Comments on noncommutative quantum mechanical systems associated with Lie
algebras

Andrei Smilga

SUBATECH, Université de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307,
France.

We consider quantum mechanics on the noncommutative spaces characterized by
the commutation relations

$[x_{a},x_{b}]\ =\ i\theta f_{abc}x_{c}\,,$

where $f_{abc}$ are the structure constants of a Lie algebra. We note that
this problem can be reformulated as an ordinary quantum problem in a commuting
momentum space. The coordinates are then represented as linear differential
operators $\hat{x}_{a}=-i\hat{D}_{a}=-iE_{ab}(p)\,\partial/\partial p_{b}$.
Generically, the matrix $E_{ab}(p)$ represents a certain infinite series over
the deformation parameter $\theta$: $E_{ab}=\delta_{ab}+\ldots$. The deformed
Hamiltonian, $\hat{H}\ =\ -\frac{1}{2}\hat{D}_{a}^{2}\,,$ describes the motion
along the corresponding group manifolds with the characteristic size of order
$\theta^{-1}$. Their metrics are also expressed into certain infinite series
in $\theta$, with $E_{ab}$ having the meaning of vielbeins.

For the algebras $su(2)$ and $u(N)$, it has been possible to represent the
operators $\hat{x}_{a}$ in a simple finite form. A byproduct of our study are
new nonstandard formulas for the metrics on all the spheres $S^{n}$, on the
corresponding projective spaces $RP^{n}$ and on $U(2)$.

##  1 Introduction

The quantum mechanical systems defined on noncommutative spaces [1] have
recently attracted a considerable interest [2].111Also noncommutative field
theories have been intensely studied [3], but they are not the subject of our
present discussion. The simplest such system involves only two dynamic
variables $x,y$ with a nontrivial commutator

$\displaystyle[x,y]\ =\ i\theta\,.$ (1.1)

Probably, the best way to deal with such a system [4] is to go over to the
momentum space, $(p_{x},p_{y})\equiv(X,Y)$, in which case $X$ and $Y$ commute,
while the original noncommuting coordinates $x,y$ can be interpreted as
differential operators, which are quantum conterparts of the velocity
components of the moving particle of unit mass.

$\displaystyle x$ $\displaystyle\equiv$ $\displaystyle\hat{v}_{X}\ =\
-i\frac{\partial}{\partial X}+\frac{\theta}{2}Y\,,$ $\displaystyle y$
$\displaystyle\equiv$ $\displaystyle\hat{v}_{Y}\ =\ -i\frac{\partial}{\partial
Y}-\frac{\theta}{2}X\,.$

A naturally defined Hamiltonian,

$H\ =\ \frac{1}{2}(\hat{v}_{X}^{2}+\hat{v}_{Y}^{2})\,,$

describes then the motion on the plane $(X,Y)$ with a homogeneous orthogonal
magnetic field $\theta$ of a particle of unit mass. The canonical momenta are
$\hat{P}_{X}=\hat{v}_{X}-\theta Y/2$ and $\hat{P}_{Y}=\hat{v}_{Y}+\theta X/2$.

##  2 Noncommutative systems involving the Lie algebra commutators

The R.H.S. of the commutator (1.1) is constant. In this case,
$\partial_{x}\,[x,y]=\partial_{y}\,[x,y]=0$, which is true if the derivative
operator satisfies the ordinary Leibnitz rules. More general, for a space of
arbitrary dimension with the constant commutators
$[x_{a},x_{b}]=i\theta_{ab}$, one can still consistently postulate

$[\partial_{a},x_{b}]=\delta_{ab},\qquad[\partial_{a},\partial_{b}]=0$

and observe that the Jacobi identities still hold.

But it is not generically true for more complicated noncommutative spaces when
$[x_{a},x_{b}]$ exhibit a nontrivial coordinate dependence. In this paper, we
consider a particular class of such spaces characterised by the commutators

$\displaystyle[x_{a},x_{b}]\ =\ i\theta f_{abc}\,x_{c}\,,$ (2.1)

where $f_{abc}$ are structure constants of an arbitrary Lie algebra. Such
spaces were considered in [5] where it was shown that, to satisfy the Jacobi
identities, one has to postulate the following nontrivial commutators:

$\displaystyle[\tilde{\partial}_{a},x_{b}]\ =\ \left[\frac{i\theta
F(\tilde{\partial})}{e^{i\theta F(\tilde{\partial})}-1}\right]_{ab}\ =\
\delta_{ab}+\sum_{n=1}^{\infty}(-i\theta)^{n}\frac{B_{n}^{+}}{n!}[F^{n}(\tilde{\partial})]_{ab},\qquad[\tilde{\partial}_{a},\tilde{\partial}_{b}]=0\,,$
(2.2)

where

$\displaystyle F_{ab}(\tilde{\partial})=f_{acb}\,\tilde{\partial}_{c}$ (2.3)

and $B_{n}^{+}$ are the Bernoulli numbers.222 The Bernoulli numbers are
defined from the Taylor expansion,  $\frac{t}{1-e^{-t}}\ =\
\sum_{n=0}^{\infty}\frac{B^{+}_{n}t^{n}}{n!}\,.$ The first few Bernoulli
numbers are $B_{0}^{+}=1,B_{1}^{+}=1/2,B_{2}^{+}=1/6,B_{3}^{+}=0$ and
$B_{4}^{+}=-1/30$. Note that $B^{+}_{2m+1}$ all vanish for $m>0$.

