Timeline for Gauge invariance of the quadratic graviton Lagrangian
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| Nov 6, 2017 at 22:42 | comment | added | Will | @AccidentalFourierTransform Do you know of any good notes that go into detail on the spin-2 derivation? | |
| Nov 6, 2017 at 22:32 | comment | added | AccidentalFourierTransform | Yes. (Well, there are some exceptions, such as topological field theories...) | |
| Nov 6, 2017 at 22:30 | comment | added | Will | @AccidentalFourierTransform So in the covariant approach does one simply require that the action is gauge invariant such that the Lagrangian is invariant up to boundary terms? | |
| Nov 6, 2017 at 22:15 | comment | added | AccidentalFourierTransform | Unfortunately, that question is kind of meaningless. In the approach of your notes, there is no gauge-fixing. The object $h$ is constructed from first principles, and is shown to satisfy the Coulomb condition a posteriori. Such a condition is not imposed, but it appears naturally from the properties of the little group of massless particles. On the other hand, in covariant approaches one introduces a different field $h'$, which is postulated to transform as a true rank-2 tensor. Only if the theory is gauge invariant do both approaches agree. But in principle they are different. | |
| Nov 6, 2017 at 22:09 | comment | added | Will | @AccidentalFourierTransform Ah apologies, I somehow managed to gloss over that bit. So how does a spin 2 field transform under Lorentz transformations when it's not gauge fixed? Do you know of any notes that go into detail on this subject? | |
| Nov 6, 2017 at 22:07 | comment | added | AccidentalFourierTransform | Yep, that document has "Weinberg" written all over it. See e.g. the references. | |
| Nov 6, 2017 at 22:01 | comment | added | Will | @AccidentalFourierTransform I've actually been following these notes: google.co.uk/url?sa=t&source=web&rct=j&url=http://… but this may be equivalent to Weinberg. I didn't intend to gauge fix $h_{\mu\nu}$. Does the spin-2 field only transform like this under Lorentz transformations if it is gauge fixed? | |
| Nov 6, 2017 at 21:56 | comment | added | AccidentalFourierTransform | Oh, now I see what you mean. You are going Weinberg style, aren't you? E.g., $h^{0\mu}=\partial_i h^{i\mu}=0$, etc. | |
| Nov 6, 2017 at 21:42 | comment | added | Will | @AccidentalFourierTransform If one studies how the spin-2 field transforms under Lorentz transformations though, one finds that $U(\Lambda)h^{\mu\nu}U^{-1}(\Lambda)=(\Lambda^{-1})^{\mu}_{\;\;\sigma}(\Lambda^{-1})^{\nu}_{\;\;\rho}h^{\sigma\rho} +2\partial^{(\mu}\xi_{_{\Lambda}}^{\nu)}$, so isn't such a transformation needed to negate the non-covariant part of the transformation? | |
| Nov 6, 2017 at 21:39 | answer | added | DanielC | timeline score: 1 | |
| Nov 6, 2017 at 21:30 | comment | added | AccidentalFourierTransform | "...the requirement of Lorentz invariance requires..." no, that transformation corresponds to diffeomorphism invariance. Your Lagrangian is Lorentz invariant regardless of this latter invariance. | |
| Nov 6, 2017 at 21:27 | history | asked | Will | CC BY-SA 3.0 |