Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity

May combine two claims.

A scalar product can be introduced in the loop representation with respect to which area and volume are symmetric operators and the trivalent expansions of spin network states are orthonormal.

“We introduce a scalar product with respect to which area and volume are symmetric operators, and (the trivalent expansions of) the spin network states are orthonormal.”

• Also stated in: § I. INTRODUCTION · p. 2 · plain

  “Finally, the technique introduced allows us to define a scalar product in the loop representation, by requiring that area and volume be symmetric, and that spin network states be orthogonal to each other - whatever the trivalent decomposition of high valence vertices we use.”

May combine two claims.

The spinor identity (one + and two -) becomes a binor identity (all +); the former is non-local, the latter local, and the latter is the basic identity of binor calculus and A = -1 recoupling theory.

“namely the spinor identity (one + and two -) has become a binor identity (all +) (see Penrose [43]). While the first is non local, the second is local, and is the basic identity at the roots of binor calculus and A = -1 recoupling theory.”

• cites [43]Not checked yet

  Not checked in the free run.

May combine two claims.

In quantum gravity, equation (4.1) does not hold for false intersections; it holds for intersections of lines within R_Φ.

“Contrary to this, in quantum gravity equation (4.1) does not hold for the false intersections. It holds for the intersections of lines within R Φ .”

May combine two claims.

Acting on an edge with color 1, the grasp creates two virtual trivalent vertices (inside the same circle, corresponding to the intersection point) - one on the spin-network state and one on the loop of the operator; the two vertices are joined by a virtual strip of color 2, and the overall multiplicative factor is determined as follows.

“acting on an edge with color 1, the grasp creates two virtual trivalent vertices (inside the same circle, corresponding to the intersection point) - one on the spin-network state and one the loop of the operator. The two vertices are joined by a virtual strip of color 2, and the overall multiplicative factor is determined as follows.”

Probably not a substantial claim: may not pass S4 (based).

The spin network states form a basis in V.

“The spin network states form a basis in V .”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The definition of the grasp is consistent and independent from the choice of the positive term.

“A moment of reflection shows that the definition is consistent, and independent from the choice of the positive term.”

May combine two claims.

Spin networks with a vertex and one or more adjacent edges lying on Σ are eigenstates of the area with eigenvalues not included in the spectrum computed in [23], so the spectrum of the area given in [23] is not complete.

“It was pointed out by A. Ashtekar that spin networks with a vertex and one -or more- of its adjacent edges lying on Σ are eigenstates of the area with eigenvalues that are not included in the spectrum of the operator computed in [23] -and derived again below. Therefore the spectrum of the area given in [23] is not complete.”

• cites [23]Not checked yet

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Classified as probably not a substantial claim.

The ε^6 factor cancels crucially in the definition of the quantum volume operator.

“Notice the crucial cancellation of the ǫ 6 factor.”

May combine two claims.

The absolute value and square root in the definition of the volume operator are well defined; arguments of the absolute value are finite dimensional matrices diagonalizable with real eigenvalues; arguments of the square root are finite dimensional matrices diagonalizable with real non-negative eigenvalues.

“We show that the absolute value and the square root that appear in the definition of the volume operator are well defined. Indeed, we show that the arguments of the absolute value are finite dimensional matrices diagonalizable and with real eigenvalues; and that the arguments of the square root are finite dimensional matrices diagonalizable and with real non-negative eigenvalues.”

Probably not a substantial claim: may not pass S3 (load-bearing).

An embedded colored trivalent graph specifies a state Φ only up to a global sign, because it does not fix the overall sign of the antisymmetrized linear combination of multiloops.

“An embedded colored trivalent graph specify a state Φ only up to a global sign, because it does not fix the overall sign of the antisymmetrized linear combination of multiloops.”

• AssumptionThe authors state this as an assumption: Ribbon-nets are oriented as graphs because they are oriented as two-surfaces; edges can be ordered clockwise.

Classified as probably not a substantial claim.