The Jacobi identities for the algebra (2.1), (2.2) are satisfied, as one can
be convinced quite directly in each order of the expansion in $\theta$, using
the recursive relations for the Bernoulli numbers. In the limit $\theta\to 0$,
the algebra elements $\tilde{\partial}_{a}$ acquire the meaning of the
ordinary partial derivatives $\partial_{a}$ with respect to the commuting
variables $x_{a}$.

In the spirit of our remark in the Introduction, we perform the quantum
canonical transformation,

$\displaystyle\hat{p}_{a}=-i\tilde{\partial}_{a}\ $ $\displaystyle\to$
$\displaystyle\ -X_{a},$ $\displaystyle x_{a}\ $ $\displaystyle\to$
$\displaystyle\ \hat{P}_{a},$ (2.4)

so that the elements $\tilde{\partial}_{a}$ are traded for the commuting
coordinates $X_{a}$, whereas the original coordinates $x_{a}$ are realized as
the linear differential operators,

$\displaystyle x_{a}\to-i\hat{D}_{a}\ =\ -i\left[\frac{F(\theta
X)}{1-e^{-F(\theta X)}}\right]_{ab}\frac{\partial}{\partial X_{b}}\,.$ (2.5)

The operators $\hat{D}_{a}$ satisfy the relations

$\displaystyle[\hat{D}_{a},\hat{D}_{b}]\ =\ -\theta f_{abc}\hat{D}_{c}\,.$
(2.6)

In other words, we have constructed a representation of the generators of an
arbitrary Lie group not in the matrix form, as is usually done, but in the
form of linear differential operators acting on the space of the dimension
coinciding with the dimension of the group. The operators $\hat{D}_{a}$ have a
transparent geometric meaning. They are none other than the generators of the
right group rotations. Indeed, one can show that

$\displaystyle e^{i\theta X_{a}t_{a}}e^{i\theta\epsilon_{a}t_{a}}\ =\
e^{i\theta t_{a}(X_{a}+\delta X_{a})}\,,$ (2.7)

where333A simple derivation of this relation (which represents a particular
case of the general Baker-Campbell-Hausdorff formula) is given in the
appendix.

$\displaystyle\delta X_{a}\ =\ \sum_{n=0}^{\infty}\frac{B_{n}^{+}[F^{n}(\theta
X)]_{ab}}{n!}\epsilon_{b}\ +o(\epsilon)\,.$ (2.8)

And this is, of course, the reason why $\hat{D}_{a}$ satisfy the Lie algebra
commutation relations.

This paper is devoted to studying the properties of the Hamiltonian

$\displaystyle\hat{H}\ =\ -\frac{1}{2}\hat{D}_{a}^{2}\,.$ (2.9)

The classical Hamiltonian corresponding to the quantum Hamiltonian (2.9) reads

$\displaystyle H^{\rm cl}\ =\ \frac{1}{2}g^{jk}P_{j}P_{k}\,,$ (2.10)

where

$\displaystyle g^{jk}\ =\ E_{a}{}^{j}E_{a}{}^{k}$ (2.11)

with

$\displaystyle E_{a}{}^{j}\ =\ \left[\frac{F(\theta X)}{1-e^{-F(\theta
X)}}\right]_{a}^{\ j}\,.$ (2.12)

This Hamiltonian describes the motion over the manifold with the inverse
metric $g^{jk}$. The objects $E_{a}{}^{j}$ have the meaning of the
vielbeins.444Having said that, we went to the middle of the alphabet to denote
the world tensor indices, while keeping its beginning for the flat tangent
space indices. The classical Lagrangian corresponding to the Hamiltonian
(2.10) depends on the covariant metric tensor and reads

$\displaystyle L\ =\ \frac{1}{2}g_{jk}\dot{X}^{j}\dot{X}^{k}\,.$ (2.13)

######  Theorem 1.

The metric $g_{jk}$ in (2.13) coincides with the invariant metric on the group
manifold.

###### Proof.

The canonical metric on a group manifold, which is invariant under the left
and right group rotations, is555It is familiar to physicists, appearing e.g.
in the effective chiral Lagrangian in QCD [6],  ${\cal L}\ \propto\ {\rm
Tr}\left\\{\partial_{\mu}\omega^{-1}\partial^{\mu}\omega\right\\}$ with
$\omega\in SU(2)$ or $SU(3)$.

