### 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 [15]. 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 notations in the loop quantum gravity
literature.)
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 [123, 80], 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 submanifold 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
nonetheless, but it will have important ramifications in the quantum theory (see the sections following
Section 5.13).