Reciprocity of the Wigner derivative for spherical tetrahedra
RReciprocity of the Wigner derivative for sphericaltetrahedra
Bruce Bartlett and V. Hosana Ranaivomanana
Abstract
The Wigner derivative is the partial derivative of dihedral angle withrespect to opposite edge length in a tetrahedron, all other edge lengthsremaining fixed. We compute the inverse Wigner derivative for sphericaltetrahedra, namely the partial derivative of edge length with respect toopposite dihedral angle, all other dihedral angles remaining fixed. Weshow that the inverse Wigner derivative is actually equal to the Wignerderivative. These computations are motivated by the asymptotics of theclassical and quantum 6j symbols for SU(2).
In his seminal book on group theory and quantum mechanics from 1959 [13],Wigner studied the classical 6j symbols for SU (2), which encode the associatordata [10] for Rep SU (2), its tensor category of representations. He related the6j symbol (cid:26) J j j (cid:48) j j j (cid:27) (1)to a Euclidean tetrahedron with side lengths given by j , j , j , J , j and j (cid:48) andgave a heuristic argument that the square of this 6j symbol should (on average,for large spins) be proportional to the partial derivative ∂θ∂j (cid:48) of the dihedral angle θ at edge j with respect to the length of the opposite edge j (cid:48) , all other lengthsbeing held fixed (see Figure 1a).In 1968 the physicists Ponzano and Regge conjectured a more refined for-mula for the asymptotics of the classical 6j symbols, which included an oscilla-tory term. This formula was first proved rigorously by Roberts in 1999, usinggeometric quantization techniques [10], and since then a number of other proofshave been given [1, 2, 4–6].In 2003 Taylor and Woodward gave a corresponding asymptotic formula forthe quantum
6j symbols, relating their asymptotics to the geometry of spherical tetrahedra [11,12]. In their outline of a possible geometric proof of their formula(this approach was later made rigorous by March´e and Paul [8]), the partialderivative of dihedral angle with respect to opposite edge length (this time fora spherical tetrahedron) again played a crucial role. Following Taylor, we callthis the
Wigner derivative (see Figure 1b).Given a spherical tetrahedron with vertices v , v , v , v and edge lengths l ij , let G be the length Gram matrix, G ij = cos( l ij ). Taylor and Woodward’s1 a r X i v : . [ m a t h . QA ] D ec a) θl (cid:48) l (b) Figure 1: (a) The Euclidean tetrahedron corresponding to the 6j symbol inequation (1). If the lengths j , j , j , j and J are held constant, then P can still traverse the indicated circle, changing j (cid:48) . The probability of a giventetrahedron occurring is proportional to ∂θ∂j (cid:48) where θ is the dihedral angle at theedge with length j . Taken from [13]; see also [3]. (b) The Wigner derivativefor a spherical tetrahedron is ∂θ∂l (cid:48) , the partial derivative of dihedral angle withrespect to opposite edge length, all other edge lengths held fixed.formula for the Wigner derivative is as follows . (We will give an independentproof in Section 3). Theorem 1.1 (Taylor-Woodward [11]) . The Wigner derivative for a sphericaltetrahedron is ∂θ ( l ij ) ∂l (cid:48) = sin l sin l (cid:48) √ det G (2) where θ is the interior dihedral angle at the edge with length l and l (cid:48) is the lengthof the opposite edge (see Figure 1b). Unlike a Euclidean tetrahedron, a spherical tetrahedron is determined up toisometry by its six edge lengths as well as by its six dihedral angles. So there isa 1-1 correspondence between edge lengths and dihedral angles( l , l , l , l , l , l ) ↔ ( θ , θ , θ , θ , θ , θ ) . See (11) in Section 3 for an explicit formula. Therefore it makes sense to askabout the inverse
Jacobian matrix ∂l ij ∂θ kl and in particular the inverse Wignerderivative ∂l (cid:48) ∂θ in Figure 1b. Indeed, in our work we were led to consider this Note that the actual statement of Proposition 2.4.1.(n) in [11] contains a typo. The lefthand side should be ∂θ ab ∂l cd not ( ∂θ ab ∂l cd ) − . Theorem 1.2.
The inverse Wigner derivative for a spherical tetrahedron (seeFigure 1b) is ∂l (cid:48) ( θ ij ) ∂θ = sin l sin l (cid:48) √ det G (3)Comparing with formula (2) for the Wigner derivative, we obtain the follow-ing corollary. Corollary 1.3 (Reciprocity of the Wigner derivative) . For spherical tetrahedra,the Wigner derivative and the inverse Wigner derivative are equal: ∂θ ( l ij ) ∂l (cid:48) = ∂l (cid:48) ( θ ij ) ∂θ (4) Remark 1.4.