$\displaystyle g_{jk}\ =\ \frac{1}{h\theta^{2}}{\rm
Tr}\left\\{\partial_{j}\omega^{-1}\partial_{k}\omega\right\\}\,,$ (2.14)

where

$\displaystyle\omega(X)\ =\ \exp\\{i\theta X^{j}t_{j}\\}$ (2.15)

and $t_{j}$ are the generators in a given representation with the Dynkin index
$h$. Then $g_{jk}(X=0)=\delta_{jk}$.

Consider the distance between two close points $\omega(X+\delta X)$ and
$\omega(X)$. For the invariant metric, this distance is the same as the
distance between $\omega^{-1}(X)\omega(X+\delta X)=\omega(\epsilon)$ and the
group unity. The latter is $ds^{2}=\epsilon_{a}\epsilon_{a}$. To find the
metric at the vicinity of $X$, we have only express $\epsilon_{a}$ in terms of
$\delta X^{j}$. To do so, we use the formula [7] (its derivation is explained
in the appendix)

$\displaystyle\frac{1}{\theta}\,\omega^{-1}(-i\partial_{j}\omega)\ =\
\int_{0}^{1}d\tau\,e^{-\tau R}t_{j}e^{\tau R}\ =\
t_{j}-\frac{1}{2}[R,t_{j}]+\frac{1}{6}[R,[R,t_{j}]]-\ldots\ =\
E_{j}{}^{a}t_{a}$ (2.16)

where $R=i\theta t_{j}X^{j}$ and

$\displaystyle E_{j}{}^{a}\ =\ \left[\frac{1-e^{-F(\theta X)}}{F(\theta
X)}\right]_{j}^{\ a}\ =\ \sum_{n=0}^{\infty}\frac{[F^{n}(-\theta
X)]_{j}{}^{a}}{(n+1)!}$ (2.17)

is the inverse vielbein. Then $\epsilon^{a}=E_{j}{}^{a}\delta X^{j}$ and the
metric is

$\displaystyle g_{jk}=E_{j}{}^{a}E_{k}{}^{a}\,,$ (2.18)

which matches (2.11).

∎

Thus, the classical Hamiltonian (2.10) describes the motion of a particle
along the group manifold.

For $SU(2)\equiv S^{3}$, the sums for the vielbeins in (2.12) and (2.17) can
be done:666This formula coincides with Eq. (4.80) in recent [8], though its
interpretation there was completely different.

$\displaystyle E_{a}{}^{j}\ $ $\displaystyle=$ $\displaystyle\
\delta_{aj}+\frac{\theta}{2}\varepsilon_{apj}X_{p}+\left(\delta_{ja}-\frac{X_{a}X_{j}}{X^{2}}\right)\left[\frac{\theta
X/2}{\tan(\theta X/2)}-1\right]$ $\displaystyle E_{j}{}^{a}\ $
$\displaystyle=$ $\displaystyle\ \delta_{ja}-\frac{2\sin^{2}(\theta
X/2)}{\theta
X^{2}}\varepsilon_{jpa}X_{p}+\left(\delta_{ja}-\frac{X_{a}X_{j}}{X^{2}}\right)\left[\frac{\sin(\theta
X)}{\theta X}-1\right]\,,$ (2.19)

where $X=\sqrt{X_{p}X_{p}}$ and the positions of the indices in the right-hand
sides do not have a tensorial meaning and are irrelevant.

The metric (2.11), (2.18) reads

$\displaystyle g^{jk}\ $ $\displaystyle=$ $\displaystyle\
A^{-1}(X)\left(\delta_{jk}-\frac{X_{j}X_{k}}{X^{2}}\right)+\frac{X_{j}X_{k}}{X^{2}},$
$\displaystyle g_{jk}\ $ $\displaystyle=$ $\displaystyle\
A(X)\left(\delta_{jk}-\frac{X_{j}X_{k}}{X^{2}}\right)+\frac{X_{j}X_{k}}{X^{2}}\,,$
(2.20)

where

$A(X)\ =\ \frac{4\sin^{2}(\theta X/2)}{\theta^{2}X^{2}}\,.$

At the vicinity of the origin,

$\displaystyle g_{jk}\ =\
\delta_{jk}+\frac{\theta^{2}}{12}(X_{j}X_{k}-X^{2}\delta_{jk})+O(X^{4})\,.$
(2.21)

The corresponding scalar curvature is $R=3\theta^{2}/2$. Bearing in mind that
$R=6/\rho^{2}$ where $\rho$ is the radius of $S^{3}$, we derive

$\displaystyle\rho\ =\ \frac{2}{\theta}\,.$ (2.22)

The metric (2.18) looks quite different from the familiar conformally flat
metric on $S^{3}$,

$\displaystyle g_{jk}\ =\ \frac{\delta_{jk}}{[1+Z^{2}/(4\rho^{2})]^{2}}\,.$
(2.23)

But these two metrics describe the same geometry, which means that the only
difference between them is the choice of coordinates $Z\to X$ [see Eqs.
(2.35),(2.36) below].

