Approaches to Quantum Gravity

Covariant loop quantum gravity? 257We start with a space-time M ∼ R × where we distinguish the time directionfrom the three space dimensions. We decompose the tetrad field e I as:e 0 = Ndt + χ i E i a dxae i = E i a N a dt + E i a dxa , (14.3)where i = 1...3 is an internal index (space components of I )anda is the spaceindex labelling the coordinates x a . N and N a are respectively the lapse and theshift. χ i indicates the deviation of the normal to the canonical hypersurface fromthe time direction: the time normal is defined as the normalised time-like 4-vectorχ = (1,χ i )/ √ 1 −|⃗χ| 2 .Let’s call X, Y,... = 1 ...6 sl(2, C)-indices labelling antisymmetric couples[IJ]. We define new connection/triad variables valued in sl(2, C) instead of thestandard su(2) of LQG. The Lorentz connection A X a is:Then we define a “rotational” triad and a boost triad,R a XA X = (φ12ω 0i ,φ12ɛ i jk ω jk ). (14.4)= (−ɛjkiE i a χ k, E i a ), Ba X = (⋆Ra ) X = (E i a ,ɛjk iE i a χ k), (14.5)where ⋆ is the Hodge operator on sl(2, C) switching the boost and rotation part ofthe algebra. We further define the actual projectors on the boost and rotation sectorsof sl(2, C), (P R )Y X = R a X Ra Y ,(P B)Y X = B a X Ba Y :P R =⎛⎝ − (δb a χ 2 −χ a χ b )1−χ 2− ɛ a bc χ c1−χ 2− ɛ a bc χ c1−χ 2δa b−χ aχ b1−χ 2⎞⎠ , P B = Id − P R , P B P R = 0.P R projects on the subspace su(2) χ generating the rotations leaving the vector χinvariant, while P B projects on the complementary subspace. The action then reads:∫S = dtd 3 x (( BX a − φ1γ X) Ra ∂t Aa X + X G X + N a H a + NH ) . (14.6)The phase space is thus defined with the Poisson bracket,{AXa (x), ( BY b − φ1γ ) } Rb Y (y) = δXY δa b δ(3) (x, y). (14.7) X , N a , N are Lagrange multipliers enforcing the first class constraints:G X = D A (B X − φ1γ R X ) ,H a = − ( BX b − φ1γ ) Rb X FXab (A),H = φ11 + 1 (B − φ1γ R)(B − φ1γ R) F(A). (14.8)γ2