Constraints at each space point leaving an eight

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Unformatted text preview: nt), leaving an eight-dimensional constraint surface. The constraints then generate four gauge invariances, so the gauge orbits are four dimensional, and the gauge invariant quotient by the orbits is four dimensional [see Henneaux and Teitelboim (1992) for an introduction to constrained Hamiltonian systems, gauge theories, and the terminology used here]. These are the two polarizations of the massless graviton, along with their conjugate momenta. In the case m Þ 0, the h0i are no longer Lagrange multipliers. Instead, they appear quadratically and are auxiliary variables. Their equations of motion yield h0i ¼ À 1 @j ij ; m2 (2.7) which can be plugged back into the action (2.5) to give Z ~2 _ S ¼ dD xij hij À H þ h00 ðr hii À @i @j hij À m2 hii Þ; (2.8) where 2 Note that canonical momenta can change under integrations by parts of the time derivatives. We fixed this ambiguity by integrating by parts such as to remove all derivatives from h0i and h00 . Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 1 11 1 2 þ @k hij @k hij À @i hjk @j hik H ¼ 2 À ij ii 2 2 DÀ2 2 1 1 þ @i hij @j hkk À @i hjj @i hkk þ m2 ðhij hij À h2 Þ ii 2 2 1 þ 2 ð@j ij Þ2 : (2.9) m The component h00 remains a Lagrange multiplier enforc~2 ing a single constraint C ¼ Àr hii þ @i @j hij þ m2 hii ¼ 0, but the Hamiltonian is no longer first class. One secondary constraint arises Rfrom the Poisson bracket with the Hamiltonian H ¼ dd xH , namely, fH; CgPB ¼½1=ðD À 2ފ m2 ii þ @i @j ij . The resulting set of two constraints is second class, so there is no longer any gauge freedom. For D ¼ 4 the 12-dimensional phase space has two constraints for a total of 10 degrees of freedom, corresponding to the five polarizations of the massive graviton and their conjugate momenta. Note that the Fierz-Pauli tuning is crucial to the appearance of h00 as a Lagrange multiplier. If the tuning is violated, then h00 appears quadratically and is an auxiliary variable and no longer enforces a constraint. There are then no constraints, and the full 12 degrees of freedom in the phase space are active. The extra 2 degrees of freedom are the scalar ghost and its conjugate momentum. B. Free solutions and graviton mode functions We now proceed to find the space of solutions of Eq. (2.1) and show that it transforms as a massive spin 2 representation of the Lorentz group, showing that the action propagates precisely one massive graviton. The equations of motion from Eq. (2.1) are S ¼ hh À @ @ h  À @ @ h  þ  @ @ h h þ @ @ h À  hh À m2 ðh À  hÞ ¼ 0: (2.10) Acting on Eq. (2.10) with @ , we find, assuming m2 Þ 0, the constraint @ h À @ h. Plugging this back into the equations of motion, we find hh À @ @ h À m2 ðh À  hÞ ¼ 0. Taking the trace of this we find h ¼ 0, which in turn implies @ h ¼ 0. This, along with h ¼ 0 applied to the equation of motion (2.10), gives ðh À m2 Þh ¼ 0. Thus the equation of...
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