3nj Morphogenesis and Semiclassical Disentangling

0.81 MB·English

by  Roger W. Anderson

3
nj morphogenesis and semiclassical disentangling
RogerW.Anderson
Department ofChemistry,UniversityofCalifornia, Santa Cruz,
California95064,USA
0
1 VincenzoAquilanti
0
Dipartimentodi Chimica,Universita` degliStudidiPerugia
2
viaElcediSotto 8,06126Perugia(Italy)
n
a
J AnnalisaMarzuoli

5 Dipartimentodi FisicaNucleareeTeorica, Universita` degliStudidi Pavia
2
and INFN, Sezione diPavia, viaA.Bassi 6, 27100Pavia(Italy);

] E-mail: [email protected]
h
p
-
t
n
a
Abstract
u
q Recouplingcoefficients(3nj symbols)areunitarytransformationsbetweenbinary
[
coupledeigenstatesofN = (n+1)mutuallycommutingSU(2)angularmomen-

1 tum operators. They have been used in a variety of applications in spectroscopy,
v
quantumchemistryandnuclearphysicsandquiterecentlyalsoinquantumgravity
6
8 and quantum computing. These coefficients, naturally associated to cubic Yutsis
3
graphs, share a numberof intriguingcombinatorial, algebraic, and analytical fea-
4
. tures that makethem fashinatingobjects to be studied on their own. In this paper
1
0 we develop a bottom–up, systematic procedure for the generation of 3nj from
0
3(n − 1)j diagrams by resorting to diagrammatical and algebraic methods. We
1
: provide also a novel approach to the problem of classifying various regimes of
v
semiclassicalexpansionsof3nj coefficients (asymptoticdisentanglingof3nj di-
i
X
agrams)forn ≥ 3 by meansofcombinatorial,analyticaland numericaltools.
r
a
Keywords: Quantum theory of angular momentum; Racah–Wigner algebra; 3nj
coefficients; Yutsisgraphs;semiclassicalanalysis
1
Introduction
In thequantumtheory ofangularmomentum–mathematicallyencoded in SU(2)
representation theory– the study of (re)coupling of pure eigenstates of several,
mutuallycommutingangularmomentumoperatorsJ ,J ,...J isconsideredan
1 2 N
advanced topic, and as such it is not widely known. As pointed out in the classic
reference [1](topic12)
We shall deal with what is appropriately called the theory of binary
coupling of angular momenta, since it is a theory in which one cou-
ples angular momenta sequentially, in pairs. (...) Accordingly, the
theorydealswiththerelationshipsbetweendifferentcouplingschemes
–that isbetween distinctsequences ofpairwisecouplings. Thisis the
contentofrecouplingtheory.
The subject is a difficult one and the literature is extensive. Several
monographs(e.g. [2,3,4,5]1)dealwithtechniquesforimplementing
”graphical”methods(...).
Thispaperismuch inthespiritoftheprogramstated soonafterthat,namely
Ourgoalistorelatecouplingmethodstostandardresultsfromgraph
theory. Thesubject dividesquitenaturallyintothreeparts:
(A)(theclassificationofcouplingschemes)
(B)theelementaryoperationsunderlyingthestructureoftransforma-
tion(recoupling)coefficients,or3nj symbols
(C) theclassification ofrecouplingcoefficients and itsrelationshipto
thetheoryofcubicgraphs.
We are going to deal first with (B) and (C) aboveby formalizing an effective,
step–by–stepmethod to build up 3nj coefficients of typeI, II from 3(n−1)js of
the same type (note that, by convention, the integer n is related to the N angular
momentaintroducedaboveby(n+1) = N).
Our second goal is to take advantage of such a recursive construction in the
searchforanovelunifyingschemeforaddressingsemiclassicallimits(asymptotic
expansions)of the 3nj themselves. This task calls into play different techniques,
both analytical and numerical, and is very important in view of practical applica-
tions(that havebeen discussedextensivelyelsewhere).
1The last two monographswere published seven years after Biedenharn–Loucktext; in par-
ticular the last one is now the most exhaustive and reliable collection of definitions of angular
momentumfunctionsandrelatedalgebraicandanalyticformulas.
2
3
1 Recursion method for nj coefficients:
general description
The diagram of any SU(2) 3nj coefficient is a cubic (regular trivalent) graph on
3n lines and 2n nodes, named Yutsisgraph from the first author of [2]. The three
edges stemming from each node are associated to a triad of angular momentum
quantum numbers satisfying triangle inequalities, i.e. to a Wigner 3j coefficient.
This property of diagrams reflects the fact that any 3nj can be always written as
a sum overall magneticquantum numbers of the product of 2n (suitablychosen)
3j coefficients (seee.g. [5], Ch.9).
The recursion method for the generation of (all) 3nj diagrams with increasing n
may besummarizedas in[2]p.65
(Recursion rule) The diagrams of the 3nj coefficients are obtained
fromthediagramsofthe3(n−1)j byinsertingtwoadditionalnodes
on any two lines of the diagram of the 3(n − 1)j coefficients and
joiningthem together.
There are some caveats underlying this combinatorial construction: it must gen-
erate diagramsthat arenot separableon fewerthan fourlinesand thisisachieved
by requiring that the initial3nj diagram is separable on no fewer than three lines
(examples are provided below); moreover, isomorphic configurations should be
recognized and ruled out2.
Algebraic expressions (”recurrence formulas”) that encode the above combi-
natorial prescriptionon diagrams are providedfor the 3nj coefficients of thefirst
(I) and second (II) type in [3] ((24.8) p.81 and (A.6.14) p.143), respectively and
read
j ... j
1 n−1
(−1)Φ (2x+1)  l1 ... ln−2 x  (1)
Xx k1 ... kn−1 
 j ... j
j k x k j x 1 n
1 n−1 1 n−1 =  l ... l ,
(cid:26)ln−1 ln kn(cid:27) (cid:26)ln−1 ln jn(cid:27) k 1 ... k n
1 n
 
