A new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group

TL;DR

This paper introduces a novel sum rule for Clebsch-Gordan coefficients derived from generalized characters of rotation group representations, utilizing integrals with Gegenbauer polynomials, potentially enabling new mathematical relations.

Contribution

The paper presents a new sum rule for Clebsch-Gordan coefficients based on generalized characters and integral identities involving Gegenbauer polynomials, expanding the mathematical tools available.

Findings

Derived a new sum rule for Clebsch-Gordan coefficients

Connected the sum rule to integrals of Gegenbauer polynomials

Suggested applicability to other polynomial integral relations

Abstract

We present a new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group. The identity is obtained from an integral involving Gegenbauer ultraspherical polynomials. A similar procedure can be applied for other types of integrals of such polynomials, and may therefore lead to the derivation of further new relations.

Equations

$${\chi }_k^j(\omega )=\sqrt{\frac{(2j+1)(2j-k)!}{(2j+k+1)!}}{\sin }^k\left(\frac{\omega }{2}\right){\left(\frac{d}{d\cos \left(\frac{\omega }{2}\right)}\right)}^k{\chi }^j(\omega ),$$

$${\chi }_k^j(\omega )=i^k\sum _m{e}^{-im\omega }⟨jmk0|jm⟩,$$

$${\chi }_k^j(\omega )=(2k)!!\sqrt{\frac{(2j+1)(2j-k)!}{(2j+k+1)!}}{\sin }^k\left(\frac{\omega }{2}\right)C_{2j-k}^{(k+1)}\left(\cos \left(\frac{\omega }{2}\right)\right),$$

$${\int }_0^{2\pi }{\chi }_k^j(\omega ){\chi }_k^{j^{\prime }}(\omega ){\sin }^2\left(\frac{\omega }{2}\right)d\omega =\pi {\delta }_{j,j^{\prime }},$$

$${\int }_{-1}^1{C}_n^{(\alpha )}(x)C_r^{(\alpha )}(x){\left(1-x^2\right)}^{\alpha -1/2}dx=\frac{\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha ){\left[\Gamma (\alpha )\right]}^2}{\delta }_{n,r},$$

$${\int }_{-1}^1{\left[C_n^{(\alpha )}(x)\right]}^2{\left(1-x^2\right)}^{\alpha +\mu -1/2}dx$$

$${\int }_{-1}^1{\left[C_n^{(\alpha )}(x)\right]}^2{\left(1-x^2\right)}^{\alpha +1/2}dx=\frac{\pi \Gamma (2\alpha +n)\left[(\alpha -1)(\alpha +1/2)+n(\alpha +n/2)\right]}{2^{2\alpha -1}(\alpha +n-1)(\alpha +n)(\alpha +n+1)n!{\left[\Gamma (\alpha )\right]}^2}.$$

$${\chi }_k^j(\eta ){\chi }_k^{j\ast }(\eta )=\sum _{m,m^{\prime }}{e}^{-2i(m-m^{\prime })\eta }⟨jmk0|jm⟩⟨j{m}^{\prime }k0|j{m}^{\prime }⟩$$

$${\chi }_k^j(\eta ){\chi }_k^{j\ast }(\eta )=(2^{k}k!{)}^2\frac{(2j+1)(2j-k)!}{(2j+k+1)!}{\left[C_{2j-k}^{(k+1)}\left(\cos \eta \right)\right]}^2{\sin }^{2k}\eta$$

$$\sum _{m,m^{\prime }}⟨jmk0|jm⟩⟨j{m}^{\prime }k0|j{m}^{\prime }⟩{\int }_0^{\pi }{e}^{-2i(m-m^{\prime })\eta }{\sin }^4\eta d\eta$$

$$={\left(2^{k}k!\right)}^2\frac{(2j+1)(2j-k)!}{(2j+k+1)!}{\int }_0^{\pi }{\left[C_{2j-k}^{(k+1)}\left(\cos \eta \right)\right]}^2{\sin }^{2k+4}\eta d\eta .$$

$${\int }_0^{\pi }{\left[C_{2j-k}^{(k+1)}\left(\cos \eta \right)\right]}^2{\sin }^{2k+4}\eta d\eta =\frac{\pi }{16}\frac{(2j+k+1)!}{(2j-k)!(2j+1)j(j+1)}\frac{\left[k(k+1)+4j(j+1)\right]}{{\left(2^{k}k!\right)}^2}$$

