, Volume 22, Issue 1, pp 19-31
Date: 08 Feb 2011

The Dyson–Feynman conjectures

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In this article, we survey recent work on a suggestion by Freeman J. Dyson in his famous “Missed Opportunities” Gibbs Lecture at the annual AMS meeting in 1972. There he offered the open problem of providing mathematical foundations for Feynman’s sum over histories (world-view) description of relativistic quantum field theory. In this view, space–time is laid out as a film, in which one becomes more and more aware of the future as more of the film unfolds. The actual physical path traveled between any two space–time points is seen as an average over all possible paths with the same beginning and end point (path integral). Feynman later developed his time-ordered operator calculus, in which the time index determines when operators operate (rather than their position on paper). In this approach, operators acting at different times commute. Feynman conjectured that the operator calculus included the sum over histories (path integral), while use of the operator calculus led Dyson to make four conjectures concerning quantum electrodynamics. We also discuss recent proofs of the Feynman conjecture and the last two remaining Dyson conjectures.