$\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)=i^{3\lambda}\sqrt{\displaystyle\frac{2j+1}{2\lambda+1}}\;\mbox{Tr}\\!\left[f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf
J})\;\hat{{\cal D}}^{(j)}\\!(\psi,{\bf\hat{n}})\right]$ (280)

Neither definition is more fundamental, but it goes without saying that
calculating traces is far simpler than calculating derivatives.

The important role that the Chebyshev polynomial operators
$f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ play in representations of the
rotation group is evident in the expansion coefficients as defined by Corio
corio:siam

$a_{\lambda}^{(j)}(\psi)=\boxed{\mbox{Tr}\\!\left[f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf
J})\;\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})\right]=i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)}$
(281)

which are proportional to the generalized characters
$\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$ varshal1:ang  of the rotation
group. The first equality of Eq.(281) is due to Corio corio:siam , but the
“boxed” relation of Eq.(281) is new, and, as we illustrate with an example in
Appendix D, can be used as another definition of the generalized characters
$\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$. Several other definitions of
$\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$ are tabulated in Varshalovich et
al. varshal1:ang . Putting $\lambda=0$ in this “boxed” relation, we recover
the character $\mbox{{\large$\chi$}}^{(j)}(\psi)$ of the irreducible
representation $\hat{{\cal D}}^{(j)}(\psi,{\bf\hat{n}})$

$\mbox{Tr}\\!\left[\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})\right]=\mbox{{\large$\chi$}}^{(j)}(\psi)$
(282)

using the facts that varshal1:ang ; corio:siam

$\displaystyle f_{0}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\mathds{1}}{\sqrt{2j+1}}$ (283)
$\displaystyle\mbox{{\large$\chi$}}_{0}^{(j)}(\psi)$ $\displaystyle\equiv$
$\displaystyle\mbox{{\large$\chi$}}^{(j)}(\psi)$ (284)

By taking the trace of the rotation operator $\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})$ expanded in the Chebyshev polynomial operator
basis $f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$, the
character $\mbox{{\large$\chi$}}^{(j)}(\psi)$ of irreducible representations
of the rotation group can be obtained. Using Corio’s expansion corio:siam  of
the rotation operator $\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})=e^{i\psi({\bf\hat{n}}\cdot{\bf J})}$, this
trace can be evaluated as

$\displaystyle\mbox{Tr}\\!\left[\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})\right]$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;\boxed{\mbox{Tr}\\!\left[f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\right]}$ (285) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;\boxed{\sqrt{2j+1}\;\delta_{\lambda
0}}$ (286) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\;\mbox{{\large$\chi$}}_{0}^{(j)}(\psi)\;\sqrt{2j+1}$
(287) $\displaystyle=$ $\displaystyle\mbox{{\large$\chi$}}^{(j)}(\psi)$ (288)
$\displaystyle\mbox{where \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{varshal1:ang}{\@@citephrase{(}}{\@@citephrase{)}}}}\;\;\;\;\mbox{{\large$\chi$}}_{0}^{(j)}(\psi)$
$\displaystyle\equiv$ $\displaystyle\mbox{{\large$\chi$}}^{(j)}(\psi)$ (289)

The “boxed” trace term in Eq.(285) has been replaced by the “boxed” term of
Eq.(286) by calculating the trace in a representation where
$({\bf\hat{n}}\cdot{\bf J})$ is diagonal. Then we find

$\displaystyle\mbox{Tr}\\!\left[f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\right]$ $\displaystyle=$ $\displaystyle\sum_{m}\langle
m|\,f_{\lambda}^{(j)}(J_{z})\,|m\rangle$ (290) $\displaystyle=$
$\displaystyle\sum_{m}f_{\lambda}^{(j)}(m)$ (291) $\displaystyle=$
$\displaystyle\sqrt{2j+1}\;\delta_{\lambda 0}\;\;\;\;\;(\mbox{using
Eq.(\ref{chebsum})}$ (292)

The Chebyshev polynomial expansions of the projection operators (see Eqs.(112)
and (113) of Section 3.3.1) are particularly effective when they are used with
Sylvester’s formula merzbacher2 ; horn . We illustrate this effectiveness with
a simple alternative derivation of the Chebyshev polynomial operator expansion
of the rotation operator corio:siam  in the version given in Eq.(273). We
begin by exploiting Sylvester’s formula merzbacher2 ; horn  to write the
exponential matrix operator as a sum of projection operators in Eq.(293), and
then use the Chebyshev polynomial expansion (see Eq.(113) of Section 3.3.1) of
the projection operator $\Pi$${}^{(j)}(m,{\bf\hat{n}})$

$\displaystyle e^{-i\psi({\bf\hat{n}}\cdot{\bf J})}$ $\displaystyle=$
$\displaystyle\sum_{m=-j}^{j}e^{-im\psi}\;\boxed{\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})}$
(293) $\displaystyle=$
$\displaystyle\sum_{m=-j}^{j}e^{-im\psi}\,\boxed{\displaystyle\sum_{\lambda=0}^{2j}f_{\lambda}^{(j)}(m)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})}$ (294) $\displaystyle=$
$\displaystyle\displaystyle\sum_{\lambda=0}^{2j}\boxed{\sum_{m=-j}^{j}e^{-im\psi}\,f_{\lambda}^{(j)}(m)}\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ (295) $\displaystyle=$
$\displaystyle\displaystyle\sum_{\lambda=0}^{2j}\boxed{i^{-\lambda}\,\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\;\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)}\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ (296)

The projection operator $\Pi$${}^{(j)}(m,{\bf\hat{n}})$ in the “boxed” term of
Eq.(293) has been replaced by its Chebyshev polynomial expansion in the
“boxed” term of Eq.(294). The sum in the “boxed” term of Eq.(295) has been
reexpressed as the “boxed” term of Eq.(296) by converting a trigonometric
series identity varshal1:ang  for the generalized characters of the rotation
group $\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$ into a Chebyshev
polynomial series as follows:

$\displaystyle\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$ $\displaystyle=$
$\displaystyle i^{\lambda}\sum_{m=-j}^{j}\;e^{-im\psi}\,\boxed{C_{jm\lambda
0}^{jm}}$ (297) $\displaystyle=$ $\displaystyle
i^{\lambda}\sum_{m=-j}^{j}\;e^{-im\psi}\,\boxed{\sqrt{\displaystyle\frac{2j+1}{2\lambda+1}}(-1)^{j-m}\;C_{jmj-m}^{\lambda
0}}$ (298) $\displaystyle=$ $\displaystyle
i^{\lambda}\sum_{m=-j}^{j}\;e^{-im\psi}\,\sqrt{\displaystyle\frac{2j+1}{2\lambda+1}}\;f_{\lambda}^{(j)}(m)$
(299)

Well-documented varshal1:ang  symmetry properties of the Clebsch-Gordan
coefficients have been used to reexpress the “boxed” term in Eq.(297) as the
“boxed” term in Eq.(298). The Clebsch-Gordan coefficient and phase factor in
the “boxed” term of Eq.(298) have been replaced by their Chebyshev polynomial
equivalent $f_{\lambda}^{(j)}(m)$ in Eq.(299). Aside from the use of
Sylvester’s formula merzbacher2 ; horn , which is not particularly unique or
novel, the novelty of this derivation is that it exploits some of the most
felicitous properties of the Chebyshev polynomials, namely the Chebyshev
polynomial operator expansion of the projection operator, and the duality of
the Chebyshev polynomials as Clebsch-Gordan coefficients.

Remarkably, it is clear in retrospect that in 1967, Albert albert:rotgroup
had anticipated Corio’s corio:siam  Chebyshev polynomial operator expansion of
the rotation operator (see Eqs.(272 and (273)). In modern notation, Albert’s
expansion of $\exp[i\psi J_{z}]$ as a finite sum of irreducible tensor
components $\hat{T}_{\lambda 0}^{(J)}$ can be expressed as

$\displaystyle e^{i\psi J_{z}}$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;\boxed{\hat{T}_{\lambda
0}^{(j)}}$ (300) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;\boxed{f_{\lambda}^{(j)}\\!(J_{z})}$
(301)

The “boxed” irreducible tensor operator term in Eq.(300) has been reexpressed
as the “boxed” Chebyshev polynomial operator term in Eq.(301) using the
operator equivalence of Eq.(96). By considering the unitary transform of both
sides of Eq.(301), and using the results of similarity transforms summarized
in Table V, we then obtain Corio’s corio:siam  Chebyshev polynomial operator
expansion of the rotation operator

$\displaystyle\hat{{\cal D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\,e^{i\psi
J_{z}}\left[\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}$ (302)
$\displaystyle=$ $\displaystyle\exp\\!\left\\{i\psi\;\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\,J_{z}\left[\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}\right\\}$ (303)
$\displaystyle=$ $\displaystyle e^{i\psi({\bf\hat{n}}\cdot{\bf J})}$ (304)
$\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\left\\{\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\,\hat{T}_{\lambda
0}^{(j)}\left[\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}\right\\}$ (305)
$\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;\left\\{\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\;f_{\lambda}^{(j)}\\!(J_{z})\;\left[\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}\right\\}$ (306)
$\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ (307)

It is astonishing that Corio’s corio:siam ; corio:correction  Chebyshev
polynomial operator $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ expansion of
the rotation operator $\hat{{\cal D}}^{(j)}\\!(\psi,{\bf\hat{n}})$ in Eq.(273)
is hardly known at all, due in no small part to the fact that this expansion
is not mentioned or discussed in any angular momentum or quantum mechanics
textbook. And yet this expansion emphasizes the fact the Chebyshev polynomial
operators $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ play a unique role in
irreducible representations of the rotation group. In Section 5.3, we will
exploit this expansion to derive a tomographic reconstruction relation for the
density operator $\hat{\rho}$ using the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$.

###  V.2 Stratonovich-Weyl Operator

In the phase-space approach to spin, the conventional quantum mechanical
operators are replaced by functions on the classical phase-space of the unit
sphere ${\bf S}^{2}$. Central to this correspondence between Hilbert space
operators and functions on the phase-space is the Stratonovich-Weyl operator
varilly2:moyal ; heissweigert , also known as the Wigner-Stratonovich-Agarwal
operator klimov:distr ; agarwal ; klimovchumakov . Beginning with the
conventional representation varilly2:moyal ; klimov:distr ; klimovchumakov ,
this operator (kernel) $\Delta^{(j)}({\bf\hat{n}})$ can be expanded in terms
of the Chebyshev polynomial operator basis
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ as follows

$\displaystyle\Delta^{(j)}({\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda}\sqrt{\displaystyle\frac{4\pi}{2j+1}}\;\boxed{\left[\hat{T}^{(j)}_{\lambda\mu}\right]^{\\!\dagger}}\;Y_{\lambda\mu}({\bf\hat{n}})$
(308) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda}\sqrt{\displaystyle\frac{4\pi}{2j+1}}\;\hat{T}^{(j)}_{\lambda\mu}\;Y^{\star}_{\lambda\mu}({\bf\hat{n}})$
(309) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\sum_{\lambda=0}^{2j}\sqrt{2\lambda+1}\;\,\boxed{{\bf
C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})}$ (310)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\sum_{\lambda=0}^{2j}\sqrt{2\lambda+1}\;\,\boxed{f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})}$ (311)

where the “boxed” term of Eq.(308) has been reexpressed using the following
relation varshal1:ang  for the adjoint of the polarization operators

$\left[\hat{T}^{(j)}_{\lambda\mu}\right]^{\\!\dagger}=(-1)^{\mu}\;\hat{T}^{(j)}_{\lambda-\mu}$
(312)

To arrive at the result of Eq.(311), we have reexpressed the canonical version
of the Stratonovich-Weyl operator varilly2:moyal ; klimov:distr ;
klimovchumakov  in terms of a direct product of spin and spatial tensors in
Eq.(310). The “boxed” tensor direct product term in Eq.(310) has been
reexpressed as the “boxed” Chebyshev polynomial operator term in Eq.(311)
using a recoupling relation which we derive in Section 6.1.

####  V.2.1 $\Delta^{(j)}({\bf\hat{n}})$ Operator Trace

Using the Chebyshev polynomial operator expansion of Eq.(311), the
Stratonovich-Weyl operator $\Delta^{(j)}({\bf\hat{n}})$ varilly2:moyal ;
heissweigert  trace is easly evaluated using the properties of the Chebyshev
polynomials. We find

$\displaystyle\mbox{Tr}\left[\Delta^{(j)}({\bf\hat{n}})\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\sum_{\lambda=0}^{2j}\sqrt{2\lambda+1}\;\,\mbox{Tr}\left[f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\right]$ (313) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\sum_{\lambda=0}^{2j}\sqrt{2\lambda+1}\;\,\mbox{Tr}\left[f_{\lambda}^{(j)}(J_{z})\right]$
(314) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\sum_{\lambda=0}^{2j}\sqrt{2\lambda+1}\;\,\sum_{m}\langle
jm|\,f_{\lambda}^{(j)}(J_{z})\,|jm\rangle$ (315) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\sum_{\lambda=0}^{2j}\sqrt{2\lambda+1}\;\,\boxed{\sum_{m}f_{\lambda}^{(j)}(m)}$
(316) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2j+1}}\sum_{\lambda=0}^{2j}\sqrt{2\lambda+1}\;\,\sqrt{2j+1}\,\delta_{\lambda
0}$ (317) $\displaystyle=$ $\displaystyle 1$ (318)

The sum over the Chebyshev polynomials in the “boxed” term of Eq.(316) was
evaluated using Eq.(119) of Section 3.2.1.

####  V.2.2 The Reproducing Kernel for the $\Delta^{(j)}({\bf\hat{n}})$
Operators

Among the most important properties of the Stratonovich-Weyl operator kernel
is the traciality condition varilly2:moyal ; heissweigert

$\displaystyle\Delta^{(j)}({\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\frac{(2j+1)}{4\pi}\int_{{\bf
S}^{2}}\mbox{Tr}\\!\left[\Delta^{(j)}({\bf\hat{n}})\;\Delta^{(j)}({\bf\hat{n}}^{\prime})\right]\Delta^{(j)}({\bf\hat{n}}^{\prime})\,d{\bf\hat{n}}^{\prime}$
(319) $\displaystyle=$ $\displaystyle\int_{{\bf
S}^{2}}\;K^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})\;\Delta^{(j)}({\bf\hat{n}}^{\prime})\;d{\bf\hat{n}}^{\prime}$
(320) $\displaystyle=$ $\displaystyle\int_{{\bf
S}^{2}}\;\delta_{\Delta}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})\;\Delta^{(j)}({\bf\hat{n}}^{\prime})\;d{\bf\hat{n}}^{\prime}$
(321)
$\displaystyle\mbox{where}\;\;\;\;K^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{(2j+1)}{4\pi}\;\mbox{Tr}\\!\left[\Delta^{(j)}({\bf\hat{n}})\;\Delta^{(j)}({\bf\hat{n}}^{\prime})\right]$
(322) $\displaystyle d{\bf\hat{n}}$ $\displaystyle=$
$\displaystyle\sin\theta\,d\theta\,d\phi$ (323)

Eqs.(320) and (321) show that for a certain subset of $(2j+1)^{2}$ functions
on the sphere ${\bf S}^{2}$ varilly2:moyal ; heissweigert ,
$K^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})\equiv\delta_{\Delta}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$
is the reproducing kernel varilly2:moyal ; heissweigert , acting as a delta
function with respect to integration over ${\bf S}^{2}$. Armed with the
Chebyshev polynomial operator expansion of Eq.(311), this delta function can
be simply evaluated using the Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ trace identity discussed in
Section 2.2.1 as follows:

$\displaystyle\delta_{\Delta}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\sum_{\lambda=0}^{2j}\;\sum_{\lambda^{\prime}=0}^{2j}\;\sqrt{2\lambda+1}\,\sqrt{2\lambda^{\prime}+1}\;\,\mbox{Tr}\\!\left[f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\,f_{\lambda^{\prime}}^{(j)}({\bf\hat{n}}^{\prime}\cdot{\bf
J})\right]\;\;\;\;\;\;$ (324) $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\sum_{\lambda=0}^{2j}\;\sum_{\lambda^{\prime}=0}^{2j}\;\sqrt{2\lambda+1}\,\sqrt{2\lambda^{\prime}+1}\;\,\delta_{\lambda\lambda^{\prime}}\;P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})$
(325) $\displaystyle=$
$\displaystyle\;\sum_{\lambda=0}^{2j}\;\boxed{\frac{2\lambda+1}{4\pi}\,P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})}$
(326) $\displaystyle=$
$\displaystyle\;\sum_{\lambda=0}^{2j}\boxed{\sum_{\mu=-\lambda}^{\lambda}Y_{\lambda\mu}({\bf\hat{n}})\,Y_{\lambda\mu}^{\star}({\bf\hat{n}}^{\prime})}$
(327)

The “boxed” term of Eq.(326) has been reexpressed as the “boxed” term of
Eq.(327) using the spherical harmonics addition theorem gottfried :

$P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})=\frac{4\pi}{2\lambda+1}\;\sum_{\mu=-\lambda}^{\lambda}Y_{\lambda\mu}({\bf\hat{n}})\,Y_{\lambda\mu}^{\star}({\bf\hat{n}}^{\prime})$
(328)

Certainly there are other methods for evaluating
$\delta_{\Delta}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$ varilly2:moyal ;
heissweigert , but this method using the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ is among the most direct. Note
that the closure relation gottfried  for the spherical harmonics
$\left\\{Y_{\lambda\mu}({\bf\hat{n}})\right\\}$ could be obtained from
Eq.(327) in the limit that $j\rightarrow\infty$ to give

