# Low-energy spectrum of $SU(3)$ Yang-Mills Quantum Mechanics

Hans-Peter Pavel
Bogoliubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research, Dubna, Russia111email:
<EMAIL_ADDRESS>
(December 12th, 2021)

###### Abstract

The SU(3) Yang-Mills Quantum Mechanics of spatially constant gluon fields is
considered in the unconstrained Hamiltonian approach using the ”flux-tube
gauge”. The Faddeev-Popov operator, its determinant and inverse, are rather
simple, but show a highly non-trivial periodic structure of six Gribov-
horizons separating six Weyl-chambers. The low-energy eigensystem of the
obtained physical Hamiltonian can be calculated (in principle with arbitrary
high precision) using the orthonormal basis of eigenstates of the
corresponding harmonic oscillator problem with the same non-trivial Jacobian
only replacing the chromomagnetic potential by the 16dimensional harmonic
oscillator potential. This turns out to be integrable and its eigenstates be
made out of orthogonal polynomials of the 45 components of eight irreducible
symmetric tensors. The calculations in this work have been carried out in all
sectors $J^{PC}$ up to spin $J=11$, and up to polynomial order $10$ for even
and $11$ for odd parity. The low-energy eigensystem of the physical
Hamiltonian of $SU(3)$ Yang-Mills Quantum Mechanics is found to converge
nicely when truncating at higher and higher polynomial order (equivalent to
increasing the resolution in functional space). Our results are in good
agreement with the results of Weisz and Zieman (1986) using the constrained
Hamiltonian approach. We find excellent agreement in the $0^{++}$ and $2^{++}$
sectors, much more accurate values in other sectors considered by them, e.g.
in the $1^{--}$ and $3^{--}$ sectors , and quite accurate ”new results” for
the sectors not considered by them, e.g. $2^{--},4^{--},5^{--},3^{++}$.

##  1 Introduction

The Yang-Mills Quantum Mechanics (YM QM) of spatially constant gluon fields
has been studied for a long time, as a toy model for the QCD vacuum[1]-[7], as
the zeroth order of a weak coupling expansion [8]-[11], and as zeroth order of
a strong coupling expansion [12]-[15]. For the case of SU(2) YM theory, the
symmetric gauge turned out to exist in the strong coupling limit and to be
very convenient for calculations, also including the quarks [16]. The reduced
gauge fields transform as symmetric tensors under spatial rotations, and the
Faddeev-Popov (FP) operator turned out to be non-trivial but managable in a
way similar to the Calogero model [17]. For the case of SU(3) YM theory,
however, the symmetric gauge can also be defined [18]-[20] and leads to
reduced fields transforming as tensors under spatial rotations, but the
corresponding FP-operator turns out to be very complicated. In previous work
[21], a new algebraic gauge for SU(3) YM theory, the flux-tube gauge, has been
proposed, which exists in the strong coupling limit and has a simple non-
trivial FP-operator. As for the case of SU(2) YM theory in the symmetric gauge
[15], the corresponding gauge reduced SU(3) YM Hamiltonian in the flux-tube
gauge can be expanded in strong coupling $\lambda=g^{-2/3}$ with the leading
order corresponding to SU(3) YM QM of spatially constant fields. The drawback
of the flux-tube gauge is however, that the reduced fields $A$ are color-
singlets, but are not transforming as tensors under spatial rotations. We
shall show in this article how this can be circumvened by forming certain
irreducible polynomials of the reduced $A$, symmetric tensors with definite
eigenvalues of J,P, and C. We shall calculate in this work the low-energy
spectrum of SU(3) YM QM in the unconstrained Hamiltonian approach using the
flux-tube gauge. The Faddeev-Popov operator, its determinant and inverse, are
rather simple, but show a highly non-trivial periodic structure of six Gribov-
horizons separating six Weyl-chambers. The low-energy eigensystem of the
obtained physical Hamiltonian can be calculated (in principle with arbitrary
high precision) using the orthonormal basis of eigenstates of the
corresponding harmonic oscillator problem with the same non-trivial Jacobian
only replacing the chromomagnetic potential by the 16-dimensional harmonic
oscillator potential. This turns out to be integrable and its eigenstates to
be made out of orthogonal polynomials of the 45 components of eight elementary
spatial tensors.

The paper is organised as follows. In Section 2 we give a short introduction
to the Hamiltonian approach of SU(3) Yang-Mills Quantum mechanics of spatially
constant gluon fields $V_{ai}$ constrained by non-Abelian Gauss laws. In
Section 3 the flux-tube gauge is defined and shown to lead to a rather simple
but non-trivial Faddeev-Popov (FP) operator. In Section 4 the corresponding
harmonic oscillator (HO) Hamiltonian, obtained by replacing the chromomagnetic
potential by an 16-dimensional harmonic oscillator potentials, is solved
analytically by polynomials in the components of eight irreducible tensors in
reduced space $A$. In Section 5 we use the obtained eigensystem of the HO-
Hamiltonian to find the eigensytem of the Hamiltonian of SU(3)-Yang-Mills QM
in dependence of truncation at higher and higher polynomial degree. The
results are compared with those of Weisz and Zieman [11] obtained in the
constrained approach, and comparison with the low glueball spectrum obtained
in [22],[23] using lattice QCD. Section 6 gives our Conclusions. Some
technical details are banned to Appendices A-F.

##  2 Constrained SU(3) Yang-Mills QM of spatially constant fields

The action of $SU(3$) Yang-Mills Quantum Mechanics of spatially constant gluon
fields $V_{\mu}(t)\equiv V_{a\mu}(t)\lambda_{a}/2$ is defined as

$\displaystyle{\cal S}[V]$ $\displaystyle:=$ $\displaystyle{\rm Vol}\int
dt\left[-\frac{1}{4}F^{\rm hom}_{a\ \mu\nu}F_{a}^{{\rm hom}\
\mu\nu}\right]={\rm Vol}\int dt\ \frac{1}{2}\left[\left(E^{\rm
hom}_{ai}\right)^{2}-\left(B^{\rm hom}_{ai}\right)^{2}\right]~{},$ (1)

with the spatially constant field strength tensor

$\displaystyle F^{\rm hom}_{a\ \mu\nu}$ $\displaystyle:=$
$\displaystyle\delta_{0\mu}\partial_{t}V_{a\nu}-\delta_{0\nu}\partial_{t}V_{a\mu}+gf_{abc}V_{b\mu}V_{c\nu}~{},\quad
a=1,..,8~{},$ (2)

or in terms of the chromoelectric and chromomagnetic parts

$\displaystyle E^{\rm hom}_{ai}\equiv F^{\rm hom}_{a\ i0}\quad,$
$\displaystyle\quad B^{\rm hom}_{ai}\equiv\frac{1}{2}\epsilon_{ijk}F^{\rm
hom}_{a\ jk}~{}.$ (3)

The action (1) is invariant under the spatially homogeneous $SU(3)$ gauge
transformations

$\displaystyle V_{a\mu}^{\omega}(t)\lambda_{a}/2$ $\displaystyle=$
$\displaystyle U[\omega(t)]\ \ V_{a\mu}(t)\lambda_{a}/2\ \
U^{-1}[\omega(t)]~{}.$ (4)

Furthermore, the action is invarinat under spatial rotations $R$

$\displaystyle R:\quad V_{ai}\rightarrow R_{ij}V_{aj}~{},$ (5)

as well as under parity transformations and charge conjugation

$\displaystyle P:\quad V_{ai}\lambda_{a}\rightarrow-
V_{ai}\lambda_{a}\quad\quad\quad C:\quad
V_{ai}\lambda_{a}\rightarrow-(V_{ai}\lambda_{a})^{*}~{}.$ (6)

In terms of the momenta $\Pi_{ai}=-E_{ai}^{\rm hom}$ canonical conjugate to
the spatial $V_{ai}$ one obtains the canonical Hamiltonian

$\displaystyle H_{C}$ $\displaystyle=$ $\displaystyle{\rm Vol}\
\Bigg{[}{1\over 2}\Pi_{ai}^{2}+{1\over 2}\left(B^{\rm
hom}_{ai}(V)\right)^{2}-gV_{a0}\left(f_{abc}V_{ci}\Pi_{bi}\right)\Bigg{]}~{}.$
(7)

Exploiting the time dependence of the gauge transformations t put

$V_{a0}=0~{},\quad\quad a=1,..,8\quad\quad({\rm Weyl\ gauge})~{},$ (8)

the dynam. vaiables $V_{ai}$, $\Pi_{ai}$ are quantized in the Schrödinger
functional approach imposing the equal-time commutation relations
$\Pi_{ai}=-i\partial/\partial V_{ai}$. The physical states $\Phi$ satisfy the
coupled system of Schrödinger Equ. and eight non-abelian Gauss law
constraints,

$\displaystyle H_{0}\,\Phi$ $\displaystyle\equiv$ $\displaystyle{\rm Vol}\
\left[{1\over 2}\Pi_{ai}^{2}+{1\over 2}\left(B^{\rm
hom}_{ai}(V)\right)^{2}\right]\Phi=E\,\Phi~{},$ (9) $\displaystyle
G_{a}\,\Phi$ $\displaystyle\equiv$ $\displaystyle g\ f_{abc}V_{ci}\Pi_{bi}\
\Phi=0~{},\quad\quad a=1,...8~{}.$ (10)

The Gauss law operators $G_{a}$ are the generators of the residual time
independent gauge transformations, satisfying $[G_{a},H]=0$ and
$[G_{a},G_{b}]=if_{abc}G_{c}$. The matrix element of an operator $O$ is given
in the Cartesian form

$\displaystyle\langle\Phi^{\prime}|O|\Phi\rangle\ \propto\int dV\
\Phi^{\prime*}(V)\,O\,\Phi(V)~{}.$ (11)

Since $H_{0}$ is invariant under spatial rotations $[H_{0},J_{i}]=0$ with

$\displaystyle J_{i}=\epsilon_{ijk}V_{aj}\Pi_{ak}\quad
i=1,2,3~{},\quad[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}~{},$ (12)

and invariant under parity $[H_{0},P]=0$ and charge conjugation $[H_{0},C]=0$
the eigenstates can be characterised by $J^{PC}$.

In their work [11] Weisz and Ziemann used the variational approach to find the
eigenvalues of the constrained Schrödinger Equ. (9) with trial functions

$\Phi^{(J)PC}(V)=P^{(J)PC}_{\rm gauge\ inv.}(V)\
\exp[-(\omega/2)\left(V_{ai}\right)^{2}]$

which are gauge invariant and hence automatically satisfy the Gauss law
constraints (10). In the sectors $0^{++}$ and $2^{++}$ they find rather
accurate eigenvalues, in other sectors first upper bounds. In the present work
we would like to demonstrate that the above constrained system becomes
integrable if one replaces

$\displaystyle\left(B^{\rm hom}_{ai}(V)\right)^{2}\ \longrightarrow\
\,\omega^{2}\left(V_{ai}\right)^{2}~{},\quad\quad\omega>0\ \ {\rm free\
parameter}$

Using an exact gauge reduction the energy-eigensystem can be found rather
accurately and used as a Hilbert-basis for the YM QM. Truncating at higher and
higher numbers of nodes, a converging low-energy eigensystem of YM QM is
obtained.

##  3 Unconstrained Hamiltonian formulation using the flux-tube gauge

###  3.1 Unconstrained Hamiltonian formulation of SU(3) Yang-Mills QM

In order to obtain an unconstrained Hamiltonian formulation, one can perform a
point transformation of the original 24 $V_{ai}$ to a new set of adapt
coordinates,

$\displaystyle V_{ai}\left(q,S\right)=O_{ab}\left(q\right)A_{bi},$ (13)

in terms of the 8 gauge angles $q_{j}$ parametrising the $O_{ab}(q)\ {\rm
orth.}\ 8\times 8\ {\rm matrix}\ {\rm adjoint\ to}\ U(q)$

$\displaystyle
O_{ab}(q)=(1/8)\mbox{Tr}\left[U^{-1}(q)\lambda_{a}U(q)\lambda_{b}\right]~{}.$
(14)

and 16 reduced $A_{ai}$ satisfying some gauge conditions

$\chi_{a}(A)=\left(\Gamma_{i}\right)_{ab}\,A_{bi}=0~{},\quad a=1,...,8~{}.$

Preserving the CCR $\rightarrow$ old canonical momenta in terms of the new
variables

$\displaystyle\Pi_{ai}(q,A,p,P)=O_{ab}\left(q\right)\left[P_{bi}-\left(\Gamma_{i}\right)_{bl}{\gamma}^{-1T}_{ls}(A)\left({1\over
g}\Omega^{-1}_{st}(q)p_{t}+T_{s}(A,P)\right)\right]~{},$ (15)

with the homogeneous part of the FP operator

$\displaystyle\gamma_{ab}(A):=\left(\Gamma_{i}\right)_{ad}\,f_{dbc}A_{ci}~{},$
(16)

and the operators

$\displaystyle T_{a}(A,P):=f_{abc}A_{bi}P_{ci}~{}.$ (17)

In terms of the new coordinates, the Gauss-laws become

$\displaystyle G_{a}\Phi\equiv
O_{ak}(q)\Omega^{-1}_{ki}(q)p_{i}\Phi=0\quad\Leftrightarrow\quad\frac{\delta}{\delta
q_{i}}\Phi=0\quad(\rm{Abelianisation})$

The unconstrained spin operator reads

$\displaystyle J_{i}=\epsilon_{ijk}A_{aj}E_{ak}~{}.$ (18)

in terms of the physical electric fields

$\displaystyle
E_{ai}:=P_{bi}-\left(\Gamma_{i}\right)_{bl}{\gamma}^{-1T}_{ls}\\!(A)\,T_{s}(A,P)~{}.$
(19)

The correctly ordered unconstrained Hamiltonian of SU(3) YM-QM takes the form
[24]

$\displaystyle H\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!{1\over 2}\Bigg{[}\frac{1}{|\gamma(A)|}\\!\\!\
P_{ai}\ |\gamma(A)|\ P_{ai}+\frac{1}{|\gamma(A)|}\\!\\!\ T_{a}\
|\gamma(A)|\left(\gamma^{-1}(\Gamma_{i}^{T}\Gamma_{i})\gamma^{-1T}\right)_{ac}T_{c}+\left(B_{ai}^{\rm
hom}(A)\right)^{2}\Bigg{]}~{},$

using the homogeneus part of the chromomagnetic field

$B^{\rm hom}_{a\,i}(A):=(1/2)g\,\epsilon_{ijk}\,f_{abc}\,A_{b\,j}A_{c\,k}~{}.$
(20)

The matrix element of a physical operator O is given by

$\displaystyle\langle\Psi^{\prime}|O|\Psi\rangle\ \propto\int dA\
{|\gamma(A)|}\ \Psi^{\prime*}[A]\ O\ \Psi[A]~{}.$

###  3.2 Unconstrained Hamiltonian formulation of SU(3) YM-QM in the flux-tube
gauge

It is our aim to find a gauge which exists and leads to a maximally simple FP-
operator. This is can be acchieved by putting the ”flux-tube-gauge”, leading
to a rather simple but non-trivial FP-operator. The drawback of the this gauge
is that the reduced fields $A_{ai}$ are color-singlets, but not spin-
eigenstates, as was the case for the SU(3) symmetric gauge [18]\- [20] leading
to a very complicated Fp-operator. We shall show that the disadvantage, that
the reduced gauge fileds in the fluxtube-gauge are not spin eigenstates , can
be circumvened by forming certain irreducible polynomials of the reduced $A$,
symmetric tensors, which have definite eigenvalues of J,P, and C.

The ”flux-tube-gauge” is defined as

$\chi_{a}(A)=0~{}:\quad\quad A_{a1}=0\quad\forall
a=1,2,4,5,6,7\quad\wedge\quad A_{a2}=0\quad\forall a=5,7~{}.$

or explicitly

$A=\,\left(\begin{array}[]{c c c}0&A_{12}&A_{13}\\\ 0&A_{22}&A_{23}\\\
A_{31}&A_{32}&A_{33}\\\ 0&A_{42}&A_{43}\\\ 0&0&A_{53}\\\ 0&A_{62}&A_{63}\\\
0&0&A_{73}\\\ A_{81}&A_{82}&A_{83}\end{array}\right)\equiv\left(X\ Y\
Z\right)~{}.$

Using finally the reparametrisation

$\displaystyle A_{31}\equiv X_{3}=r\cos[\psi]$ $\displaystyle A_{81}\equiv
X_{8}=r\sin[\psi]~{},$ (21)

the explicit expression for the homogeneous part of the FP-operator (16) is

$\gamma=\left(\\!\\!\\!\begin{array}[]{c c c c c c c
c}0&-r\cos[\psi]\\!\\!\\!\\!&0&0&0&0&0&0\\\
r\cos[\psi]\\!\\!\\!\\!&0&0&0&0&0&0&0\\\
-Y_{6}/2&0&-Y_{4}/2&-Y_{+}&0&Y_{1}/2&-Y_{2}/2&-\sqrt{3}\,Y_{4}/2\\\
0&0&0&0&r\cos[\psi+{2\pi\over 3}]\\!\\!\\!\\!&0&0&0\\\
0&0&0&\\!\\!\\!\\!-r\cos[\psi+{2\pi\over 3}]\\!\\!\\!\\!&0&0&0&0\\\
0&0&0&0&0&0&\\!\\!\\!\\!-r\cos[\psi+{4\pi\over 3}]\\!\\!\\!\\!&0\\\
0&0&0&0&0&\\!\\!\\!\\!r\cos[\psi+{4\pi\over 3}]\\!\\!\\!\\!&0&0\\\
-Y_{4}/2&0&Y_{6}/2&Y_{1}/2&Y_{2}/2&Y_{-}&0&-\sqrt{3}\,Y_{6}/2\end{array}\\!\\!\\!\right)~{},$
(22)

using the abbreviations $Y_{\pm}:=-(Y_{3}\pm\sqrt{3}Y_{8})/2$. The FP-
determinant factorises

$\displaystyle|\gamma(A)|=r^{6}\cos^{2}[3\,\psi]\ Y_{4}Y_{6}~{}.$ (23)

The inverse $\gamma^{-1}$ of the Faddeev-Popov operator is rather simple
(shown in Appendix A) and exists in the regions of non-vanishing determinant.
The matrix elements are

$\langle\Psi_{1}|O|\Psi_{2}\rangle=\int d\mu_{X}\int d\mu_{Y}\int d\mu_{Z}\
\Psi_{1}^{\dagger}O\ \Psi_{2}~{},$ (24)

with the completely factorised

$\displaystyle\int d\mu_{X}$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}dr\,r^{\,7}\int_{0}^{2\pi}d\psi\,\cos^{2}[3\,\psi]~{},$
(25) $\displaystyle\int d\mu_{Y}$ $\displaystyle:=$
$\displaystyle\int_{-\infty}^{\infty}dY_{1}\int_{-\infty}^{\infty}dY_{2}\int_{-\infty}^{\infty}dY_{3}\int_{-\infty}^{\infty}dY_{8}\int_{0}^{\infty}dY_{4}Y_{4}\int_{0}^{\infty}dY_{6}Y_{6}~{},$
(26) $\displaystyle\int d\mu_{Z}$ $\displaystyle:=$
$\displaystyle\prod_{a=1}^{8}\int_{-\infty}^{\infty}dZ_{a}~{}.$ (27)

Furthermore, for the operators $T_{a}(A,P)$ defined in (17) we find that
$T_{a}^{X}=-if_{abc}\,X_{b}\,\partial/\partial X_{c}\equiv 0~{}$, i.e.

$T_{a}(A,P)\equiv T_{a}^{Y}(Y,P_{Y})+T_{a}^{Z}(Z,P_{Z})~{}.$

where the components of the (non-reduced)
$T_{a}^{Z}=\\!-if_{abc}\,Z_{b}\,\partial/\partial Z_{c}$ satisfy the $su(3)$
algebra

$[T^{Z}_{a},T^{Z}_{b}]=i\,f_{abc}\,T^{Z}_{c}~{},$

whereas the reduced $T_{a}^{Y}=\\!-if_{abc}\,Y_{b}\,\partial/\partial Y_{c}$
do not.

