Wikipedia / Hamiltonian constraint
Wikipedia / Hamiltonian constraint
The Hamiltonian constraint arises from any theory that admits a Hamiltonian formulation and is reparametrisation-invariant. The Hamiltonian constraint of general relativity is an important non-trivial example.
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. However, most of the time the term Hamiltonian constraint is reserved for the constraint that generates time diffeomorphisms.
Simplest example: the parametrized clock and pendulum system
Parametrization
In its usual presentation, classical mechanics appears to give time a special role as an independent variable. This is unnecessary, however. Mechanics can be formulated to treat the time variable on the same footing as the other variables in an extended phase space, by parameterizing the temporal variable(s) in terms of a common, albeit unspecified parameter variable. Phase space variables being on the same footing.
Say our system comprised a pendulum executing a simple harmonic motion, which is simply a parameter of the system, possibly having no objective meaning in its own right.
can be replaced by any monotonic function but only the relationship between these quantities. Dynamics is then determined by this relationship.
Dynamics of this reparametrization-invariant system
The action
is our Hamiltonian constraint! It could also be obtained from the Euler–Lagrange equation will take.
Deparametrization
The other equations of Hamiltonian mechanics
These represent the fundamental equations governing our system.
is the independent variable:
can be deduced by
We recover the usual differential equation
and the Hamiltonian is a linear combination of constraints which generate spatial and time diffeomorphisms).
Reason why we could deparametrize here
which is the standard action for the harmonic oscillator. General relativity is an example of a physical theory where the Hamiltonian constraint isn't of the above mathematical form in general, and so cannot be deparametrized in general.
Hamiltonian of classical general relativity
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, (this tells us how the spatial slice curves with respect to spacetime and is a measure of how the induced metric evolves in time). These are the metric canonical coordinates.
Dynamics such as time-evolutions of fields are controlled by the Hamiltonian constraint.
The identity of the Hamiltonian constraint is a major open question in quantum gravity, as is extracting of physical observables from any such specific constraint.
In 1986, Abhay Ashtekar introduced a new set of canonical variables, the Ashtekar variables, to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable. The Hamiltonian was much simplified in this reformulation. This led to the loop representation of quantum general relativity and in turn loop quantum gravity.
Within the loop representation, Thiemann formulated a mathematically rigorous operator as a proposal as such a constraint. 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.
Metric formulation
The idea was to quantize the canonical variables is the scalar curvature. Being a non-polynomial expression in the canonical variables and their derivatives it is very difficult to promote to a quantum operator.
Expression using Ashtekar variables
The configuration variables of Ashtekar's variables). What do these variables have to do with gravity? The densitized triads can be used to reconstruct the spatial metric via
The densitized triads are not unique, and in fact one can perform a local in space rotation gauge invariance. The connection can be used to reconstruct the extrinsic curvature. The relation is given by
is related to the spin connection.
In terms of Ashtekar variables the classical expression of the constraint is given by,
this is non-polynomial in the Ashtekar's variables. Since we impose the condition
we could consider the densitized Hamiltonian instead,
This Hamiltonian is now polynomial the Ashtekar's variables. This development raised new hopes for the canonical quantum gravity programme. Although Ashtekar variables had the virtue of simplifying the Hamiltonian, it has the problem that the variables become complex. When one quantizes the theory it is a difficult task ensure that one recovers real general relativity as opposed to complex general relativity. Also there were also serious difficulties promoting the densitized Hamiltonian 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. One can then define what is called a generalized Wick rotation to recover the Lorentzian theory..
Expression for real formulation of Ashtekar variables
Thomas Thiemann addressed both the above problems. He used the real connection
In real Ashtekar variables the full Hamiltonian is
is the Barbero–Immirzi parameter. factor.
by using the identity
is the volume,
The first term of the Hamiltonian constraint becomes
? It actually come about from the Gauss Law
We can solve this in much the same way as the Levi-Civita connection; by rotating the various indices and then adding and subtracting them. The result is complicated and non-linear. To circumvent the problems introduced by this complicated relationship Thiemann first defines the Gauss gauge invariant quantity
, and notes that
We are then able to write
. We obtain for the second term of the Hamiltonian
can be rewritten as
where we have used that the integrated densitized trace of the extrinsic curvature is the "time derivative of the volume".