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1. Could you explain relations between independent components of the metric tensor field, its helicity, polarization and spin?

Let me specify the question. The metric tensor in 4D has 10 independent components. From dynamical analysis of the Einstein-Hilbert action we know that gravity is a theory with 2 degrees of freedom. Quantum field theory tells us that it describes a spin 2 particle, thus has five helicity modes ±2, ±1, 0. It also tells us that the graviton is massless (let us neglect massive graviton models if possible), thus only two of these modes are physical, ±2 ones. I could be wrong in this statement, but that issue looks analogously to non physical polarizations (longitudinal and time/scalar) of a photon in contrary to the massive Proca field. Finally, some theories extend dimensionality of the phase space of the gravitational field to more than 10*2 dimensions. For instance, considering Ashtekar variables, we introduce additional internal SU(2) symmetry in the spatial sector of the metric tensor, obtaining (3*3+4)*2 instead of just mentioned (6+4)*2 dimensions. I would not try even to interpret these extra non physical degrees of freedom and the corresponding gauge, but maybe someone could do it.

2. When explaining graviton quantum numbers could you refer to the specification of the problem above, giving analogous examples for photon and weak bosons states (even considering a massive gravity for the latters if necessary)?

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Helicity of a field is a synonym of a polarization in the case of bosons.

Prokopec here:

Prokopec, Lecture notes on Cosmology

gives a nice explanation for counting propagating degrees of freedom (DOF) in Einstein gravity (page 10, lower part). However, we find that there are six general DOF out of which two are physical. Hence this number does not correspond to five spin-2 particle modes. What happens with the sixth one is explained here:

Wikipedia, Linearized massive gravity

It is a ghost describing the mode with a negative kinetic energy (Prokopec calls it "the 'time-like' potential") and having negative norm. Now the analogue between graviton and photon becomes clearer. The 'time-like' potential of the former corresponds to the time/scalar polarization of the latter - they are both ghost fields equipped with a negative norm. There are 2 physical DOF, both in the case of graviton (helicity +-2, i.e. tensor polarization modes) and photon (helicity +-1, i.e. vector polarization modes). The remaining non-physical positive energy / positive norm graviton solutions are (names after Prokopec) 'vector-like' potentials (helicity +-1, i.e. vector polarization modes) and the '(spatial) Newton' potential (helicity 0, i.e. scalar polarization mode). In the case of the photon field, the non-physical positive energy / positive norm solution is the longitudinal polarization (helicity 0, i.e. scalar polarization mode).

The correspondence between the weak interactions bosons and the gravitational field remains an open issue. I could be wrong, but maybe gravity in terms of the Ashtekar variables captures this analogy - in both cases the gauge group is SU(2).

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