The physical electric fields read in the flux-tube gauge

$E(A,P)=\left(\begin{array}[]{c c c}-{\cal P}_{2}(A,P)&P_{12}&P_{13}\\\ {\cal
P}_{1}(A,P)&P_{22}&P_{23}\\\ P_{31}&P_{32}&P_{33}\\\ {\cal
P}_{5}(A,P)&P_{42}&P_{43}\\\ -{\cal P}_{4}(A,P)&-{\cal P}_{+}(A,P)&P_{53}\\\
-{\cal P}_{7}(A,P)&P_{62}&P_{63}\\\ {\cal P}_{6}(A,P)&{\cal
P}_{-}(A,P)&P_{73}\\\ P_{81}&P_{12}&P_{83}\end{array}\right)~{},$ (28)

with the Hermitean ${\cal P}_{\pm}^{\dagger}={\cal P}_{\pm}$

$\displaystyle{\cal P}_{+}={1\over
Y_{4}}\left[T^{Y}_{3}+\left(T^{Z}_{3}+{1\over\sqrt{3}}T^{Z}_{8}\right)\right]~{},$
$\displaystyle{\cal P}_{-}={1\over
Y_{6}}\left[T^{Y}_{3}+\left(T^{Z}_{3}-{1\over\sqrt{3}}T^{Z}_{8}\right)\right]~{},$
(29)

and

$\displaystyle{\cal P}_{1}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!{1\over
r\,\cos[\psi]}\left(\widetilde{T}^{Y}_{1}+\widetilde{T}^{Z}_{1}\right)~{},\quad\quad\quad\quad\quad\quad\quad{\cal
P}_{2}={1\over
r\,\cos[\psi]}\left(\widetilde{T}^{Y}_{2}+\widetilde{T}^{Z}_{2}\right)~{},$
$\displaystyle{\cal P}_{4}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!{1\over
r\,\cos[\psi+2\pi/3]}\left(\widetilde{T}^{Y}_{4}+\widetilde{T}^{Z}_{4}\right)~{},\quad\quad\quad{\cal
P}_{5}={1\over
r\,\cos[\psi+2\pi/3]}\left(\widetilde{T}^{Y}_{5}+\widetilde{T}^{Z}_{5}\right)~{},$
$\displaystyle{\cal P}_{6}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!{1\over
r\,\cos[\psi+4\pi/3]}\left(\widetilde{T}^{Y}_{6}+\widetilde{T}^{Z}_{6}\right)~{},\quad\quad\quad{\cal
P}_{7}={1\over
r\,\cos[\psi+4\pi/3]}\left(\widetilde{T}^{Y}_{7}+\widetilde{T}^{Z}_{7}\right)~{},$
(30)

with the definition of the tilded operators $\widetilde{T}^{Y}$ and
$\widetilde{T}^{Z}$ given in Appendix B.

Hence, the components of the spin angular momentum operators in reduced space

$\displaystyle J_{i}=\epsilon_{ijk}A_{aj}E_{ak}~{},$ (31)

read explicitly in the flux-tube gauge

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle-i\\!\\!\\!\\!\sum_{a=1,2,3,4,6,8}\left[Y_{a}{\partial\over\partial
Z_{a}}-Z_{a}{\partial\over\partial Y_{a}}\right]+\left(Z_{5}{\cal
P}_{+}-Z_{7}{\cal P}_{-}\right)~{},$ $\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle-i\sum_{a=3,8}\left[Z_{a}{\partial\over\partial
X_{a}}-X_{a}{\partial\over\partial Z_{a}}\right]-\left(Z_{1}{\cal
P}_{2}-Z_{2}{\cal P}_{1}\right)+\left(Z_{4}{\cal P}_{5}-Z_{5}{\cal
P}_{4}\right)-\left(Z_{6}{\cal P}_{7}-Z_{7}{\cal P}_{6}\right)~{},$
$\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle-i\sum_{a=3,8}\left[X_{a}{\partial\over\partial
Y_{a}}-Y_{a}{\partial\over\partial X_{a}}\right]+\left(Y_{1}{\cal
P}_{2}-Y_{2}{\cal P}_{1}\right)-Y_{4}{\cal P}_{5}+Y_{6}{\cal P}_{7}~{},$ (32)

and the physical Hamiltonian reads

$\displaystyle H$ $\displaystyle=$ $\displaystyle{1\over 2}{\cal
J}^{-1}_{X}\sum_{a=3,8}\left({\partial\over\partial X_{a}}{\cal
J}_{X}{\partial\over\partial X_{a}}\right)+{1\over 2}{\cal
J}_{Y}^{-1}\\!\\!\\!\\!\\!\\!\\!\\!\sum_{a=1,2,3,4,6,8}\\!\\!\\!\left({\partial\over\partial
Y_{a}}{\cal J}_{Y}{\partial\over\partial Y_{a}}\right)+{1\over
2}\sum_{a=1}^{8}\left({\partial\over\partial Z_{a}}{\partial\over\partial
Z_{a}}\right)$ (33) $\displaystyle\quad\quad+{1\over 2}{\cal
J}_{Y}^{-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\alpha=1,2,4,5,6,7}\\!\\!\\!\\!\\!\\!{\cal
P}^{\dagger}_{\alpha}\,{\cal J}_{Y}\,{\cal P}_{\alpha}+{1\over 2}\left({\cal
P}_{-}^{2}+{\cal P}_{+}^{2}\right)+{1\over
2}\sum_{a=1}^{8}\sum_{i=1}^{3}\left(B^{\rm hom}_{ai}[X,Y,Z]\right)^{2}~{},$

using the abbreviations $|\gamma(A)|={\cal J}_{X}\,{\cal J}_{Y}$ of (23) with
${\cal J}_{X}:=r^{6}\cos^{2}[3\,\psi]$ and ${\cal J}_{Y}:=Y_{4}Y_{6}$.

###  3.3 Hamiltonian of SU(3) YM-QM in the flux-tube gauge

The Hamiltonian of SU(3) YM-QM in the flux-tube gauge (33) can be written in
the form

$\displaystyle H[A,P]$ $\displaystyle=$ $\displaystyle
K_{X}+K_{Y}+K_{Z}+{1\over 2\,r^{2}}\Bigg{[}\
{\left(I_{1}^{Y\\!Z}+I_{2}^{Y\\!Z}\right)\over\cos^{2}{\psi}}+\
{\left(I_{4}^{Y\\!Z}+I_{5}^{Y\\!Z}\right)\over\cos^{2}{[\psi+2\pi/3]}}+\
{\left(I_{6}^{Y\\!Z}+I_{7}^{Y\\!Z}\right)\over\cos^{2}{[\psi+4\pi/3]}}\Bigg{]}$
(34) $\displaystyle+{1\over 2\,Y_{4}^{2}}I_{+}^{Y\\!Z}+{1\over
2\,Y_{6}^{2}}I_{-}^{Y\\!Z}\ +\ \ {1\over 2}\left(B^{\rm
hom}_{ai}[X,Y,Z]\right)^{2}~{}.$

The single-direction kinetic terms read

$\displaystyle K_{X}$ $\displaystyle=$ $\displaystyle-{1\over
2}\left[{\partial^{2}\over\partial r^{2}}+{7\over r}{\partial\over\partial
r}+{1\over
r^{2}}\left(-6\tan[3\psi]{\partial\over\partial\psi}+{\partial^{2}\over\partial\psi^{2}}\right)\right]~{},$
$\displaystyle K_{Y}$ $\displaystyle=$ $\displaystyle-{1\over
2}\Bigg{[}\sum_{a=1,2,3,8}{\partial^{2}\over\partial
Y_{a}^{2}}+\sum_{a=4,6}\left({\partial^{2}\over\partial Y_{a}^{2}}+{1\over
Y_{a}}{\partial\over\partial Y_{a}}-{1\over
Y_{a}^{2}}\left(Y_{1}{\partial\over\partial Y_{2}}-Y_{2}{\partial\over\partial
Y_{1}}\right)^{2}\right)\Bigg{]}~{},$ $\displaystyle K_{Z}$ $\displaystyle=$
$\displaystyle-{1\over 2}\sum_{a=1}^{8}{\partial^{2}\over\partial Z_{a}^{2}},$

and the interations $I_{m}^{Y\\!Z}$ and $I_{\pm}^{Y\\!Z}$ given in Appendix B.

It has been proven a long time ago by Simon [3], that the spectrum is discrete
although the chromomagnetic potential owns three flat valleys narrowing down.
Although there exist classical zero energy trajectories to infinity, the
quantum fluctuations in the narrowing valleys confine the wavefunction and
lead to a discret spectrum. It is therefore reasonable to replace in an
intermediate step the chromomagnetic potential by a separable 16-dimensional
harmonic oscillator potential, and then use the obtained eigensystem to find
the eigensystem of SU(3) YM QM.

###  3.4 The corresponding harmonic oscillator problem $H_{h.o.}$

Replacing in $H(A,P)$ the magnetic potential by the separable harmonic
oscillator potential with free parameter $\omega>0$

$\displaystyle{1\over 2}\left(B^{\rm hom}_{ai}(A)\right)^{2}\ \longrightarrow\
{1\over 2}\,\omega^{2}\left(A_{ai}\right)^{2}\equiv{1\over
2}\,\omega^{2}\left[r^{2}+Y_{1}^{2}+Y_{2}^{2}+Y_{3}^{2}+Y_{4}^{2}+Y_{6}^{2}+Y_{8}^{2}+Z_{a}^{2}\right]$
(35)

we obtain the corresponding harmonic oscillator problem (with the same measure
!!!),

$\displaystyle H_{h.o.}[A,P]$ $\displaystyle=$ $\displaystyle
H_{X}+H_{Y}+H_{Z}+{1\over 2\,Y_{4}^{2}}I_{+}^{Y\\!Z}+{1\over
2\,Y_{6}^{2}}I_{-}^{Y\\!Z}+\ $ (36) $\displaystyle+{1\over 2\,r^{2}}\Bigg{[}\
{\left(I_{1}^{Y\\!Z}+I_{2}^{Y\\!Z}\right)\over\cos^{2}{\psi}}+\
{\left(I_{4}^{Y\\!Z}+I_{5}^{Y\\!Z}\right)\over\cos^{2}{[\psi+2\pi/3]}}+\
{\left(I_{6}^{Y\\!Z}+I_{7}^{Y\\!Z}\right)\over\cos^{2}{[\psi+4\pi/3]}}\Bigg{]}$

The single-direction Hamiltonions read

$\displaystyle H_{X}\\!\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!{1\over 2}\left[-{\partial^{2}\over\partial
r^{2}}-{7\over r}{\partial\over\partial r}+{1\over
r^{2}}\left(6\tan[3\psi]{\partial\over\partial\psi}-{\partial^{2}\over\partial\psi^{2}}\right)+\omega^{2}r^{2}\right]~{},$
$\displaystyle H_{Y}\\!\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!{1\over
2}\Bigg{[}\sum_{a=1,2,3,8}\left(-{\partial^{2}\over\partial
Y_{a}^{2}}+\omega^{2}Y_{a}^{2}\right)+\sum_{a=4,6}\left(-{\partial^{2}\over\partial
Y_{a}^{2}}-{1\over Y_{a}}{\partial\over\partial Y_{a}}+{1\over
Y_{a}^{2}}\left(Y_{1}{\partial\over\partial Y_{2}}-Y_{2}{\partial\over\partial
Y_{1}}\right)^{2}+\omega^{2}Y_{a}^{2}\right)\Bigg{]}~{},$ $\displaystyle
H_{Z}\\!\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!{1\over
2}\sum_{a=1}^{8}\left[-{\partial^{2}\over\partial
Z_{a}^{2}}+\omega^{2}Z_{a}^{2}\right],$

As stated already in our earlier work [21], this system is integrable and can
be solved analytically in terms of orthogonal polynomials. We shall
demonstrate this in more detail in the following pragraphs.

##  4  Exact solution of the corresponding harmonic oscillator problem

The operators $T^{Y}_{a}$ and $T^{Z}_{a}$ lead to the coupling between the
three spatial directions.

###  4.1 Solutions of the $H_{h.o.}$ Schrödinger equation separable in
$X,Y,Z$.

First looking for solutions for the case where the kinetic terms decouple for
all three directions:

$\displaystyle\Phi_{X|Y|Z}=\Phi_{X}[X]\Phi_{Y}[Y]\Phi_{Z}[Z]~{},$
$\displaystyle H_{h.o.}\Phi_{X|Y|Z}$ $\displaystyle=$
$\displaystyle\left(H_{X}\\!+\\!H_{Y}\\!+\\!H_{Z}\right)\Phi_{X|Y|Z}\
=\left(\epsilon_{X}+\epsilon_{Y}+\epsilon_{Z}\right)\Phi_{X|Y|Z}~{}.$ (37)

with the single-direction functionals $\Phi_{X}$, $\Phi_{Y}$, and $\Phi_{Z}$
satisfying the separate Schrödinger equations

$\displaystyle H_{X}\Phi_{X}$ $\displaystyle=$
$\displaystyle\epsilon_{X}\Phi_{X}~{},$ (38) $\displaystyle H_{Y}\Phi_{Y}$
$\displaystyle=$ $\displaystyle\epsilon_{Y}\Phi_{Y}\quad\wedge\quad
T^{Y}_{a}\Phi_{Y}=0~{},\quad a=1,...,8~{},$ (39) $\displaystyle H_{Z}\Phi_{Z}$
$\displaystyle=$ $\displaystyle\epsilon_{Z}\Phi_{Z}\quad\wedge\quad
T^{Z}_{a}\Phi_{Z}=0~{},\quad a=1,...,8~{}.$ (40)

Note that the X-equation is unconstrained, whereas the Y- and Z-equations are
constrained.

####  4.1.1 Solution of the X-equation

Consider first the X-equation (38)

$H_{X}\Phi_{X}\equiv-{1\over 2}\left[{\partial^{2}\over\partial r^{2}}+{7\over
r}{\partial\over\partial r}-\omega^{2}r^{2}+{1\over
r^{2}}\left(-6\tan[3\,\psi]{\partial\over\partial\psi}+{\partial^{2}\over\partial\psi^{2}}\right)\right]\Phi_{X}=\epsilon_{X}\Phi_{X}$
(41)

and the matrix elements

$\langle\Phi^{\prime}_{X}|O_{X}|\Phi_{X}\rangle=\int_{0}^{\infty}\\!\\!\\!\\!dr\,r^{7}\int_{0}^{2\pi}\\!\\!\\!\\!d\psi\,\cos^{2}[3\,\psi]\
\ \Phi^{\prime}_{X}\ O_{X}[X]\ \Phi_{X}$ (42)

In terms of the new coordinates

$s_{11}=X_{a}X_{a}=r^{2}~{},\quad\quad\quad\quad
s_{111}=d_{abc}X_{a}X_{b}X_{c}={1\over\sqrt{3}}r^{3}\sin[3\psi]$ (43)

which are the $s_{11}$ and $s_{111}$ components of the 6- and 10-component
symmetric tensors

$s^{++}_{[2]ij}[A]:=A_{ai}A_{aj}~{},\quad\quad\quad\quad
s^{--}_{[3]ijk}[A]:=d_{abc}A_{ai}A_{bj}A_{ck}$ (44)

and using the scaled

$\overline{y}:=s_{111}/s_{11}^{3/2}={1\over\sqrt{3}}\sin[3\,\psi]$

Equ. (41) reads

$-{1\over 2}\left[4s_{11}{\partial^{2}\over\partial
s_{11}^{2}}+16{\partial\over\partial s_{11}}-s_{11}\ \omega^{2}+{1\over
s_{11}}\left(3(1-3\overline{y}^{2}){\partial^{2}\over\partial\overline{y}^{2}}-27\overline{y}{\partial\over\partial\overline{y}}\right)\right]\Phi_{X}=\epsilon_{X}\Phi_{X}$

which can easily be solved by separation of variables. The solutions can be
written

$\Phi^{X}_{n_{1},n_{2}}[s_{11},s_{111}]={\omega^{2}\over\sqrt{6\pi}}\
p^{(1)}_{n_{1},n_{2}}\left(\omega\,s_{11},\omega^{3/2}\,s_{111}\right)\exp{[-\omega\,s_{11}/2]}$
(45)

with the energy eigenvalues

$\epsilon_{n_{1},n_{2}}=\left(4+2n_{1}+3n_{2}\right)\omega=:\left(4+n_{X}\right)\omega$
(46)

The lowest polynomials read

$\displaystyle p^{(1)}_{0,0}(x,y)=1~{},\quad\quad p^{(1)}_{1,0}(x,y)={1\over
2}(-4+x)~{},\quad\quad p^{(1)}_{0,1}(x,y)={1\over\sqrt{10}}\,y~{},$ (47)
$\displaystyle p^{(1)}_{2,0}(x,y)={1\over
2\sqrt{10}}(20-10x+x^{2})~{},\quad\quad
p^{(1)}_{1,1}(x,y)={1\over\sqrt{70}}(-7+x)y~{},\quad...$

The number $n_{X}:=2n_{1}+3n_{2}$ is the degree of the corresponding
polynomial in the components of $X$ for each solution. After Gram-Schmidt-
orthogonalisation of those degenerate in energy, they an ONB with respect to
the measure (42).

We mention here, that the X-equation (38), due to the special form of the FP-
Jacobian (16), can be considered also for the case of singular boundary
conditions. The lowest solutions for this singular case and their energy
eigenvalues are shown in Appendix C. Although very interesting, we shall limit
ourselves in this article on the regular case.

####  4.1.2 Solution of the Y-equation

For functonals $\Phi_{Y}[s_{22},s_{222}]$ depending only on the components

$\displaystyle s_{22}=Y_{a}Y_{a}~{},\quad\quad\quad\quad
s_{222}=d_{abc}Y_{a}Y_{b}Y_{c}~{},$ (48)

of the symmetric tensors (44), satisfying
$T^{Y}_{a}s_{22}=T^{Y}_{a}s_{222}=0$, i.e
$T^{Y}_{a}\Phi_{Y}[s_{22},s_{222}]=0~{},a=1,...,8,$ and the Y-Equ.(39)

$\displaystyle
H_{Y}\Phi_{Y}[s_{22},s_{222}]=\epsilon_{Y}\Phi_{Y}[s_{22},s_{222}]$ (49)

is solved by the same functionals (45), and in particular polynomials, and
energy eigenvalues as for the X-Equ.(38), but with $s_{11}$ and $s_{111}$
replaced by $s_{22}$ and $s_{222}$ respectively. They form an ONB with respect
to the measure

$\langle\Phi^{\prime}_{Y}|O_{Y}|\Phi_{Y}\rangle=\int_{0}^{\infty}dY_{4}Y_{4}\int_{0}^{\infty}dY_{6}Y_{6}\prod_{a=1,2,3,8}\left[\int_{0}^{\infty}dY_{a}\right]\
\Phi^{\prime}_{Y}\,O_{Y}[Y]\,\Phi_{Y}~{}.$ (50)

####  4.1.3 Solution of the Z-equation

Similarly for the Z-equation with $\Phi_{Z}[s_{33},s_{333}]$ depending only on
the components

$\displaystyle s_{33}=Z_{a}Z_{a}~{},\quad\quad\quad\quad
s_{333}=d_{abc}Z_{a}Z_{b}Z_{c}~{},$ (51)

of the symmetric tensors (44), satisfying
$T^{Z}_{a}s_{33}=T^{Z}_{a}s_{333}=0$, i.e.
$T^{Z}_{a}\Phi_{Z}[s_{33},s_{333}]=0~{},a=1,...,8~{},$ and the Z-Equ.(40)

$\displaystyle
H_{Z}\Phi_{Z}[s_{33},s_{333}]=\epsilon_{Z}\Phi_{Z}[s_{33},s_{333}]~{},$ (52)

solved by the same functionals (45), and in particular polynomials, and energy
eigenvalues as for the X-Equ.(38), but with $s_{11}$ and $s_{111}$ replaced by
$s_{33}$ and $s_{333}$ respectively. They form an ONB with respect to the
measure

$\langle\Phi^{\prime}_{Z}|O_{Z}|\Phi_{Z}\rangle=\prod_{a=1,...,8}\left[\int_{0}^{\infty}dZ_{a}\right]\
\Phi^{\prime}_{Z}\,O_{Z}[Z]\,\Phi_{Z}~{}.$ (53)

####  4.1.4 Trigonal form of the one-direction Hamiltonian

The polynomials can be easily obtained using the equation

$\displaystyle D^{(1)}\ p^{(1)}_{n_{1},n_{2}}(x,y)=(\epsilon/\omega)\
p^{(1)}_{n_{1},n_{2}}(x,y)~{},$ (54)

with the differential operator $D^{(1)}:=D_{-2}^{(1)}+D_{0}^{(1)}$ consisting
of two parts

$\displaystyle D_{-2}^{(1)}:=-{3\over
2}x^{2}\partial_{y}^{2}-2\left(x\partial_{x}+3y\partial_{y}+4\right)\partial_{x}~{},$
$\displaystyle D_{0}^{(1)}:=\left(2x\partial_{x}+3y\partial_{y}+4\right)~{},$
(55)

acting in the space of monomials $x^{n_{1}}y^{n_{2}}$. The operator
$D_{0}^{(1)}$ reproduces the given monomial with eigenvalue
$(2n_{1}+3n_{2}+4)$ where $(2n_{1}+3n_{2})$ is its power seen as a homogeneous
polynomial in the $A$ (”A-power”), attributing $x$ the A-power $2$ and $y$ the
A-power $3$, as is the case e.g. for $x=\omega s_{11}(A)$ and
$y=\omega^{3/2}s_{111}(A)$ in Equ.(43). On the other hand, the operator
$D_{=2}^{(1)}$ transforms one $x^{n_{1}}y^{n_{2}}$ into another one,
$x^{m_{1}}y^{m_{2}}$, with an by two lowered A-power
$(2m_{1}+3m_{2})=(2n_{1}+3n_{2})-2$. Hence the operator $D^{(1)}$ is
triangular in the space of monomials $x^{n_{1}}y^{n_{2}}$ and can easily be
diagonalised with eigenvalues

$\displaystyle\epsilon=(2n_{1}+3n_{2}+4)\omega~{},$ (56)

and the eigenfunctions the polynomials

$\displaystyle p^{(1)}_{n_{1},n_{2}}(x,y)={\cal
N}\left(\sum_{m_{1},m_{2}}^{2m_{1}+3m_{2}<2n_{1}+3n_{2}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[{\cal
A}^{(1)}_{n_{1},n_{2}}(m_{1},m_{2})\
x^{m_{1}}y^{m_{2}}\right]+x^{n_{1}}y^{n_{2}}\right)~{},$ (57)

with the leading (”defining”) monomial $x^{n_{1}}y^{n_{2}}$ and a ”tail” of
monomials of decreasing, lower powers with some definite coefficients ${\cal
A}^{(1)}_{n_{1},n_{2}}(m_{1},m_{2})$.

