how the symplectic Clifford algebra, the symplectic analogue of the orthogonal Clifford algebra, plays a central role in this approach. This structure has links with the ideas of Ernst Binz and Maurice de Gosson that will be discussed in this meeting. I will also show how space-time and phase space can be abstracted from these algebras forming the basis of a non-commutative geometry. My approach is through the algebra but I will indicate how some of these ideas become much clearer in the monoidal tensor category, a topic that will be discussed in detail by Bob Coecke.

Ernst Binz

Field Theoretic Weyl Quantization for Vector Fields in Three Space

Given a manifold with or without boundary in three space and a vector field on it, we delete its singularities and associate a canonical bundle of Heisenberg groups to the singularity free field. The module of its Schwartz sections determines an infinite dimensional Heisenberg group and in turn two C*-algebras namely a Heisenberg C*-algebra and a Weyl C*-algebra (depending on Planck‚s constant) linked by a surjective *- homomorphism. Some unitary *-representations of the Weyl C*-algebra on a Hilbert space yield quantum field operators satisfying the CCR. The variation of the parameter causes a deformation quantization and a classical limit.

Nick Monk

Models for Space-Time and Pre-Space-Time

I will discuss some implications of quantumanics and general relativity for the understanding of space-time. There are strong indications that Classical notions of space-time are inadequate as a basis for these theories. I will outline an approach based on Bohm’s framework for the description of natural phenomena based on the notion of process. In this approach, space-time is no longer taken as basic; rather it is a derived feature of the underlying process. This approach opens up the possibility of investigating space-time problems from a new angle, where the nature of space-time is determined by the system under study.

Bob Coecke

In The Beginning God Created the Tensors … as a Picture

We show that the quantum mechanical structure can be fully expressed in terms of tensor structure alone. This includes entanglement (of course), scalar, unitarity, spectra, trace, superposition, projective measurements and the Born rule, POVMs, density operators and completely positive maps (etc.) [1, 2, 3, 4, 6, 7]. What we mean by tensor structure is an operational axiomatization of `combining systems’, which has the Hilbert space tensor product as one particular instance. In particular, notions of sum (cf. superposition), orthogonality and even probability do not need to be taken as primitive but turn out to be both mathematically and conceptually derivable from the more primitive notion of “conceiving two systems as one” [5].

The appropriate mathematical background to perform this axiomatisation is category theory, but, in fact, it also admits a very intuitive graphical calculus which is such that any equational statement which is algebraically provable is also provable in the graphical calculus and vice versa [3, 4]. This calculus is in fact an extension and refinement of Dirac’s notation, hence the categorical algebra provides a formal justification/semantics for the Dirac notation.

The graphical calculus proves to be extremely useful for the design and analysis of quantum information protocols, both qualitatively and quantitatively. For example, it turns several sophisticated quantum informatic protocols into trivial undergraduate exercises [1, 4]. Also theorems such as Naimark’s theorem admit extremely elegant proves in the graphical calculus [6]. As compared to so-called quantum logic (Birkhoff-von Neumann, Mackey, Piron, Foulis-Randall), this setting does come with logical mechanisms such as deduction — in fact, it turns out to be some kind of `super logic’ as compared to the Birkhoff-von Neumann non-logic.

[1] S. Abramsky and B. Coecke (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp.415-425. IEEE Computer Science Press. quant-ph/0402130.

[2] S. Abramsky and B. Coecke (2005) Abstract physical traces. Theory and Applications of Categories 14: 111-124.

[3] P. Selinger (2006) Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science (To appear). http://www.mathstat.dal.ca/~selinger/papers.html#dagger

[4] B. Coecke (2005) Kindergarten quantum mechanics – lecture notes. In: Quantum Theory: Reconsiderations of the Foundations III, pp.81–98. AIP Press. quant-ph/0510032

[5] B. Coecke (2005) Introducing categories to the practising physicist. In: What is category theory? Polimetrica Publishing. Advanced Studies in Mathematics and Logic 30, pp.45-74. Polimetrica Publishing.

[6] B. Coecke and D. Pavlovic (2005) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Taylor and Francis. (to appear)

[7] B. Coecke and E. O. Paquette (2006) Generalized measurements and Naimark‚s theorem without sums. Submitted.

Andreas Doering

Topos Quantum Physics

In this talk, I will present some aspects of current work with Chris Isham on a possible reformulation of quantum physics in the language of topos theory. The Kochen-Specker theorem rules out all naive approaches to a realistic version of quantum theory and emphasizes the need for contextuality. When trying to formulate a contextual form of quantum logic/theory retaining a realist Œflavour‚, one is quite naturally led to presheaves over a suitable category of contexts. The fact that such presheaves form a topos delivers a natural distributive, albeit intuitionistic logic, given as a part of the internal structure of the topos. In our work, the so-called spectral presheaf plays a central role. It is a contextual replacement for phase-space. Quantum propositions can be shown to correspond to certain subobjects of the spectral presheaf. These subobjects form a Heyting algebra and hence give a contextualized, non-standard form of quantum logic. I will also report on first steps toward a description of states and observables internally in a topos. The long-term goal is an axiomatic generalization of this topos-theoretic formulation of quantum theory.