where Φ = j −k −j +k , and
1 1 n−1 n−1
2As an aside remark recall that the isomorphism problem in graph theory is a typical NP–
complete problem, actually a #P problem, even for regular graphs: this means that, given two
instancesofgraphs,onecandecideiftheyareisomorphicornotinpolynomialtimebutthegeneral
solution–theenumerationofallgraphsisomorphictoagivenone–requiresanexponentialtime
asthecomplexityofthegraphsgrows,i.e. asthenumberofverticesincreases(seee.g.[6]).
3
j ... j
1 n−1
(−1)Ψ (2x+1)  l1 ... ln−2 x  (2)
Xx k1 ... kn−1
 
j ... j
k k x j j x 1 n
1 n−1 1 n−1 =  l ... l ,
(cid:26)ln−1 ln kn(cid:27) (cid:26)ln−1 ln jn(cid:27) k 1 ... k n
1 n
 
whereΨ = j −k +j −k . Heretheangularmomentavariables{k ,...,l ,
1 1 n−1 n−1 1 1
...,j ,...} run over either {0,1,2,...} or {1, 3,...} and satisfy suitable trian-
1 2 2
gle inequalities. The summation variable x is also constrained by triangle rela-
tions.
Therolesplayedbytheexpressions(1)and(2)aremultipleandinterconnected
i) at facevalue they provide a straightforward decomposition of a 3nj coeffi-
cientintoasinglesuminvolvingone3(n−1)j ofthesametypeandexacly
two Wigner6j symbols.
(Notehoweverthatthesedecompositionsarenot thebasiconesforcompu-
tationalpurposes,sinceit iswell knownthatbothtypes ofcoefficients may
beexpressedas singlesumsofproductsofn6j symbols[5], Eqns. 1 and2
p. 361.)
ii) from the structural point of view they encode also generalizations of the
basicBiedenharn–Elliottidentity(seethecaseof9j generationinsection2
where thesetwo aspectsof(1)and (2)areworked outexplicitly).
iii) taking advantage of symmetries of (any kind of) 3nj coefficients one can
equate two different decompositions related by a symmetry of a same 3nj
to get relations for the associated 3(n − 1)j coefficients. For instance the
relations derived from (1) have the following general structure (dropping
phasesandomittingalltheargumentsnotrelevantinthepresentdiscussion)
• ···
• • x • • x
 • ··· x (3)
(cid:26)• λ •(cid:27) (cid:26)• λ •(cid:27)
Xx • ··· 
 • ··· 
• • y • • y
=  • ··· y .
(cid:26)• λ •(cid:27) (cid:26)• λ •(cid:27)
Xy • ··· 
 