Consider now the quantum Hamiltonian (2.9) implying a particular way of
ordering of the coordinates and canonical momenta.

######  Theorem 2.

The Hamiltonian (2.9) coincides up to the factor $-1/2$ with the Laplace-
Beltrami operator on the group manifold,

$\displaystyle\triangle\ =\
\frac{1}{\sqrt{g}}\partial_{j}(\sqrt{g}g^{jk})\partial_{k}\,,$ (2.24)

with the invariant metric (2.18).

###### Proof.

Clearly the operator $(\hat{D}_{a})^{2}$ commutes with $\hat{D}_{a}$. This
means that the Hamiltonian (2.9) is invariant under right group rotations. But
it must also be invariant under left group rotations,

$\displaystyle e^{i\theta t_{j}X^{j}}\ \to\ e^{i\theta
t_{j}\epsilon^{j}}e^{i\theta t_{j}X^{j}}\,.$ (2.25)

Indeed, right and left rotations commute and their generators $\hat{D}_{a}$
and $\hat{D}^{\prime}_{a}$ must also commute. Both of them can be represented
as infinite sums

$\displaystyle\hat{D}_{a}\ $ $\displaystyle=$ $\displaystyle\
\sum_{n=0}^{\infty}\frac{B_{n}^{+}\,\theta^{n}[F^{n}(X)]_{a}{}^{p}}{n!}\,\partial_{p}\,,$
$\displaystyle\hat{D}^{\prime}_{a}\ $ $\displaystyle=$ $\displaystyle\
\sum_{n=0}^{\infty}\frac{B_{n}^{-}\,\theta^{n}[F^{n}(X)]_{a}{}^{p}}{n!}\,\partial_{p}\,.$
(2.26)

The only difference is that the sum for $\hat{D}_{a}$ involves the Bernoulli
numbers $B_{n}^{+}$, while the sum for $\hat{D}^{\prime}_{a}$ involves the
Bernoulli numbers $B_{n}^{-}$. The latter coincide with $B_{n}^{+}$ for all
$n$ except $n=1$ where $B_{1}^{-}=-B_{1}^{+}=-1/2$.

Left and right rotations commute as well as their generators. The vanishing of
the commutator $[\hat{D}^{\prime}_{a},\hat{D}_{b}]\ =\ 0$ entails the
vanishing of $[\hat{D}^{\prime}_{a},\hat{H}^{\rm qu}]$. The only second order
differential operator that is invariant under both left and right group
rotations is, up to a constant, the Laplace-Beltrami operator.

∎

Note that the operators $\hat{D}^{\prime}_{a}$ satisfy the commutation
relations

$\displaystyle[\hat{D}^{\prime}_{a},\hat{D}^{\prime}_{b}]\ =\ \theta
f_{abc}\hat{D}^{\prime}_{c}$ (2.27)

with the opposite sign, compared to (2.6). The origin of this extra sign is
quite clear bearing in mind (2.7) and (2.25). The composition of two right
rotations $g_{1}$ and $g_{2}$ is $g_{1}g_{2}$, while for left rotations it is
$g_{2}g_{1}$.

Another remark is the following. The Hamiltonian (2.24) seems to be not
Hermitian. But it is nothing but an “optical illusion”. It is Hermitian in the
Hilbert space where the wave functions are normalized with the covariant
measure,

$\displaystyle\int d^{d}X\sqrt{g}\left|\Psi(X)\right|^{2}\ =\ 1\,,$ (2.28)

$g=\det\|g_{jk}\|$. In the Hilbert space with the flat measure $d^{d}X$, the
Hamiltonian would include extra wrapping factors: $H^{\rm flat}=g^{1/4}H^{\rm
flat}g^{-1/4}$.

The same concerns the operator $\hat{D}_{a}$ defined in (2.5): it is anti-
Hermitian in the Hilbert space including the factor $\sqrt{g}$ in the measure.

We hasten to comment, however, that the representation (2.5) for $\hat{D}_{a}$
is not unique. For example, for $su(2)$, the following nice representation is
possible:777Cf. Eq. (37) in Ref. [9].

$\displaystyle\hat{D}_{a}\ =\ \frac{\partial}{\partial
Y_{a}}-\frac{\theta}{2}\varepsilon_{abc}Y_{b}\,\frac{\partial}{\partial
Y_{c}}+\frac{\theta^{2}}{4}Y_{a}Y_{b}\,\frac{\partial}{\partial Y_{b}}\,.$
(2.29)

The algebra (2.6) still holds.

The operator $(\hat{D}_{a})^{2}$ describes then the motion of a particle over
the 3-dimensional manifold with the metric

$\displaystyle g^{jk}$ $\displaystyle=$
$\displaystyle\delta^{jk}+\kappa(Y^{2}\delta^{jk}+Y^{j}Y^{k})+\kappa^{2}Y^{2}Y^{j}Y^{k}\,,$
$\displaystyle g_{jk}$ $\displaystyle=$
$\displaystyle\frac{\delta_{jk}}{1+\kappa Y^{2}}-\frac{\kappa
Y_{j}Y_{k}}{(1+\kappa Y^{2})^{2}}\,,$ (2.30)

where $\kappa=\theta^{2}/4$.

