In both spin-1 and spin-2 gauge theories, the gauge bosons (e.g. the photon & gluon and the graviton respectively) have two physical degrees of freedom, which can be observed quantum mechanically via arguments from Wigner's little group of massless particles. This naively would seem to be in conflict with the apparent number of degrees of freedom in the field theories for each respective particle, which are typically given by a four-vector $A_\mu(x)$ or a two-tensor $g_{\mu\nu}(x)$ respectively.

The normal way of resolving this tension is to postulate that some of these degrees of freedom are 'gauge' degrees of freedom. In the case of the photon the gauge group is $A_\mu \to A_\mu + \partial_\mu f$, which introduces one local degree of freedom. Naively, this would seem to only have eliminated one local degree of freedom. This is discussed more at Gauge theory and eliminating unphysical degrees of freedom. This type of argument works for the graviton as well, but with the 10 components of a symmetric tensor reduced by the 4 parameters of the diffeomorphism group as well as 4 non-dynamical degrees of freedom.

While reducing the number of degrees of freedom by $n$ by counting the generators of an $n$-parameter gauge group seems quite simple, deriving the other $n$ nondynamical degrees of freedom in each seems to require particular manipulations that depend on the particular equations of motion for the particle, even though it seems to hold for both the graviton and the photon which have different EOMs. Presumably some version of this argument must also hold for higher-spin massless particles, or alternative gauge-invariant EOMs for spin-1 and spin-2 particles (although I am unclear on whether the latter is possible at lowest order).

My question is thu whether there is a more general argument for why each degree of freedom removed by each gauge generator should also be matched by a nondynamical degree of freedom. Can such a fact be reasoned about without appealing to the equations of motion specifically?


1 Answer 1


Yes, one can argue that [1]

  • 19:38 YM (GR) is the unique local Poincare-covariant unitary low-energy theory for interacting massless spin-1 (spin-2) particles with an $1/r^2$-dependence at long distances, respectively.

In particular, they are gauge theories with non-dynamical DOF. The list of arguments involves, among other things,


  1. Nima Arkani-Hamed, Robustness of GR. Attempts to Modify Gravity, part 1, Talk at Institute for Advanced Study (IAS), 2011, YouTube video.
  • $\begingroup$ Wow this is a fantastic talk. Thank you! I think this also roughly resolves the other question I asked to my satisfaction so I will close it. $\endgroup$
    – Panopticon
    Aug 25 at 17:40

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