The Evolution of the Idea of Causality [1942d]

B. F. McGuinness, Friedrich Waismann

Gunji YP, Taichi Haruna

Causal and Acausal Sets Based on a Quotient Lattice The spacetime viewed from the inside by using a lattice is here described as a causal set, and a quotient lattice is described as a causal history. It is shown that this description is consistent with general relativity and quantum theory. In particular, the influence of measurement on a causal set is considered and the interaction between a causal set and history is defined. The transitions from a causal set to a causal history and vice versa are both defined in a category theory and they are shown to be an adjunction. This adjunction leads to the idea of a relative casual set, and it results in the evolution and superposition of causal sets and histories. It is shown that the superposed casual set is a bicontinuous poset equipped with a compact interval. Relative causal sets are also equipped with the operation of self-embedding. A causal set is continually transformed into a degenerate distributive lattice due to the self-embedding. When an acausal set in a causal set can be transformed into a tensor product of underlying Hilbert spaces, the distributive lattice reveals a unitary evolution of composite states that is consistent with quantum theory.

Henry Margenau

Lawrence Paul Horwitz

A consistent classical and quantum relativistic mechanics can be constructed if Einstein's covariant time is considered as a dynamical variable. The evolution of a system is then parametrized by a universal invariant identified with Newton's time. This theory, originating in the work of Stueckelberg in 1941, contains many questions of interpretation, reaching deeply into the notions of time, localizability, and causality. Some of the basic ideas are discussed here, and as an example, the solution of the two body problem with invariant action-at-a-distance potential is given. A proper generalization of the Maxwell theory of electromagnetic interaction, implied by the Stueckelberg-Schrödinger dynamical evolution equation and its physical implications are also discussed. It is also shown that a similar construction occurs, applying similar ideas, in the theory of gravitation.

Philipp G. Frank, Robert S. Cohen

R. E. Wagner, B. T. Shields, Matthew R. Ware +2 more

We compare the relativistic time evolution of an initially localized quantum particle obtained from the relativistic Schr\"odinger, the Klein-Gordon and the Dirac equations. By computing the amount of the spatial probability density that evolves outside the light cone we quantify the amount of causality violation for the relativistic Schr\"odinger Hamiltonian. We comment on the relationship between quantum field theoretical transition amplitudes, commutators of the fields and their bilinear combinations outside the light cone as indicators of a possible causality violation. We point out the relevance of the relativistic localization problem to this discussion and comment on ideas about the supposed role of quantum field theory as a vehicle of making a theory causal by introducing antiparticles.