Approaches to Quantum Gravity

258 E. Livine
However, in contrast to the usual LQG framework, we also have second class
constraints:
φ ab = (⋆R a ) X R b X = 0, ψab ≈ RRD A R. (14.9)
The constraint φ = 0 is the simplicity constraint. The constraint ψ = 0comes
from the Poisson bracket {H,φ} and is required in order that the constraint φ = 0
is preserved under gauge transformations (generated by G, H a , H) and in particular
under time evolution. ψ corresponds to the reality constraint Re E∇[E, E] =0of
complex LQG.
To solve the second class constraints, we define the Dirac bracket { f, g} D =
{ f, g} −{f,ϕ r }rs −1{ϕ
s, g} where the Dirac matrix rs ={ϕ r ,ϕ s } is made of the
Poisson brackets of the constraints ϕ = (φ, ψ). Following [8; 11], one then checks
that the algebra of the first class constraints is not modified. Defining smeared
constraints, we find the following Dirac brackets:
∫
∫
∫
G() = X G X , H(N) = NH, D( ⃗N) = N a (H a + Aa X G X),
{
}
{
}
G( 1 ), G( 2 ) = G([ 1, 2 ]), D( ⃗N), D( ⃗M) =−D([ ⃗N, ⃗M]),
{
} D {
} D
D( ⃗N), G() =−G(N a ∂ a ), D( ⃗N), H(N) =−H(L ⃗N
{
N),
D {
} D
H(N), G()
}D = 0, H(N), H(M) = D( ⃗K ) − G(K b A b ),
D
[ 1 , 2 ] X = fYZ X 1 Z 2 , [ ⃗N, ⃗M] a = N b ∂ b M a − M b ∂ b N a ,
LN ⃗ N = N a ∂ a N − N∂ a N a , K b = (N∂ a M − M∂ a N)R a X Rb Y g XY ,
where fYZ X are the structure constant of the algebra sl(2, C). With A ∈{1, 2, 3}
boost indices and B ∈{4, 5, 6} ∼{1, 2, 3} rotation indices, we have f
AA A = f BB A =
f
AB B = 0and f AA A =−f AB A =−f BB B given by the antisymmetric tensor ɛ.
The Gs generate SL(2, C) gauge transformations. The vector constraint H a generates
spatial diffeomorphisms on the canonical hypersurface invariant . Finally,
the scalar constraint H is called the Hamiltonian constraint and generates the (time)
evolution of the canonical variables.
14.2.2 The choice of connection and the area spectrum
As shown in [8; 11; 12], although the triad field R is still commutative for the Dirac
bracket, the properties of the connection A change drastically: it is not canonically
conjugated to the triad and it does not commute with itself. Nevertheless, one
should keep in mind that when using the Dirac bracket the original canonical variables
lose their preferred status and we should feel free to identify better suited