The following are some new ideas about physics. I have most of these ideas partially written up, but I am so busy with other work at present that it may be some time before I can put them into a respectable form. Until then, I offer them as food for thought. To avoid making this post too long, I will give only very brief summaries of these ideas. Subsequent posts will treat them one at a time, in somewhat more detail.
1. Shape/Causal Duality
Shape dynamics, invented by Julian Barbour, is a theory of spacetime that applies an action principle and a procedure called best matching to a configuration space of conformal 3-manifolds to recover the Hamiltonian formulation of general relativity. Shape dynamics is a “space-first” theory, with time arising in a secondary fashion. Causal theory, pioneered by Rafael Sorkin, is a “time-first” theory. Time is viewed as a proxy for causality, which is taken to be fundamental. Spatial separation is viewed as indicative of causal disjunction, and is taken to be secondary. Shape dynamics is based on symmetric relations between elements, while causal theory is based on antisymmetric relations. The following question occurred to me while discussing these two theories with Daniel Alves, Lawrence Crowell, Sean Gryb, and Flavio Mercati during the 2012 FQXi essay contest: is there a shape/causal duality principle relating appropriate versions of shape dynamics and causal theory? Continuum theories such as standard shape dynamics are not directly comparable to locally finite theories such as causal set theory, but these theories can be reformulated in comparable contexts. Shape/causal duality would lend greater scope and credence to both theories, while opening up interesting lines of research.
2. Causal Theory in Power Space
George Ellis and a few other physicists have recently introduced the idea of top-down causation in fundamental physics. In the context of classical spacetime, the hypothesis of top-down causation is that causal relationships among subsets of spacetime are not completely reducible to causal relations among their constituent events. Sorkin’s causal set theory, by contrast, is purely bottom-up at the classical level. In causal set theory, causality is modeled as an irreflexive, acyclic, locally finite binary relation on a set, whose elements are viewed as spacetime events. However, if the hypothesis of top-down causation is true, then causal structure exists not at the level of spacetime itself, but at the level of the power set of spacetime. Causal graphs may still be employed to describe such a structure, but the vertices now correspond to families of events. This approach is attractive both because of the convincing arguments for top-down causation, and because power-space relations add potentially useful structure without changing the types of mathematical objects and techniques involved in the theory. There is considerable precedent for this type of generalization, such as Grothendieck’s scheme-theoretic approach to algebraic geometry. The holistic aspects of causal theory in power space might also aid in establishing shape/causal duality.
3. Causal Atomic Resolution
If spacetime manifold structure degenerates near the Planck-scale, the number of fundamental spacetime elements involved in observable interactions might be very large. Organizing the information associated with such interactions is an important preamble to calculation and prediction. Causal atomic resolution is an attractive way of organizing such information in the context of pure causal theory. A causal atomic resolution is a sequence of causal atomic decompositions that progressively simplify the original graph by combining “causal atoms” at each level of the sequence into larger atoms at the next level, thereby imposing a fractal structure. Causal atomic resolution is similar to the theory of categorification in abstract algebra, in which objects correspond to categories at different “algebraic levels.” Categorification may be generalized in a graph-theoretic setting, with causal atomic resolution as a special case. In quantum causal theory, causal atomic resolution may be combined with power space and path integral methods to carry the theory in promising and unexplored directions. Causal atomic resolution also has the potential for important applications to existing areas of computer science, economics, biology, physics, and mathematics.
4. Order-Theoretic Representation Theory
Group representation theory lies at the heart of modern physics as the mathematical expression of symmetry. Covariance in relativity is expressed locally in terms of the Poincare group of symmetries of four-dimensional Minkowski spacetime. The same group constrains particle states in the standard model. The gauge theories describing electromagnetism, the weak interaction, and the strong interaction are also based on group representation theory. However, more general notions are needed. 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. Representation theory has interesting analogues in the context of pure causal theory. The active viewpoint on group actions is replaced by the theory of order automorphisms, while the passive viewpoint is replaced by the theory of refinements. Both theories are interesting and important.