Specifying λ = 1,1, 3,2, etc. and using explicit forms of the resulting 6j
2 2
symbols, various recursion relations may be obtained (note that the sums
4
on both sides of (3) reduce to a same number of terms, e.g. for λ = 1 one
2
would get x,y = 0,1). The case of the 9j symbol and associated 5–term
recursion relationwillbeworked outinsection2.2.
Thecaseofthe6j symbolisquitesubtlesincethewellknown3–termrecur-
sion relation in one variable – used in [7, 8], [1], topic9) as a second order
difference equation to be analyzed within the WKB framework– actually
derives directly from (2) which is nothing but the standard Biedenharn–
Elliottidentity(seealso(5)inthenextsection). Thisremarkhastodowith
a deep property of hypergeometric polynomials in the Askey scheme [9]:
the defining difference or differential equation and the recursion relation
areusually”dual”toeachotherbutthe6j happenstobe”self–dual”sothat
thetwoviewpointscan beused equivalently.
iv) Decompositionslikethosegivenin (1) and (2), as well as associated recur-
rence relations, can be also employed in the opposite direction, namely to
showhowa3nj symbolfallsback intoa3(n−1)j when oneofitsentryis
settozero. Thiswellknownfeature[5]willbeusedinremark(i)attheend
ofsection2.2.
Thebottom–uprecursionmethod,fullydevelopedinthispaperbyresortingto
bothdiagrammaticalandalgebraictools[2,3,5],representsnotonlyasystematic
procedure for the generation of 3nj’s from 3(n − 1)j diagrams (section 3), but
will provide also a novel approach to the classification of the (various regimes
of)asymptoticdisentanglingofthecoefficientsthemselves(section4),improving
and extendingpreviousresultsforthe9j found in[10].
To make the reader familiar with diagrammatic and algebraic tools, in the next
section wearegoingtoaddress thecasestudygivenby the9j (I,II).
9
2 Genesis of j coefficients and associated diagrams
The 9j(I) coefficient (the ”standard” 9j) can be written as a single sum of three
6j symbols(notationsas in [5], Ch. 10 and App. 12). Notethat thefollowingex-
pressioncanbelookedatasthefirstinstanceoftherecursionformula(1),namely
this coefficient arises as a 3nj(I) for n = 3 starting from the unique 3(n − 1)j
withn = 2,i.e. fromthe6j symbolitself
a f r
a b x c d x e f x
(−)2x(2x+1) = d q e, (4)
(cid:26)c d p(cid:27)(cid:26)e f q(cid:27)(cid:26)a b r(cid:27)
Xx p c b
 
5
where max{|a−b|,|f −e|,|c−d|} ≤ x ≤ min{a+b,f +e,c+d}.
Theassociated Yutsisdiagramis
a
• •
2 2
(cid:12) 2 (cid:12) 2
(cid:12) 2 (cid:12) 2
•2(cid:12)2(cid:12)2(cid:12)de(cid:12)2(cid:12)22(cid:12)2(cid:12)2(cid:12)2(cid:12) (cid:12)2r(cid:12)2(cid:12)2(cid:12)2(cid:12)2p(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2(cid:12)2q (cid:12)2(cid:12)2(cid:12)2(cid:12)2fc(cid:12)2(cid:12)2(cid:12)2(cid:12)2•
2 (cid:12) 2 (cid:12)
2 (cid:12) 2 (cid:12)
•(cid:12) •(cid:12)
b
from which the triad structure as well as the symmetries of the coefficient are
easily inferred (the six triads are associated with 3–valent nodes and correspond
to columns and rows of the array; symmetries, up to phases, are implemented by
oddorevenpermutationsofcolumnsorrowsandthevalueofthecoefficientdoes
not changeundertransposition).
On applying typeII recurrence formula(2) one shouldget a 9j(II) coefficient
but it can be easily shownthat the resultingconfiguration is a ”trivial”one, being
theproduct(withoutsum)oftwo6j’ssharing acommontriad
a b x c d x e f x
(−)R+x(2x+1) (5)
(cid:26)c d p(cid:27)(cid:26)e f q(cid:27)(cid:26)b a r(cid:27)
Xx
a d e
. p q r p q r
=  p q r = .
(cid:26)e a d(cid:27)(cid:26)f b c(cid:27)
b c f
 
