Hamiltonian constraint
In the context of general relativity, the Hamiltonian constraint technically refers to a linear combination of spatial and time diffeomorphism constraints reflecting the reparametrizability of the theory under both spatial as well as time coordinates.
In its usual presentation, classical mechanics appears to give time a special role as an independent variable.
Say our system comprised a pendulum executing a simple harmonic motion and a clock.
We put these variables on the same footing by introducing a fictitious parameter
takes us continuously through every possible correlation between the displacement and reading on the clock.
are their conjugate momenta respectively and represent our extended phase space (we will show that we can recover the usual Newton's equations from this expression).
are constrained to take values on this constraint-hypersurface of the extended phase space.
The 'smeared' Hamiltonian constraint tells us how an extended phase space variable (or function thereof) evolves with respect to
In the case of the parametrized clock and pendulum system we can of course recover the usual equations of motion in which
We recover the usual differential equation for the simple harmonic oscillator,
Deparametrization and the identification of a time variable with respect to which everything evolves is the opposite process of parametrization.
General relativity being a prime physical example (here the spacetime coordinates correspond to the unphysical
and the Hamiltonian is a linear combination of constraints which generate spatial and time diffeomorphisms).
The underlining reason why we could deparametrize (aside from the fact that we already know it was an artificial reparametrization in the first place) is the mathematical form of the constraint, namely,
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,
, on the spatial slice (the metric induced on the spatial slice by the spacetime metric), and its conjugate momentum variable related to the extrinsic curvature,
, (this tells us how the spatial slice curves with respect to spacetime and is a measure of how the induced metric evolves in time).
[3] Within the loop quantum gravity representation Thiemann formulated 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[by whom?]
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.
, making them into operators acting on wavefunctions on the space of 3-metrics, and then to quantize the Hamiltonian (and other constraints).
However, this program soon became regarded as dauntingly difficult for various reasons, one being the non-polynomial nature of the Hamiltonian constraint:
Being a non-polynomial expression in the canonical variables and their derivatives it is very difficult to promote to a quantum operator.
The densitized triads are not unique, and in fact one can perform a local in space rotation with respect to the internal indices
This development raised new hopes for the canonical quantum gravity programme.
, that is Euclidean instead of Lorentzian, then one can retain the simple form of the Hamiltonian for but for real variables.
One can then define what is called a generalized Wick rotation to recover the Lorentzian theory.
[6] Generalized as it is a Wick transformation in phase space and has nothing to do with analytical continuation of the time parameter
To circumvent the problems introduced by this complicated relationship Thiemann first defines the Gauss gauge invariant quantity
where we have used that the integrated densitized trace of the extrinsic curvature is the "time derivative of the volume".