The corresponding Christoffel symbols are

$\displaystyle\Gamma^{p}_{kl}\ =\ -\frac{\kappa}{1+\kappa
Y^{2}}(Y_{k}\delta_{pk}+Y_{l}\delta_{pk})\,.$ (2.31)

The Riemann tensor reads

$\displaystyle R^{i}_{kpl}\ =\
\partial_{p}\Gamma^{i}_{kl}-\partial_{l}\Gamma_{kp}^{i}+\Gamma_{np}^{i}\Gamma_{kl}^{n}-\Gamma_{nl}^{i}\Gamma_{kp}^{n}$
$\displaystyle=\ \frac{\kappa}{1+\kappa
Y^{2}}(\delta_{kl}\delta_{ip}-\delta_{kp}\delta_{il})+\frac{\kappa^{2}Y_{k}(Y_{p}\delta_{il}-Y_{l}\delta_{ip})}{(1+\kappa
Y^{2})^{2}}\ =\ \kappa(\delta^{i}_{p}g_{kl}-\delta^{i}_{l}g_{kp})\,.$ (2.32)

Hence $R_{ikpl}\ =\ \kappa(g_{ip}g_{kl}-g_{il}g_{kp})$.

It follows that all the curvature invariants are constant:

$\displaystyle R\ =\ 6\kappa=\frac{3\theta^{2}}{2},\qquad R_{kl}R^{kl}\ =\
12\kappa^{2}\ =\ \frac{3\theta^{4}}{4},\qquad R_{kp}R^{pn}R_{n}^{k}\ =\
24\kappa^{3}\ =\ \frac{3\theta^{6}}{8},$ $\displaystyle R_{ikpl}R^{ikpl}\ =\
12\kappa^{2}\ =\ \frac{3\theta^{4}}{4}\,,\quad\ldots$ (2.33)

One is then tempted to say that our manifold represents the “round” 3-sphere
of radius (2.22), but it is not exactly so. The determinant of the metric is

$g\ =\ \frac{1}{(1+\kappa Y^{2})^{4}}.$

The volume of the manifold,

$\displaystyle V\ =\ \iiint_{-\infty}^{\infty}\sqrt{g}\,d^{3}x\ =\
\pi^{2}\rho^{3}\,,$ (2.34)

is two times smaller than the volume of $S^{3}$, which means that we are
dealing with the projective space $RP^{3}=S^{3}/Z_{2}\equiv SO(3)$. In fact,
one can confirm it, writing an explicit coordinate change [10], that brings
the metric (2) in the form (2):

$\displaystyle Y^{p}\ =\ X^{p}\,\frac{2\tan\left(\theta X/2\right)}{\theta
X}\,.$ (2.35)

Then $Y^{p}=\infty$ corresponds to $X=\sqrt{X^{p}X^{p}}=\pi/\theta$. And the
latter value corresponds to the equator on $S^{3}$ in the exponential
parameterization (2.15).

Another variable change,

$\displaystyle Y^{p}\ =\ Z^{p}\,\frac{1}{1-\kappa Z^{2}/4}\,,$ (2.36)

brings the metric (2) in the habitual conformally flat form.

###  2.1 Higher spheres

The expressions (2) and (2) describe the metric of $S^{n}$ in any dimension.
Indeed, the stereographic projection leading to the metric (2.23) can be
performed in any dimensions, and the variable changes (2.36) (2.35) bring it
in the forms (2), (2) in any dimension.

However, the manifolds $S^{3}$ and $S^{7}$ are special. These spheres are
parallelizable and admit quaternion (resp. octonion) algebraic structure. The
case of $S^{3}$ we already discussed. It is just a group manifold and the
Hamiltonian describing its motion is related to the Lie algebra (2.6). But the
motion over $S^{7}$ can also be associated with a not so simple nonlinear
algebra.

Define the operators $\hat{D}_{A}=E_{A}^{J}\partial_{J}$ with the vielbeins

$\displaystyle E_{A}{}^{J}\ =\
\delta_{AJ}+\frac{\theta}{2}\eta_{APJ}X_{P}+\left(\delta_{AJ}-\frac{X_{A}X_{J}}{X^{2}}\right)\left[\frac{\theta
X/2}{\tan(\theta X/2)}-1\right]\,.$ (2.37)

They have the same form as in (2), but $X_{J}$ are now the coordinates on
$S^{7}$ and the antisymmetric tensor $\eta_{ABC}$ defines the rules of
octonion multiplication,888Different choices for $\eta_{ABC}$ are possible.
One of them is
$\eta_{123}=\eta_{435}=\eta_{147}=\eta_{165}=\eta_{246}=\eta_{123}=\eta_{367}\
=\ 1\,,$ the other nonzero components being restored by antisymmetry.

