With manifest lorentz symmetry and locality for

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Unformatted text preview: orentz symmetry and locality. For helicity 1, if we choose a vector field A to carry the particle, its action is fixed to be the Maxwell action, so even without Maxwell, we could have discovered electromagnetism via these arguments. If we now ask for consistent self-interactions of such massless particles, we are led to the problem of deforming the action (and possibly the form of the gauge transformations), in such a way that the linear form of the gauge transformations is preserved. These requirements are enough to lead us essentially uniquely to the non-Abelian gauge theories, two of which describe the strong and weak forces (Henneaux, 1998). Moving on to helicity 2, the required gauge symmetry is linearized general coordinate invariance. Asking for consistent self-interactions leads essentially uniquely to GR and full general coordinate invariance (Gupta, 1954; Kraichnan, 1955; Weinberg, 1965; Deser, 1970; Boulware and Deser, 1975; Fang and Fronsdal, 1979; Wald, 1986) [see also Chapter 13 of Weinberg (1995), which shows how helicity 2 implies the equivalence principle]. For helicity ! 3, the story ends, because there are no self-interactions that can be written (Berends, Burgers, and van Dam, 1984) [see also Chapter 13 of Weinberg (1995), which shows that the scattering amplitudes for helicity ! 3 particles vanish]. This path is straightforward, starting from the principles of special relativity (Lorentz invariance) to the classification of particles and fields that describe them, and finally to their possible interactions. The path Einstein followed, on the other hand, is a leap of insight and has logical gaps; the equivalence principle and general coordinate invariance, although they suggest GR, do not lead uniquely to GR. General coordinate invariance is a gauge symmetry, and gauge symmetries are redundancies of description, not fundamental properties. In any system with gauge symmetry, one can always fix the gauge and eliminate the gauge symmetry, without breaking the physical global symmetries (such as Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Lorentz invariance) or changing the physics of the system in any way. One often hears that gauge symmetry is fundamental, in electromagnetism, for example, but the more correct statement is that gauge symmetry in electromagnetism is necessary only if one demands the convenience of linearly realized Lorentz symmetry and locality. Fixing a gauge will not change the physics, but the price paid is that the Lorentz symmetries and locality are not manifest. On the other hand, starting from a system without gauge invariance, it is always possible to introduce gauge symmetry by putting in redundant variables. Often this can be very useful for studying a system and can elucidate properties which are otherwise difficult to see. This is the essence of ¨ the Stuckelberg trick, which we make use of extensively in our study of massive gravity. In fact, as we will see, this trick can be used to make...
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This document was uploaded on 09/28/2013.

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