We shall see in the folling paragraphs, that such behaviour will also for the
case of solutions separable in only one space direction and the non separable
general case. Before discussing these we shall give in the following
subsection some examples of solution build from those separable in all
$X,Y,Z$.

####  4.1.5 Putting the solutions of the X-,Y, and Z-equations together

Together, we have

$H_{h.o.}\Phi_{X|Y|Z}=\left(\epsilon_{X}+\epsilon_{Y}+\epsilon_{Z}\right)=\epsilon_{h.o.}\,\Phi_{X|Y|Z}~{}.$
(58)

with the energy eigenvalues

$\displaystyle\epsilon_{h.o.}=\left(12+n_{X}+n_{Y}+n_{Z}\right)\omega=\left(12+n\right)\omega$
(59)

We find the lowest solutions $\
\Phi_{[n]X|Y|Z}[A]=P_{[n]X|Y|Z}\exp[-\omega\left(s_{11}+s_{22}+s_{33}\right)/2]$
with

$\displaystyle\epsilon_{h.o.}=12\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[0]\,X|Y|Z}\propto
p_{0,0}[X]\,p_{0,0}[Y]\,p_{0,0}[Z]=1~{},$
$\displaystyle\epsilon_{h.o.}=14\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[2]\,X|Y|Z}\propto
p_{1,0}[X]\,p_{0,0}[Y]\,p_{0,0}[Z]=\left(-2+\omega
s_{11}/2\right)~{},\quad{\rm and\ perm.}$
$\displaystyle\epsilon_{h.o.}=15\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[3]\,X|Y|Z}\propto
p_{0,1}[X]\,p_{0,0}[Y]\,p_{0,0}[Z]=\omega^{3/2}s_{111}/\sqrt{10}~{},\quad{\rm
and\ perm.}$ $\displaystyle\epsilon_{h.o.}=16\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[4]1\,X|Y|Z}\propto
p_{2,0}[X]\,p_{0,0}[Y]\,p_{0,0}[Z]=\left(10-5\,\omega
s_{11}+\omega^{2}s_{11}^{2}/2\right)/\sqrt{10}~{},\ {\rm and\ perm.}$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[4]2\,X|Y|Z}\propto
p_{0,0}[X]\,p_{1,0}[Y]\,p_{1,0}[Z]=4-\omega(s_{22}+s_{33})+\omega^{2}s_{22}s_{33}/4~{},\
{\rm and\ perm.}$ $\displaystyle\epsilon_{h.o.}=17\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[5]1\,X|Y|Z}\propto
p_{1,1}[X]\,p_{0,0}[Y]\,p_{0,0}[Z]=\left(-7+\omega
s_{11}\right)\omega^{3/2}\,s_{111}/\sqrt{70}~{},\ {\rm and\ perm.}$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[5]2\,X|Y|Z}\propto
p_{0,1}[X]\,p_{1,0}[Y]\,p_{0,0}[Z]=\left(-2+\omega
s_{22}/2\right)\omega^{3/2}s_{111}/\sqrt{10}~{},\ {\rm and\ perm.}$
$\displaystyle...$ (60)

The number $n$ is the maximal power of the polynomial $P_{[}n]$, seen as a
polynomial in the reduced gauge field $A$. Superposing (60), and denoting

$s_{[2]}^{(0)}:=s_{[2]ii}=s_{11}+s_{22}+s_{33}=\left(A_{ai}\right)^{2}$

we can build the $0^{++}$ eigenstates
$\Phi_{[n]}^{(0)++}[A]=P_{[n]}^{(0)++}[A]\exp[-\omega\left(A_{ai}\right)^{2}/2],$
with

$\displaystyle\epsilon^{(0)++}_{h.o.}=12\,\omega~{}:\quad
P_{[0]}^{(0)++}\\!\\!\\!$ $\displaystyle\propto$ $\displaystyle\\!\\!\\!1~{},$
$\displaystyle\epsilon^{(0)++}_{h.o.}=14\,\omega~{}:\quad
P_{[2]}^{(0)++}\\!\\!\\!$ $\displaystyle\propto$
$\displaystyle\\!\\!\\!-12+\omega\,s^{(0)}_{[2]}~{},$
$\displaystyle\epsilon^{(0)++}_{h.o.}=16\,\omega~{}:\quad
P_{[4]}^{(0)++}\\!\\!\\!$ $\displaystyle\propto$
$\displaystyle\\!\\!\\!108-18\,\omega\,s^{(0)}_{[2]}+\,\omega^{2}\\!\left(s^{(0)}_{[2]}\right)^{2}.$
(61)

We find rotational invariant solutions although the flux-tube gauge is not
rotational invariant.

We mention here, that for the case of singular solutions of the the X-equation
(38), discussed in Appendix C, rotational invariance is broken in one
direction, here the x-direction, leaving only a cylindrical symmetry.

###  4.2 Separation of one of the three directions

####  4.2.1 Separation of the Y-equation

Starting with the case where the y-direction decouples

$\displaystyle\Phi_{XZ|Y}[X,Y,Z]=\Phi_{Y}[Y]\ \Phi_{XZ}[X,Z]~{},$
$\displaystyle
H_{h.o.}\Phi_{Y|XZ}[X,Y,Z]=\left(H_{Y}+H_{XZ}\right)\Phi_{Y|XZ}[X,Y,Z]=(\epsilon_{Y}+\epsilon_{XZ})\Phi_{Y|XZ}[X,Y,Z]~{}.$
(62)

where $\Phi_{Y}$ satisfies the above discussed constrained Y-equation (39) and
$\Phi_{XZ}$ is to solve the constrained x-z-direction Schroedinger equation

$\displaystyle\left[H_{X}+H_{Z}+{1\over
2\,r^{2}}\\!\\!\left[{\left(T_{1}^{Z}\right)^{2}+\left(T_{2}^{Z}\right)^{2}\over\cos^{2}{\psi}}+{\left(T_{4}^{Z}\right)^{2}+\left(T_{5}^{Z}\right)^{2}\over\cos^{2}{[\psi+2\pi/3]}}+{\left(T_{6}^{Z}\right)^{2}+\left(T_{7}^{Z}\right)^{2}\over\cos^{2}{[\psi+4\pi/3]}}\right]\\!\right]\\!\\!\Phi_{XZ}$
$\displaystyle=$ $\displaystyle\epsilon_{XZ}\Phi_{XZ}~{},$
$\displaystyle\wedge\quad T_{3}^{Z}\Phi_{XZ}=0\quad\wedge\quad
T_{8}^{Z}\Phi_{XZ}=0~{}.\quad\quad\quad\quad\quad\quad\quad\quad$ (63)

In terms of the components $s_{13}$, $s_{113}$, and $s_{133}$ of the symmetric
tensor (44),

$\displaystyle s_{13}=X_{a}Z_{a}~{},\quad\quad
s_{113}=d_{abc}X_{a}X_{b}Z_{c}~{},\quad\quad
s_{133}=d_{abc}X_{a}Z_{b}Z_{c}~{},$ (64)

we find the $\epsilon_{XZ}=10\,\omega$ solution

$\displaystyle\Phi^{XZ}_{[2]}[X,Z]$ $\displaystyle\propto$
$\displaystyle\big{[}\omega\,s_{13}\big{]}\exp{[-\omega(s_{11}+s_{33})/2]}~{},$
(65)

and the $\epsilon_{XZ}=11\,\omega$ solutions

$\displaystyle\Phi^{XZ}_{[3]1}[X,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}\omega^{3/2}\,s_{113}\Big{]}\exp{[-\omega(s_{11}+s_{33})/2]}~{},$
$\displaystyle\Phi^{XZ}_{[3]2}[X,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}\omega^{3/2}\,s_{133}\Big{]}\exp{[-\omega(s_{11}+s_{33})/2]}~{},$
(66)

We have the $\epsilon^{(2)}=12\,\omega$ solutions

$\displaystyle\Phi^{XZ}_{[4]1}[X,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}2-{1\over
2}\omega(s_{11}+s_{33})+\omega^{2}s_{13}^{2}\Big{]}\exp{[-\omega(s_{11}+s_{33})/2]}~{},$
(67) $\displaystyle\Phi^{XZ}_{[4]2}[X,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}6-{3\over
2}\omega(s_{11}+s_{33})+\omega^{2}b_{22}\Big{]}\exp{[-\omega(s_{11}+s_{33})/2]}~{},$
(68)

using additionally the $22$-component

$\displaystyle b_{22}[X,Z]=f_{abc}\,f_{ade}\,X_{b}\,Z_{c}\,X_{d}\,Z_{e}~{},$
(69)

of the symmetric tensor

$\displaystyle b^{++}_{[4]ij}[A]$ $\displaystyle:=$ $\displaystyle
g^{-2}B^{\rm hom}_{a\,i}[A]\,B^{\rm hom}_{a\,j}[A]~{},$ (70)

It appears in the tail of the $\epsilon^{(2)}=14\,\omega$ solution

$\displaystyle\Phi^{XZ}_{[6]}[X,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}\omega^{2}\left({1\over
2}s_{11}s_{33}-s_{13}^{2}-b_{22}\right)+\omega^{3}s_{111}s_{133}\Big{]}\exp{[-\omega(s_{11}+s_{33})/2]}~{},$
(71)

with the product $s_{111}s_{133}$ as leading term, and is irreducible in
A-space.

In general, the X-Z-solutions are of the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Phi^{XZ}_{n_{1},...,n_{8}}[X,Z]\\!\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\\!{2\over
5}{\omega^{4}\over\sqrt{11}\pi^{3/2}}p^{(2)}_{n_{1},...,n_{8}}\left(\omega
s_{11},\omega s_{33},\omega
s_{13},\omega^{3/2}s_{111},\omega^{3/2}s_{333},\omega^{3/2}s_{113},\omega^{3/2}s_{133},\omega^{2}b_{22}\right)$
(72)
$\displaystyle\quad\quad\quad\quad\quad\quad\times\exp{[-\omega(s_{11}+s_{33})/2]}$

with energy

$\displaystyle\epsilon^{(2)}_{n_{1},...,n_{8}}:=\left[8+2(n_{1}+n_{2})+3(n_{3}+n_{4}+n_{5}+n_{6}+n_{7})+4n_{8}\right]\omega$
(73)

where $p^{(2)}_{n_{1},...,n_{8}}\left(x_{1},...,x_{8}\right)$ is a polynomial
in the eight variables $x_{1},...,x_{8}$ where the monomial with the maximal
order reads $\prod_{i=1}^{8}x_{i}^{n_{i}}$. After orthogonalisation of those
degenerate in energy) they form an ONB with respect to the measure

$\langle\Phi^{\prime}_{XZ}|O|\Phi_{XZ}\rangle=\int d\mu_{X}\int d\mu_{Z}\ \
\Phi^{\prime}_{XZ}\,O[X,Z]\,\Phi_{XZ}~{}.$ (74)

####  4.2.2 Separation of the Z-equation

Next we consider the case where the z-direction decouples

$\displaystyle\Phi_{Z|XY}[X,Y,Z]=\Phi_{Z}[Z]\ \Phi_{XY}[X,Y]~{},$
$\displaystyle
H_{h.o.}\Phi_{Z|XY}[X,Y,Z]=\left(H_{Z}+H_{XY}\right)\Phi_{Z|XY}[X,Y,Z]=(\epsilon_{Z}+\epsilon_{XY})\Phi_{Z|XY}[X,Y,Z]~{}.$
(75)

where $\Phi_{Z}$ satisfies the above discussed constrained Z-Schrödinger-
equation (40) and functional $\Phi_{XY}$ is to solve the unconstrained x-y-
direction Schrödinger equation

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{XY}\Phi_{XY}\\!\\!\\!\\!$
$\displaystyle\equiv$ $\displaystyle\\!\\!\\!\\!\Bigg{[}H_{X}+H_{Y}+{1\over
2\,r^{2}(Y_{4}Y_{6})}\Bigg{[}{\widetilde{T}_{1}^{Y\dagger}\,Y_{4}Y_{6}\,\widetilde{T}^{Y}_{1}+\widetilde{T}_{2}^{Y\dagger}\,Y_{4}Y_{6}\,\widetilde{T}^{Y}_{2}\over\cos^{2}{\psi}}+$
(76)
$\displaystyle+{\widetilde{T}_{4}^{Y\dagger}\,Y_{4}Y_{6}\,\widetilde{T}^{Y}_{4}+\widetilde{T}_{5}^{Y\dagger}\,Y_{4}Y_{6}\,\widetilde{T}^{Y}_{5}\over\cos^{2}{[\psi+2\pi/3]}}+{\widetilde{T}_{6}^{Y\dagger}\,Y_{4}Y_{6}\,\widetilde{T}^{Y}_{6}]+\widetilde{T}_{7}^{Y\dagger}\,Y_{4}Y_{6}\,\widetilde{T}^{Y}_{7}\over\cos^{2}{[\psi+4\pi/3]}}\Bigg{]}\Bigg{]}\Phi_{XY}=\epsilon_{XY}\Phi_{XY}$

Using also the additional components

$\displaystyle s_{12}=X_{a}Y_{a}~{},\quad
s_{112}=d_{abc}X_{a}X_{b}Y_{c}~{},\quad
s_{122}=d_{abc}X_{a}Y_{b}Y_{c}~{},\quad
b_{33}[X,Y]=f_{abc}\,f_{ade}\,X_{b}\,Y_{c}\,X_{d}\,Y_{e}~{},$ (77)

the x-y-solutions are of the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Phi^{XY}_{n_{1},...,n_{8}}[X,Y]\\!\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\\!{2\over
5}{\omega^{4}\over\sqrt{11}\pi^{3/2}}p^{(2)}_{n_{1},...,n_{8}}\left(\omega
s_{11},\omega s_{22},\omega
s_{12},\omega^{3/2}s_{111},\omega^{3/2}s_{222},\omega^{3/2}s_{112},\omega^{3/2}s_{122},\omega^{2}b_{33}\right)$
(78)
$\displaystyle\quad\quad\quad\quad\quad\quad\times\exp{[-\omega(s_{11}+s_{22})/2]}$

and energy

$\displaystyle\epsilon^{(2)}_{n_{1},...,n_{8}}:=\left[8+2(n_{1}+n_{2})+3(n_{3}+n_{4}+n_{5}+n_{6}+n_{7})+4n_{8}\right]\omega$
(79)

where $p^{(2)}_{n_{1},...,n_{8}}$ are the same polynomials as for the x-z
case. After orthogonalisation of those degenerate in energy) they form an ONB
with respect to the measure

$\langle\Phi^{\prime}_{XY}|O|\Phi_{XY}\rangle=\int d\mu_{X}\int d\mu_{Y}\ \
\Phi^{\prime}_{XY}\,O[X,Y]\,\Phi_{XY}~{}.$ (80)

####  4.2.3 Separation of the X-equation

Finally, we consider the case where the x-direction decouples

$\displaystyle\Phi_{X|YZ}[X,Y,Z]=\Phi_{X}[X]\ \Phi_{YZ}[Y,Z]~{},$
$\displaystyle
H_{h.o.}\Phi_{X|YZ}[X,Y,Z]=\left(H_{X}+H_{YZ}\right)\Phi_{X|YZ}[X,Y,Z]=\left(\epsilon_{X}+\epsilon_{YZ}\right)\Phi_{X|YZ}[X,Y,Z]~{}.$
(81)

where $\Phi_{X}$ satisfies the above discussed unconstrained X-equation (38)
and the functionals $\Phi_{YZ}$ have to solve the constrained Y-Z- equation

$\displaystyle H_{YZ}\Phi_{YZ}\equiv\Bigg{[}H_{Y}+H_{Z}\\!\\!\\!\\!$
$\displaystyle+$ $\displaystyle\\!\\!\\!\\!{1\over 2Y_{4}^{2}}\left[{1\over
2}\left(T^{Z}_{3}+{1\over\sqrt{3}}T^{Z}_{8}\right)-T^{Y}_{3}\right]\left(T^{Z}_{3}+{1\over\sqrt{3}}T^{Z}_{8}\right)$
$\displaystyle+{1\over 2Y_{6}^{2}}\left[{1\over
2}\left(T^{Z}_{3}-{1\over\sqrt{3}}T^{Z}_{8}\right)-T^{Y}_{3}\right]\left(T^{Z}_{3}-{1\over\sqrt{3}}T^{Z}_{8}\right)\Bigg{]}\Phi_{YZ}=\epsilon_{YZ}\,\Phi_{YZ}~{},$
$\displaystyle\wedge\quad\left(\widetilde{T}^{Y}_{a}+\widetilde{T}^{Z}_{a}\right)\Phi_{YZ}$
$\displaystyle=$ $\displaystyle 0~{},\quad a=1,2,4,5,6,7~{}.$ (82)

In terms of the components

$\displaystyle s_{23}=Y_{a}Z_{a}~{},\quad
s_{223}=d_{abc}Y_{a}Y_{b}Z_{c}~{},\quad
s_{233}=d_{abc}Y_{a}Z_{b}Z_{c}~{},\quad
b_{11}[Y,Z]=f_{abc}\,f_{ade}\,Y_{b}\,Z_{c}\,Y_{d}\,Z_{e}~{},$ (83)

the y-z-solutions are of the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Phi^{YZ}_{n_{1},...,n_{8}}[Y,Z]\\!\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\\!{2\over
5}{\omega^{4}\over\sqrt{11}\pi^{3/2}}p^{(2)}_{n_{1},...,n_{8}}\left(\omega
s_{22},\omega s_{33},\omega
s_{23},\omega^{3/2}s_{222},\omega^{3/2}s_{333},\omega^{3/2}s_{223},\omega^{3/2}s_{233},\omega^{2}b_{11}\right)$
(84)
$\displaystyle\quad\quad\quad\quad\quad\quad\times\exp{[-\omega(s_{22}+s_{33})/2]}$

and energy

$\displaystyle\epsilon^{(2)}_{n_{1},...,n_{8}}:=\left[8+2(n_{1}+n_{2})+3(n_{3}+n_{4}+n_{5}+n_{6}+n_{7})+4n_{8}\right]\omega$
(85)

where $p^{(2)}_{n_{1},...,n_{8}}$ are the same polynomials as for the x-z
case. After orthogonalisation of those degenerate in energy) they form an ONB
with respect to the measure

$\langle\Phi^{\prime}_{YZ}|O|\Phi_{YZ}\rangle=\int d\mu_{Y}\int d\mu_{Z}\ \
\Phi^{\prime}_{YZ}\,O[Y,Z]\,\Phi_{YZ}~{}.$ (86)

####  4.2.4 Trigonal form of the two-direction Hamiltonian

The polynomials

$p_{n_{1},...,n_{8}}^{(2)}(x_{11},x_{33},x_{13},x_{111},x_{333},x_{113},x_{133},b_{22})$
(87)

can be easily obtained using the equation

$\displaystyle\Big{[}D^{(1)}_{1}+D^{(1)}_{3}+D^{(2)}_{13}\Big{]}p^{(2)}=(\epsilon_{13}/\omega)\
p^{(2)}$ (88)

with the one-direction differential operators

$D^{(1)}_{1}:=D^{(1)}[x_{11},x_{111}]~{},\quad\quad\quad
D^{(1)}_{3}:=D^{(1)}[x_{33},x_{333}]$ (89)

and

$D^{(2)}_{13}:=D^{(2)}[x_{11},x_{33},x_{13},x_{111},x_{333},x_{113},x_{133},b_{22}]~{},$
(90)

with the two-direction operator $D^{(2)}$ shown explicitly in Appendix D. As
for the one-direction operators $D^{(1)}$, the two-direction operator
$D^{(2)}$ contains an A-power conserving part $D^{(2)}_{0}$ and a the A-power
by two lowering operator $D^{(2)}_{-2}$. Hence, the eigenvalue problem (88) is
triangular in the space of monomials $M(x_{1},...,x_{8})$ in $x_{1},...,x_{8}$
and can easily be diagonalised with eigenvalues

$\displaystyle\epsilon=\left(8+2(n_{1}+n_{2})+3(n_{3}+n_{4}+n_{5}+n_{6}+n_{7})+4n_{8}\right)\omega\equiv\left(8+n\right)\omega$
(91)

The eigenfunctions are polynomials

$\displaystyle p^{(2)}_{n_{1},...,n_{8}}={\cal
N}\left(\sum_{m_{1},...,m_{2}}^{m<n}\\!\\!\\!\\!\ \left[{\cal
A}^{(1)}_{n_{1},...,n_{8}}(m_{1},...,m_{8})\ x_{1}^{m_{1}}\cdots
x_{8}^{m_{8}}\right]+x_{1}^{n_{1}}\cdots x_{8}^{n_{8}}\right)$ (92)

with the leading ”defining” monomial $\prod_{i=1}^{8}x_{i}^{n_{i}}$ of maximal
order and a ”tail” of monomials of decreasing, lower powers with some definite
coefficients ${\cal A}^{(1)}_{n_{1},...,n_{8}}(m_{1},...,m_{8})$.