The ribbon-net construction allows us to 'open up' the intersection point and represent it by means of two vertices (one over α and one over β) and a ('zero length') edge connecting the two vertices, all in the same point of M.

“Notice that it is the ribbon-net construction that allows us to 'open up' the intersection point and represent it by means of two vertices (one over α and one over β ) and a ('zero length') edge connecting the two vertices. These two vertices and this edge are all in the same point of the three-manifold M.”

Classified as probably not a substantial claim.

The quantum volume operator is defined by [formula] following [23].

“We have then immediately a definition of the quantum volume operator [23]”

• cites [23]Not checked yet

  Not checked in the free run.

• AssumptionRests on an assumption: Classical fields are smooth, so they can be approximated by their value at a point to zeroth order in ε.

May combine two claims.

P_Φ can be interpreted as a linear combination of tangles in the sense of reference [32]; the tangles obey the axioms of recoupling theory.

“Thus P Φ can be interpreted as a linear combination of tangles in the sense of reference [32]. The tangles obey the axioms of recoupling theory.”

• cites [32]Not checked yet

  Not checked in the free run.

May combine two claims.

The two operators commute with the Mandelstam relations and are therefore well defined on V.

“The two operators commute with the Mandelstam relations and are therefore well defined on V .”

Classified as probably not a substantial claim.

For one term, by definition of the T loop operators and the grasp operation, it is obtained by inserting two trivalent intersections on the spin network edge ei (inside its ribbon), connected by a new edge of color 2.

“By definition of the ˆ T loop operators and of the grasp operation (Section 3), this is obtained by inserting two trivalent intersections on the spin network edge e i (inside its ribbon), connected by a new edge of color 2.”

Classified as probably not a substantial claim.

A novel sign factor can be inserted into the definition of the loop representation, short-cutting sign complications of the previous formulation.

“This allows us to insert a novel sign factor into the very definition of the loop representation, short-cutting sign complications of the previous formulation.”

May combine two claims.

A spin network state can be represented as a colored trivalent graph over the ribbon-net RvS (with a single edge along each ribbon), and this representation satisfies the identities of recoupling theory.

“We can represent a spin network state as a colored trivalent graph over the ribbon-net R v S (with a single edge along each ribbon). This representation satisfies the identities of recoupling theory.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The element 〈α| does not depend on the parameterization of α.

“The element 〈 α | does not depend on the parameterization of α : [ α ] ∼ [ β ] if β a ( τ ) = α a ( f ( τ )).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The multiloop states 〈α| span (actually, overspan) the state space V.

“Clearly, the multiloop states 〈 α | span (actually, overspan) the state space V .”

May combine two claims.

W_I is a non-local quantity that approximates det g(x_I) for small ε.

“Thus, W I is a non-local quantity that approximates det g ( x I ) for small ǫ .”

• AssumptionRests on an assumption: Classical fields are smooth, so they can be approximated by their value at a point to zeroth order in ε.

Probably not a substantial claim: may not pass S4 (based).

For small enough ε, each intersection i will lie inside a distinct ΣIε surface.

“For small enough ǫ , each intersection i will lie inside a distinct Σ I ǫ surface.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The element 〈α| does not depend on the orientation of α.

“The element 〈 α | does not depends on the orientation of α : [ α ] ∼ [ α -1 ].”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The representation (ΓΦ, PΦ) of a state |Φ> can be expanded in terms of a 'virtual' trivalent representation.

“The representation (Γ Φ , P Φ ) of a state 〈 Φ | can be expanded in terms of a 'virtual' trivalent representation as follows.”

May combine two claims.

If there are only two intersection points between the boundary of the cube and the spin network, then contributions cancel, so the sum reduces to cubes whose boundary has at least three distinct intersections with the spin network.

“If there are only two intersection points between the boundary of the cube and the spin network, then there are always two hands grasping in the same point; contributions have to be summed before taking the absolute value, and thus they cancel. Thus, the sum in (7.9) reduces to a sum over the cubes I i ǫ whose boundary has at least three distinct intersections with the spin network”

May combine two claims.

