Post by kensetoThis is pure double talk. You said the charge is resided outside the
electron. This would eliminate the infinity problem in QED.
There is no infinity problem in QED. The infinity prioblem is in ALL
field theory, both classical and quantum. The correct designation,
therefore, is "infinity problem in field theory".
So, whatever explanation one casts for the "infinity problem in [any
quantum theory]" is wrong before it gets out of the starting gate,
because the issue has nothing specifically to do with quantum theory
in the first place! The resolution of the issue is something that has
to apply across the board at the very foundation of field theory,
itself. And "across the board" means, specifically, independently of
what setting it is in, be it classical or quantum.
Indeed, the consistency problem for CLASSICAL theory was posed as a
challenge in the American Mathematical Association monthly back in the
1980's and headway had only begun to be made on it by around 2000.
The infinities of field theory are those of (a) the self-energy of a
concentrated source and (b) the "self-force" (the force a concentrated
source sees). These are the poblems that lie at the root of the issue.
In quantum theory, the infinities are directly inherited from these as
follows. Corresponding to the self-force infinity is the singularity
in the propagators on the light cone. The quantized fields,
themselves, are defined in terms of these propagators. Therefore, they
become singular operators in the same sense that the delta function is
a singular function. When applied in non-linear combinations, this
leads to products of singular operators by singular operators, which
(like for singular functions) is ill-defined. The two places this
occurs are in the expressions for the energy & stress tensor (which
are quadratic, generally) and in the interactions (which involve non-
linear expressions in the field equations).
Likewise in the classical theory, the two places where the double-
singularity occur are in the force law (hence the self-force infinity0
and the energy & stress tensor (also quadratic).
The resolution of the problem is clear -- once it's been pointed out.
Maxwell, himself, had resolved the issue, but his resolution was lost
by the time of Lorentz and was only partially rediscovered (more
properly: stumbled back onto) by the time of Tomomaga, Dyson and the
others who had engaged in the renormalization programme (thus leading
to the false impression that both the infinity and its resolution
pertained specifically to quantum theory, rather than to classical
theory).
The fields EXPERIENCED by sources are not identical to the fields
PRODUCED by sources. The link between the two is what is known as the
"constitutive laws". When Lorentz posed the trivial relation (D =
epsilon_0 E, B = mu_0 H), he also did a serious disservice in re-
introducing the very plague that Maxwell had gone out of his way to
remove. It was by identifying these two sets of fields that the
infinity came to be introuced.
It is introduced in the following way:
(1) Self-force. For a singular or concentrated source, rho, through
Gauss's law (div D = rho), one has a singularity or near-singularity
in D. In order for the force to be well-defined (F = rho E) as a
density, it perforce obtains that only one of (rho) and (E) can be
singular, but never both!
Ipso facto, this precludes any such relation as (D = epsilon_0 E) near
the source! Given such a relation, one has the self-force infinity.
Similar expressions for the other parts of the force law all involve
quadratic expressions that combine one element out of the set (E, B)
and one out of the set (rho, J) the latter being derived from (D, H).
The kinematic fields (E, B) are those experienced by sources, while
the dynamic fields -- their conjugates -- (D, H) are those produced by
sources.
To equate the two sets by trivial linear relations with constant
coefficients introduces an infinity in the force law. It is a
violation of the First Commandment of Field Theory:
What God hath sundered let no man reunite.
(2) Self-energy. The stress tensor and energy all involve quadratic
expressions of similar construction -- one element from the set (E, B,
A, phi) and one from the set (D, H, J, rho). The same argument
pertains as for self-force.
In a more general Lagrangian theory, the roles of (A, phi) are played
by the "configuraton coordinates" q, while the roles of (E, B) are
played by various distinguished combinations of the gradients of q
(their respective covariant derivatives) that together give you the
field "velocities" v. Together, these are the kinematic variables.
The conjugates are those constructed as the derivatives of the
Lagrangian. For Maxwell, these are D = dL/dE, H = -dL/dB, J = dL/dA,
rho = -dL/d(phi). For a more general field theory, one may have the
conjugate "momentum" p = dL/dv, and conjugate "source" j = dL/dq.
The stress tensor is always constructed out of combinations of (q,v)
with (p,j), similarly for the force law (which, actually, is derived
from the stress tensor).
For field theories where the Lagrangian is quadratic in the field
velocities, one usually sees relations of the form p = e.v (in a
"matrix-vector" form) with the coefficients e being constant. It is
their constancy that lies at the root of the problem.
If the field law is Lorentz invariant, the coefficients e will be such
that the resulting field equations for (q) will be either a Klein-
Gordon or wave equation. The corresponding Green's functions will be
singular on the light cone and the propagators (and, in turn, the
quantized fields) will likewise be singular. Thus the chain of events
that leads to the self-force and self-energy problems (both
classically and quantum mechanically) is set into motion.
Ipso facto that means that the coefficients e are NOT constant. This
resolution, in fact, is stumbled onto in a quantum setting through the
device of renormalization theory. One finds there, as a result of the
need to have self-consistency, that the coefficients "e" must be
endowed with "scale dependency". What that means is that in the
setting of a scattering experiment where one probes a concentrated
source, the effective values of "e" will be dependent on the scale of
resolution (i.e. how close to the source you get). That is, e becomes
a function of position.
For all practical purposes in quantum theory, e is promoted to an
external field with unknown or unresolved dynaics.
Maxwell, in fact, had done that very thing in electromagnetism. There,
the coefficients are just those arising in the Lagrangian L =
epsilon_0 (E^2 - B^2 c^2)/2 -- i.e., the vacuum permittivity. Hence,
arose the idea that the vacuum, itself, was a polarizing medium. He
made it a point not only to explain that this device was specifically
to resolve the field-theoretic infinity (via 2 thought experiments in
chapter 1 of his treatise), but to finish up the opening chapters by
stressing that we needed to find the actual DYNAMICS of this extra
field, epsilon.
In the more general setting of Lagrangian theory, the coefficients "e"
of a Lagrangian quadratic in the field velocities essentially comprise
the METRIC for the vector space where the field's configuration
variables (q) lie. Thus, the direction quantum theory stumbled onto
was nothing less than that which points back to CLASSICAL theory,
calling on "e" to be promoted to a field.
When "e" is made variable, the Green's functions and propagators are
no longer generally singular on the light cone. They can become non-
singular; likewise for the quantum fields. One begins to see their
actual construction when pulling back from a quantum field theory to
the corresponding classical field theory through the device of
"effective Lagrangians". For electrodynamics, for instance, one
constructs a constitutive law of the form
D = epsilon E + theta B, H = epsilon c^2 B - theta E
where both epsilon AND theta have to be variable.
The other major kind of field theory is the one whose Lagrangians are
linear in the field velocities (gravity, fermions, Chern-Simons,
topologial field theories). That I won't say anything more about
here...
