Novel mathematical structures in physics – Non- commutative geometry.

The Discussion of possible relationships between the foundations of mathematics and the foundations of physics was also encouraged.

Titles of Talks and Abstracts

Ronnie Brown.

1. Higher dimensional algebra: origins and prospects

ABSTRACT: In the second half of the 1960s I convinced myself that the use of groupoids rather than groups in 1-dimensional homotopy theory led to more powerful theorems with simpler proofs. The objects of a groupoid add a spatial element to group theory which enables wider applications.
This led to the question of the putative uses of groupoids in higher homotopy theory, and the existence of a higher dimensional group(oid) theory, for example higher order symmetry. Now we can see `higher dimensional algebra’ (HDA) as being about partially defined algebraic operations whose domains of definitions are defined by geometric conditions. For this reason, HDA gives a range of noncommutative, higher dimensional methods for local-to-global problems, and can give precise results seemingly unobtainable by other means.
The talk will give some of the intuitions, background and ideas essential for the theory. The main gap in the methods seems to be the link to analogues of representation theory.

2. Category theory, higher dimensional algebra, groupoid atlases: prospective descriptive tools in theoretical neuroscience
In neuroscience, and systems theory, one deals with hierarchical systems, with problems of describing their structure and the interactions and communications at local, global and hierarchical levels. This speculative presentation will discuss what seems available, and what might be needed in the future. There will be a mention of the Ehresmann-Vanbremeersch theory of Memory Evolutive Systems.

References:

136. (with T. Porter), `Category theory and higher dimensional algebra: potential descriptive tools in neuroscience’, Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92. arXiv:math/0306223

149. (with Bak, A., Minian, G., and Porter, T.), `Global actions, groupoid atlases and applications’, J. Homotopy and Related Structures, 1 (2006) 101-167. `Memory Evolutive Systems; Hierarchy, Emergence, Cognition’, by A Ehresmann and J.-P. Vanbremeersch; Elsevier, 2009.

Bob Coecke.

What can we learn from relations?

Among other things I will talk about hidden variable theories and properties of quantum correlations under time reversal.

Andreas Doring.

States and measures

In classical physics, the most general kind of state of a physical system is given by a probability measure on the state space of the system. This corresponds to an integral on the commutative algebra of physical quantities. Quantum systems, on the other hand, do not have a state space description. Quantum states are positive linear functionals (i.e., integrals) on the non-commutative algebra of physical quantities. In this talk, it will be shown that quantum states can also be understood as probability measures which live not on a (non-existent) state space, but
on a presheaf associated to the quantum system. This presheaf is a generalised space. Its non-commutative character is expressed by the fact that it has no points.

Basil Hiley.

1. The Geometric Structure of Quantum Mechanics.

The traditional approach to quantum mechanics using Hilbert space obscures the geometric structure of the theory, yet the appearance of a Clifford algebraic structure in the Dirac theory. David Bohm proposed an approach based on a notion of ‘structure process’. I will present some of the background ideas which motivates an approach to quantum theory based entirely within the Clifford structure thus exposing the geometric background.

2. The Bohm Model of the Dirac Particle.

Using the ideas discussed in the first talk I will present the complete Bohm model of the Dirac particle, thus showing for the first time that this model can be made fully Lorentz invariant. Bob Callaghan and I have obtained an expression for the quantum potential which reduces to the corresponding expression for the Pauli particle in the non-relativistic limit, and to that for the Schroedinger particle when the spin is zero. I am informed by Olival Freire, a science historian form Sao Paulo, that Bohm once wrote that “a good treatment of Dirac equation in his own approach would deserve a champagne case”! Why didn’t we complete this before he passed on!

Yuri Manin

1. Infinity and renormalization in quantum physics and computer science.

The main observable quantities in Quantum Field Theory, correlation functions, are expressed by the celebrated Feynman path integrals which are not well defined mathematical objects. Perturbation formalism interprets such an integral as a formal series of finite–dimensional but generally divergent integrals, indexed by Feynman graphs, the list of which is determined by the Lagrangian of the theory. Renormalization is a prescription that allows one to systematically “subtract infinities’’ from these divergent terms producing an asymptotic series for quantum correlation functions. On the other hand, graphs treated as “flowcharts’’, also form a
combinatorial skeleton of the abstract computation theory and various operadic formalisms in abstract algebra. In this role of descriptions of various (classes of) computable functions, such as recursive functions, functions computable by a Turing machine with oracles etc., graphs can be used to replace standard formalisms having linguistic flavor, such as Church’s $\lambda$–calculus and various programming languages.
The functions in question are generally not everywhere defined due to potentially infinite loops and/or necessity to search in an infinite haystack for a needle which is not there. In this paper I argue that such infinities in classical computation theory can be addressed in the same way as Feynman divergences, and that meaningful versions of renormalization in this context can be devised. Connections with quantum computation are also touched upon.

2. Foundations of Mathematics: from discrete to continuous and backwards

Abstract: In this talk, the scope of which will be more of a round table contribution, I will discuss some historical and philosophical issues related to the continuous/ discrete dichotomy. I will also argue that an emerging shift in foundations is related to homotopy theory and proceeds from the continuous to the discrete rather than in the reverse direction that has dominated the consciousness of working mathematicians for the last century or so.

