Effective Field Theory & Curing UV Divergence

Rock Brentwood

Effective field theory arrives at the right conclusions, that I think
are for the wrong reasons.

Its point of view is that because that the quantum theory is broken
which is rooted in a classical field theory that possesses the Lorentz
relations (for electromagnetism) or, more generally, rooted in the
classical field theory with a Lagrangian quadratic in the field
velocities with constant quadratic coefficients; then this broken
condition means that it is actually an effective low energy limit of a
high energy theory (and one that may not even be a field theory at
all), when all is said and done. Nonetheless, we can seal off the low
energy level from the high energy level and treat the effective field
theory in a quasi-self-contained fashion by simply asking the
question: what is the most general Lagrangian consistent with
spacetime and gauge symmetries?

In contrast, Maxwell's resolution of the divergence problem -- and its
modern-day descendant -- pays no need to the issue of quantum theory
at all. It says: let's pose the same question of the CLASSICAL field
theory, instead. The modern-day extension of this idea goes one step
further: it says that maybe if we just start out with the right
classical theory, in the first place, then we can resolve the
divergence issue at its root -- at the classical level -- and THEN
move on to a quantum theory that is in a proper state of repair. The
final point reached is precisely the same as that of effective field
theory: we are once again at the point of asking what the most general
Lagrangian is that is consistent with spacetime and gauge symmetries.
Except that: (a) we're reaching this point along the right path now
from the right direction; and (b) we no longer regard the result as an
"effective field theory", but rather the ACTUAL field theory.

In the case of electromagnetism, this leads to a Lagranian L = L(I, J)
as a function of the Lorentz invariants I = 1/2 (E^2 - B^2 c^2) and J
= E.B. The coefficients, epsilon = dL/dI and theta = dL/dJ then play a
central role in the field theory. The first coefficient, epsilon, is
none other than the cornerstone of Maxwell's theory of the universal
dielectric medium. Not only does he employ the device of epsilon to
remove the self-energy and self-force divergence from the classical
theory, but he bases his whole idea on it of the vacuum as an active
medium with a structure described by epsilon. At the end of Chapter 1
in his treatise he points out that we should now be seeking to find
what structure and dynamics epsilon conforms to, if we want to get a
deeper understanding of electromagnetism that goes beyond the field
law.

Unfortunately, theta never entered Maxwell's considerations because --
when all is said and done -- Maxwell did NOT explore the issue of what
the most general field theory is that is consistent with the
underlying spacetime symmetry. Under the Galilei group, the most
general field theory that possesses Maxwell's universal velocity
vector G is given by the following constitutive law:
D = epsilon (E + G x B) + theta B, B/mu = H - G x D + theta E
where epsilon, mu and theta are indepdendent coefficients. In the
absence of G, the most general law that can be stated is only of the
form:
D = theta B, B - theta E = mu H.
One needs G in order to account for the dielectric coefficient. Thus,
Maxwell's theory of the dielectric medium was inextricably mixed up
with an otherwise independent theory of the vacuum as a universal
reference medium.

So, when Lorentz substituted the Lorentz group for the Galilei group,
he not only threw out the velocity vector G, but also eliminated
epsilon and wrote:
D = epsilon_0 E, B = mu_0 H, epsilon_0 mu_0 = 1/c^2.
But, if you were to ask what the MOST GENERAL Lorentz-invariant
constitutive law for the vacuum is, you get the following:
D = epsilon E + theta B, H = epsilon c^2 B - theta E
where epsilon = epsilon(I, J) and theta = theta(I, J) are otherwise
arbitrary functions of the Lorentz invariants.

In it, light propagation is already accounted for. The null field is
characterized by I = 0 = J.
Without loss of generality, we can set theta(0, 0) = 0, since theta is
already only given up to an additive constant (a residue of complexion
symmetry). Defining epsilon_0 = epsilon(0, 0), one arrives at the
following dynamics for a null field:
E^2 = B^2 c^2, E.B = 0, D = epsilon_0 E, B = mu_0 H, epsilon_0 mu_0
= 1/c^2.
Substituting in source-free Maxwell's equations
del.D = 0 = del.B, del x E + dB/dt = 0 = del x H - dD/dt
yields the wave equations for all the fields:
((1/c)^2 (d/dt)^2 - del^2) { E, B, D, H } = 0.

This applies generally for ABELIAN gauge fields, where the only
Lorentz invariants are I and J ... except that here I and J are GxG
matrices, where G is the dimension of the underlying gauge group. the
coefficients epsilon = dL/dI then become, up to factors of c, the
components of the gauge group metric k. In particular, k = epsilon c.

It is not necessarily the case, however, for NON-ABELIAN gauge fields,
since one also has cubic Lorentz invariants in spacetimes of dimension
4 or greater. These need not be 0, so that the constitutive
coefficients need not be at their asymptotic values, even for null
fields.

There is a much more complicated classification by invariants even for
the SU(2) field, and one is no longer talking about a mere 3-way split
into (0 vs. null vs. non-null).