Re: gauge transformations: local, global, large and asymptotically non-trivial

Darth Sidious skrev:

> >
> > [local, global] = local
> > [local, local] = local
> >
> If I understand you correctly you want to put
>
> Local |state> = 0,
>
> and this is compatible with the algebra.

In standard treatments of gauge theories, one wants to construct the
Hilbert space of physical states. In Dirac's approach, a physical
state |phys> is characterized by

Gauge |phys> = 0

|phys> ~ |phys> + Gauge |any state>

This evidently cannot apply to global gauge transformations, since
they include the charge operators. But it does apply to local
transformations, as long as no divergent ones are around.

> > whereas the local+global+divergent algebra reads
> >
> > [J^a_m, J^b_n] = f^ab_c J^c_m+n + k m delta^ab delta_m+n,
>
> How do you get this one?

The bracket above defines a Lie algebra (check anti-symmetry and
Jacobi identities), and its restriction to 0 >= m,n >= -infinity
doesn't remember the last term. So any value of k is as natural as
k = 0. In fact, this is the unique central extension of the algebra of
Laurent-polynomial gauge transformations. It is called an affine
Kac-Moody algebra.

Moreover, the only unitary irrep of the algebra with k = 0 is the
trivial one. If you want non-trivial unitary irreps, you must set
k > 0; with suitable normalization, the level k is a positive integer.

>
> > This is a significant difference between how gauge transformations are
> > treated in perturbative string theory and in YM theory. In string
> > theory, we consider the local+global+divergent version of the gauge
> > algebra, e.g. the Virasoro algebra, but in YM theory one restrict
> > attention to the local+global version.
>
> Why are they treated so differently? I was under the impression that
> the main idea should be the same.

Yes, this was my point. For some time, I have been trying to
understand why anomalies arise in string theory but not in YM theory,
and I think that this difference is the reason. If we do consider
divergent transformations in YM theory, we can no longer assume that
local transformations annihilate physical states. As a result,
classical gauge degrees of freedom become physical at the quantum
level.

Consider e.g. the electromagnetic field, which is described by a gauge
potential A_u with four components. Classically, the longitudinal and
temporal polarizations are gauge, so only the two transverse
polarizations are physical; the photon has two polarizations. This
remains true in the free Maxwell theory, because the photon field is
uncharged. However, in QED we couple the photon to charged particles,
and the argument above applies. This means that three photon
polarization are physical in the interacting theory, which may be
phrased as virtual photons may have unphysical polarizations. At least,
this
is my understanding of the situation.

.

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