Gravity as a Goldstone-Higgs field, instead of a Gauge Field
- From: markwh04@xxxxxxxxx
- Date: Sun, 5 Nov 2006 17:07:01 +0000 (UTC)
Jack Sarfatti wrote:
"The Question is: What is The Question?" John Archibald Wheeler
OK I have been reading the "philofawzy" of the quantum gravity
literature and I dimly realize a few things:
1) When The Pundits talk about "gauge invariance" in GR they do not mean
what I mean (local gauging of Poincare group to get renormalizable SPIN
1 curved tetrads and torsion spin connections as compensating gauge
potentials like in Yang-Mills internal symmetries). They work at the
spin 2 geometrodynamic level of the ADM formalism and get stuck on all
kinds of physical nonsense such as:
The concept of gravity as a Poincare' gauge field is problematic, at
best. This is best explained as follows, which reflects the
introductory passage of "Gauge Gravitation Theory"; G. Sardanashvily
and O. Zakharov, World Scientific 1992
The geometric nature of gravity (as first discovered by Einstein)
arises as a direct result of the Equivalence Principle. In the context
of Galilean symmetry, it would lead to the notion of Newton-Cartan
spacetimes. In the context of Lorentz symmetry, it leads to General
Relativity. Reformulating the latter in terms of fibre bundles as a
gauge theory helps to further elucidate the nature of gravity field as
being a Higgs-Goldstone field corresponding to the spontaneous
breakdown of world symmetries. Specifically, it is the fermions that
brings about and makes necessary this extra element.
The main problem with thinking of gravity as a gauge theory is that the
field involves the metric/tetrad, whereas gauge potentials in gauge
theory are connections. The conventional scheme is to equate the tetrad
components h^m_a to the gauge potentials A^m_n associated with the
translation generators in the Poincaré group and to try and fit
gravity within the mould of a Poincaré gauge theory.
Needless to say, there are a lot of problems with this idea. First and
foremost, the Poincaré group isn't compact or semi-simple. So, you
lose much of the important machinery associated with gauge theory,
which relies on this property. Second, the whole exercise loses sight
of the analogous role that Higgs-Goldstone fields play in spontaneous
symmetry breaking, in gauge theory, and fails to note the analogy
present here. Third, the fit isn't all that good: holonomic
transformations fail to be reproduced within the framework, and
different types of gauge transformations (atlas transformations,
principal morphisms, gauge freedom transformations, etc.) cannot be
discerned.
Continuing on from there...
The authors go on to develop a general framework based on the notion of
generalized connection. An ordinary connection resides within the
well-known theory of principal bundles; or within a bundle associated
with a principal bundle. The latter are called principal connections.
A matter field resides within an associated bundle. However, one also
has a general concept of "connection" for bundles, independent of any
prior notion of principal connections. This is seen as follows.
A bundle Q over a base space M provides you, locally, with a set of
coordinates (x^m, q^a) = (x^1,...,x^n, q^1,...,q^N) describing (for
instance) a system of N degrees of freedom within a spacetime of n
dimensions. As soon as one writes down a first order law for the system
configuration, one is also bringing in the "velocities" (v^a_m =
@q^a/@x^m) -- the resulting space J^1(Q) with coordinats (x, q, v) is
called the "first jet" and is a bundle in two ways:
(1) over the base space J^1(Q) --> M
(2) over the configuration space J^1(Q) --> Q.
The latter is particularly of interest. For a velocity field A = v(x,q)
when thought of as a function of both the coordinates and configuration
is then represented as a SECTION A:: Q -> J^1(Q) in the latter bundle.
This is the general concept of a connection.
The key result related to these concepts is that if the matter field
has a gauge symmetry and Q is thereby associated with a principal
bundle P that represents that symmetry, then there will also be a
"principal connection" Gamma on Q inherited from P. But it won't
generally coincide with a general connection A on Q. Instead, there
will be a decomposition
A = Gamma + Sigma,
the latter being called a "soldiering form".
In the particular case of interest, the fields are fermion fields, the
gauge symmetry is the local frame SO(3,1) symmetry of the Lorentz
group, and the principal connection is the "spin connection" out of
which ultimately the connection coefficients relating to gravity arise.
But the general connection also has a soldiering form ... and it is out
of this that the tetrad ultimately arises.
The tetrad does NOT provide the gauge potentials of a set of
translation generators of the Poincaré group. It provides the
parameters for the Goldstone-Higgs field associated with the breaking
of the global GL(4) symmetry down to the local SO(3,1) symmetry of
fermion fields.
The tetrad part of gravity is neither a gauge field, nor a part
thereof; but a Goldstone-Higgs field. Among other things, it is
essentially a classical field, does not admit quantization in any of
the usual ways and (in fact) parametrizes separate state spaces and
separate "vacuum phases" associated with each state space. Each
different setting of the tetrad corresponds to an inequivalent state
space, and between any two of these spaces there are no coherent
superpositions.
Despite the book's dating from 1992, this whole line of investigation
represents the outgrowth of a long-standing thread of research going
back to the 1970's or before and it is still active at present, as a
search on arXiv will show. It is quite friendly to the somewhat-related
line of development that's taking place with the "covariant
Hamiltonian" or "polymomentum" or "deDonder-Weyl/Lepagean" approach
which replaces quantization by a 3+1 Hamiltonian by one with respect to
a general covariant "Hamiltonian".
.
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