The corresponding area spectrum is labeled by multiplets j=(j1,...,jn) of positive half integers, with arbitrary n, and given by [formula].

“The corresponding spectrum is labeled by multiplets ⃗ j = ( j 1 , ..., j n ) of positive half integers, with arbitrary n , and given by”

May combine two claims.

The operator i W^(4)[012] may be represented by a purely imaginary antisymmetric matrix i~W with non-vanishing matrix elements only for k = l ± 2; such a matrix is diagonalizable and has real eigenvalues.

“Thus, we have show that the operator i ˆ W (4) [012] may be represented by a purely imaginary antisymmetric matrix i ˜ W l k with non-vanishing matrix elements only for k = l ± 2. Such matrix is diagonalizable and has real eigenvalues.”

Classified as probably not a substantial claim.

The paper has modified the definition of the theory by inserting a minus sign in the definition of the loop observables.

“We have modified the definition of the theory by inserted a minus sign in the definition of the loop observables.”

May combine two claims.

The definition of the scalar product does not depend on the trivalent expansion chosen, and the volume and area operators are symmetric with respect to this scalar product.

“We have immediately from the discussion above that the definition does not depend on the trivalent expansion chosen, and that the volume and area operators are symmetric with respect to this scalar product.”

May combine two claims.

The area and volume operators A and V do not correspond to physical observables: they are not gauge invariant and do not commute with GR's constraints.

“Notice that the area and volume operators ˆ A and ˆ V do not correspond to physical observables: they are not gauge invariant and do not commute with GR's constraints.”

May combine two claims.

For an n-valent vertex expanded in trivalent vertices, two requirements can be satisfied: the orthogonal basis works independently of the expansion, and the volume is hermitian in this basis.

“We have two highly non-trivial requirements. First, that this works independently from the way the n -valent vertex is expanded in trivalent ones. Second, that the volume be hermitian in this basis. Rather remarkably, we believe, both requirements can be satisfied.”

May combine two claims. The quote only partly states this claim.

For the four-valent vertex, the formula for the recoupling matrix is obtained, and the other 3 matrices in the volume action are deduced from identities (7.30).

“and the other 3 matrix that appear in the definition of the action of the volume operator are easily deduced from the identities (7.30).”

May combine two claims.

The paper has proven that the square root in the volume operator is well defined, because the relevant operator is hermitian.

“We have proven that the square root in the volume operator is well defined, because the relevant operator is hermitian.”

May combine two claims.

The eigenvectors of V are the common eigenvectors of all V_i, labeled by a multi-index β = (β1...βp), where p is the number of vertices in the region.

“Therefore the eigenvectors of ˆ V are the common eigenvectors of all ˆ V i . They are labeled by one β i for every vertex i , namely by a multi-index ⃗ β = ( β 1 ...β p ), where p is the number of vertices in the region.”

Classified as probably not a substantial claim.

The paper has defined a scalar product by a suitable normalization of the trivalent spin networks.

“We have defined a scalar product by a suitable normalization of the trivalent spin networks.”

May combine two claims.

The matrix W3 vanishes, and the trivalent vertices give no contribution to the volume.

“Therefore the matrix W 3 vanishes, and the trivalent vertices give no contribution to the volume.”

May combine two claims.

A unitary transformation preserves the property of a matrix of being diagonalizable and having real eigenvalues; therefore the results obtained for four-valent vertices hold in general.

“A unitary transformation preserves the property of a matrix of being diagonalizable and having real eigenvalues. It follows that the results we have obtained for the four-valent vertices hold in general.”

Classified as probably not a substantial claim.

Equations (7.13) and (7.15) completely define the volume operator.

“Equations (7.13) and (7.15) completely define the volume operator.”

May combine two claims.

The absolute value of the matrices i W^(4)[i1 i2 i3] is well defined and is given by a non-negative antisymmetric matrix.

“Therefore, the absolute value of the matrices i ˆ W (4) [ i 1 i 2 i 3 ] is well defined. It is given by a non-negative (i.e., having real eigenvalues equal or greater than zero) antisymmetric matrix.”