$\displaystyle\delta({\bf\hat{n}}-{\bf\hat{n}}^{\prime})$
$\displaystyle\equiv$
$\displaystyle\frac{\delta(\theta-\theta^{\prime})\;\delta(\phi-\phi^{\prime})}{\sin\,\theta}$
(329) $\displaystyle=$
$\displaystyle\;\sum_{\lambda=0}^{\infty}\sum_{\mu=-\lambda}^{\lambda}Y_{\lambda\mu}({\bf\hat{n}})\,Y_{\lambda\mu}^{\star}({\bf\hat{n}}^{\prime})$
(331)

Substituting the expression for the reproducing kernel
$\delta_{\Delta}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$ from Eq.(326) in
Eq.(321), and using the Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ expansion of the Stratonovich-
Weyl operator given in Eq.(311), we easily obtain the reproducing kernel
$\delta_{f}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$

$\delta_{f}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})=\frac{2\lambda+1}{4\pi}\,P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})$
(332)

which for the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$, acts as a delta function with
respect to integration over ${\bf S}^{2}$:

$\displaystyle f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle\int_{{\bf
S}^{2}}\;\delta_{f}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})\;f_{\lambda}^{(j)}({\bf\hat{n}}^{\prime}\cdot{\bf
J})\;d{\bf\hat{n}}^{\prime}$ (333) $\displaystyle=$
$\displaystyle\frac{2\lambda+1}{4\pi}\,\int_{{\bf
S}^{2}}\;P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})\;f_{\lambda}^{(j)}({\bf\hat{n}}^{\prime}\cdot{\bf
J})\;d{\bf\hat{n}}^{\prime}$ (334) $\displaystyle=$
$\displaystyle\frac{2\lambda+1}{4\pi}\,\int_{{\bf
S}^{2}}\;\mbox{Tr}\\!\left[f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf
J})\;f^{(j)}_{{\lambda}}({\bf\hat{n}}^{\prime}\\!\cdot{\bf
J})\right]\;f_{\lambda}^{(j)}({\bf\hat{n}}^{\prime}\cdot{\bf
J})\;d{\bf\hat{n}}^{\prime}$ (335)

The reproducing kernel in Eq.(334) has been rewritten as a trace in Eq.(335)
to elicit the analogy with the traciality condition varilly2:moyal ;
heissweigert  of Eq.(319) for the Stratonovich-Weyl operators. If in Eq.(334),
we put ${\bf\hat{n}}={\bf\hat{z}}$ and ${\bf\hat{n}}^{\prime}={\bf\hat{n}}$,
so that
${\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime}\equiv{\bf\hat{z}}\cdot{\bf\hat{n}}=\cos\theta$
then what at first sight is certainly not a familiar relation becomes

$\displaystyle f^{(j)}_{{\lambda}}({\bf\hat{z}}\\!\cdot{\bf J})\equiv
f^{(j)}_{{\lambda}}(J_{z})$ $\displaystyle=$
$\displaystyle\frac{2\lambda+1}{4\pi}\,\int_{{\bf
S}^{2}}\;P_{\lambda}(\cos\theta)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\;d{\bf\hat{n}}$ (336)

But since $f^{(j)}_{{\lambda}}(J_{z})\equiv\hat{T}^{(j)}_{\lambda 0}$, and
$P_{\lambda}(\cos\theta)\equiv C_{\lambda 0}({\bf\hat{n}})$, Eq.(336) can be
reexpressed as

$\hat{T}^{(j)}_{\lambda 0}=\frac{2\lambda+1}{4\pi}\,\int_{{\bf
S}^{2}}\;C_{\lambda 0}({\bf\hat{n}})\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\;d{\bf\hat{n}}$ (337)

which for the spin polarization operators $\hat{T}^{(j)}_{\lambda\mu}$ is a
particular version $(\mu=0)$ of a decomposition on the Chebyshev polynomial
operators $f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$ (see
Eq.(57) in Section 2.2.2).

In the next section, we will use this reproducing kernel
$\delta_{f}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$ to derive a tomographic
reconstruction formula for the density operator from the conventional
statistical tensor expansion of the density operator.

###  V.3 Tomographic Reconstruction of the Density Operator

As an alternative to the phase-space approach to spin, the tomographic map of
spin states onto a probability distribution filippov4 ; filippov2:thesis ;
fillipov1:qubit ; filippov3 ; manko:spintomo ; klimovchumakov  represents
another approach to mapping spin operators onto functions. In the first two
parts of this section, specific examples of these distributions are
considered, and in each case, these mappings are shown to lead to a novel
tomographic reconstruction formula for the density operator $\hat{\rho}$
expressed exclusively in terms of Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$. In the concluding part, without
considering a specific probability distribution, this tomographic
reconstruction formula is recovered using more fundamental approaches.

####  V.3.1 Spin Tomogram Distributions

One example of tomographic mapping is the approach developed by Man’ko and
colleagues filippov4 ; filippov2:thesis ; fillipov1:qubit ; filippov3 ;
manko:spintomo , who have made very effective use of the following Chebyshev
polynomial operator $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ expansions
of the dequantizer (alias projection) operators
$\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})$ and quantizer operators
$\mbox{{\boldmath$\Xi$}}^{(j)}(m,{\bf\hat{n}})$

$\displaystyle\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}f_{\lambda}^{(j)}(m)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ $\displaystyle\mbox{{\boldmath$\Xi$}}^{(j)}(m,{\bf\hat{n}})$
$\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}(2\lambda+1)\,f_{\lambda}^{(j)}(m)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ (338)

The spin tomogram $w^{(j)}(m,{\bf\hat{n}})$ of a state determined by the
density operator $\hat{\rho}$ is filippov4 ; filippov2:thesis ;
fillipov1:qubit ; filippov3 ; manko:spintomo

$w^{(j)}(m,{\bf\hat{n}})=\mbox{Tr}\\!\left[\hat{\rho}\;\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})\right]$
(339)

whereas the inverse mapping of the spin tomogram $w^{(j)}(m,{\bf\hat{n}})$
onto the density operator $\hat{\rho}$ was expressed through the quantizer
operator $\mbox{{\boldmath$\Xi$}}^{(j)}(m,{\bf\hat{n}})$ as the tomographic
reconstruction filippov4 ; filippov2:thesis ; fillipov1:qubit ; filippov3 ;
manko:spintomo

$\displaystyle\hat{\rho}$ $\displaystyle=$
$\displaystyle\sum_{m^{\prime}=-j}^{j}\;\frac{1}{4\pi}\int_{{\bf
S}^{2}}\,w^{(j)}(m^{\prime},{\bf\hat{n}}^{\prime})\;\,\mbox{{\boldmath$\Xi$}}^{(j)}(m^{\prime},{\bf\hat{n}}^{\prime})\;d{\bf\hat{n}}^{\prime}$
(340) $\displaystyle\mbox{where}\;\;\;\;d{\bf\hat{n}}^{\prime}$
$\displaystyle\equiv$ $\displaystyle d\Omega=\sin\theta\,d\theta\,d\phi$

Upon substituting this integral representation for the density operator in
Eq.(339), we obtain

$\displaystyle w^{(j)}(m,{\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\mbox{Tr}\\!\left[\hat{\rho}\;\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})\right]$
(341) $\displaystyle=$ $\displaystyle\sum_{m=-j}^{j}\;\frac{1}{4\pi}\int_{{\bf
S}^{2}}\,\;\mbox{Tr}\\!\left[\mbox{{\boldmath$\Xi$}}^{(j)}(m^{\prime},{\bf\hat{n}}^{\prime})\;\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})\right]w^{(j)}(m^{\prime},{\bf\hat{n}}^{\prime})\,d{\bf\hat{n}}^{\prime}$
(342)

This implies that for the set of tomograms $w^{(j)}(m,{\bf\hat{n}})$ on the
sphere ${\bf S}^{2}$, the trace in Eq.(342) must act like a delta function
with respect to integration over ${\bf S}^{2}$. This delta function can easily
be evaluated using the Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ trace identity discussed in
Section 2.2.1 as follows:

$\displaystyle\delta_{w}^{(j)}(m,{\bf\hat{n}};m^{\prime},{\bf\hat{n}}^{\prime})$
(343) $\displaystyle=$
$\displaystyle\delta_{m,m^{\prime}}\;\mbox{Tr}\\!\left[\mbox{{\boldmath$\Xi$}}^{(j)}(m^{\prime},{\bf\hat{n}}^{\prime})\;\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})\right]$
(344) $\displaystyle=$
$\displaystyle\delta_{m,m^{\prime}}\;\sum_{\lambda=0}^{2j}\;\sum_{\lambda^{\prime}=0}^{2j}\;(2\lambda{{}^{\prime}}+1)\,f_{\lambda}^{(j)}(m)\;f_{\lambda^{\prime}}^{(j)}(m^{\prime})\;\mbox{Tr}\\!\left[f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\,f_{\lambda^{\prime}}^{(j)}({\bf\hat{n}}^{\prime}\cdot{\bf J})\right]$
(345) $\displaystyle=$
$\displaystyle\delta_{m,m^{\prime}}\;\sum_{\lambda=0}^{2j}(2\lambda+1)\,f_{\lambda}^{(j)}(m)\;f_{\lambda}^{(j)}(m^{\prime})\;P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})$
(346)

As with the evaluation of
$\delta_{\Delta}^{(j)}({\bf\hat{n}},{\bf\hat{n}}^{\prime})$ in Section 5.2,
evaluating $\delta_{w}^{(j)}(m,{\bf\hat{n}};m^{\prime},{\bf\hat{n}}^{\prime})$
is straightforward using the the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$.

A much simpler expression for the tomographic reconstruction of the density
operator can be obtained from Eq.(340) just by replacing the quantizer and
dequantizer operators with their corresponding Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ expansions given in Eq.(338). By
means of these replacements, the density operator tomographic reconstruction
formula can be expressed exclusively in terms of Chebyshev polynomial
operators $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$

$\displaystyle\hat{\rho}$ $\displaystyle=$
$\displaystyle\sum_{m^{\prime}=-j}^{j}\;\frac{1}{4\pi}\int_{{\bf
S}^{2}}\mbox{Tr}\\!\left[\hat{\rho}\;\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})\right]\;\mbox{{\boldmath$\Xi$}}^{(j)}(m,{\bf\hat{n}})\;d{\bf\hat{n}}$
(347) $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{\lambda,\lambda^{\prime}=0}^{2j}(2\lambda+1)\;\boxed{\sum_{m^{\prime}=-j}^{j}f_{\lambda^{\prime}}^{(j)}(m)\;f_{\lambda}^{(j)}(m)}\int_{{\bf
S}^{2}}f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\;\;\mbox{Tr}\\!\left[\hat{\rho}\,f_{\lambda^{\prime}}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\right]d{\bf\hat{n}}\;\;\;\;$ (348) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}\frac{(2\lambda+1)}{4\pi}\int_{{\bf
S}^{2}}f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\;\mbox{Tr}\\!\left[\hat{\rho}\,f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\right]d{\bf\hat{n}}$ (349)

The “boxed” term in Eq.(348) has been simplified using the Chebyshev
polynomial orthogonality relation of Eq.(73), which collapses the double sum
in Eq.(348) to a single sum in Eq.(349).

####  V.3.2 Other Distributions

Other examples of probability distributions are Husimi’s husimi
$Q({\bf\hat{n}})$ function klimovchumakov  and the Stratonovich-Weyl
distribution $W({\bf\hat{n}})$ klimovchumakov , defined as

$\displaystyle Q({\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\langle{\bf\hat{n}},j|\;\hat{\rho}\;|{\bf\hat{n}},j\rangle=\mbox{Tr}\\!\left[\hat{\rho}\;\mbox{{\boldmath$\Pi$}}^{(j)}(j,{\bf\hat{n}})\right]$
(350) $\displaystyle W({\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\mbox{Tr}\\!\left[\hat{\rho}\;\Delta^{\\!(j)}({\bf\hat{n}})\right]$
(351)

For a spin state of well-defined angular momentum, Husimi’s husimi
$Q({\bf\hat{n}})$ function klimovchumakov  is defined as the average value of
the density matrix in the coherent state $|{\bf\hat{n}},j\rangle$, while the
Stratonovich-Weyl distribution $W({\bf\hat{n}})$ is defined in terms of the
Stratonovich-Weyl operator varilly2:moyal ; heissweigert
$\Delta^{\\!(j)}({\bf\hat{n}})$ of Section 5.2 by the trace of Eq.(351). Table
VII compares these probability distributions and their tomographic
reconstruction relations with those of the spin tomogram distributions
$w^{(j)}(m,{\bf\hat{n}})$ filippov4 ; filippov2:thesis ; fillipov1:qubit ;
filippov3 ; manko:spintomo . All the distributions and tomographic
reconstruction relations in this table are expressed only in terms of
Chebyshev polynomials $f_{\lambda}^{(j)}(m)$ or Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$, so that substitution of each
distribution in the corresponding tomographic reconstruction relation
immediately yields the tomographic reconstruction relation of Eq.(349). All
the distributions $D({\bf\hat{n}})\;\;[\equiv
w^{(j)}(m,{\bf\hat{n}}),Q({\bf\hat{n}})$ or $W({\bf\hat{n}})]$ of Table VI are
normalized klimovchumakov  so that

$\frac{2j+1}{4\pi}\int_{{\bf S}^{2}}D({\bf\hat{n}})\;d{\bf\hat{n}}=1$ (352)

Just as measurements of the spin tomograms $w^{(j)}(m,{\bf\hat{n}})$ enable
the reconstruction of the density operator $\hat{\rho}$ via the tomographic
reconstruction relation of Eq.(340) filippov4 ; filippov2:thesis ;
fillipov1:qubit ; filippov3 ; manko:spintomo , so do the corresponding
tomographic reconstruction relations for $\hat{\rho}$ in Table VII also permit
the reconstruction of all the density matrix elements simply by measuring the
corresponding $Q({\bf\hat{n}})$ or $W({\bf\hat{n}})$ function distributions
klimovchumakov .

####  V.3.3 More Fundamental Perspectives

In the previous two sections, specific examples of tomographic reconstruction
formulae for the density operator were considered. Regardless of the
probability distribution under consideration, the tomographic map of spin
states lead to a novel tomographic reconstruction formula expressed
exclusively in terms of Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$. From a more fundamental
perspective, can this formula be derived without considering a specific
probability distribution? Two approaches are now considered which demonstrate
that such a derivation is possible.

##### Statistical tensor expansion of the density operator

In this approach, we begin with a consideration of the statistical tensor
expansion of the density operator given in Table IV

$\hat{\rho}=\displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda}\mbox{Tr}\\!\left[\hat{\rho}\,\left[\hat{T}_{\lambda\mu}^{(j)}\right]^{\\!\dagger}\right]\hat{T}_{\lambda\mu}^{(j)}$
(353)

Viewing the spin polarization operators $\hat{T}^{(j)}_{\lambda\mu}$ in
Eq.(353) as the following integral transformation of the Chebyshev polynomial
operators $f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$ (see
Eq.(57) in Section 2.2.2)

$\displaystyle\hat{T}^{(j)}_{\lambda\mu}$ $\displaystyle=$
$\displaystyle\frac{2\lambda+1}{4\pi}\int
C_{\lambda\mu}({\bf\hat{n}})\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)\,d{\bf\hat{n}}$ (354)

the density operator can be written as

$\displaystyle\hat{\rho}$ (355) $\displaystyle\\!\\!\\!=$
$\displaystyle\displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda}\\!\mbox{Tr}\\!\left[\hat{\rho}\,\left[\frac{2\lambda+1}{4\pi}\int_{{\bf
S}^{2}}\\!C_{\lambda\mu}({\bf\hat{n}})\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)\,d{\bf\hat{n}}\right]^{\\!\dagger}\right]\frac{2\lambda+1}{4\pi}\\!\int_{{\bf
S}^{2}}\\!C_{\lambda\mu}({\bf\hat{n}}^{\prime})\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}^{\prime}\cdot{\bf
J}\right)\,d{\bf\hat{n}}^{\prime}\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\\!\\!\\!=$
$\displaystyle\displaystyle\sum_{\lambda=0}^{2j}\frac{2\lambda+1}{4\pi}\int\\!\mbox{Tr}\\!\left[\hat{\rho}\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)\right]\int_{{\bf
S}^{2}}\frac{2\lambda+1}{4\pi}\boxed{\sum_{\mu=-\lambda}^{\lambda}C_{\lambda\mu}^{\star}({\bf\hat{n}})\;C_{\lambda\mu}({\bf\hat{n}}^{\prime})}\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}^{\prime}\cdot{\bf
J}\right)\,d{\bf\hat{n}}^{\prime}\;d{\bf\hat{n}}\;$ (356)
$\displaystyle\\!\\!\\!=$
$\displaystyle\displaystyle\sum_{\lambda=0}^{2j}\frac{2\lambda+1}{4\pi}\int_{{\bf
S}^{2}}\\!\mbox{Tr}\\!\left[\hat{\rho}\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)\right]\boxed{\int_{{\bf
S}^{2}}\frac{2\lambda+1}{4\pi}P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}^{\prime}\cdot{\bf
J}\right)\,d{\bf\hat{n}}^{\prime}}\;d{\bf\hat{n}}\;$ (357)
$\displaystyle\\!\\!\\!=$
$\displaystyle\displaystyle\sum_{\lambda=0}^{2j}\frac{2\lambda+1}{4\pi}\int_{{\bf
S}^{2}}\\!\mbox{Tr}\\!\left[\hat{\rho}\;f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)\right]f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)d{\bf\hat{n}}$ (358)

The “boxed” term of Eq.(356) has been simplified using the spherical harmonics
addition theorem gottfried  of Eq.(328), and the “boxed” term of Eq.(357) has
been simplified using Eq.(334), recognizing
$\left[(2\lambda+1)/4\pi\right]\;P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})$
as the reproducing kernel for the Chebyshev polynomial operators
$f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$.

