Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum-gravity model is the loop representation basis

The physical length of the i-th link in an Ooguri state is precisely equal to its coloring in Planck units.

“Thus, the physical length of the i-th link in Ooguri state is precisely equal to its coloring.”

• Also stated in: § Abstract · p. 6 · plain

  “the state Ψ ∆ ,c ( w ) , which Ooguri associates to the colored triangulation (∆ , c ) is a quantum state in which the physical length of the links of the triangulation is precisely equal to their coloring, in Planck units.”

If the gravitational field is in the state <α|, the length of a curve C is given by a formula in which n[C,α] is the number of times α crosses C.

“ii. if the gravitational field is in the state 〈 α | , the length of a curve C is given (reinserting conventional units) by”

• Also stated in: § Abstract · p. 6 · plain

  “where n [ C, α ] is the number of times α crosses C .”

The length of every curve C is quantized in units of 1/2 in Planck units.

“i. the length of every curve C is quantized in units of 1 / 2”

• Also stated in: § Abstract · p. 14 · plain

  “the spectrum of the length operator has discrete half-integer eigenvalues.”

The above construction defines a combinatorial quantum theory in 4-d for a fixed triangulation.

“The above construction defines a combinatorial quantum theory in 4-d (for a fixed triangulation).”

In Witten's 3-d gravity, the lengths of the arcs of a Regge triangulation can be computed up to diffeomorphisms.

“We show how to compute lengths in Witten's 3-d gravity and how to reconstruct the 2-d geometry from a state of Witten's theory.”

• Also stated in: § Abstract · p. 13 · plain

  “In Witten's theory, we can compute the lengths of the arcs of a Regge traingulation, (up to diffeomorphisms).”

The states of the 4-d combinatorial theory are given by induced colorings of the induced triangulation on the 3-d boundary of the 4-d triangulated spacetime.

“The states of this theory are given by the induced colorings of the induced triangulation on the 3-d boundary of the 4-d triangulated spacetime.”

In the state |α,β> on a torus, the minimal length of any curve wrapping once around the torus is 1/2 L_Planck.

“In this state, the minimal length of any curve wrapping once around the torus is 1 2 L Planck .”

Eq. (3) provides an identification between the Loop representation basis states |α> and the Ooguri states |∆,c>.

“Eq. (3) provides an identification between the Loop representation basis states | α 〉 and the Ooguri states | ∆ , c 〉 .”

• Also stated in: § Abstract · p. 5 · plain

  “This relation is our first result.”

The scalar product defined by summing over triangulations is invariant under independent diffeomorphisms on each of the two loops.

“The important point is that scalar product defined in this way is invariant under independent diffeomorphisms on each of the two loops, because so is the sum.”

The boundaries of the spacetime manifold carry an infinite dimensional number of degrees of freedom.

“At the same time, however, these boundaries still carry an infinite dimensional number of degrees of freedom.”

These 4-d states are in correspondence with the states of the Loop Representation.

“These states are therefore in correspondence with the states of the Loop Representation.”

The Regge action of a 4-simplex can be expressed as a function of the areas of its 10 faces.

“The Regge action of the simplex can be expressed as a function of these areas: S Regge ( s ) = S Regge ( a 1 ... a 10 ).”

The relation between Loop representation basis states and Ooguri states is many-to-one.

“The relation is many-to-one because the loop states are not independent (they form an overcomplete basis).”

• AssumptionRests on an assumption: The loop states are not independent; they form an overcomplete basis.

The Hilbert space basis that defines the Ponzano-Regge-Turaev-Viro-Ooguri combinatorial definition of 3-d quantum gravity is the same as the basis that defines the Loop Representation.

“We show that the Hilbert space basis that defines the Ponzano- ReggeTuraev-Viro- Ooguri combinatorial definition of 3-d Quantum Gravity is the same as the one that defines the Loop Representation.”

• Also stated in: § Abstract · p. 13 · plain

  “The 'colored triangulation' basis of the Ponzano-Regge-Turaev-ViroOoguri quantization of 3-d gravity is precisely the Loop Representation basis.”

Non-degenerate geometries are contained in Witten's Hilbert space for 2+1 gravity.

“We show that the non-degenerate geometries are contained in the Witten's Hilbert space.”

• Also stated in: § Abstract · p. 6 · plain

  “the non-degenerate geometries live in the same Hilbert space as the completely degenerate state.”

