Editing Loop quantum gravity (section)

== Improved dynamics and the master constraint ==
{{main|Hamiltonian (quantum mechanics)|Hamiltonian constraint of LQG|Friedrichs extension}}

=== The master constraint ===
Thiemann's Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Hamiltonian constraint equations in terms of a single master constraint <math>M</math>, which involves the square of the constraints in question. An initial objection to the use of the master constraint was that on first sight it did not seem to encode information about the observables; because the Master constraint is quadratic in the constraint, when one computes its Poisson bracket with any quantity, the result is proportional to the constraint, therefore it vanishes when the constraints are imposed and as such does not select out particular phase space functions. However, it was realized that the condition 
<math display="block">\{ O , \{ O , M \} \}_{M = 0} = 0,</math>
is where <math>O</math> is at least a twice differentiable function on phase space is equivalent to <math>O</math> being a weak Dirac observable with respect to the constraints in question. So the master constraint does capture information about the observables. Because of its significance this is known as the master equation.{{sfn|Thiemann|2006a|pp=2211–2247}}

That the master constraint Poisson algebra is an honest Lie algebra opens the possibility of using a method, known as group averaging, in order to construct solutions of the infinite number of Hamiltonian constraints, a physical inner product thereon and [[Dirac observables]] via what is known as [[refined algebraic quantization]], or RAQ.<ref>{{cite arXiv |last=Thiemann |first=Thomas |title=Introduction to Modern Canonical Quantum General Relativity |date=2001-10-05 |eprint=gr-qc/0110034}}</ref>

=== The quantum master constraint ===
Define the quantum master constraint (regularisation issues aside) as
<math display="block">\hat{M} := \int d^3x \widehat{\left( \frac{H}{\sqrt[4]{\det (q(x))}} \right)}^\dagger(x) \widehat{\left(\frac{H}{\sqrt[4]{\det (q(x))}} \right)} (x). </math>

Obviously,
<math display="block">\widehat{\left( \frac{H}{\sqrt[4]{\det (q(x))}} \right)} (x) \Psi = 0</math>
for all <math>x</math> implies <math>\hat{M} \Psi = 0</math>. Conversely, if <math>\hat{M} \Psi = 0</math> then
<math display="block">0 = \left \langle \Psi , \hat{M} \Psi \right \rangle = \int d^3x \left\| \widehat{\left( \frac{H}{\sqrt[4]{\det (q(x))}} \right)} (x) \Psi \right\|^2 \qquad Eq \; 4</math>
implies
<math display="block">\widehat{\left( \frac{H}{\sqrt[4]{\det (q(x))}} \right)} (x) \Psi = 0.</math>

First compute the matrix elements of the would-be operator <math>\hat{M}</math>, that is, the quadratic form <math>Q_M</math>. <math>Q_M</math> is a graph changing, diffeomorphism invariant quadratic form that cannot exist on the kinematic Hilbert space <math>H_{Kin}</math>, and must be defined on <math> H_{Diff}</math>. Since the master constraint operator <math>\hat{M}</math> is [[densely defined]] on <math>H_{Diff}</math>, then <math>\hat{M}</math> is a positive and [[symmetric operator]] in <math>H_{Diff}</math>. Therefore, the quadratic form <math>Q_M</math> associated with <math>\hat{M}</math> is [[closable]]. The closure of <math>Q_M</math> is the quadratic form of a unique [[self-adjoint operator]] <math>\hat{\overline{M}}</math>, called the [[Friedrichs extension]] of <math>\hat{M}</math>. We relabel <math>\hat{\overline{M}}</math> as <math>\hat{M}</math> for simplicity.

Note that the presence of an inner product, viz Eq 4, means there are no superfluous solutions i.e. there are no <math>\Psi</math> such that
<math display="block">\widehat{\left( \frac{H}{\sqrt[4]{\det (q(x))}} \right)} (x) \Psi \not= 0,</math>
but for which <math>\hat{M} \Psi = 0</math>.

It is also possible to construct a quadratic form <math>Q_{M_E}</math> for what is called the extended master constraint (discussed below) on <math>H_{Kin}</math> which also involves the weighted integral of the square of the spatial diffeomorphism constraint (this is possible because <math>Q_{M_E}</math> is not graph changing).

The spectrum of the master constraint may not contain zero due to normal or factor ordering effects which are finite but similar in nature to the infinite vacuum energies of background-dependent quantum field theories. In this case it turns out to be physically correct to replace <math>\hat{M}</math> with <math>\hat{M}' := \hat{M} - \min (spec (\hat{M})) \hat{1}</math> provided that the "normal ordering constant" vanishes in the classical limit, that is,
<math display="block">\lim_{\hbar \to 0} \min (spec(\hat{M})) = 0,</math>
so that <math>\hat{M}'</math> is a valid quantisation of <math>M</math>.

