Ja n 20 05 Feynman ’ s Path Integrals and Bohm ’ s Particle Paths

Both Bohmian mechanics, a version of quantum mechanics with trajectories, and Feynman's path integral formalism have something to do with particle paths in space and time. The question thus arises how the two ideas relate to each other. In short, the answer is, path integrals provide a re-formulation of Schrödinger's equation, which is half of the defining equations of Bohmian mechanics. I try to give a clear and concise description of the various aspects of the situation. (nonrelativistic) quantum theory without observers; in a world governed by this theory, quantum particles have precise trajectories, and observers find the statistics of outcomes of their experiments in agreement with quantum mechanics. For a system of N particles, their positions Q i (t) ∈ R 3 change according to the equation of motion, dQ i (t) dt = m i Im ψ * t ∇ i ψ t ψ * t ψ t Here, m i is the mass of particle i, φ * ψ denotes the scalar product in C k , and ψ t : R 3N → C k is the wave function of non-relativistic quantum mechanics, defined on the configuration space and evolving according to the (nonrelativistic) Schrödinger equation. In a typical Bohmian universe, the positions Q i (t) appear random, at any time t, with joint distribution |ψ t | 2 [3, 4]. The method of path integrals [6, 7, 8], invented by R. P. Feynman in 1942, is nowadays widely used in quantum physics. I briefly recall the key idea. Let us