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.
Contents
- Parametrization
- Dynamics of this reparametrization invariant system
- Deparametrization
- Reason why we could deparametrize here
- Hamiltonian of classical general relativity
- Metric formulation
- Expression using Ashtekar variables
- Expression for real formulation of Ashtekar variables
- References
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.
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 and a clock. Whereas the system could be described classically by a position x=x(t), with x defined as a function of time, it is also possible to describe the same system as x(
The system would be described by the position of a pendulum from the center, denoted
whose `evolution' with respect to
Dynamics of this reparametrization-invariant system
The action for the parametrized Harmonic oscillator is then
where
we identify the
Hamilton's equations for
which gives a constraint,
(these are actually the other Hamilton's equations). These equations describe a flow or orbit in phase space. In general we have
for any phase space function
Deparametrization
The other equations of Hamiltonian mechanics are
Upon substitution of our action these give,
These represent the fundamental equations governing our system.
In the case of the parametrized clock and pendulum system we can of course recover the usual equations of motion in which
Now
We recover the usual differential equation for the simple harmonic oscillator,
We also have
Our Hamiltonian constraint is then easily seen as the condition of constancy of energy! Deparametrization and the identification of a time variable with respect to which everything evolves is the opposite process of parametrization. It turns out in general that not all reparametrisation-invariant systems can be deparametrized. General relativity being a prime physical example (here the spacetime coordinates correspond to the unphysical
Reason why we could deparametrize here
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,
Substitute the Hamiltonian constraint into the original action we obtain
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,
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, 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 quantum gravity 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
where
Expression using Ashtekar variables
The configuration variables of Ashtekar's variables behave like an
The densitized triads are not unique, and in fact one can perform a local in space rotation with respect to the internal indices
where
In terms of Ashtekar variables the classical expression of the constraint is given by,
where
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.
A way of addressing the problem of reality conditions was noting that if we took the signature to be
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
where the constant
Thiemann was able to make it work for real
where
The first term of the Hamiltonian constraint becomes
upon using Thiemann's identity. This Poisson bracket is replaced by a commutator upon quantization. It turns out that a similar trick can be used to teat the second term. Why are the
We can solve this in much the same way as the Levi-Civita connection can be calculated from the equation
where
We are then able to write
and as such find an expression in terms of the configuration variable
Why is it easier to quantize
where we have used that the integrated densitized trace of the extrinsic curvature is the``time derivative of the volume".