Hinterbichler theoretical aspects of massive gravity

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Unformatted text preview: spects of massive gravity 678 ap; ¼ ðup; ; hÞ; h ðxÞ ¼ Z X dd p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ap;  ðp; ÞeipÁx ð2Þd 2!p  þ aà ; à ðp; ÞeÀipÁx : p The solution is a general linear combination of the following mode functions and their conjugates  ¼ 1; 2; ... ;d: (2.29) These are the solutions representing gravitons, and they have ´ the following Poincare transformation properties: u ðx p; À aÞ ¼ u ðxÞeÀipÁa ; p; (2.38) In the quantum theory, the a and aà become creation and annihilation operators which satisfy the usual commutation relations and produce massive spin 2 states. The fields hij and their canonical momenta ij , constructed from the a and aà , will then automatically satisfy the Dirac algebra and constraints of the Hamiltonian analysis of Sec. II.A, providing a quantization of the system. Once interactions are taken into account, external lines of Feynman diagrams will get a factor of the mode functions (2.29). (2.28) 1 u ðxÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðp;ÞeipÁx ; p; ð2Þd 2!p (2.37) aà ; ¼ Àðuà ; ; hÞ: p p The general solution to Eq. (2.10) thus reads C. Propagator (2.30) Integrating by parts, we can rewrite the Fierz-Pauli action (2.1) as sffiffiffiffiffiffiffiffiffi 00 ! Ãp  à u  ðÃÀ1 xÞ ¼ W ðÃ; pÞ0  u;0 ðxÞ; à 0 0 p; Ãp !p S¼ Z (2.31) where W ðÃ; pÞ ¼ is the Wigner rotation, and W ðÃ; pÞ0  is its spin 2 rep, R  ! ðBÀ1 RBÞ0  .4 Thus the gravitons are spin 2 solutions. In terms of the modes, the general solution reads Z X h ðxÞ ¼ dd p ½ap; u ðxÞ þ aà ; uà ðxފ: (2.32) p p; p; O ¼ ðð Þ À   Þðh À m2 Þ À 2@ð @ð Þ Þ þ @ @  þ @ @  ; (2.40) is a second order differential operator satisfying  O; ¼ O; ¼ O; ¼ O ; : The inner (symplectic) product on the space of solutions to the equations of motion is Z $ ðh; h0 Þ ¼ i dd xhà ðxÞ @ 0 h0 ðxÞjt¼0 : (2.33)  ðup; ; up0 ;0 Þ ¼ À 0 ; (2.41) In terms of this operator, the equation of motion (2.10) can be written simply as S=h ¼ O; h ¼ 0. To derive the propagator, we go to momentum space, The u functions are orthonormal with respect to this product, p0 Þ (2.39) where LÀ1 ðÃpÞÃLðpÞ d ðp 1 dD x h O; h ; 2 O ð@ ! ipÞ ¼ Àðð Þ À   Þðp2 þ m2 Þ (2.34) þ 2pð pð Þ Þ À p p  ðuà ; ; uà 0 ;0 Þ ¼ Àd ðp À p0 Þ0 ; p p (2.35) À p p  : ðup; ; uà 0 ;0 Þ ¼ 0; p (2.36) The propagator is the operator D ; with the same symmetries Eq. (2.41) which satisfies and we can use the product to extract the a and aà coefficients from any solution h ðxÞ, 4 i O; D ; ¼ ð  þ   Þ:   2 We show the Lorentz transformation property as follows: 00 à 0 à 0   ðp; ÞeipÁà À1 x 00 ¼ ½ÃLðpފ 0 ½ÃLðpފ 0   ðk; ÞeiÃpÁx 00 ¼ ½L...
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