##### Group theory operator identity

In this altenative approach, we demonstrate that the Chebyshev polynomial
operator tomographic reconstruction formula of Eqs.(349) and (358) is a
consequence of much deeper and more general result in the form of a single
operator identity based on group theory ariano . Using group theory, D’Ariano
et al. ariano  have derived the following fundamental tomographic
reconstruction formula for the density operator

$\hat{\rho}=\int\mbox{Tr}\\!\left[\hat{\rho}\;{\cal R}(g)\right]{\cal
R}^{\dagger}(g)\,dg$ (359)

valid for an irreducible unitary representation ${\cal R}(g)$ on the Hilbert
space ${\cal H}$ of the physical system. The appropriate operators of such a
unitary irreducible representation in the case of a single spin physical
system are ariano

${\cal R}(g)\equiv{\cal R}({\bf\hat{n}},\psi)=e^{i\psi({\bf\hat{n}}\cdot{\bf
J})}$ (360)

In this parametrization, the invariant measure is given by ariano

$dg({\bf\hat{n}},\psi)=\frac{(2j+1)}{4\pi^{2}}\sin^{2}\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\sin\theta\,d\theta\,d\phi\,d\psi$ (361)

so that the density operator according to Eq.(359) can be written ariano

$\hat{\rho}=\frac{(2j+1)}{4\pi^{2}}\int_{0}^{2\pi}d\psi\,\sin^{2}\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\;\int_{{\bf
S}^{2}}\mbox{Tr}\\!\left[\hat{\rho}\,e^{i\psi({\bf\hat{n}}\cdot{\bf
J})}\right]e^{-i\psi({\bf\hat{n}}\cdot{\bf J})}\,d{\bf\hat{n}}$ (362)

D’Ariano et al. ariano  have also described an experimental apparatus designed
to reconstruct the density operator according to Eq.(362).

Taking advantage of Corio’s corio:siam ; corio:correction  Chebyshev
polynomial operator expansion of the operator $e^{i\psi({\bf\hat{n}}\cdot{\bf
J})}$ given in Eq.(273) of Section 5.1, and using the orthogonality relation
varshal1:ang  for the generalized character functions
$\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$

$\int_{0}^{2\pi}d\psi\,\sin^{2}\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\;\mbox{{\large$\chi$}}_{\lambda_{1}}^{(j_{1})}\\!(\psi)\;\mbox{{\large$\chi$}}_{\lambda_{2}}^{(j_{2})}\\!(\psi)=\pi\,\delta_{j_{1}j_{2}}\,\delta_{\lambda_{1}\lambda_{2}}$
(363)

which define the expansion coefficients, the spin tomographic reconstruction
formula of Eq.(359) derived by D’Ariano et al. from group theory ariano  can
be expressed exclusively in terms of the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$

$\hat{\rho}=\sum_{\lambda=0}^{2j}\frac{(2\lambda+1)}{4\pi}\int_{{\bf
S}^{2}}f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\;\mbox{Tr}\\!\left[\hat{\rho}\,f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})\right]d\Omega$ (364)

This is the tomographic reconstruction formula for the density operator
obtained in Sections 5.3.1 and 5.3.2 from specific examples of tomographic
reconstruction formulae using different spin tomography probability
distributions. Recovering this formula from the group theoretical operator
identity derived by D’Ariano et al. ariano  indicates that for the angle-axis
$(\psi,{\bf\hat{n}})$ parameterization of the $SU(2)$ group, the Chebyshev
polynomial operators $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ have a
unique role to play in tomographic reconstruction of the density operator.

##  VI  The recoupling of spin and spatial tensors via
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$

From a comparison of two different operator expansions corio:siam ; happer
for both the rotation operator $\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})=e^{i\psi({\bf\hat{n}}\cdot{\bf J})}$ in
Section 6.1.1, and for the coherent state projector
$|{\bf\hat{n}},j\rangle\langle{\bf\hat{n}},j|=\mbox{{\boldmath$\Pi$}}^{(j)}(j,{\bf\hat{n}})$
in Section 6.1.2, we derive a novel recoupling expression for the Chebyshev
polynomial operators $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$. In Section
6.2, we discuss a distinctly different approach filippov2:thesis  to deriving
the same recoupling expression. We conclude with specific examples of this
recoupling expression in the case of first- and second-rank tensors in Section
6.3.

###  VI.1 Operator expansion comparisons

####  VI.1.1 Rotation operator expansions

Using the orthonormal basis functions $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$, Corio corio:siam  derived the following expansion of the rotation
operator $\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})=e^{i\psi({\bf\hat{n}}\cdot{\bf J})}$

$\displaystyle e^{i\psi({\bf\hat{n}}\cdot{\bf J})}$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}a_{\lambda}^{(j)}\\!(\psi)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ (365) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2J}A(\lambda,j)\,i^{\lambda}\,s^{\lambda}\,\mbox{
C}_{2j-\lambda}^{\lambda+1}(c)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$
(366) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ (367) $\displaystyle\mbox{where}\;\;A(\lambda,j)$ $\displaystyle=$
$\displaystyle(2\lambda)!!\sqrt{2\lambda+1}\sqrt{\displaystyle\frac{(2j-\lambda)!}{(2j+\lambda+1)!}}$
(368) $\displaystyle s$ $\displaystyle=$ $\displaystyle\sin(\psi/2)$ (369)
$\displaystyle c$ $\displaystyle=$ $\displaystyle\cos(\psi/2)$ (370)
$\displaystyle\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$ $\displaystyle=$
$\displaystyle\sqrt{\displaystyle\frac{2j+1}{2\lambda+1}}\,A(\lambda,j)\,s^{\lambda}\,\mbox{
C}_{2j-\lambda}^{\lambda+1}(c)$ (371)

In Eq.(367), we have introduced the generalized character functions
varshal1:ang  $\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$ to reexpress the
operator expansion in Eq.(366) that Corio corio:siam  first presented. These
generalized character functions are defined in terms of the Gegenbauer
polynomials tem:bk ; askey  $\mbox{ C}_{2j-\lambda}^{\lambda+1}(c)$ as

$\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)=(2\lambda)!!\sqrt{2j+1}\sqrt{\frac{(2j-\lambda)!}{(2j+\lambda+1)!}}\;\;s^{\lambda}\mbox{
C}_{2j-\lambda}^{\lambda+1}(c)$ (372)

Using the direct product of spin and spatial tensors expressed in terms of the
spherical harmonics brinksatch:ang  $Y_{\lambda\mu}(\theta,\phi)$ and the spin
polarization tensor operators varshal1:ang  $\hat{T}_{\lambda\mu}^{(j)}$ as

$\displaystyle{\bf Y}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$
$\displaystyle\equiv$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}(-1)^{\mu}\;Y_{\lambda-\mu}({\bf\hat{n}})\;\hat{T}_{\lambda\mu}^{(j)}=\sum_{\mu=-\lambda}^{\lambda}Y_{\lambda\mu}^{\star}({\bf\hat{n}})\;\hat{T}_{\lambda\mu}^{(j)}$
(373)

Happer happer  derived an alternative rotation operator expression. Making use
of the generalized character functions varshal1:ang , Varshalovich et al.
varshal1:ang ; varshal2:expansion  subsequently reexpressed Happer’s happer
partial-wave expansion of the rotation operator $\hat{{\cal
D}}^{(j)}\\!(\psi,{\bf\hat{n}})=e^{i\psi({\bf\hat{n}}\cdot{\bf J})}$ as

$\displaystyle e^{i\psi({\bf\hat{n}}\cdot{\bf J})}$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}b_{\lambda}^{(j)}\\!(\psi)\;{\bf
Y}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ (374)
$\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}B(\lambda,j)\;i^{\lambda}\,s^{\lambda}\mbox{
C}_{2j-\lambda}^{\lambda+1}(c)\;{\bf Y}_{\lambda}({\bf\hat{n}})\cdot{\bf
T}_{\lambda}({\bf J})$ (375) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}i^{\lambda}\sqrt{\displaystyle\frac{2\lambda+1}{2j+1}}\,\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)\;{\bf
C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ (376)
$\displaystyle\mbox{where}\;\;B(\lambda,j)$ $\displaystyle=$
$\displaystyle\sqrt{4\pi}\;(2\lambda)!!\;\sqrt{\displaystyle\frac{(2j-\lambda)!}{(2j+\lambda+1)!}}$
(377) $\displaystyle C_{\lambda\mu}({\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\sqrt{\displaystyle\frac{4\pi}{2\lambda+1}}\;Y_{\lambda\mu}({\bf\hat{n}})$
(378)

Comparing Eq.(366) with Eq.(375), we see that

$A(\lambda,j)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})=B(\lambda,j)\;{\bf
Y}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ (379)

so that

$\displaystyle f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{B(\lambda,j)}{A(\lambda,j)}\;{\bf
Y}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ (380)
$\displaystyle=$
$\displaystyle\sqrt{\displaystyle\frac{4\pi}{2\lambda+1}}\;{\bf
Y}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ (381)
$\displaystyle=$ $\displaystyle{\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf
T}_{\lambda}({\bf J})$ (382) $\displaystyle=$
$\displaystyle\sum_{\mu=-{\lambda}}^{{\lambda}}(-1)^{\mu}\,C_{{\lambda}-{\mu}}({\bf\hat{n}})\;\hat{T}^{(j)}_{{\lambda}{\mu}}$
(383) $\displaystyle=$
$\displaystyle\sum_{\mu=-{\lambda}}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf\hat{n}})\;\hat{T}^{(j)}_{\lambda\mu}$
(384) $\displaystyle=$
$\displaystyle(-1)^{\lambda}\sqrt{2\lambda+1}\,\Big{\\{}{\bf T}_{\lambda}({\bf
J})\circledast{\bf C}_{\lambda}({\bf\hat{n}})\Big{\\}}_{\\!0}^{\\!0}$ (385)

where in Eq.(385), we have introduced the rank-zero composite product tensor
(see Appendix E) defined in terms of Clebsch-Gordan coefficients as

$\Big{\\{}{\bf R}^{(k)}\circledast{\bf
S}^{(k)}\Big{\\}}_{\\!0}^{\\!0}=\sum_{\begin{subarray}{c}q,q^{\prime}\\\
q+q^{\prime}=0\end{subarray}}{\bf R}^{(k)}_{q}\;{\bf
S}^{(k)}_{q^{\prime}}\;C^{00}_{kqkq^{\prime}}$ (386)

In his expansion of the rotation operator using the Chebyshev polynomial
operators $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$, Corio corio:siam  did
not express these basis operators as the direct product in Eq.(382) of
rank-$\lambda$ spatial and spin tensors. On the other hand, Happer happer  did
not identify or recognize this direct product ${\bf
C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ in his expansion of
the rotation operator as the Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$. The essential conclusion of
this section, embodied in Eq.(382), is that the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ and the direct product ${\bf
C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ of rank-$\lambda$
spatial and spin tensors are identical, and this identity leads to the
recoupling expression of Eq.(385). In Section 6.2, our discussion is centred
around an alternative approach taken by Filippov filippov2:thesis  which leads
to the same recoupling expression.

####  VI.1.2 Coherent state projection operator expansions

The coherent state projector $|{\bf\hat{n}},j\rangle\langle{\bf\hat{n}},j|$
has the following Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ expansion (see Eq.(124) of
Section 3.3.2):

$|{\bf\hat{n}},j\rangle\langle{\bf\hat{n}},j|=\mbox{{\boldmath$\Pi$}}^{(j)}(j,{\bf\hat{n}})=\sum_{\lambda=0}^{2j}f_{\lambda}^{(j)}(j)\;f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf
J})$ (387)

In the notation of this article, Ducloy ducloy  had already obtained the
following operator expansion for the coherent state projector

$\displaystyle|{\bf\hat{n}},j\rangle\langle{\bf\hat{n}},j|$ $\displaystyle=$
$\displaystyle\sqrt{4\pi}\,\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda}\frac{(2j)!}{\sqrt{(2j+\lambda+1)!(2j-\lambda)!}}\;Y^{\;\star}_{\lambda\mu}({\bf\hat{n}})\;\hat{T}^{(j)}_{\lambda\mu}$
(388) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}\boxed{\frac{(2j)!\sqrt{2\lambda+1}}{\sqrt{(2j+\lambda+1)!(2j-\lambda)!}}}\;\sum_{\mu=-\lambda}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf\hat{n}})\;\hat{T}^{(j)}_{\lambda\mu}$
(389) $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{2j}f_{\lambda}^{(j)}(j)\;\boxed{\sum_{\mu=-\lambda}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf\hat{n}})\;\hat{T}^{(j)}_{\lambda\mu}}$
(390)

The original expansion obtained by Ducloy ducloy  in Eq.(388) has been
rewritten in Eqs.(389) and (390) in order to facilitate a comparison with
Eq.(387). After using Eq.(71) to replace the “boxed” coefficient term in
Eq.(389) with the Chebyshev polynomial $f_{\lambda}^{(j)}(j)$ in Eq.(390), it
is clear from a comparison of the operator expansions in Eq.(387) and Eq.(390)
that the Chebyshev polynomial operator
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ can be expressed as the
following direct product or recoupling expressions

$\displaystyle f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle\sum_{\mu=-{\lambda}}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf\hat{n}})\;\hat{T}^{(j)}_{\lambda\mu}$
(391) $\displaystyle=$ $\displaystyle{\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf
T}_{\lambda}({\bf J})$ (392) $\displaystyle=$
$\displaystyle(-1)^{\lambda}\sqrt{2\lambda+1}\,\Big{\\{}{\bf T}_{\lambda}({\bf
J})\circledast{\bf C}_{\lambda}({\bf\hat{n}})\Big{\\}}_{\\!0}^{\\!0}$ (393)

in agreement with Eqs.(384) and (385) in Section 6.1.1.

###  VI.2 Exploiting the transformation properties of the Chebyshev polynomial
operators $f_{\lambda}^{(j)}(J_{z})$

Independently established by many workers meckler:angular ; filippov2:thesis ;
corio:ortho ; NormRay ; werb:tensor ; fillipov1:qubit  over the last 50 years,
the relation of Eq.(55)

$f_{\lambda}^{(j)}(J_{z})=\hat{T}_{\lambda 0}^{(j)}$ (394)

is an important and useful operator equivalent. In this section, we show how
it can provide another independent proof of the operator equivalence between
the Chebyshev polynomials $f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$, and
${\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf
J})=\sum_{\mu=-{\lambda}}^{\lambda}C^{\;\star}_{\lambda\mu}({\bf\hat{n}})\;\hat{T}^{(j)}_{\lambda\mu}$,
the direct product of rank-$\lambda$ spatial and spin tensors, namely the
(renormalized) Racah spherical harmonics brinksatch:ang
$C_{\lambda\mu}({\bf\hat{n}})$ and the spin polarization tensor operators
varshal1:ang  $\hat{T}^{(j)}_{\lambda\mu}$, respectively.

