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Virtual particles and Coulomb interaction
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In quantum field theory, a static field is a stationary e/m field
whose expectation is independent of time. Photon states arise from the
quantization of the fluctuations of the e/m field and are always
transverse and real.
Longitudinal or virtual photons never enter the picture. Both are
fictions stemming from an inadequate interpretation of the QFT
formalism that takes perturbative expansions literally.
Virtual objects have strange properties. For example,
the Coulomb interaction between two electrons is mediated by
virtual photons faster than the speed of light, with imaginary masses.
If one assumes that the electron remains real, on-shell, throughout,
as in nonrelativistic QM, by 4-momentum conservation.
This is often made palatable by invoking a time-energy uncertainty
relation, which would allow particles to go off-shell.
But there is no time operator in QFT, so the analogy to Heisenberg's
uncertainty relation for position and momentum is highly dubious.
(But formally, the electron becomes virtual during the interaction,
and the photons would be virtual too, because a static external field
in QED in its textbook definition cannot be formulated from first
principles without having it emanate from its source - far away
electrons. Thus everything becomes virtual, except at times +-infinity.)
Strictly speaking, the Coulomb interaction is simply the Fourier
transform of the photon propagator 1/q^2, followed by a nonrelativistic
approximation. It has nothing at all to do with virtual particle
exchanges --- except if one does perturbation theory. (And even there
only in the covariant formalism, not in the older Hamiltonian approach
as in the books by Bjorcken and Drell.) But then there is no surprise
that it must influence already the tree level. By a hand waving
argument (equate the Born approximations) this gives the
nonrelativistic correspondence.
But to get the Coulomb interaction as part of the Schroedinger equation,
one needs to sum all ladder diagrams with 0,1,2,3,...,n,... exchanged
photons arranged in form of a ladder. Then one needs to approximate
the resulting Bethe-Salpeter equation. These are nonperturbative
techniques. (The computations are still done at few loops only,
which means that questions of convergence never enter.)
Virtual photons mediating the Coulomb repulsion between electrons
have spacelike momenta and purely imaginary mass, and hence would
proceed faster than light if there were any reality to them.
(This is _very_ unlike real, short-living particles, which travel
slower than light and have a nearly real mass, with an imaginary part
much smaller than the real part.) But there cannot be; one'd need
infinitely many of them, and infinitely many virtual
electron-positron pairs (and then superpositions of any numbers of
these) to match exactly even a single real, dressed object or
interaction.