A length operator for canonical quantum gravity

The regularized length operator does not require any renormalization.

“it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization.”

The computed part of the length operator's spectrum is purely discrete.

“We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length.”

The length operator contains full and direct information about all components of the metric tensor.

“The length operator contains full and direct information about all the components of the metric tensor which faciliates the construction of a new type of weave states which approximate a given classical 3-geometry.”

The length operator's spectrum is roughly quantized in units of the Planck length.

“We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length.”

The length operator admits self-adjoint extensions.

“We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length.”

In the representation in which an SU(2) connection is diagonal, the regularized length operator is densely defined.

“it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization.”

Part of the spectrum of the length operator has been computed.

“We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length.”

The paper constructs an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity.

“We construct an operator that measures the length of a curve in fourdimensional Lorentzian vacuum quantum gravity.”

The regularized length operator does not suffer from factor ordering singularities.

“it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization.”

Every projection L_γ(c) defines a symmetric and positive semi-definite operator on H with domain Dγ := Cyl^3_γ(A/G).

“Lemma 3.1 Every projection ˆ L γ ( c ) defines a symmetric and positive semi-definite operator on H with domain D γ := Cyl 3 γ ( A / G ) .”

Every eigenvalue of the length, area, or volume operators is of uncountably infinite multiplicity when considering all of H.

“On the other hand we see that every eigenvalue of the length, area or volume operators is of uncountably infinite multiplicity when we consider all of H .”

Since the tube is prescribed to be a congruence, each vertex lies on at most one curve.

“Notice that since we have prescribed the tube to be a congruence, it follows that each vertex lies on at most one curve.”

The regularized quantity (3.7) converges classically to the classical length expression (3.3) in the limit n→∞.

“Therefore (3.7) converges classically to (3.3) in the limit n → ∞ .”

The corresponding eigenvalues of the length operator on trivalent graphs are non-vanishing in general.

“We will see that the corresponding eigenvalues are non-vanishing in general.”

In Case A, the spin-network state is an eigenfunction of the length operator L_v(c) with an eigenvalue expressed in terms of lambda_plus, lambda_minus, and n3.

“It follows then from (4.4) and h -1 ˆ V 2 h = ( h -1 ˆ V h ) 2 that our spin-network state is eigenfunction of the length operator ˆ L v ( c ) with eigenvalue ℓ p 2 √ 1 n 3 +1 [( n 3 +2) λ 2 + + n 3 λ 2 -]. 2 2”

The operator (3.12) is consistently defined.

“This furnishes the proof that (3.12) is consistently defined.”

The family (L_γ(c), Dγ) defines a positive semi-definite symmetric operator on D := Cyl^3(A/G).

“Theorem 3.1 The family ( ˆ L γ ( c ) , D γ ) defines a positive semi-definite symmetric operator on D := Cyl 3 ( A / G ) .”

There are weave states Psi(q_ab := delta_ab, j0; delta(j0)) which approximate q_ab = delta_ab, with j0 considered a free parameter.

“So we see that there are weave states Ψ( q ab := δ ab , j 0 ; δ ( j 0 )) which approximate q ab = δ ab where we have considered j 0 as a free parameter.”

On trivalent graphs, the first three terms in the volume operator expression vanish identically.

“and the simplification that occurs on trivalent graphs is that the first three terms vanish identically.”

It follows that delta = Planck length times the cube root of pi times lambda(j0).

“It then follows that δ = ℓ p 3 √ πλ ( j 0 ).”

The limit delta approaching infinity gives a zero eigenvalue because Sigma becomes more and more empty.

“The limit δ →∞ gives a zero eigenvalue because Σ becomes more and more empty.”

For trivalent graphs, all spin-network states are eigenvectors of the length operator.

“Since for trivalent graphs the vertex contractors of a spin-network are unique and since the result of applying the length operator to a spin-network state is a spin-network state on the same graph and with the same spin, it follows that all spin-network states on trivalent graphs are eigenvectors of the length operator.”

Since the length operators L(s_v) commute for distinct vertices, all contributing length operators can be simultaneously diagonalized.

“Since the length operators ˆ L ( s v ) commute for distinct vertices, it follows that all contributing length operators can be simultaneously diagonalized.”

The spectrum attained on D_gamma and D_phi(gamma) is identical for any analyticity-preserving diffeomorphism phi.

“Therefore, the spectrum attained on D γ and D ϕ ( γ ) , ϕ ∈ Diff(Σ) an analyticity preserving diffeomorphism, is identical.”

