Kiefer C. Quantum gravity

6
QUANTUM GRAVITY WITH CONNECTIONS AND LOOPS
6.1 Connection and loop variables
In Section 4.3, we encountered a Hamiltonian formulation of GR alternative to
geometrodynamics, using the concepts of connections and loops. In the present
chapter, we shall review approaches to formulate a consistent quantum theory
with such variables, leading to ‘quantum connection dynamics’ or ‘quantum loop
dynamics’. More details and references can be found in Rovelli (2004) and Thiemann
(2001, 2003), Ashtekar and Lewandowski (2004), and Nicolai et al. (2005).
The popular name ‘loop quantum gravity’ stems from the use of the loop variables
to be defined in Section 6.1.2.
In the present section, we shall introduce the quantum versions of the connection
and loop variables and discuss the Gauss constraints (4.122) and the
diffeomorphism constraints (4.127). These constraints already give a picture of
the way space might look like on the smallest scales. At least on a kinematical
level (using the Gauss constraints only), a major result can be obtained:
the spectrum of geometric operators representing area or volume in the classical
limit turns out to be discrete (Section 6.2). This has a direct bearing on
the interpretation of black-hole entropy (see Section 7.1). The more complicated
implementation of the Hamiltonian constraint is relegated to Section 6.3. We
treat here only pure gravity, but various results can be extended to the case of
standard-model matter (Thiemann 2001).
6.1.1 Connection representation
The connection representation is characterized classically by the Poisson bracket
(4.120) between the densitized tetrad Ej b(y) and the SU(2)-connection Ai a(x).
These variables are then formally turned into operators obeying the commutation
relation ]
[Âi a (x), Êb j (y) =8πβiδj i δb aδ(x, y) . (6.1)
In the functional Schrödinger representation, one can implement this relation
formally through
 i a (x)Ψ[A] =Ai a (x)Ψ[A] , (6.2)
Êj b (y)Ψ[A] =8πβ δ
i δA j Ψ[A] , (6.3)
b
(y)
where the A in the argument of the wave functional is a shorthand for A i a (x).
As in Chapter 5, the constraints are implemented as conditions on allowed wave
functionals. The Gauss constraints (4.122) then becomes
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