=== Testing the master constraint ===
The constraints in their primitive form are rather singular, this was the reason for integrating them over test functions to obtain smeared constraints. However, it would appear that the equation for the master constraint, given above, is even more singular involving the product of two primitive constraints (although integrated over space). Squaring the constraint is dangerous as it could lead to worsened ultraviolet behaviour of the corresponding operator and hence the master constraint programme must be approached with care.

In doing so the master constraint programme has been satisfactorily tested in a number of model systems with non-trivial constraint algebras, free and interacting field theories.{{sfn|Dittrich|Thiemann|2006a|pp=1025–1066}}{{sfn|Dittrich|Thiemann|2006b|pp=1067–1088}}{{sfn|Dittrich|Thiemann|2006c|pp=1089–1120}}{{sfn|Dittrich|Thiemann|2006d|pp=1121–1142}}{{sfn|Dittrich|Thiemann|2006e|pp=1143–1162}} The master constraint for LQG was established as a genuine positive self-adjoint operator and the physical Hilbert space of LQG was shown to be non-empty,{{sfn|Thiemann|2006b|pp=2249–2265}} a consistency test LQG must pass to be a viable theory of quantum general relativity.

=== Applications of the master constraint ===
The master constraint has been employed in attempts to approximate the physical inner product and define more rigorous path integrals.{{sfn|Bahr|Thiemann|2007|pp=2109–2138}}{{sfn|Han|Thiemann|2010a|p=225019}}{{sfn|Han|Thiemann|2010b|p=092501}}{{sfn|Han|2010|p=215009}}

The Consistent Discretizations approach to LQG,{{sfn|Gambini|Pullin|2009|p=035002}}{{sfn|Gambini|Pullin|2011|loc=Section 10.2.2}} is an application of the master constraint program to construct the physical Hilbert space of the canonical theory.

=== Spin foam from the master constraint ===
The master constraint is easily generalized to incorporate the other constraints. It is then referred to as the extended master constraint, denoted <math>M_E</math>. We can define the extended master constraint which imposes both the Hamiltonian constraint and spatial diffeomorphism constraint as a single operator,
<math display="block">M_E = \int_\Sigma d^3x {H (x)^2 - q^{ab} V_a (x) V_b (x) \over \sqrt{\det (q)}} .</math>

Setting this single constraint to zero is equivalent to <math>H(x) = 0</math> and <math>V_a (x) = 0</math> for all <math>x</math> in <math>\Sigma</math>. This constraint implements the spatial diffeomorphism and Hamiltonian constraint at the same time on the Kinematic Hilbert space. The physical inner product is then defined as
<math display="block">\langle\phi, \psi\rangle_{\text{Phys}} = \lim_{T \to \infty} \left\langle\phi, \int_{-T}^T dt e^{i t \hat{M}_E} \psi\right\rangle</math>
(as <math display="inline">\delta (\hat{M_E}) = \lim_{T \to \infty} \int_{-T}^T dt e^{i t \hat{M}_E}</math>). A spin foam representation of this expression is obtained by splitting the <math>t</math>-parameter in discrete steps and writing
<math display="block">e^{i t \hat{M}_E} = \lim_{n \to \infty} \left [e^{i t \hat{M}_E / n} \right]^n = \lim_{n \to \infty} [1 + i t \hat{M}_E / n]^n.</math>

The spin foam description then follows from the application of <math>[1 + i t \hat{M}_E / n]</math> on a spin network resulting in a linear combination of new spin networks whose graph and labels have been modified. Obviously an approximation is made by truncating the value of <math>n</math> to some finite integer. An advantage of the extended master constraint is that we are working at the kinematic level and so far it is only here we have access semiclassical coherent states. Moreover, one can find none graph changing versions of this master constraint operator, which are the only type of operators appropriate for these coherent states.

=== Algebraic quantum gravity (AQG) ===
The master constraint programme has evolved into a fully combinatorial treatment of gravity known as algebraic quantum gravity (AQG).{{sfn|Giesel|Thiemann|2007a|pp=2465–2498}} The non-graph changing master constraint operator is adapted in the framework of algebraic quantum gravity. While AQG is inspired by LQG, it differs drastically from it because in AQG there is fundamentally no topology or differential structure – it is background independent in a more generalized sense and could possibly have something to say about topology change. In this new formulation of quantum gravity AQG semiclassical states always control the fluctuations of all present degrees of freedom. This makes the AQG semiclassical analysis superior over that of LQG, and progress has been made in establishing it has the correct semiclassical limit and providing contact with familiar low energy physics.{{sfn|Giesel|Thiemann|2007b|pp=2499–2564}}{{sfn|Giesel|Thiemann|2007c|pp=2565–2588}}