The Screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and Hamiltonian dynamics
Roger W. Anderson, Vincenzo Aquilanti, Ana Carla P. Bitencourt, Dimitri Marinelli, Mirco Ragni
TThe Screen representation of spin networks: 2Drecurrence, eigenvalue equation for 6 j symbols,geometric interpretation and Hamiltoniandynamics. Roger W. Anderson , Vincenzo Aquilanti , , Ana Carla P. Bitencourt , DimitriMarinelli , , and Mirco Ragni Department of Chemistry, University of California, Santa Cruz, CA 95064, U.S.A. [email protected] Dipartimento di Chimica, Universit`a di Perugia, Italy [email protected] Istituto Metodologie Inorganiche e Plasmi CNR, Roma, Italy Departamento de F´ısica, Universidade Estadual de Feira de Santana, Brazil Dipartimento di Fisica, Universit`a degli Studi di Pavia, Italy INFN, sezione di Pavia, 27100 Pavia, Italy
Abstract.
This paper treats 6 j symbols or their orthonormal forms asa function of two variables spanning a square manifold which we call the“screen”. We show that this approach gives important and interestinginsight. This two dimensional perspective provides the most natural ex-tension to exhibit the role of these discrete functions as matrix elementsthat appear at the very foundation of the modern theory of classicaldiscrete orthogonal polynomials. Here we present 2D and 1D recursionrelations that are useful for the direct computation of the orthonormal6 j , which we name U . We present a convention for the order of the ar-guments of the 6 j that is based on their classical and Regge symmetries,and a detailed investigation of new geometrical aspects of the 6 j symbols.Specifically we compare the geometric recursion analysis of Schulten andGordon with the methods of this paper. The 1D recursion relation, writ-ten as a matrix diagonalization problem, permits an interpretation as adiscrete Shr¨odinger-like equations and an asymptotic analysis illustratessemiclassical and classical limits in terms of Hamiltonian evolution. Continuing and extending previous work [1,2,3,4] on 6 j symbols, (or on the equiv-alent Racah coefficients), of current use in quantum mechanics and recently alsoof interest as the elementary building blocks of spin networks [5,6,7], in this pa-per we (i) - adopt a representation (the “screen”) accounting for exchange andRegge symmetries; (ii) - introduce a recurrence relationship in two variables, al-lowing not only a computational algorithm for the generation of the 6 j symbolsto be plotted on the screen, but also representing a partial difference equation a r X i v : . [ qu a n t - ph ] A p r allowing us to interpret the dynamics of the two dimensional system. (iii) - in-troduce a recurrence relationship as an equation in one variable, extending theknown ones which are also computationally interesting; (iv) - give a formula-tion of the difference equation as a matrix diagonalization problem, allowing itsinterpretation as a discrete Schr¨odinger equation; (v) - discuss geometrical anddynamical aspects from an asymptotic analysis. We do not provide here detailedproofs of these results, but give sufficient hints for the reader to work out thederivations. For some of the topics we refer to a recent problem recently tackled[8]; numerical and geometrical illustrations are presented on a companion paper[9]. A concluding section introduces aspects of relevance for the general spinnetworks by sketching some features of the 9 j symbols. The Wigner 6 j symbols (cid:26) j j j j j j (cid:27) are defined as a matrix element beetweenalternative angular momentum coupling schemes [10] by the relation (cid:104) j j ( j ) j jm | j j j ( j ) j (cid:48) m (cid:48) (cid:105) = ( − j + j + j + j δ jj (cid:48) δ mm (cid:48) U ( j j jj ; j j ) , where the orthonormal transformation U is U ( j j jj ; j j ) = (cid:112) (2 j + 1) (2 j + 1) (cid:26) j j j j j j (cid:27) (1)For given values of j , j , j , and j the U will be defined over a range for both j and j . These ranges are given by j min =max ( | j − j | , | j − j | ) , j max = min ( j + j , j + j ) ,j min =max ( | j − j | , | j − j | ) , j max = min ( j + j, j + j ) , and j min ≤ j ≤ j max , j min ≤ j ≤ j max . (2)The screen corresponds to the 6 j or, as we specify below, the U values for allpossible values of j and j .The range for j and j is determined by the values of the independentvariables: j , j , j , and j . In the remainder of this paper we make this clear byintroducing new variables a, b, c, d, x and y to replace the j values. We specifythe new variables by establishing a correspondence: (cid:26) a b xc d y (cid:27) ⇔ (cid:26) j j j j j j (cid:27) (3)Assuming that x and y remain respectively in the upper and lower right side ofthe 6 j symbols, there are four classical and one Regge relevant symmetries: (cid:26) a b xc d y (cid:27) = (cid:26) b a xd c y (cid:27) = (cid:26) d c xb a y (cid:27) = (cid:26) c d xa b y (cid:27) = (cid:26) s − a s − b xs − c s − d y (cid:27) , (4) where s = ( a + b + c + d ) / x max − x min = y max − y min = 2 min ( a, b, c, d, s − d, s − c, s − b, s − a ) = 2 κ . The square screen willcontain (2 κ + 1) values. The canonical ordering for 6 j screens can now be spec-ified by considering the two sets of values: a , b , c , d and its Regge transform a (cid:48) = s − a , b (cid:48) = s − b , c (cid:48) = s − c , and d (cid:48) = s − d . Take the set with the smallestentry and use the classical 6 j symmetries to place this smallest value in theupper left corner of the 6 j symbol. The placement of the other 6 j arguments aredetermined by the symmetry relations. The resulting symbol has the propertythat x min = b − a ≤ x ≤ b + a = x max and y min = d − a ≤ y ≤ d + a = y max .Furthermore we require that a ≤ b ≤ d for the Canonical form. This may requireusing Eq. 5 to ”orient” the screen in this way. (cid:26) a b xc d y (cid:27) = (cid:26) a d yc b x (cid:27) (5)It can be shown that any symbol to be studied as a function of two entries canbe reduced to the canonical form of Eq. 5 where a ≤ b ≤ d ≤ b + c − a and c min = d − a + b ≤ c ≤ d + a − b = c max .Regge transformation for the parameters of the screen is a linear O (4) trans-formation: 12 − − − − abcd = s − as − bs − cs − d . (6)It can be checked that several functions appearing below (caustics, ridges, etc.)are invariant under such symmetry and also when represented on the screen (See[9]). j symbols In the following when we consider 6 j properties as correlated to those of thetetrahedron of Figure 1a [3], we use the substitutions A = a + 1 / B = b + 1 / C = c + 1 / D = d + 1 / X = x + 1 / Y = y + 1 / F ( A, B, C ) = 14 (cid:112) ( A + B + C )( − A + B + C )( A − B + C )( A + B − C ) (7)where A , B , C are the sides of the face. Upper case letters are used here tostress that geometric lengths are used in the equation. The square of the areacan be also expressed as a Cayley-Menger determinant. Similarly, the square ofthe volume of an irregular tetrahedron, can also be written as a Cayley-Menger determinant (Eq. 8 or as a Gramian determinant [11]. The latter determinantembodies a clearer relationship with a vectorial picture but with partial spoilingof the symmetry. V = 1288 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C D Y C X B D X A Y B A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (8)An explicit formula, due to Piero della Francesca, will be used in the compan-ion paper [9]. Additionally mirror symmetry [1], can be used to extend screensto cover a larger range of arguments. The appearance of squares of tetrahedronedges entails that the invariance with respect to the exchange X ↔ − X impliesformally x ↔ − x − j symbol. Although this isphysically irrelevant when the j s are pseudo-vectors, such as physical spins ororbital angular momenta, it can be of interest for other (e.g. discrete algorithms)applications. Regarding the screen, it can be seen that actually by continuationof X and Y to negative values, one can have replicas that can be glued by cuttingout regions shaded in Fig. in [12], allowing mapping onto the S manifold.Figure 1b illustrates V for values of a, b, c , and d used later in this paper. (a) Ponzano-Regge tetra-hedron built with the sixangular momenta in the6 j symbol.