$$\sum _{m,m^{\prime }}⟨jmk0|jm⟩⟨j{m}^{\prime }k0|j{m}^{\prime }⟩{\int }_0^{\pi }{e}^{-2i(m-m^{\prime })\eta }{\sin }^4\eta d\eta =\frac{\pi \left[k(k+1+4j(j+1)\right]}{16j(j+1)},$$

$${\int }_0^{\pi }{e}^{-2i(m-m^{\prime })\eta }{\sin }^4\eta d\eta$$

$$\sum _{m,m^{\prime }}\frac{(-1{)}^{m^{\prime }-m}\sin \left[\left(m^{\prime }-m\right)\pi \right]}{\left(m^{\prime }-m\right)\pi \left[1-{\left(m^{\prime }-m\right)}^2\right]\left[4-{\left(m^{\prime }-m\right)}^2\right]}⟨jmk0|jm⟩⟨j{m}^{\prime }k0|j{m}^{\prime }⟩$$

$$=\frac{\left[k(k+1)+4j(j+1)\right]}{24j(j+1)}\delta (jkj),$$

$$\Gamma (x)\Gamma (1-x)=\frac{\pi }{\sin (\pi x)},$$

$$\sum _{m,m^{\prime }}\frac{(-1{)}^{m^{\prime }-m}}{\Gamma \left(3+m-m^{\prime }\right)\Gamma \left(3-m+m^{\prime }\right)}⟨jmk0|jm⟩⟨j{m}^{\prime }k0|j{m}^{\prime }⟩$$

$$=\frac{\left[k(k+1)+4j(j+1)\right]}{24j(j+1)}\delta (jkj).$$

$$\langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle=(-1)^{j_{1}-j_{2}+m_{3}}\sqrt{2j_{3}+1}\left(\begin{array}[]{ccc}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{array}\right),$$

$$\left(\begin{array}[]{ccc}j_{3}&j_{2}&j_{1}\\ -m_{3}&m_{2}&m_{1}\end{array}\right)=(-1)^{j_{1}+j_{2}+j_{3}}\left(\begin{array}[]{ccc}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{array}\right),$$

$$\displaystyle\sum_{m,m^{\prime}}\frac{\sin\left[\left(m^{\prime}-m\right)\pi\right]}{\left(m^{\prime}-m\right)\pi\left[1-\left(m^{\prime}-m\right)^{2}\right]\left[4-\left(m^{\prime}-m\right)^{2}\right]}\left(\begin{array}[]{ccc}j&k&j\\ -m&0&m\end{array}\right)\left(\begin{array}[]{ccc}j&k&j\\ -m^{\prime}&0&m^{\prime}\end{array}\right)$$

$$=\frac{\left[k(k+1)+4j(j+1)\right]}{24j(j+1)(2j+1)}\delta (jkj).$$

$$\displaystyle\sum_{m,m^{\prime}}\frac{1}{\Gamma\left(3+m-m^{\prime}\right)\Gamma\left(3-m+m^{\prime}\right)}\left(\begin{array}[]{ccc}j&k&j\\ -m&0&m\end{array}\right)\left(\begin{array}[]{ccc}j&k&j\\ -m^{\prime}&0&m^{\prime}\end{array}\right)$$

