Fock theories and quantum logics (Q702988)

A quantum logic or physical theory is here identified with ``yes/no'' experiments or ``tests'', and considered to be an atomic, orthomodular lattice with the covering property. States of the theory are mappings from \(L\) to [0,1], and pure states those where just one atom is assigned the value 1. Observables are morphisms over a Borel algebra of subsets and here regarded as Boolean algebras in \(L\). Lastly time evolutions are automorphisms of \(L\), which for example take any pure state to another pure state of the theory. Fock theories are introduced to provide a ``mathematical structure for studying systems containing particles whose number varies in time'' (p. 86). Let \(L\) be a theory as above, \(\mathbb N\) the natural numbers and \(\alpha = (k_1,\dots,k_n)\) an \(n\)-tuple in \(\mathbb N^n\). Then for a set \(S = \{s_1,\dots,s_n\}\) ``of \(n\) species of particles'', \(\alpha\) is considered ``a possible composition in particles of the species \(s_1,\dots,s_n\) respectively'' (p. 79). It is assumed that for any \(\alpha\) there is a ``counting test'' \(c(\alpha)\), and a Fock Theory is defined for the set \(S\), as ``a set of counting tests'' \(F(S) = \{c(\alpha)\}\) such that the join over \(\alpha\) of all \(c(\alpha)\) is 1, and a null or ``vacuum'' test is orthogonal to any other tests in the theory (ibid.). Thus Fock theories associate ``sets of particles'' of different kinds, with elements of \(L\), the ``counting tests'' for these particles. Further definitions introduce the notion of admissible states of a Fock theory, and particle Fock theories, which are designed ``from a physical point of view to describe sharply defined particles'' (p. 80). Time evolutions are introduced, and evolutions with a Hamiltonian become the special case where not only are lattice operations preserved, but a Boolean subspace of the lattice is picked out, i.e. ``\(V\) has a Hamiltonian if there exists a Boolean algebra \(H\subset L\) having the property (EH): \(V(a) = a \Leftrightarrow (a,H)C\)'' (p. 81). Further refinements of the definition as well as further results are then derived. Later in the section, the notion of a many particle theory and its Hamiltonian are discussed. Two conclusions are claimed for this analysis: first that Fock spaces ``give a precise mathematical possibility for distinguishing between sharply defined particles and other particles'' and second that time evolutions are of interest because ``sharply defined particles do not change their number in conservative conditions'', while the non-sharply defined do, ``possibly because they are composed of other particles'' (ibid.). There is a brief defence of this ``purely algebraic'' approach at the end of the paper and a short reference section that cites only six works, two papers from the author (1987 and 1995), and four classics: those of Jauch, Piron, Sikorski and Stone.