There exists a countable basis of eigenvectors for Hγ.

“Since spin-networks span D γ and provide a countable basis for its completion H γ (with respect to µ 0 ,γ ) it follows that there exists a countable basis of eigenvectors for H γ .”

The weave states used necessarily involve intersecting and overlapping loops with j0 > 1/2, rather than only linked loops.

“Also observe that we use here weave states which necessarily involve intersecting and overlapping ( j 0 > 1 / 2), rather than only linked, loops [25].”

• cites [25] [25] A. Ashtekar, C. Rovelli, 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.

Each intersection makes a contribution no matter whether the intersection point is a vertex or not.

“Each intersection makes a contribution no matter whether the intersection point is a vertex or not.”

The extended spin-network function h_e4 f is represented on the abstract angular momentum Hilbert space by a vector proportional to psi = |j12 = j3, j123 = 0, j = 1/2; j1, j2, j3, j4 = 1/2>.

“The extended spin-network function h e 4 f is represented on the abstract angular momentum Hilbert space by a vector proportional to ψ := | j 12 = j 3 , j 123 = 0 , j = 1 / 2; j 1 , j 2 , j 3 , j 4 = 1 / 2 >”

The spectrum of L_γ(c) on Hγ is discrete.

“Thus, the spectrum on H γ is discrete in the”

The operator (L(c), D) admits self-adjoint extensions on H.

“Theorem 3.2 The operator ( ˆ L ( c ) , D ) admits self-adjoint extensions on H .”

On each finite-dimensional subspace associated with a fixed graph and fixed edge colouring, the length operator is a symmetric, positive semi-definite, finite-dimensional matrix.

“Since the number of spin-network states associated with a fixed graph and a fixed colouring of its edges by irreducible representations is finite-dimensional [18] it follows that the length operator on each such subspace of D is just a symmetric, positive semi-definite, finite dimensional matrix.”

• cites [18] [18] T. Thiemann, 'The inverse Loop Transform', preprint HUTMP-95/B-346, hepth/9601105 -Not checked yet

  Not checked in the free run.

The eigenvalue of L_v(c) on a trivalent spin-network state T_gamma with vertex v such that c and gamma do not share a segment incident at v is given by a formula where a, b, and c are given through (4.10)-(4.14).

“Theorem 4.2 The eigenvalue of ˆ L v ( c ) on a trivalent spin-network state T γ with vertex v such that c and γ do not share a segment incident at v is given by”

• Also stated in: § 4.2.2 Case B · p. 18 · plain

  “where a, b, c are given through (4.10)-(4.14).”

The spectrum does not depend on whether two curves touch each other in a C^m or C^n fashion for any 1 <= m < n.

“Moreover, the spectrum does not depend on whether two curves touch each other in a C m or C n fashion for any 1 ≤ m&lt;n .”

The operator (3.12) is finite for each n because there are at most |V(γ)| terms in the sum.

“This demonstrates that ˆ L n ( c ) is a finite operator for each n because there are at most | V ( γ ) | terms in the sum (3.12).”

The classical volume of the region filled by the tube is approximately pi Delta^2 L(c), provided that q_ab is slowly varying there.

“Notice that there is an uncountable sum involved in (5.1). In order to see that this makes sense we first of all notice that the classical volume of the region filled by the tube is given approximately by π ∆ 2 L ( c ), provided that q ab is slowly varying there.”

L_γ(c) has pure point spectrum when restricted to Dγ.

“Summarizing, we have shown that ˆ L γ ( c ) has pure point spectrum when restricted to D γ .”

All spin-network states on graphs with valence not bigger than three are eigenvectors of the length operator with, in general, non-vanishing eigenvalue.

“In particular, all spin-network states on graphs with valence not bigger than three are eigenvectors of the length operator with, in general, non-vanishing eigenvalue.”

One can build intersecting and overlapping weave states which reproduce the classical length of the tubes given a classical geometry.

“One can build intersecting and overlapping weave states which are such that they reproduce the classical length of the tubes given a classical geometry.”

In the geometrodynamic representation, the Wheeler-DeWitt constraint operator adopts an algebraic form that is still not rigorously defined or solved almost half a century after its discovery.

“However, in this so-called geometrodynamic representation [1] one of the most important operators, the Wheeler-DeWitt constraint operator, adopts an algebraic form which is so difficult that almost half a century after its discovery it is still unknown how to rigorously define it and much less how to solve it (compare [2] for a detailed analysis).”

