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Natural Science Forum / Physics / Research / June 2008Quantization of gravitational momenta
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? Thanks in advance, Henrique On Jun 17, 12:33 pm, Henrique de Andrade Gomes <gomes...@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 [quoted text clipped - 12 lines] > 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, 11:09 pm, to...@cc.usu.edu wrote: > The quantization strategy is actually the same in both > cases. One way to think about this is that (for typical fields) [quoted text clipped - 4 lines] > the shift vector is actually there, too (unless you've > set it to zero with a special choice of foliation). I didn't mean to refer to EM fields in 3+1 curved space, I meant pure gravity. And my question is also not about where the shift comes from, which I understand. > The shift vector term in the gravitational momentum > is also very analogous to the gradient of the scalar [quoted text clipped - 4 lines] > > charlie torre This is more towards what I meant. If the electromagnetic potential has the form of a gradient, it can be altogether ignored (in simply connected space). Now, supposing one could view the shift term as a connection as well, would that mean that we could not have curvature of that connection? For classical space-time this would always be the case, as curvature needs at least two independent directions to be non null (in this case, directions are global metric velocities) . On Jun 17, 11:09 pm, to...@cc.usu.edu wrote: > On Jun 17, 12:33 pm, Henrique de Andrade Gomes <gomes...@gmail.com> > wrote: [quoted text clipped - 24 lines] > the shift vector is actually there, too (unless you've > set it to zero with a special choice of foliation). I didn't mean the quantization of electromagnetism in Hamiltonian GR, but the analogy between pure gravity in 3+1 canonical quantization and the quantization of an electromagnetic dynamical system. > 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... Yes, I think this comment is more in line with what I was thinking. In electromagnetism, if you have a gradient potential, it does not affect at all observability (suppposing we have a simply connected space). Is it the same with the gravitational momenta? How do we know? We would only observe such a deviation quantum mechanically as well... |
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