In the proof of [11, Proposition 2.4.1.(n)] and in [12, Proposition2.2.0.5] Taylor and Woodward show that ∂l (cid:48) ∂θ = √ det G sin l sin l (cid:48) which is the reciprocal of our formula (3) in Theorem 1.2 and thus seems tocontradict it. What is going on? The answer is that they are different partialderivatives as different sets of variables are being held constant. In our formula (3) , θ is changing while keeping the five remaining dihedral angles constant,while in Taylor and Woodward’s formula (4) , θ is changing while all lengthsexcluding l (cid:48) are being held constant. It is interesting that these two differentpartial derivatives are reciprocals of each other. To the best of our knowledge,formula (3) and its corollary (4) are new (see [9] for related work). Overview of paper
In Section 2 we show, as a warm-up result, that reci-procity of the Wigner derivative holds for spherical triangles. In Section 3we consider spherical tetrahedra. By relating the dihedral angles to the edgelengths via the links of the vertices, we can apply the reasoning from Section 1and hence prove our main results, Theorem 3.5 and Corollary 3.6.
In this section we review some elementary spherical trigonometry, and provereciprocity of the Wigner derivative for spherical triangles. This serves as awarm-up example before tackling spherical tetrahedra.Consider a spherical triangle ∆ ⊆ S as in Figure 2 with vertices v , v , v ∈ S , ∆ := { t v + t v + t v : t , t , t ≥ } ∩ S . The sine law says that sin a sin A = sin b sin B = sin c sin C . (5)3 v v c baAB C Figure 2: A spherical triangle.The cosine law expresses the interior angles in terms of the side lengths,cos A = cos a − cos b cos c sin b sin c , (6)while the dual cosine law expresses the side lengths in terms of the interiorangles: cos a = cos A + cos B cos C sin B sin C (7)For the sine and cosine laws, see [14]. From (6) and (7) we obtain ∂A∂a = sin a sin A sin b sin c , ∂a∂A = sin A sin a sin B sin C (8)To write these formulas in the form of (2) and (3), we introduce the lengthGram matrix G ij = cos l ( v i , v j ), G = c cos b cos c a cos b cos a . The following fact is fairly well known, but we include a proof in order to beself-contained and because Lemma 3.3 uses similar manipulations.
Lemma 2.1. √ det G = sin A sin b sin c Proof.
By row operations we obtaindet G = det c cos b − cos c cos a − cos b cos c a − cos b cos c − cos b = sin b sin c − (cos a − cos b cos c ) = sin b sin c − sin b sin c (cid:18) cos a − cos b cos c sin b sin c (cid:19) = sin b sin c (1 − cos A )= sin b sin c sin A where we have used the cosine law in the second last step.4 v v v α βγ λσ ρδ (cid:15) κ AC E DBF ac e dbf (a) n n n α βγ Γ a a (b) Figure 3: (a) A spherical tetrahedron ∆. (b) The link of v .This allows us to prove the formula for the Wigner derivative and its inversefor spherical triangles, and show that they are equal. Theorem 2.2 (Wigner reciprocity for spherical triangles) . For spherical trian-gles, we have ∂A ( a, b, c ) ∂a = sin a √ det G = ∂a ( A, B, C ) ∂A . Proof.
The first equation follows directly from (8a) and Lemma 2.1. The secondequation follows from: ∂A∂a∂a∂A = sin a sin B sin C sin A sin b sin c (by 8)= 1 (by the sine rule) In this section we prove reciprocity of the Wigner derivative for spherical tetra-hedra. Our method is to use the links of the vertices (as in [7]) to express thedihedral angles as explicit functions of the edge lengths, and then to use Freideland Louapre’s formula [5] for the determinant of the Gram matrix.Consider a spherical tetrahedron ∆ ⊆ S with vertices v , v , v , v ∈ S ,∆ := { t v + t v + t v + t v : t , t , t , t ≥ } ∩ S . In Figure 3a, the edge lengths a, b, c, d, e, f and interior dihedral angles
A, B, C, D, E, F are shown, as well as the inner angles at v , v and v .5he link Lk( v ) of a vertex v is the spherical triangle with edge lengths givenby the inner angles at v . In Lk( v ), let Γ be the interior angle opposite the edgewith length γ , as in Figure 3b. The following fact is used in [7]; here we give anexplicit proof. Lemma 3.1.