May combine two claims.

The matrix S can be considered as a change of basis in the space of the 4-valent vertices, and i·W^(4)[012] can be rewritten as eq. (7.23).

“The matrix S j i can be consider as a change of basis in the space of the 4-valent vertices and the matrix i · W (4) [012] j i can be rewritten as:”

Classified as probably not a substantial claim.

If both bases were orthonormal, the transformation between them had to be given by a unitary matrix.

“If we wanted both of them to be orthonormal, the transformation between the two had to be given by a unitary matrix.”

Classified as probably not a substantial claim. The quote only partly states this claim.

By repeated application of the recoupling theorem, eq. (7.13) can be rewritten as eq. (7.32).

“By repeated application of the recoupling theorem, eq. (7.13) can be rewritten as”

May combine two claims.

The loop states obey the axioms of recoupling theory, and the corresponding graphical formalism provides a powerful tool for computing the action of geometrical operators.

“We have shown that the loop states obey the axioms of recoupling theory, and the corresponding graphical formalism provides a powerful tool for computing the action of geometrical operators.”

May combine two claims.

The eigenvalues of the four operators i W^(4)[i1 i2 i3] are real, and if x is an eigenvalue, so is -x.

“The eigenvalues of the four operators i ˆ W (4) [ i 1 i 2 i 3 ] are real and, if x is an eigenvalue, so is -x .”

May combine two claims.

The methods introduced in this paper allow completing the process, suggesting a norm for the spin network states, and yielding a complete definition for a scalar product 〈|〉.

“The methods introduced in this paper allow us to complete the process, suggest a norm for the spin network states, and thus yield a complete definition for a scalar product 〈 | 〉 .”

May combine two claims.

Being the sum of hermitian non-negative matrices, V_i^2 is diagonalizable with real non-negative eigenvalues λ^2_βi and spectral projectors P_βi.

“Being the sum of hermitian non-negative matrices, ˆ V 2 i as well is diagonalizable with real non-negative eigenvalues, which we denote as λ 2 β i , and spectral projectors P β i :”

Classified as probably not a substantial claim. May combine two claims.

The paper defines a scalar product 〈|〉 and motivates the choice; it has no compelling argument for uniqueness, but shows it satisfies all consistency requirements so far considered, so it is reasonable as a first ansatz.

“Here, we define a scalar product 〈 | 〉 , and motivate the choice. We have no compelling argument for the uniqueness of this scalar product, but we will show that it satisfies all consistency requirements so far considered. Therefore, it is reasonable to take it as a first ansatz.”

May combine two claims.

Since the matrices i W^(n_i)[rts] are diagonalizable with real eigenvalues, the spectral theorem allows writing them as eq. (7.42), where αλ are real quantities and αP are spectral projectors.

“Since the matrices i ˆ W ( n i ) [ rts ] ¯ K I i K I i are diagonalizable with real eigenvalues, from the spectral theorem we can write them as:”

May combine two claims. The quote only partly states this claim.

In the basis defined, the action of the operator W^(4)[012] is given by eq. (7.25), where ~W^(4)[012] is a real antisymmetric matrix.

“where ˜ W (4) [012] j i a real antisymmetric matrix.”

May combine two claims.

The action of the volume operator is described by the sum (7.15) extended over all vertices of the spin network, where the explicit form for the recoupling matrix (7.13) is given by the paper's formula and where i1 = k1 = P0 and i_{n-1} = k_{n-1} = ̄k_{n-1} = ̃k_{n-1} = P_{-1}.

“Bringing all together, we have show that the action of the volume operator is described by the sum (7.15) extended over all vertices of the spin network, where the explicit form for the recoupling matrix (7.13) is given by and where i 1 = k 1 = P 0 and i n -1 = k n -1 = ¯ k n -1 = ˜ k n -1 = P -1 .”

May combine two claims.

For higher valence vertices, the transformation from one trivalent expansion to another can be obtained by repeated applications of the recoupling theorem transformation, and therefore by a product of orthogonal matrices; thus the argument extends immediately to higher valence.

