Gravity and spin 2 representation

[jk22]

I'm not at all involved in QG but from far away I noticed :

Spin 2 representations are 5x5 matrices.

But in gravity what mathematical objects are quantized ? If it's the metric then it's a 4x4 matrix so that cannot be that.

Or : how does quantization reveal a 5x5 matrix ?

 

[samalkhaiat]

Spin 2 representations are 5x5 matrices.

This is true if, and only if, the object which carries the spin-2 representation has non-zero mass. In this case, the (massive) irreducible spin-2 representation (of the Poincare group) can be represented by a conserved, traceless and symmetric rank-2 tensor field: Consider the symmetric tensor field (i.e., we have 10 components) $h^{\mu\nu} = h^{\nu\mu}$, take its trace $h = \eta_{\rho \sigma}h^{\rho\sigma}$ then form the following traceless symmetric tensor field $$G^{\mu\nu} = h^{\mu\nu} - \frac{1}{4} \eta^{\mu\nu} h .$$ Now $G^{\mu\nu}$ has $10 - 1= 9$ components because $G^{\mu\nu} = G^{\nu\mu}$ and $\eta_{\mu\nu}G^{\mu\nu} = 0$. So, to reduce the number of components to 5, we need to impose 4 more conditions: Usually, in field theory $G^{\mu\nu}$ is generated by a conserved source $T^{\mu\nu} \ , \partial_{\mu}T^{\mu\nu} = 0$, and satisfies the second-order equation $$( \partial^{2} + m^{2} ) G^{\mu\nu} = \alpha T^{\mu\nu} \ .$$ Therefore, the required 4 conditions on $G^{\mu\nu}$ are given by $\partial_{\mu}G^{\mu\nu} = 0$. So, when a tensor field $G^{\mu\nu}$ satisfies the conditions,$G^{\mu\nu} = G^{\nu\mu} \ , \eta_{\mu\nu}G^{\mu\nu} = 0$ and $\partial_{\mu}G^{\mu\nu} = 0$, we say that $G^{\mu\nu}$ carries a massive irreducible spin-2 representation of the Poincare group.

In the massless case, we can show that the appropriate tensor is given by $$G^{\mu\nu} = h^{\mu\nu} - \frac{1}{2} \eta^{\mu\nu} h \ .$$ Notice, in this case, that $G^{\mu\nu}$ is still symmetric, but not traceless. So this tensor has 10 components. Again, we require $G^{\mu\nu}$ to be identically conserved, i.e., $\partial_{\mu}G^{\mu\nu} = 0$ (because, in this case the equation of motion has the form $\partial^{2}G^{\mu\nu} = \beta T^{\mu\nu}$). Therefore, the number of components of $G^{\mu\nu}$ has been reduced to $10 - 4 = 6$. However, we can show that theory is invariant under the following “gauge” transformations $$h^{\mu\nu} \to h^{\mu\nu} - \partial^{\mu} \chi^{\nu} - \partial^{\nu} \chi^{\mu} \ ,$$where $\chi^{\mu}$ is an arbitrary 4-vector field. This allows us to fix 4 out of the above 6 components of $G^{\mu\nu}$. Thus, there are 2 and only 2 independent components left in $G^{\mu\nu}$ as it should be for massless fields. So, the massless “spin”-2 irreducible representation of the Poincare group is carried by a symmetric rank-2 tensor with only 2 independent components.