20 30 40 50 60 70304050607080 y x V / (b) V (contours for Eq. 8), caustics Eq. 12 (grayboundary), ridges (solid white Eq. 9, dashed Eq. 13)for a = 30, b = 45, c = 60, and d = 55. Fig. 1The following equations were first introduced in Refs. [1] and [3], but theyare rewritten here with changed notation. When the values of A , B , C , D and X are fixed, the maximum value for the volume as a function of Y is given bythe “ridge” curve Y V max = (cid:18) ( A − B )( C − D ) + ( A + B + C + D ) X − X X (cid:19) / , (9)the corresponding volume is V max ( A, B, C, D, X ) = (cid:112) Λ A,B,X Λ C,D,X X , (10)where Λ α,β,γ = (cid:0) α − β (cid:1) − γ (cid:0) α + β (cid:1) + γ . (11)Therefore the two values of Y for which the volume is zero are: Y z = (cid:32)(cid:0) Y V max (cid:1) ± (cid:112) Λ A,B,X Λ C,D,X X (cid:33) / . (12)The values for Y z mark the boundaries between classical and nonclassical re-gions, and therefore called “caustics”.Also when the values of A , B , C , D and Y are fixed, the maximum value forthe volume as a function of X is given by the other “ridge” curve: X V max = (cid:18) ( A − D )( C − B ) + ( A + B + C + D ) Y − Y Y (cid:19) / . (13) The U values that are represented on the screen must be calculated by efficientand accurate algorithms, and we employed several methods that we have previ-ously discussed and tested. Explicit formulas are available either as sums overa single variable and series, and we have used such calculations with multipleprecision arithmetic in previous work [13],[3], [14], [15] . These high accuracycalculations are entirely reliable for all U that we have considered in the past,and the results provide a stringent test for other methods. However recourse torecursion formulas appears most convenient for fast accurate calculations and-as we will emphasize- also for semiclassical analysis, in order to understandhigh j limit and in reverse to interpret them as discrete wavefunctions obeyingSchr¨odinger type of difference (rather than differential) equations.The goal is to determine the elements of the ortho-normal transformationmatrix: U ( x, y ) = (cid:112) (2 x + 1) (2 y + 1) (cid:26) a b xc d y (cid:27) . (14)Two approaches can be used to evaluate U ( x, y ): evaluate the 6 j from recursionformulas and then apply the normalization or to use direct calculation fromexplicit formulas. x , y ) recursion for U In this work, we first derive and computationally implement a two variable recur-rence that permits construction of the whole orthonormal matrix The derivationfollows our paper in [14] and is also of interest for other 3 nj symbols.By setting h = 0 in the formula 43 in section 6 , we obtain a five termrecurrence relation for U ( x, y ):( − x (cid:114) x − y + 1 (cid:26) b x − a a x (cid:27) (cid:26) d x − c c x (cid:27) U ( x − , y )+( − x (cid:114) x + 12 y + 1 (cid:26) b x a a x (cid:27) (cid:26) d x c c x (cid:27) U ( x, y )+( − x (cid:114) x + 32 y + 1 (cid:26) b x + 1 a a x (cid:27) (cid:26) d x + 1 c c x (cid:27) U ( x + 1 , y )= ( − y (cid:114) y − x + 1 (cid:26) b y − c c y (cid:27) (cid:26) d y − a a y (cid:27) U ( x, y − − y (cid:114) y + 12 x + 1 (cid:26) b y c c y (cid:27) (cid:26) d y a a y (cid:27) U ( x, y )+( − y (cid:114) y + 32 x + 1 (cid:26) b y + 1 c c y (cid:27) (cid:26) d y + 1 a a y (cid:27) U ( x, y + 1) (15)This recurrence relation Eq. 15 will yield the entire set of U ( x, y ) that con-stitute the screen. Replacing the 6 j symbols of unit argument with the algebraicexpressions in Varshalovich [10], we obtain an effective method to calculate thescreen. x ) symmetric recursion for U Starting with the recurrence relation in Neville [16] and Schulten and Gordon[17] for the 6 j and carefully converting it into a recurrence relation for U , we canwrite a three term symmetric recursion relationship, which is here convenientlyrepresented as an eigenvalue equation: p + ( x ) U ( x + 1 , y ) + w ( x ) U ( x, y ) + p − ( x ) U ( x − , y ) = λ ( y ) U ( x, y ) , (16)where p + ( x ) = { ( a + b + x + 2) ( a + b − x ) ( a − b + x + 1) ( − a + b + x + 1) } ×{ ( d + c + x + 2) ( d + c − x ) ( d − c + x + 1) ( − d + c + x + 1) } × ( x + 1) − [(2 x + 1) (2 x + 3)] − (17) p − ( x ) = p + ( x −
