Order-Theoretic Representation Theory

Group representation theory lies at the heart of modern physics as the mathematical expression of symmetry, remaining perhaps the most promising vehicle for initial progress beyond relativity and the standard model.  Covariance in relativity is expressed locally in terms of the Poincaré group of symmetries of four-dimensional Minkowski spacetime.   This spacetime model is taken for granted in the quantum field theory underlying the standard model, thereby constraining particle states to correspond to representations of the Poincaré group.  The gauge theories describing electromagnetism, the weak interaction, and the strong interaction are based on the representation theory of the gauge groups U(1), SU(2), and SU(3), respectively.   Despite the success of group representation theory in fundamental physics, the need for more general notions is now becoming apparent.

Nonmanifold models of spacetime microstructure arising in quantum gravity require different interpretations of covariance, based on structures such as partial orders.  The properties of these structures alter the constraints on quantum states.  Gauge theory also takes a different form in this context.  Modification of group representation theory in such models is often viewed in a negative sense, as “Lorentz invariance violation,” or “covariance breaking.”  However, such modifications should be pursued positively as a refinement, not abandonment, of the representation-theoretic tradition.

Group representation theory generalizes in interesting ways in the context of pure causal theory.   Before discussing this, I will briefly review how elements of the Poincaré group P may be applied to Minkowski spacetime M.   From the active viewpoint, an element of P induces an automorphism of M, which rearranges its elements.  From the passive viewpoint, an element of P induces a change of coordinates on M, which merely rearranges physically insignificant labels on its elements.  The active and passive viewpoints are virtually interchangeable in this context, but they are completely different in more general settings.

The relativity of simultaneity provides a good context for understanding the passive viewpoint.  Given two causally unrelated events in M, a coordinate system may be chosen in which either event precedes the other. Coordinate systems may therefore be viewed as refinements of the causal order on M.   Each element of P exchanges one refinement for another, so these elements may be identified with special ordered pairs of refinements of the causal order on M.

Now replace Minkowski space with a partially ordered set G.   An automorphism of G is an order-preserving bijection.   Meanwhile, a refinement of G is an inclusion of G into a finer order; i.e., an order with the same elements but more relations.   A pair of refinements of G is equivalent to an appropriate pair of inclusions from G into two finer orders.   The important point is that such a pair of inclusions is in general totally different from an automorphism of G!

The reason why the active and passive viewpoints seem so similar in the context of Minkowski space M is because operations on M are usually described in terms of some coordinate system.   This is true, in particular, for automorphisms of M, which are usually described by specifying the coordinates of an arbitrary element and its automorphic image.    This relates a pair of coordinate systems to each automorphism, thereby connecting the active and passive viewpoints.   However, this relationship is merely an artifact of the special structure of M.  It survives to some degree for general Lie group actions on manifolds, but usually disappears in the absence of manifold structure.

The question then is, which viewpoint is the “right” one in general, the active viewpoint or the passive viewpoint?  The answer is both!  The active viewpoint generalizes to the theory of automorphisms, while the passive viewpoint generalizes to the theory of augmented structures over a fixed base.   Both theories are important in physics, but in different ways.   In particular, theories involving spacetime microstructure generally do not have exact symmetries on large scales, unless these symmetries are imposed artificially.  Automorphisms in such theories will generally leave large-scale structures fixed, while permuting small-scale structures locally and independently.   By contrast, such theories will often exhibit pairs of “refinements” closely approximating any given coordinate transformation on an appropriate “smoothing.”  I will not attempt to make these notions more precise at present.

This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Your email address will not be published. Required fields are marked *