Hidden Metrics, Field Divergences & Finite Quantum Field Theories

markwh04@yahoo.com

Dated 2001 January 24
http://groups.google.com/group/sci.p...e=source&hl=en

semorrison@hotmail.com wrote:
> I've been wondering about the usual prescription for the Lagrangian in
> Yang-Mills theory, defined in terms of the Lie algebra valued 2-form
> curvature. With some coefficient, this is written as the integral of
> Tr (F^{m n} F_{m n}).
>
> So - some questions.
> 1) Why is it that we take the _trace_?

When written out explicitly in component form, the Lagrangian takes on
the form
L = -1/4 F^{mn}_a F_{mn}^a.
The fields F_{mn}^a generalize the Maxwell (E,B) fields; while F^{mn}_a
generalize (D,H).

The relation normally assumed between the two sets is ultimately
derived historically from the hypothesis Lorentz asserted regarding the
respective fields: the Lorentz relations D = epsilon_0 E, B = mu_0 H.

In explicit component form, the corresponding relations for a
Yang-Mills field would then read D^a = epsilon^{ab} E_b; B_a = mu_{ab}
H^b. Here, one sees that despite their superficial similarity, one is
dealing with two very different kinds of objects -- a distinction that
was not as clearly seen for the Maxwell fields because of the extra
index not being explicitly written out for a U(1) field. The (E,B)
objects are Lie valued, while (D,H) lie in the *dual* of the Lie
algebra.

To relate the two, thus, requires a metric -- k_{ab} with inverse
k^{ab}. This is what generalizes epsilon and mu. Under the assumption
of linear duality, then, the relation is written out explicitly as
F^{mn}_a = g^{mr} g^{ns} k_{ab} (det(g))^{1/2} F_{mn}^a
(noting that D and H are actually tensor densities).

What you're calling the "trace" is actually a metric. Generally,
whenever one sees a "trace", there is always a metric of some form.

This, of course, leads to the question: is the metric part of the
background, or does it, too, participate in the dynamics as a field?
Part of the point of working out the 5-D point-like singularity for a
U(1) field with a variable metric k (note the recently posted thread
describing the 5-D black hole metric) is to show off the split between
the two approaches to a Kaluza-Klein theory. In one, k is fixed,
essentially as the Killing Metric. The higher dimensional field is then
forced to not be a Ricci vacuum. In the other, the higher dimensional
field is a Ricci vacuum, but k varies.

In the latter, one sees the beginnings of what looks like the classical
version of the very type of smoothing, "running of the couplings", a
non-trivial dielectric structure to the vacuum, that had been
hypothesized from the time of Maxwell (starting with Maxwell), onwards
to its revival in the 1940's under the 20th century guise of
"renormalization theory".

Going back, it's not to hard to see where the ultimate problem lay ...
both with classical theory and its quantized version. IF ... (a) D and
E are linearly related, even near point-like sources (as Lorentz had
hypothesized), then in the presence of the other relations, (b) the
Gauss law (rho = div D), and (c) the force law (F = rho E for force
density), one has a problem that amounts to a no-go.

Essentially the argument Maxwell had made is that point and line like
singularities in the sources cannot occur and would be smeared out by
the polarization in the surrounding space, leading to a screening of
the actual charge. What happens in field theory is that the global form
of the force law F_V = q_V <E>_V, for a force acting on a volume V of
charge q_V only relates the AVERAGE field <E>_V acting on the volume
from without. As soon as you localize it to a force density law, F =
rho E, you also end up adding an extra term corresponding to the
force of the source on itself. The result is an infinity where there
had been none before.

The product rho E, when localized, therefore is actually more along the
lines of a distributional form with (rho D/epsilon_0) only as an
approximate kernel, but one which is smeared out near point-like
sources.

So, the argument is quite simple. Wherever there is a point or line
like singularity in the source rho (or a concentration that
approximates such a singularity), then in virtue of (b) there must also
be one for D, there too. But in order for the force law (c) to be
well-defined kernel yielding well-defined integrals F_V, then per
force, E must be regular wherever rho goes singular.