Lou Kauffman.

1. Topological Quantum Information Theory

Abstract: This talk is about using topological structures such as unitary representations of the braid group and unitary solutions to the Yang-Baxter Equation in studying entanglement, quantum computing and quantum algorithms. A motivation for this project is the question: Is there a relationship betweem topological entanglement (linking and knotting) and quantum entanglement (indecomposable states, non-locality)? We will give numerous examples and speculate about the eventual role of topological structures in this aspect of physics and communication.

2. Reflexivity, Eigenforms, Iterants and Discrete Physics

Abstract: A reflexive domain is a domain D such that there is a 1-1 correspondence between D and the mappings [D,D] from D to D. Letting I:D —–> [D,D] be this 1-1 correspondence. We have the Fixed Point Theorem: For any F:D —-> D, there is an element E of D such that F(E) =E.
Proof. Define Gx = F(I(x)x). Then G = I(g) for some g in D and so I(g)x = F(((x)x), whence I(g)g =F(I(g)g) and I(g)g is the desired fixed point. Q.E.D. (The reader may note that this arguement is a generalization of Cantor’s Diagonal argument, so that if we were in a classical logical algebra and F(x) = ~ x represented negation, then we would conclude the existence of fixed points for negation; non-standard or imaginary logical values must appear in a reflexive logical space). This talk will explore the analogies of fixed points, observations and observables, eigenvectors and recursive processes in relation to foundations of physics. In particular we shall re-open the books on the complex numbers and view them in terms of recursion and reflexivity, finding new and natural ways to think about their roles in physical theory. To give a hint, think of the oscillatory process generatoed by R(x) = -1/x. The fixed point is i with i^2 = -1, but the processes generated over the real numbers must be directly related to the idealized i. We shall let {+1,-1} stand for an undisclosed alternation or ambiguity between +1 and -1 and call {+1,-1} an <iterant>. There are two <iterant views>: [+1,-1] and [-1,+1]. These, seen as points of view of alternating process will become the square roots of negative unity. We introduce a temporal shift operator # such that [a,b]# = #[b,a] and ## = 1 so that contcatenated observations can include a time step of one-half period of the process …abababab… . We combine iterant views term-by-term as in [a,b][c,d] = [ac,bd]. Then we have, with i = [1,-1]# (i is view/operator), ii = [1,-1]#[1,-1]# = [1,-1] [-1,1]## = [-1,-1] = -1. This gives rise to a new process-oriented construction of the complex numbers, quaternions, and in fact of all of matrix algebra. We relate this point of view to thinking about the role of complex numbers in quantum mecahnics, the role of temporal shift operators in discrete physics and our previous work on Non- Commutative Worlds that begins with the understanding that temporal shift operators in discrete physics allow the corresponding calculus to be represented in a non- commutative (Lie algebraic) context where all derivatives are represented by commutators. We also relate these ideas of reflexivity and fixed points to left or right distributive non-associative algebras and their relationships with knot theory and with approoaches to wholeness in physics such as the work of Barbara Piechosinska. A <magma> is a non-associative algebra with a single binary operation that is left- associative: a*(b*c) = (a*b)*(a*c). Note that this axiom says that every element A of the magma is a structure preserving mapping of the magma to itself: A(x*y) = (A*x)* (A*y). So the notion of a magma is another view of what should be a self-reflexive domain. We will raise questions about the relationship of magmas and reflexive domains, illustrate the remarkable and deep relationships between magmas and knots and braids and raise the question of the relationships of left distributivity and the Jacobi identity where a(bc) = (ab)c + b(ac) so that left distributivity is replaced by derivation satisfying the Leibniz rule. It is, at the very least, mysterious, that this level
of reflexivity (after all a Lie algebra is an algebra satisfying the Jacobi identity and anticommuativity) is so fundamental to mathematics and physics.

Freddy van Oystaeyen

Sheaves on the moment space

Maurice de Gosson

The Schroedinger equation, classical or quantum?

ABSTRACT: For the vast majority of quantum physicists Schroedinger’s equation cannot be derived mathematically from first principles; it should be seen as a postulate, in fact a physical law governing the evolution of de Broglie’s matter waves. We will show that Schroedinger’s equation can be rigorously derived using the notion of density of trajectories introduced in 1928 by Van Vleck. The physical content of the solutions to this equation is then made clear applying the principle of the symplectic camel, which is a topological reformulation of the uncertainty principle of quantum mechanics. The relation with the so-called Bohmian trajectories is briefly discussed.

Isar Stubbe

Sheaves as modules

abstract: A quantale is a monoid in the monoidal category Sup of complete lattices and supremum-preserving morphisms, so by general categorical principles we can study modules on a quantale. But each hom-set of Sup is itself a complete lattice, making Sup a so-called 2- category. Using this 2-dimensional structure I shall define “principally generated modules” on Q, to then show that these provide a non- commutative generalisation of sheaf theory.Georg

Georg Wikman. Twins: What sort of Category ?