As irreducible tensor operators, the spin polarization operators
$\hat{T}^{(j)}_{\lambda\nu}$ transform as varshal1:ang

$\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\,\hat{T}^{(j)}_{\lambda\nu}\left[\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}=\sum_{\mu=-\lambda}^{\lambda}{\cal{D}}_{\mu\nu}^{(\lambda)}(\theta,{\bf\hat{n}}_{\bot})\;\hat{T}^{(j)}_{\lambda\mu}$
(395)

and in particular, the basis functions of Eq.(95) can be rewritten as
filippov2:thesis

$\displaystyle f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$
$\displaystyle=$ $\displaystyle\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\,f_{\lambda}^{(j)}(J_{z})\left[\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}$ (396)
$\displaystyle=$ $\displaystyle\hat{{\cal
D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\,\hat{T}^{(j)}_{\lambda
0}\left[\hat{{\cal D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}$
(397) $\displaystyle=$ $\displaystyle\sum_{\mu=-\lambda}^{\lambda}{\cal
D}^{(\lambda)}_{\mu
0}(\theta,{\bf\hat{n}}_{\bot})\;\hat{T}^{(j)}_{\lambda\mu}$ (398)
$\displaystyle=$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}C_{\lambda\mu}^{\star}(\theta,\phi)\;\hat{T}^{(j)}_{\lambda\mu}$
(399) $\displaystyle=$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}(-1)^{\mu}\;C_{\lambda-\mu}(\theta,\phi)\;\hat{T}^{(j)}_{\lambda\mu}$
(400) $\displaystyle=$ $\displaystyle{\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf
T}_{\lambda}({\bf J})$ (401) $\displaystyle=$
$\displaystyle(-1)^{\lambda}\sqrt{2\lambda+1}\,\Big{\\{}{\bf T}_{\lambda}({\bf
J})\circledast{\bf C}_{\lambda}({\bf\hat{n}})\Big{\\}}_{\\!0}^{\\!0}$ (402)

where as shown in Appendix B, the rotation matrix elements ${\cal
D}^{(\lambda)}_{\mu 0}(\theta,{\bf\hat{n}}_{\bot})\equiv
C_{\lambda\mu}^{\,\star}(\theta,\phi)$ brinksatch:ang  of Eq.(398) are the
Racah spherical harmonics of Eq.(378). Eqs.(397) and (398), originally derived
by Filippov filippov2:thesis  using a Euler angle parametrization, have been
reexpressed using an angle-axis parametrization. In Eqs.(399 \- 402), we
demonstrate that Filippov’s filippov2:thesis  transformation equation for the
tensor operators in Eq.(398) can then be expressed as a recoupling of spin and
spatial tensors. The final expressions for the basis functions
$f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$ in Eqs.(402) and
(385) demonstrate that a Chebyshev polynomial of degree $\lambda$ in the
variable $({\bf\hat{n}}\cdot{\bf J})$ can be recoupled as a rank-zero
irreducible composite tensor defined by the product of two rank-$\lambda$
tensors, one the spin tensor ${\bf T}_{\lambda}({\bf J})$, and the other, the
spatial Racah tensor ${\bf C}_{\lambda}({\bf\hat{n}})$:

$f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)={\bf
C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf
J})=(-1)^{\lambda}\sqrt{2\lambda+1}\,\Big{\\{}{\bf T}_{\lambda}({\bf
J})\circledast{\bf C}_{\lambda}({\bf\hat{n}})\Big{\\}}_{\\!0}^{\\!0}$ (403)

Alternatively, by determining the Euler angles $(\alpha,\beta,\gamma)$ of the
rotation $R\equiv R(\theta,{\bf\hat{n}}_{\bot})\equiv R(\alpha,\beta,\gamma)$
discussed in Section 3.3.1, an Euler angle parametrization of the rotation
operator $\hat{{\cal D}}^{(J)}\\!(\alpha,\beta,\gamma)$ can be used to verify
Eq.(401). If ${\bf\hat{n}}_{\bot}=(-\sin\phi,\cos\phi,0)$, then the angle-axis
parameters are
$(\psi;\Theta,\Phi)=\left(\theta;\displaystyle\frac{\pi}{2},\phi+\displaystyle\frac{\pi}{2}\right)$.
Since the Euler angles $(\alpha,\beta,\gamma)$ can be expressed in terms of
the angle-axis parameters $(\psi;\Theta,\Phi)$ using the following relations
varshal1:ang

$\displaystyle\sin\frac{\beta}{2}$ $\displaystyle=$
$\displaystyle\sin\Theta\;\sin\frac{\psi}{2}$
$\displaystyle\tan\frac{\alpha+\gamma}{2}$ $\displaystyle=$
$\displaystyle\cos\Theta\;\tan\frac{\psi}{2}$
$\displaystyle\frac{\alpha-\gamma}{2}$ $\displaystyle=$
$\displaystyle\Phi-\frac{\pi}{2}$ (404)

the Euler angles corresponding to the rotation operator
$R(\theta,{\bf\hat{n}}_{\bot})=e^{-i\theta({\bf\hat{n}}_{\bot}\cdot{\bf J})}$
are therefore $(\phi,\theta,-\phi)$, determined as a solution of the following
relations

$\displaystyle\sin\frac{\beta}{2}$ $\displaystyle=$
$\displaystyle\sin\frac{\theta}{2}\rightarrow\boxed{\beta=\theta}$
$\displaystyle\tan\frac{\alpha+\gamma}{2}$ $\displaystyle=$ $\displaystyle
0\rightarrow\boxed{\alpha+\gamma=0}$ $\displaystyle\frac{\alpha-\gamma}{2}$
$\displaystyle=$ $\displaystyle\phi\rightarrow\boxed{\alpha-\gamma=2\phi}$
(405)

Since irreducible spherical tensors $\hat{T}_{LM}$ transform as

$\left[\hat{{\cal
D}}^{(J)}(\phi,\theta,-\phi)\right]^{\\!\dagger}\;\hat{T}_{LM}\;\hat{{\cal
D}}^{(J)}\\!(\phi,\theta,-\phi)\;=\sum_{m}{\cal{D}}_{mM}^{(L)}(\phi,\theta,-\phi)\;\hat{T}_{LM}$
(406)

then in particular

$\displaystyle f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$
$\displaystyle=$ $\displaystyle\hat{{\cal
D}}^{(j)}\\!(\phi,\theta,-\phi)\;\hat{T}_{\lambda 0}^{(j)}\;\left[\hat{{\cal
D}}^{(j)}(\phi,\theta,-\phi)\right]^{\\!\dagger}$ (407) $\displaystyle=$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}{\cal D}^{(\lambda)}_{\mu
0}(\phi,\theta,-\phi)\;\hat{T}^{(j)}_{\lambda\mu}$ $\displaystyle=$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}C_{\lambda\mu}^{\star}(\theta,\phi)\;\hat{T}^{(j)}_{\lambda\mu}\;\;(\mbox{using
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{brinksatch:ang}{\@@citephrase{(}}{\@@citephrase{)}}}}\;\;{\cal{D}}_{\mu
0}^{(\lambda)}(\alpha,\beta,\gamma)=C_{\lambda\mu}^{\star}(\beta,\alpha))$
$\displaystyle=$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}(-1)^{\mu}\;C_{\lambda-\mu}(\theta,\phi)\;\hat{T}^{(j)}_{\lambda\mu}$
$\displaystyle=$ $\displaystyle{\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf
T}_{\lambda}({\bf J})$ $\displaystyle=$
$\displaystyle(-1)^{\lambda}\sqrt{2\lambda+1}\;\Big{\\{}{\bf T}_{\lambda}({\bf
J})\circledast{\bf C}_{\lambda}({\bf\hat{n}})\Big{\\}}_{\\!0}^{\\!0}$

where ${\bf C}_{\lambda}({\bf\hat{n}})$ is the renormalized Racah spherical
harmonics (tensor) brinksatch:ang .

###  VI.3 Specific recoupling examples: first- and second-rank tensors

In order to elicit the recoupling feature of the relation in Eq.(407), we
consider the more familiar and commonly encountered cases of first-rank
$(\lambda=1)$ and second-rank $(\lambda=2)$ spherical tensors.

####  VI.3.1 First-rank tensors

To illustrate our approach, we begin with the trivial case of first-rank
spherical tensors. The spin polarization operators ${\bf T}_{1}({\bf J})$ in
Eq.(401) can be replaced with their spherical operator equivalents
$\mathcal{T}$$\\!{}_{1}={\bf J}$ using the following relation varshal1:ang ;
ambler:traces

$\displaystyle{\bf T}_{1}({\bf J})$ $\displaystyle=$ $\displaystyle
a_{1}(j)\;\mbox{\boldmath$\mathcal{T}$}\\!_{1}=a_{1}(j)\;{\bf J}$ (408)
$\displaystyle\mbox{where}\;\;\;\;a_{1}(j)$ $\displaystyle=$
$\displaystyle\sqrt{\displaystyle\frac{3}{j(j+1)(2j+1)}}$ (409)

Component-wise, the relation of Eq.(408) can be expressed as the following
proportionality between the spin polarization operators $\hat{T}^{(j)}_{1M}$
and the spherical components of the spin operator $J_{M}$ as

$\hat{T}^{(j)}_{1M}=a_{1}(j)\,J_{M}\;\;\;\;\;(M=\pm 1,0)$ (410)

The Chebyshev polynomial operator $f_{1}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ can
be replaced with $({\bf\hat{n}}\cdot{\bf J})$ using the following relation
corio:ortho

$\displaystyle f_{1}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle a_{1}(j)\;({\bf\hat{n}}\cdot{\bf J})$ (411)

After these replacements, in this special case of first-rank spherical
tensors, Eq.(403) can then be written as

$\displaystyle({\bf\hat{n}}\cdot{\bf J})={\bf J}\cdot{\bf
C}_{1}({\bf\hat{n}})$ $\displaystyle=$ $\displaystyle({\bf\hat{n}}\cdot{\bf
J})\;\;\;\;\;\;\;\;$ (412) $\displaystyle\mbox{where
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{brinksatch:ang}{\@@citephrase{(}}{\@@citephrase{)}}}}\;\;\;\;{\bf
C}_{1}({\bf\hat{n}})$ $\displaystyle=$ $\displaystyle{\bf\hat{n}}$ (413)

####  VI.3.2 Second-rank tensors

While the result of Eq.(412) is self-evident, the case of second-rank tensors
is much more revealing. The spin polarization operators ${\bf T}_{2}({\bf J})$
in Eq.(401) can be replaced with their spherical operator equivalents
$\mathcal{T}$$\\!{}_{2}=\sqrt{6}\;\left\\{{\bf J}\circledast{\bf
J}\right\\}^{2}$ using the following relation ambler:traces

$\displaystyle{\bf T}_{2}({\bf J})$ $\displaystyle=$ $\displaystyle
a_{2}(j)\;\mbox{\boldmath$\mathcal{T}$}\\!_{2}=a_{2}(j)\sqrt{6}\left\\{{\bf
J}\circledast{\bf J}\right\\}^{2}$ (414)
$\displaystyle\mbox{where}\;\;\;\;a_{2}(j)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\sqrt{5}}{\sqrt{j(j+1)(2j+3)(2j-1)(2j+1)}}$
(415)

Component-wise, the relation of Eq.(414) can be expressed as the following
proportionalities between the spin polarization operators $\hat{T}^{(j)}_{2M}$
and the spherical components of the spin operator $\left\\{{\bf
J}\circledast{\bf J}\right\\}_{\\!M}^{2}$ as

$\displaystyle\hat{T}^{(j)}_{20}$ $\displaystyle=$
$\displaystyle\sqrt{6}\left\\{{\bf J}\circledast{\bf J}\right\\}_{0}^{2}$
(416) $\displaystyle\hat{T}^{(j)}_{2\pm 1}$ $\displaystyle=$
$\displaystyle\sqrt{6}\left\\{{\bf J}\circledast{\bf J}\right\\}_{\pm 1}^{2}$
(417) $\displaystyle\hat{T}^{(j)}_{2\pm 2}$ $\displaystyle=$
$\displaystyle\sqrt{6}\left\\{{\bf J}\circledast{\bf J}\right\\}_{\pm 2}^{2}$
(418)

The Chebyshev polynomial operator $f_{2}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ can
be replaced with $\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-({\bf J}\cdot{\bf
J})\right]$ using the following relation corio:ortho

$\displaystyle f_{2}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle a_{2}(j)\\!\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-({\bf
J}\cdot{\bf J})\right]$ (419) $\displaystyle\mbox{where}\;\;\;\;({\bf
J}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle\kappa\,\mathds{1}=j(j+1)\,\mathds{1}$ (420)

After these replacements, in this special case of second-rank spherical
tensors, Eq.(403) can be written as

$\displaystyle\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-({\bf J}\cdot{\bf
J})\right]=\sqrt{6}\left\\{{\bf J}\circledast{\bf J}\right\\}^{2}\cdot{\bf
C}_{2}({\bf\hat{n}})$ $\displaystyle=$ $\displaystyle
3\sqrt{5}\;\Big{\\{}\left\\{{\bf J}\circledast{\bf
J}\right\\}^{2}\circledast\left\\{{\bf\hat{n}}\circledast{\bf\hat{n}}\right\\}^{2}\Big{\\}}_{\\!0}^{\\!0}\;\;\;\;\;\;\;\;$
(421) $\displaystyle\mbox{where \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{brinksatch:ang}{\@@citephrase{(}}{\@@citephrase{)}}}}\;\;\;\;{\bf
C}_{2}({\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\sqrt{\displaystyle\frac{3}{2}}\left\\{{\bf\hat{n}}\circledast{\bf\hat{n}}\right\\}^{2}$
(422)

This recoupling of two irreducible second-rank spherical tensors in Eq.(421)
is exactly analogous to the recoupling of the magnetic dipolar interaction
abragamtext  or the tensor force brinksatch:ang

$\left[3({\bf\hat{n}}\cdot{\bf J}_{1})({\bf\hat{n}}\cdot{\bf J}_{2})-({\bf
J}_{1}\cdot{\bf J}_{2})\right]=3\sqrt{5}\;\Big{\\{}\left\\{{\bf
J}_{1}\circledast{\bf
J}_{2}\right\\}^{2}\circledast\left\\{{\bf\hat{n}}\circledast{\bf\hat{n}}\right\\}^{2}\Big{\\}}_{\\!0}^{\\!0}$
(423)

Normally, the recoupling relationships of Eqs.(421) or (423) can only be
established by a laborious expansion of both sides of these equations, or by
using the formalism of angular momentum theory brinksatch:ang  to recouple the
four rank-1 tensors on the right-hand side of these equations via a
9$j$-symbol as outlined in Appendix E. Instead, the recoupling relation of
Eq.(421) arises very simply and directly just from the definition of the
Chebyshev polynomial operator $f_{2}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)$, examples of which are provided in Table I for $j=1/2,1$ and $3/2$.

####  VI.3.3 Matrix elements of the truncated homonuclear dipolar Hamiltonian

From the recoupling expression in Eq.(399) applied to
$f_{2}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$, the general matrix
elements are

$\displaystyle\langle jm|\,f_{\lambda}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)|jm^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}C_{\lambda\mu}^{\star}(\theta,\phi)\;\langle
jm|\,\hat{T}^{(j)}_{\lambda\mu}|jm^{\prime}\rangle$ (424) $\displaystyle=$
$\displaystyle\sum_{\mu=-\lambda}^{\lambda}C_{\lambda\mu}^{\star}(\theta,\phi)\;\sqrt{\frac{2\lambda+1}{2j+1}}\;C_{jm^{\prime}\lambda\mu}^{jm}$
(425) $\displaystyle=$ $\displaystyle
C_{\lambda\,(m-m^{\prime})}^{\star}(\theta,\phi)\;C_{jmj-m^{\prime}}^{\lambda\,(m-m^{\prime})}\;(-1)^{j-m^{\prime}}$
(426)

The diagonal matrix elements $(m=m^{\prime})$ for
$f_{2}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf J}\right)$ are therefore

$\displaystyle\langle jm|\,f_{2}^{(j)}\\!\left({\bf\hat{n}}\cdot{\bf
J}\right)|jm\rangle$ $\displaystyle=$ $\displaystyle
C_{20}^{\star}(\theta,\phi)\;C_{jmj-m}^{20}\;(-1)^{j-m}$ (427)
$\displaystyle=$ $\displaystyle P_{2}(\cos\theta)\;f_{2}^{(j)}(m)$ (428)
$\displaystyle=$ $\displaystyle
P_{2}(\cos\theta)\;\frac{3m^{2}-2}{\sqrt{6}}\;\;\;\;\;\;(j=1)$ (429)

But then using Eqs.(415) and (419),

$\displaystyle f_{2}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle a_{2}(j)\\!\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-({\bf
J}\cdot{\bf J})\right]$ (430) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\;\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-({\bf
J}\cdot{\bf J})\right]\;\;\;\;\;\;(j=1)$ (431)

we find that the diagonal matrix elements of $\left[3({\bf\hat{n}}\cdot{\bf
J})^{2}-({\bf J}\cdot{\bf J})\right]$ for $j=1$ are given by

$\langle m|\,\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-({\bf J}\cdot{\bf
J})\right]\,|m\rangle=P_{2}(\cos\theta)\left[3m^{2}-2\right]$ (432)

The homonuclear dipolar Hamiltonian ${\cal H}_{D}^{\mbox{intra}}$ abragamtext
; keller , which describes the through-space interaction between the magnetic
moments of two adjacent protons separated by a distance $r$ is given by

$\displaystyle{\cal H}_{D}^{\mbox{intra}}$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}\gamma^{2}\hbar}{4\pi
r^{3}}\;\left[3({\bf\hat{n}}\cdot{\bf J}_{1})({\bf\hat{n}}\cdot{\bf
J}_{2})-({\bf J}_{1}\cdot{\bf J}_{2})\right]$ (433)
$\displaystyle\mbox{where}\;\;\;\;{\bf\hat{n}}$ $\displaystyle=$
$\displaystyle{\bf\hat{r}}$ (434)

This interaction Hamiltonian can also be expressed in terms of the total spin
operator keller  ${\bf J}={\bf J}_{1}+{\bf J}_{2}$ for each pair of protons as

${\cal H}_{D}^{\mbox{intra}}=\frac{\mu_{0}\gamma^{2}\hbar}{4\pi
r^{3}}\;\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-({\bf J}\cdot{\bf J})\right]$
(435)

High-field approximation of this Hamiltonian leaves only the truncated part
abragamtext

$\displaystyle\left[{\cal
H}_{D}^{\mbox{intra}}\right]^{\prime}=\frac{\mu_{0}\gamma^{2}\hbar}{4\pi
r^{3}}\;P_{2}(\cos\theta)\left[3J_{z}^{2}-({\bf J}\cdot{\bf J})\right]$ (436)

where $\theta$ is the angle between the internuclear vector ${\bf r}$ and the
magnetic field ${\bf B}_{0}$. The form of $\left[{\cal
H}_{D}^{\mbox{intra}}\right]^{\prime}$ is identical to that of spin-1 systems,
provided the “quadrupole frequency” $\nu_{Q}$ is redefined as keller

$\nu_{Q}=\frac{3\hbar\gamma^{2}}{4\pi r^{3}}$ (437)

If a basis of the total angular momentum eigenstates of ${\bf J}={\bf
J}_{1}+{\bf J}_{2}$ is used, it is evident from Eq.(432) that the diagonal
elements of the truncated Hamiltonian $\left[{\cal
H}_{D}^{\mbox{intra}}\right]^{\prime}$ of Eq.(436) match the diagonal elements
of the diagonal elements of the dipolar Hamiltonian ${\cal
H}_{D}^{\mbox{intra}}$ of Eq.(433). The latter elements are just what we
expect from the secular approximation abragamtext , which retains only that
part of the dipolar Hamiltonian $\left[{\cal
H}_{D}^{\mbox{intra}}\right]^{\prime}$ which commutes with the Zeeman
Hamiltonian ${\cal H}_{Z}$

$\left[\left[{\cal H}_{D}^{\mbox{intra}}\right]^{\prime}\\!\\!,{\cal
H}_{Z}\right]=0$ (438)

##  VII Operator Equivalents

Finding suitable operator equivalents for irreducible tensor operators in
terms of the familiar angular momentum operators $J_{z},J_{\pm}=J_{x}\pm
iJ_{y}$, and $({\bf J}\cdot{\bf J})=J^{2}\equiv J(J+1)$ has been a long-
standing challenge in fields such as magnetic resonance. As a result, there is
an extensive literature over more than seven decades devoted to this problem
in NMR, EPR and ENDOR. A reasonably comprehensive list of the salient
literature references on operator equivalents can be found in a recent
publication by Ryabov ryabov:equiv . In their studies, Ryabov ryabov:equiv
himself, as well as Ambler et al. ambler:traces , Ohlsen ohlsen , and
Biedenharn and Louck biedenharn , for example, have tabulated a significant
number of these operator equivalents. Few however are those studies that focus
on the operator equivalences between the spin polarization operators
varshal1:ang  $\hat{T}_{\lambda\mu}^{(j)}$ and the Chebyshev polynomial
operators $f_{\lambda}^{(j)}(J_{z})$. Work by Meckler meckler:angular , Corio
corio:ortho , Marinelli et al. werb:tensor  and by Normand and Raynal NormRay
constitutes the essential core of the research that has elucidated the role of
Chebyshev polynomials in developing operator equivalents, and in this section,
we provide a brief summary of this work.

