The physical Hamiltonian in nonperturbative quantum gravity

It is possible to explicitly implement the proposal of using a matter degree of freedom as a physical clock in the full theory of quantum general relativity using the nonperturbative approach based on Ashtekar variables and the loop representation.

“In this letter we show that it is possible to explicitly implement this proposal in the full theory of quantum general relativity, using the nonperturbative approach [3, 4, 5] based on the Ashtekar variables [6] and the loop representation [7, 8].”

• cites [3] [3] C. Rovelli, Classical and Quantum Gravity, 8 (1991) 1613-1676; A. Ashtekar, Non-perturbative canonical gravity , Lecture notes prepared in collaboration with R. S. Tate. (World Scientific, Singapore,1991). - sourceNot checked yet

  Not checked in the free run.

• cites [4] [4] L. Smolin, Recent developments in nonperturbative quantum gravity in Quantum gravity and cosmology , the Proceedings of the 1991 GIFT International Seminar on Theoretical Physics: Quantum Gravity and Cosmology , (World Scientific, Singapore, 1992). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [5] [5] A. Ashtekar, in the Proceedings of the 1992 Les Houches lectures. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [6] [6] A. V. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986) ; Phys. Rev. D 36 (1987) 1587. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [7] [7] C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B133 (1990) 80. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [8] [8] R. Gambini and A. Trias, Phys. Rev. D23 (1981) 553, Lett. al Nuovo Cimento 38 (1983) 497; Phys. Rev. Lett. 53 (1984) 2359; Nucl. Phys. B278 (1986) 436; R. Gambini, L. Leal and A. Trias, Phys. Rev. D39 (1989) 3127; R. Gambini, Phys. Lett. B 255 (1991) 180; R. Gambini and L. Leal, Loop space coordinates, linear representations of the diffeomorphism group and knot invariants preprint, University of Montevideo,1991. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The problem of constructing the Hamiltonian H is reduced to a combinatorial and algebraic problem involving rearrangements of lines through the vertices of arbitrary graphs.

“The problem of constructing this Hamiltonian is reduced to a combinatorial and algebraic problem which involves the rearrangements of lines through the vertices of arbitrary graphs.”

• Also stated in: § Abstract · p. 2 · boosted

  “Thus, our main result is the reduction of the problem of computing the physical evolution of the quantum gravitational field with respect to a clock to a problem in graph theory and combinatorics .”

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.”

• cites [3] [3] C. Rovelli, Classical and Quantum Gravity, 8 (1991) 1613-1676; A. Ashtekar, Non-perturbative canonical gravity , Lecture notes prepared in collaboration with R. S. Tate. (World Scientific, Singapore,1991). - sourceNot checked yet

  Not checked in the free run.

• cites [4] [4] L. Smolin, Recent developments in nonperturbative quantum gravity in Quantum gravity and cosmology , the Proceedings of the 1991 GIFT International Seminar on Theoretical Physics: Quantum Gravity and Cosmology , (World Scientific, Singapore, 1992). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [12] [12] C. Rovelli, A generally covariant quantum field theory and a prediction on quantum measurements of geometry , in print on Nucl. Phys. B (1993). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [13] [13] L. Smolin Finite, diffeomorphism invariant observables in quantum gravity , Syracuse preprint, January (1993). - sourceNot checked yet

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The procedure provides a construction of the operator corresponding to the spatial volume of the universe.

“as well as a construction of the operator corresponding to the spatial volume of the universe.”

• Also stated in: § Abstract · p. 9 · plain

  “with the Einstein term left out, our construction provides a definition of the diffeomorphism invariant operator for the volume of the universe.”

The Hamiltonian H is diffeomorphism invariant.

“and is shown to be finite and diffeomorphism invariant.”

• Also stated in: § Abstract · p. 7 · plain

  “The action indicated by (17) is finite and diffeomorphism invariant.”

The Hamiltonian H is finite.

“and is shown to be finite and diffeomorphism invariant.”

• Also stated in: § Abstract · p. 7 · plain

  “The action indicated by (17) is finite and diffeomorphism invariant.”

It is possible to construct the Hamiltonian operator H that gives evolution in the clock time provided by a scalar field.

