### 4.2 Isotropy: Connection variables

Isotropic connections and triads, as discussed in Appendix B.2, are analogously described by single
components and , respectively, related to the scale factor by
for the densitized triad component and
for the connection component . Both components are canonically conjugate:
It is convenient to absorb factors of into the basic variables, which is also suggested by the
integrations in holonomies and fluxes on which background-independent quantizations are built [16]. We
thus define

together with . The symplectic structure is then independent of and so are integrated
densities such as total Hamiltonians. For the Hamiltonian constraint in isotropic Ashtekar variables we have
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.
Note that here seems to differ from in (18) by a factor of 8. This happens due to
different normalizations of the coordinate volume , 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 and thus plays a special role since this is where
isotropic classical singularities are located. On this subset the evolution equation (17) with standard matter
choices is singular in the sense that , e.g.,

for a scalar with momentum and potential , 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 [178, 114], any classical
trajectory must intersect the subset 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 is a sub-manifold in the classical phase space of
triad variables, where 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 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).