The Resolution of the UV Divergence Problem II
- From: markwh04@xxxxxxxxx
- Date: Wed, 2 May 2007 02:40:51 +0000 (UTC)
Part 2: The Generalized Principle of Relativity
Continuing from the previous article...
Here, we'll focus on one of the more important consequences that is
necessarily entailed by the standard mathematical formulation
underlying the general notion of velocities (Jet bundles) -- the
complete relativity of velocities and the necessity of generalized
connections.
The upshot of what's to follow is that the solution to the ultraviolet
divergence problem is that:
* There are no quadratic couplings in the field Lagrangian at all.
* Correspondingly, the field law is not linear, except
asymptotically.
* The couplings are all at least cubic and only asymptotically
quadratic.
* The free field propagators are therefore only valid as asymptotic
approximations.
* Combined with the generalized connection that mediates velocity-
momenta relations, this produces a total space metric in the spirit of
Kaluza-Klein.
A more interesting exercise working off the general framework
developed here will be to systematically translate the mathematics of
the microlocal-Causal-Epstein-Glaser approach into the formalism
below. For instance, the effective dielectric vacuum produced by the
Heisenberg-Euler Lagrangian, for instance, will be of the following
form, in MKS units:
D = epsilon E + lambda B
H = B/mu - lambda E
where
epsilon/epsilon_0 = mu_0/mu = 1 + 2 Z I1
lambda = 7 Z I2
and
Z = 4/45 e^4 h-bar/(4 pi epsilon_0)^2 (1/m_e^4 c^7)
I1 = epsilon_0 E^2 - B^2/mu_0
I2 = epsilon_0/mu_0 E.B,
involving the electron mass m_e.
However, I don't know how this links up to the counterterms arrived at
in microlocal analysis, other than that the Minkowski-space limit of
the dynamics spelled out below will clamp down on *some* set of actual
values for the counterterms and entirely remove the ambiguities (as
well as giving you the effective field classical limit of actual sum
that the perturbation expansion approximates).
========
(1) General Connections and the Relativity of Field Velocities
The velocities of fields resides in an affine space and is therefore
"relative". As such, *any* velocity-momentum relation in field theory
must involve a connection. When this missing piece of the big picture
is combined with the other missing piece (raised in the previous
article) of the field metric e_{ab}, the result is that a total space
metric can be defined, much in the spirit of Kaluza-Klein, but
generalizing from "gauge fields" to all fields.
Hence, the foundation may arise for a bona fide unified field theory
at the classical level specifically defined to address and resolve the
UV divergence problem at both the classical and quantum level.
Fields live in a bundle (Q->X) over a base space X, which plays the
role of the space-time manifold. The field velocities live in the
bundle J1(Q). A connection is a section of the bundle (J1(Q) -> Q).
This bundle is an affine space! Inevitably, this means the appearance
of a connection.
The affine nature can be seen by its coordinate transformation
properties. Locally, the coordinates are given by
X: (x^m); Q: (x^m, q^a); J1(Q): (x^m, q^a, v^a_m).
The connection is locally given as a function of the coordinates and
velocities
A: (x^m,q^a) |-> (x^m,q^a,A^a_m(x,q)),
which generalizes the connections of principal and associated bundles.
The transformation law is that derived from the transformation of the
fields and their velocities. Thus, given a field (i.e., a section over
(Q->X)),
q: (x^m) |-> (x^m, q^a(x)),
one has the corresponding extension
j1(q): (x^m) |-> (x^m, q^a(x), d_m q^a = dq^a/dx^m).
The bundle transformation
x -> X = A(x), q -> Q = B(x,q)
yields the transformed field
Q: (X^n) |-> (X^n, Q^b(X))
given by
Q(A(x)) = B(x, q(x)).
The corresponding velocity components V^b_n are then given by
dQ^b(X) = V^b_n dX^n
hence
d(B^b(x,q(x)) = dQ^b(A(x)) = V^b_n dA^n/dx^m dx^m
and
dB^b/dx^m dx^m + dB^b/dq^a dq^a/dx^m dx^m = V^b_n dA^n/dx^m dx^m
or
dB^b/dx^m + dB^b/dq^a v^a_m = V^b_n dA^n/dx^m.
dQ^b/dx^m + dQ^b/dq^a v^a_m = V^b_n dX^n/dx^m,From this, the general transformation property is read off:
which is an affine transformation.
Thus (J1(Q) -> Q) is the "affinization" of the vector bundle given by
the bundle tensor product
T*M x_Q VQ.
The space of velocities is therefore NOT a vector space! There is no
absolute zero for velocities -- not even field velocities!
This, of course, is none other than the statement of the principle of
Galilean relativity generalized to field components: "velocities are
relative".
