Consider a classical system and write a lagrangian for it.
To find the classical behavior of the system, find the path q which has an extreme value of the action S (usually a minimum). The action is the integral of the lagrangian density L over the path q:
S = ∫ L dt
q
How to add wave nature to this?
Use
exp( i / h-bar ∫ L dt )
q
and calculate sums over paths. That is the Feynman path integral.
Paths q which have an extreme value of the action will have constructive interference. Other paths will have destructive interference.
We converted a problem of classical mechanics into a Feynman path integral problem.
The wavelength of the wave nature is determined by the Planck constant h. When h approaches zero, the wavelength approaches zero. The constructive interference in the path integral is concentrated to sharper and sharper points. In the limit, the Feynman path integral reduces to the classical optimization problem.
Our photograph model of quantum field theory is clarified by the above explanation. At the low level, it is classical physics. The wavelength, which is determined by the Planck constant, adds a wave nature over the classical world, through the Feynman path integral.
We have been wondering how Feynman diagrams can calculate classical things, like the reduced mass of the electron under acceleration. Above we have a probable explanation. Feynman diagrams are an approximate way of calculating Feynman path integrals. And Feynman path integrals are built on classical physics.
Paul Dirac's 1933 paper seems to contain similar ideas.
This still does not explain details, like why a virtual photon line in a Feynman diagram has a certain propagator and a certain coupling constant. Maybe they calculate some type of classical action approximately right and therefore reproduce the correct classical behavior? We will look into this specific question in coming blog postings.
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