Constraint in loop quantum gravity
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, formula_1, 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, formula_2, (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, 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 formula_3 and formula_4, 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:
formula_5
where formula_6 is the scalar curvature of the three metric formula_1. 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 formula_8 gauge field or connection formula_9. Its canonically conjugate momentum formula_10 is the densitized "electric" field or triad (densitized as formula_11). Their connection with gravity is that the densitized triads can be used to reconstruct the spatial metric via
formula_12.
The densitized triads are not unique, and in fact one can perform a local in space rotation with respect to the internal indices formula_13. This is actually the origin of the formula_8 gauge invariance. The connection can be used to reconstruct the extrinsic curvature. The relation is given by
formula_15
where formula_16 is related to the spin connection, formula_17, by formula_18 and formula_19.
In terms of Ashtekar variables, the classical expression of the constraint is given by
formula_20.
where formula_21 field strength tensor of the gauge field formula_9 . Due to the factor formula_23 this is non-polynomial in the Ashtekar's variables. Since we impose the condition
formula_24,
we could consider the densitized Hamiltonian formula_25 instead:
formula_26.
This Hamiltonian is now polynomial in the Ashtekar's variables. This development raised new hopes for the canonical quantum gravity programme. 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 formula_27, 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 formula_28.
Expression for real formulation of Ashtekar variables.
Thomas Thiemann was able to address both the above problems. He used the real connection
formula_29
In real Ashtekar variables the full Hamiltonian is
formula_30.
where the constant formula_31 is the Barbero-Immirzi parameter. The constant formula_32 is -1 for Lorentzian signature and +1 for Euclidean signature. The formula_16 have a complicated relationship with the desitized triads and causes serious problems upon quantization. Ashtekar variables can be seen as choosing formula_34 to make the second more complicated term was made to vanish (the first term is denoted formula_35 because for the Euclidean theory this term remains for the real choice of formula_36). Also we still have the problem of the formula_23 factor.
Thiemann was able to make it work for real formula_31. First he could simplify the troublesome formula_23 by using the identity
formula_40
where formula_41 is the volume,
formula_42.
The first term of the Hamiltonian constraint becomes
formula_43
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 formula_16 given by the densitized triads formula_45? It comes about from the compatibility condition
formula_46.
We can solve this in much the same way as the Levi-Civita connection can be calculated from the equation formula_47; 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 formula_48. The result is complicated and non-linear, but a homogeneous function of formula_10 of order zero,
formula_50.
To circumvent the problems introduced by this complicated relationship Thiemann first defines the Gauss gauge invariant quantity
formula_51
where formula_19, and notes that
formula_53.
(this is because formula_54 which comes about from the fact that formula_55 is the generator of the canonical transformation of constant rescaling, formula_56, and formula_16 is a homogeneous function of order zero). We are then able to write
formula_58
and as such find an expression in terms of the configuration variable formula_9 and formula_60 for the second term of the Hamiltonian
formula_61.
Why is it easier to quantize formula_60? This is because it can be rewritten in terms of quantities that we already know how to quantize. Specifically formula_60 can be rewritten as
formula_64
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
formula_65.
where formula_66 are spacetime indices. We define the conjugate momentum of the scalar field with the usual formula_67, the Hamiltonian can be rewritten as,
formula_68,
where formula_69 and formula_70 are the lapse and shift. In Ashtekar variables this reads,
formula_71
As usual the (smeared) spatial diffeomorphisn constraint is associated with the shift function formula_70 and the (smeared) Hamiltonian is associated with the lapse function formula_69. So we simply read off the spatial diffeomorphism and Hamiltonian constraint,
formula_74
formula_75.
These should be added (multiplied by formula_76) 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 formula_77 are contracted onto vierbiens,
formula_78.
We wish to construct a generally covariant Dirac equation. Under a flat tangent space Lorentz transformation transforms the spinor as
formula_79
We have introduced local Lorentz transformations on flat tangent space, so formula_80 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 formula_81 that allows us to gauge the Lorentz group. The covariant derivative defined with the spin connection is,
formula_82,
and is a genuine tensor and Dirac's equation is rewritten as
formula_83.
The Dirac action in covariant form is
formula_84
where formula_85 is a Dirac bi-spinor and formula_86 is its conjugate. The covariant derivative formula_87 is defined to annihilate the tetrad formula_88.
Coupling to Electromagnetic field.
The action for an electromagnetic field in curved spacetime is
formula_89
where
formula_90
is the field strength tensor, in components
formula_91
and formula_92
where the electric field is given by
formula_93
and the magnetic field is.
formula_94.
The classical analysis with the Maxwell action followed by canonical formulation using the time gauge parametrisation results in:
formula_95
formula_96
with formula_97 and formula_98 being the canonical coordinates.
Coupling to Yang–Mills field.
The action for a Yang–Mills field for some compact gauge group formula_99 in curved spacetime is
formula_100
where formula_101 is the curvature of some formula_102connection. For the standard model formula_103.
formula_104
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 
formula_105.
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
formula_106.
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
formula_107
where formula_108 indicates a path ordering so that factors for smaller values of formula_109 appear to the left, and where the formula_110 satisfy the formula_111 algebra,
formula_112.
It is easy to see from this that,
formula_113.
implies that formula_114.
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,
formula_115.
This is called the inverse loop transform. The loop transform is given by
formula_116
and is analogous to what one does when one goes over to the momentum representation in quantum mechanics,
formula_117.
The loop transform defines the loop representation. Given an operator formula_118 in the connection representation,
formula_119,
we define formula_120 by the loop transform,
formula_121.
This implies that one should define the corresponding operator formula_122 on formula_123 in the loop representation as
formula_124,
or
formula_125,
where by formula_126 we mean the operator formula_118 but with the reverse factor ordering. We evaluate the action of this operator on the spin network as a calculation in the connection representation and rearranging the result as a manipulation purely in terms of loops (one should remember that when considering the action on the spin network one should choose the operator one wishes to transform with the opposite factor ordering to the one chosen for its action on wavefunctions formula_128). This gives the physical meaning of the operator formula_122. For example, if formula_126 were a spatial diffeomorphism, then this can be thought of as keeping the connection field formula_131 of the formula_132 where it is while performing a spatial diffeomorphism on formula_133 instead. Therefore, the meaning of formula_122 is a spatial diffeomorphism on formula_133, the argument of formula_123.
The holonomy operator in the loop representation is the multiplication operator,
formula_137
Promotion of the Hamiltonian constraint to a quantum operator.
We promote the Hamiltonian constraint to a quantum operator in the loop representation. One introduces a lattice regularization procedure. we assume that space has been divided into tetrahedra formula_138. One builds an expression such that the limit in which the tetrahedra shrink in size approximates the expression for the Hamiltonian constraint.
For each tetrahedron pick a vertex and call formula_139. Let formula_140 with formula_141 be three edges ending at formula_139. We now construct a loop
formula_143
by moving along formula_140 then along the line joining the points formula_145 and formula_146 that are not formula_139 (which we have denoted formula_148) and then returning to formula_139 along formula_146. The holonomy
formula_151
along a line in the limit the tetrahedron shrinks approximates the connection via
formula_152
where formula_153 is a vector in the direction of edge formula_154. It can be shown that
formula_155.
(this expresses the fact that the field strength tensor, or curvature, measures the holonomy around `infinitesimal loops'). We are led to trying
formula_156
where the sum is over all tetrahedra formula_138. Substituting for the holonomies,
formula_158.
The identity will have vanishing Poisson bracket with the volume, so the only contribution will come