Tim Palmer

The Mysteries of Entanglement

As Penrose has emphasised recently, there are two fundamental mysteries about the phenomenon of entanglement: how can we make sense of it in terms of ideas that we can comprehend, and why is it something that we barely notice in our direct experience of the world. I want to discuss these two mysteries from a very non-orthodox perspective. I will firstly put forward the argument that deterministic non-periodic flow (aka chaos), initialised on the system’s attractor, provides the means to make sense of the first mystery – the key idea being that hypothetical counterfactual perturbations almost certainly take the system off its measure-zero attractor, ie to a dynamically disjoint part of state space. A number-theoretic representation of the Cantor-set structure of state space is related to the conventional state space of quantum theory, the complex Hilbert space, though a set of self-similar permutation operators with complex and hyper-complex structure, acting on elements of the Cantor set. In terms of this relationship, quantum entanglement, and in particular macro-measurement of the quantum micro-state, would be associated with a deterministic dynamical reduction in the dimension of the underlying attractor. By reformulating quantum theory in this deterministic way, a solution to the second mystery of entanglement is proposed: we do indeed notice the phenomenon of entanglement in our direct experience of the world – we call it gravity!

Maurice de Gosson

Symplectic Quantum Cells and Indistinguishability

The time is gone where physicists daring to mention quantum phase space were viewed as defrocked friars. Not so long time ago physicists (beginning with P. A. M. Dirac) considered the sole existence of a quantum phase space as a heresy because of the uncertainty principle; the situation is reminiscent of Oppenheimer’s outburst about Bohm’s formulation of QM (“if we cannot disprove Bohm, let us ignore him!). The situation is however slowly changing: bona-fide quantum mechanics in phase space (and not just its souped-down version, semiclassical mechanics) is the daily bread of people working in quantum optics, nuclear physics, or the study of mesoscopic systems, to name a few. On the other hand, the tradition of working in quantum phase space has been quite long in quantum chemistry, and the last taboo seems to have been transgressed in the 90’s when physicists began to study Schrödinger equations in phase space.

The aim of this talk is to give a mathematically rigorous account, based on a refined notion of the uncertainty principle due to Schrödinger and Robertson, of the quantum phase space. This quantum phase space is defined by a coarse graining of the ordinary phase space, not in cubic quantum cells of volume h? as is customary in, for instance, thermodynamics, but rather in symplectic quantum cells (which we have called “quantum blobs” elsewhere), which are sets with symplectic capacity equal to one-half of the quantum of action. This definition emphasize the role played by the notion of area in both classical and quantum mechanics, as has been highlighted by many authors. In addition, this definition does not impose any boundedness condition on the symplectic quantum cells: they can have infinite volume and spatial extension, and might therefore be particularly well-adapted to various non-locality questions arising in quantum mechanics.

Tim Palmer

Global Warming in a Chaotic Climate – Can We Be Sure?

Michael Crighton’s recent blockbuster, “State of Fear” attacked the scientific orthodoxy about global warming, and provided a rallying point for the so-called climate sceptics. For example, he asked, if tomorrow’s weather forecasts can be so unreliable, why should we believe in the predictions of climate change a hundred years from now? I discuss the basic science underlying global warming, including uncertainties in the science, try to address Crighton’s concerns, and talk about the methods climate scientists use to make predictions including estimates of uncertainty in these predictions.

Stephen Wood

Complexity and Adjointness

Classically, physics has taken the result of acting from the left to be the same as acting from the right. But only simple, symmetrical structures can be built by processes that follow such a rule. Complex processes, involving

real change, require an asymmetry between left and right. This fact has been recognised in quantum physics as noncommutativity and more generally in category theory as adjointness. That left and right are adjoint rather than equal has led to important conclusions in biology as well as physics: (1) the complementarity of structure and process (Varela); (2) the necessity of noncomputable models to describe life (Rosen).

Malcolm Coupland

Where Would Gravity Be without Newton’s Third Law ?

I propose that the application of Newton‚s Third Law to classical gravity is not justified from a modern perspective, either theoretically or observationally. If the requirement is dropped then straightforward explanations are found for many mysterious phenomena, such as the apparent rarity of antimatter, the smallness of the cosmological constant, dark energy, dark matter, and the anomalous accelerations of the Pioneer spacecraft. The current evidence that any violation of the third law in gravity is negligibly small (one laboratory experiment and observations on the gravity and orbit of the Moon) is shown to be in fact inconclusive. Possible means to adapt General Relativity so as to include this principle are explored within the context of questioning conventional approaches to building field equations. The consequences for quantum mechanics are touched upon.