In considering the history of how Chebyshev polynomial operators
$f_{\lambda}^{(j)}(J_{z})$ were used to develop operator equivalents, two
developments stand out. First, the operator equivalents for projection-zero
tensor operators $\hat{T}_{\lambda 0}^{(j)}$ were identified as the Chebyshev
polynomial operators $f_{\lambda}^{(j)}\\!(J_{z})$ meckler:angular ;
corio:ortho ; werb:tensor ; NormRay . Second, an intriguing relationship
between irreducible (spherical) tensor operators of arbitrary projection
$T^{(j)}_{\lambda\mu}$ and successive partial derivatives of the Chebyshev
polynomial operators $f^{(j)}_{\lambda}\\!(J_{z})$ werb:tensor  was discovered
werb:tensor . In this section, we begin in Section 7.1 by revisiting a
Chebyshev polynomial operator equivalent for projection-zero tensor operators
$\hat{T}_{\lambda 0}^{(j)}$ that we had used in Section 3.1. Then, in Section
7.2, we use Table VIII to illustrate that Chebyshev polynomial operators
$f^{(j)}_{\lambda}(J_{z})$ have been used to develop operator equivalents for
any irreducible tensor operator $\hat{T}_{\lambda\mu}^{(j)}$.

###  VII.1 Operator equivalent matrix elements

As we illustrated in Section 3.1, among the most useful operator equivalences
involving the Chebyshev polynomial operators is

$\hat{T}_{L0}^{(j)}\equiv f_{L}^{(j)}\\!(J_{z})$ (439)

Meckler’s meckler:angular  recognition of this equivalence between the
Chebyshev polynomial operators $f_{L}^{(j)}(J_{z})$ and the projection-zero
spin polarization operators varshal1:ang  $\hat{T}_{L0}^{(j)}$ anticipated
subsequent work by several investigators filippov2:thesis ; corio:ortho ;
NormRay ; werb:tensor ; fillipov1:qubit , who independently established this
operator equivalence, and who provided specific and extensive tables
illustrating this operator equivalence corio:ortho ; NormRay ; werb:tensor .

Ambler et al. ambler:traces  tabulated operator equivalents for the
irreducible tensor operators $T_{q}^{(k)}\\!(J)$ (equivalent to the spin
polarization operators $\hat{T}_{kq}^{(J)}$ in our notation), as did Corio
corio:ortho  for his orthonormal operator basis $U^{(n)}_{r}\\!(J)$, whose
relation to the irreducible tensor operators $T_{q}^{(k)}\\!(J)$ was
determined by these relations corio:ortho

$\displaystyle n+r$ $\displaystyle=$ $\displaystyle k$ $\displaystyle r=q$
(440)

In addition to showing how to construct operator equivalents for his
orthonormal basis $U^{(n)}_{r}\\!(J)$, Corio corio:ortho , unaware of
Meckler’s earlier work meckler:angular , also independently identified the
$U^{(k)}_{0}\\!(J)\equiv\hat{T}_{k0}^{(J)}$ operators, the tensors of rank $k$
and projection $0$, as the Chebyshev polynomial operator equivalents
$f_{k}^{(J)}\\!(J_{z})$. Although a few others werb:tensor ; NormRay  have
also independently made this identification of the $\hat{T}_{k0}^{(J)}$
operators with the Chebyshev polynomial operators $f_{k}^{(J)}\\!(J_{z})$,
neither they nor others biedenharn ; ryabov:equiv ; ohlsen  who have developed
operator equivalents for $\hat{T}_{k0}^{(J)}$ tensors have cited the original
work by Meckler meckler:angular  and Corio corio:ortho  .

###  VII.2 A comparison of operator equivalents

In Table VIII, we compare specific examples of tensor operator equivalents
developed by Corio corio:ortho  (top row) with those developed by Marinelli et
al. werb:tensor  (bottom row). The examples have been chosen to demonstrate
that all the equivalents have the same form (to within $j$-dependent
normalization constants). The first column compares operator equivalents for
the irreducible tensor operators $\hat{T}_{60}^{(j)}$, which both Corio
corio:ortho  and Marinelli et al. werb:tensor  recognized as equivalent to the
Chebyshev polynomial operator $f^{(j)}_{6}(J_{z})$. The second column compares
operator equivalents for the irreducible tensor operators
$\hat{T}_{41}^{(j)}$. Although both equivalents have the same form, Marinelli
et al. werb:tensor  went one step further than Corio corio:ortho  did to show
that any irreducible (spherical) tensor operator $T^{(k)}_{L}\\!(S)$ of rank
$L$ and projection $k>0$ could be represented by suitable linear combinations
of successive partial derivatives (with respect to $S_{z}$) of the Chebyshev
polynomial operators $f^{(S)}_{L}(S_{z})$ as follows

$\displaystyle T^{(k)}_{L}\\!(S)$ $\displaystyle\sim$
$\displaystyle\left(S_{+}\right)^{|k|}\displaystyle\sum_{n=|k|}^{L}\;A_{Ln}^{k}\\!\left(\displaystyle\frac{\partial}{\partial
S_{z}}\right)^{\\!\\!n}\\!\left[f^{(S)}_{L}(S_{z})\right]$ (441)

A table of the required coefficients $A_{Ln}^{k}$ was provided by Marinelli et
al. werb:tensor . As an example of this linear combination of successive
partial derivatives, the table entry in Table VIII for Marinelli’s werb:tensor
operator equivalent of $T^{(1)}_{4}\\!(S)$ we have calculated as follows

$\displaystyle\displaystyle\sum_{n=1}^{4}\;A_{4n}^{1}\\!\left(\displaystyle\frac{\partial}{\partial
S_{z}}\right)^{\\!\\!n}\\!\left[f^{(S)}_{4}(S_{z})\right]$ (442)
$\displaystyle=$
$\displaystyle\frac{1}{1!}[14\,S_{z}^{3}-(6K-5)S_{z}]+\frac{1}{2!}[42\,S_{z}^{2}-(6K-5)]+\frac{1}{3!}[84\,S_{z}]+\frac{1}{4!}[84]\,\mathds{1}\;\;\;\;\;\;\;\;$
(443) $\displaystyle=$ $\displaystyle
14S_{\\!z}^{3}+21S_{z}^{2}+(19-6K)S_{z}+3(2-K)\,\mathds{1}$ (444)

$\mbox{where}\;\;\;\;f^{(S)}_{4}(S_{z})\sim
35S^{4}_{z}-5S^{2}_{z}(6K-5)+3K(K-2)$. Each summand in Eq.(443) is the product
of a partial-derivative of $f^{(S)}_{4}\\!(S_{z})$ (in square brackets), and
coefficients $A_{4n}^{1}$ tabulated in Marinelli et al. werb:tensor .

In conclusion, the examples discussed in Table VIII illustrate the fact that
Chebyshev polynomial operators $f^{(S)}_{L}(S_{z})$ can be used to develop an
operator equivalent for any irreducible tensor operator.

##  VIII Conclusion

The Chebyshev polynomials $f_{\lambda}^{(j)}(m)$ first introduced in a physics
application by Meckler meckler:majorana ; meckler:angular , are special
functions askey , a particular case of one member of the family of classical
orthogonal polynomials of a discrete variable known as the Hahn polynomials
bateman ; nikiforov2 ; Nikiforov ; vilenkin:specfuncbook ; olver . Beginning
with a close examination of the Meckler formula meckler:majorana ;
meckler:angular  for spin transition probabilities, we have described how
Chebyshev polynomials of a discrete variable can be applied in physics.
Applications of these very special special functions include spin physics,
spin tomography, and the development of operator expansions and operator
equivalents.

Beyond their role as special functions, the Chebyshev polynomials
$f_{\lambda}^{(j)}(m)$ double as the Clebsch-Gordan coupling coefficients
(3$j$-symbols brinksatch:ang ) $C_{jmj-m}^{\lambda\,0}$ of angular momentum
theory:

$f_{\lambda}^{(j)}(m)=(-1)^{j-m}\;C_{jmj-m}^{\lambda\,0}$ (445)

We have in this article often taken advantage of this duality of the Chebyshev
polynomials to prove identities from their Clebsch-Gordan coupling coefficient
homologs. The Chebyshev polynomial operators $f_{\lambda}^{(j)}(J_{z})$ are
identical to the projection-zero spin polarization operators $\hat{T}_{\lambda
0}^{(j)}$

$f_{\lambda}^{(j)}\\!(J_{z})\equiv\hat{T}_{\lambda 0}^{(j)}$ (446)

a relationship first recognized by Meckler meckler:angular . The similarity
transform of this equivalence defines the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ as the direct product of two
rank-$\lambda$ tensors, one the spin tensor ${\bf T}_{\lambda}({\bf J})$, and
the other, the spatial Racah tensor ${\bf C}_{\lambda}({\bf\hat{n}})$
brinksatch:ang :

$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})={\bf
C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ (447)

Whether we consider the Chebyshev polynomial scalars $f_{\lambda}^{(j)}(m)$,
or the Chebyshev polynomial operators $f_{\lambda}^{(j)}(J_{z})$ and
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$, the Chebyshev polynomials are
distinguished as special functions by their very close connection to angular
momentum theory and spin physics. As we have described in Section 4, a vivid
reminder of this connection is the Meckler formula meckler:majorana ;
meckler:angular  for the spin transition probability $\mbox{
P}^{(j)}_{mm^{\prime}}(t)$, whose calculation relies on angular momentum
theory meckler:angular ; schwinger:majorana . As we have described in Section
5, additional reminders of this connection are the Chebyshev polynomial
operator expansions of projection operators, the rotation operator, the
Stratonovich-Weyl operator, and the tomographic reconstruction of the density
operator.

##  IX Appendix A

###  IX.1 Legendre polynomial operators

In this section, two versions of Legendre polynomial operators are discussed,
one version due to Zemach zemach , and the other due to Schwinger
schwinger:majorana . The Legendre polynomial operators
$\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ defined by Zemach zemach
are closely related to the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$. On the other hand, the matrix
elements of the Legendre polynomial operators $P_{\lambda}({\bf J})$ defined
by Schwinger schwinger:majorana  are closely related to the Chebyshev
polynomials $f_{\lambda}^{(j)}(m)$.

####  IX.1.1 Legendre polynomial operators
$\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$

From the addition theorem for spherical harmonics, it is well-known that the
direct product of rank-$\lambda$ spherical harmonics tensors can be expressed
in scalar form as the following Legendre polynomial of order $\lambda$:

${\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf
C}_{\lambda}({\bf\hat{n}}^{\prime})=P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})$
(448)

Not so well-known is how the direct product of rank-$\lambda$ spatial and spin
tensors ${\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ could
be expressed. Certainly the direct product of rank-$\lambda$ spatial and spin
tensors ${\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$ could
not be a scalar function of $({\bf\hat{n}}\cdot{\bf J})$, but an operator
function of $({\bf\hat{n}}\cdot{\bf J})$. Could it be expressed as a Legendre
polynomial operator? In developing tensor representations for application to
angular-momentum problems in elementary-particle reactions, Zemach zemach  was
the first to develop such an expression. In the notation of this article, this
direct product takes the following form

$\displaystyle{\bf C}_{\lambda}({\bf\hat{n}})\cdot{\bf T}_{\lambda}({\bf J})$
$\displaystyle=$
$\displaystyle\sqrt{\displaystyle\frac{2\lambda+1}{(2j+1)\left[{\bf
J}^{2}\right]^{l}}}\;\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ (449)
$\displaystyle\mbox{where}\;\;\;\;\left[{\bf J}^{2}\right]^{l}$
$\displaystyle=$ $\displaystyle\prod_{n=0}^{l}\left[{\bf
J}^{2}-\tfrac{1}{4}(n^{2}-1)\right]$ (450)

Just as the addition theorem of Eq.(448) can be taken as the definition zemach
of the Legendre polynomials
$P_{\lambda}({\bf\hat{n}}\cdot{\bf\hat{n}}^{\prime})$, the direct product
expression of Eq.(449) can be used to define zemach  the Legendre polynomial
operators $\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$. Comparing this
expression with the corresponding direct product relations of Eq.(382) or
(401) also defines the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$ in terms of the Legendre
polynomial operators $\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$:

$\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf
J})=\sqrt{\displaystyle\frac{(2j+1)\left[{\bf
J}^{2}\right]^{l}}{2\lambda+1}}\,f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$
(451)

The recursion relation for the Legendre polynomials $P_{\lambda}(x)$ can be
compared with its counterpart for the Legendre polynomial operators
$\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ zemach :

$\displaystyle(2\lambda+1)\,x\,P_{\lambda}(x)$ $\displaystyle=$
$\displaystyle(\lambda+1)\,P_{\lambda+1}(x)+\lambda\,P_{\lambda-1}(x)$ (452)
$\displaystyle(2\lambda+1)({\bf\hat{n}}\cdot{\bf
J})\,\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle(\lambda+1)\,\overline{P}_{\lambda+1}({\bf\hat{n}}\cdot{\bf
J})+\lambda\left[{\bf
J}^{2}-\tfrac{1}{4}(\lambda^{2}-1)\right]\overline{P}_{\lambda-1}({\bf\hat{n}}\cdot{\bf
J})\;\;\;\;\;\;\;\;\;\;\;\;$ (453)

The recursion relation of Eq.(453) was used to generate the Legendre
polynomial operators $\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ zemach
tabulated in Table IX. Inspection of this table confirms Zemach’s observation
zemach  that although the functional forms of the Legendre polynomial
operators $\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ agree with those
of the Legendre polynomials $P_{\lambda}(\cos\theta)$ for $\lambda=0,1,2$, the
non-commutivity of components of ${\bf J}$ makes a difference for $\lambda\geq
3$. As we shall see in the next section, the same issue arises for the same
reasons when comparing the Legendre polynomial operators $P_{\lambda}({\bf
J})$ introduced by Schwinger schwinger:majorana  with the Legendre polynomials
$P_{\lambda}(\cos\theta)$.