“In particular, we use a scalar field as a clock as suggested in several recent papers [9, 10], and we show that it is possible to construct the hamiltonian operator ˆ H that gives the evolution in this clock time.”

• cites [9] [9] K. Kuchar, in the Brill Feschrift Proceedings, ed. B-l Hu and T. Jacobson, (CUP,Cambridge,1993). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [10] [10] L. Smolin, in the Brill Feschrift Proceedings, op. cit. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A quantum Hamiltonian H that evolves the gravitational field according to time measured by constant surfaces of a scalar field is defined through a regularization procedure based on the loop representation.

“A quantum Hamiltonian which evolves the gravitational field according to time as measured by constant surfaces of a scalar field is defined through a regularization procedure based on the loop representation”

The procedure provides a construction of the Hamiltonian constraint as a finite operator on the space of diffeomorphism invariant states.

“This procedure also provides a construction of the Hamiltonian constraint as a finite operator on the space of diffeomorphism invariant states”

• Also stated in: § Abstract · p. 9 · plain

  “the techniques described here also provide a finite expression for the (full) hamiltonian constraint; in fact, we can define a finite operator that corresponds to H ( f ) = ∫ Σ f √ C for any f .”

The operator M_i has the form of an infinite dimensional matrix.

“The operator M i then has the form of an infinite dimensional matrix”

If there are four or less distinct tangent vectors at the point of intersection, V_i is finite dimensional.

“If there are four or less distinct tangent vectors at the point of intersection V i is finite dimensional, as the only diffeomorphism invariant information is contained in the rooting and in the possible coincidences of three or more tangent vectors in planes.”

For fixed α, and for small enough δ, the action of the operator becomes independent of δ, because the added loop does not link any other component of α.

“For fixed α , and for small enough δ the action of the operator becomes independent of δ , because the added loop doesn't link any other component of α .”

The limits δ → 0, A → 0 and L → 0 can be taken, respecting δ ≪ L and δ^2 ≪ A, so that the resulting operator in (6) is finite.

“We can accomplish this task if we pose L = κδ and A = κ 3 δ 2 and take first the limit δ → 0 at fixed κ , followed by the limit κ →∞ .”

These states satisfy a Schroedinger-like equation.

“These states then satisfy a Schroedingerlike equation”

The subspace S[K_P, m_1, ..., m_n] is isomorphic to the tensor product of a linear space V_i for every intersection i.

“This subspace is isomorphic to the tensor product of a linear space V i for every intersection i .”

If there are five or more distinct lines entering the intersection, the diffeomorphism equivalence classes form a finite dimensional configuration space which in general is not discrete.

“However, if there are five or more distinct lines entering the intersection, the diffeomorphism equivalence classes form a finite dimensional configuration space which in general is not discrete 2 , because in general the diffeomorphism equivalence classes of loops with intersections are also labelled by continuous parameters.”

The limit taken in (6) does not define an area derivative.

“Second, as the A -1 in (7) is cancelled against other factors, the limit taken in (6) does not define an area derivative.”

To zeroth order in κ^-1 and δ, the action of O^A_I is diffeomorphism covariant, in the sense that O^A_I Ψ[φ∘α] = O^A_{φ^-1∘I} Ψ[α].

“To zeroth order in κ -1 and δ , the action of O A I is diffeomorphism covariant, in the sense that O A I Ψ[ φ ◦ α ] = O A φ -1 ◦ I Ψ[ α ]”

The dependence on the regularization parameters cancels.

“As the dependence on the regularization parameters cancels, we may go on to discuss the diffeomorphism invariance of the operator.”

The sum over faces leads to the diffeomorphism covariant quantum action (12).

“At the same time, this sum leads (see [11]) to the diffeomorphism covariant quantum action”

• cites [11] [11] A. Ashtekar, C. Rovelli and L. Smolin, Phys. Rev. Lett. 69 (1992) 237. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Einstein term is nonvanishing only in the first upper diagonal in the indices m_i and K_P: more precisely, it has the form δ_{m_i+1}^{ ̃m_i} δ_{K_P+2}^{ ̃K_P}, times a matrix in the a_i space.

