Editing Loop quantum gravity (section)

=== 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>.