Theory e quantum corrections in the 3 theory ix brane

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Unformatted text preview: E. Quantum corrections in the Ã3 theory IX. Brane worlds and the Resonance Graviton A. The DGP action B. Linear expansion C. Resonance gravitons X. Conclusions and Future Directions Appendix: Total Derivative Combinations 698 699 700 700 701 701 703 705 706 707 I. INTRODUCTION Our goal is to explore what happens when one tries to give the graviton a mass. This is a modification of gravity, so we first discuss what gravity is and what it means to modify it. A. General relativity is massless spin 2 General relativity (GR) (Einstein, 1916) is by now widely accepted as the correct theory of gravity at low energies or large distances. The discovery of GR was in many ways ahead of its time. It was a leap of insight, from the equivalence principle and general coordinate invariance, to a fully nonlinear theory governing the dynamics of spacetime itself. It provided a solution, one more elaborate than necessary, to the problem of reconciling the insights of special relativity with the nonrelativistic action at a distance of Newtonian gravity. Had it not been for Einstein’s intuition and years of hard work, general relativity would likely have been discovered anyway, but its discovery may have had to wait several more decades, until developments in field theory in the 1940s and 1950s primed the culture. But in this hypothetical world without Einstein, the path of discovery would likely have been very different and in many ways more logical. Ó 2012 American Physical Society 672 Kurt Hinterbichler: Theoretical aspects of massive gravity This logical path starts with the approach to field theory espoused in the first volume of Weinberg’s field theory text (Weinberg, 1995). Degrees of freedom in flat fourdimensional spacetime are particles, classified by their spin. These degrees of freedom are carried by fields. If we wish to describe long-range macroscopic forces, only bosonic fields will do, since fermionic fields cannot build up classical coherent states. By the spin statistics theorem, these bosonic fields must be of integer spin s ¼ 0, 1, 2, 3, etc. A field c , which carries a particle of mass m, will satisfy the KleinGordon equation ðh À m2 Þ c ¼ 0, whose solution a distance r from a localized source is similar to $rÀ1 eÀmr . Long-range forces, those without exponential suppression, must therefore be described by massless fields m ¼ 0. Massless particles are characterized by how they transform under rotations transverse to their direction of motion. The transformation rule for bosons is characterized by an integer h ! 0, which we call the helicity. For h ¼ 0, such massless particles can be carried most simply by a scalar field . For a scalar field, any sort of interaction terms consistent with Lorentz invariance can be added, and so there are a plethora of possible self-consistent interacting theories of spin 0 particles. For helicities s ! 1, the field must carry a gauge symmetry if we are to write interactions with manifest L...
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This document was uploaded on 09/28/2013.

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