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4.2 Isotropy: Connection variables

Isotropic connections and triads, as discussed in Appendix B.2, are analogously described by single components &tidle;c and &tidle;p, respectively, related to the scale factor by
2 |&tidle;p| = &tidle;a2 = a-- (19 ) 4
for the densitized triad component &tidle;p and
&tidle;c = &tidle;Γ + γ ˙&tidle;a = 1-(k + γ ˙a) (20 ) 2
for the connection component &tidle;c. Both components are canonically conjugate:
8πγG-- {c&tidle;,p&tidle;} = 3V . (21 ) 0

It is convenient to absorb factors of V0 into the basic variables, which is also suggested by the integrations in holonomies and fluxes on which background-independent quantizations are built [16Jump To The Next Citation Point]. We thus define

p = V02∕3p&tidle;, c = V01∕3&tidle;c (22 )
together with 1∕3&tidle; Γ = V0 Γ. The symplectic structure is then independent of V0 and so are integrated densities such as total Hamiltonians. For the Hamiltonian constraint in isotropic Ashtekar variables we have
3 −2 2 2 ∘ --- H = − -----(γ (c − Γ ) + Γ ) |p| + Hmatter(p) = 0, (23 ) 8πG
which is exactly the Friedmann equation. (In most earlier papers on loop quantum cosmology some factors in the basic variables and classical equations are incorrect due, in part, to the existence of different and often confusing notation in the loop quantum gravity literature.1 Note that H here seems to differ from HADM in (18View Equation) by a factor of 8. This happens due to different normalizations of the coordinate volume V0, which is a unit 3-sphere for a closed model in Friedmann–Robertson–Walker form, while isotropic connection variables, as reduced in Appendix B.2, are based on a 2-fold covering space obtained from isotropizing a Bianchi IX model.)

The part of phase space where we have p = 0 and thus a = 0 plays a special role since this is where isotropic classical singularities are located. On this subset the evolution equation (17View Equation) with standard matter choices is singular in the sense that Hmatter, e.g.,

1- −3∕2 2 3∕2 Hφ (a, φ,pφ) = 2 |p| pφ + |p| V(φ ) (24 )
for a scalar φ with momentum pφ and potential V(φ ), diverges and the differential equation does not pose a well-defined initial-value problem there. Thus, once such a point is reached, the further evolution is no longer determined by the theory. Since, according to singularity theorems [178114], any classical trajectory must intersect the subset a = 0 for the matter we need in our universe, the classical theory is incomplete.

This situation, certainly, is not changed by introducing triad variables instead of metric variables. However, the situation is already different since p = 0 is a sub-manifold in the classical phase space of triad variables, where p can have both signs (the sign determining whether the triad is left or right handed, i.e., the orientation). This is in contrast to metric variables where a = 0 is a boundary of the classical phase space. There are no implications in the classical theory since trajectories end there anyway, but it will have important ramifications in the quantum theory (see Sections 5.14, ??, 5.17 and 5.19).


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