Rock Brentwood
05.28.08, 05:00 AM
> > On May 15, 4:51pm, yy...@fas.harvard.edu wrote:
> > > As far as I understand, the gauge group for Ashtekar variable theory for General Relativitiy is SU(2), because its group is three-
> > > dimensional.
> >
> > Ashtekar's formulation is just the Palantini formulation (i.e.
> > Einstein-Hilbert) with an additional term. The extra term is a local
> > Lorentz invariant, as is the Palantini term. Everything takes place
> > within an Einstein-Cartan geometry, so has local Lorentz invariance.
> > The symmetry group is the connected part of SO(3,1).
> >
> > > But, why is this SU(2) instead of SO(3)?
> >
> > SO(3,1) splits into SU(2) + i SU(2), via the well-known (Spin + i
> > Boost) decomposition. The +/- i signals parity. So, taking the left or
> > right helicity part (respectively, anti-self dual and self dual) of
> > the Lagrangian gives you the Lagrangian the "new variables" work with.
> >
> > It's not SU(2), ultimately, but SO(3,1).
> > > As far as I understand, the gauge group for Ashtekar variable theory for General Relativitiy is SU(2), because its group is three-
> > > dimensional.
> >
> > Ashtekar's formulation is just the Palantini formulation (i.e.
> > Einstein-Hilbert) with an additional term. The extra term is a local
> > Lorentz invariant, as is the Palantini term. Everything takes place
> > within an Einstein-Cartan geometry, so has local Lorentz invariance.
> > The symmetry group is the connected part of SO(3,1).
> >
> > > But, why is this SU(2) instead of SO(3)?
> >
> > SO(3,1) splits into SU(2) + i SU(2), via the well-known (Spin + i
> > Boost) decomposition. The +/- i signals parity. So, taking the left or
> > right helicity part (respectively, anti-self dual and self dual) of
> > the Lagrangian gives you the Lagrangian the "new variables" work with.
> >
> > It's not SU(2), ultimately, but SO(3,1).