Expressing the metric as $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$, assuming $h_{mu \nu} \ll 1$ we can write the Einstein Hilbert action to leading order in $h_{\mu \nu}$ and quantize the linearized Einstein Hilbert action to construct the graviton field. Gravitons are spin 2 particles, which is easiest to see by noting that $h_{\mu \nu}$ has two indices. These enjoy a "gauge symmetry" corresponding to diffeomorphisms.

However, classically, gravity can be understood to be largely analogous to a gauge theory. The chrisoffel symbol $\Gamma^{\alpha}_{\beta \mu}$ takes the place of $A^a_\mu T^a$ as the gauge field.

Note that $\Gamma^{\alpha}_{\beta \mu}$ has three indices, however the $\alpha,\beta$ indices can be understood as a matrix, much like the lie algebra elements $T^a$ in Yang Mills theory.

If we quantize this field instead shouldn't we not be able to realize gravity as a theory mediated by spin 1 gauge particles?

(This should be especially true if we take the action to be the Kretschmann scalar, which seems to be equivalent to the Yang Mills Lagrangian.

$$ \mathrm{Tr}(F_{\mu \nu} F^{\mu \nu}) \leftrightarrow R^a_{b \mu \nu} {R^b_{a}}^{\mu \nu} $$

However, this would obviously give a different theory than Einstein gravity.)

  • $\begingroup$ Related. $\endgroup$ – Cosmas Zachos May 12 '19 at 18:24
  • $\begingroup$ Related: physics.stackexchange.com/q/108230/2451 , physics.stackexchange.com/q/263572/2451 and links therein. $\endgroup$ – Qmechanic May 12 '19 at 18:31
  • $\begingroup$ see my answer here physics.stackexchange.com/q/11542 .charges play a role in attraction and repulsion together with the spin. $\endgroup$ – anna v May 12 '19 at 18:38
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    $\begingroup$ It sure feels like 3 questions: a) Physically, coupling to the energy-momentum tensor dictates spin 2. b) Formally, the weak field expansion of Einstein's equations produce a metric perturbation field which is spin 2. c) Indeed, the intuitive gauge-theory simulacrum for gravity is the gauged tangent space Lorentz group, effected by the spin connection ω, related to Γ, which is not a tensor, and which is utilized in supergravity; however, possibly counterintuitively, it demonstrably leads to spin 2. You apparently want to focus on 3)? $\endgroup$ – Cosmas Zachos May 12 '19 at 19:32
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    $\begingroup$ ...perhaps (3.3) here might help--unless it doesn't. $\endgroup$ – Cosmas Zachos May 12 '19 at 20:27

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