In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric,
, (this tells us how the spatial slice curves with respect to spacetime and is a measure of how the induced metric evolves in time).
Within the loop quantum gravity representation Thomas Thiemann was able to formulate a mathematically rigorous operator as a proposal as such a constraint.
[4] Although this operator defines a complete and consistent quantum theory, doubts have been raised as to the physical reality of this theory due to inconsistencies with classical general relativity (the quantum constraint algebra closes, but it is not isomorphic to the classical constraint algebra of GR, which is seen as circumstantial evidence of inconsistencies definitely not a proof of inconsistencies), and so variants have been proposed.
However, this program soon became regarded as dauntingly difficult for various reasons, one being the non-polynomial nature of the Hamiltonian constraint: where
Being a non-polynomial expression in the canonical variables and their derivatives it is very difficult to promote to a quantum operator.
, that is Euclidean instead of Lorentzian, then one can retain the simple form of the Hamiltonian for but for real variables.
; by rotating the various indices and then adding and subtracting them (see article spin connection for more details of the derivation, although there we use slightly different notation).
of order zero, To circumvent the problems introduced by this complicated relationship Thiemann first defines the Gauss gauge invariant quantity where
can be rewritten as where we have used that the integrated densitized trace of the extrinsic curvature is the``time derivative of the volume".
In Ashtekar variables this reads, As usual the (smeared) spatial diffeomorphisn constraint is associated with the shift function
This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime.
The classical analysis with the Maxwell action followed by canonical formulation using the time gauge parametrisation results in:
Wilson loops are not independent of each other, and in fact certain linear combinations of them called spin network states form an orthonormal basis.
It can be shown that (this expresses the fact that the field strength tensor, or curvature, measures the holonomy around `infinitesimal loops').
Substituting for the holonomies, The identity will have vanishing Poisson bracket with the volume, so the only contribution will come from the connection.
; the identity term doesn't contribute as the Poisson bracket is proportional to a Pauli matrix (since
The triangulation is chosen to so as to be adapted to the spin network state one is acting on by choosing the vertices an lines appropriately.
Due to the presence of the volume the Hamiltonian constraint will only contribute when there are at least three non-coplanar lines of a vertex.
If the functions were not diffeomorphism invariant, the added line would have to be shrunk to the vertex and possible divergences could appear.
In order to recover the correct semi classical theory these extra terms need to vanish, but this implies additional constraints and reduces the number of degrees of freedom of the theory making it unphysical.
This problem can be circumvented with the use of the Master constraint (see below) allowing the just mentioned results to be applied to obtain the physical Hilbert space
In fact, repeated action of the Hamiltonian generates more and more new edges ever closer to the vertex never intersecting each other.
This implies, for instance, that for surfaces that enclose a vertex (diffeomorphically invariantly defined) the area of such surfaces would commute with the Hamiltonian, implying no "evolution" of these areas as it is the Hamiltonian that generates "evolution".
As usual, before quantisation, we need to express the constraints (and other observables) in terms of the holonomies and fluxes.
Apart from the non-Abelian nature of the gauge field, in form, the expressions proceed in the same manner as for the Maxwell case.
The conjugate variable to the point holonomy which is promoted to an operator in the quantum theory, is taken to be the smeared field momentum
In the quantum theory one looks for a representation of the Poisson bracket as a commutator of the elementary operators,
That is, Thus we proceed exactly as for the Hamiltonian constraint and introduce a partition into tetrahedra, splitting both integrals into sums, where the meaning of
However, it can be shown that graph-changing, spatially diffeomorphism invariant operators such as the Master constraint cannot be defined on the kinematic Hilbert space