When we perform a 3+1 decomposition of GR (let us assume the sppatial
part is compact without boundary to make it simpler), we obtain
something which is the so called gravitational momenta. This momentum
has two components, one is the bare metric velocity [/tex]\dot g[tex]
and the other involves the so called shift vector, and is basically
the Lie derivative of the metric in the direction of this shift
vector. So we have [\tex]\dot g-L_\xi g[tex]. Now, it looks a lot like
the electromagnetic momenta, where we have an extra term to the
original momenta, which is the electromagnetic potential. But
apparently, unlike in electromagnetism (if I am not mistaken), under
the  quantization procedure for the  canonical coordinates, one simply
quantizes the whole thing  [\tex]\dot g-L_\xi g[tex] as the functional
derivative along the metric coordinate [\tex]g_{ij}[tex]. So, does one
set the shift to zero before quantizing or is this the way it should
be done even with the shift? If so, why is it different from
electromagnetism?

The quantization strategy is actually the same in both
cases. One way to think about this is that (for typical fields)
the canonical momenta are always defined as the
Lie derivative of the field along the *unit normal* to
the hypersurfaces of your spacetime foliation.
So, for the EM field, or for a scalar field, or whatever,
the shift vector is actually there, too (unless you've
set it to zero with a special choice of foliation).

The shift vector term in the gravitational momentum
is also very analogous to the gradient of the scalar
potential term in the EM momentum if you make
the analogy between spatial diffeomorphisms in GR
and gauge transformations in EM. But that's another
story...

charlie torre

On Jun 17, 12:33 pm, Henrique de Andrade Gomes <gomes... at (no spam) gmail.com
wrote:



When we perform a 3+1 decomposition of GR (let us assume the sppatial
part is compact without boundary to make it simpler), we obtain
something which is the so called gravitational momenta. This momentum
has two components, one is the bare metric velocity [/tex]\dot g[tex]
and the other involves the so called shift vector, and is basically
the Lie derivative of the metric in the direction of this shift
vector. So we have [\tex]\dot g-L_\xi g[tex]. Now, it looks a lot like
the electromagnetic momenta, where we have an extra term to the
original momenta, which is the electromagnetic potential. But
apparently, unlike in electromagnetism (if I am not mistaken), under
the quantization procedure for the canonical coordinates, one simply
quantizes the whole thing [\tex]\dot g-L_\xi g[tex] as the functional
derivative along the metric coordinate [\tex]g_{ij}[tex]. So, does one
set the shift to zero before quantizing or is this the way it should
be done even with the shift? If so, why is it different from
electromagnetism?

The quantization strategy is actually the same in both
cases. One way to think about this is that (for typical fields)
the canonical momenta are always defined as the
Lie derivative of the field along the *unit normal* to
the hypersurfaces of your spacetime foliation.
So, for the EM field, or for a scalar field, or whatever,
the shift vector is actually there, too (unless you've
set it to zero with a special choice of foliation).

The shift vector term in the gravitational momentum
is also very analogous to the gradient of the scalar
potential term in the EM momentum if you make
the analogy between spatial diffeomorphisms in GR
and gauge transformations in EM. But that's another
story...