Hi
Heisenberg said that particles were not fundamental, because every
particle in some sense contained all others. It appears that the
same should be true of Bekenstein's atoms for spacetime. There is
no physical difficulty in thinking of spacetime degrees of freedom
in a quantum manner. The difficulty arises in coupling matter and
spacetime degrees of freedom in a mathematically sensible way. It
appears that this is not at all possible unless one addresses some
basic issues in quantum logic.
Categorical internalisation is an essential element here. There is
mounting support for this point of view from studies of, for
instance, the Hopf algebra structure of renormalisation (see
Connes and Marcolli) and its connection with non-commutative
geometry.
Twistor theory is one investigation that attempted to respect
background independence, and which played an important role in the
development of state sum models for quantum gravity.
The first interesting step towards a modern category theoretic
understanding of mass is perhaps the study of the Klein-Gordon
equation in the Hughston and Hurd paper, in which they combine two
solutions to the massless equations for spin s particles thought
of as elements of a sheaf cohomology group on a twistor space. The
Klein-Gordon equation solutions then belong to a second cohomology
group.
Naively at least, therefore, a quantisation of this origin of mass
involves a non-Abelian sheaf theoretic second cohomology group. And
an understanding of such an object leads one inexorably in the
direction of topos cohomology.
The first cocycle condition may be thought of as a triangle. Such
triangles make sense in any category, so the coefficients for H1
may be generalised, in particular to non-Abelian groups. The
difficulty arises in understanding categories deeply enough to
develop a sufficiently subtle higher dimensional analogue.
The interplay of categories and logic (ie. topos theory) in
physics has already been carefully considered by Markopoulou in
the context of causal sets, and Isham and others in the context of
quantum theory.
A topological space is a category of objects the open sets, with
inclusions for arrows. For example, the celestial sphere of the
twistor correspondence is considered as such a category.
Already in two dimensions, Yang-Mills theory involves some
beautiful combinatorics (see Witten's work). This uses a
generalisation of the Abelian localisation principle from
equivariant cohomology. Localisation reaches a pinnacle of
abstraction in an adjunction between the inclusion of a topos of
sheaves into the presheaf category and the so-called
sheafification functor (see Mac Lane and Moerdijk). Sheaves are
defined with respect to a topology on the base category.
As the String theorists like to tell us, path integrals are
heinously complex and unsmooth things. They are now telling us
that maybe 4D Yang-Mills is pretty amazing all on its own. And
they seem to be saying that twistors are cool too.
In other words, we want a higher categorical analogue of the
evaluation of path integrals like 2D Yang-Mills.
The intended interpretation of pieces of categories is that they
are geometric entities. Objects are zero dimensional and arrows
are one dimensional etc. I won't go into this now.
Objects in a category such as Rep(SU(2)) are representation spaces
rather than 'particle states', so to capture the notion of a state
properly in category theoretic terms it is necessary to
internalise this picture further than is normally considered and
to replace the Mac Lane pentagon by at least its tricategorical
analogue. The truly fascinating thing is that tensor products in
higher dimensional categories are no longer stable dimensionally.
For the pentagon this leads to a sort of symmetry breaking. This
has already been used to explain confinement RIGOROUSLY (see
Joyce).