####  4.2.5 Some examples of solutions

Together, we have

$H_{h.o.}\Phi_{X|YZ}=\left(\epsilon_{X}+\epsilon_{YZ}\right)\Phi_{X|YZ}=\epsilon_{h.o.}\,\Phi_{X|YZ}~{},\quad{\rm
and\ cycl.\ perm.}$ (93)

with the energy eigenvalues

$\displaystyle\epsilon_{h.o.}=\left(12+n_{X}+n_{YZ}\right)\omega=\left(12+n\right)~{},\omega\quad{\rm
and\ cycl.\ perm.}$ (94)

The lowest solutions are
$\Phi_{[n]\,X|YZ}[A]=P_{[n]\,X|YZ}[A]\exp[-\omega\left(A_{ai}\right)^{2}/2]$
and cycl. perm., with

$\displaystyle\epsilon_{h.o.}=14\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[2]\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{0,0,1,0,0,0,0,0}[Y,Z]\propto\omega
s_{23}~{},\quad{\rm and\ cycl.\ perm.}$
$\displaystyle\epsilon_{h.o.}=15\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[3]\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{0,0,0,0,0,1,0,0}[Y,Z]\propto\omega^{3/2}s_{223}~{},\quad{\rm
and\ cycl.\ perm.}$ $\displaystyle\\!\\!\\!\\!\\!\\!P_{[3]\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{0,0,0,0,0,0,1,0}[Y,Z]\propto\omega^{3/2}s_{233}~{},\quad{\rm
and\ cycl.\ perm.}$ $\displaystyle\epsilon_{h.o.}=16\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[4]1\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{0,0,0,0,0,0,0,1}[Y,Z]\propto
6-3\omega(s_{22}+s_{33})/2+\omega^{2}b_{11}~{},\ {\rm and\ cycl.\ perm.}$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[4]2\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{0,0,2,0,0,0,0,0}[Y,Z]\propto
2-\omega(s_{22}+s_{33})/2+\omega^{2}s_{23}^{2}~{},\ {\rm and\ cycl.\ perm.}$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[4]3\,X|YZ}\propto
p^{(1)}_{1,0}[X]\,p^{(2)}_{0,0,1,0,0,0,0,0}[Y,Z]\propto(-2+\omega
s_{11}/2)\,\omega s_{23}~{},\ {\rm and\ cycl.\ perm.}$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[4]4\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{1,0,1,0,0,0,0,0}[Y,Z]\propto-5\,\omega
s_{23}+\omega^{2}s_{22}\,s_{23}~{},\ {\rm and\ perm.}$
$\displaystyle\epsilon_{h.o.}=17\,\omega~{}:$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[5]1\,X|YZ}\propto
p^{(1)}_{1,0}[X]\,p^{(2)}_{0,0,0,0,0,1,0,0}[Y,Z]\propto(-2+\omega
s_{11}/2)\,\omega^{3/2}s_{223}~{},\ {\rm and\ cycl.\ perm.}$
$\displaystyle\\!\\!\\!\\!\\!\\!P_{[5]2\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{1,0,0,0,0,1,0,0}[Y,Z]\propto-6\,\omega^{3/2}\,s_{223}+\omega^{5/2}s_{22}\,s_{223}~{},\
{\rm and\ cycl.\ perm.}$ $\displaystyle\\!\\!\\!\\!\\!\\!P_{[5]3\,X|YZ}\propto
p^{(1)}_{0,0}[X]\,p^{(2)}_{0,1,0,0,0,1,0,0}[Y,Z]\propto-5\,\omega^{3/2}\,s_{223}+\omega^{5/2}s_{33}\,s_{223}~{},\
{\rm and\ cycl.\ perm.}$ $\displaystyle...$ (95)

By superposiition, we obtain from (95) and (60), all spin-0 solutions up to
polynomial order $n=4$, namely
$\Phi^{(0)++}_{[n]}[A]=P^{(0)++}_{[n]}[A]\exp[-\omega\left(A_{ai}\right)^{2}/2]$
, with

$\displaystyle\epsilon^{(0)++}_{h.o.}=12\,\omega~{}:$ $\displaystyle
P_{[0]}^{(0)++}[A]\propto 1~{},$
$\displaystyle\epsilon^{(0)++}_{h.o.}=14\,\omega~{}:$ $\displaystyle
P_{[2]}^{(0)++}[A]\propto\left[-12+\omega\,s^{(0)}_{[2]}\right]~{},$
$\displaystyle\epsilon^{(0)++}_{h.o.}=16\,\omega~{}:$ $\displaystyle
P_{[4]1}^{(0)++}[A]\propto\left[108-18\,\omega\,s^{(0)}_{[2]}+\,\omega^{2}\\!\left(s^{(0)}_{[2]}\right)^{2}\right]~{},$
(96) $\displaystyle P_{[4]2}^{(0)++}[A]\propto\left[72-12\,\omega
s^{(0)}_{[2]}+\omega^{2}s_{[2]ij}s_{[2]ij}\right]~{},$ $\displaystyle
P_{[4]3}^{(0)++}[A]\propto\left[18-3\,\omega\,s^{(0)}_{[2]}+\omega^{2}b^{(0)}_{[4]}\right]~{},$

completing those of (61) up to polynomial order $n=4$ in $A$.

Furthermore, using the notation
$s^{(2)}_{[2]\,ij}:=(s_{[2]\,ij}-\delta_{ij}s_{[2]kk}/3)$, we can build first
spin-2 solutions
$\Phi^{(2)++}_{ij}[A]=P^{(2)++}_{ij}[A]\exp[-\omega\left(A_{ak}\right)^{2}/2]$
with

$\displaystyle\epsilon_{h.o.}=14\,\omega~{}:$ $\displaystyle\quad
P^{(2)++}_{[2]\,ij}[A]\propto\left[\omega\,s^{(2)}_{[2]\,ij}\right]~{},$
$\displaystyle\epsilon_{h.o.}=16\,\omega~{}:$ $\displaystyle\quad
P^{(2)++}_{[4]\,ij}[A]\propto\left[-14\,\omega\,s^{(2)}_{[2]\,ij}+\omega^{2}s^{(0)}_{[2]}\,s^{(2)}_{[2]\,ij}\right]~{},$
(97)

and first vector solutions
$\Phi^{(1)--}_{i}[A]=P^{(1)--}_{i}[A]\exp[-\omega\left(A_{ak}\right)^{2}/2]$,
denoting $v_{[3]i}:=s_{[3]ijj}$,

$\displaystyle\epsilon_{h.o.}=15\,\omega~{}:$ $\displaystyle\quad
P^{(1)--}_{[3]\,i}[A]\propto\left[\omega^{3/2}\,v_{[3]i}\right]~{},$
$\displaystyle\epsilon_{h.o.}=17\,\omega~{}:$ $\displaystyle\quad
P^{(1)--}_{[5]\,i}[A]\propto\left[-15\,\omega^{3/2}\,v_{[3]i}+\,\omega^{5/2}\,s^{(0)}_{[2]}\,v_{[3]i}\right]~{}.$
(98)

In order to obtain all solutions, we have to consider the general case of non-
separable solutions, discussed in the next paragraph.

###  4.3 Solutions depending on all three directions

Finally we consider the completely non-saparable case

$H\Phi[X,Y,Z]=\epsilon\,\Phi[X,Y,Z]~{}.$ (99)

####  4.3.1 Inclusion of the components $s_{123}$ and
$(b_{23},b_{13},b_{12})$.

Including the component $s_{123}$ of the symmetric 3-tensor $s_{[3]\,ijk}[A]$,
defined in (44),

$\displaystyle s_{123}$ $\displaystyle=$ $\displaystyle
d_{abc}X_{a}Y_{b}Z_{c}~{},$

depending on all three space directions, we obtain the further
$\epsilon_{h.o.}=15\,\omega$ solution

$\displaystyle\Phi_{[3]}[X,Y,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}\omega^{3/2}\,s_{123}\Big{]}\exp{[-\omega(s_{11}+s_{22}+s_{33})/2]}~{}.$

From this, (60), (95), we have the spin-3 solution
$\Phi^{(3)--}_{ijk}[A]=P^{(3)--}_{ijk}[A]\exp[-\omega\left(A_{ak}\right)^{2}/2]$
with

$\displaystyle\epsilon_{h.o.}=15\,\omega~{}:$ $\displaystyle\quad
P^{(3)--}_{[3]\,ijk}[A]\propto\left[\omega^{3/2}\,s^{(3)}_{[3]\,ijk}\right]~{},$
(100)

with the spin-3 part $s^{(3)}_{[3]ijk}$ of the symmetric 3-tensor
$s_{[3]ijk}$, see Appendix E.

Including finally the components $b_{23},b_{13},b_{12}$ of the symmetric
4-tensor $b_{[4]\,ij}[A]$, defined in (70),

$\displaystyle\quad\quad
b_{23}=f_{abc}\,f_{ade}\,X_{b}\,Z_{c}\,X_{d}\,Y_{e}~{},\quad{\rm and\ cycl.\
perm.}\ \ b_{13},b_{12}~{},$

irreducible in A-space, we obtain the triplet of $\epsilon_{h.o.}=16\,\omega$
solutions

$\displaystyle\Phi_{[4]}[X,Y,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}-3\,\omega
s_{23}+\omega^{2}\,b_{23}\Big{]}\exp{[-\omega(s_{11}+s_{22}+s_{33})/2]}~{},\quad{\rm
and\ cycl.\ perm.}$

Noting also the triplet of $\epsilon_{h.o.}=16\,\omega$ solutions

$\displaystyle\Phi_{[4]}[X,Y,Z]$ $\displaystyle\propto$
$\displaystyle\Big{[}\omega
s_{23}+\omega^{2}\,s_{12}s_{13}\Big{]}\exp{[-\omega(s_{11}+s_{22}+s_{33})/2]}~{},\quad{\rm
and\ cycl.\ perm.}$

we obtain all spin-2 solutions
$\Phi^{(2)++}_{[n]ij}[A]=P^{(2)++}_{[n]ij}[A]\exp[-\omega\left(A_{ak}\right)^{2}/2]$
up to $n=4$ with

$\displaystyle\epsilon_{h.o.}=14\,\omega~{}:$ $\displaystyle\quad
P^{(2)++}_{[2]\,ij}[A]\propto\left[\omega\,s^{(2)}_{[2]\,ij}\right]~{},$
$\displaystyle\epsilon_{h.o.}=16\,\omega~{}:$ $\displaystyle\quad
P^{(2)++}_{[4]\,ij}[A]\propto\left[-14\,\omega\,s^{(2)}_{[2]\,ij}+\omega^{2}s^{(0)}_{[2]}\,s^{(2)}_{[2]\,ij}\right]~{},$
(101) $\displaystyle\quad
P^{(2)++}_{[4]\,ij}[A]\propto\left[\omega\,s^{(2)}_{[2]\,ij}+\omega^{2}\left(s_{[2]ik}\,s_{[2]kj}\right)^{(2)}\right]~{},$
$\displaystyle\quad
P^{(2)++}_{[4]\,ij}[A]\propto\left[\omega\,s^{(2)}_{[2]\,ij}+\omega^{2}\,b^{(2)}_{[4]\,ij}\right]~{}.$

completing (97) up to polynomial order $n=4$ in A.

####  4.3.2 Inclusion of the further irrreducible vector $b_{[5]i}^{--}[A]$.

Similarly, we have the vector solutions
$\Phi^{(1)--}_{i}[A]=P^{(1)--}_{i}[A]\exp[-\omega\left(A_{ak}\right)^{2}/2]$,
with

$\displaystyle\epsilon_{h.o.}=15\,\omega~{}:$ $\displaystyle\quad
P^{(1)--}_{[3]\,i}[A]\propto\left[\omega^{3/2}\,v_{[3]i}\right]~{},$
$\displaystyle\epsilon_{h.o.}=17\,\omega~{}:$ $\displaystyle\quad
P^{(1)--}_{[5]\,i}[A]\propto\left[-15\,\omega^{3/2}\,v_{[3]i}+\,\omega^{5/2}\,s^{(0)}_{[2]}\,v_{[3]i}\right]~{},$
(102) $\displaystyle\quad
P^{(1)--}_{[5]\,i}[A]\propto\left[-8\,\omega^{3/2}\,v_{[3]i}+\omega^{5/2}\,s_{[2]ij}\,v_{[3]j}\right]~{},$
$\displaystyle\quad
P^{(1)--}_{[5]\,i}[A]\propto\left[-9\,\omega^{3/2}\,v_{[3]i}+\omega^{5/2}\,s_{[3]ijk}\,s_{[2]jk}\right]~{},$
$\displaystyle\quad
P^{(1)--}_{[5]\,i}[A]\propto\left[\omega^{5/2}\,b_{[5]i}\right]~{},$

completing (98) up to polynomial order $n=5$ in $A$. The vector $b_{[5]i}$ in
(102) is defined as

$\displaystyle b_{[5]i}^{--}[A]\\!:=\\!d_{abc}B^{\rm hom}_{ai}\\!B^{\rm
hom}_{bi}\\!A_{ci}\\!+\\!{1\over
4}\\!\left(2s_{jk}s_{123}\\!-\\!s_{jj}s_{ikk}\\!-\\!s_{kk}s_{ijj}\right),\
(i\neq j\neq k)\\!\\!~{}.$ (103)

It appears e.g. in the tail of $\epsilon_{h.o.}=19\,\omega$ solution
$b_{[4]ij}\,v_{[3]j}$

$\displaystyle\\!\\!\\!\\!\\!P^{(1)--}_{[7]\,i}[A]\propto\Big{[}-{9\over
2}\,\omega^{3/2}\,v_{[3]i}+\omega^{5/2}\Big{(}-{5\over
4}\,s_{[2]}^{(0)}\,v_{[3]i}+{3\over 2}\,s_{[2]ij}\,v_{[3]j}-{1\over
4}\,s_{[3]ijk}\,s_{[2]jk}-5\,b_{[5]i}\Big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\omega^{7/2}\,b_{[4]ij}\,v_{[3]j}\Big{]}~{}.$
(104)

and is irreducible, i.e. not representable as the product of the components of
$s_{[2]},s_{[3]}$, and $b_{[4]}$. Considering the solutions of similar
products of components up to $n=10$, we find no further elementary tensors
beyond the four already included, the 6 components of the symmetric 2-tensor
$s_{[2]}$, the 10 components of the symmetric 3-tensor $s_{[3]}$, the 6
components of the further symmetric 2-tensor $b_{[4]}$, and the 3 components
of the vector $b_{[5]}$, not considering the axial sector so far.

####  4.3.3 Inclusion of the axial states

Indeed, using the axial scalar

$\displaystyle a_{[3]}^{-+}[A]:={1\over
6}\epsilon_{ijk}f_{abc}A_{ai}A_{bj}A_{ck}\equiv f_{abc}X_{a}Y_{b}Z_{c}~{},$
(105)

we obtain the $\epsilon=15\,\omega$ solution

$\displaystyle\Phi_{[3]}^{-+}[A]$ $\displaystyle:=$
$\displaystyle{\omega^{9/2}\over\sqrt{6\pi}}\
\Big{[}\omega^{3/2}\,a_{[3]}\Big{]}\exp{[-\omega(s_{11}+s_{22}+s_{33})/2]}~{}.$
(106)

Now, considering the tail of the $a_{[3]}v_{[3]i}$ solution with $n=6$,

$\displaystyle\Phi^{+-}_{[6]i}[A]$ $\displaystyle\propto$
$\displaystyle\Big{[}-{5\over
2}\omega^{2}\,a_{[4]i}+\omega^{3}\,a_{[3]}v_{[3]i}\Big{]}\exp{[-\omega\left(A_{ak}\right)^{2}/2]}~{}.$
(107)

we find that we have to inclued also the axial $n=4$ vector the axial vetor

$\displaystyle a_{[4]i}^{+-}[A]:=d_{abc}\,B^{\rm
hom}_{ai}A_{bi}A_{ci}~{},\quad(i=1,2,3)$ (108)

which is an irreducible polynomial $\epsilon=16\,\omega$ solution of 4-th
order in A. Furthermore, considering the tails of the $a_{[3]}b_{[4]}$
solutions with $n=7$,

$\displaystyle\Phi^{(0)-+}_{[7]}[A]$ $\displaystyle\propto$
$\displaystyle\Big{[}30\,\omega^{3/2}\,a_{[3]}+\omega^{5/2}\left(-4\,a_{[3]}s^{(0)}_{[2]}+3\,a^{(0)}_{[5]}\right)+\omega^{7/2}\,a_{[3]}b^{(0)}_{[4]}\Big{]}\exp{[-\omega\left(A_{ak}\right)^{2}/2]}~{}.$
(109) $\displaystyle\Phi^{(2)-+}_{[7]ij}[A]$ $\displaystyle\propto$
$\displaystyle\Big{[}\omega^{5/2}\left(2\,a_{[3]}s^{(2)}_{[2]ij}+{3\over
2}\,a^{(2)}_{[5]ij}\right)+\omega^{7/2}\,a_{[3]}b^{(2)}_{[4]ij}\Big{]}\exp{[-\omega\left(A_{ak}\right)^{2}/2]}~{}.$
(110)

we find that we have also to include the symmetric axial 2-tensor

$\displaystyle a_{[5]ij}^{-+}[A]:=d_{abc}\,B^{\rm
hom}_{ak}A_{bk}(d_{cde}\,A_{di}A_{ej})~{},\quad(i\leq j\wedge k\neq i,j)$
(111)

into the list of irreducible polynomials. Finally, considering the tails of
the $a_{[3]}b_{[5]}$ and $a_{[4]}b_{[4]}$ solutions with $n=8$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Phi^{+-}_{[8]i}[A]\propto\Big{[}\omega^{2}\,a_{[4]i}+\omega^{3}\left(-{1\over
4}\,s^{(0)}_{[2]}\,a_{[4]i}+{1\over 4}\,s_{[2]ij}\,a_{[4]j}-{1\over
2}\,a^{(1)}_{[6]i}\right)+\omega^{4}\,a_{[3]}b_{[5]i}\Big{]}\exp{[-\omega\left(A_{ak}\right)^{2}\\!\\!/2]}~{}.$
(112)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Phi^{(3)+-}_{[8]ijk}[A]\propto\Big{[}\omega^{3}\left({1\over
2}\,a_{[3]}\,s^{(3)}_{[3]ijk}+{7\over
4}\left(a_{[4]i}s_{[2]ij}\right)_{S}^{(3)}-a^{(3)}_{[6]ijk}\right)+\omega^{4}\left(a_{[4]i}b_{[4]ij}\right)_{S}^{(3)}\Big{]}\exp{[-\omega\left(A_{ak}\right)^{2}\\!\\!/2]}~{}.$
(113)

that we have, last but not least, to include also the symmetric axial 3-tensor

$\displaystyle a_{[6]ijk}^{+-}[A]:=d_{abc}B^{\rm hom}_{ai}B^{\rm
hom}_{bj}B^{\rm hom}_{ck}~{},\quad(i\leq j\leq k)~{},$ (114)

containing spin-1 and spin-3 parts, into the list of irreducible polynomials.
In addition to the four irreducible tensors $s_{[2]},s_{[3]},b_{[4]}$, and
$b_{[5]}$, the axial scalar $a_{[3]}$, the 3 componenst of the axial vector
$a_{[4]}$, the 6 components of the symmetric axial 2-tensor $a_{[5]}$,
containing another axial scalar and an axial spin-2 part, and finally the
symmetrix axial 3-tensor $a_{[6]}$ have to be included into the list of all
irreducible polynomials, in terms of which all polynomial solutions can be
represented. The complete list is shown in Table 1. Their transformation
properties under spatial rotations are summarised in Appendix E. Considering
the tails of all solutions from products of components of these eight
irreducible symmetric tensors, up to maximal order $n=12$, no further
irreducible polynomials have been found.