$e_{A}e_{B}=-\delta_{AB}+\eta_{ABC}e_{C}$. Using the identity

$\displaystyle\eta_{ABC}\eta_{DEC}\ =\
\delta_{AD}\delta_{BE}-\delta_{AE}\delta_{BD}-\eta_{ABDE}$ (2.38)

with

$\displaystyle\eta_{ABDE}\ =\ \frac{1}{6}\epsilon_{ABDEFGH}\,\eta_{FGH},$
(2.39)

one can be convinced that the convolution of the vielbeins (2.37) gives the
metric on $S^{7}$, which, as we have mentioned, has the same form as in (2),
but with seven coordinates.

The difference is that the operators $\hat{D}_{A}$ do not form a Lie algebra
as in (2.6), but their commutators have a more complicated form with the
coordinate dependence in the right hand side:999In different terms, this
result was derived in [11] using (2.38) and the identity
$\eta_{ABC}\,\eta_{AFDE}\ =\
\delta_{CF}\,\eta_{BDE}+\delta_{CD}\,\eta_{BEF}+\delta_{CE}\,\eta_{BFD}-(B\leftrightarrow
C)\,,$ but we translated it in our notation.

$\displaystyle[\hat{D}_{A},\hat{D}_{B}]\ $ $\displaystyle=$ $\displaystyle\
-\theta\eta_{ABC}\hat{D}_{C}+\frac{\theta\sin(\theta
X)}{X}\,\eta_{ABCP}X_{P}\hat{D}_{C}$ (2.40) $\displaystyle+$ $\displaystyle
2\frac{\theta\sin^{2}(\theta
X/2)}{X^{2}}\,\eta_{ABCP}\eta_{CDQ}X_{P}X_{Q}\hat{D}_{D}\,.$

##  3 Gurevich-Saponov model.

###  3.1 $u(2)$ model.

In Refs. [12, 13], a noncommutative quantum mechanical system was studied with
the following nontrivial commutation relations:

$\displaystyle\,[x_{a},x_{b}]\ =\
i\theta\epsilon_{abc}x_{c},\qquad[x_{a},\tau]\ =\ 0\,,$ (3.1)

supplemented by

$\displaystyle[\tilde{\partial}_{a},x_{b}]\ =\
\delta_{ab}\left(1+\frac{i\theta}{2}\tilde{\partial}_{\tau}\right)+\frac{i\theta}{2}\epsilon_{abc}\tilde{\partial}_{c},\qquad[\tilde{\partial}_{\tau},x_{a}]\
=\ -\frac{i\theta}{2}\tilde{\partial}_{a},$
$\displaystyle[\tilde{\partial}_{\tau},\tau]\ =\
1+\frac{i\theta}{2}\tilde{\partial}_{\tau},\qquad[\tilde{\partial}_{a},\tau]\
=\
\frac{i\theta}{2}\tilde{\partial}_{a}\,,\qquad[\tilde{\partial}_{a},\tilde{\partial}_{b}]\
=\ [\tilde{\partial}_{a},\tilde{\partial}_{\tau}]\ =\ 0\,.$ (3.2)

It is easy to check that the Jacobi identities hold. This model belongs to the
large class of the models (2.1), but the algebra here is $u(2)$, which is not
simple. Note that the R.H.S. of the commutators in (3.1) does not represent an
infinite series as in (2.2), but has a simple form involving only the first
generalized derivatives [cf. Eq. (2.29), which, in contrast to (2.5), involves
only the linear and quadratic terms in $\tilde{\partial}_{a}\to-iY_{a}$ ].

Proceeding in the same way as before, we may trade the coordinates for momenta
and vice versa to arrive at the ordinary QM system with four commuting dynamic
variables $X_{a},T$ and former coordinates and now momenta being realized as
the differential operators101010We added a constant in $P_{0}$ to make it
Hermitian with the flat measure $d\mu=d^{3}XDT$.

$\displaystyle P_{a}\ =\ -i\hat{D}_{a}\ $ $\displaystyle=$ $\displaystyle\
-i\left[\left(1+\frac{\theta T}{2}\right)\frac{\partial}{\partial
X_{a}}-\frac{\theta}{2}X_{a}\frac{\partial}{\partial
T}+\frac{\theta}{2}\epsilon_{abc}X_{b}\frac{\partial}{\partial
X_{c}}\right]\,,$ $\displaystyle P_{0}\ =\ -i\hat{D}_{0}\ $ $\displaystyle=$
$\displaystyle\ -i\left[\left(1+\frac{\theta
T}{2}\right)\frac{\partial}{\partial
T}+\frac{\theta}{2}X_{a}\frac{\partial}{\partial X_{a}}+\theta\right]\,.$
(3.3)

The algebra

$[\hat{D}_{a},\hat{D}_{b}]=-\theta\epsilon_{abc}\hat{D}_{c},\qquad[\hat{D}_{a},\hat{D}_{0}]=0$

holds.

