Loop space representation of quantum fermions and gravity

The gravitational connection is real only when fermions are represented by half-density variables; using scalar variables makes the connection complex.

“As we will see in the appendix, the connection is real only if we use the quantities ξ with density weight 1 2 , if we would use the scalar variables η as in [9] then the gravitational connection becomes by the argument given in [18] A”

The fermionic excitations produced by this regularization differ greatly from those in the cited earlier work, because the reality conditions are faithfully implemented.

“Quite surprisingly, these fermionic excitations are very different from those discussed in [8,9], the underlying reason being the faithful implementation of the reality conditions.”

The strategy of working with ψ as advertised in prior work is not suitable for quantizing the Einstein-Dirac theory, so the framework developed in another prior work is forced.

“Therefore the strategy of working with ψ as advertised in [9] is not suitable for quantizing the Einstein-Dirac theory and we are forced to adapt the framework developed in [7].”

The kinematical framework developed in this paper is applied in [22], which extends the dynamical framework of [16,17,18,19,20] to the standard model coupled to gravity and other kind of matter.

“The kinematical framework developed in this paper is applied in [22] which extends the dynamical framework of [16, 17, 18, 19, 20] to the standard model coupled to gravity and other kind of matter.”

The same techniques can be employed to quantize Euclidean 2+1 gravity in the real-valued Ashtekar formulation, giving expected results, and the ADM energy.

“Moreover, the same techniques can be employed to quantize a) Euclidean 2+1 gravity in the real-valued Ashtekar- (rather than Witten-) formulation [19] giving the expected results and b) the ADM energy [20].”