“For higher valence vertices, the transformation from one trivalent expansion to another can be obtained by a repeated applications of the recoupling theorem transformation, and therefore by a product of orthogonal matrices. Thus, the argument above extends immediately to higher valence.”

May combine two claims.

For the complex Ashtekar connection, an extra i in the factor associated to each grasp destroys the reality of the eigenvalues of area and volume, which is a main result here.

“This additional imaginary factor destroies the reality of the eigenvalues of area and volume, which is a main result here.”

May combine two claims.

With this convention, the spinor identity is transformed into the binor identity, allowing immediately a local graphical calculus for the grasping operation and the use of recoupling theory.

“With this convention, the spinor identity is transformed into the binor identity, allowing immediately a local graphical calculus for the grasping operation and the use of recoupling theory.”

Classified as probably not a substantial claim. The quote only partly states this claim.

In the rescaled basis, the recoupling theorem becomes eq. (8.3).

“In this basis, the recoupling theorem becomes”

May combine two claims.

The matrix i·W^(4)[012] is diagonalizable with real eigenvalues, and consequently its absolute value is well defined.

“We now prove that the matrix i · W (4) [012] j i its diagonalizable with real eigenvalues and, as a consequence, that its absolute values is well defined.”

Probably not a substantial claim: may not pass S3 (load-bearing).

These properties of the Kauffman bracket are sufficient to generate the entire formalism of recoupling theory.

“These properties of the Kauffman bracket are sufficient to generate the entire formalism of recoupling theory.”

Classified as probably not a substantial claim.

Looking at eq. (E2) and (E4), one can compute the sign of the argument of the square root.

“Looking at eq. (E2) and (E4) we can easily compute the sign of the argument of the square root, which is:”

May combine two claims.

The sum of non-negative matrices is a non-negative matrix; therefore the sum of the absolute values of the four matrices i W^(4)[i1 i2 i3] is a non-negative antisymmetric matrix.

“But the sum of non-negative matrices is a non-negative matrix. Therefore the sum of the the absolute values of the four matrices i ˆ W (4) [ i 1 i 2 i 3 ] is a non-negative antisymmetric matrix as well.”

May combine two claims.

The paper has shown that the scalar product is well defined and independent from the trivalent expansion chosen, and that the volume is symmetric with respect to this scalar product.

“We have shown that that the scalar product is well defined and independent from the trivalent expansion chosen, and that the volume is symmetric with respect to this scalar product.”

Classified as probably not a substantial claim.

The dimension N of V is N = ∏_i N_i, where N_i is the number of compatible colorings of the vertex n_i.

“Let N i be the number of compatible colorings of the vertex n i . The dimension N of the subspace V we are considering is N = i N i .”

Probably not a substantial claim: may not pass S1 (asserted), S4 (based).

Projectors acting on different vertices commute: P_βi P_β'j = P_β'j P_βi if i ≠ j.

“Now, the projectors acting on different vertices commute among themselves: ˆ P β i ˆ P β ′ j = ˆ P β ′ j ˆ P β i if i = j .”

Probably not a substantial claim: may not pass S3 (load-bearing).

A closed loop in Penrose's binor calculus has value -2.

“A closed loop (with this convention) has value ( -2), because”

May combine two claims.

The limit in the definition of the area operator cannot be taken in the Hilbert space topology, where in general it does not exist; it must be taken in a topology that 'remembers' the topology in which the corresponding classical limit is taken.

“As it is usual for limits involved in the regularization of quantum field theoretical operators, the limit cannot be taken in the Hilbert space topology where, in general, it does not exist. The limit must be taken in a topology that 'remembers' the topology in which the corresponding classical limit (6.7) is taken.”

May combine two claims.

A basis of the Temperley-Lieb algebra is obtained by using the Jones-Wenzl projectors Π(e)n; since A=-1, the Π(e)n are just normalized antisymmetrizers.

“A basis of this algebra is obtained by using the Jones-Wenzl projectors Π ( e ) n . Since we are here in the case A = -1, the Π ( e ) n are just normalized antisymmetrizers.”