$\begin{array}[]{|c|c|c|}\hline\cr\quad\ {\rm sym.\
2\,tensor}\quad\\![2]\\!\\!&s_{[2]ij}^{++}[A]:=A_{ai}A_{aj}~{},\quad(i\leq
j)&\\!0^{++}\\!,2^{++}\\!\\\ \hline\cr\quad\ {\rm sym.\
3\,tensor}\quad\\![3]\\!\\!&s_{[3]ijk}^{--}[A]:=d_{abc}\,A_{ai}A_{bj}A_{ck}~{},\quad(i\leq
j\leq k)&\\!1^{--}\\!,3^{--}\\!\\!\\\ \hline\cr\quad\ {\rm sym.\
2\,tensor}\quad\\![4]\\!\\!&b_{[4]ij}^{++}[A]:=B^{\rm hom}_{ai}B^{\rm
hom}_{aj}~{},\quad(i\leq j)~{},\quad\ B^{\rm
hom}_{ai}:=(1/2)\epsilon_{ijk}\,f_{abc}\,A_{bj}A_{ck}&\\!0^{++}\\!,2^{++}\\!\\!\\\
\hline\cr\quad\quad\ \ {\rm
vector}\quad\quad\,[5]\\!\\!\\!\\!&\\!b_{[5]i}^{--}[A]\\!:=\\!d_{abc}B^{\rm
hom}_{ai}\\!B^{\rm hom}_{bi}\\!A_{ci}\\!+\\!{1\over
4}\\!\left(2s_{jk}s_{123}\\!-\\!s_{jj}s_{ikk}\\!-\\!s_{kk}s_{ijj}\right),(i\\!\neq\\!j\\!\neq\\!k)\\!\\!&1^{--}\\\
\hline\cr\quad\ \ {\rm axial\
scalar}\quad\,[3]\\!\\!\\!&a_{[3]}^{-+}[A]:=f_{abc}\,A_{a1}A_{b2}A_{c3}=B^{\rm
hom}_{a1}A_{a1}=B^{\rm hom}_{a2}A_{a2}=B^{\rm hom}_{a3}A_{a3}&0^{-+}\\\
\hline\cr\quad\ \ {\rm axial\
vector}\quad\,[4]\\!\\!&a_{[4]i}^{+-}[A]:=d_{abc}\,B^{\rm
hom}_{ai}A_{bi}A_{ci}~{},\quad(i=1,2,3)&1^{+-}\\\ \hline\cr\\!\\!{\rm
sym.\,axial\ 2\,tensor}\,[5]\\!\\!&a_{[5]ij}^{-+}[A]:=d_{abc}\,B^{\rm
hom}_{ak}A_{bk}(d_{cde}\,A_{di}A_{ej})~{},\quad(i\leq j\wedge k\neq
i,j)&\\!0^{-+}\\!,2^{-+}\\!\\!\\\ \hline\cr\\!\\!{\rm sym.\,axial\
3\,tensor}\,[6]\\!\\!&a_{[6]ijk}^{+-}[A]:=d_{abc}B^{\rm hom}_{ai}B^{\rm
hom}_{bj}B^{\rm hom}_{ck}~{},\quad(i\leq j\leq k)&\\!1^{+-}\\!,3^{+-}\\!\\!\\\
\hline\cr\end{array}$

Table 1: Definition of the complete set of eight elementary $SU(3)$-invariant
spatial tensors on gauge-reduced A-space. The color indices $a,b,c$ are summed
over, but the spatial indices $i,j,k$ are not, in all lines of the table. Note
that for the case $i=j$ in the seventh line one can choose any of the two
$k\neq i,j$, both give the same $a_{[5]}$. The second column shows the degree
$[n]$ of the tensor (as a polynomial in A). The last column shows the spin
components into which the tensor can be decomposed.

####  4.3.4 Triangular from of the complete Hamiltonian

We would like to point out here, that as for the 1-dimensional and the
2-dimensional cases, the polynomial solutions

$\displaystyle\\!\\!\\!\\!\\!\\!P_{[n]}\left[\omega
s_{[2]},\omega^{3/2}s_{[3]},\omega^{2}b_{[4]},\omega^{5/2}b_{[5]},\omega^{3/2}a_{[3]},\omega^{2}a_{[4]},\omega^{5/2}a_{[5]},\omega^{3}a_{[6]}\right]$
(115)

of the general 3-dimensional case, seen as polynomial in the 45 components of
the 8 irreducible tsymmeric tensors can be also obtained using a trigonal
differential equation

$\displaystyle\Big{[}D^{(1)}_{1}+D^{(1)}_{2}+D^{(1)}_{3}+D^{(2)}_{23}+D^{(2)}_{13}+D^{(2)}_{12}+D^{(3)}_{123}\Big{]}P_{[n]}=(12+n)\
P_{[n]}$ (116)

with the three-direction-differential operator $D^{(3)}_{123}$. As the 1-dim.
$D^{(1)}=D_{0}^{(1)}+D_{-2}^{(1)}$ given in (55) and the 2-dim.
$D^{(2)}=D_{0}^{(2)}+D_{-2}^{(2)}$ given in (89) and shown explicitely in
Appendix D, the 3-dim Differential operator $D^{(3)}=D^{(3)}_{0}+D^{(3)}_{-2}$
has a diagonal and a by an power-of-2-lowering part and hence is
diagonalisable analytically. The explicit expression of $D^{(3)}_{123}$ takes
several pages and is therefore not show here explicitly.

####  4.3.5 Comment on reducibility of polynomials in original constrained and
reduced spaces

As discussed in the work of Dittner [25], for $SU(3)$ in original constrained
functional space $\\{V_{i}^{a},i=1,..,D\\}$ there are two irreducible
$SU(3)$-invariant tensors in $D=1$ spatial dimension,

$\displaystyle V_{1}^{a}V_{1}^{a}\quad{\rm and}\quad
d_{abc}V_{a1}V_{b1}V_{c1}~{},\quad(D=1)~{},$ (117)

nine irreducible $SU(3)$-invariant tensors for $D=2$,

$\displaystyle V_{1}^{a}V_{1}^{a}~{},\ d_{abc}V^{a}_{1}V^{b}_{1}V^{c}_{1}~{},\
V_{2}^{a}V_{2}^{a}~{},\ d_{abc}V^{a}_{2}V^{b}_{2}V^{c}_{2}~{},\
V_{1}^{a}V_{2}^{a}~{},\ d_{abc}V^{a}_{1}V^{b}_{1}V^{c}_{2}~{},\
d_{abc}V^{a}_{1}V^{b}_{2}V^{c}_{2}~{},$ $\displaystyle\quad\
C^{a}[V]C^{a}[V]~{},\
d_{abc}\,C^{a}[V]\,C^{b}[V]\,C^{c}[V]~{},\quad\quad(D=2)~{},$ (118)

where $C^{a}[V]:=d_{abc}\,V^{b}_{1}\,V^{c}_{2}~{}.$ The nineth is independent
of the first 8, in the sense, that it cannot be represented as a sum of
products of them. It is, however, not primitive because it is related to them
via outer products.

Furthermore, Dittner proved that for $D\geq 3$ there are maximally 35
independent irreducible $SU(3)$-invariant (”primitive”) tensors of maximally
6th rank. For tensors of rank higher than 6, the number of constraints due to
outer products exceeds the number of irreducible tensors.

In reduced functional space $\\{A_{i}^{a},i=1,..,D\\}$, considered here, we
have also two $SU(3)$-invariant irreducible tensors in $D=1$ spatial
dimension,

$\displaystyle s_{11}=A_{1}^{a}A_{1}^{a}\quad{\rm and}\quad
s_{111}=d_{abc}A^{a}_{1}A^{b}_{1}A^{c}_{1}~{},\quad(D=1)~{}.$ (119)

For reduced functional space in $D=2$ we have different to the constrained
case, only 8 $SU(3)$-invariant irreducible tensors 222the eighth is related to
the eighth in (118) via the identity
$f_{abc}f_{ade}=(2/3)(\delta_{bd}\delta_{ce}-\delta_{be}\delta_{cd})+d_{abd}d_{ace}-d_{abe}d_{acd}$,

$\displaystyle
s_{11}~{},s_{111}~{},s_{22}~{},s_{222}~{},s_{12}~{},s_{112}~{},s_{122}~{},\
{\rm and}\ B_{3}^{2}~{},\quad(D=2)~{}.$ (120)

In contrast to the case of original constrained functional space, the last 9th
tensor, which is of rank 6, is reducible in reduced space

$\displaystyle d_{abc}C^{a}[A]C^{b}[A]C^{c}[A]={1\over 18}s_{12}^{3}-{1\over
6}s_{12}\,s_{11}\,s_{22}-{1\over 12}s_{111}\,s_{222}+{3\over
4}s_{112}\,s_{122}+{1\over 6}s_{12}\,B^{2}_{3}~{},\ (D=2)~{}.$ (121)

For the case of 3-dimensional reduced space here we find eigth irreducible
tensors of maximally 6th polynomial order in $A$. Since the $SU(3)$ gauge is
reduced completely, outer products in color space are absent in the reduced
approch, in contrast to the constrained approach.

###  4.4 All solutions of the corresponding harmonic oscillator problem

As demonstrated in the preceeding paragraphs, the corresponding harmonic
oscillator problem

$H_{h.o.}(A,P)\,|\Phi_{i,M}^{(J)PC}\rangle=\epsilon^{(J)PC}_{h.o.}[\omega]\,|\Phi_{i,M}^{(J)PC}\rangle~{},$

with the same measure as for the case of Yang-Mills Quantum mechanics (24),

$\langle\Phi_{1}|O|\Phi_{2}\rangle=\int d\mu_{X}\int d\mu_{Y}\int d\mu_{Z}\
\Phi_{1}^{\dagger}O\ \Phi_{2}~{},$ (122)

only replacing the chromomagnetic potential by the 16-dim. harmonic oscillator
potential with parameter $\omega$, see (35), turns out to be trigonal in the
space of the monomial functionals

$M[\omega
s_{[2]},\omega^{3/2}s_{[3]},\omega^{2}b_{[4]},\omega^{5/2}b_{[5]},\omega^{3/2}a_{[3]},\omega^{2}a_{[4]},\omega^{5/2}a_{[5]},\omega^{3}a_{[6]}]\,\exp[-{1\over
2}\omega\left(A_{ai}\right)^{2}]~{},$ (123)

and hence integrable. The $M$ are monomials in the 45 components of eight
elementary $SU(3)$-invariant spatial tensors in reduced A-space shown in Table
1. Note $\left(A_{ai}\right)^{2}\equiv\left(s_{11}+s_{22}+s_{33}\right)$.

Organising the monomial functionals according to the degree $n$ (as a
polynomial in the $A$) and the conserved quantum numbers J,M,P,C and applying
a Gram-Schmidt orthogonalisation with respect to the measure, we obtain all
exact solutions

$\displaystyle\Phi_{[n]\,i,M}^{(J)PC}[A]=P_{[n]\,i,M}^{(J)PC}[\omega
s_{[2]},\omega^{3/2}s_{[3]},\omega^{2}b_{[4]},\omega^{5/2}b_{[5]},\omega^{3/2}a_{[3]},\omega^{2}a_{[4]},\omega^{5/2}a_{[5]},\omega^{3}a_{[6]}]\,\times\exp[-\omega\left(A_{ai}\right)^{2}/2]~{},$

of the corresponding harmonic oscillator problem with energies

$\epsilon^{(J)PC}_{h.o.}=\left(12+n\right)\omega~{},$

where $n$ is the degree of $P_{[n]}$ as a polynom in the $A$.

####  4.4.1 Lowest order monomials for all symmetry sectors $J^{PC}$

The lowest order monomials for each symmetry sector $J^{PC}$ are listed in
Appendix F, Tables 6a-6d. In the upper part of Table 6a one finds the
monomials for the case $J^{++}$ for even $J=0,2,4,..$, the first
$0_{[n]}^{++}$ and $2_{[n]}^{++}$ monomials up to $n=4$ can be read of from
Equ.(96) and (101) in the last paragraphs. Similarly, the upper half of Table
6b lists the lowest monomials for the case $J^{--}$ for odd $J=1,3,5,..$, the
first $1_{[n]}^{--}$ monomials up to $n=5$ and $3_{[n]}^{--}$ monomials up to
$n=3$ can be read of from Equ.(102) and (100) respectively.

Next we note that for the sector $2_{[6]}^{++}$ in the upper part of table 6a,
there appears the symmetric part $\left(b_{[4]ik}\ s_{[2]kj}\right)^{(2)}_{\rm
sym}$. The corresponding antisymmetric part $\left(\epsilon_{ist}\ b_{[4]sj}\
s_{[2]jt}\right)$ also of order $n=6$ describes the lowest spin-1 monomial
$1_{[6]}^{++}$ shown in the lower part of Table 6a listing the case $J^{++}$
for odd $J=1,3,5,..$ . Similarly, for the sector $3_{[5]}^{--}$ in the upper
part of table 6b, there appears the symmetric parts $\left(s_{[3]ijp}\
s_{[2]pk}\right)_{\rm sym}^{(3)}$ and $\left(v_{[3]i}\ s_{[2]jk}\right)_{\rm
sym}^{(3)}$, where the symmetrisation is over the open indices $i,j,k$. The
corresponding antisymmetric parts $\left(\epsilon_{ist}\ s_{[3]sjk}\
s_{[2]kt}\right)_{\rm sym}^{(2)}$ and $\left(\epsilon_{ist}\ v_{[3]s}\
s_{[2]jt}\right)_{\rm sym}^{(2)}$, also of order $n=5$, describes the lowest
spin-2 monomials $2_{[5]}^{--}$ appearing in the lower part of Table 6b
listing $J^{--}$ for even $J=0,2,4,..$.

In the Tables 6a and 6b the number of axial components in a monomial has to be
even, e.g. in the case of the $0_{[6]}^{++}$ state
$(a^{-+}_{[3]}a^{-+}_{[3]})$. The upper and lower parts of Tables 6c and 6d
list the monomials for the case of $J^{-+}$ and $J^{+-}$ where an odd number
of axial components appear in the monomials.

####  4.4.2 Energy spectrum of the harmonic oscillator problem $H_{h.o.}$

Fig.1 shows the lowest energy eigenvalue of the harmonic oscillator problem
$H_{h.o.}$ in each symmetry sector $J^{PC}$ for the (maximal) polynomial order
of 10/11 nodes for even/odd parity.

Figure 1: The lowest energy eigenvalue of the corresponding harmonic
oscillator problem $H_{h.o.}$ in each symmetry sector $J^{PC}$ for the
(maximal) polynomial order of 10/11 nodes for even/odd parity.

The spectrum reflects the gauge invariant constructability of the monomials
from the components of the irreducible symmetric tensors for a given symmetry
sector. Hence in sectors where the monomials are build from antisymmetric
parts of products of tensors, lie relatively high in energy. For example, the
lowest $2^{--}$ state appears only at $n=5$ and therefore energy
$\epsilon=19\,\omega$, the lowest $1^{++}$ state only at $n=6$ and therefore
energy $\epsilon=20\,\omega$. Similarly, their axial colleagues, the lowest
$2^{+-}$ state appears only at $n=6$ and therefore energy
$\epsilon=20\,\omega$, the lowest $1^{-+}$ states only at $n=7$ and therefore
energy $\epsilon=21\,\omega$.

Very important for the use of a orthogonal basis for the case of SU(3) Yang-
Mills Quantum Mechanics is the analytical construction of the eigensystem of
the corresponding harmonic oscillator Hamiltonian $H_{\rm h.o.}$ . This is
most effectively done using Gram-Schmidt orthogonalisation described in the
next paragraph.

####  4.4.3 Eigenstates of $H_{\rm h.o.}$ from Gram-Schmidt orthogonalisation

Enumerate all possible monomials for given $J^{PC}$ in increasing order $n$
and multiplicity $m$

$M^{(J)PC}_{r}[A]:=\omega^{n/2}M_{[n]m}^{(J)PC}\left(s_{[2]}[A],s_{[3]}[A],b_{[4]}[A],b_{[5]}[A],a_{[3]}[A],a_{[4]}[A],a_{[5]}[A],a_{[6]}[A]\right)~{},$

e.g. for the $0^{++}$ sector

$\displaystyle M^{(0)++}_{1}$ $\displaystyle=$ $\displaystyle
M_{[0]}^{(0)++}\equiv 1~{},$ $\displaystyle M^{(0)++}_{2}$ $\displaystyle=$
$\displaystyle\omega\,M_{[2]}^{(0)++}\equiv\omega\,s_{[2]ii}~{},$
$\displaystyle M^{(0)++}_{3}$ $\displaystyle=$
$\displaystyle\omega^{2}\,M_{[4]1}^{(0)++}\equiv\omega^{2}\,s_{[2]ii}s_{[2]jj}~{},$
$\displaystyle M^{(0)++}_{4}$ $\displaystyle=$
$\displaystyle\omega^{2}\,M_{[4]2}^{(0)++}\equiv\omega^{2}\,s_{[2]ij}s_{[2]ij}~{},$
$\displaystyle M^{(0)++}_{5}$ $\displaystyle=$
$\displaystyle\omega^{2}\,M_{[4]3}^{(0)++}\equiv\omega^{2}\,b_{[4]ii}~{}.$
$\displaystyle...$

Consider now the Gram matrix

$\displaystyle G_{rs}^{(J)PC}:=\langle\langle
M_{r}^{(J)PC}M_{s}^{(J)PC}\rangle\rangle_{A}~{},$ (124)

with the measure

$\displaystyle\langle\langle\ \rangle\rangle_{A}$ $\displaystyle:=$
$\displaystyle\int_{0}^{2\pi}d\psi\,\cos^{2}[3\,\psi]\int_{0}^{\infty}dr\,r^{\,7}\exp[-\omega
r^{2}]\left[\prod_{a=1}^{8}\int_{-\infty}^{\infty}dZ_{a}\exp[-\omega
Z_{a}^{2}]\right]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\left[\prod_{a}^{1,2,3,8}\int_{-\infty}^{\infty}dY_{a}~{}.\exp[-\omega
Y_{a}^{2}]\right]\int_{0}^{\infty}dY_{4}Y_{4}\exp[-\omega
Y_{4}^{2}]\int_{0}^{\infty}dY_{6}Y_{6}\exp[-\omega Y_{6}^{2}]~{}.$

Here it is very usful that the integration completely factorises into simple
1-dimensional integrations, which is due to the choice of the flux-tube gauge.
Since the functionals to be integrated, are polynomials, the integrations can
be carried out as replacements.

Gram-Schmidt orthogonalisation corresponds to finding a lower-triangular
matrix $T$ such that

$T\,G^{(J)PC}T^{T}=1~{},$

obtaining the orthogonal polynomials

$P^{(J)PC}_{n}[A]:=\sum_{k=1}^{n}T_{nk}M^{(J)PC}_{k}[A]\quad\quad\quad~{}.$

Then the functionals

$\Phi^{(J)PC}_{n}[A,\omega]\equiv P^{(J)PC}_{n}[A]\ \exp[-{1\over
2}\omega\left(A_{ai}\right)^{2}]$

form an ONB of solutions of the corresponding harmonic oscillator problem.
Note that during the orthogonalisation procedure, the linear dependent states
appear as zero-eigenvalues of the Gram matrix $G^{(J)PC}$ Equ.(124) and can
therefore systematically be removed.

The matrix elements of the harmonic potential can then be easily obtained
using

$\displaystyle\langle\Phi_{m}^{(J)PC}[A,\omega]\left(A_{ai}\right)^{2}\Phi_{n}^{(J)PC}[A,\omega]\rangle=T_{mr}\,T_{ns}\,\langle\langle
M_{r}^{(J)PC}\left(A_{ai}\right)^{2}\ M_{s}^{(J)PC}\rangle\rangle_{A}=-{1\over
2}\omega^{2}{\partial\over\partial\omega}\left(...\right)~{}.$ (125)

Finally, the magnetic matrix elements can be calculated using

$\displaystyle\langle\Phi_{m}^{(J)PC}[A,\omega]\left(B_{ai}^{2}[A]\right)\Phi_{n}^{(J)PC}[A,\omega]\rangle=T_{mr}\,T_{ns}\,\langle\langle
M_{r}^{(J)PC}B_{ai}^{2}[A]\ M_{s}^{(J)PC}\rangle\rangle_{A}~{}.$ (126)

These will be, as shown in the next section, the main steps to obtain the low
energy eigensystem of SU(3) YM QM.

