### 2.8 A note on the constraints

We have focused in the above presentation on the dynamical equations of motion. The constraints were
only briefly mentioned, with no discussion, except for the Hamiltonian constraint. This is legitimate because
the constraints are first class and hence preserved by the Hamiltonian evolution. Thus, they need only be
imposed at some “initial” time. Once this is done, one does not need to worry about them any more.
Furthermore the momentum constraints and Gauss’ law constraints are differential equations relating the
initial data at different spatial points. This means that they do not constrain the dynamical variables at a
given point but involve also their gradients – contrary to the Hamiltonian constraint which
becomes ultralocal. Consequently, at any given point, one can freely choose the initial data on
the undifferentiated dynamical variables and then use these data as (part of) the appropriate
boundary data necessary to integrate the constraints throughout space. This is why one can
assert that all the walls described above are generically present even when the constraints are
satisfied.
The situation is different in homogeneous cosmologies where the symmetry relates the values of the fields
at all spatial points. The momentum and Gauss’ law constraints become then algebraic equations and might
remove some relevant walls. But this feature (removal of walls by the momentum and Gauss’ law
constraints) is specific to some homogeneous cosmologies and does not hold in the generic case where spatial
gradients are non-zero.

A final comment: How the spatial diffeomorphism constraints and Gauss’ law fit in the conjectured
infinite-dimensional symmetry is a point that is still poorly understood. See, however, [52] for recent
progress in this direction.