Conformal divergence
- From: DRLunsford <antimatter33@xxxxxxxxx>
- Date: Sun, 20 Nov 2005 22:45:12 +0000 (UTC)
I've been thinking about conservation laws in conformal space and the
following idea came up.
In Riemannian geometry, given a vector Jm, one can always interpret
Jm;m = 0
as a conservation law, because of the special character of the
connection, for which
Crm,r = d/dxm log sqrt det(g) = d/dxm S
This is the "magic formula" that allows one to write
Jm;m = 1/S d/dxm (S Jm)
If we now bang this against S dV in an integral over a volume, we get
something that can be converted into a surface integral by Stoke's
theorem. So we can make a conservation law out of Jm;m only because it
can be reduced to ordinary derivatives.
What about in Weyl space? There is an analogous magic formula
Cr,mr = d/dxm log sqrt det(g) + N/2 Am
where Am is the Weyl gauge field and N is the dimension of the space.
Now the natural operation of differentiation in Weyl space is the
"conformal covariant derivative"
Dm Tab.. = (dm + N Am) Tab..
where dm is the ordinary covariant derivative with respect to C, and N
is the conformal weight of Tab. Now the J formula reads
Dm Jm = 1/S d/dxm (S Jm) + (N/2 + M) JmAm
where M is the weight of Jm. Thus we can convert an integral of this
"conformal divergence" over an N-volume into a surface integral, so
making a conservation law, only if
N/2 + M = 0
that is, the current Jm must be weight -N/2. Thus onty in 4-d is a
Lagrangian like
1/4 FmnFmn + JmAm
conformally justified, because the terms have equal weight. The
interaction term JmAm then represents a "conformal deficit" (difference
between conformal and ordinary covariant divergence) and is unique to
4-dimensions. Also, the form of Maxwell's equations
dF = J
is thus only strictly conformally possible in 4-d. In 6-d one would
need to bring in an extra factor of weight -1 on the first term, but
the only meaningful one at hand is the Weyl-Ricci scalar R, and so
write
1/4 R Fmn Fmn + Jm Am
etc.
In 10 dimensions one needs to make a term of weight -3 to bang against
F^2 - you could try
1/4 R^3 F^2 + J.A
or
1/4 R F^4 + J.A
Needless to say, the field equations of these theories are a disaster,
and they get worse with each step up the ladder of dimensions! Note
that odd dimensions are ruled out completely, because the weight of J
is non-integral.
The above analysis may seem good for 4-d, but in fact it's fatal to it
because there is no hope of going beyond the simple interaction term Jm
Am. This is the mathematical basis of the failure of his theory in 4-d.
-drl
.
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