Of necessity, this entails the involvement of a connection in any
relation that links velocities to momenta, since momentum components
reside in the cotangent bundle, which IS a vector bundle. Thus, the
correct relation for an otherwise linear field must be of the form
like the following:
p^m_a = root(|g|) g^{mn} e_{ab} (v^b_n + A^b_n),
where A is the connection; and is NOT of the form
p^m_a = root(|g|) g^{mn} e_{ab} v^b_n
as is asserted in the theories of linear free fields.
A more general linear relation may be of the form
p^m_a = e^{mn}_{ab} (v^b_n + A^b_n),
where it need not factor cleanly. But the connection must still be
present.
========
(2) The Combined Metric
The argument made in the previous article generalizes Maxwell's
observations. Maxwell made a clear distinction between the fields
produced by a source (D, H) versus those felt by sources (E, B) (as
did later observers, like Einstein in his earliest days and Hehl et.
al. nowadays). In the more general setting here, (D, H) have lower
indices (D_a, H_a); and (E, B) have upper indices (E^a, B^a), being
derived from the potentials (A^a,phi^a). Here, (D,H) become (p^m_a),
(E,B) become (v^a_m) and the (A,phi) become (q^a).
A source does not directly experience its own (D,H) field,
particularly since these approach infinity. It experiences (E,B),
which remains well-defined at the source. Thus, no such relation of
the form
(D = epsilon_0 E; B = mu_0 H)
can exist near the source! Instead, the ratios |D|/|E|, |H|/|B| must
approach infinity.
Indeed, this is already seen to the first order in QED through the
approximation known as the Heisenberg-Euler Lagrangian, where one
finds relations of the form
D = (epsilon_0 + a h-bar E^2) E + b h-bar (E.B) B
H = (1/mu_0 + c h-bar B^2) B - d h-bar (E.B) E
for certain constants a, b, c and d.
In the more general case of fields, the same reasoning applies. In
order for the stress tensor
T^m_n = p^m_a v^a_n - delta^m_n L
to remain well-defined near sources, the constant-coefficient linear
relation that holds asymptotically between p and v must break down and
become a variable linear relation with the coefficients (e_ab) and/or
(g^mn) showing up as fields. In the Minkowski limit, if one wants a
consistent field theory, this necessarily entails making (e_ab) a
field!
These two points complement each other, each providing the missing
piece for the other, to give you a unified metric for the fields:
h_mn = g_mn + e_ab A^a_m A^b_n
h_mb = e_ab A^a_m
h_an = e_ab A^b_n
h_ab = e_ab
Hence, the ultimate tie-in to the original article the previous
article was in reply to: Kaluza-Klein as a basis for a unified field
theory.
========
(3) A Generalization of Kaluza-Klein
The question of dynamics remains to be addressed...
*IF* one assumes that the Einstein-Hilbert Lagrangian applies in this
context, the result will be a dynamics for the extra fields (A^a_m)
and (e_ab). Taking the Minkowski space limit, one will arrive at an
effective flat-space dynamics for what will generalize Maxwell's
theory of the dielectric vacuum to a corresponding theory of
"dielectric" vacuua for general fields, in which the quadratic
relations between velocities and momenta are smeared out along with
sharp singular propagators in whatever field theory results from
quantization.
The upshot of all this is that the solution to the ultraviolet
divergence problem is that:
* There are no quadratic couplings in the field Lagrangian at all.
* Correspondingly, the field law is not linear, except
asymptotically.
* The couplings are all at least cubic and only asymptotically
quadratic.
* The free field propagators are therefore only valid as asymptotic
approximations.
Thus, Einstein was dead-on right all along,
"At the present time the opinion prevails that a field theory must
first,
by 'quantization', be transformed into a statistical theory of
field
probabilities according to more or less established rules. I see
in this
method only an attempt to describe relationships of an essentially
nonlinear character by linear methods."
-- The Meaning of Relativity, 5th Edition (1956)
As of yet, I don't know if an Einstein-Hilbert Lagrangian on the total
metric will produce a classical theory possessing a quantization that
yields the correct results on near-field probes, such as the Delbrueck
scattering relation.
A more interesting exercise is to systematically translate the
mathematics of the microlocal-Causal-Epstein-Glaser approach into the
framework of the theory of "effective dielectric vacuua" or "e_ab
metrics", along the lines spelled out above. The effective dielectric
vacuum produced by the Heisenberg-Euler Lagrangian, for instance, will
be of the following form, in MKS units:
D = epsilon E + lambda B
H = B/mu - lambda E
where
epsilon/epsilon_0 = mu_0/mu = 1 + 2 Z I1
lambda = 7 Z I2
and
Z = 4/45 e^4 h-bar/(4 pi epsilon_0)^2 (1/m_e^4 c^7)
I1 = epsilon_0 E^2 - B^2/mu_0
I2 = epsilon_0/mu_0 E.B
with Z being the first-order approximation that takes into account the
contribution of the dielectric vacuum brought about by electron-
positron polarization; with m_e being the electron mass.
.
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