# Spin-2 Lagrangian

I can't seem to find this anywhere. I would like to know what the Lagrangian density for a general spin-2 particle is.

Note that I am not talking about gravitons. What I am asking is for something along the lines of the Lagrangian for, say, spin-1 or 1/2 or 0, which are well known.

• presumably, of interest: physics.stackexchange.com/search?q=Fierz+Pauli – AccidentalFourierTransform Aug 6 '17 at 20:38
• Please accept that I am out of my depth on this, and that you probably have this document. If you Google "1305.1513.pdf spin 2" there is a pdf which, although it does use the g word, seems to concentrate on massive spin 2 particles and the various Langranians that describe claas As you have excluded the graviton, I don't know if mass is therefore implicit in what you are looking for. But the title is " Massive spin-2 particle from a rank-2 tensor" Apologies if this is a dead end. – user163104 Aug 6 '17 at 22:52
• SRS- But for spin-2. There's lagrangians for spin 0,1/2,3/2, but I haven't seen a single general one for a spin-2 particle. Countto10-That link did help a bit but I'm still uncertain. thanks! – Ringo Hendrix Aug 7 '17 at 17:57
• @RingoHendrix I think this is a good question. If I understand correctly, you're asking whether it possible to write down the Lagrangian of a spin-2 particle in flat space and if yes, which objects should be used to construct the Lorentz invariant Lagrangian? – SRS Aug 8 '17 at 13:26
• Yes that's what I'm asking for., but I'd like to see it. – Ringo Hendrix Aug 8 '17 at 15:13

To construct a general spin 2 Lagrangian, you might do the following.

To begin with, we notice that Wigners classification tells you that spin-2 particles must be described by a rank-2 tensor field, i.e. $h_{\mu\nu}$.

Unitarity demands that we have a theory with at most two derivatives, and Lorentz invariance tells us that a theory containing spin two fields must have either zero or two derivatives.

Starting at second order, we would write $$\mathcal{L} = a_1h^{\mu\nu}\partial_\mu\partial_\nu h + a_2 h^{\mu\nu}\partial_\lambda\partial_\nu h^\lambda_\nu + a_3 h\partial_\nu\partial_\mu h^{\mu\nu} + a_4 h^{\mu\nu}\partial^2 h_{\mu\nu} + a_5 h\partial^2 h + a_6h_{\mu\nu}h^{\mu\nu} + a_7 h^2$$ We also note that since this Lagrangian is part of an action, there are many more possible terms we could write down, however the ones above form a minimal basis, since all others can be related via integration by parts.

If we now further demanding that our theory is gauge invariant, we would yield the Fierz-Pauli Lagrangian, fixing the coefficients to be $$\left\{a_1 = 0, a_2 = x,a_3 = -x, a_4 = -\frac{x}{2}, a_5 = \frac{x}{2}, a_6 = a_7 = 0\right\}$$ Where $x$ is a free parameter. However, if gauge invariance isn't a requirement then you can be content with the Lagrangian density above. Depending on what is required of your theory, you can fix some coefficients by demanding Newtonian gravity like limits, or make specific demands of the spin-1 and spin-0 modes of $h_{\mu\nu}$.

Additionally, you can also play the same game at third order if you want to consider self-interacting spin-2 fields, and construct a third order Lagrangian in exactly the same was as before, by considering \begin{align*} \mathcal{L} &= a_1 h^{\alpha \beta } \partial_{\alpha }h_{\delta \lambda } \partial_{\beta }h^{\delta \lambda } + \frac12 a_2 \left(h^{\alpha \beta } \partial_{\alpha }h_{\delta \lambda } \partial^{\lambda }h^{\delta }{}_{\beta } + h^{\alpha \beta } \partial_{\beta }h_{\delta \lambda } \partial^{\lambda }h^{\delta }{}_{\alpha }\right)+a_3 h^{\alpha \beta } \partial_{\lambda }h_{\alpha \delta } \partial^{\lambda }h^{\delta }{}_{\beta }\\ &~~+ a_4 h^{\alpha \beta } \partial_{\lambda }h_{\alpha \delta } \partial^{\delta }h^{\lambda }{}_{\beta } + \frac{1}{2} a_5 \left(h^{\alpha \beta } \partial_{\delta }h^{\delta \lambda } \partial_{\alpha }h_{\beta \lambda }+h^{\alpha \beta } \partial_{\delta }h^{\delta \lambda }\partial_{\beta }h_{\alpha \lambda }\right)+a_6 h^{\alpha \beta } \partial_{\delta }h^{\delta \lambda } \partial_{\lambda }h_{\alpha \beta }\\&~~+\frac{1}{2} a_7 \left(h^{\alpha \beta } \partial^{\lambda }h \partial_{\alpha }h_{\beta \lambda }+h^{\alpha \beta } \partial^{\lambda }h\partial_{\beta }h_{\alpha \lambda }\right) + a_8 h^{\alpha \beta } \partial^{\lambda }h \partial_{\lambda }h_{\alpha \beta } \\&~~+\frac{1}{2} a_9 \left(h^{\alpha \beta } \partial_{\beta }h \partial_{\lambda }h^{\lambda }{}_{\alpha }+h^{\alpha \beta }\partial_{\alpha }h \partial_{\lambda }h^{\lambda }{}_{\beta }\right) + a_{10} h^{\alpha \beta } \partial_{\alpha }h \partial_{\beta }h+a_{11} h^{\alpha \beta } \partial_{\lambda }h^{\lambda }{}_{\alpha } \partial_{\delta }h^{\delta }{}_{\beta }\\ &~~+a_{12} \partial^{\gamma }h^{\delta \lambda } \partial_{\gamma }h_{\delta \lambda }+a_{13} \partial^{\delta }h^{\gamma \lambda } \partial_{\gamma }h_{\delta \lambda }+a_{14} \partial_{\gamma }h^{\gamma }{}_{\delta } \partial_{\lambda }h^{\delta \lambda }+a_{15} \partial_{\delta }h \partial_{\gamma }h^{\gamma \delta }+ a_{16} h\partial_{\gamma }h \partial^{\gamma }h \end{align*}

Playing the same game with Gauge invariance again fixes all your coefficients, up to some redundancies, and you recover the weak-field GR third order Lagrangian, however again you can choose how to fix your coefficients depending on the theory under consideration.