Knot Theory and Quantum Gravity

The Hamiltonian H is background independent.

“is nevertheless diffeomorphism invariant, background independent and (as we have argued elsewhere [3, 4, 12, 13] is implied by these conditions) finite.”

Replacing G with the quantum group G_q at q=1 a root of unity, which has a finite set of inequivalent irreducible representations, and colouring spin foams with G_q representations and intertwiners while replacing A by its natural generalization yields a finite sum.

“This can be accomplished by replacing G with the quantum group G q with q = 1 a root of unity, which has a finite set of inequivalent irreducible representations. The spin foams are coloured with the representations and intertwiners of G q , and A is replaced by its natural generalization to such representations [16][18] yielding a finite sum.”

The spectrum of geometrical operators in the coherent-state formulation turns out to be quite different from the standard one of loop quantum gravity.

“Furthermore, the spectrum of the geometrical operators in this formulation turns out to be quite different from the standard one of loop quantum gravity [14].”

The Barrett-Crane vertex W(j) is a function of the ten spins alone.

“where W ( j ) is the Barrett-Crane vertex, which a functions of the ten spins alone.”

The transition amplitude matching between coherent states includes a phase factor exactly balancing those phases, as in the free propagator of a non-relativistic quantum particle and in the calculation illustrated in [4, 5].

“This is the case for instance for the free propagator of a non-relativistic quantum particles, as well as for the phases associated to the angles between tetrahedra in the calculation illustrated in [4, 5].”

There is a structural difficulty with the definition (64) of the amplitude.

“Second, as already mentioned, there is in fact a structural difficulty, already pointed out in [4, 5], with the definition (64) of the amplitude, and we think that this difficulty is at the roots of the problem.”

In loop quantum gravity, space geometry has no fixed structural property at all.

“Space geometry, indeed, has no fixed structural property at all in this approach.”

The apparent conflict between Lorentz contraction and Planck scale discreteness is resolved in loop quantum gravity.

“Here, we show how the apparent conflict between Lorentz contraction and Planck scale discreteness is resolved in loop quantum gravity [3]”

In loop quantum gravity, the geometry of space comes from a quantum field, the quantum gravitational field.

“The geometry of space comes from a quantum field, the quantum gravitational field.”

The kinematics of loop quantum gravity is rather well understood, while its dynamics is not understood as cleanly.

“While the kinematics of loop quantum gravity (LQG) [1] is rather well understood [2, 3], its dynamics is not understood as cleanly.”

A spin foam picture emerges from considering the time evolution of spin networks.

“A spin foam picture emerges when considering the evolution in time of spin networks [16,17], which were discovered to represent states of quantum general relativity at the kinematical level [18][19][20][21][22], from the canonical side.”

In canonical quantum gravity, the matrix elements of the proper-time evolution operator between spin network states are transition amplitudes, and computing them is equivalent to solving the theory by imposing the Hamiltonian and diffeomorphism constraints.

“Now we want to calculate the matrix elements of this operator between two spin network states (two s-knots).These are to be interpreted as transition amplitudes between quantum states of the gravitational field, and computing them is equivalent to having solved the theory imposing both the Hamiltonian and diffeomorphism constraints.”

To first order, the computation produces a 4d version of the 'nutshell' 3d model studied in [6].

“To this order, the computation produces a 4d version of the 'nutshell' 3d model studied in [6].”

It was shown in [15] that one can derive the Bekenstein-Hawking entropy formula Entropy = const × A from loop quantum gravity, with a constant const that is finite but has an incorrect value.

“It was shown in [15] that one can derive the Bekenstein-Hawking entropy formula Entropy = const × A from loop quantum gravity - with a constant const that turns out to be finite, but with an incorrect value.”

The loop approach to quantum general relativity seems to be leading to a realization of a combinatorial space-time, deriving it solely from a strict application of conventional quantum ideas to standard general relativity.

“The loop approach to quantum general relativity [2,3] seems to be leading precisely to a realization of such a vision of a combinatorial space-time, deriving it solely from a strict application of conventional quantum ideas to standard general relativity.”

The expectation value of the volume operator in GFT is given by the sum over spin representations of V_j ρ_j^2, as constructed from the first-quantized counterpart in LQG.

“Since the volume operator is diagonal in the basis of spin representations, one has for its expectation value”