The important point is that such a pair of inclusions is in general totally different from an automorphism of G! A pair of refinements of G is equivalent to an appropriate pair of inclusions from G into two finer orders. 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. An automorphism of G is an order-preserving bijection. Now replace Minkowski space with a partially ordered set G. 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. Coordinate systems may therefore be viewed as refinements of the causal order on M. Given two causally unrelated events in M, a coordinate system may be chosen in which either event precedes the other. The relativity of simultaneity provides a good context for understanding the passive viewpoint. The active and passive viewpoints are virtually interchangeable in this context, but they are completely different in more general settings. From the passive viewpoint, an element of P induces a change of coordinates on M, which merely rearranges physically insignificant labels on its elements. From the active viewpoint, an element of P induces an automorphism of M, which rearranges its elements. Before discussing this, I will briefly review how elements of the Poincaré group P may be applied to Minkowski spacetime M. Group representation theory generalizes in interesting ways in the context of pure causal theory. 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. Gauge theory also takes a different form in this context. The properties of these structures alter the constraints on quantum states. Nonmanifold models of spacetime microstructure arising in quantum gravity require different interpretations of covariance, based on structures such as partial orders. Despite the success of group representation theory in fundamental physics, the need for more general notions is now becoming apparent. 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. 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. Covariance in relativity is expressed locally in terms of the Poincaré group of symmetries of four-dimensional Minkowski spacetime.
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.
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