This gauge symmetry the relative coefcient of 1

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Unformatted text preview: ry. The relative coefficient of À1 between the h2 and h h contractions is called the Fierz-Pauli tuning, and it is not enforced by any known symmetry. However, the only thing that needs to be said about this action is that it describes a single massive spin 2 degree of freedom of mass m. We show this explicitly in what follows. Any deviation from the form (2.1) and the action will no longer describe a single massive spin 2. For example, violating the Fierz-Pauli tuning in the mass term by changing to À 1 m2 ½h h À ð1 À aÞh2 Š for a Þ 0 gives an action de2 scribing a scalar ghost (a scalar with negative kinetic energy) of mass m2 ¼ ½ð3 À 4aÞ=2aŠm2 , in addition to the massive g spin 2. For small a, the ghost mass squared goes like $1=a. It goes to infinity as the Fierz-Pauli tuning is approached, rendering it nondynamical. Violating the tuning in the kinetic terms similarly alters the number of degrees of freedom; see van Nieuwenhuizen (1973a) for a general analysis. There is a method of constructing Lagrangians such as Eq. (2.1) to describe any given spin. See, for example, the first few chapters of Weinberg (1995), the classic papers on higher spin Lagrangians by Singh and Hagen (1974) and Fronsdal (1978), and the reviews by Bouatta, Compere, and Sagnotti (2004) and Sorokin (2005). A. Hamiltonian and degree of freedom count We begin our study of the Fierz-Pauli action (2.1) by casting it into Hamiltonian form and counting the number of degrees of freedom. We show that it propagates DðD À 1Þ=2 À 1 degrees of freedom in D dimensions (5 degrees of freedom for D ¼ 4), the right number for a massive spin 2 particle. Kurt Hinterbichler: Theoretical aspects of massive gravity 676 We start by Legendre transforming Eq. (2.1) only with respect to the spatial components hij . The canonical momenta are2 ij ¼ @L _ _ ¼ hij À hkk ij À 2@ði hjÞ0 þ 2@k h0k ij : _ @hij (2.3) Inverting for the velocities, we have _ hij ¼ ij À 1   þ 2@ði hjÞ0 : D À 2 kk ij (2.4) In terms of these Hamiltonian variables, the Fierz-Pauli action (2.1) becomes Z _ S ¼ dD xij hij À H þ 2h0i ð@j ij Þ þ m2 h2i 0 ~2 þ h00 ðr hii À @i @j hij À m2 hii Þ; (2.5) where 1 11 1 H ¼ 2 À 2 þ @ h @ h À @i hjk @j hik 2 ij 2 D À 2 ii 2 k ij k ij 1 1 þ @i hij @j hkk À @i hjj @i hkk þ m2 ðhij hij À h2 Þ: ii 2 2 (2.6) First consider the case m ¼ 0. The timelike components h0i and h00 appear linearly multiplied by terms with no time derivatives. We interpret them as Lagrange multipliers en~2 forcing the constraints @j ij ¼ 0 and r hii À @i @j hij ¼ 0. It is straightforward to check that these are first class constraints, and that the Hamiltonian (2.6) is first class. Thus Eq. (2.5) is a first class gauge system. For D ¼ 4, the hij and ij each have six components, because they are 3  3 symmetric tensors, so together they span a 12-dimensional (for each space point) phase space. We have four constraints (at each space poi...
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This document was uploaded on 09/28/2013.

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