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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, [52Jump To The Next Citation Point] for recent progress in this direction.


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