$$=\frac{\left[k(k+1)+4j(j+1)\right]}{24j(j+1)(2j+1)}\delta (jkj).$$

Full text

1 Introduction

Several special relations (identities, sum rules) involving Clebsch-Gordan coefficients or Wigner $3jm$ symbols have been discovered in connection with atomic, molecular and nuclear spectroscopy (see the non-exhaustive list of references [9, 19, 23, 25, 20, 7, 16, 8, 3, 11, 4, 15, 13, 18, 5, 21, rowe97, speziale17, ibort17]). Applications concern for instance the hydrogen molecular ion [9], the non-relativistic helium atom [19, 23, 25, 20, 7], the high-order radiative transitions in hydrogenic ions [16], the stability properties of some special classical solutions of the $O(n)$ non-linear $\sigma$-model in two dimensions [8, 3], the non-trivial zeros of $3jm$ and $3nj$ coefficients [4], the pion double charge exchange cross-sections in the nuclear shell model [13], the Stark effect of hydrogenic systems [5], or the statistical modeling of anomalous Zeeman effect [21]. Some of these identities are sometimes referred to as “unusual sum rules”, in the sense that they can not be reduced to orthogonality relations, or that they do not include the $(2j+1)$ weighting factor in a summation over angular momentum $j$ [11]. Sum rules can be of great interest for checking numerical calculations involving Clebsch-Gordan or Wigner $3jm$ symbols. The reference book of Varshalovich, Moskalev and Khersonskii [27] presents (section 8.7.7, p. 262) three unusual sum rules involving $3jm$ symbols [9, 19, 8] and only one consisting of a summation over projection of angular momentum $j$ ($-j\leq m\leq j$), a relation obtained by Dunlap and Judd [9]. In this work we present, using the connections between generalized characters of irreducible representations and a particular integral involving a product of ultraspherical Gegenbauer polynomials, a new sum rule for Clebsch-Gordan coefficients. To the best of our knowledge, such a result has not been reported before. The sum rule is also given in terms of $3jm$ symbols.

2 Derivation of the new sum rule

The generalized character (of order $k$) of the irreducible representation of rank $j$ of the rotation group is defined as

[TABLE]

where $k$ is integer ($0\leq k\leq 2j$) and $\chi^{j}$ is the character of the irreducible representation of rank $j$. The generalized character can be written [24, 27] as

[TABLE]

where $\langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle$ is the usual Clebsch-Gordan coefficient [6]. $\chi_{k}^{j}(\omega)$ can also be expressed in terms of Gegenbauer (or ultraspherical) [1] polynomials $C_{n}^{(\alpha)}$:

[TABLE]

where $(2k)!!=2k\times(2k-2)\times(2k-4)\times\cdots\times 4\times 2=2^{k}k!$. Gazeau and Kibler [12] obtained a sum rule on $3jm$ symbols using Bander-Itzykhson polynomials. Their expression can be obtained from the orthogonality relation for generalized characters

[TABLE]

$\delta_{a,b}$ being Kronecker’s symbol. The latter equation is equivalent to the orthogonality relation for Gegenbauer poynomials:

[TABLE]

where $\Gamma$ represents the Gamma function. Integrals involving products of Gegenbauer polynomials [14] is a powerful tool for finding identities which may be of great interest for atomic, molecular and nuclear physics [22, 26]. Laursen and Mita obtained, for $\alpha\geq 0$, the following expression [17]:

[TABLE]

where $~{}_{4}F_{3}$ represents the hypergeometric function. The latter expression becomes, in the particular case $\mu$=1,

[TABLE]

Setting $\eta=\omega/2$, one gets, from Eqs. (2) and (3):

[TABLE]

and

[TABLE]

respectively. Equating the two right-hand sides of Eqs. (10) and (11) yields, after multiplication by $\sin^{4}\eta$ and integration from $\eta$=0 to $\eta$=$\pi$:

[TABLE]

Setting $x=\cos\eta$, $\alpha=k+1$ and $n=2j-k$, Eq. (9) becomes

[TABLE]

and thus

[TABLE]

with

[TABLE]

yielding to the new sum rule for Clebsch-Gordan coefficients

[TABLE]

where the notation $\delta(abc)$ (see Ref. [10]) means that $a$, $b$ and $c$ satisfy triangular relations. Using Euler’s sine product formula

[TABLE]

Eq. (2) can also be put in the form

[TABLE]

Using the relation between Clebsch-Gordan coefficients and $3j$ symbols [27]:

[TABLE]

as well as the symmetry property

[TABLE]

we find that Eq. (2) becomes

[TABLE]

Eq. (25) can also be expressed as

[TABLE]

The identities (2), (2), (25) and (31) have been tested numerically.

3 Conclusion

A new sum rule for Clebsch-Gordan coefficients or $3j$ Wigner symbols was derived using generalized characters of irreducible representations of the rotation group. Beyond this simple but non-trivial result, which may be of interest for angular-momentum calculations, we hope that the technique presented here shall stimulate the generation of new identities.