## Re : The Higgs is necessary (was: Higgs found?)

Originally from, s.p.r. 2000 September 19:

John Baez wrote:
> In article <8q2vm5$34c$1@uwm.edu>,
> Mark William Hopkins <whopkins@alpha2.csd.uwm.edu> wrote:
> >I think the actual mechanism is much simpler: it's nothing more than
> >the effects of the g_55 component of a 5-dimensional vacuum metric
> >being seen in terms of an effective field theory representation in
> >4 dimensions.

>
> So: have you found a scenario where you get a dilaton that interacts
> with fermions and the W and Z in a way that could mimic the Higgs?

You could write down the generators for GL(5) in a more revealing form
by taking K^a = x L^a_5, P_b = L^5_b, M = -x L^5_5. Then the
commutators
[L^a_b, L^c_d] = delta^b_c L^a_d - delta^a_d L^b_c
ranging over (a,b,c,d = 1,2,3,4,5) yield
[L^a_b, K^c] = delta^c_b K^a
[L^a_b, P_d] = -delta^a_d P_b
[L^a_b, M] = 0
[K^a, K^c] = 0 = [P_b, P_d]
[K^a, P_d] = x (L^a_b) + delta^a_b M
[K^a, M] = -x K^a
[P_b, M] = x P_b
This is very similar to the conformal group.

In the limit (as x -> 0), you get a representation for an inhomogeneous
GL(4). The M generator becomes a Casimir invariant which might then be
identified as mass. The above is also similar to the conformal group
with M playing the role of the dilation generator and K^a the generator
of conformal transformations.

One interesting question that also arises is what does the gauge theory
for GL(5) or for the inhomogeneous GL(4) look like? The L generators
give you the spin connection, and P the tetrad. If you identify the
co-tetrad with K, you can actually build up a metric (provided you
write in constraints on the gauge fields by hand) because its world
index is already lowered. The gauge potentials for M are somewhat of a
mystery.

> I don't see how to do this. There are lots of different strategies,
> but all of them seem to have obstacles. For starters, where's your
> SU(2) x U(1) gauge theory coming from?

An idea I've been having recently is to move away from SU(2)_I x U(1)_Y
to SU(2)_I x U(1)_Y x U(1)_{B-L} and then to consider, instead, its
subgroup SU(2)_I x U(1)_{Y-(B-L)/2}, which fits real nicely into a
SU(2)_I x SU(2)_R ~~ SL(2,C).

So, now comes in that rarely-discussed Wigner class corresponding to
0-momentum and 0-energy, which apparently doesn't even have a name
("vacuon" seems obvious for a name). Its residual symmetry group is
just SL(2,C).

Maybe there's a way of explaining away an SU(2)_I x SU(2)_R as
dynamically generated, analogous to the way that SU(2) x SU(2) symmetry
is dynamically generated (with constraints) from the Kepler problem.

Alternatively, maybe the Higgs vacuum, itself, can be considered as a
true vacuum, rather than a false vacuum, but one that is represented as
a "vacuon", rather than as the Poincare-invariant vacuum state that the
Wightmann formalism postulates.

These two ideas are not mutually exclusive.

> (I'm ignoring QCD for now to keep life simple

That, then, has to be extended from SU(3) to U(3) = SU(3) x U(1)_{B-L}
in order to recover Y.

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