1) (18) w ( x ) = [ b ( b + 1) − a ( a + 1) + x ( x + 1)] × [ d ( d + 1) − c ( c + 1) − x ( x + 1)] / [ x ( x + 1)] (19) λ ( y ) = 2 [ y ( y + 1) − b ( b + 1) − c ( c + 1)] . (20)For convenience we can also define: w λ = w ( x ) − λ ( y ) (21)A row of the screen may be efficiently and accurately calculated from theseequations. Diagonalization of the symmetric tridiagonal matrix given by the p + ( x ) , w ( x ) , p − ( x ) provides an accurate check: the eigenvalues of the tridiago-nal matrix precisely match those expected from Eq. 20 and eigenvectors generate U ( x, y ) . Stable results are obtained with double precision arithmetic. For the eigenvalue equation (Eq. 16), interpreted as discrete Schr¨odinger-likeequation, two potentials W + ( x ) and W − ( x ) can be defined: W ± ( x ) = w ( x ) ± | p ( x ) | , (22)where [18] p ( x ) = 12 ( p + ( x ) + p − ( x )) (23)or [19] p ( x ) = (cid:112) ( p + ( x ) p − ( x )) . (24)The two definitions agree well except for x near the limits x min or x max . Withthe second choice for ¯ p ( x ) the values for W ± are the same at the limits, butthere are differences with the first choice. See the figures 2a and 2b. Comparewith Ref. [8] where Hamiltonian dynamics is developed for a similar system.Braun’s potential functions are closely related to the caustics illustrated in [1]and [9]. The geometrical interpretations of the 6 j symbols provide fundamental under-standing and important semiclassical limits. This approach originates from Pon-zano and Regge [20] and elaborated by others, notably Schulten and Gordon[17].The three-term recursion relationship (Eq. 16), for U admits an illustrationin terms of a geometric interpretation: with some approximations to be detailedbelow one has finite difference equations (see Ref.[16], Eq.(67) for relationships
10 20 30 40 50 60 70 800200040006000800010000120001400016000 P o t en t i a l f un c t i on s x W + W - (a) Angular momenta corresponding toFigure 1b
100 200 300 400 500 600 700 800 P o t en t i a l f un c t i on s x W - W + (b) Angular momenta: a = 300, b = 450, c = 600, and d = 550. Fig. 2: Potential functions corresponding to Eq. 23 (dashed blue and black lines)and Eq.24 (thin solid orange and red lines)between recursions and finite difference), Consider the Schulten-Gordon rela-tionships Eq.(66) and Eq.(67)(Ref. [17]). Here we show new geometric represen-tations of the recursion relationships.By setting a = A − , b = B − , c = C − , d = D − , x = X − , and y = Y − one can write Eq. 16 in terms of triangle areas, a length X (cid:48) , and thecosine of a dihedral angle θ . The accuracy of this approximation is excellent,and depends slightly on the choice for X (cid:48) . F ( X − , A, B ) F ( X − , C, D ) (cid:0) X − (cid:1) U ( x − , y )+ F ( X + , A, B ) F ( X + , C, D ) (cid:0) X + (cid:1) U ( x + 1 , y ) − θ F ( X (cid:48) , A, B ) F ( X (cid:48) , C, D ) X (cid:48) U ( x, y ) ≈ cos θ = 2 X (cid:48) Y − X (cid:48) (cid:0) − X (cid:48) + D + C (cid:1) − B (cid:0) X (cid:48) + D − C (cid:1) − A (cid:0) X (cid:48) − D + C (cid:1) F ( X (cid:48) , B, A ) F ( X (cid:48) , D, C ) , (26) where F ( a, b, c ) is “area” of abc triangle (Eq. 7). (This recursion relation Eq. 25must be multiplied through by 8 to compare precisely with Eq. 16.Here we consider two choices for X (cid:48) in Eq. 19: X (cid:48) = (cid:18) X − (cid:19) (cid:18) X + 12 (cid:19) = X − , (27) X (cid:48) = X (28) The first choice (Eq. 27) provides an almost exact