The colored triangulation (∆,c) uniquely determines an ensemble E∆,c of multiple loops such that precisely 2j single loops cross a link of the triangulation with color j.

“the colored triangulation (∆ , c ) uniquely determines an ensemble E ∆ ,c = { α 1 , α 2 , ... } of multiple loops (sets of closed loops) α i = ( α i 1 , α i 2 , ...α iN ), where α ij are (single) loops; each multiple loop α i having the property that precisely 2 j single loops cross a link of the triangulation with color j .”

The scalar product of two loop states |γ> and |γ'> can be written by a sum over colorings in 3-d.

“the relation between the Loop Representation theory and the PRTVO model allows us to write the scalar product of two loop states | γ 〉 and | γ ′ 〉 by means of a sum over colorings”

In the state |α,β>, the two variables characterizing a flat torus have a well-defined non-vanishing value.

“In the state | α, β 〉 , the two variables have a well-defined non-vanishing value.”

The scalar product defined in this way defines a scalar product <K|K'> between knot states.

“Therefore it defines a scalar product 〈 K | K ′ 〉 between knot states”

The no-loop state |0> gives a vanishing metric tensor.

“The 'no-loop' state | 0 〉 gives a vanishing metric tensor.”

The double integral is precisely the positive intersection number between C and α.

“The double integral is precisely the (positive) intersection number between C and α .”

All general relativistic physics can be read from the boundaries of the spacetime manifold.

“All the general relativistic physics can be read from the boundaries of this manifold.”

The quantum state |α,β> corresponds to the classical non-degenerate configuration in which space is a flat torus with two radii each 1/2 L_Planck.

“Thus the quantum state | α, β 〉 corresponds to the classical non-degenerate configuration in which space is formed by a flat torus in which the two radii are long 1 2 L Planck each.”

A 4-d simplex has the same number (10) of faces and links.

“A key observation is that a 4-d simplex has the same number (10) of faces (2-d symplices) and links (1-d simplicies).”

In 4-d, areas of surfaces, not lengths, are naturally quantized in 1/2 the Planck unit.

“In 4-d, they are the areas of surfaces, not the lengths, that are naturally quantized in 1/2 the Planck unit.”

• Also stated in: § Abstract · p. 8 · plain

  “the physical area of any surface is 1/2 the number of loops that cross the surface.”

• cites [7] [7] A. Ashtekar, C. Rovelli, L. Smolin, Phys Rev Lett 69 (1992) 237Not matched

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

The Ooguri representation Φ∆(c) is essentially equivalent to the Loop Representation.

“The Ooguri representation Φ ∆ ( c ) is therefore essentially equivalent to the Loop Representation.”

The topological quantum field theories studied by Witten and Atiyah provide a framework in terms of which quantum gravity itself may be framed, in spite of the infinite degrees of freedom.

“The topological quantum field theories studied by Witten and Atiyah provide a framework in terms of which quantum gravity itself may be framed, in spite of the infinite degrees of freedom.”

• AssumptionRests on an assumption: The theory with action S[A,E]=∫F∧E∧E has a physical phase space that is infinite dimensional.
• AssumptionRests on an assumption: The theory with action S[A,E]=∫F∧E∧E is standard general relativity in Ashtekar's form.

In classical general relativity, no experiment can distinguish a Minkowskian spacetime metric from a non-Minkowskian flat metric.

“in classical general relativity no experiment can distinguish a Minkowskian spacetime metric from a non-Minkowkian flat metric.”

The correct general axiomatic scheme for a physical quantum theory of gravity is Atiyah's set of axioms up to finite dimensionality of the Hilbert state space.

“These considerations leads us to propose that the correct general axiomatic scheme for a physical quantum theory of gravity is simply Atiyah's set of axioms [3] up to finite dimensionality of the Hilbert state space.”

• cites [3] [3] MF Atiyah, The Geometry and Physics of Knots , Accademia Nazionale dei Lincei, Cambridge University Press 1990; Publ Math Inst hautes Etudes Sci Paris 68 (1989) 175 sourceNot checked yet

  Not checked in the free run.

The Ooguri relation (1) is nothing but the Loop Transform.

