Ernst Binz.
Quantization and Field Quantization

Abstract.

Any singularity free vector field in oriented, Euclidean three-space defines a canonical complex line bundle C as well as a Heisenberg algebra bundle H containing C. Information preserving transmission along the vector field causes Fresnel optics and quantum mechanics at each point. The field quantization as a part of a deformation quantization emanates out of an infinite dimensional Heisenberg group constructed out of a space of sections of H. The Planck parameter is the deformation parameter.

Ray Brummelhuis.
To be announced.

Bob Coecke
Compatibility vs complementarity: algebra, diagrammatics, information flows, and toy models.

(Part of this is joint work with Ross Duncan, and Bill Edwards respectively)

Abstract.

Within a high-level category-theoretic framework we axiomtise an essential feature of quantum mechanics: complementary quantum observables, their interaction, and the information flows mediated by them. More specifically, we show that pairs of mutually unbiased quantum observables form bialgebra-like structures. We also provide an abstract account on phase data, which, for very general reasons, always constitutes a group, and which is subject to a normal form theorem.

Together the bialgebra-like structure on complementary observables and the group of phases enable us to describe all states, operations and observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform.

All these computations moreover happen within an intuitive diagrammatic calculus. The bialgebraic laws, when conceived as re-writes, cause radical changes in the topology and the connectedness of the pictures, hence providing valuable insights in the way information flows in complex networks. With the purely diagrammatic evaluation of the quantum Fourier transform, due to its importance for quantum algorithms, we reached an important milestone of the research program which aims to provide a purely diagrammatic quantum mechanical formalism.

Besides the standard one, an interesting model which exposes these features is the category Spek, which in a concise compositional manner encodes Rob Spekkens’ toy quantum theory. It is striking how non of the vector-space-like additive structure survives when passing to this model, but that our axiomatization captures its quantum-like features on-the-nose.

Andreas Döring
Neo-realist quantum theory in a topos of presheaves.

Abstract.

I will report on recent work with Chris Isham on the application of topos theory to physics. In particular, the representation of quantum theory in a topos of presheaves will be discussed, including the definition of a state object, which generalizes state space, a quantity-value object, which generalizes the real numbers, and arrows between them, which represent physical quantities. I will present some aspects of the resulting new form of quantum logic and will briefly discuss open questions and related work.”

Maurice de Gosson
About the h-dependence of the density matrix.

Abstract.

Mixed quantum states are described by density matrices (or operators) which are positive trace class operators with trace one. It has been known for some time (Narcowich, Werner) that the positivity property is sensitive to the value of Planck’s constant h. Thus, an operator may be a density matrix for one value of h and not for another value of. We discuss this property from the point of view of the symplectic uncertainty principle we have introduced in previous work. We also discuss the density matrix from the point of vie of deformation quantization by introducing the notion of Bopp pseudodifferential operator.

Basil Hiley
From Process to Clifford Algebras: a different approach to Quantum Phenomena.

Abstract.

Starting from a basic notion of process, we construct two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schrödinger, Pauli and Dirac formalisms, describe the classical light-cone structure of space-time, as well as providing a basis for the description of quantum phenomena. By constructing an orthogonal Clifford bundle with a Dirac connection, we make contact with quantum mechanics through the Bohm formalism, which emerges quite naturally from the connection, showing that it is a structural feature of the mathematics. We then generalize the approach to include the symplectic Clifford algebra, which leads us to a non-commutative geometry with projections onto shadow manifolds. These shadow manifolds are none other than examples of the phase space constructed by Bohm. We also argue that this provides us with a mathematical structure that fits the implicate-explicate order proposed by Bohm.

Lou Kauffman
Knots and Physics.

Abstract.

These talks will discuss the relationship of knot theory to aspects of physics (quantum gravity, quantum information, quantum computing, statistical mechanics) and how these relationships influence the evolution of knot theory as a topological subject (state sums, Jones polynomials, virtual knot theory, quantum link invariants, spin networks, Khovanov homology).

