New Hamiltonian formulation of general relativity

For the limiting procedure in [13] to be well-defined, an implicit assumption on continuity properties of loop states is needed; that assumption is not satisfied by the states of interest, so an alternative limiting procedure is needed.

“Second, for the limiting procedure which removes the regulator to be well-defined, there is an implicit assumption on the continuity properties of loop states (spelled out in detail in [30]). A careful examination shows that this assumption is not satisfied by the states of interest and hence an alternative limiting procedure, analogous to that discussed in section 3.1, is needed.”

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].”

The calculation shows c is identified with A1 and (a,b) with (A2,A3) up to a rescaling by √2 used in [24].

“The calculation above shows that c has to be identified with A 1 and (up to a rescaling with √ 2 used in [24] in order to obtain the standard symplectic structure) (a, b) with (A 2 , A 3 ).”

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 k=1 sectors describe degenerate sectors different from the degenerate sector found in [24].

“Thus, the sectors with k = 1 describe degenerate sectors which are, however, different from that found in [24].”

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 ).”

A solution to problem 1 has been suggested: one can define real Ashtekar variables which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to the Lorentzian regime where complex Ashtekar variables are needed to simplify the Lorentzian constraint.

“A solution to problem 1) has been suggested in [4] (see also [5]) : namely, one can define real Ashtekar variables [6] which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to the Lorentzian regime where the complex Ashtekar variables are needed to simplify the Lorentzian constraint.”

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.”

For the Einstein-Dirac theory in variables (A^i_a, E^a_i), the symplectic structure for each fermionic degree of freedom is Θ = i/2 ∫ d^3x √det(q)[η ̇η - ̇ηη] with scalar-valued η.

“The second reason becomes evident when recalling that for the Einstein-Dirac theory, formulated in terms of the variables ( A i a , E a i ) [22] which have proved to be successful for quantizing the Wheeler-DeWitt constraint [16, 17], the symplectic structure for each of the fermionic degrees of freedom is given by Θ = i 2 ∫ Σ d 3 x √ det( q )[ η ˙ η -˙ ηη ] with scalar-valued η .”