##  5  Low-energy spectrum of $SU(3)$ YM QM

###  5.1 The energy spectrum of $SU(3)$ YM QM from the corresponding harmonic
oscillator problem

Consider the basis of energy eigenstates of the correponding unconstrained
harmonic oscillator Schrödinger equation orthonormal with respect to the Yang-
Mills measure

$\displaystyle H_{\rm h.o.}\Phi_{n}[A,\omega]\equiv\left[T_{\rm kin}+{1\over
2}\omega^{2}A_{ai}^{2}\right]\Phi_{n}[A,\omega]=\epsilon^{\rm
h.o.}_{n}\Phi_{n}[A,\omega]~{}.$

Then the matrix elements of the unconstrained Yang-Mills Hamiltonian are given
as

$\displaystyle{\cal M}_{mn}\\!\\!\\!\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!\\!\\!\\!\langle\Phi^{\dagger}_{m}[A,\omega]\left(T_{\rm
kin}+{1\over 2}B_{ai}^{2}[A]\right)\Phi_{n}[A,\omega]\rangle_{A}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\left[\delta_{nm}\epsilon^{\rm
h.o.}_{n}-\langle\Phi^{\dagger}_{m}[A,\omega]\\!\left({1\over
2}\omega^{2}A_{ai}^{2}\\!\right)\\!\Phi_{n}[A,\omega]\rangle_{A}\right]\\!+{1\over
2}\langle\Phi^{\dagger}_{m}[A,\omega]\left(B_{ai}^{2}[A]\right)\Phi_{n}[A,\omega]\rangle_{A}$

since the kinetic terms $T_{\rm kin}$ are the same for the Yang-Mills and the
corresponding harmonic oscillator problem. These are calculated analytically
using formulae (125) and (126). We treat $\omega$ as a variational parameter,
which in each symmetry sector can be choose to minimize the lowest eigenvalue
of the matrix ${\cal M}$, which is easily diagonalised numerically with high
accuracy. The most time consuming part is the calculation of the expectation
values of the chromomagnetic potential according to formula (126), it takes
about a week on 36-kernel micro-supercomputers for truncations at polynomial
order 10 or 11 for higher spins, and 2-3 months for truncation at polynomial
order 12, which is work in progress.

###  5.2 Results

We have calculated the low energy spectrum of SU(3) Yang-Mills Quantum
Mechanics for all symmetry sectors $J^{PC}$ up to spin $J=11$, including
polynomials up to 10th order (10-node resolution) for even parity and up to
11th order (11-node resolution) for odd parity.

####  5.2.1 Lowest level in each symmetry sector $J^{PC}$

Fig.2 shows the lowest energy eigenvalue in each symmetry sector $J^{PC}$ as a
function of the polynomial order of truncation up to 10 for even parity and up
to 11 for odd parity. The values are listed in Tables 2-5.

Figure 2: Energy-eigenvalues as a function of polynomial order of truncation.

Fig.3 shows the lowest energy eigenvalue of SU3 YM-QM in each symmetry sector
$J^{PC}$ including all states up to polynomial order of 10 for even parity and
up to polynomial order 11 for odd parity.

Figure 3: The lowest energy eigenvalue of SU3 YM-QM in each symmetry sector
$J^{PC}$ including all states up to polynomial order of 10 for even parity and
up to polynomial order 11 for odd parity.

The spectrum is purely discrete in accordance with the proof of Simon [3] and
the groundstate energy is obtained to be $\epsilon_{0}^{++}=12.5868~{}$ (when
truncating at 12 nodes). The lowest states correspond to the $16$ components
of $0^{*++},2^{++}$ and $1^{--},3^{--}$ which show good convergence as a
function of increasing polynomial order. These $16$ states correspond to the
spins of the elementary dynamical variables of the ”symmetric gauge”
[18]-[20].

The higher the polynomial order of truncation, the less the dependence on the
variational parameter $\omega$. At 10-th or 11th order of truncation the
results for the spectrum is practically independent of the arbitrarily
introduced parameter $\omega$.

Also good convergence as a function of increasing polynomial order show the
states $4^{++}$, $0^{-+}$, $5^{--},4^{--},2^{--}$, $1^{+-}$, $2^{-+}$.

For higher states one can use the flow of the energy values with increasing
resolution, indicating that at higher energies the energy levels might be
almost equidistant in each symmetry sector as for the corresponding harmonic
oscillator spectrum Fig.1.

Deviations of the spectra of SU(3) Yang-Mills QM Fig.3 with spectra of the
corresponding harmonic oscillator Fig.1 show the effect of the chromomagnetic
potential. All aspects and constraints due to gauge invariance are already
included in the corresponding harmonic oscillator spectrum.

####  5.2.2 The lowest few levels in each symmetry-sector $J^{PC}$

Fig.4 shows the lowest few energy eigenvalues of SU3 YM-QM in each symmetry
sector $J^{PC}$ as a function of the polynomial order of truncation up to
10/11 nodes for even/odd parity.

Figure 4: The lowest few energy eigenvalues of SU3 YM-QM in each symmetry
sector $J^{PC}$ as a function of the polynomial order of truncation up to
10/11 nodes for even/odd parity.

Consideration of the lowest few energy eigenvalues of SU3 YM-QM in each
symmetry sector $J^{PC}$ shows, that higher excitations e.g. the first excited
state of spin-2 are close to degenerate to the lowest state of spin-4 similar
to the integrable harmonic oscillator problem.

In order to even improve the convergence of the lowest states and to get
convergent results also at higher energy, polynomials in orders of $14/15$ are
necessary. Hence we need very effective computer algorithms to cope with very
large polynomials. This will be subject of future work.

####  5.2.3 Comparison with the results of Weisz and Ziemann in the
constrained approach

Theresults found in the present work are in good agreement with the results of
Weisz and Zieman [11], shown in the last column denoted by ”WZ” in Tables 2-5,
using the constrained Hamiltonian approach. The agreement is excellent in the
$0^{++}$ and $2^{++}$, where theire results are already quite accurate.,
although their error estimates turned out to be too optimistic. Our results
are much more accurate values in other sectors considered by them wit only few
trial states, e.g. in $1^{--}$ and $3^{--}$ sectors, and we give quite
accurate ”new results” for the states not considered by them, as e.g.
$2^{--},4^{--},5^{--},3^{++}.$

####  5.2.4 Comparison with the results of Lattice QCD

Comparing the results of the low energy spectrum of SU(3) Yang-Mills Quantum
Mechanics with those obtained in Lattice QCD using asymetric lattices, eg. by
Morningstar and Peardon [22] and Chen et al. [23], using dimensionless results
obtained by dividing by the lowest (spin-0) mass, show reasonable overall
agreement for the $0^{++},2^{++},3^{++}$, $0^{-+},2^{=+}$, and
$1^{+-},3^{+-},2^{+-},0^{+-}$ glueball states considered by them. Their
$1^{--},2^{--},3^{--}$ results , however, are much higher in energy then those
of Yang-Mills Quantum Mechanics.

##  6 Conclusions

It has been shown in this work, that an unconstrained Hamiltonian formulation
of SU(3) Yang-Mills Quantum Mechanics of spatially constant fields, which
corresponds to the lowest order in an strong coupling expansion of SU(3) YM
theory, can be carried out in a rather practical way using the flux-tube
gauge. The corresponding Faddeev-Popov operator is simple but non-trivial. The
drawback, that the reduced gauge fields in the fluxtube-gauge themselves are
not tensors under spatial rotations, as was the case for the symmetric gauge
on cost of a very complicated FP-operator, can be circumvened by forming
certain irreducible polynomials of the reduced $A$, eight symmetric tensors,
four of which are axial, which have definite eigenvalues of J,P, and C.

The spectrum of the Hamiltonian of SU(3) Yang-Mills QM of spatially constant
fields can be determined in an effective way using the exact solutions of the
corresponding harmonic oscillator problem only replacing the chromomagnetic
potential by a 16-dimensional harmonic oscillator potential parametrised by
one parameter $\omega$, but leaving the non-trivial FP-operator unchanged.
This model has been demonstrated in this work to be integrable. The
eigensystem turned out to be orthogonal polynomials of the 45 components of
the eigth irreducible tensors, four of them axial, multiplied by a
16-dimensional Gaussian. Its energy spectrum depends only on the polynomial
order with respect to the $A$ of the eigenstate, and is highly degenerate.
Using the Gram-Schmidt orthogonalisation we could find the eigensystem of the
corresponding harmonic oscillator Hamiltonian up to 10th polynomial order for
even, and 11th order for odd parity states. This eigensystem could then be
used to find the corresponding eigensystem of SU(3) Yang-Mills Quantum
Mechanics with relatively high accuracy for the low lying states, and the
dependence on the variational parameter $\omega$ became very small. Very
helpful for the analytical calculations is here the fact, that uing the flux-
tube gauge, the integrations in functional space for calculating matrix
elements completely factorise, and since the eigenstates are polynomials
multiplied by a 16-dim Gaussian, the integrations can be substituted by
replacements.

The results are in good agreement with the results of Weisz and Zieman (1986)
using the constrained Hamiltonian approach in the $0^{++}$ and $2^{++}$
sectors, much more accurate values in other sectors considered by them, e.g.
in $1^{--}$ and $3^{--}$ sectors , and give quite accurate ”new results” for
the states not considered by them, as e.g. $2^{--}$, $3^{++}$ .

By considering the corresponding harmonic oscillator problem, which includes
already all effects of gauge invariance, as an intermediate step, the
comparison of its energy spectrum with the final Yang-Mills spectrum shows the
effect of the chromomagnetic potential. In order to further investigate the
emerging structures in the spectrum, even higher accuracy results and
polynomials up to order $14/15$ and hence even more effective programs and
algorithms are necessary. An accurate knowledge of the eigensystem of SU(3)
Yang-Mills QM is also necessary for strong coupling perturbation theory in
small $\lambda=g^{-2/3}$ proposed in earlier work [21] analogous to the SU(2)
approach [15]. Analogous to the case of SU(2) Dirac-Yang-Mills QM [16], the
calculation can also be generalised to the inclusion of quarks to study the
masses of mesons.

## Acknowledgements

I would like to thank A.B. Arbusov, A. Pilloni ,Y. Buistritskij, A. Dorokhov,
V. Gerdt, A. Khvedelidze, S. Nedelko, F. Niedermayer, Y. Palii, M. Staudacher,
O.V. Teryaev, J. Wambach, and P. Weisz for interesting discussions. This work
was partly financed by the SFB 647 ”Raum-Zeit-Materie: Analytische und
Geometrische Structuren.”

##  Appendix A Explicit form of the inverse of the FP-operator in the flux-
tube gauge

The inverse $\gamma^{-1}$ of the homogeneous part of the Faddeev-Popov
operator exists in the regions of non-vanishing determinant (23), and its non-
vanishing matrix elements are rather simple,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\gamma^{-1}\right)_{12}=-\left(\gamma^{-1}\right)_{21}={1\over
r\cos[\psi]}~{},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\gamma^{-1}\right)_{32}={1\over
2\,r\cos[\psi]}\left(Y_{4}/Y_{6}-Y_{6}/Y_{4}\right)~{},\quad\left(\gamma^{-1}\right)_{82}=-{1\over
2\sqrt{3}\,r\cos[\psi]}\left(Y_{4}/Y_{6}+Y_{6}/Y_{4}\right)~{},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!---------------------------------------$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\gamma^{-1}\right)_{45}=-\left(\gamma^{-1}\right)_{54}=-{1\over
r\cos[\psi+2\pi/3]}~{},\quad\quad\left(\gamma^{-1}\right)_{34}=-\sqrt{3}\left(\gamma^{-1}\right)_{84}=-{1\over
2\,r\cos[\psi+2\pi/3]}(Y_{2}/Y_{6})~{},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\gamma^{-1}\right)_{35}={1\over
2\,r\cos[\psi+2\pi/3]}\left(2\,Y_{+}/Y_{4}+Y_{1}/Y_{6}\right)~{},\quad\left(\gamma^{-1}\right)_{85}={1\over
2\sqrt{3}\,r\cos[\psi+2\pi/3]}\left(2\,Y_{+}/Y_{4}-Y_{1}/Y_{6}\right)~{},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!---------------------------------------$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\gamma^{-1}\right)_{67}=-\left(\gamma^{-1}\right)_{76}={1\over
r\cos[\psi+4\pi/3]}~{},\quad\quad\left(\gamma^{-1}\right)_{36}=\sqrt{3}\left(\gamma^{-1}\right)_{86}={1\over
2\,r\cos[\psi+4\pi/3]}(Y_{2}/Y_{4})~{},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\gamma^{-1}\right)_{37}={1\over
2\,r\cos[\psi+4\pi/3]}\left(Y_{1}/Y_{4}-2\,Y_{-}/Y_{6}\right)~{},\quad\quad\left(\gamma^{-1}\right)_{87}={1\over
2\sqrt{3}\,r\cos[\psi+4\pi/3]}\left(Y_{1}/Y_{4}+2\,Y_{-}/Y_{6}\right)~{},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!---------------------------------------$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\gamma^{-1}\right)_{33}=\sqrt{3}\left(\gamma^{-1}\right)_{83}=Y_{4}^{-1}~{},\quad\quad\left(\gamma^{-1}\right)_{38}=-\sqrt{3}\left(\gamma^{-1}\right)_{88}=Y_{6}^{-1}~{},$

grouped into those proportional to $\cos^{-1}[\psi]$,
$\cos^{-1}[\psi+2\pi/3]$, and $\cos^{-1}[\psi+4\pi/3]$, and those independent
of $r$ and $\psi$. Such a ”Weyl-decomposition” leads to a considerable
simplification of the non-local potential.

##  Appendix B Explicit forms of the $I_{m}^{YZ}\ (m=1,2,4,5,6,7)$ and
$I_{\pm}^{YZ}$

The explicit expressions of the interations $I_{m}^{Y\\!Z}$ and
$I_{\pm}^{Y\\!Z}$ in (34 ) read

$\displaystyle I_{m}^{Y\\!Z}$ $\displaystyle:=$ $\displaystyle\left({1\over
Y_{4}Y_{6}}\widetilde{T}_{m}^{Y\dagger}Y_{4}\,Y_{6}+\widetilde{T}_{m}^{Z}\right)\left(\widetilde{T}^{Y}_{m}+\widetilde{T}_{m}^{Z}\right)$
$\displaystyle I_{\pm}^{Y\\!Z}$ $\displaystyle:=$
$\displaystyle\left[\left(T^{Z}_{3}\pm{1\over\sqrt{3}}T^{Z}_{8}\right)+2T^{Y}_{3}\right]\left(T^{Z}_{3}\pm{1\over\sqrt{3}}T^{Z}_{8}\right)$

with the shifted non-Hermitean $\widetilde{T}^{Y}_{a}$ and Hermitean
$\widetilde{T}^{Z}_{a}$, defined as

$\displaystyle\widetilde{T}^{Y}_{1}:=T^{Y}_{1}-{1\over 2}\left({Y_{6}\over
Y_{4}}-{Y_{4}\over Y_{6}}\right){T}^{Y}_{3}~{},\quad$
$\displaystyle\quad\widetilde{T}^{Z}_{1}:=T^{Z}_{1}-{Y_{6}\over
2Y_{4}}\left({T}^{Z}_{3}+{1\over\sqrt{3}}{T}^{Z}_{8}\right)+{Y_{4}\over
2Y_{6}}\left({T}^{Z}_{3}-{1\over\sqrt{3}}{T}^{Z}_{8}\right)~{},$
$\displaystyle\widetilde{T}^{Y}_{2}:=T^{Y}_{2}~{},\quad\quad\quad\quad\quad\quad\quad\quad\
\ \quad$ $\displaystyle\quad\widetilde{T}^{Z}_{2}:=T^{Z}_{2}~{},$
$\displaystyle\widetilde{T}^{Y}_{4}:=T^{Y}_{4}-\left({Y_{1}\over
2Y_{6}}+{Y_{+}\over Y_{4}}\right){T}^{Y}_{3}~{},\,\quad$
$\displaystyle\quad\widetilde{T}^{Z}_{4}:=T^{Z}_{4}-{Y_{1}\over
2Y_{6}}\left({T}^{Z}_{3}-{1\over\sqrt{3}}{T}^{Z}_{8}\right)-{Y_{+}\over
Y_{4}}\left({T}^{Z}_{3}+{1\over\sqrt{3}}{T}^{Z}_{8}\right)~{},$
$\displaystyle\widetilde{T}^{Y}_{5}:=T^{Y}_{5}-{Y_{2}\over
2Y_{6}}{T}^{Y}_{3}~{},\quad\quad\quad\quad\ \quad$
$\displaystyle\quad\widetilde{T}^{Z}_{5}:=T^{Z}_{5}-{Y_{2}\over
2Y_{6}}\left({T}^{Z}_{3}-{1\over\sqrt{3}}{T}^{Z}_{8}\right)~{},$
$\displaystyle\widetilde{T}^{Y}_{6}:=T^{Y}_{6}+\left({Y_{1}\over
2Y_{4}}-{Y_{-}\over Y_{6}}\right){T}^{Y}_{3}~{},\,\quad$
$\displaystyle\quad\widetilde{T}^{Z}_{6}:=T^{Z}_{6}+{Y_{1}\over
2Y_{4}}\left({T}^{Z}_{3}+{1\over\sqrt{3}}{T}^{Z}_{8}\right)-{Y_{-}\over
Y_{6}}\left({T}^{Z}_{3}-{1\over\sqrt{3}}{T}^{Z}_{8}\right)~{},$
$\displaystyle\widetilde{T}^{Y}_{7}:=T^{Y}_{7}-{Y_{2}\over
2Y_{4}}{T}^{Y}_{3}~{},\quad\quad\quad\quad\ \quad$
$\displaystyle\quad\widetilde{T}^{Z}_{7}:=T^{Z}_{7}-{Y_{2}\over
2Y_{4}}\left({T}^{Z}_{3}+{1\over\sqrt{3}}{T}^{Z}_{8}\right)~{}.$

##  Appendix C The case of singular solutions of the X-equation (38)

The X-equation (38) is solved also by the singular functionals

$\Phi^{\rm
sing}_{n_{1},n_{2}}[r,\psi]:={2\omega^{2}\over\sqrt{6}\pi^{3/4}(\omega\,r^{2})^{3/4}}\sqrt{{1+\sin[3\psi]\over\cos^{2}[3\psi]}}\
p^{\rm sing}_{n_{1},n_{2}}\left(\omega\,r^{2},\sin[3\psi]\right)e^{-\omega
r^{2}/2}$

with the energy eigenvalues

$\epsilon^{\rm
sing}_{\nu,\mu}=\left(5/2+2n_{1}+3n_{2}\right)\omega=\left(5/2+n\right)\omega$

with the lowest polynomials $\quad p^{\rm sing}_{0,0}(x,y)=1~{},\quad p^{\rm
sing}_{1,0}(x,y)={1\over\sqrt{10}}(5-2x)~{},\quad...$.

Due to the $\cos^{2}[3\,\psi]$ factor in the measure (16), absent in the Y-
and Z-equations, they are nevertheless finite and normalisable. Hence, when
including the regular solutions in other two directions, y and z, we obtain
solutions, breaking rotational invariance in one direction, the x-direction,
leaving only a cylindrical symmetry.