####  IX.1.2 Legendre polynomial operators $P_{\lambda}({\bf J})$

In Section 4.1.2, we discussed how Schwinger schwinger:majorana  calculated
the spin transition probability $\mbox{ P}^{(j)}_{mm^{\prime}}(t)$. Using
Schwinger’s notation schwinger:majorana , the final outcome of his calculation
of the spin transition probability took the following form

$\mbox{ P}^{(j)}_{mm^{\prime}}(t)=\left|{\cal
D}_{mm^{\prime}}^{(j)}(\psi,{\bf\hat{n}})\right|^{2}=\frac{1}{(2j+1)}\sum_{l=0}^{2j}(2l+1)\,P_{l}(j,m)\;P_{l}(j,m^{\prime})\;P_{l}(\cos\beta)$
(454)

in which the $P_{l}(j,m)$ functions were defined by Schwinger
schwinger:majorana  as the matrix elements of the Legendre polynomial
operators $P_{l}({\bf J})$:

$P_{l}(j,m)=\langle jm|\,P_{l}({\bf J})\,|jm\rangle$ (455)

Schwinger schwinger:majorana  introduced the Legendre polynomial operators
$P_{l}({\bf J})$ by expressing them in terms of the solid harmonic functions
of the angular momentum vector ${\bf J}$:

$\displaystyle P_{l}({\bf J})$ $\displaystyle=$
$\displaystyle\left[\frac{2l+1}{4\pi}\left[{\bf
J}^{2}\right]^{\\!l}\right]^{-1/2}\\!\\!\mathcal{Y}_{l0}({\bf J})$ (456)
$\displaystyle\mbox{where \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ournote}{\@@citephrase{(}}{\@@citephrase{)}}}}\;\;\;\;\;\left[{\bf
J}^{2}\right]^{\\!l}$ $\displaystyle=$
$\displaystyle\prod_{n=0}^{l-1}\left[{\bf
J}^{2}-\mathchoice{\frac{\raisebox{-1.72218pt}{$n$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle
n$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{n}{2}}{\frac{n}{2}}\left(\mathchoice{\frac{\raisebox{-1.72218pt}{$n$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle
n$}}{\raisebox{0.43057pt}{$\scriptstyle 2$}}}{\frac{n}{2}}{\frac{n}{2}}\
+1\right)\right]$ (457)

The $\mathcal{Y}_{lm}({\bf J})$ are operator analogues ligarg  of the solid
harmonics $r^{l}\,Y_{lm}({\bf\hat{n}})$ varshal1:ang , which include an extra
factor $r^{l}$ over the surface harmonics $Y_{lm}({\bf\hat{n}})$. They are in
fact irreducible tensor operators, proportional to the spin polarization
operators $\hat{T}^{(j)}_{lm}$ according to the following relation

$\mathcal{Y}_{lm}({\bf J})=\left[\frac{2j+1}{4\pi}\left[{\bf
J}^{2}\right]^{\\!l}\right]^{\\!1/2}\hat{T}^{(j)}_{lm}$ (458)

It is evident from this proportionality that $Y_{lm}({\bf\hat{n}})$ and
$\mathcal{Y}_{lm}({\bf J})$ transform identically under rotations. They
satisfy the following trace relation ligarg

$\displaystyle\mbox{Tr}\\!\left[\mathcal{Y}_{lm}({\bf
J})\;\mathcal{Y}^{\dagger}_{l^{\prime}m^{\prime}}({\bf J})\right]$
$\displaystyle=$
$\displaystyle\frac{2j+1}{4\pi}(a_{jl})^{2}\;\delta_{ll^{\prime}}\;\delta_{mm^{\prime}}$
(459) $\displaystyle\mbox{where}\;\;\;\;(a_{jl})^{2}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{l}\left[\left(j+\tfrac{1}{2}\right)^{\\!2}-\tfrac{1}{4}n^{2}\right]\equiv\left[{\bf
J}^{2}\right]^{\\!l}$ (460)

The solid harmonic operator functions $\mathcal{Y}_{l0}({\bf J})$ in Eq.(456)
were defined schwinger:majorana  by the generating function schwinger:majorana
; schwingerosti ; schwingerbiedenvandam ; schwingerbook

$\displaystyle\frac{1}{2^{l}l!}\left[\frac{2l+1}{4\pi}\right]^{\\!1/2}\left[-z_{+}^{2}(J_{x}+iJ_{y})+z_{-}^{2}(J_{x}-iJ_{y})+2z_{+}z_{-}J_{z}\right]^{l}$
(461) $\displaystyle=$
$\displaystyle\sum_{m=-l}^{l}\;\frac{z_{+}^{l+m}\;z_{-}^{l-m}}{\left[(l+m)!(l-m)!\right]^{1/2}}\;\mathcal{Y}_{lm}({\bf
J})$

After evaluating $\mathcal{Y}_{l0}({\bf J})$ with this generating function,
Eq.(456) was used to tabulate examples of the Legendre polynomial operators
$P_{l}({\bf J})$ in Table IX. Recognizing that the $\mathcal{Y}_{lm}({\bf J})$
are operator analogues of the solid harmonics $r^{l}\,Y_{lm}({\bf\hat{n}})$
ligarg , we should expect to see a difference in the functional forms of
$P_{l}({\bf J})\sim\mathcal{Y}_{l0}({\bf J})$ and $P_{l}(\cos\theta)\sim
Y_{l0}({\bf\hat{n}})$ because the order of factors in the operator case is
significant. Indeed, as this table demonstrates, for $\lambda\geq 3$, we can
expect such a difference taking into account the angular momentum commutation
relations

$\left[J_{i},J_{j}\right]=J_{k}\;\;\;\;(i,j,k=x,y,z)$ (462)

Using the relation of Eq.(458), and Eq.(456), the matrix element of Eq.(455)
which defines the $P_{l}(j,m)$ functions which appear in the transition
probability formula of Eq.(454) can be related to the Chebyshev polynomials of
a discrete variable $f^{(j)}_{l}(m)$ as follows

$\displaystyle P_{l}(j,m)=\langle jm|\,P_{l}({\bf J})\,|jm\rangle$
$\displaystyle=$ $\displaystyle\sqrt{\frac{2j+1}{2l+1}}\langle
jm|\,\hat{T}^{(j)}_{l0}\,|jm\rangle$ (463) $\displaystyle=$
$\displaystyle\sqrt{\frac{2j+1}{2l+1}}\langle
jm|\,f^{(j)}_{l}(J_{z})\,|jm\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2j+1}{2l+1}}\;f^{(j)}_{l}(m)$

The Legendre polynomial operators schwinger:majorana  $P_{l}({\bf J})$ and the
Legendre polynomial operators zemach
$\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ described in the previous
section are related to the Chebyshev polynomial operators
$f_{\lambda}^{(j)}(J_{z})$ according to

$P_{\lambda}({\bf J})=\left[\left[{\bf
J}^{2}\right]^{\\!l}\right]^{\\!-1/2}\,\overline{P}_{\lambda}(J_{z})=\sqrt{\displaystyle\frac{2j+1}{2\lambda+1}}\,f_{\lambda}^{(j)}(J_{z})$
(464)

This relation shows that the Chebyshev polynomials $f_{\lambda}^{(j)}(m)$ can
be calculated from the diagonal matrix elements of the Legendre polynomial
operators $\overline{P}_{\lambda}(J_{z})$ or $P_{\lambda}({\bf J})$.

Although Schwinger schwinger:majorana  did not cite a reference for the
generating function of Eq.(461), Schwinger himself had derived this generating
function. His derivation was included in a 1952 technical report schwingerosti
, later published in a book collection of reprints and original papers
schwingerbiedenvandam , and as a book schwingerbook  with the same title as
the technical report. Additional discussions of this generating function can
be found in books by Schwinger et al. schwingerbook2  and by Garg garg .

As we shall demonstrate, Schwinger’s generating function schwinger:majorana ;
schwingerosti ; schwingerbiedenvandam ; schwingerbook  is equivalent to the
Herglotz generating function couranth ; herglot ; ligarg  for the solid
harmonic operators $\mathcal{Y}_{lm}({\bf J})$

$\displaystyle e^{\zeta{\bf\hat{a}}\cdot{\bf J}}$ $\displaystyle=$
$\displaystyle\sum_{lm}\sqrt{\frac{4\pi}{2l+1}}\;\frac{\zeta^{l}\lambda^{m}}{\sqrt{(l+m)!(l-m)!}}\;\mathcal{Y}_{lm}({\bf
J})$ (465) $\displaystyle\mbox{where }\;\;\;\;\;{\bf\hat{a}}$ $\displaystyle=$
$\displaystyle{\bf\hat{z}}-\frac{\lambda}{2}({\bf\hat{x}}+i{\bf\hat{y}})+\frac{1}{2\lambda}({\bf\hat{x}}-i{\bf\hat{y}})$
(466)

Introducing the definitions

$\displaystyle{\hat{A}}$ $\displaystyle=$ $\displaystyle
J_{z}-\frac{\lambda}{2}(J_{x}+iJ_{y})+\frac{1}{2\lambda}(J_{x}-iJ_{y})\equiv({\bf\hat{a}}\cdot{\bf
J})$ $\displaystyle\lambda$ $\displaystyle=$ $\displaystyle z_{+}/z_{-}$ (467)

and after a slight rearrangement, Schwinger’s generating function
schwinger:majorana ; schwingerosti ; schwingerbook  of Eq.(461) can be
rewritten as

$\frac{{\hat{A}}^{l}}{l!}=\sum_{m=-l}^{l}\;\sqrt{\frac{4\pi}{2l+1}}\;\frac{\lambda^{m}}{\left[(l+m)!(l-m)!\right]^{1/2}}\;\mathcal{Y}_{lm}({\bf
J})$ (468)

After multiplying both sides of Eq.(468) by $\zeta^{l}$, and summing both
sides over $l$, we obtain the Herglotz generating function couranth ; herglot
; ligarg  of Eq.(466):

$\sum_{l=1}^{\infty}\frac{[\zeta{\hat{A}}]^{l}}{l!}=\sum_{l=1}^{\infty}\frac{(\zeta{\bf\hat{a}}\cdot{\bf
J})^{l}}{l!}\equiv e^{\zeta{\bf\hat{a}}\cdot{\bf
J}}=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\;\sqrt{\frac{4\pi}{2l+1}}\;\frac{\zeta^{l}\lambda^{m}}{\left[(l+m)!(l-m)!\right]^{1/2}}\;\mathcal{Y}_{lm}({\bf
J})$ (469)

Appendix A of the first English edition of Courant and Hilbert couranth ;
herglot  published in 1953 cited Herglotz for his formula, but none of the
pre-war German editions had this appendix. This meant that the generating
function schwinger:majorana ; schwingerosti ; schwingerbook  documented by
Schwinger in a technical report schwingerosti  a year earlier in 1952
anticipated the Herglotz generating function couranth ; herglot ; ligarg , and
so there is some justification for renaming this generating function the
Schwinger-Herglotz generating function.

###  IX.2 Unit tensor (Wigner) operators

In their comments on Schwinger’s interpretation schwinger:majorana  of the
Majorana formula emajorana  , Biedenharn and Louck biedenharn  observe that
$P_{l}(j,m)$ denotes the matrix element of the unit tensor operator (alias
Wigner operators biedenharn )

$P_{l}(j,m)\equiv\langle jm|\left\langle\begin{array}[]{ccc}&l&\\\ 2l&&0\\\
&l&\end{array}\right\rangle|jm\rangle$ (470)

The Wigner operators can be defined by their action on the angular momentum
basis $|jm\rangle$ biedenharn , and in particular, the shift action is defined
by biedenharn

$\left\langle\begin{array}[]{ccc}&J+\Delta&\\\ 2J&&0\\\
&J+M&\end{array}\right\rangle|jm\rangle=C_{jmJM}^{j+\Delta\;m+M}\;|j+\Delta,m+M\rangle$
(471)

and so in particular

$\displaystyle\left\langle\begin{array}[]{ccc}&l&\\\ 2l&&0\\\
&l&\end{array}\right\rangle|jm\rangle$ $\displaystyle=$ $\displaystyle
C_{jml0}^{jm}\;|jm\rangle$ (475) $\displaystyle=$
$\displaystyle\sqrt{\frac{2j+1}{2l+1}}\;\boxed{(-1)^{j-m}\;C_{jmj-m}^{l0}}\;|jm\rangle$
(476) $\displaystyle=$
$\displaystyle\sqrt{\frac{2j+1}{2l+1}}\;f_{l}^{(j)}(m)\;|jm\rangle$ (477)

The “boxed” term in Eq.(476) defines the Chebyshev polynomial $f_{l}^{(j)}(m)$
in Eq.(477). Then, using the result of Eq.(477), the unit tensor matrix
element of Eq.(470) is evaluated as

$P_{l}(j,m)\equiv\langle jm|\left\langle\begin{array}[]{ccc}&l&\\\ 2l&&0\\\
&l&\end{array}\right\rangle|jm\rangle=\sqrt{\frac{2j+1}{2l+1}}\;f_{l}^{(j)}(m)$
(478)

in agreement with the matrix element $P_{l}(j,m)=\langle jm|\,P_{l}({\bf
J})\,|jm\rangle$ of Schwinger’s schwinger:majorana  Legendre polynomial
operator $P_{l}({\bf J})$ in Eq.(463).

##  X Appendix B

As noted by Varshalovich et al. varshal1:ang , an alternative explicit form of
${\cal D}^{J}_{MM^{\prime}}(\psi,{\bf\hat{n}})\equiv{\cal
D}^{J}_{MM^{\prime}}(\psi;\Theta,\Phi)$ can be obtained directly from ${\cal
D}^{J}_{MM^{\prime}}(\alpha,\beta,\gamma)$ by changing variables
$(\psi;\Theta,\Phi)\rightarrow(\alpha,\beta,\gamma)$ with the aid of the
following relations (and the corresponding inverse relations):

$\displaystyle\sin\mathchoice{\frac{\raisebox{-1.72218pt}{$\beta$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\beta$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\beta}{2}}{\frac{\beta}{2}}$ $\displaystyle=$
$\displaystyle\sin\Theta\;\sin\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}$
$\displaystyle\tan\frac{\alpha+\gamma}{2}$ $\displaystyle=$
$\displaystyle\cos\Theta\;\tan\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}$ $\displaystyle\frac{\alpha-\gamma}{2}$
$\displaystyle=$
$\displaystyle\Phi-\mathchoice{\frac{\raisebox{-1.72218pt}{$\pi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\pi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\pi}{2}}{\frac{\pi}{2}}$ (479)

The result of this variable change is the following varshal1:ang

$\displaystyle{\cal D}^{J}_{MM^{\prime}}(\psi;\Theta,\Phi)$ $\displaystyle=$
$\displaystyle
i^{M-M^{\prime}}\;e^{-i(M-M^{\prime})\Phi}\left(\displaystyle\frac{1-i\tan\frac{\psi}{2}\cos\Theta}{\sqrt{1+\tan^{2}\frac{\psi}{2}\cos^{2}\Theta}}\right)^{\\!\\!M+M^{\prime}}d^{J}_{MM^{\prime}}(\xi)\;\;\;\;$
(480) $\displaystyle\mbox{where}\;\;\;\;\sin\frac{\xi}{2}$ $\displaystyle=$
$\displaystyle\sin\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\;\sin\Theta$ (481)

The expression for ${\cal D}^{J}_{MM^{\prime}}(\psi;\Theta,\Phi)$ given in
Eq.(480) will now be used to evaluate ${\cal
D}^{J}_{MM^{\prime}}(\psi,{\bf\hat{n}})$ in two cases described in the next
sections.

###  X.1 Case I: ${\cal D}_{00}^{L}(\psi,{\bf\hat{n}})=d_{00}^{\,L}(\xi)\equiv
P_{L}(\cos\xi)=P_{L}(\cos\beta)$

Since $M=M^{\prime}=0$, from Eq.(480) we find

$\displaystyle{\cal D}_{00}^{L}(\psi,{\bf\hat{n}})$ $\displaystyle=$
$\displaystyle d_{00}^{\,L}(\xi)\equiv P_{L}(\cos\xi)$ (482)
$\displaystyle\mbox{where \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{brinksatch:ang}{\@@citephrase{(}}{\@@citephrase{)}}}}\;\;\;\;d_{00}^{\,L}(\xi)$
$\displaystyle=$ $\displaystyle P_{L}(\cos\xi)$ (483)

Using Eq.(481), the argument of the Legendre polynomial $P_{L}(\cos\xi)$ in
Eq.(482) can therefore be rewritten as

$\cos\xi\equiv
1-2\sin^{2}\displaystyle\frac{\xi}{2}=\boxed{1-2\sin^{2}\displaystyle\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\sin^{2}\theta=\cos\beta}$ (484)

so that

${\cal D}_{00}^{L}(\psi,{\bf\hat{n}})=P_{L}(\cos\beta)$ (485)

The “boxed” equivalence of Eq.(484) follows from this relation abragamtext

$\cos\beta=1-2\sin^{2}\displaystyle\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\sin^{2}\theta$ (486)

where $\psi=|\gamma{\bf H}_{e}|t$, and $\theta$ is the angle between the
direction of the effective field He and the static applied field H0.

###  X.2 Case II: ${\cal D}^{\lambda}_{\mu
0}(\theta,{\bf\hat{n}}_{\bot})\equiv{\cal D}^{\lambda}_{\mu
0}(\theta;\frac{\pi}{2},\phi+\frac{\pi}{2})\equiv
C_{\lambda\mu}^{\star}(\theta,\phi)$

Since $\Theta=\pi/2$, $\sin\Theta=1$, $\cos\Theta=0$, and $\xi=\theta$, and so
using the expression of Eq.(480), we find

$\displaystyle{\cal D}^{\lambda}_{\mu 0}(\theta,{\bf\hat{n}}_{\bot})$
$\displaystyle\equiv$ $\displaystyle{\cal D}^{\lambda}_{\mu
0}(\theta;\mathchoice{\frac{\raisebox{-1.72218pt}{$\pi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\pi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\pi}{2}}{\frac{\pi}{2}},\phi+\mathchoice{\frac{\raisebox{-1.72218pt}{$\pi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\pi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\pi}{2}}{\frac{\pi}{2}})$ (487) $\displaystyle=$ $\displaystyle
i^{\mu}\,e^{-i\mu\phi}\,e^{-i\mu\frac{\pi}{2}}\;\boxed{d^{\lambda}_{\mu
0}(\theta)}$ (488) $\displaystyle=$ $\displaystyle
e^{-i\mu\phi}\,\boxed{(-1)^{\mu}\sqrt{\displaystyle\frac{(\lambda-\mu)!}{(\lambda-\mu)!}}\;P_{\lambda}^{\mu}(\theta)}$
(489) $\displaystyle=$ $\displaystyle C_{\lambda\mu}^{\star}(\theta,\phi)$
(490)

The reduced matrix element in the “boxed” term of Eq.(488) has been replaced
by its equivalent brinksatch:ang  in the “boxed” term of Eq.(489).

