k 0 where b is any unitary matrix given a

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Unformatted text preview: k; Þ ¼ B   ðk; 0 Þ; where B is any unitary matrix. ^ Given a particular spatial direction, with unit vector ki , there is an SOðd À 1Þ subgroup of the little group SOðdÞ ^ which leaves ki invariant, and the symmetric traceless representation (rep) of SOðdÞ breaks up into three reps of SOðd À 1Þ, a scalar, a vector, and a symmetric traceless tensor. The scalar mode is called the longitudinal graviton and has spatial components sffiffiffiffiffiffiffiffiffiffiffiffi  d ij ^ ^i kj À 1 ij : (2.20) k L ¼ dÀ1 d ^ After a large boost in the ki direction, it goes similar to L $ p2 =m2 . As we see later, in the massless limit, or large boost limit, this mode is carried by a scalar field, which generally becomes strongly coupled once interactions are taken into account. The vector modes have spatial components pffiffiffi ^ (2.21) ij ¼ 2kði jÞ ; V;k k ^ and after a large boost in the ki direction, they go similar to L $ p=m. In the massless limit, these modes are carried by a vector field, which decouples from conserved sources. The remaining linearly independent modes are symmetric traceless ^ tensors with no components in the ki directions, and they form the symmetric traceless mode of SOðd À 1Þ. They are invari^ ant under a boost in the ki direction, and in the massless limit, they are carried by a massless graviton. In the massless limit, we therefore expect that the extra degrees of freedom of the massive graviton organize themselves into a massless vector and a massless scalar. We see later explicitly how this comes about at the Lagrangian level. Upon boosting to p, the polarization tensors satisfy the following properties: they are transverse to p and traceless, p  ðp; Þ ¼ 0;   ðp; Þ ¼ 0; We choose the standard boost to be 1 ð À 1Þpi pj ; jpj2 pi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À 1; Li 0 ðpÞ ¼ L0 i ðpÞ ¼ jpj Li j ðpÞ ¼ ij þ (2.14) L0 0 ðpÞ ¼ ; where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p0 =m ¼ jpj2 þ m2 =m is the usual relativistic . See Chapter 2 of Weinberg (1995) for discussions of this standard boost and general representation theory ´ of the Poincare group. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (2.22) and they satisfy orthogonality and completeness relations  ðp; Þà ðp; 0 Þ ¼ 0 ;  X  (2.23) 1  ðp; Þà ðp; Þ ¼ ðP P þ P P Þ 2 À 3 (2.19) 1 P P ; DÀ1 (2.24) where P   þ p p =m2 . The right-hand side of the completeness relation (2.24) is the projector onto the symmetric and transverse traceless subspace of tensors, i.e., the identity on this space. We also have the following symmetry properties in p, which can be deduced from the form of the standard boost (2.14): ij ðÀp; Þ ¼ ij ðp; Þ; 0i ðÀp; Þ ¼ À0i ðp; Þ; 00 ðÀp; Þ ¼ 00 ðp; Þ: i; j ¼ 1; 2; . . . ; d; (2.25) i ¼ 1; 2; . . . ; d; (2.26) (2.27) Kurt Hinterbichler: Theoretical a...
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