Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the ‘quantum logic’ of subspaces of a general vector space—which is then specialized to the closed subspaces of a Hilbert space. But there is a ‘dual’ progression. The set notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions (i.e. the vector space version of a set partition) of a general vector space—which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.
We construct generalized quantum Cauchy pre-measures that correspond to the analytic continuation of the transition probability of the Cauchy process to imaginary time. We show that these complex pre-measures of time translations extend to a measure on the space dual to a real Hilbert space whose support is locally compact in the uniform convergence topology and with velocities in the Hilbert space. At that the quantum Cauchy-Dirac and Cauchy-Maxwell pre-measures of time translations of electrons and photons, that correspond to the retarded Green's functions of the Dirac and Maxwell equations with no sources viewed as generalized functions on bump functions, are unitary equivalent to quantum Cauchy pre-measures Therefore these pre-measures on the space dual to a real Hilbert space are
σ-additive as well, but their support on the electron (respectively, photon) trajectories are compact in the uniform convergence topology with velocities in the Hilbert space. We study the way the classical relativistic mechanics of particle comes from the quantum mechanics of the free Dirac particle.