$\begin{array}[]{c|c|c|c|c|c|c|c||c|}aE&[0]&\leq[2]&\leq[4]&\leq[6]&\leq[8]&\leq[10]&\leq[12]&WZ\\\
\hline\cr 0_{1}^{++}\\!\\!&\\!\\!12.980\ (1)\\!\\!&\\!\\!12.951\
(2)\\!\\!&\\!\\!12.632\ (5)\\!\\!&\\!\\!12.620\ (13)\\!\\!&\\!\\!12.591\
(30)\\!\\!&\\!\\!12.589\ (69)\\!\\!&\\!\\!12.5868\
(157)\\!\\!&\\!\\!12.5887\,(106)\\!\\!\\\ \hline\cr
0_{2}^{++}\\!\\!&\--&\\!\\!16.879\ (2)\\!\\!&\\!\\!15.995\
(5)\\!\\!&\\!\\!15.705\ (13)\\!\\!&\\!\\!15.448\ (30)\\!\\!&\\!\\!15.390\
(69)\\!\\!&\\!\\!15.351\ (157)\\!\\!&15.38\ (106)\\\ \hline\cr
0_{3}^{++}\\!\\!&\--&\--&\\!\\!18.164\ (5)\\!\\!&\\!\\!17.893\
(13)\\!\\!&\\!\\!17.342\ (30)\\!\\!&\\!\\!17.238\ (69)\\!\\!&\\!\\!17.137\
(157)\\!\\!&17.23\ (106)\\\ \hline\cr 0_{4}^{++}\\!\\!&\--&\--&\\!\\!19.548\
(5)\\!\\!&\\!\\!18.806\ (13)\\!\\!&\\!\\!18.425\ (30)\\!\\!&\\!\\!18.095\
(69)\\!\\!&\\!\\!17.989\ (157)\\!\\!&\--\\\ \hline\cr
2_{1}^{++}\\!\\!&\--&\\!\\!15.496\ (1)\\!\\!&\\!\\!15.333\
(4)\\!\\!&\\!\\!14.914\ (13)\\!\\!&\\!\\!14.861\ (42)\\!\\!&\\!\\!14.808\
(122)\\!\\!&14.800\ (328)\\!\\!&\\!\\!14.854\,(69)\\!\\!\\\ \hline\cr
2_{2}^{++}\\!\\!&\--&\--&\\!\\!18.187\ (4)\\!\\!&\\!\\!17.819\
(13)\\!\\!&\\!\\!17.291\ (42)\\!\\!&\\!\\!17.176\ (122)\\!\\!&17.091\
(328)\\!\\!&17.26\ (69)\\\ \hline\cr 2_{3}^{++}\\!\\!&\--&\--&\\!\\!19.063\
(4)\\!\\!&\\!\\!18.362\ (13)\\!\\!&\\!\\!17.825\ (42)\\!\\!&\\!\\!17.604\
(122)\\!\\!&17.506\ (328)\\!\\!&\--\\\ \hline\cr
4_{1}^{++}\\!\\!&\--&\--&\\!\\!18\ (1)\\!\\!&\\!\\!17.668\
(6)\\!\\!&\\!\\!17.128\ (25)\\!\\!&\\!\\!17.023\ (86)\\!\\!&{\rm
in\,cal.}\,(273)\\!\\!&18\ (3)\\\ \hline\cr
4_{2}^{++}\\!\\!&\--&\--&\--&\\!\\!20.585\ (6)\\!\\!&\\!\\!20.095\
(25)\\!\\!&\\!\\!19.491\ (86)\\!\\!&{\rm in\,cal.}\,(273)\\!\\!&\--\\\
\hline\cr 6_{1}^{++}\\!\\!&\--&\--&\--&\\!\\!20.497\ (2)\\!\\!&\\!\\!19.977\
(9)\\!\\!&\\!\\!19.311\ (38)\\!\\!&&\--\\\ \hline\cr
6_{2}^{++}\\!\\!&\--&\--&\--&\\!\\!20.497\ (2)\\!\\!&\\!\\!20.101\
(9)\\!\\!&\\!\\!19.506\ (38)\\!\\!&&\--\\\ \hline\cr
6_{3}^{++}\\!\\!&\--&\--&\--&\--&\\!\\!23.071\ (9)\\!\\!&\\!\\!22.395\
(38)\\!\\!&&\--\\\ \hline\cr 8_{1}^{++}\\!\\!&\--&\--&\--&\--&\\!\\!22.989\
(2)\\!\\!&\\!\\!22.269\ (11)\\!\\!&&\--\\\ \hline\cr
8_{2}^{++}\\!\\!&\--&\--&\--&\--&\\!\\!22.989\ (2)\\!\\!&\\!\\!22.367\
(11)\\!\\!&&\--\\\ \hline\cr
8_{3}^{++}\\!\\!&\--&\--&\--&\--&\--&\\!\\!25.508\ (11)\\!\\!&&\--\\\
\hline\cr 10_{1}^{++}\\!\\!&\--&\--&\--&\--&\--&\\!\\!25.478\
(2)\\!\\!&&\--\\\ \hline\cr
10_{1}^{++}\\!\\!&\--&\--&\--&\--&\--&\\!\\!25.478\ (2)\\!\\!&&\--\\\
\hline\cr\hline\cr 1_{1}^{++}&\--&\--&\--&\\!\\!22.334\
(1)\\!\\!&\\!\\!21.929\ (8)\\!\\!&\\!\\!21.523\ (34)\\!\\!&&\--\\\ \hline\cr
3_{1}^{++}&\--&\--&\--&\\!\\!20.638\ (3)\\!\\!&\\!\\!20.139\
(15)\\!\\!&\\!\\!19.560\ (61)\\!\\!&&\--\\\ \hline\cr
3_{2}^{++}&\--&\--&\--&\\!\\!22.337\ (3)\\!\\!&\\!\\!21.758\
(15)\\!\\!&\\!\\!21.364\ (61)\\!\\!&&\--\\\ \hline\cr
5_{1}^{++}&\--&\--&\--&\--&\\!\\!23.141\ (6)\\!\\!&\\!\\!22.438\
(34)\\!\\!&&\--\\\ \hline\cr 7_{1}^{++}&\--&\--&\--&\--&\\!\\!22.989\
(1)\\!\\!&\\!\\!22.414\ (10)\\!\\!&&\--\\\ \hline\cr
9_{1}^{++}&\--&\--&\--&\--&\--&\\!\\!25.478\ (1)\\!\\!&&\--\\\
\hline\cr\end{array}$ Table 2: The lowest few energy eigenvalues of SU3 YM-QM
in the symmetry sector $J^{++}$ as a function of the polynomial order of
truncation up to 10 nodes. In brackets the number of states.
$\begin{array}[]{c|c|c|c|c|c||c|}aE_{1}[g^{2/3}]&[3]&\leq[5]&\leq[7]&\leq[9]&\leq[11]&WZ\\\
\hline\cr 1^{--}&17.555\ (1)&17.090\ (5)&16.742\ (17)&16.580\ (52)&16.528\
(143)&17.05\ (29)\\\ \hline\cr 3^{--}&16.749\ (1)&16.624\ (4)&16.144\
(17)&16.083\ (61)&16.024\ (193)&16.5\ (7)\\\ \hline\cr 5^{--}&\--&19.249\
(1)&18.870\ (6)&18.293\ (27)&18.162\ (104)&\--\\\ \hline\cr
7^{--}&\--&\--&21.744\ (1)&21.153\ (8)&20.442\ (40)&\--\\\ \hline\cr
9^{--}&\--&\--&\--&24.234\ (2)&23.429\ (11)&\--\\\ \hline\cr
11^{--}&\--&\--&\--&\--&26.722\ (2)&\--\\\ \hline\cr\hline\cr
0^{--}&\--&\--&\--&25.560\ (3)&25.204\ (12)&\--\\\ \hline\cr
2^{--}&\--&19.455\ (2)&19.037\ (10)&18.528\ (37)&18.366\ (122)&\--\\\
\hline\cr 4^{--}&\--&19.249\ (1)&18.921\ (6)&18.376\ (28)&18.261\ (107)&\--\\\
\hline\cr 6^{--}&\--&\--&21.744\ (1)&21.215\ (10)&20.544\ (49)&\--\\\
\hline\cr 8^{--}&\--&\--&\--&24.234\ (1)&23.501\ (13)&\--\\\ \hline\cr
10_{1,2}^{--}&\--&\--&\--&\--&26.722\ (2)&\--\\\ \hline\cr\end{array}$ Table
3: The lowest few energy eigenvalues of SU3 YM-QM in the symmetry sector
$J^{--}$ as a function of the polynomial order of truncation up to 11 nodes.
In brackets the number of states.
$\begin{array}[]{c|c|c|c|c|c||c|}aE_{1}[g^{2/3}]&[3]&\leq[5]&\leq[7]&\leq[9]&\leq[11]&WZ\\\
\hline\cr 0^{-+}&18.573\ (1)&18.076\ (3)&17.864\ (9)&17.759\ (26)&17.744\
(66)&17.8\ (15)\\\ \hline\cr 2^{-+}&\--&20.685\ (2)&20.304\ (10)&19.950\
(37)&19.838\ (124)&21\ (4)\\\ \hline\cr 4^{-+}&\--&\--&23.133\ (3)&22.610\
(18)&22.133\ (81)&\--\\\ \hline\cr 6^{-+}&\--&\--&\--&25.591\ (5)&24.901\
(32)&\--\\\ \hline\cr 8^{-+}&\--&\--&\--&\--&28.061\ (6)&\--\\\
\hline\cr\hline\cr 1^{-+}&\--&\--&23.698\ (2)&23.451\ (11)&22.939\ (46)&23\
(4)\\\ \hline\cr 3^{-+}&\--&\--&23.712\ (2)&23.379\ (15)&22.895\ (69)&\--\\\
\hline\cr 5^{-+}&\--&\--&\--&26.098\ (4)&25.593\ (36)&\--\\\ \hline\cr
7^{-+}&\--&\--&\--&\--&28.589\ (7)&\--\\\ \hline\cr\end{array}$ Table 4: The
lowest few energy eigenvalues of SU3 YM-QM in the symmetry sector $J^{-+}$ as
a function of the polynomial order of truncation up to 11 nodes. In brackets
the number of states.
$\begin{array}[]{c|c|c|c|c||c|}aE_{1}[g^{2/3}]&[4]&\leq[6]&\leq[8]&\leq[10]&WZ\\\
\hline\cr 1^{+-}&19.390\ (1)&19.192\ (5)&18.817\ (19)&18.768\
(59)&18(\rightarrow 19\,?)\ (4)\\\ \hline\cr 3^{+-}&\--&21.862\ (3)&21.573\
(16)&21.144\ (68)&\--\\\ \hline\cr 5^{+-}&\--&\--&24.319\ (4)&23.768\
(26)&\--\\\ \hline\cr 7^{+-}&\--&\--&\--&26.811\ (5)&\--\\\ \hline\cr\hline\cr
0^{+-}&\--&\--&\--&28.552\ (2)&28\ (4)\\\ \hline\cr 2^{+-}&\--&22.425\
(1)&22.147\ (9)&21.694\ (43)&22.1\ (17)\\\ \hline\cr 4^{+-}&\--&\--&24.930\
(4)&24.338\ (27)&\--\\\ \hline\cr 6^{+-}&\--&\--&\--&27.428\ (6)&\--\\\
\hline\cr\end{array}$ Table 5: The lowest few energy eigenvalues of SU3 YM-QM
in the symmetry sector $J^{+-}$ as a function of the polynomial order of
truncation up to 10 nodes. In brackets the number of states. The $1^{+-}$
estimate ${}^{\prime}18^{\prime}$ given in Tab.3 in WZ seems to be a misprint,
in their Fig.1 it looks more like ${}^{\prime}19^{\prime}$.

##  Appendix D Some explicit expressions for the 2-dim case

###  D.1 Explicit form of the operator $D^{(2)}=D^{(2)}_{0}+D^{(2)}_{-2}$

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!D^{(2)}_{0}:=2\,x_{12}\partial_{x_{12}}+3\,x_{112}\partial_{x_{112}}+3\,x_{122}\partial_{x_{122}}+4\,b_{33}\partial_{b_{33}}~{},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!D^{(2)}_{-2}:=-{2\over
3}\left({1\over
4}x_{11}^{2}+x_{11}x_{22}-b_{33}\right)\partial_{x_{112}}^{2}-{2\over
3}\left({1\over 4}x_{22}^{2}+x_{11}x_{22}-b_{33}\right)\partial_{x_{122}}^{2}$
$\displaystyle-{2\over
3}(x_{11}+x_{22})\,x_{12}\partial_{x_{112}}\partial_{x_{122}}-2\,x_{12}\left(x_{11}\partial_{x_{111}}\partial_{x_{112}}+x_{22}\partial_{x_{222}}\partial_{x_{122}}\right)$
$\displaystyle+2\left({1\over
2}x_{11}x_{22}-b_{33}-x_{12}^{2}\right)\left(\partial_{x_{111}}\partial_{x_{122}}+\partial_{x_{222}}\partial_{x_{112}}\right)-{1\over
2}(x_{11}+x_{22})\left(\partial_{x_{12}}^{2}+6\,\partial_{b_{33}}\right)$
$\displaystyle-3\left(x_{112}\partial_{x_{111}}+x_{122}\partial_{x_{222}}\right)\partial_{x_{12}}-\left((x_{111}+2x_{122})\partial_{x_{112}}+(x_{222}+2x_{112})\partial_{x_{122}}\right)\partial_{x_{12}}$
$\displaystyle-3\left((x_{22}x_{111}+x_{11}x_{122}-2x_{12}x_{112})\partial_{x_{111}}+(x_{11}x_{222}+x_{22}x_{112}-2x_{12}x_{122})\partial_{x_{222}}\right)\partial_{b_{33}}$
$\displaystyle-\left((x_{22}x_{112}+x_{11}x_{222}-2x_{12}x_{122})\partial_{x_{112}}+(x_{11}x_{122}+x_{22}x_{111}-2x_{12}x_{112})\partial_{x_{122}}\right)\partial_{b_{33}}$
$\displaystyle-{1\over
2}\Big{[}(x_{11}+x_{22})(-x_{11}x_{22}+x_{12}^{2}+5b_{33})+3(-(x_{222}x_{112}+x_{111}x_{122})+(x_{112}^{2}+x_{122}^{2}))\Big{]}\partial_{b_{33}}^{2}$
$\displaystyle-\left(2\,x_{12}\partial_{x_{12}}+4\,b_{33}\partial_{b_{33}}\right)\left(\partial_{x_{11}}+\partial_{x_{22}}\right)-2\,x_{112}\partial_{x_{112}}\left(2\partial_{x_{11}}+\partial_{x_{22}}\right)-2\,x_{122}\partial_{x_{122}}\left(\partial_{x_{11}}+2\partial_{x_{22}}\right)~{}.$

###  D.2 Explicit expression for the 2-dim Jacobian in terms of gauge
invariant functions

In the 2-dim case we can perform a coordinate transformation of the eight
gauge fields to the eight irreducible gauge invariant polynomials

$(X_{1},X_{2},Y_{1},Y_{2},Y_{3},Y_{4},Y_{6},Y_{8})\quad\longrightarrow\quad
x:=(x_{11},x_{22},x_{12},x_{111},x_{222},x_{112},x_{122},b_{33})~{}.$

Then the matrix elements take the form

$\langle\Phi^{\prime}_{XY}|O[X,Y]|\Phi_{XY}\rangle\propto\int
d^{8}x{\exp[-(x_{11}+x_{22})]\over\sqrt{m(x)}}\
p_{1}^{(2)}(x)\,O[x]\,p^{(2)}_{2}(x)~{},$ (127)

with the polynomial

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m(x_{11},x_{22},x_{12},x_{111},x_{222},x_{112},x_{122},b_{33})=$
$\displaystyle(x_{11}^{3}-3x_{111}^{2})(x_{22}^{3}-3x_{222}^{2})+16\,b_{33}^{3}+36\,b_{33}^{2}\left(x_{12}^{2}-x_{11}x_{22}\right)+12\,b_{33}\Big{[}2\left(x_{12}^{2}-x_{11}x_{22}\right)^{2}$
$\displaystyle+3\,x_{11}\left(x_{112}x_{222}-x_{122}^{2}\right)+3\,x_{22}\left(x_{111}x_{122}-x_{112}^{2}\right)+3\,x_{12}\left(x_{112}x_{122}-x_{111}x_{222}\right)\Big{]}$
$\displaystyle+4\left(x_{12}^{2}-x_{11}x_{22}\right)^{3}-x_{11}^{3}x_{22}^{3}+9x_{11}^{2}x_{22}\left(x_{122}^{2}-2x_{112}x_{222}\right)+9x_{11}x_{22}^{2}\left(x_{112}^{2}-2x_{111}x_{122}\right)$
$\displaystyle+12x_{12}^{3}(3x_{112}x_{122}-x_{111}x_{222})-36x_{12}^{2}(x_{11}x_{122}^{2}+x_{22}x_{112}^{2})$
$\displaystyle+18x_{12}(x_{11}x_{122}+x_{22}x_{111})(x_{11}x_{222}+x_{22}x_{112})+$
$\displaystyle+36x_{112}^{3}x_{222}-54x_{112}x_{122}x_{111}x_{222}-27x_{112}^{2}x_{122}^{2}+36x_{111}x_{122}^{3}~{}.$
(128)

The angular momentum operator takes the form

$\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle-i\Bigg{[}2\,x_{12}\left(\partial_{x_{11}}-\partial_{x_{22}}\right)+(x_{22}-x_{11})\partial_{x_{12}}$
(129)
$\displaystyle\quad\quad+3\left(x_{112}\,\partial_{x_{111}}-x_{122}\,\partial_{x_{222}}\right)+\left(2x_{122}-x_{111}\right)\partial_{x_{112}}+\left(x_{222}-2\,x_{112}\right)\partial_{x_{122}}\Bigg{]}~{}.$

##  Appendix E Spin representation of the symmetric tensors

The simultaneous eigenfunctions of $J_{3}$ and $J^{2}\equiv
J_{1}^{2}+J_{2}^{2}+J_{3}^{2}$ , given explicitly in (32), for the symmetric
2-tensor $s^{++}_{[2]ij}$ are determined as the spin-0 component

$\displaystyle
s^{(0)++}_{[2]\,0}={1\over\sqrt{3}}\left(s_{11}+s_{22}+s_{33}\right)~{},$
(130)

and the five spin-2 components
$s^{(2)++}_{[2]\,ij}:=s^{++}_{[2]\,ij}-\delta_{ij}s^{++}_{[2]\,kk}/3$ ,

$\displaystyle s^{(2)++}_{[2]\,0}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left(s_{11}+s_{22}-2\,s_{33}\right)~{},$
$\displaystyle s^{(2)++}_{[2]\,\pm 1}$ $\displaystyle=$
$\displaystyle\pm\,s_{13}+i\,s_{23}~{},$ $\displaystyle s^{(2)++}_{[2]\,\pm
2}$ $\displaystyle=$ $\displaystyle-{1\over 2}\left(s_{11}-s_{22}\right)\mp
is_{12}~{}.$ (131)

For the symmetric 3-tensor $s^{--}_{[3]\,ijk}$ we have the three spin-1
components $v^{(1)--}_{[3]\,i}:=s^{--}_{[3]\,ijj}$ ,

$\displaystyle v^{(1)--}_{[3]\,0}$ $\displaystyle=$
$\displaystyle\left(s_{113}+s_{223}+s_{333}\right)~{},$ $\displaystyle
v^{(1)--}_{[3]\,\pm 1}$ $\displaystyle=$
$\displaystyle\mp{1\over\sqrt{2}}\left((s_{111}+s_{122}+s_{133})\pm
i(s_{112}+s_{222}+s_{233})\right)~{},$ (132)

as well as the seven spin-3 components
$s^{(3)--}_{[3]\,ijk}:=s^{--}_{[3]\,ijk}-{1\over
5}\left(\delta_{jk}v^{--}_{[3]\,i}+\delta_{ik}v^{--}_{[3]\,j}+\delta_{ij}v^{--}_{[3]\,k}\right)$
,

$\displaystyle s^{(3)--}_{[3]\,0}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{30}}\left(3(s_{113}+s_{223})-2\,s_{333}\right)~{},$
$\displaystyle s^{(3)--}_{[3]\,\pm 1}$ $\displaystyle=$
$\displaystyle\mp{1\over 2\sqrt{10}}\left((s_{111}+s_{122}-4\,s_{133})\pm
i(s_{112}+s_{222}-4\,s_{233})\right)~{},$ $\displaystyle s^{(3)--}_{[3]\,\pm
2}$ $\displaystyle=$ $\displaystyle-{1\over 2}\left(s_{113}-s_{223}\right)\mp
i\,s_{123}~{},$ $\displaystyle s^{(3)--}_{[3]\,\pm 3}$ $\displaystyle=$
$\displaystyle\pm{1\over 2\sqrt{6}}\left((s_{111}-3\,s_{122})\mp
i(s_{222}-3\,s_{112})\right)~{}.$ (133)

Analogous expressions hold for the other irreducible symmteric tensors
$b^{++}_{[4]ij}$, $b^{(1)--}_{[5]i}$, as well as the irreducible symmetric
axial tensors $a^{(0)-+}_{[3]}$, $a^{(1)+-}_{[4]i}$, $a^{-+}_{[5]ij}$ and
$a^{+-}_{[6]ijk}$.

##  Appendix F Lowest monomials for all symmetry sectors $J^{PC}$

This Appendix presents the Tables 6a-6d showing the monomials of lowest
polynomial order in $A$ for each symmetry sector $J^{PC}$. Here the following
conventions are used. Double indices are summed over. The subscript ”sym”
indicates symmetrization over all open indices. Furthermore only linearly
independent monomials are shown, that is those that are not expressible as
sums of others in the same symmetry sector. For example, the 6th order spin-2
monomial

$\displaystyle\left(s_{[2]ik}s_{[2]kl}s_{[2]lj}\right)^{(2)}$
$\displaystyle\equiv$ $\displaystyle
s_{[2]ll}\left(s_{[2]ik}s_{[2]kj}\right)^{(2)}+(1/2)\left(s_{[2]kl}s_{[2]kl}-s_{[2]kk}s_{[2]ll}\right)s_{[2]ij}^{(2)}$
(134)

is linear dependent and not shown in Table 6a.

