Editing Loop quantum gravity (section)

=== 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.