The content of this formula is however highly non–trivial since it can be recog-
nized as the Biedenharn–Elliott identity relating five 6j coefficients. (Recall that
itrepresents,togetherwiththeorthogonalityconditions,the”definingrelation”of
the hypergeometric polynomial of type F associated with the 6j, see [5] p.295
4 3
and [9].)
Theassociated Yutsisdiagramis clearly ”separable”on thethreelinesp,q,r
p
• •
2
(cid:12) 2
(cid:12) 2
(cid:12) 2
d(cid:12)(cid:12) 22c •? •?
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a b 22222 (cid:127)d(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ?q??p??? (cid:127)p(cid:127)(cid:127)q(cid:127)(cid:127)(cid:127) ????c??
•(cid:12) • −→ •(cid:127) • •(cid:127) •
222e22222 q (cid:12)(cid:12)(cid:12)f(cid:12)(cid:12)(cid:12)(cid:12) ??e????a?•(cid:127)(cid:127)(cid:127)(cid:127)r(cid:127)(cid:127)(cid:127) ??r?????•(cid:127)b(cid:127)(cid:127)(cid:127)f(cid:127)(cid:127)(cid:127)
2 (cid:12)
2 (cid:12)
2 (cid:12)
• •(cid:12)
r
6
where on the right there appear two Yutsisdiagrams of the 6j symbol, each to be
thought of as a complete quadrilateral whose trivalent nodes correspond to triads
ofangularmomentumvariablesaccording totheconvention
•
?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
a (cid:127)(cid:127) ??e
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127)(cid:127) c ?? a b c
•(cid:127) • ←→
?
?? (cid:127)(cid:127) (cid:26) d e f (cid:27)
? (cid:127)
? (cid:127)
?? f (cid:127)(cid:127)
? (cid:127)
b ?? (cid:127)(cid:127)d
? (cid:127)
? (cid:127)
? (cid:127)
? (cid:127)
? (cid:127)
•(cid:127)
The two inequivalent ways of applying the Yutsis rule stated at the beginning
of section 1, namely ”inserting two nodes and joining them”, generate in this
simplestcasetwographsisomorphictothe9j(I)(left)and9j(II)(right)diagrams
• • • ⊚ •
⊚????????????(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)⊚ ????????????(cid:127)??(cid:127)??(cid:127)??(cid:127)??(cid:127)??(cid:127)??(cid:127)??(cid:127)??(cid:127)??(cid:127)??(cid:127)⊚
(cid:127)? (cid:127)?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
(cid:127) ? (cid:127) ?
•(cid:127) • •(cid:127) •
Theexplicitrelationshipbetweensuch acombinatorialprocedureand thecontent
of the algebraic recursion formulas (4) and (5) is postposed to section 3, where
the insertion operations will be addressed in the 3nj(I,II) (any n ≥ 3) cases and
related to thegeneral expressionsgivenin(1) and(2).
2.1 Combinatorial properties of 6j and 9j graphs
In this section we are going to discuss properties of the simplest Yutsis graphs
which will be recovered in an improved way when addressing recursively 3nj
(I,II) diagramsin section3.
a) The diagrams considered so far possessHamiltonian circuits, as shown be-
low.
7
••?••••••••••••••• ••0••••••••• •••••••••••
••••••?????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)•••••• •••••••00000000 (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)••••••• ••••••• •••••••
•••••••••(cid:127)(cid:127)•(cid:127)•(cid:127)(cid:127)•(cid:127)(cid:127)•(cid:127)•(cid:127)(cid:127)•(cid:127)•(cid:127)?(cid:127)?•??•?•??•??•?•??••••••••• •••••••••••(cid:14)(cid:14)•(cid:14)(cid:14)•(cid:14)(cid:14)(cid:14)•(cid:14)(cid:14)•(cid:14)0(cid:14)00•00•00•000••••••••••• •••••••••••••••••••••••••••••
