Infinite Permutation Groups and the Origin of Quantum Mechanics

We propose an interpretation for the meets and joins in the lattice of experimental propositions of a physical theory, answering a question of Birkhoff and von Neumann in [1]. When the lattice is atomistic, it is isomorphic to the lattice of definably closed sets of a finitary relational structure in First Order Logic. In terms of mapping experimental propositions to subsets of the atomic phase space, the meet corresponds to set intersection, while the join is the definable closure of set union. The relational structure is defined by the action of the lattice automorphism group on the atomic layer. Examining this correspondence between physical theories and infinite group actions, we show that the automorphism group must belong to a family of permutation groups known as geometric Jordan groups. We then use the classification theorem for Jordan groups to argue that the combined requirements of probability and atomicism leave uncountably infinite Steiner 2-systems (of which projective spaces are standard examples) as the sole class of options for generating the lattice of particle Quantum Mechanics.Comment: 23 page

Infinite Jordan Permutation Groups

Abstract If G is a transitive permutation group on a set X, then G is a Jordan group if there is a partition of X into non-empty subsets Y and Z with |Z| > 1, such that the pointwise stabilizer in G of Y acts transitively on Z (plus other non-degeneracy conditions). There is a classification theorem by Adeleke and Macpherson for the infinite primitive Jordan permutation groups: such group preserves linear-like structures, or tree-like structures, or Steiner systems or a ‘limit’ of Steiner systems, or a ‘limit’ of betweenness relations or D-relations. In this thesis we build a structure M whose automorphism group is an infinite oligomorphic primitive Jordan permutation group preserving a limit of D-relations. In Chapter 2 we build a class of finite structures, each of which is essentially a finite lower semilinear order with vertices labelled by finite D-sets, with coherence conditions. These are viewed as structures in a relational language with relations L,L',S,S',Q,R. We describe possible one point extensions, and prove an amalgamation theorem. We obtain by Fra¨ıss´e’s Theorem a Fra¨ıss´e limit M. In Chapter 3, we describe in detail the structure M and its automorphism group. We show that there is an associated dense lower semilinear order, again with vertices labelled by (dense) D-sets, again with coherence conditions. By a method of building an iterated wreath product described by Cameron which is based on Hall’s wreath power, we build in Chapter 4 a group K < Aut(M) which is a Jordan group with a pre-direction as its Jordan set. Then we find, by properties of Jordan sets, that a pre-D-set is a Jordan set for Aut(M). Finally we prove that the Jordan group G = Aut(M) preserves a limit of D-relations as a main result of this thesis