• cites [32]Not checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

The loop approach to quantum general relativity seems to be leading to a realization of a combinatorial space-time, deriving it solely from a strict application of conventional quantum ideas to standard general relativity.

“The loop approach to quantum general relativity [2,3] seems to be leading precisely to a realization of such a vision of a combinatorial space-time, deriving it solely from a strict application of conventional quantum ideas to standard general relativity.”

• cites [2] sourceNot checked yet

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• cites [3] sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The area operator can be defined as a limit of regularized operators Aε[Σ] that do not contain operator products, and a regularization that does not break general covariance can be achieved by a geometrical regularization.

“Following conventional quantum field theoretical techniques, we deal with operator products by defining ˆ A [Σ] as a limit of regularized operators ˆ A ǫ [Σ] that do not contain operator products. The difficulty in the present context is to find a regularization that does not break general covariance. This can be achieved by a geometrical regularization [22,10].”

• cites [22] sourceNot checked yet

  Not checked in the free run.

• cites [10]Not checked yet

  Not checked in the free run.

• AssumptionRests on an assumption: Classical fields are smooth, so they can be approximated by their value at a point to zeroth order in ε.

Classified as probably not a substantial claim.

Trivalent spin networks form a basis.

“The last equality in the last equation follows from the fact that trivalent spin network form a basis (see Sec. V).”

• cites Sec. VNot checked yet

  Not checked in the free run.

May combine two claims.

The convergence of the limit to the integral, and its independence from the details of the construction, are assured by the Riemann theorem for all bounded smooth fields.

“The convergence of the limit to the integral, and its independence from the details of the construction, are assured by the Riemann theorem for all bounded smooth fields.”

May combine two claims.

Reidemeister moves can be used at the tangles' intersections within the ribbon-net as well as at the false intersections, but a skein equation (4.1) holds in the first case and not in the second.

“Thus, as far as diff-invariant states are concerned, Reidemeister moves can be used at the tangles' intersections within the ribbon-net as well as at the false intersections. But in the first case a skein equation (equation (4.1)) holds, in the second it doesn't.”

May combine two claims.

The original definition of the loop representation of quantum general relativity left the problem of fixing the scalar product undetermined; the scalar product had to be determined by requiring quantum observables to be hermitian.

“The original definition of the loop representation of quantum general relativity left the problem of fixing the scalar product undetermined: the scalar product had to be determined by requiring quantum observables to be hermitian [3].”

• cites [3] sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

Two such products represent tangles equivalent under the Kauffman brackets if and only if they can be transformed into each other by the relations (B3)-(B5), which are at the basis of the Temperley-Lieb algebra.

“Two such products represent tangles equivalent under the Kauffman brackets if and only they can be transformed into each other by the relations”

Probably not a substantial claim: may not pass S3 (load-bearing).

Every planar non intersecting n-tangle with n inputs and n outputs is equivalent to the product of elementary tangles 1^n, U1, ..., Un-1.

“In [59] it is shown that every planar non intersecting n -tangle with n inputs and n outputs is equivalent to the product of elementary tangles 1 1 n , U 1 , . . . , U n -1 , given by”

• cites [59] L. H. Kauffman, Trans. Amer. Math. Soc. 318 , 417 (1990). sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

The Temperley-Lieb algebra T_n is the free additive algebra over Z[A,A^{-1}] with multiplicative generators 1^n, U1, ..., Un-1.

“The Temperley-Lieb algebra T n is the free additive algebra over ˜ Z [ A,A -1 ] with multiplicative generators 1 1 n , U 1 , . . . , U n -1 .”

Probably not a substantial claim: may not pass S3 (load-bearing).

The precise relation between the scalar product and the loop transform is discussed by Reisenberger [33] and in [18].

“The precise relation is discussed by Reisenberger [33] and in [18].”

• cites [33] sourceNot checked yet

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• cites [18]Not checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The Kauffman bracket satisfies properties (B1) and (B2) and is completely determined by them.