Recall that a Hamiltonian path in an undirected graph is a path which visits
each vertex exactly once. A Hamiltonian cycle, or circuit, is a Hamiltonian path
that returns tothestartingvertex. Determiningwhethersuchpaths and cyclesex-
istin graphsis atypicalinstanceofNP–completeproblem[6].
Aswillbeshowninsection3,both3nj graphsoftypeIandIIadmitbyconstruc-
tion a Hamiltonian circuit of length 2n for all n, bounding a polygonal region
(”plaquette”) with thesame numberof sides (this is not necessarily true for other
types ofdiagrams encounteredforn > 4).
b) Apart from Hamiltonian circuits, the diagrams possess other cycles bound-
ing characteristic polygonal plaquettes. The drawings below display, for
eachdiagram,atypicalcycleofthiskind(ofcourse,owingtosymmetriesof
thecoefficients,othercyclesofthesameshapemighthavebeen depicted).
••••••••••••••••• ••••••••••• •••••••••••
(cid:127)•(cid:127)•(cid:127)(cid:127)•(cid:127)•(cid:127)•(cid:127)•(cid:127)•(cid:127)(cid:127)•(cid:127)•(cid:127)•(cid:127)•(cid:127)•(cid:127)(cid:127)•(cid:127)•(cid:127)•(cid:127)•(cid:127)(cid:127)•(cid:127)•(cid:127)••••••••••••••• (cid:13)1(cid:13)1(cid:13)1(cid:13)1(cid:13)1(cid:13)1(cid:13)1(cid:13)1(cid:13)1(cid:13)1(cid:14)•(cid:14)•(cid:14)•(cid:14)•(cid:14)(cid:14)•(cid:14)•(cid:14)•(cid:14)•(cid:14)•(cid:14)(cid:14)•(cid:14)•(cid:14)•(cid:14)•(cid:14)•(cid:14)(cid:14)•(cid:14)•••••••••••••••••• ••1•1••1•11••1•1••1•11••••••••(cid:13)•(cid:13)••(cid:13)•(cid:13)(cid:13)••(cid:13)•(cid:13)••(cid:13)•(cid:13)(cid:13)••
The number of edges N(n) of the ”largest”, non–Hamiltoniancircuit in a 3nj
max
diagram depends only on n (there are only triangular plaquettes for n = 2 while
thereappearquadrilateralsforn = 3,pentagonsforn = 4etc.,sothatN(n) = n+
max
1). The 9j (II) graph actually contains also triangular circuits (not highlighted in
theabovepicture)andthisfeatureisrelatedtothe”separability”ofthisparticular
diagram as discussed above. Since the girth of a graph is defined as the length of
the shortest cycle contained in the graph, the 9j(II) diagram is characterized by
girth3 whilethe9j(I)has girth4 = N(3).
max
Generally speaking, the presence of cycles with a number of edges strictly less
than N(n) isrelatedtotheseparabilityof3nj coefficientsintoaproduct(nosum)
max
oflowerordercoefficients,namelyto ”trivial”configurations.
8
c) Theexistenceofcircuits(Hamiltonianornot)canbeassumedasa”guiding
principle” in the search for asymptotic expansions of the 3nj coefficients
by letting the edges belonging to the circuit to become ”large” while keep-
ing ”quantum” the others. This is what has been already done for the 6j in
[11], for the9j in [10] and general results on the so–called phenomenonof
”disentangling”ofnetworks willbediscussedin section4below.
d) By resorting to notions from topological graph theory (see section 3) it is
possible to classify Yutsis graphs associated with 3nj(I, II) diagrams ac-
cording tothepossibilityof”embedding”themonclosed surfaces.
Consider in particular the graphs of the 6j (to be classified as type II!) and of the
9j (II) (the separable one). They are both embeddable on a 2–sphere since they