First talk – This talk will discuss the bracket state summation model for the Jones polynomial, how it is related to a generalization of Penrose spin networks and how this structure can be used to give sufficiently rich unitary braid group representations to generate quantum computing.

Second Talk – How the Witten functional integral is related to Vassiliev invariants in knot theory and to loop quantum gravity.

Tim Palmer
Bell Inequalities, Free Variables and the Undecidability of Fractal Invariant Sets.

Abstract.

As emphasised by recent “Free-Will” theorems, the notion that quantum theory is not locally causal and cannot be embedded in a locally-causal theory, depends on treating experimental parameters as free variables. Whilst “superdeterminism” – there are no free variables – may provide a way out of this conceptual difficulty, it is hard to see how such a strong constraint could ever emerge from the sort of primitive notions of invariance and symmetry that underlie theoretical physics. By contrast, based on the notion of dynamical invariance, a new class of locally-causal physical theory is proposed which imposes weaker (but nevertheless important) constraints on the notion of free variables. Specifically, we consider a class of physical theory whose states are constrained to lie on dynamically-invariant subsets of state space: the Invariant Set Conjecture. It is shown that any nontrivial property of fractal invariant sets is algorithmically undecidable. We discuss how quantum theory might be embedded within such a class of theory – leading to the notion of the Schrödinger equation as a singular limit. As such, it is claimed that it is unprovable that quantum theory cannot be embedded into one of the members of this class of locally-causal theory.

These ideas are used to put forward a new type of locally-causal quantum ontology, relevant to some of the problems of quantum theory, such as preferred bases, delayed-choice experiments and of course nonlocality. Since the Invariant Set Conjecture is geometric and hence general relativistically covariant, these results may in turn provide a new conceptual basis for the role of gravity in quantum theory.

Roger Penrose
THURSDAY Conformal Cyclic Cosmology: Ideas and Current Observational Status.

Abstract.

The second law of thermodynamics, together with the nature of the cosmic microwave background (CMB), imply that the Big Bang must have had a very particular structure. Conformal cyclic cosmology (CCC) posits that this structure comes about from an assumption that our Big Bang is the conformal continuation of the remote future of a previous aeon of the universe, this process continuing indefinitely. The geometrical consistency of this picture is made possible only by the presence of a positive cosmological constant (or equivalent) and some scalar material (dark matter?) produced at the Big Bang. Physical consistency also demands that there is very significant loss of information (in the sense of phase-space volume) in the Hawking evaporation and ultimate disappearance of large black holes. Galactic black-hole encounters in the previous aeon, involving much energy loss in the form of gravitational radiation, should, according to CCC, lead to statistically discernable circles in the CMB sky, either of slightly higher than average temperature or else of slightly lower than average (depending upon the location of the source). The observational situation is currently under examination.

SATURDAY Twistor Theory in a Cosmological Setting.

Abstract.

Twistor theory provides a non-local formalism for physics

which is particularly well suited (though not exclusively) to conformally invariant physics. Thus, massless particles in Minkowski space and, for example, the high-energy limit of strong-interaction physics (of considerable relevance to LHC) are situations where twistors are particularly valuable. Yet there remain deep problems (to do with mass, gravitational interactions, etc.). The observational input from cosmology that there appears to be a positive cosmological constant (or equivalent) suggests a modification of the standard Poincare-invariant twistor theory to one in which there is a complex symplectic structure. This leads to a new perspective on asymptotic twistors and the “googly problem” in a general cosmological setting.

Barbara Piechocinska
Physics from Wholeness.

Abstract.

By changing the perspective from which we view a certain subject new patterns often become visible. Motivated by reductionisms inability to encompass the quantum theory we explore a dynamical, indivisible wholeness as a perspective from which to start when addressing physics. By doing so we ask if entropy increase could not be such an emerging pattern. We present a suggestion for a mathematical description for the dynamics of wholeness and show a pattern that could potentially provide a fundamental description of entropy increase if given a physical interpretation.

Georg Wikman.
To be announced.