The combination of (b) and (c) therefore precludes any such relation
(a) holding near a point source. Linear duality is unphysical! Either
the fields are related in a more complex way (i.e. non-linear
electrodynamics) or the extra hidden element that had been present
there all along (k or epsilon) is actually a dynamic quantity and not
part of the background. These two cases are not mutually exclusive.

The issue of hidden metrics, as was briefly mentioned, is far more
prevalent than just with Yang-Mills theory. It occurs ANYWHERE you have
a bilinear form or trace entering in the picture. So, what other places
do you see this occur?

A scalar field is coupled through a bilinear form to yield the
Lagrangian. When dealing with a multicomponent scalar field, there is
therefore a hidden metric. Explicitly writing out the field as phi^a,
one has a Lagrangian
L = 1/2 P^m_a A^a_m - 1/2 m^2 F_a phi^a
where, here, the roles analogous to (D,H) are now played by the dual
fields (P,F), with the spin 0 analogue to Maxwell's equations relating
the potential phi^a to the "velocities" A^a_m:
@_m phi^a = A^a_m
(@ being used to denote partial derivatives), and the homogeneous
equations analogous to the homogeneous Maxwell equations
@_m A^a_n - @_n A^a_m = 0
(in 3+1 form: curl A = 0, -@A/@t - grad phi = 0).

The dual fields are now
P^m_a = g^{mr} (det g)^{1/2} e_{ab} A^b_r, F_a = e_{ab} (det
g)^{1/2} phi^b.

The same consideration applies with respect to the field infinity -- as
well as the same conclusion. Either (P,F) are related by a more complex
non-linear or even distributional relation to (phi, v) or the scalar
field metric e_{ab} is a dynamic variable.

One also finds a bilinear form appearing with the fermion field,
largely hidden through the notation of the Dirac conjugate, with the
Lagrangian written as
L = (det g)^{1/2} psi^- (i D - m) psi.
For gravity, the bilinear form is even more in disguised form, with the
Lagrangian taking on the form
L = p^{mn} R_{mn}
with
p^{mn} = k (det g)^{1/2} g^{mn}.

In all these places, you will see a hidden metric of some sort
involved, relating the dual sets of fields. In each case, the naive
assumption of this being a simple linear relation with the metric fixed
-- which is the modern equivalent of Lorentz's hypothesis -- introduces
the field infinity in the very same way that Lorentz did with the
localization of the field law to form (c).

In each case, this simple linear relation can only be an approximation
of what's really going on. The field theory the effectively emerges as
a product of the renormalization process tries to compensate for this,
recovering a semblance of the non-linear relation. If one starts out,
instead, at the classical level with a modified field theory that
explicitly incorporates either (d) a non-linear relation between the
dual sets of fields (or even a distributional relation) or (e) a
dynamic metric (or both (d) and (e)), so as to remove the field
infinity from the classical theory; then this should also provide a
more consistent starting ground that will result in a quantized field
theory where the infinity is not present either and where the result
that renormalization attempts to arrive at emerges automatically
without any further need for correction.

Matej Pavsic

markwh04@yahoo.com wrote:
>
> Dated 2001 January 24
> http://groups.google.com/group/sci.p...e=source&hl=en
>
>
> In each case, this simple linear relation can only be an approximation
> of what's really going on. The field theory the effectively emerges as
> a product of the renormalization process tries to compensate for this,
> recovering a semblance of the non-linear relation. If one starts out,
> instead, at the classical level with a modified field theory that
> explicitly incorporates either (d) a non-linear relation between the
> dual sets of fields (or even a distributional relation) or (e) a
> dynamic metric (or both (d) and (e)), so as to remove the field
> infinity from the classical theory; then this should also provide a
> more consistent starting ground that will result in a quantized field
> theory where the infinity is not present either and where the result
> that renormalization attempts to arrive at emerges automatically
> without any further need for correction.

A discussion that might be relevant to that proposal, although
not quite the same, can be found at:

http://arXiv.org/abs/gr-qc/0610061
http://arXiv.org/abs/hep-th/0501222
http://arXiv.org/abs/gr-qc/0507053

I think that these works could be useful for those who would
like to pursue the program raised above.

Matej Pavsic
________________________________________
Home Page http://www-f1.ijs.si/~pavsic/