represent (thesurfaces of)atetrahedronand atriangularprism,respectively
•
2
(cid:15)(cid:15)(cid:15)•????? (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 222222
(cid:15)(cid:15)(cid:15)(cid:15) ????? (cid:12)(cid:12)(cid:12)(cid:12) (cid:11)(cid:11)•333 2222
•(cid:15)?(cid:15)(cid:15)?(cid:15)?(cid:15)????•ooooooooo??o?o?o?• •(cid:12)r(cid:12)(cid:12)r(cid:12)r(cid:12)(cid:12)r(cid:12)r(cid:12)(cid:12)r(cid:12)•(cid:12)(cid:12)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) 3333333•22L2L22L2L22L2L22•
On the other hand, the 9j(I) diagram is a genus–1 or ”toroidal” graph, as can be
inferred by noting that this graph is isomorphic to K (the bipartite graph on 6
3,3
nodes, or”utilitygraph”). To makethispointexplicit,recall that thetorus T2 can
be thought of as a rectangle in the plane with opposite edges pairwise identified
without twisting. The existenceof a Hamiltonian circuit connecting all of the six
nodes is crucial here: the circuit with its nodes is drawn as an horizontal straight
line inside the rectangle and represents a longitudinal path on the torus (which is
topologicallyacircleowingtoidentificationofoppositepointsonverticalbound-
aries). Theotherconnectionsareestablishedinsuchawaythattheconnectivityof
the original diagram is recovered (points to be identified pairwise have the same
9
horizontalcoordinateson theupperand lowerboundaries).
(cid:31) ⋄ ⋄ ⋄ (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127) (cid:127) (cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127) (cid:127) (cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:31)
◦ •(cid:127) •(cid:127) •(cid:127) • • • ◦
(cid:31) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) (cid:127)(cid:127) (cid:127)(cid:127) (cid:127)(cid:127) (cid:31)
(cid:127) (cid:127) (cid:127)
(cid:31) ⋄(cid:127)(cid:127) ⋄(cid:127)(cid:127) ⋄(cid:127)(cid:127) (cid:31)
In section 3.2 the embedding of both 9j(I) and any other 3nj(I) diagram into
theprojectivespace willbeaddressed.
2.2 Recursion relation for the 9j symbol
According to remark iii) of section 1, the starting point is the generating formula
for a type I 3nj coefficient given in (1). It can be used to get an expression
involving(sums of) products of two 3(n−1)j and four 6j as in (3). The explicit
form ofthelatterin thepresent casereads ([5] p. 345;thesamein [2]p.175)
a b x
a b x j f x
(−)a+f+b+j (−)2x(2x+1)d e f (6)
(cid:26)c λ a′(cid:27) (cid:26)λ c f′(cid:27)
Xx g h j
 a′ bc
= (−)a′+f′+g+e (−)2y(2y +1)y e f′ f′ e y g a′ y
(cid:26)d λ f(cid:27) (cid:26)λ d a(cid:27)
Xy g h j
Specifying λ = 1, a′ = a and f′ =f and using explicit expressions for the
6j symbols, the above formula gives a 5–term recursion relation for the 9j ([5],
(10.5.3)p. 347;thesamein[3]p.177),namely
Ac+1(ab,fj) c+1 a b Ac(ab,fj) c−1 a b c a b
f d e + f d e +C(a,b,c,f,j) f d e
(c+1)(2c+1) n j g ho c(2c+1) n j g ho nj g ho
(7)
Ad+1(ef,ag) c a b Ad(ef,ag) c a b c a b
= f d+1 e + f d−1 e +D(a,d,g,e,f) f d e ,
(d+1)(2d+1) nj g ho d(2d+1) nj g ho nj g ho
where symmetries of the 9j have been used. The functions in front of the 9j
coefficients are givenexplicitlyby
A (pr,st) = [(−p+r+q)(p−r +q)(p+r−q +1)(p+r+q+1) (8)
q
1
×(−s+t+q)(s−t+q)(s+t−q +1)(s+t+q +1)]2
10