“The Einstein term is nonvanishing only in the first upper diagonal in these indices: more precisely, it has the form δ m i +1 ˜ m i δ K P +2 ˜ K P , times a matrix in the a i space.”

The term coming from det(q) is diagonal in the indices m_i and K_P, because it only rearranges the lines at the intersection.

“The term coming from det( q ) is diagonal in the indices m i and K P , because it only rearranges the lines at the intersection.”

This structure gives M_i a relatively simple block diagonal and upper-diagonal form.

“This structure gives M i a relatively simple 'block diagonal and upper-diagonal' form”

On the diffeomorphism invariant states the limit exists trivially because close to δ = 0, O^{κ^3δ^2}_I Ψ[α] is constant in δ, provided that the intersection remains in the box as the box is scaled down.

“It follows that on the diffeomorphism invariant states the limit exists trivially because close to δ = 0 we have that O ( κ 3 δ 2 ) I Ψ[ α ] is constant in δ (provided that the intersection remains in the box as the box is scaled down).”

The three-surfaces of constant values of the scalar field T may be taken, under appropriate circumstances, to represent time.

“We begin by introducing the scalar field T ( x ), whose three-surfaces of constant values may be taken, under appropriate circumstances, to represent time [9, 10].”

• cites [9] [9] K. Kuchar, in the Brill Feschrift Proceedings, ed. B-l Hu and T. Jacobson, (CUP,Cambridge,1993). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [10] [10] L. Smolin, in the Brill Feschrift Proceedings, op. cit. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Hamiltonian H is represented by an infinite dimensional matrix in knot space.

“ˆ H is represented by an infinite dimensional matrix in knot space.”

The gauge fixing ∂_a T = 0 implies that the lapse is N(x) = a/π(x) for some constant a.

“It is not difficult to show [10] that this implies that the lapse is N ( x ) = a/π ( x ) for some constant a”

• cites [10] [10] L. Smolin, in the Brill Feschrift Proceedings, op. cit. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Hamiltonian H is well defined on the space V of spatially diffeomorphism invariant states of the gravitational field.

“It follows that ˆ H is well defined on the space V of the spatially diffeomorphism invariant states of the gravitational field.”

If the construction developed here is consistent, the geometrical action of the gravitational Hamiltonian H, which is just to add loops and rearrange rootings at the intersections of the knot states, should code the full content of the Einstein equations in a diffeomorphism invariant form.

“If the construction developed here is consistent, this geometrical action of the gravitational hamiltonian ˆ H , which is just to add loops and rearrange rootings at the intersections of the knots states, should code the full content of the Einstein equations, in a diffeomorphism invariant form 3 .”

The effect of a diffeomorphism on α can be simply compensated by moving the box accordingly.

“Thus, the effect of a diffeomorphism on α can be simply compensated by moving the box accordingly.”

It is necessary to compute at least the second order term in the expansion in time of the evolution operator to see whether the sums over virtual states converge.

“It is necessary to compute at least the second order term in the expansion in time of the evolution operator, to see whether the sums over virtual states converge [16].”

• cites [16] [16] S. Coleman, personal communication.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The finiteness of the Hamiltonian is not sufficient for the evolution operator to be finite.

“Finally, we note that the finiteness of the Hamiltonian is not sufficient for the evolution operator to be finite.”

In the quantum theory, the diffeomorphism invariant states are of the form Ψ[{α}, T], where {α} is a generalized knot class and T is the constant value of the time.

“If we go over to the quantum theory the diffeomorphism invariant states are then of the form Ψ[ { α } , T ], where { α } indicates a generalized knot class, namely a diffeomorphism equivalence class of (multiple) loops α , and the real number T is the constant value of the time.”

The gauge fixing ∂_a T = 0 implies that all of the infinite number of Hamiltonian constraints C(x) are gauge fixed except one, which is an integral over the three manifold Σ.

“and that all of the infinite number of hamiltonian constraints C ( x ) turn out to be gauge fixed, except one, which is an integral over the three manifold Σ:”

• cites [10] [10] L. Smolin, in the Brill Feschrift Proceedings, op. cit. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.