Consider now the operator

$\displaystyle\hat{H}\ =\
-\frac{1}{2}(\hat{D}_{0}\hat{D}_{0}+\hat{D}_{a}\hat{D}_{a})\,.$ (3.4)

Its classical counterpart is

$\displaystyle H^{\rm cl}\ =\ \frac{P_{0}^{2}+P_{a}^{2}}{2}\left[1+\theta
T+\frac{\theta^{2}}{4}(T^{2}+X_{a}^{2})\right]\,.$ (3.5)

This Hamiltonian describes the motion of a particle over a 4-dimensional
manifold with the conformally flat metric

$\displaystyle g_{\mu\nu}=\ \frac{\delta_{\mu\nu}}{F(T,X_{a})}\ =\
\frac{\delta_{\mu\nu}}{1+\theta T+\theta^{2}(T^{2}+X_{a}^{2})/4}\,.$ (3.6)

The Ricci tensor has the following components:

$\displaystyle R_{00}\ =\ \frac{\theta^{4}}{8F^{2}},\qquad R_{0a}\ =\
-\frac{\theta^{3}(1+\theta T/2)X_{a}}{4F^{2}},\qquad R_{ab}\ =\
\frac{\theta^{2}}{2F}\left(\delta_{ab}-\frac{\theta^{2}X_{a}X_{b}}{4F}\right)\,.$
(3.7)

An explicit calculation shows that all the curvature invariants made from the
Ricci tensor are constants:

$\displaystyle R=R_{\mu\nu}g^{\mu\nu}\ =\ \frac{3\theta^{2}}{2}\,,\qquad
R_{\mu\nu}R^{\mu\nu}\ =\ \frac{3\theta^{4}}{4},\qquad
R_{\mu\nu}R^{\nu\rho}R_{\rho}{}^{\mu}\ =\ \frac{3\theta^{6}}{8},\ \ {\rm
etc.}$ (3.8)

They are the same as for a 3-dimensional “round” sphere of radius (2.22) [see
Eq. (2)].

In other words, our manifold is111111It is not $S^{3}\times S^{1}$ or
$SO(3)\times S^{1}$ because the integral  $\int
d^{3}XdT\sqrt{g}=\int\frac{d^{3}XdT}{F^{2}}$ diverges and the manifold is not
compact. $S^{3}\times\mathbb{R}$ or maybe $SO(3)\times\mathbb{R}$ — it is
difficult to say based solely on the form (3.6) of the metric where the
coordinates $X_{a}$ and $T$ are intertwined. This result is quite natural as
the Gurevich-Saponov model involves the nontrivial commutators (3.1) of the
$u(2)$ algebra. After interchanging the coordinates and momenta, we are
arriving at the system that describes the motion along $U(2)$ in a non-compact
realization.

###  3.2 Higher $N$

One can also consider [13] the $u(N)$ generalization of the algebra (3.1),
(3.1). Introduce $N^{2}+N^{2}$ elements $l_{j}^{n}$,
$\tilde{\partial}_{j}^{n}$ and postulate the following commutators:

$\displaystyle\,[l_{j}^{n},\l_{k}^{m}]\ =\
\theta(\delta_{j}^{m}l_{k}^{n}-\delta_{k}^{n}l_{j}^{m}),\qquad[\tilde{\partial}_{j}^{n},l_{k}^{m}]\
=\
\delta_{j}^{m}\delta_{k}^{n}+\theta\delta_{j}^{m}\tilde{\partial}_{k}^{n}\,.$
(3.9)

The first relation is the habitual $u(N)$ algebra. It is supplemented by the
commutation laws for the quasiderivatives $\tilde{\partial}_{i}^{n}$. It is
not difficult to check that the Jacobi identities are satisfied.

To make contact with (3.1), (3.1), we introduce the notation

$\displaystyle x_{A}\ =\
(t_{A})_{n}^{j}\,l_{j}^{n},\qquad\tilde{\partial}_{A}=2(t_{A})_{n}^{j}\,\tilde{\partial}_{j}^{n}\,,$
(3.10)

where $A=(a,0)$ and $t_{A}$ are the generators of $U(N)$ in the fundamental
representation — the Hermitian $N\times N$ matrices normalized in a habitual
way

${\rm Tr}\\{t_{A}t_{B}\\}\ =\ \frac{1}{2}\delta_{AB}\,.$

In particular, $t_{0}=\mathbb{1}/\sqrt{2N}$, so that $x_{0}\equiv
t=l_{j}^{j}/\sqrt{2N}$ and
$\tilde{\partial}_{t}=\sqrt{2/N}\,\tilde{\partial}_{j}^{j}$. It is also
convenient to introduce
$\tau=it,\tilde{\partial}_{\tau}=-i\tilde{\partial}_{t}$. In these terms, the
algebra (3.9) reads

