Duality in Logic and Physics
Duality has many manifestations in logic and physics. In
classical logic, propositions form a partially ordered set and
negation is an order-reversing involution which switches "true"
and "false". The same holds in quantum logic, with propositions
corresponding to closed subspaces of a Hilbert space. But the
full structure of quantum physics involves more: at the very
least, the category of Hilbert spaces and bounded linear
operators. This category has another kind of duality, a
contravariant involution that switches "preparation" and
"observation". Other closely related dualities in quantum
physics include "time reversal" (switching future and past) and
"charge conjugation" (switching matter and antimatter). The
quest to find a unified mathematical framework for dualities in
logic and physics leads to a fascinating variety of structures:
star-autonomous categories, n-categories with duals, and more.
We give a tour of these, with an effort to focus on conceptual
rather than technical issues.
State sum models, induced gravity and the spectral action
I will give a new proposal for the spectral action for gravity coupled
to matter, taking into account the phenomenon of induced gravity.
This is then used to give some perspectives for the construction of a quantum
theory of gravity coupled to matter using state sum models based on a tricategory.
The category of spacetime regions
The purpose of this talk is to propose a connection between the spin foam
approach to quantum gravity and a branch of abstract homotopy theory called
model category theory.
The spin foam approach models regions in spacetime by four dimensional
simplicial complexes. The quantum theory of the metric on them is constructed
by putting representations of the Lorentz group on the triangulation and combining
them in a specific way, called a categorical state sum. Recently a technical
advance has removed the problems that plagued the theory, so that a finite
computational theory of quantum general relativity seems to be available.
The deeper interpretational problems remain: what do the specific complexes
mean? Are they triangulations of an underlying point set, or Feynman diagrams
of a fundamental theory? If they are just approximations, how does the exact
theory emerge? How does the Bekenstein bound relate to this picture? How
does the classical limit of the models approach general relativity?
What we are going to propose is that the state sum models on different
simplicial complexes can be modified to fit together to form what is called a
model category. We can then incorporate the Bekenstein bound into the model
by using a technique from model category theory called localization.
The flow of information from a system to its exterior would be a map in the
category, and the localization of a model of the system with respect to this map
would give a simplicial complex on which the exact effective theory would be
What is information? Reversibility and simulation
Over the last two decades, quantum physics has introduced new concepts
of information and computation based on physical laws. This has required
a re-examination of our old notions of information and information processing.
In what sense can we regard quantum entanglement as a kind of "information",
or the dynamical evolution of a quantum system (a computer) as a kind of
"computation"? This talk will draw on the lessons of the quantum realm to
suggest a general framework for thinking about information and physics.
We will argue that the idea of "information" describes the reversible and
irreversible transformations of a physical state, while "computation" addresses
how one physical process may act as a simulation of another.