• cites [1] [1] R. M. Wald, 'General Relativity', The University of Chicago Press, Chicago, 1989 -Not matched

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

• cites [2] [2] B. S. DeWitt, Phys. Rev. 160 (1967) 1113, Phys. Rev. 162 (1967) 1195, Phys. Rev. 162 (1967) 1239 -Not matched

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

The same novel technique introduced in [6,7] can be used to derive a completely well-defined operator corresponding to the length of a curve.

“It turns out that the same novel technique introduced in [6, 7] can be used to derive a completely well-defined operator corresponding to the length of a curve.”

• cites [6] [6] T. Thiemann, 'Anomaly-free formulation of non-perturbative, fourdimensional, Lorentzian quantum gravity', Harvard Preprint HUTMP-96/B350, Physics Lett. B (in press) -Not matched

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

• cites [7] [7] T. Thiemann, 'Quantum Spin Dynamics (QSD)', Harvard Preprint HUTMP96/B-351 T. Thiemann, 'Quantum Spin Dynamics (QSD) II', Harvard Preprint HUTMP-96/B-352 -Not matched

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

It is possible to obtain a finite, densely defined, symmetric operator corresponding to the original Wheeler-DeWitt constraint directly in the Lorentzian regime without multiplying by a power of sqrt(det(q)).

“The second possibility arises from the recent discovery [6, 7] that by employing a certain novel technique it is indeed possible to obtain a finite, densely defined (and symmetric) operator corresponding to the original Wheeler-DeWitt constraint directly in the Lorentzian regime and without multiplying by a power of √ det( q ).”

• cites [6] [6] T. Thiemann, 'Anomaly-free formulation of non-perturbative, fourdimensional, Lorentzian quantum gravity', Harvard Preprint HUTMP-96/B350, Physics Lett. B (in press) -Not matched

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

• cites [7] [7] T. Thiemann, 'Quantum Spin Dynamics (QSD)', Harvard Preprint HUTMP96/B-351 T. Thiemann, 'Quantum Spin Dynamics (QSD) II', Harvard Preprint HUTMP-96/B-352 -Not matched

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

Before this work, only volume and area operators could be constructed in the real connection representation.

“In fact, that is why only volume and area operators could be constructed so far [10, 11].”

• cites [10] [10] C. Rovelli, L. Smolin, 'Discreteness of volume and area in quantum gravity' Nucl. Phys. B 442 (1995) 593, Erratum : Nucl. Phys. B 456 (1995) 734 - sourceNot checked yet

  Not checked in the free run.

• cites [11] [11] A. Ashtekar, J. Lewandowski, 'Quantum Theory of Geometry I: Area Operators', Preprint CGPG-96/2-4, gr-qc/9602046 -Not matched

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

The length operator facilitates the construction of a new type of weave states that approximate a given classical 3-geometry.

“The length operator contains full and direct information about all the components of the metric tensor which faciliates the construction of a new type of weave states which approximate a given classical 3-geometry.”

The complications of the complex connection can be overcome by starting with Euclidean gravity and Wick rotating to the Lorentzian regime.

“the first possibility is to start with Euclidean gravity, in which the Wheeler-DeWitt constraint is also a polynomial [4], and to Wick rotate the Euclidean theory into the Lorentzian regime [5].”

• cites [4] [4] A. Ashtekar, in 'Mathematics and general relativity', AMS, Providence, 1987 -Not matched

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

• cites [5] [5] T. Thiemann, Acta Cosmologica 21 (1995) 145-67 'Reality conditions inducing transforms for quantum gauge field theory and quantum gravity', preprint HUTMP-95/B-348, gr-qc/9511057 - sourceNot checked yet

  Not checked in the free run.

The matrix elements of the volume operator can be given purely in terms of polynomials of 6j-symbols.

“the matrix elements of the volume operator can be given purely in terms of polynomials of 6 j -symbols [23].”

• cites [23] [23] T. Thiemann, 'Closed formula for the matrix elements of the volume operator for canonical quantum gravity', Harvard Preprint HUTMP-96/B-353 -Not matched

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

A state that approximates a fixed classical 3-geometry has to be defined on an infinite graph filling all of Sigma in the sense that a parameter delta characterizes the average distance between neighbouring vertices.