“we show, indeed, that (1) is nothing but the Loop Transform [6]”

• cites [6] [6] C. Rovelli, L. Smolin, Phys Rev Lett 61 (1988) 1155; Nucl Phys B331 (1990) 80. For a review, see: C Rovelli, Class Quant Grav 8 (1991) 1613. For the 2+1 dimensional theory see: A Ashtekar, V Husain, C Rovelli, J Samuel, L Smolin, Class and Quant Grav 6 (1989) L185Not matched

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

The relation between lengths and areas of a 4-simplex can be generically inverted, expressing the lengths of the 10 links as functions of the areas of the 10 faces.

“Therefore we can generically invert the relation between lengths and areas, and express the lengths of the 10 links of each 4-simplex as functions of the areas of the 10 faces.”

In classical general relativity, there is no gauge-invariant observable which is local in the coordinates.

“In classical general relativity, there is no gauge-invariant obervable which is local in the coordinates.”

The Regge calculus action can be written as a sum over tetrahedra of the 6-j symbols of the 6 half-integer lengths of the links of each tetrahedron.

“The Regge calculus action can then be written as a very simple expression, which is essentially a sum over the tetrahedra of the triangulation of the 6-j symbols of the 6 (half- integer) lengths l i = j i of the links of each tetrahedron.”

The problem of existence of an unbroken phase and a broken phase does not subsist for 2+1 gravity.

“The considerations above show that as far as 2+1 gravity is concerned, the problem does not sussist.”

This is indirect support of our expectation that Eq.(5) projects on the solutions of the hamiltonian constraint.

“This is an indirect support of our expectation that Eq.(5) projects on the solutions of the hamiltonian constraint.”

Ooguri related the quantization of 3-d quantum gravity based on this model to the Witten quantization of the same theory.

“Ooguri [11] has related the quantization of 3-d quantum gravity based on this model to the Witten quantization of the same theory.”

• cites [11] [11] H Ooguri, Nucl Phys B382 (1992) 276 - [12] H Ooguri, Mod Phys Lett A7 (1992) 2799 - [13] T Regge, Nuovo Cimento 19 (1961) 551Not matched

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

Bigger spaces are obtained by wrapping the loops that define the state more times around the torus.

“Bigger spaces are obtained by wrapping the loops that define the state more times around the torus.”

The length l of a curve C in 3-d gravity is given by the formula involving E and ε.

“The length l of a curve C in 3-d gravity is given by”

• cites [6] [6] C. Rovelli, L. Smolin, Phys Rev Lett 61 (1988) 1155; Nucl Phys B331 (1990) 80. For a review, see: C Rovelli, Class Quant Grav 8 (1991) 1613. For the 2+1 dimensional theory see: A Ashtekar, V Husain, C Rovelli, J Samuel, L Smolin, Class and Quant Grav 6 (1989) L185Not matched

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

The partition function of Quantum Gravity can be constructed by fixing a sufficiently thin triangulation ∆ and summing over its colorings c.

“The partition function of Quantum Gravity can then be constructed by fixing a sufficiently thin triangulation ∆, and summing over its colorings c (assignments of half integers to every link).”

In the continuum case, Hawking gives a formal derivation that the functional integral defines a projection on the solutions of the Wheeler DeWitt equation.

“In this continuum case, Hawkings does indeed give a formal derivation of the fact that the functional integral defines a projection on the solutions of the Wheeler DeWitt equation.”

• cites [17] [17] SW Hawking, in General Relativity, an Einstein centenary Survey , eds. SW Hawking, W Israel, Cambridge University Press 1979Not matched

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

The quantum states of the Ponzano-Regge theory have to be taken as quantum combinations Φ∆(c) of the colored triangulations induced on the 2-d boundary ∂M of the 3-d manifold M.

“The quantum states of the Ponzano-Regge theory have to be taken, following Atiyah's general formulations of topological quantum field theories, as quantum combinations Φ ∆ ( c ) of the colored triangulations (∆ , c ) induced on the 2-d boundary ∂M of the 3-d manifold M .”

• cites [3] [3] MF Atiyah, The Geometry and Physics of Knots , Accademia Nazionale dei Lincei, Cambridge University Press 1990; Publ Math Inst hautes Etudes Sci Paris 68 (1989) 175 sourceNot checked yet

  Not checked in the free run.

The extrinsic curvature of these spacelike surfaces will be completely undetermined, due to the Heisenberg principle.

“The extrinsic curvature of these spacelike surfaces, then, will of course be completely indetermined, due to Heisenberg principle.”