Real Ashtekar variables for Lorentzian signature space-times

The classically reduced Schwarzschild system has been quantized with Ashtekar variables, though not with loop quantization techniques.

“The classically reduced Schwarzschild system has been quantized using Ashtekar variables (but not using loop quantization techniques) in [24,25].”

In [24,25] the Hamiltonian constraint was solved, but together with the diffeomorphism constraint it had to be rephrased into a form not well suited for loop quantization.

“In [24,25] it was solved, but it, together with the diffeomorphism constraint, had to be rephrased into a form not well suited for loop quantization.”

The reconstructed connection form (25) differs from the connection form (26) used in [24,25] only by a gauge rotation g1 := exp(-θ n_i φ τ_i) followed by g2 := exp(-φ n_i x τ_i).

“The two connection forms differ, however, only by a gauge rotation g 1 := exp(-ϑn i ϕ τ i ) followed by a second gauge rotation g 2 := exp(-ϕn i x τ i ).”

The term cosθ dφ τ3 in (25) leads automatically to the appearance of A3 as the combination A3-√2 in the connection form (26) of [24,25].

“We note that the term cos ϑ dϕ τ 3 in equation (25), which is the only term independent of the fields a, b and c, in the connection form A(x, ϑ, ϕ) derived above leads automatically to the appearance of A 3 as the combination A 3 -√ 2 in the connection form (26) of [24,25].”

The Gauss and diffeomorphism constraints can be solved.

“Moreover, Gauss and diffeomorphism constraints can be solved (see [11] and references therein for an introduction to these concepts).”

The calculus on generalized connections modulo gauge transformations for compact gauge groups provides a rigorous kinematic framework for regularizing constraint operators unambiguously.

“This calculus provides a rigorous kinematic framework by means of which constraint operators can be regularized in a well defined and unambiguous manner.”

All versions of Hamiltonian constraint operators for canonical quantum gravity in the continuum constructed so far in the literature, whether in the connection representation or in the loop representation, provide quantizations only of the Euclidean theory.

“It should be stressed at this point that all versions of Hamiltonian constraint operators for canonical quantum gravity in the continuum that have been constructed so far in the literature, whether in the connection representation [9] or in the loop representation [8, 10], suffer from the fact that they provide quantizations only of the Euclidean theory.”

It is expected that Euclidean solutions found already in the literature for operators defined in [10,8] are far from being annihilated by our operator.

“It is therefore to be expected that the Euclidean solutions that have been found already in the literature for the operators defined in [10, 8] are far from being annihilated by our operator.”

The Euclidean operator (0.9) and the ones proposed in [10] and [8,9] are entirely different; they only share that their classical limits or the square thereof are proportional to each other.

“Not even our Euclidean operator (0.9) and the ones proposed in [10] and [8, 9] have anything to do with each other, they are entirely different . The only thing they share is that their classical limits or the square thereof are proportional to each other.”

None of the operators in [8,9,10] can be used to define K along the lines proposed here because it was crucial that the classical identity K = -{V, H_E[1]} holds.

“none of the operators in [8, 9, 10] can be used to define ˆ K along the lines proposed here because it was crucial that the classical identity K = -{ V, H E [1] } holds”

LQG and spin foams are based on the possibility of formulating Hamiltonian general relativity in terms of Yang-Mills-like connection variables.

“LQG and spin foams are based on the possibility of formulating Hamiltonian general relativity in terms of Yang-Mills-like connection variables.”

Modern LQG and spin foams are based on a relative of Ashtekar's variables known as Ashtekar-Barbero variables.

“Modern LQG and spin foams are based on a certain relative of these variables, that is often referred to as Ashtekar-Barbero variables [56, 227].”

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 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.”