##  XI Appendix C

###  XI.1 A closed-form expression for the Fourier-Legendre series expansion
of the spin transition probability $\mbox{P}^{(j)}_{j,-j}(t)$

Consider the following Fourier-Legendre series expansion (identity 5.10.1(17)
in Prudnikov et al. prudnikov )

$(a-1)\left[\frac{1-x}{2}\right]^{\\!(a-2)}=\;\sum_{k=0}^{\infty}(2k+1)\frac{(2-a)_{k}}{(a)_{k}}\,P_{k}(x)\;\;\;\;\;\;\;\;(|x|<1;\;\;a>5/4)$
(491)

where the Pochhammer symbol $(a)_{n}$ is defined in terms of the Gamma
function $\Gamma(z)$ as follows prudnikov

$(a)_{n}=\frac{\Gamma(a+n)}{\Gamma(a)}$ (492)

If we let $a=n+2$ in the identity of Eq.(491), where $n\geq 0$ is an integer,
then we obtain

$\left[\frac{1-x}{2}\right]^{\\!n}=\frac{1}{n+1}\sum_{k=0}^{\infty}(2k+1)\frac{(-n)_{k}}{(n+2)_{k}}\,P_{k}(x)$
(493)

Then using the following Pochhammer symbol expressions prudnikov

$\displaystyle(-n)_{k}$ $\displaystyle=$
$\displaystyle(-1)^{k}\frac{n!}{(n-k)!}$ (494) $\displaystyle(n+2)_{k}$
$\displaystyle=$
$\displaystyle\frac{\Gamma(n+2+k)}{\Gamma(n+2)}=\frac{(n+1+k)!}{(n+1)!}$ (495)

we see that the Fourier-Legendre series expansion of Eq.(493) terminates when
$k=n$, and that this expansion now simplifies to read

$\left[\frac{1-x}{2}\right]^{\\!n}=\left[n!\right]^{2}\sum_{k=0}^{n}\frac{(-1)^{k}(2k+1)}{(n-k)!(n+1+k)!}\,P_{k}(x)$
(496)

Mathematica® wolfram  can be also used to verify this expansion. For example,
by using the following input command to implement the summation of Eq.(496)
with $n=101$

⬇

In[1] = Simplify[2^(101) Sum[(-1)^L (2 L + 1)(101!)^2

LegendreP[L, x]/((101 - L)!(101 + L + 1)!),{L, 0, 101}]]

the expected output is obtained as

⬇

Out[2] = -(-1 + x)^101

An amusing consequence of Eq.(496) is the following Fourier-Legendre series
expansion of $[a+b]^{n}$:

$[a+b]^{n}=[2a]^{n}\;\\!\\!\left[n!\right]^{2}\sum_{k=0}^{n}\frac{(2k+1)}{(n-k)!(n+1+k)!}\,P_{k}(b/a)$
(497)

##  XII Appendix D

###  XII.1 Using $f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ to evaluate the
generalized characters $\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)$

Using the “boxed” trace relation of Eq.(281), the generalized character
functions can be written as

$\mbox{{\large$\chi$}}_{\lambda}^{(j)}(\psi)=i^{3\lambda}\sqrt{\displaystyle\frac{2j+1}{2\lambda+1}}\;\mbox{Tr}\\!\left[f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf
J})\;\hat{{\cal D}}^{(j)}\\!(\psi,{\bf\hat{n}})\right]$ (498)

Then, putting $\lambda=1$ and $J=1/2$, the generalized character function
$\chi$${}_{1}^{1/2}(\psi)$ can be evaluated as

$\displaystyle\mbox{{\large$\chi$}}_{1}^{(1/2)}(\psi)$ $\displaystyle=$
$\displaystyle-i\sqrt{\frac{2}{3}}\;\mbox{Tr}\\!\left[f^{(1/2)}_{1}({\bf\hat{n}}\cdot{\bf
J})\;\hat{{\cal D}}^{(1/2)}(\psi,{\bf\hat{n}})\right]$ (499) $\displaystyle=$
$\displaystyle-i\sqrt{\frac{2}{3}}\;\mbox{Tr}\\!\left[\sqrt{2}\;J_{z}\left(\mathds{1}\cos\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}+2i\,J_{z}\,\sin\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\right)\right]$ (500) $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{3}}\,\sin\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\;\mbox{Tr}\\!\left[J_{z}^{2}\right]$
(501) $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{3}}\,\sin\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}$ (502)

a result which agrees with $\mbox{{\large$\chi$}}_{1}^{(1/2)}(\psi)$ tabulated
in Varshalovich et al. varshal1:ang . Because the trace is invariant with
respect to a change in basis, the trace of Eq.(500) was evaluated in a
representation in which $({\bf\hat{n}}\cdot{\bf J})$ is diagonal. In addition,
the following identities have been used to arrive at the final result in
Eq.(502):

$\displaystyle f^{(1/2)}_{1}({\bf\hat{n}}\cdot{\bf J})$ $\displaystyle=$
$\displaystyle\sqrt{2}\;J_{z}\;\;\;\;\;\;\;\;\;\;\;\;\;\ \;\;\;\;\;\;\;\;\;\;\
\;\;\;\;\;\;\;\mbox{(see Table II)}$ (503) $\displaystyle\hat{{\cal
D}}^{(1/2)}(\psi,{\bf\hat{n}})$ $\displaystyle=$
$\displaystyle\mathds{1}\cos\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}+2i\,J_{z}\,\sin\mathchoice{\frac{\raisebox{-1.72218pt}{$\psi$}}{\raisebox{1.72218pt}{$2$}}}{\frac{\raisebox{-1.72218pt}{$\scriptstyle\psi$}}{\raisebox{0.43057pt}{$\scriptstyle
2$}}}{\frac{\psi}{2}}{\frac{\psi}{2}}\;\;\;\;\;\;\;\;\;\;\mbox{(Reference
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{varshal1:ang}{\@@citephrase{(}}{\@@citephrase{)}}})}$
(504) $\displaystyle\mbox{Tr}\\!\left[J_{z}^{2}\right]$ $\displaystyle=$
$\displaystyle\sum_{m=-1/2}^{1/2}\\!m^{2}=\frac{1}{2}$ (505)
$\displaystyle\mbox{Tr}\\!\left[J_{z}\right]$ $\displaystyle=$ $\displaystyle
0$ (506)

##  XIII Appendix E

The composite irreducible product tensor ${\bf X}^{K}_{Q}$ of rank $K$ is
defined by the following combination of irreducible tensors ${\bf T}^{k}_{q}$
and ${\bf U}^{k^{\prime}}_{q^{\prime}}$ of rank $k$ and $k^{\prime}$,
respectively:

${\bf X}^{K}_{Q}=\left\\{{\bf T}^{k}_{q}\circledast{\bf
U}^{k^{\prime}}_{q^{\prime}}\right\\}^{K}_{Q}=\sum_{\begin{subarray}{c}q,q^{\prime}\\\
q+q^{\prime}=Q\end{subarray}}{\bf T}^{k}_{q}\;{\bf
U}^{k^{\prime}}_{q^{\prime}}\;C^{KQ}_{kqk^{\prime}q^{\prime}}$ (507)

where $C^{KQ}_{kqk^{\prime}q^{\prime}}$ is a Clebsch-Gordan coefficient which
vanishes unless $q+q^{\prime}=Q$. In general,

$\left\\{{\bf R}^{(k)}\circledast{\bf
S}^{(k)}\right\\}^{0}_{0}=\sum_{\begin{subarray}{c}q,q^{\prime}\\\
q+q^{\prime}=0\end{subarray}}{\bf R}^{(k)}_{q}\;{\bf
S}^{(k)}_{q^{\prime}}\;C^{00}_{kqkq^{\prime}}$ (508)

But since the Clebsch-Gordan coefficient is evaluated as brinksatch:ang

$C^{00}_{kqkq^{\prime}}=\frac{(-1)^{k-q}}{\sqrt{2k+1}}\;\delta_{q,-q^{\prime}}$
(509)

the double sum of Eq.(508) reduces to a single sum

$\displaystyle\left\\{{\bf R}^{(k)}\circledast{\bf S}^{(k)}\right\\}^{0}_{0}$
$\displaystyle=$
$\displaystyle\frac{(-1)^{k}}{\sqrt{2k+1}}\sum_{q}(-1)^{-q}\;{\bf
R}^{(k)}_{q}\;{\bf S}^{(k)}_{-q}$ (510) $\displaystyle=$
$\displaystyle\frac{(-1)^{k}}{\sqrt{2k+1}}\;{\bf R}^{(k)}\cdot{\bf S}^{(k)}$

where ${\bf R}^{(k)}\cdot{\bf S}^{(k)}=\displaystyle\sum_{q}(-1)^{-q}\;{\bf
R}^{(k)}_{q}\;{\bf S}^{(k)}_{-q}$ defines the scalar product of two tensors.

A more general result for the recoupling of four arbitrary commuting tensors
can be defined in terms of a 9$j$-symbol brinksatch:ang  as follows

$\displaystyle\left\\{\left\\{{\bf S}^{k_{1}}_{q_{1}}\circledast{\bf
T}^{k_{2}}_{q_{2}}\right\\}^{k_{12}}_{q_{12}}\circledast\left\\{{\bf
U}^{k_{3}}_{q_{3}}\circledast{\bf
V}^{k_{4}}_{q_{4}}\right\\}^{k_{34}}_{q_{34}}\right\\}^{K}_{Q}$ (514)
$\displaystyle=$
$\displaystyle\sum_{k_{13},k_{24}}\Pi_{k_{13}k_{24}k_{12}k_{34}}\left\\{\begin{array}[]{ccc}k_{1}&k_{2}&k_{12}\\\
k_{3}&k_{4}&k_{34}\\\ k_{13}&k_{24}&K\end{array}\right\\}\left\\{\left\\{{\bf
S}^{k_{1}}_{q_{1}}\circledast{\bf
U}^{k_{3}}_{q_{3}}\right\\}^{k_{13}}_{q_{13}}\circledast\left\\{{\bf
T}^{k_{2}}_{q_{2}}\circledast{\bf
V}^{k_{4}}_{q_{4}}\right\\}^{k_{24}}_{q_{24}}\right\\}^{K}_{Q}\;\;\;\;\;\;\;\;\;\;\;\;$

where as a matter of notation, in this equation, and all that follow, we find
it convenient to define varshal1:ang :

$\Pi_{abc\ldots d}=\sqrt{(2a+1)(2b+1)(2c+1)\ldots(2d+1)}$ (515)

In Eq.(514), set $K=Q=0$, from which it follows that $k_{12}=k_{34}$ and
$k_{13}=k_{24}$. In addition, set $k_{12}=k_{34}=0$, and consider the
particular case where we recouple four irreducible tensors of rank 1, so that
$k_{1}=k_{2}=k_{3}=k_{4}=1$. In particular, we suppose

$\displaystyle{\bf S}^{k_{1}}$ $\displaystyle\equiv$ $\displaystyle{\bf
I}_{1}$ $\displaystyle{\bf U}^{k_{3}}$ $\displaystyle\equiv$
$\displaystyle{\bf I}_{2}$ $\displaystyle{\bf T}^{k_{2}}$ $\displaystyle=$
$\displaystyle{\bf V}^{k_{4}}\equiv{\bf r}$ (516)

Then, from Eq.(514), we have

$\Big{\\{}\left\\{{\bf I}_{1}\circledast{\bf
r}\right\\}^{0}\circledast\left\\{{\bf I}_{2}\circledast{\bf
r}\right\\}^{0}\Big{\\}}^{0}_{0}=\sum_{k_{13},k_{24}}\Pi_{k_{13}k_{24}}\left\\{\begin{array}[]{ccc}1&1&0\\\
1&1&0\\\ k_{13}&k_{24}&0\end{array}\right\\}\Big{\\{}\left\\{{\bf
I}_{1}\circledast{\bf I}_{2}\right\\}^{k_{13}}\circledast\left\\{{\bf
r}\circledast{\bf r}\right\\}^{k_{24}}\Big{\\}}^{0}_{0}$ (517)

Now since $k_{13}$ and $k_{24}$ satisfy the triangle inequalities

$\displaystyle|k_{1}-k_{3}|$ $\displaystyle\leq$ $\displaystyle
k_{13}\leq|k_{1}+k_{3}|$ $\displaystyle|k_{2}-k_{4}|$ $\displaystyle\leq$
$\displaystyle k_{24}\leq|k_{2}+k_{4}|$ (518)

and $k_{1}=k_{2}=k_{3}=k_{4}=1$, $k_{13}$ and $k_{24}$ are summed from 0 to 2.
Note however that when $k_{24}=1$, $\left\\{{\bf r}\circledast{\bf
r}\right\\}^{1}$ vanishes since $\left\\{{\bf r}\circledast{\bf
r}\right\\}^{1}\propto({\bf r}\times{\bf r})=0$. Therefore, in the sum of
Eq.(517), only two terms contribute:

$\displaystyle\Big{\\{}\left\\{{\bf I}_{1}\circledast{\bf
r}\right\\}^{0}\circledast\left\\{{\bf I}_{2}\circledast{\bf
r}\right\\}^{0}\Big{\\}}^{0}_{0}$ (525) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ccc}1&1&0\\\ 1&1&0\\\
0&0&0\end{array}\right\\}\Big{\\{}\left\\{{\bf I}_{1}\circledast{\bf
I}_{2}\right\\}^{0}\circledast\left\\{{\bf r}\circledast{\bf
r}\right\\}^{0}\Big{\\}}^{0}_{0}+5\left\\{\begin{array}[]{ccc}1&1&0\\\
1&1&0\\\ 2&2&0\end{array}\right\\}\Big{\\{}\left\\{{\bf I}_{1}\circledast{\bf
I}_{2}\right\\}^{2}\circledast\left\\{{\bf r}\circledast{\bf
r}\right\\}^{2}\Big{\\}}^{0}_{0}$

The 9$j$-symbols in Eq.(LABEL:sum9j) can be evaluated by expressing them in
terms of 6$j$-symbols as follows varshal1:ang

$\left\\{\begin{array}[]{ccc}a&b&e\\\ c&d&e\\\
f&f&0\end{array}\right\\}=\frac{(-1)^{b+c+e+f}}{\sqrt{(2e+1)(2f+1)}}\left\\{\begin{array}[]{ccc}a&b&e\\\
d&c&f\end{array}\right\\}$ (527)

and in turn, all the 6$j$-symbols required can then be evaluated using tables
roten:3j6j . Replacing all the recoupling coefficients in Eq.(LABEL:sum9j)
then yields the following relation

$\Big{\\{}\left\\{{\bf I}_{1}\circledast{\bf
r}\right\\}^{0}\circledast\left\\{{\bf I}_{2}\circledast{\bf
r}\right\\}^{0}\Big{\\}}^{0}_{0}=\frac{1}{3}\Big{\\{}\left\\{{\bf
I}_{1}\circledast{\bf I}_{2}\right\\}^{0}\circledast\left\\{{\bf
r}\circledast{\bf
r}\right\\}^{0}\Big{\\}}^{0}_{0}+\frac{\sqrt{5}}{3}\Big{\\{}\left\\{{\bf
I}_{1}\circledast{\bf I}_{2}\right\\}^{2}\circledast\left\\{{\bf
r}\circledast{\bf r}\right\\}^{2}\Big{\\}}^{0}_{0}$ (528)

Then, after expressing the rank zero recoupled tensors in Eq.(LABEL:sum9j) as
scalar products using the general relation

$\left\\{{\bf a}\circledast{\bf b}\right\\}^{0}_{0}=-\frac{{\bf a}\cdot{\bf
b}}{\sqrt{3}}$ (529)

and using the following relations brinksatch:ang

$\displaystyle\left\\{{\bf r}\circledast{\bf r}\right\\}^{2}_{q}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{2}{3}}\;{\bf r}^{2}\;{\bf
C}^{2}_{q}$ $\displaystyle\mbox{where}\;\;{\bf C}^{2}_{q}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{4\pi}{5}}\;{\bf Y}^{2}_{q}$ (530)

the following recoupling expression for the classical dipolar Hamiltonian is
obtained

$\displaystyle W_{12}$ $\displaystyle=$ $\displaystyle a\left[{\bf
I}_{1}\cdot{\bf I}_{2}-3\frac{({\bf I}_{1}\cdot{\bf r})({\bf I}_{2}\cdot{\bf
r})}{r^{2}}\right]$ (531) $\displaystyle=$
$\displaystyle-\sqrt{6}a\left\\{{\bf I}_{1}\circledast{\bf
I}_{2}\right\\}^{2}\cdot{\bf C}^{2}_{q}$ $\displaystyle=$
$\displaystyle-\sqrt{6}a\left\\{{\bf I}_{1}\circledast{\bf
I}_{2}\right\\}^{2}\cdot\sqrt{\frac{4\pi}{5}}{\bf Y}^{2}_{q}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{24\pi}{5}}\sum_{q}(-1)^{q}\left\\{{\bf
I}_{1}\circledast{\bf I}_{2}\right\\}^{2}_{q}{\bf Y}^{2}_{-q}$

Table I

Title: Chebyshev polynomial basis operators
$f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})\;\;(\lambda\leq
2j;\;\;\kappa\equiv j(j+1))$
Caption: As described in Section 2.2, all elements were generated following
the procedure described by Corio corio:siam .