Γ = E. Proof.
By acting with an appropriate element of SO (4), we can rotate ∆ sothat v = (1 , , , , v i = (cos θ i , sin θ i n i ) i = 1 , · · · , n i ∈ S are the vertices of Lk ( v ) as in Figure 3b. By definition,cos E = − w · w where w , w ∈ R are the outward unit normals to the faces v v v and v v v of ∆ respectively. Evidently we have w = (0 , a ) , w = (0 , a )where a , a ∈ R are the outward unit normals to the edges n n and n n of Lk ( v ) respectively (see Figure 3b). But by definition,cos Γ = − a · a which shows that Γ = E . Lemma 3.2.
In the spherical tetrahedron ∆ , the Wigner derivative and inverseWigner derivative are: ∂E ( a, b, c, d, e, f ) ∂f = sin f sin E sin α sin β sin a sin b (9) ∂f ( A, B, C, D, E, F ) ∂E = sin E sin f sin κ sin σ sin A sin B (10) Proof.
For the Wigner derivative, E = E ( α, β, γ ) (cid:18) by cosine rulefor Lk( v ), see Fig. 4 (cid:19) = E ( α ( a, c, e ) , β ( b, d, e ) , γ ( a, b, f )) (cid:18) by cosine rule for triangles v v v , v v v and v v v (cid:19) (11)and so by the chain rule, ∂E∂f = ∂E∂γ ∂γ∂f = (cid:18) sin γ sin E sin α sin β (cid:19) (cid:18) sin f sin γ sin a sin b (cid:19) = sin f sin E sin α sin β sin a sin b . A Bα βγLk ( v ) (cid:15) κ AδCF Lk ( v ) λ σ BρDF Lk ( v )Figure 4: The links of v , v and v .For the inverse Wigner derivative, f = f ( κ, σ, γ ) (cid:18) by dual cosine rulefor triangle v v v (cid:19) = f ( κ ( A, C, F ) , σ ( B, D, F ) , γ ( A, B, E )) (cid:18) by dual cosine rule for Lk( v ), Lk ( v ) and Lk ( v ), see Fig. 4 (cid:19) and so by the chain rule, ∂f∂E = ∂f∂γ ∂γ∂E = ( sin γ sin f sin κ sin σ )( sin E sin γ sin A sin B )= sin E sin f sin κ sin σ sin A sin B .
To write these derivatives in the form (2) and (3), we will need Freideland Louapre’s formula for the determinant of the 4 × G ij =cos( l ( v i , v j )), for which we give our own proof. Lemma 3.3 (See [5]) . √ det G = sin a sin b sin e sin α sin β sin E .Proof. det G = det e cos a cos b cos e c cos d cos a cos c f cos b cos d cos f = det e cos a cos b − cos e cos c − cos a cos e cos d − cos b cos e c − cos a cos e − cos a cos f − cos a cos b d − cos b cos e cos f − cos a cos b − cos b = sin a sin b sin e det α cos β cos α γ cos β cos γ = sin a sin b sin e det G (cid:48) G (cid:48) is the Gram matrix of Lk ( v ). Now use Lemma 2.1.Note that Lemmas 3.2 and 3.3 combine to give a different proof of Taylorand Woodward’ s formula for the Wigner derivative. Theorem 3.4 (Taylor-Woodward [11]) . The Wigner derivative for a sphericaltetrahedron (see Figure 3a) is ∂E ( a, b, c, d, e, f ) ∂f = sin e sin f √ det G Moreover, we can now prove our main results.
Theorem 3.5.
The inverse Wigner derivative for a spherical tetrahedron (seeFigure 3a) is ∂f ( A, B, C, D, E, F ) ∂E = sin e sin f √ det G Corollary 3.6 (Reciprocity of the Wigner derivative) . For spherical tetrahedra,the Wigner derivative is equal to the inverse Wigner derivative.Proof.
Both results follow from: ∂E∂f∂f∂E = sin f sin A sin B sin κ sin σ sin E sin β sin α sin b sin a = sin f sin E sin E sin γ sin γ sin f (cid:18) using sine law for Lk ( v )and triangle v v v (cid:19) = 1 References [1] Vincenzo Aquilanti, Hal M Haggard, Austin Hedeman, Nadir Jeevan-jee, Robert G Littlejohn, and Liang Yu. Semiclassical mechanics of theWigner 6j-symbol.
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