“It is clear that a state that approximates such a geometry has to be defined on an (infinite, in case Σ is not compact) graph filling all of Σ in the following sense : consider a parameter δ of which we may think as a lattice spacing and which is to characterize the average distance (as measured by q ab ) between neighbouring vertices of the graph.”

Cyl(A/G) is dense in H.

“One can show that Cyl( A / G ) is dense in H .”

Since the tube operator is to be state-independent and there is no other state-independent length scale in the problem, one is forced to choose Delta = Planck length.

“Since we want the tube operator to be state-independent and since there is no other state-independent length scale in the problem, we are forced to choose ∆ = ℓ p (which is also physically motivated as above; we have chosen a fixed factor of proportionality to equal unity).”

• AssumptionRests on an assumption: There is no other state-independent length scale in the problem besides the Planck length.

There is a σ-additive, faithful and diffeomorphism invariant probability measure μ0 on A/G which equips us with a Hilbert space structure H := L2(A/G, dμ0).

“Moreover, there is a σ - additive, faithful and diffeomorphism invariant probability measure µ 0 on A / G which equips us with a Hilbert space structure H := L 2 ( A / G , dµ 0 ).”

If one fixes delta and varies j0, the corresponding classical geometry must be warped compared to delta_ab, because the same curve appears longer the more spin the lattice carries.

“On the other hand we can fix δ , vary j 0 and ask which classical geometry is being approximated. Since one and the same curve appears the longer, according to (5.2), the more spin the lattice carries, we see that that the corresponding q ab must be warped as compared to δ ab .”

The volume operator measures the difference between recoupling schemes of n angular momenta based on J_IJ and J_JK.

“Expression (4.1) captures a neat interpretation of the volume operator : it measures the difference between recoupling schemes of n angular momenta based on J IJ and J JK respectively.”

The elements of A/G are in one-to-one correspondence with the set of all homomorphisms from the group of holonomically equivalent loops in Σ into SU(2)/Ad.

“One can show that the elements of A / G are in one-to-one correspondence with the set of all homomorphisms from the group of holonomically equivalent loops in Σ into SU (2) / Ad [12].”

• cites [12] [12] A. Ashtekar, C.J. Isham, Class. &amp; Quan. Grav. 9 , (1992) 1433 A. Ashtekar, J. Lewandowski, 'Representation theory of analytic holonomy C ⋆ algebras', in Knots and quantum gravity , J. Baez (ed), (Oxford University Press, Oxford 1994). -Not matched

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

The length operator constructed cannot be directly associated with the length of a given classical curve.

“This is obviously not what we want and so the length operator constructed cannot be directly associated with the length of a given classical curve.”

For a good approximation, the parameter delta needs to be small compared to the scales of macroscopic objects.

“In order to serve as a good approximation, the parameter needs to be small as compared to the scales of macroscopic objects.”

Since lambda is a monotonously growing function of j0, the lattice can be chosen the coarser the more spin it carries in order to still approximate the same geometry.

“Since λ is a monotonously growing function of j 0 it follows that the lattice can be chosen the coarser the more spin it carries in order to still approximate the same geometry.”

The volume functional involves the square root of a polynomial in E^a_i.

“We see that this functional involves the square root of a polynomial in E a i .”

The triad is integrable with generating functional V, the total volume of Σ.

“The triad is integrable with generating functional V := V (Σ), the total volume of Σ.”

The characterization of a graph relevant to the spectrum depends only on discrete labels: the number of vertices N, the valence n(v) of each vertex, and the finite number of topologically different ways of connecting incident edges in a C-infinity manner.

“In other words all we need to know about a graph is a) the number N of its vertices b) the valence n ( v ) of each vertex v c) the topologically different ways of connecting edges incident at different vertices in a C ∞ manner which is a finite number.”

The spin characterizes the eigenvalue of the ADM Hamiltonian.

“Indeed, the spin characterizes the eigenvalue of the ADM Hamiltonian [7, 26].”

• cites [7] [7] T. Thiemann, 'Quantum Spin Dynamics (QSD)', Harvard Preprint HUTMP96/B-351 T. Thiemann, 'Quantum Spin Dynamics (QSD) II', Harvard Preprint HUTMP-96/B-352 -Not checked yet

  Not checked in the free run.

• cites [26] [26] T. Thiemann, 'The ADM Hamiltonian operator for canonical quantum gravity', Harvard University PreprintNot checked yet

  Not checked in the free run.

A satisfactory quantization of the length of a piecewise smooth curve was given.

“A satisfactory quantization of the length of a piecewise smooth curve was given.”