$\lambda$ | $j=1/2$ | $j=1$ | $j=3/2$
---|---|---|---
0 | $f^{(\frac{1}{2})}_{0}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{\sqrt{2}}\,\mathds{1}$ | $f^{(1)}_{0}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{\sqrt{3}}\,\mathds{1}$ |  $f^{(\frac{3}{2})}_{0}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{\sqrt{4}}\,\mathds{1}$
1 |  $f^{(\frac{1}{2})}_{1}({\bf\hat{n}}\cdot{\bf J})=\sqrt{2}\,\left[({\bf\hat{n}}\cdot{\bf J})\right]$ | $f^{(1)}_{1}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{\sqrt{2}}\,\left[({\bf\hat{n}}\cdot{\bf J})\right]$ | $f^{(\frac{3}{2})}_{1}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{\sqrt{5}}\,\left[({\bf\hat{n}}\cdot{\bf J})\right]$
2 | — |  $f^{(1)}_{2}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{\sqrt{6}}\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-\kappa\,\mathds{1}\right]$ | $f^{(\frac{3}{2})}_{2}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{6}\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-\kappa\,\mathds{1}\right]$
3 | — | — |  $f^{(\frac{3}{2})}_{3}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\frac{1}{3\sqrt{5}}\left[5({\bf\hat{n}}\cdot{\bf J})^{3}-(3\kappa-1)({\bf\hat{n}}\cdot{\bf J})\right]$

Table II

Title: A comparison of Chebyshev polynomial scalars $f^{(j)}_{\lambda}(m)$ and
operators $f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$.
Caption: First column compares the definitions of the Chebyshev polynomial
scalars adopted by various authors, while the second column compares the
definitions of the Chebyshev polynomial operators adopted by the same authors.
Note that Corio corio:ortho ; corio:siam  suppressed the explicit dependence
on the spin angular momentum $j$ in his definitions of both the Chebyshev
polynomial scalars $Z_{n}(x)$ and operators $U_{n}({\bf\hat{n}}\cdot{\bf J})$,
while Meckler meckler:angular  suppressed the explicit dependence on the spin
angular momentum $S$ in his definition of the Chebyshev polynomial operators
$A^{(n})/g_{n}$, where
$g_{n}=[n!]^{2}\sqrt{2S+1+n}\left[2^{n}(2n-1)!!\sqrt{(2n+1)(2S-n)!}\right]^{-1}$.
Using Filippov’s notation filippov2:thesis  for the Chebyshev polynomial
operators, Meckler’s operators meckler:angular  can be written as
$A^{(n})/g_{n}\equiv f^{(S)}_{n}({\bf\hat{a}}\cdot{\bf S})$. Just as the
$(2j+1)$ Chebyshev polynomial scalars $p_{n}(S,-j),\ldots,p_{n}(S,j)$ defined
by Meckler meckler:angular  are the diagonal matrix elements of
$A^{(n)}/g_{n}$ in a representation where $({\bf\hat{a}}\cdot{\bf S})$ is
diagonal, the $(2j+1)$ Chebyshev polynomial scalars
$Z_{n}(0),\ldots,Z_{n}(2j)$ defined by Corio corio:ortho ; corio:siam  are the
diagonal matrix elements of $U_{n}({\bf\hat{n}}\cdot{\bf J})$ in a
representation where $({\bf\hat{n}}\cdot{\bf J})$ is diagonal.

Scalars | Operators | Authors
---|---|---
$f^{(j)}_{\lambda}(m)$ | $f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ |  Filippov filippov2:thesis
$p_{n}(S,m)$ | $\displaystyle\frac{A^{(n)}}{g_{n}}$ |  Meckler meckler:majorana ; meckler:angular
$(n\equiv\lambda;S\equiv j)$ | $(n\equiv\lambda)$ |
$Z_{n}(x)$ | $U_{n}({\bf\hat{n}}\cdot{\bf J})$ |  Corio corio:ortho ; corio:siam
$(n\equiv\lambda;x\equiv j+m)$ | $(n\equiv\lambda)$ |

Table III

Title: Chebyshev polynomial definitions and recursion relations.

Caption: The top box in the first column gives Filippov’s definition
filippov2:thesis  of the Chebyshev polynomials $f_{L}^{(j)}(m)$ in terms of
Bateman’s bateman  Chebyshev polynomials $t_{L}(j+m,2j+1)$, which are defined
in the bottom box of the first column using finite differences footnote2 . Two
versions of the recursion relations are given in the second column. The form
of the first version is that given by Filippov filippov2:thesis  for the
Chebyshev polynomials $f_{L}^{(j)}(m)$, although equivalent forms were derived
previously by Meckler meckler:majorana ; meckler:angular  and by Corio
corio:ortho ; corio:siam . The second version of the recursion relation is
that given by Varshalovich et al. varshal1:ang  for the Clebsch-Gordan
coefficients $C^{L\,0}_{jmj-m}$, which are identical to the Chebyshev
polynomials $f_{L}^{(j)}(m)$ to within a phase-factor. The bottom boxes also
define functions $F(L,j)$ and $G(a,b)$ used for the matrix elements (top box,
first column) and recursion relations (top box, second column), respectively.

Matrix Element Definition via Bateman’s bateman  |  Recursion Relations
---|---
Chebyshev Polynomials $\boxed{t_{L}(j+m,2j+1)}$ |
$f_{L}^{(j)}(m)=\langle jm|\;f_{L}^{(j)}(J_{z})\;|jm\rangle$ | $G(L+1,j)\,f_{L+1}^{(j)}(m)-2m\,f_{L}^{(j)}(m)+G(L,j)\,f_{L-1}^{(j)}(m)=0$
$=F(L,j)\;t_{L}(j+m,2j+1)$ | $G(L+1,j)\,C^{L+1\,0}_{jmj-m}-2m\,C^{L\,0}_{jmj-m}+G(L,j)\,C^{L-1\,0}_{jmj-m}=0$
$=(-1)^{j-m}\;C^{L\,0}_{jmj-m}$ |
$(1)\;\;\;F(L,j)=\left[\displaystyle\frac{(2L+1)(2j-L)!}{(2j+L+1)!}\right]^{1/2}$ | $G(a,b)=\left[\displaystyle\frac{a^{2}((2b+1)^{2}-a^{2})}{4a^{2}-1}\right]^{1/2}$
$(2)\;\;\;t_{L}(j+m,2j+1)=L!\,\Delta^{L}\\!\left[H_{L}^{j}(m)\right]$ |
where $H_{L}^{j}(m)=\left[\displaystyle{j+m\choose L}{m-j-1\choose L}\right]$ |
$\Delta h(m)=h(m+1)-h(m)$ |
$\Delta^{k+1}h(m)=\Delta[\Delta^{k}h(m)]$ |

Table IV

Title: A comparison of expansions, traces, matrix elements, Hermitian
conjugates and density operators $\hat{\rho}$ for the spin polarization
operators $\hat{T}^{(j)}_{\lambda\mu}$ and the Chebyshev polynomial operators
$f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$.

Operator | Spin Polarization | Chebyshev Polynomial
---|---|---
$\hat{O}$ | $\hat{O}=\hat{T}_{\lambda\mu}^{(j)}$ | $\hat{O}=f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})=\displaystyle\sum_{\mu=-\lambda}^{\lambda}C_{\lambda\mu}^{\star}({\bf\hat{n}})\;\,\hat{T}^{(j)}_{\lambda\mu}$
Expansions | $\hat{A}=\displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda}A_{\lambda\mu}^{(j)}\;\hat{T}_{\lambda\mu}^{(j)}$ | $\hat{B}=\displaystyle\sum_{\lambda=0}^{2j}B_{\lambda}^{(j)}\,f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$
| $A_{\lambda\mu}^{(j)}=\mbox{Tr}\\!\left[\left[\hat{T}_{\lambda\mu}^{(j)}\right]^{\\!\dagger}\\!\hat{A}\right]$ | $B_{\lambda}^{(j)}=\mbox{Tr}\\!\left[f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})\,\hat{B}\right]$
Traces | $\mbox{Tr}\\!\left[\hat{T}_{\lambda\mu}^{(j)}\left[\hat{T}_{\lambda^{\prime}\mu^{\prime}}^{(j)}\right]^{\\!\dagger}\right]=\delta_{\lambda\lambda^{\prime}}\;\delta_{\mu\mu^{\prime}}$ | $\mbox{Tr}\\!\left[f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})\;f^{(j)}_{{\lambda}^{\prime}}({\bf\hat{n}}\cdot{\bf J})\right]=\delta_{\lambda\lambda^{\prime}}$
$\langle m|\;\hat{O}\;|m^{\prime}\rangle$ | $C_{jmj-m^{\prime}}^{\lambda\,(m-m^{\prime})}\;(-1)^{j-m^{\prime}}$ | $C_{\lambda\,(m-m^{\prime})}^{\star}({\bf\hat{n}})\;C_{jmj-m^{\prime}}^{\lambda\,(m-m^{\prime})}\;(-1)^{j-m^{\prime}}$
Hermitian | $\left[\hat{T}_{\lambda\mu}^{(j)}\right]^{\\!\dagger}=(-1)^{\mu}\;\,\hat{T}_{\lambda-\mu}^{(j)}$ | $\left[f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})\right]^{\\!\dagger}=f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})$
Conjugate |  |
$\hat{\rho}$ | $\displaystyle\sum_{\lambda=0}^{2j}\sum_{\mu=-\lambda}^{\lambda}\mbox{Tr}\\!\left[\hat{\rho}\,\left[\hat{T}_{\lambda\mu}^{(j)}\right]^{\\!\dagger}\right]\hat{T}_{\lambda\mu}^{(j)}$ | $\;\;\;\displaystyle\sum_{\lambda=0}^{2j}\frac{(2\lambda+1)}{4\pi}\\!\\!\int_{{\bf S}^{2}}f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})\;\mbox{Tr}\\!\left[\hat{\rho}\,f_{\lambda}^{(j)}({\bf\hat{n}}\cdot{\bf J})\right]d{\bf\hat{n}}\;\;\;$

Table V

Title: Similarity Transforms

Caption: Similarity transforms of $J_{z}$ and $f^{(j)}_{\lambda}(J_{z})$.

Operator | Similarity Transform
---|---
$\hat{O}$ | $\hat{{\cal D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\,\hat{O}\left[\hat{{\cal D}}^{(j)}(\theta,{\bf\hat{n}}_{\bot})\right]^{\\!\dagger}$
$J_{z}$ | $({\bf\hat{n}}\cdot{\bf J})$
$f^{(j)}_{\lambda}(J_{z})$ | $f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$

Table VI

Title: A comparison of the relative orientations of Meckler’s instantaneous
axis ${\bf\hat{b}}(t)$ meckler:majorana  and Abragram’s magnetic moment
${\bf\hat{m}}(t)$ abragamtext .

| Meckler meckler:majorana  | Abragam abragamtext
---|---|---
$t=0$ | ${\bf\hat{b}}(0)\parallel{\bf\hat{a}}$ | ${\bf\hat{m}}(0)\parallel\mbox{H}_{0}\,{\bf\hat{z}}$
$t>0$ | ${\bf\hat{b}}(t)\cdot{\bf\hat{a}}=1-\displaystyle\frac{\lambda^{2}}{u^{2}}(1-\cos ut)$ | ${\bf\hat{m}}(t)\cdot{\bf\hat{m}}(0)=1-\displaystyle\frac{\omega^{2}_{1}}{a^{2}}(1-\cos at)$
| $\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\equiv Z$ | $\\!\\!\\!\\!\\!\\!\\!\equiv\cos\alpha$
Nutation | $u=\sqrt{\lambda^{2}+(\omega-\omega_{0})^{2}}$ | $a=\sqrt{\omega_{1}^{2}+(\omega-\omega_{0})^{2}}$
frequency |  |
Excitation | $\lambda$ | $\omega_{1}$
radiofrequency |  |

Table VII

Title: Probability distributions and tomographic reconstruction relations.
Caption: Probability distributions defined in the left-hand column are used to
define the corresponding tomographic reconstructions of the density matrix
$\hat{\rho}$ in the right-hand column. The function
$F_{\lambda}({\bf\hat{n}})$ used in the definitions of the probability
distributions is defined at the bottom of the left-hand column, and the
function ${\cal F}_{\lambda}({\bf\hat{n}})$ used in the definitions of the
tomographic reconstruction relations is defined at the bottom of the right-
hand column.

Probability Distributions |  Tomographic Reconstruction Relations
---|---
$w^{(j)}(m,{\bf\hat{n}})=\mbox{Tr}\\!\left[\hat{\rho}\;\mbox{{\boldmath$\Pi$}}^{(j)}(m,{\bf\hat{n}})\right]$ |  $\hat{\rho}=\displaystyle\sum_{\lambda=0}^{2j}\displaystyle\sum_{m=-j}^{j}f^{(j)}_{\lambda}(m)\int_{{\bf S}^{2}}w^{(j)}(m,{\bf\hat{n}})\;{\cal F}_{\lambda}({\bf\hat{n}})\;d{\bf\hat{n}}$
$=\displaystyle\sum_{\lambda=0}^{2j}f^{(j)}_{\lambda}(m)\;F_{\lambda}({\bf\hat{n}})$ |
$Q({\bf\hat{n}})=\mbox{Tr}\\!\left[\hat{\rho}\;\mbox{{\boldmath$\Pi$}}^{(j)}(j,{\bf\hat{n}})\right]$ |  $\hat{\rho}=\displaystyle\sum_{\lambda=0}^{2j}\left[f^{(j)}_{\lambda}(j)\right]^{-1}\int_{{\bf S}^{2}}Q({\bf\hat{n}})\;{\cal F}_{\lambda}({\bf\hat{n}})\;d{\bf\hat{n}}$
$=\displaystyle\sum_{\lambda=0}^{2j}f^{(j)}_{\lambda}(j)\;F_{\lambda}({\bf\hat{n}})$ |
$W({\bf\hat{n}})=\mbox{Tr}\\!\left[\hat{\rho}\;\Delta^{\\!(j)}({\bf\hat{n}})\right]$ |  $\hat{\rho}=\displaystyle\sum_{\lambda=0}^{2j}\sqrt{\frac{2j+1}{2\lambda+1}}\;\displaystyle\int_{{\bf S}^{2}}W({\bf\hat{n}})\;{\cal F}_{\lambda}({\bf\hat{n}})\;d{\bf\hat{n}}$
$=\displaystyle\sum_{\lambda=0}^{2j}\sqrt{\frac{2\lambda+1}{2j+1}}\;F_{\lambda}({\bf\hat{n}})$ |
$\;\;F_{\lambda}({\bf\hat{n}})=\mbox{Tr}\\!\left[\hat{\rho}\;f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})\right]$ | $\;\;{\cal F}_{\lambda}({\bf\hat{n}})=\left(\displaystyle\frac{2\lambda+1}{4\pi}\right)f^{(j)}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$

Table VIII

Title: A comparison of tensor operator equivalents expressed in terms of
Chebyshev polynomials.
Caption: First row is work done by Corio corio:ortho , and second row is work
done by Marinelli et al. werb:tensor .

$\hat{T}_{60}^{(j)}\equiv f^{(j)}_{6}(J_{z})$ |  $\hat{T}_{41}^{(j)}$
---|---
$U^{(6)}_{0}\\!(J)\sim\Big{\\{}231J_{\\!z}^{6}-105(3\kappa-7)J_{\\!z}^{4}$ |  $U^{(3)}_{1}\\!(J)\sim J_{+}\Big{\\{}14\,J_{\\!z}^{3}+21J_{z}^{2}+(19-6\kappa)\,J_{z}$
$\;\;\;\;\;\;\;\;\;\;\;+21(5\kappa^{2}-25\kappa+14)J_{\\!z}^{2}$ |  $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+3(2-\kappa)\mathds{1}$ }
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-5\kappa(\kappa^{2}-8\kappa+12)\mathds{1}\Big{\\}}$ |
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{\kappa\equiv J(J+1)}$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{\kappa\equiv J(J+1)}$
$T^{(0)}_{6}\\!(S)\sim\Big{\\{}231S_{\\!z}^{6}-105(3K-7)S_{\\!z}^{4}$ |  $T^{(1)}_{4}\\!(S)\sim S_{+}\Big{\\{}14S_{\\!z}^{3}+21S_{z}^{2}+(19-6K)S_{z}$
$\;\;\;\;\;\;\;\;\;\;\;+21(5K^{2}-25K+14)S_{\\!z}^{2}$ |  $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+3(2-K)\,\mathds{1}$ }
$\;\;\;\;\;\;\;\;\;\;\;\;\;-5K(K^{2}-8K+12)\mathds{1}\Big{\\}}$ |  $\;\;\;\;\;\;\;\;\;\;\sim S_{+}\\!\displaystyle\sum_{n=1}^{4}\;A_{4n}^{1}\,\left(\displaystyle\frac{\partial}{\partial S_{z}}\right)^{\\!\\!n}\left[f^{(S)}_{4}(S_{z})\right]$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{K\equiv S(S+1)}$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boxed{K\equiv S(S+1)}$

Table IX

Title: Legendre polynomial operators.
Caption: A comparison of the Legendre polynomial operators defined by
Schwinger schwinger:majorana  ($P_{\lambda}({\bf J})$ in the second column)
and by Zemach zemach  ($\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ in
the third column) with the Legendre polynomials ($P_{\lambda}(\cos\theta)$ in
the fourth column). ($\kappa\equiv{\bf J}\cdot{\bf J}=j(j+1))$

$\lambda$ | $P_{\lambda}({\bf J})$ |  $\overline{P}_{\lambda}({\bf\hat{n}}\cdot{\bf J})$ | $P_{\lambda}(\cos\theta)$
---|---|---|---
0 | $\mathds{1}$ | $\mathds{1}$ | $1$
1 | $\left[\kappa^{2}\right]^{-1/2}J_{z}$ | $({\bf\hat{n}}\cdot{\bf J})$ | $\cos\theta$
2 | $\left[\kappa^{2}\\!\left(\kappa^{2}-\tfrac{3}{4}\right)\right]^{-1/2}\,\tfrac{1}{2}\\!\left[3J_{z}^{2}-\kappa\mathds{1}\right]$ | $\tfrac{1}{2}\\!\left[3({\bf\hat{n}}\cdot{\bf J})^{2}-\kappa\,\mathds{1}\right]$ | $\tfrac{1}{2}(3\cos^{2}\theta-1)$
3 |  $\left[\kappa^{2}\\!\left(\kappa^{2}-\tfrac{3}{4}\right)\\!\left(\kappa^{2}-2\right)\right]^{-1/2}\,\tfrac{1}{2}\\!\left[5J_{z}^{3}-\left(3\kappa-1\right)J_{z}\right]$ |  $\tfrac{1}{2}\\!\left[5({\bf\hat{n}}\cdot{\bf J})^{3}-(3\kappa-1)({\bf\hat{n}}\cdot{\bf J})\right]$ | $\tfrac{1}{2}(5\cos^{3}\theta-\cos\theta)$

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