approximation to Eqns. 17,18,19, and 20 Coefficients in Eq. 16. The second Eq. 28 uses only integer or halfinteger arguments, and for most X works as well as the first. The figures 3a and3b show the errors and their significance. In these figures w λ ( approx ) is specifiedas: w λ ( approx ) = − θ F ( X (cid:48) , A, B ) F ( X (cid:48) , C, D ) X (cid:48) (29)For either choice of X (cid:48) , the recursion coefficients are connected to the geom-etry of tetrahedra [20]:32 V X (cid:48) = F ( X (cid:48) , A, B ) F ( X (cid:48) , C, D ) sin θ , (30)where V is the tetrahedral volume.Equations 25 can be recast by the geometric mean approximation: F ( X ± , A, B ) (cid:0) X ± (cid:1) (cid:39) (cid:112) F ( X ± , A, B ) F ( X, A, B ) (cid:112) X ( X ± , (31)where A and B can be also replaced by C and D .With Eq. 31 Eq. 25 becomes: (cid:112) F ( X − , A, B ) F ( X, A, B ) F ( X − , C, D ) F ( X, C, D ) X ( X − U ( x − , y )+ (cid:112) F ( X + 1 , A, B ) F ( X, A, B ) F ( X + 1 , C, D ) F ( X, C, D ) X ( X + 1) U ( x + 1 , y ) − θ F ( X, A, B ) F ( X, C, D ) X U ( x, y ) ≈ , (32)This equation is useful, but definitely less accurate than Eq. 25 (See Figures4a and 4b).With cancellation of terms in X , this Eq. 32 becomes: (cid:112) F ( X − , A, B ) F ( X − , C, D )( X − U ( x − , y )+ (cid:112) F ( X + 1 , A, B ) F ( X + 1 , C, D )( X + 1) U ( x + 1 , y ) − θ (cid:112) F ( X, A, B ) F ( X, C, D ) X U ( x, y ) ≈ . (33)This is equivalent to the recursion relation of Schulten and Gordon [17], thatthey use to establish their semiclassical approximations for 6 j symbols. Theirequation is accurate enough for x min (cid:28) x (cid:28) x max , but not so accurate near thelimits.In terms of the finite difference operator, Eq. 32 becomes after using Eq. 30: ∆ ( x ) f ( x ) = f ( x + 1) − f ( x ) + f ( x − ∆ ( X ) + 2 − θ ] f ( X ) (cid:39) , (34)
10 20 30 40 50 60 70 80-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0 w l - w l ( app r o x ) x X' = X - 1/4 X' = X (a) Error in w λ
20 30 40 50 60 70304050607080 y x -7000-6000-5000-4000-3000-2000-10000.0001000200030004000500060007000 w l (b) w λ values Fig. 3: Parameters a, b, c, d of Figure 1b
20 30 40 50 60 701E-51E-41E-30.010.1 F r a c t i ona l E rr o r i n p + x (a) Angular momenta of figure 1b
200 300 400 500 600 7001E-71E-61E-51E-41E-30.010.1 F r a c t i ona l E rr o r i n p + x (b) Angular momenta of figure 2b Fig. 4: Fractional errors in p + using Eq. 25, solid black line and Eq. 32, dashedblue line. where f ( X ) = (cid:112) F ( X, A, B ) F ( X, C, D ) X U ( x, y ) = (cid:114) VX sin θ U ( x, y ) . (35)We have, explicitlycos θ = ± (cid:115) − (cid:18) V X F ( X, A, B ) F ( X, C, D ) (cid:19) . (36)Our Eq. 35 is only slightly different from that of Schulten and Gordon, be-cause we have an extra X in the denominator of the definition of f ( X ). Thisoccurs because we use the recursion for U instead of that for 6 j . The following developments parallel those in [3]. From the above formulas, andfrom that of the volume, we have that– V = 0 implies cos θ = ± X min and X max – F ( X, A, B ) = 0 or F ( X, C, D ) = 0 establish the definition limits x min and x max .For a Schr¨odinger type equation d ψdx + p ψ = 0 , (cid:126) / m = 1 , (37)its discrete analog in a grid having one as a step, ψ n +1 + ( p − ψ n + ψ n − = 0 , (38)and we then have after comparing Eq. 38 with Eq. 34 f ( X + 1) − θ f ( X ) + f ( X −
1) = 0 . (39)The identification p = ± (2 − θ ) / (40)is then evident. Here we present a x,y plot Fig. 5 of 1 − cos θ that clearly showsthis definition of the classical region.Evidentially, on the closed loop, we can enforce Bohr-Sommerfeld phase spacequantization: (cid:73) p d x = ( n + 1 / π . (41)The eigenvalues n obtained in this way may be easily related to the allowed y . These formulas are illustrated in Fig. 6 and 7 of Ref [3].
