1994/01/06 by Kostas Skenderis, K. Skenderis, Skenderis, K. +2
Physics and Astronomy · #Cosmology and Gravitation Theories #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9401024
35 pages, ITP-SB-93-86
arxiv created 1994/01/06 · arxiv updated 2009/11/30
We \em derive the exact configuration space path integral, together with the way how to evaluate it, from the Hamiltonian approach for any quantum mechanical system in flat spacetime whose Hamiltonian has at most two momentum operators. Starting from a given, covariant or non-covariant, Hamiltonian, we go from the time-discretized path integral to the continuum path integral by introducing Fourier modes. We prove that the limit N → ∞ for the terms in the perturbation expansion (``Feynman graphs'') exists, by demonstrating that the series involved are uniformly convergent. \em All terms in the expansion of the exponent in <x| exp (- Δt H / ℏ) |y> contribute to the propagator (even at order Δt!). However, in the time-discretized path integral the only effect of the terms with H2 and higher is to cancel terms which naively seem to vanish for N → ∞, but, in fact, are nonvanishing. The final result is that the naive correspondence between the Hamiltonian and the Lagrangian approach is correct, after all. We explicitly work through the example of a point particle coupled to electromagnetism. We compute the propagator to order (Δt)2 both with the Hamiltonian and the path integral approach and find agreement.