“In the context of Knot theory [32], the appearance of recoupling theory is based on the observation that the Kauffman bracket satisfies the properties (and is completely determined by the properties)”

• cites [32]Not checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

The explicit calculation of the Temperley-Lieb trace of a trivalent spin network is generally based on a generalization of the chromatic method of spin-network evaluation.

“The explicit calculation of the trace is generally based on a generalization of the chromatic method of spin-network evaluation [60].”

• cites [60] Moussoris, in Advances in Twistor Theory , Research Notes in Mathematics , edited by Huston and Ward (Pitman, 1979), pp. 308-312. sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

The trace on the Temperley-Lieb algebra satisfies tr(x + y) = tr(x) + tr(y).

“(ii) tr ( x + y ) = tr ( x ) + tr ( y ).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The trace on the Temperley-Lieb algebra T_n is defined by tr(x) = 〈 ̄x〉, where 〈〉 denotes the Kauffman bracket, or the recursive evaluation of ̄x using (B1) and (B2).

“(i) If x is an n -tangle then tr ( x ) = 〈 ¯ x 〉 where 〈 〉 denotes the Kauffman bracket, or, which is the same, the recursive evaluation of ¯ x using (B1) and (B2) .”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The normalization factor 'norm' of Brink and Satcheler corresponds to the choice given in the paper, which is also the standard normalization of the Clebsh-Gordon coefficient giving the usual normalization of the Wigner 3nJ-Symbol.

“where the normalization factor 'norm' of [52] corresponds to the choice”

• Also stated in: § APPENDIX D: GRAPHICAL CALCULUS OF ANGULAR MOMENTUM AND ITS RELATION WITH THE TANGLE-THEORETICAL RECOUPLING THEORY · p. 32 · plain

  “This is also the standard normalization of the Clebsh-Gordon coefficient that gives the usual normalization of the Wigner 3 nJ -Symbol.”

• cites [52] sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

It is possible to use a topological invariant calculus for writing generic SL(2,C) invariant tensor expressions.

“An important point that emerges from this brief discussion is the possibility of using a topological invariant calculus for writing generic SL (2 , C ) invariant tensor expressions.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The A = -1 case of recoupling theory is equivalent to the graphical calculus of the algebra of the SU(2) representations.

“Finally, the A = -1 case of recoupling theory is equivalent to the graphical calculus of the algebra of the SU (2) representations.”

• AssumptionRests on an assumption: In the A = -1 case, the Jones-Wenzel projectors reduce to antisymmetrizers.

Probably not a substantial claim: may not pass S3 (load-bearing).

The most used graphical methods of computation in su(2) representation theory are those of Levinson, developed by Yutsis, Levinson and Vanagas, and the modified version of Brink and Satcheler.

“To our knowledge the most used graphical method of computation in the representation theory of su (2) are the one due to Levinson [62] and developed by Yutsis, Levinson and Vanagas [63] and the slightly modified version of Brink and Satcheler [52].”

• cites [62] I. Levinson, Liet. TSR Mokslu Acad. Darbai B Ser. 2 , 17 (1956). sourceNot checked yet

  Not checked in the free run.

• cites [63] A. P. Yutsin, J. B. Levinson, and V. V. Vanagas, Mathematical Apparatus of the Theory of angular momentum (Israel program for Scientific Translation, Jerusalem, 1962).Not checked yet

  Not checked in the free run.

• cites [52] sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

The additional rules of Penrose's binor calculus make the calculus topological invariant, so graphical expressions can be smoothly deformed without changing their meaning.

“The advantage of these additional rules is that they make the calculus topological invariant, namely one can arbitrarily smoothly deform a graphical expression without changing its meaning.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Any SL(2,C) Mandelstam identities (3.3) can be written graphically, and in particular expressed in spin-network-like graphical relations in which each edge is the antisymmetrization of the holonomies along the edge.

“It is possible to write any SL (2 , C ) Mandelstam identities (3.3) in a graphical way and in particular we can express these identities in spin-network-like graphical relations, in which each edge it is the antisymmetrization of the holonomies along the edge.”