The Language of Quanta in the Vacuum
Nick Rossiter & Michael Heather, Northumbria University

Mathematics over the last three hundred years has had great success in modelling phenomena of the physical world with in turn geometry, algebra, differential calculus, set theory, tensors, etc, often with new mathematics to support new physics. However, this success sometimes obscures that there are also spectacular failures. The world of real number, Boolean logic and Hilbert spaces cannot deal adequately with turbulence, quantum theory, information systems, consciousness studies and natural language. In the last 50 years Category Theory has appeared in North America as a unifying mathematics of geometry, algebra and topology and in Europe as an integration of the Bourbakian tripartite algebra topology and order. It appears it may well extend to parts of physics that classical methods cannot reach.

An example is particle condensation where the U(1) x SU(2) x SU(3)  Standard Model that seems to have outgrown its origins in group and gauge theory for the classical symmetry of Landau cannot explain fractional quantum Hall states at zero temperature. Topological orders are needed but these only apply to states with a finite energy gap. For gapless quantum states it is necessary to resort to orders of quantum theory.  Ying Xiao-Gang Wen has observed that the Standard Model of elementary particles in condensate phase in a vacuum is as primitive and as inadequate as the atomic theory was for the nucleus at the end of the 19th Century. 

This work examines Rosen’s contention that Category Theory extends mathematics with the power of natural language. However, for Category Theory to present formally the informal content of natural language requires for applied mathematics a constructive rather than axiomatic approach found in the pure subject.  The main advantage is that concepts like geodesic, variational principles, Lagrangians, string nets, gauge coupling and many other necessary components are inherent and therefore unified in Category Theory.