$\begin{array}[]{|c|l|l}J^{++}&M_{i}^{(J)++}\\\ \hline\cr 0^{++}_{[0]}&1\\\
\hline\cr 0^{++}_{[2]}&s^{(0)}_{[2]}\equiv s_{[2]ii}\\\ \hline\cr
0^{++}_{[4]}&\left(s^{(0)}_{[2]}\right)^{2},\ \ \ s_{[2]ij}s_{[2]ij},\ \ \
b^{(0)}_{[4]}\equiv b_{[4]ii}\\\ \hline\cr
0^{++}_{[6]}&\left(s^{(0)}_{[2]}\right)^{3},\
s^{(0)}_{[2]}\,s_{[2]jk}s_{[2]jk},\ s^{(0)}_{[2]}\,b^{(0)}_{[4]},\
a_{[3]}a_{[3]},\ s_{[2]ij}b_{[4]ij},\ s_{[3]ijk}s_{[3]ijk},\
v_{[3]i}v_{[3]i},\ s_{[2]ij}s_{[2]jk}s_{[2]ki}\\\ \hline\cr
2^{++}_{[2]}&s_{[2]ij}^{(2)}\\\ \hline\cr
2^{++}_{[4]}&s^{(0)}_{[2]}\,s_{[2]ij}^{(2)},\
\left(s_{[2]ik}s_{[2]kj}\right)^{(2)}\\!\\!\\!,\ \ b_{[4]ij}^{(2)}\\\
\hline\cr 2^{++}_{[6]}&\left(s^{(0)}_{[2]}\right)^{2}\\!s_{[2]ij}^{(2)},\
s^{(0)}_{[2]}\left(s_{[2]ik}s_{[2]kj}\right)^{(2)}\\!\\!\\!,\ \
s^{(0)}_{[2]}\,b_{[4]ij}^{(2)},\ \ s_{[2]kl}s_{[2]kl}\,s_{[2]ij}^{(2)},\
b^{(0)}_{[4]}\,s_{[2]ij}^{(2)},\\\
&\left(\\!s_{[2]ik}\,b_{[4]kj}\\!\right)_{\rm sym}^{\\!(2)},\ \
\left(v_{[3]i}\,v_{[3]j}\right)^{\\!(2)}\\!\\!\\!,\ \
\left(s_{[3]ijk}\,v_{[3]k}\right)^{\\!(2)}\\!\\!\\!,\ \
\left(s_{[3]ikl}\,s_{[3]jkl}\right)^{\\!(2)}\\\ \hline\cr
4^{++}_{[4]}&\left(s_{[2]ij}\ s_{[2]kl}\right)_{\rm sym}^{(4)}\\\ \hline\cr
6^{++}_{[6]}&\left(s_{[2]ij}\ s_{[2]kl}\ s_{[2]mn}\right)_{\rm
sym}^{(6)},\quad\left(s_{[3]ijk}\ s_{[3]lmn}\right)_{\rm sym}^{(6)}\\\
\hline\cr 8^{++}_{[8]}&\left(s_{[2]ij}\ s_{[2]kl}\ s_{[2]mn}\
s_{[2]st}\right)_{\rm sym}^{(8)},\quad\left(s_{[3]ijk}\ s_{[3]lmn}\
s_{[2]st}\right)_{\rm sym}^{(8)}\\\ \hline\cr
10^{++}_{[10]}\\!\\!&\left(s_{[2]ij}\ s_{[2]kl}\ s_{[2]mn}\ s_{[2]st}\
s_{[2]uv}\right)_{\rm sym}^{(10)},\quad\left(s_{[3]ijk}\ s_{[3]lmn}\
s_{[2]st}\ s_{[2]uv}\right)_{\rm sym}^{(10)}\\\ \hline\cr\hline\cr
1^{++}_{[6]}&\epsilon_{ist}\ s_{[2]sj}\,b_{[4]jt}\\\ \hline\cr
3^{++}_{[6]}&\left(\epsilon_{ist}\ s_{[2]sj}\ b_{[4]tk}\right)_{\rm
sym}^{(3)},\quad\left(\epsilon_{ist}\ s_{[3]sjk}\ v_{[3]t}\right)_{\rm
sym}^{(3)},\quad\left(\epsilon_{ist}\ s_{[2]sj}\ s_{[2]tl}\
s_{[2]lk}\right)_{\rm sym}^{(3)}\\\ \hline\cr
5^{++}_{[8]}&\left(\epsilon_{ist}\ b_{[4]sj}\ s_{[2]tk}\ s_{[2]lm}\right)_{\rm
sym}^{\\!(5)},\ \left(\epsilon_{ist}\ s_{[3]sjk}\ s_{[3]tlp}\
s_{[2]pm}\right)_{\rm sym}^{\\!(5)},\ \left(\epsilon_{ist}\ s_{[3]sjk}\
s_{[3]lmp}\ s_{[2]tp}\right)_{\rm sym}^{\\!(5)},\\\ &\left(\\!\epsilon_{ist}\
s_{[3]sjp}\ s_{[3]klp}\ s_{[2]tm}\\!\right)_{\rm sym}^{\\!(5)}\\!,\
\left(\\!\epsilon_{ist}\ s_{[2]sjk}\ v_{[3]t}\ s_{[2]lm}\\!\right)_{\rm
sym}^{\\!(5)}\\!,\ \left(\\!\epsilon_{ist}\ s_{[2]sj}\ s_{[2]tp}\ s_{[2]pk}\
s_{[2]lm}\\!\right)_{\rm sym}^{\\!(5)}\\\ \hline\cr
7^{++}_{[8]}&\left(\epsilon_{ist}\ s_{[3]sjk}\ s_{[3]lmn}\
s_{[2]rt}\right)_{\rm sym}^{(7)}\\\ \hline\cr
9^{++}_{[10]}&\left(\epsilon_{ist}\ s_{[2]sj}\ s_{[3]tkl}\ s_{[3]mnr}\
s_{[2]uv}\right)_{\rm sym}^{(9)}{}\\\ \hline\cr\end{array}$

Table 6a: Lowest order monomials for the $J^{++}$ sector, for even and odd
spin $J$.

$\begin{array}[]{|c|l|l}J^{--}&M^{(J)--}\\\ \hline\cr
1^{--}_{[3]}&v_{[3]i}\equiv s_{[3]ijj}\\\ \hline\cr
1^{--}_{[5]}&s^{(0)}_{[2]}v_{[3]i},\quad s_{[2]ij}v_{[3]j},\quad
s_{[3]ijk}s_{[2]jk},\quad b_{[5]i}\\\ \hline\cr
3^{--}_{[3]}&s_{[3]ijk}^{(3)}\\\ \hline\cr
3^{--}_{[5]}&s^{(0)}_{[2]}s_{[3]ijk}^{(3)},\ \
\left(s_{[2]il}s_{[3]ljk}\right)_{\rm sym}^{(3)},\ \
\left(s_{[2]ij}v_{[3]k}\right)_{\rm sym}^{(3)}\\\ \hline\cr
5^{--}_{[5]}&\left(s_{[3]ijk}\ s_{[2]lm}\right)_{\rm sym}^{(5)}\\\ \hline\cr
7^{--}_{[7]}&\left(s_{[3]ijk}\ s_{[2]lm}\ s_{[2]ns}\right)_{\rm sym}^{(7)}\\\
\hline\cr 9^{--}_{[9]}&\left(s_{[3]ijk}\ s_{[2]lm}\ s_{[2]ns}\
s_{[2]tu}\right)_{\rm sym}^{(9)},\ \ \left(s_{[3]ijk}\ s_{[3]lmn}\
s_{[3]stu}\right)_{\rm sym}^{(9)},\\\ \hline\cr
11^{--}_{[11]}&\left(s_{[3]ijk}\ s_{[2]lm}\ s_{[2]ns}\ s_{[2]tu}\
s_{[2]vw}\right)_{\rm sym}^{(11)},\ \ \left(s_{[3]ijk}\ s_{[3]lmn}\
s_{[3]stu}\ s_{[2]vw}\right)_{\rm sym}^{(11)},\\\ \hline\cr\hline\cr
0^{--}_{[9]}&\epsilon_{ijk}\ b_{[4]il}\ s_{[3]jlm}\ s_{[2]km},\ \
\epsilon_{ijk}\ b_{[4]il}\ v_{[3]j}\ s_{[2]kl},\ \ \epsilon_{ijk}\ s_{[3]ilm}\
s_{[2]jm}\ s_{[2]kn}\ s_{[2]ln},\\\ \hline\cr
2^{--}_{[5]}&\left(\epsilon_{ist}\ s_{[3]sjk}\ s_{[2]kt}\right)_{\rm
sym}^{(2)},\ \ \left(\epsilon_{ist}\ v_{[3]s}\ s_{[2]jt}\right)_{\rm
sym}^{(2)}\\\ \hline\cr 4^{--}_{[5]}&\left(\epsilon_{ist}\ s_{[3]sjk}\
s_{[2]lt}\right)_{\rm sym}^{(4)}\\\ \hline\cr
6^{--}_{[7]}&\left(\epsilon_{ist}\ s_{[3]sjk}\ s_{[2]lt}\
s_{[2]mn}\right)_{\rm sym}^{(6)}\\\ \hline\cr
8^{--}_{[9]}&\left(\epsilon_{ist}\ s_{[3]sjk}\ s_{[2]lt}\ s_{[2]mn}\
s_{[2]ru}\right)_{\rm sym}^{(8)}\\\ \hline\cr
10^{--}_{[11]}&\left(\epsilon_{ist}\ s_{[3]sjk}\ s_{[2]lt}\ s_{[2]mn}\
s_{[2]ru}\ s_{[2]vw}\right)_{\rm sym}^{(10)},\ \ \left(\epsilon_{ist}\
s_{[3]sjk}\ s_{[2]lt}\ s_{[3]mnr}\ s_{[3]uvw}\right)_{\rm sym}^{(10)}\\\
\hline\cr\end{array}$

Table 6b: Lowest order monomials for the $J^{--}$ sector, for odd and even
spin $J$.

$\begin{array}[]{|c|l|l}J^{-+}&M^{(J)-+}\\\ \hline\cr 0^{-+}_{[3]}&a_{[3]}\\\
\hline\cr 0^{-+}_{[5]}&s^{(0)}_{[2]}a_{[3]},\quad a^{(0)}_{[5]}\equiv
a_{[5]ii}\\\ \hline\cr 2^{-+}_{[5]}&a_{[3]}\ s_{[2]ij},\quad a_{[5]ij}\\\
\hline\cr 4^{-+}_{[7]}&a_{[3]}\left(s_{[2]ij}\ s_{[2]kl}\right)_{\rm
sym}^{(4)},\ \left(a_{[4]i}\ s_{[3]jkl}\right)_{\rm sym}^{(4)},\
\left(a_{[5]ij}\ s_{[2]kl}\right)_{\rm sym}^{(4)}\\\ \hline\cr
6^{-+}_{[9]}&a_{[3]}\left(s_{[2]ij}\ s_{[2]kl}\ s_{[2]mn}\right)_{\rm
sym}^{(6)},\ a_{[3]}\left(s_{[3]ijk}\ s_{[3]lmn}\right)_{\rm sym}^{(6)},\
\left(a_{[4]i}\ s_{[3]jkl}\ s_{[2]mn}\right)_{\rm sym}^{(6)},\\\
&\left(a_{[5]ij}\ s_{[2]kl}\ s_{[2]mn}\right)_{\rm sym}^{(6)},\
\left(a_{[6]ijk}\ s_{[3]lmn}\right)_{\rm sym}^{(6)}\\\ \hline\cr
8^{-+}_{[11]}&a_{[3]}\\!\left(\\!s_{[2]ij}\ s_{[2]kl}\ s_{[2]mn}\
s_{[2]st}\\!\right)_{\rm sym}^{(8)}\\!\\!,\ a_{[3]}\\!\left(\\!s_{[3]ijk}\
s_{[3]lmn}\ s_{[2]st}\\!\right)_{\rm sym}^{(8)}\\!\\!,\
\left(\\!a_{[4]i}s_{[2]jk}\ s_{[3]lmn}\ s_{[2]st}\\!\right)_{\rm
sym}^{(8)}\\!\\!,\\\ &\left(a_{[5]ij}\ s_{[2]kl}\ s_{[2]mn}\
s_{[2]st}\\!\right)_{\rm sym}^{(8)}\\!,\ \left(a_{[5]ij}\ s_{[3]klm}\
s_{[3]nst}\\!\right)_{\rm sym}^{(8)}\\!\\!,\ \left(\\!a_{[6]ijk}\ s_{[3]lmn}\
s_{[2]st}\\!\right)_{\rm sym}^{(8)}\\\ \hline\cr\hline\cr
1^{-+}_{[7]}&\epsilon_{ist}\ a_{[5]sj}\ s_{[2]jt},\quad\epsilon_{ist}\
a_{[4]s}\ v_{[3]t}\\\ \hline\cr 3^{-+}_{[7]}&\left(\epsilon_{ist}\ a_{[5]sj}\
s_{[2]kt}\right)_{\rm sym}^{(3)},\ \left(\epsilon_{ist}\ a_{[4]s}\
s_{[3]tjk}\right)_{\rm sym}^{(3)}\\\ \hline\cr
5^{-+}_{[9]}&\left(\epsilon_{ist}\ a_{[4]s}\ s_{[3]tjk}\ s_{[2]lm}\right)_{\rm
sym}^{(5)},\ \left(\epsilon_{ist}\ a_{[4]s}\ s_{[3]jkl}\ s_{[2]tm}\right)_{\rm
sym}^{(5)},\ \left(\epsilon_{ist}\ a_{[5]sj}\ s_{[2]kt}\ s_{[2]lm}\right)_{\rm
sym}^{(5)},\\\ &\left(\epsilon_{ist}\ a_{[6]sjk}\ s_{[3]tlm}\right)_{\rm
sym}^{(5)}\\\ \hline\cr 7^{-+}_{[11]}&a_{[3]}\left(\epsilon_{ist}\ s_{[3]sjk}\
s_{[3]lmn}\ s_{[2]rt}\right)_{\rm sym}^{(7)},\left(\epsilon_{ist}\ a_{[4]s}\
s_{[3]tjk}\ s_{[2]lm}\ s_{[2]nr}\right)_{\rm sym}^{(7)},\left(\epsilon_{ist}\
a_{[4]s}\ s_{[3]jkl}\ s_{[2]tm}\ s_{[2]nr}\right)_{\rm sym}^{(7)},\\\
&\left(\epsilon_{ist}\ a_{[5]sj}\ s_{[2]kl}\ s_{[2]mn}\ s_{[2]rt}\right)_{\rm
sym}^{(7)},\ \left(\epsilon_{ist}\ a_{[5]sj}\ s_{[3]tkl}\
s_{[3]mnr}\right)_{\rm sym}^{(7)},\\\ &\left(\epsilon_{ist}\ a_{[6]sjk}\
s_{[3]lmn}\ s_{[2]rt}\right)_{\rm sym}^{(7)},\ \left(\epsilon_{ist}\
a_{[6]jkl}\ s_{[3]smn}\ s_{[2]rt}\right)_{\rm sym}^{(7)}\\\
\hline\cr\end{array}$

Table 6c: Lowest order monomials for the $J^{-+}$ sector, for even and odd
spin $J$.

$\begin{array}[]{|c|l|l}J^{+-}&M^{(J)+-}\\\ \hline\cr 1^{+-}_{[4]}&a_{[4]i}\\\
\hline\cr 3^{+-}_{[6]}&a_{[3]}\ s_{[3]ijk}^{(3)}~{},\quad\left(a_{[4]i}\
s_{[2]jk}\right)_{\rm sym}^{(3)},\ a_{[6]ijk}^{(3)}\\\ \hline\cr
5^{+-}_{[8]}&a_{[3]}\left(s_{[3]ijk}\ s_{[2]lm}\right)_{\rm
sym}^{(5)},\quad\left(a_{[4]i}\ s_{[2]jk}\ s_{[2]lm}\right)_{\rm
sym}^{(5)},\quad\left(a_{[5]ij}\ s_{[3]klm}\right)_{\rm sym}^{(5)},\
\left(a_{[6]ijk}\ s_{[2]lm}\right)_{\rm sym}^{(5)}\\\ \hline\cr
7^{+-}_{[10]}&a_{[3]}\left(s_{[3]ijk}\ s_{[2]lm}\ s_{[2]ns}\right)_{\rm
sym}^{(7)},\ \left(a_{[4]i}\ s_{[2]jk}\ s_{[2]lm}\ s_{[2]ns}\right)_{\rm
sym}^{(7)},\ \left(a_{[4]i}\ s_{[3]jkl}\ s_{[3]mns}\right)_{\rm sym}^{(7)},\\\
&\left(\ a_{[5]ij}s_{[3]klm}\ s_{[2]ns}\right)_{\rm sym}^{(7)},\
\left(a_{[6]ijk}\ s_{[2]lm}\ s_{[2]ns}\right)_{\rm sym}^{(7)}\\\
\hline\cr\hline\cr 0^{+-}_{[10]}&\epsilon_{ist}\ a_{[5]il}\ s_{[3]slm}\
s_{[2]tm},\quad\epsilon_{ist}\ a_{[5]il}\ v_{[3]s}\
s_{[2]tl},\quad\epsilon_{ist}\ b_{[4]il}\ a_{[4]s}\ s_{[2]tl}\\\ \hline\cr
2^{+-}_{[6]}&\left(\epsilon_{ist}\ a_{[4]s}\ s_{[2]jt}\right)_{\rm
sym}^{(2)}\\\ \hline\cr 4^{+-}_{[8]}&a_{[3]}\left(\epsilon_{ist}\ s_{[3]sjk}\
s_{[2]lt}\right)_{\rm sym}^{(4)}~{},\left(\epsilon_{ist}\ a_{[4]s}\ s_{[2]lt}\
s_{[2]kl}\right)_{\rm sym}^{(4)},\left(\epsilon_{ist}\ a_{[2]sj}\
s_{[3]klm}\right)_{\rm sym}^{(4)},\left(\epsilon_{ist}\ a_{[6]sjk}\
s_{[2]lt}\right)_{\rm sym}^{(4)}\\\ \hline\cr
6^{+-}_{[10]}&a_{[3]}\left(\epsilon_{ist}\ s_{[3]sjk}\ s_{[2]lt}\
s_{[2]mn}\right)_{\rm sym}^{(6)},\left(\epsilon_{ist}\ a_{[4]s}\ s_{[3]tjk}\
s_{[3]lmn}\right)_{\rm sym}^{(6)},\left(\epsilon_{ist}\ a_{[4]s}\ s_{[2]tj}\
s_{[2]kl}\ s_{[2]mn}\right)_{\rm sym}^{(6)},\\\ &\left(\epsilon_{ist}\
s_{[3]sjk}\ a_{[5]lt}\ s_{[2]mn}\right)_{\rm sym}^{(6)},\left(\epsilon_{ist}\
s_{[3]sj}\ a_{[5]kt}\ s_{[3]lmn}\right)_{\rm sym}^{(6)},\left(\epsilon_{ist}\
a_{[6]sjk}\ s_{[2]lt}\ s_{[2]mn}\right)_{\rm sym}^{(6)}\\\
\hline\cr\end{array}$

Table 6d: Lowest order monomials for the $J^{+-}$ sector, for odd and even
spin $J$.

## References

  * [1] G.Z. Basean, S.G. Matinyan and G.K. Savvidi, JETP Let. 29 (1979) 587.
  * [2] H.M. Asatryan and G. K. Savvidy, Phys. Lett. A99 (1983) 290.
  * [3] B. Simon, Ann. Phys. 146 (1983) 209.
  * [4] G. K. Savvidy, Phys. Lett. 159B (1985) 325.
  * [5] B.V. Medvedev, Theor. Math. Phys. 60 (1985) 782.
  * [6] Yu. Simonov, Sov. J. Nucl. Phys. 41 (1985) 835.
  * [7] C. Martin and D. Vautherin, Nucl. Phys. B (Proc. Suppl.) 39 B,C (1995) 231.
  * [8] M. Lüscher, Nucl. Phys. B219 (1983) 233.
  * [9] M. Lüscher and G. Münster, Nucl. Phys. B232 (1984) 445.
  * [10] J. Koller and P. van Baal, Nucl. Phys. B273 (1986) 387; Nucl. Phys. B302 (1988) 1; P. van Baal and J. Koller, Ann. of Phys. (N.Y.) 174 (1987) 299.
  * [11] P. Weisz and V. Ziemann, Nucl. Phys. B284 (1987) 157.
  * [12] A.M. Khvedelidze and H.-P. Pavel, Phys. Rev. D 59 (1999) 105017.
  * [13] A.M. Khvedelidze, D. M. Mladenov, H.-P. Pavel, and G. Röpke, Phys. Rev. D 67 (2003) 105013.
  * [14] H.-P. Pavel, Phys. Lett. B 648 (2007) 97-106.
  * [15] H.-P. Pavel, Phys. Lett. B 685 (2010) 353-364.
  * [16] H.-P. Pavel, Phys. Lett. B 700 (2011) 265-276.
  * [17] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197; J. Math. Phys. 12 (1971) 419.
  * [18] H.-P. Pavel, Unconstrained Hamiltonian formulation of low-energy SU(3) Yang-Mills quantum theory, arXiv: 1205.2237v1 [hep-th] (2012).
  * [19] H.-P. Pavel, PoS Confinement X (2012) 071, arXiv: 1303.3763 v1 [hep-th] (2013).
  * [20] H.-P. Pavel, EPJ Web of Conferences 71 (2014) 00104 , arXiv: 1405.1970v1 [hep-th] (2014).
  * [21] H.-P. Pavel, ”SU(3) Yang-Mills Hamiltonian in the flux-tube gauge: Strong coupling expansion and glueball dynamics.” arXiv:1611.06542 [hep-th] .
  * [22] C. J. Morningstar and M.J. Peardon, Phys. Rev. D 60 (1999) 034509 [hep-lat/9901004].
  * [23] Y. Chen et al., Phys. Rev. D 73 (2006) 014516 [hep-lat/0510074].
  * [24] N.H. Christ and T.D. Lee, Phys. Rev. D 22 (1980) 939.
  * [25] P. Dittner, Commun. Math. Phys. 22 (1971) 238; ibidem 27 (1972) 44.