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 distance function 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).Gravitation by Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, published by W. H. Freeman and company. New York. 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, 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 Thomas Thiemann was able to formulate 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.
Classical expressions for the Hamiltonian
Metric formulation
The idea was to quantize the canonical variables and , 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:
where is the scalar curvature of the three metric . 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 behave like an gauge field or connection . Its canonically conjugate momentum is the densitized "electric" field or triad (densitized as ). Their connection with gravity is that 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 with respect to the internal indices . This is actually the origin of the gauge invariance. The connection can be used to reconstruct the extrinsic curvature. The relation is given by
where is related to the spin connection, , by and .
In terms of Ashtekar variables, the classical expression of the constraint is given by
.
where field strength tensor of the gauge field . Due to the factor 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 in the Ashtekar's variables. This development raised new hopes for the canonical quantum gravity programme.See the book Lectures on Non-Perturbative Canonical Gravity for more details on this and the subsequent development. First published in 1991. World Scientific Publishing Co. Pte. LtD. Although Ashtekar variables have the virtue of simplifying the Hamiltonian, it has the problem that the variables become complex numbers. When one quantizes the theory, it is a difficult task to ensure that one recovers real general relativity as opposed to complex general relativity. There are 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 , 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. It is a Wick transformation in phase space and has nothing to do with analytical continuation of the time parameter .
Expression for real formulation of Ashtekar variables
Thomas Thiemann was able to address both the above problems. He used the real connection
In real Ashtekar variables the full Hamiltonian is
.
where the constant is the Barbero-Immirzi parameter. The constant is -1 for Lorentzian signature and +1 for Euclidean signature. The have a complicated relationship with the densitized triads and causes serious problems upon quantization. Ashtekar variables can be seen as choosing to make the second more complicated term was made to vanish (the first term is denoted because for the Euclidean theory this term remains for the real choice of ). Also we still have the problem of the factor.
Thiemann was able to make it work for real . First he could simplify the troublesome by using the identity
where is the volume,
.
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 given by the densitized triads ? It comes about from the compatibility condition
.
We can solve this in much the same way as the Levi-Civita connection can be calculated from the equation ; 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). We then rewrite this in terms of the densitized triad using that . The result is complicated and non-linear, but a homogeneous function of of order zero,
.
To circumvent the problems introduced by this complicated relationship Thiemann first defines the Gauss gauge invariant quantity
where , and notes that
.
(this is because which comes about from the fact that is the generator of the canonical transformation of constant rescaling, , and is a homogeneous function of order zero). We are then able to write
and as such find an expression in terms of the configuration variable and for the second term of the Hamiltonian
.
Why is it easier to quantize ? This is because it can be rewritten in terms of quantities that we already know how to quantize. Specifically can be rewritten as
where we have used that the integrated densitized trace of the extrinsic curvature is the``time derivative of the volume".
Coupling to matter
Coupling to scalar field
The Lagrangian for a scalar field in curved spacetime
.
where are spacetime indices. We define the conjugate momentum of the scalar field with the usual , the Hamiltonian can be rewritten as,
,
where and are the lapse and shift. In Ashtekar variables this reads,
As usual the (smeared) spatial diffeomorphisn constraint is associated with the shift function and the (smeared) Hamiltonian is associated with the lapse function . So we simply read off the spatial diffeomorphism and Hamiltonian constraint,
.
These should be added (multiplied by ) to the spatial diffeomorphism and Hamiltonian constraint of the gravitational field, respectively. This represents the coupling of scalar matter to gravity.
Coupling to Fermionic field
There are problems coupling gravity to spinor fields: there are no finite-dimensional spinor representations of the general covariance group. However, there are of course spinorial representations of the Lorentz group. This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime. The Dirac matrices are contracted onto vierbeins,
.
We wish to construct a generally covariant Dirac equation. Under a flat tangent space Lorentz transformation transforms the spinor as
We have introduced local Lorentz transformations on flat tangent space, so is a function of space-time. This means that the partial derivative of a spinor is no longer a genuine tensor. As usual, one introduces a connection field that allows us to gauge the Lorentz group. The covariant derivative defined with the spin connection is,
,
and is a genuine tensor and Dirac's equation is rewritten as
.
The Dirac action in covariant form is
where is a Dirac bi-spinor and is its conjugate. The covariant derivative is defined to annihilate the tetrad .
Coupling to Electromagnetic field
The action for an electromagnetic field in curved spacetime is
where
is the field strength tensor, in components
and
where the electric field is given by
and the magnetic field is.
.
The classical analysis with the Maxwell action followed by canonical formulation using the time gauge parametrisation results in:
with and being the canonical coordinates.
Coupling to Yang–Mills field
The action for a Yang–Mills field for some compact gauge group in curved spacetime is
where is the curvature of some connection. For the standard model .
Total Hamiltonian of matter coupled to gravity
The dynamics of the coupled gravity-matter system is simply defined by the adding of terms defining the matter dynamics to the gravitational hamiltonian. The full hamiltonian is described by
.
Quantum Hamiltonian constraint
In this section we discuss the quantization of the hamiltonian of pure gravity, that is in the absence of matter. The case of inclusion of matter is discussed in the next section.
The constraints in their primitive form are rather singular, and so should be `smeared' by appropriate test functions. The Hamiltonian is the written as
.
For simplicity we are only considering the "Euclidean" part of the Hamiltonian constraint, extension to the full constraint can be found in the literature. There are actually many different choices for functions, and so what one then ends up with an (smeared) Hamiltonians constraints. Demanding them all to vanish is equivalent to the original description.
The loop representation
The Wilson loop is defined as
where indicates a path ordering so that factors for smaller values of appear to the left, and where the satisfy the algebra,
.
It is easy to see from this that,
.
implies that .
Wilson loops are not independent of each other, and in fact certain linear combinations of them called spin network states form an orthonormal basis. As spin network functions form a basis we can formally expand any Gauss gauge invariant function as,
.
This is called the inverse loop transform. The loop transform is given by
and is analogous to what one does when one goes over to the momentum representation in quantum mechanics,
.
The loop transform defines the loop representation. Given an operator in the connection representation,
,
we define by the loop transform,
.
This implies that one should define the corresponding operator on in the loop representation as
,
or
,