20 30 40 50 60 70 y x q Fig. 5: x,y plot of cos θ for angular momenta of Fig 1(b)The Ponzano-Regge formula for the 6 j in the classical region is (cid:26) a b xc d y (cid:27) ≈ (cid:112) π | V | cos ( Φ ) , (42)where the Ponzano-Regge phase is: Φ = Aθ + Bθ + Xθ + Cη + Dη + Y η + π .The angles are determined by rearranged equations: Eq. 30. The various dihedralangles are found from the equations in [20].To the extent that the Ponzano-Regge approximation is valid we see that the6 j symbols have a magnitude envelop given by V and a phase that is a functionof X and Y determined by Φ . Eq. 42 works quite well for X and Y well withinthe classical region. However its use near the caustics is limited because of twofactors:1. The approximate recursion relation given by Eq. 32 differs most from theexact recursion Eqs. 17,18,19,20 near the caustics.2. The semiclassical approximation for the 6 j also breaks down near the caus-tics.For piece-wise extensions , see [20] and for uniformly valid formulas see [17]. j and higher spin networks In this work, we first have derived and computationally implemented a twovariable recurrence that permits construction of the whole orthonormal matrixThe derivation follows our paper in [14] and is also of interest for other 3 nj symbols.We find in [14]; see also [10], the following 2D recurrence relationship for 9 j symbols: A c +1 ( ab,fj )( c +1)(2 c +1) a b c + 1 d e fg h j + A c ( ab,fj ) c (2 c +1) a b c − d e fg h j − A d +1 ( ef,ag )( d +1)(2 d +1) a b cd + 1 e fg h j − A d ( ef,ag ) d (2 d +1) a b cd − e fg h j = (cid:104) B d ( ag,fe ) d ( d +1) − B c ( ab,jf ) c ( c +1) (cid:105) a b cd e fg h j A q ( pr, st ) = [( − p + r + q ) ( p − r + q ) ( p + r − q + 1) ( p + r + q + 1)] × [( − s + t + q ) ( s − t + q ) ( s + t − q + 1) ( s + t + q + 1)] B q ( pr, st ) = [ q ( q + 1) − p ( p + 1) + r ( r + 1)] [ q ( q + 1) − s ( s + 1) + t ( t + 1)](43)Geometrical interpretations of A ’s as proportional to products of areas of trian-gular faces and of B ’s as angular functions of associated structures, will serve forfurther work on the dynamical description of general spin networks. As noted in[14], Eq. 15 can be derived by setting h = 0 in Eq. 43, and using the propertythat a 3 nj symbol downgrades to a (3 n − j symbol when one of its entries iszero. In conclusion, expanding the discussion of Eq. 43 in [14], we suggest thatthe “screen” for the above 9 j symbols is three-dimensional, and generalizationto higher spin networks should be straight forward. Acknowledgement.
We thank Professor Annalisa Marzuoli for many produc-tive discussions during this research.
References
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