$\displaystyle[x_{a},x_{b}]\ =\ i\theta f_{abc}x_{c},\qquad[x_{a},\tau]\ =\
0,$ $\displaystyle[\tilde{\partial}_{a},x_{b}]\ =\
\delta_{ab}\left(1+\frac{i\theta}{\sqrt{2N}}\,\tilde{\partial}_{\tau}\right)+\frac{i\theta}{2}(f_{abc}-id_{abc})\,\tilde{\partial}_{c},$
$\displaystyle[\tilde{\partial}_{a},\tau]\ =\
-[\tilde{\partial}_{\tau},x_{a}]\ =\
\frac{i\theta}{\sqrt{2N}}\tilde{\partial}_{a},\qquad[\tilde{\partial}_{\tau},\tau]\
=\ 1+\frac{i\theta}{\sqrt{2N}}\,\tilde{\partial}_{\tau}\,,$ (3.11)

where $f_{abc}$ are the $su(N)$ structure constants and $d_{abc}$ is a
symmetric tensor entering the relation

${\rm Tr}\\{t_{a}t_{b}t_{c}\\}\ =\ \frac{1}{4}(if_{abc}+d_{abc})\,.$

For $N=2$, $d_{abc}$ vanishes and (3.2) coincides with (3.1), (3.1).

We perform now our quantum canonical transformation to represent this algebra
in terms of the commuting coordinates (former momenta) $X_{a},T$ and the
differential operators

$\displaystyle x_{a}\to P_{a}\ =\ -i\hat{D}_{a}\ $ $\displaystyle=$
$\displaystyle\ -i\left[\left(1+\frac{\theta
T}{\sqrt{2N}}\right)\frac{\partial}{\partial
X_{a}}-\frac{\theta}{\sqrt{2N}}X_{a}\frac{\partial}{\partial
T}+\frac{\theta}{2}(f_{abc}+id_{abc})X_{b}\frac{\partial}{\partial
X_{c}}\right]\,,$ $\displaystyle\tau\to P_{0}\ =\ -i\hat{D}_{0}\ $
$\displaystyle=$ $\displaystyle\ -i\left[\left(1+\frac{\theta
T}{\sqrt{2N}}\right)\frac{\partial}{\partial
T}+\frac{\theta}{\sqrt{2N}}X_{a}\frac{\partial}{\partial X_{a}}\right]\,.$
(3.12)

The algebra

$[\hat{D}_{a},\hat{D}_{b}]=-\theta
f_{abc}\hat{D}_{c},\qquad[\hat{D}_{a},\hat{D}_{0}]=0$

holds.121212It can be checked explicitly using the identities (see e.g.
Appendix A of Ref. [14])  $\displaystyle
f_{abe}f_{cde}+f_{cae}f_{bde}+f_{bce}f_{ade}\ =\ 0,$ $\displaystyle
f_{abe}d_{cde}+f_{ace}d_{dbe}+f_{ade}d_{bce}\ =\ 0,$ $\displaystyle
f_{abe}f_{cde}\ =\
\frac{2}{N}(\delta_{ac}\delta_{bd}-\delta_{bc}\delta_{ad})+d_{ace}d_{bde}-d_{bce}d_{ade}\,.$
(3.13) Note now the intrinsic complexity $\sim f_{abc}+id_{abc}$ in the
expression for $\hat{D}_{a}$. That means that the would-be vielbein

$\displaystyle E^{j}_{a}\ =\ \left(1+\frac{\theta
T}{\sqrt{2N}}\right)\delta_{aj}+\frac{\theta}{2}(f_{abj}+id_{abj})X_{b}$
(3.14)

is complex, and the same concerns the metric $g^{jk}=E^{j}_{A}E^{k}_{A}$. The
determinant of the latter is also complex.

Complex metric and complex volume have little geometric sense. Also the
canonical momenta $\hat{D}_{a}$ and the Hamiltonian (3.4) are not Hermitian
for $N>2$. There are some Hamiltonians that do not seem to be Hermitian, but
are in fact Hermitian in disguise and have a real spectrum [15]. We do not
think that it is so in our case, but a special study of this question would be
interesting.

## Acknowledgements

I am deeply indebted to Dmitry Gurevich for many illuminating discussions and
the collaboration at the initial stage of this work. I thank Vladislav
Kupriyanov for important comments, for bringing my attention to Refs. [8, 9,
11] and for informing me about the paper [10] prior to publication. I also
acknowledge useful discussions with Maxim Kontsevich, Zoran Škoda and Arkady
Vainshtein.

## Appendix

To make the paper self-contained, we will explain here how the identities
(2.16) and (2.8) are derived.

The formula (2.16) has a nice physical interpretation [7]. Consider the
Hamiltonian $\hat{H}=\hat{H}_{0}+\hat{V}$ and the corresponding evolution
operator

$\displaystyle U\ =\ e^{i(\hat{H}_{0}+\hat{V})t}\ =\
\lim_{N\to\infty}\left[1+\frac{it}{N}(\hat{H}_{0}+\hat{V})\right]^{N}\,.$
(A.1)

Suppose that the perturbation $\hat{V}$ is small and keep only the linear in
$\hat{V}$ terms. The perturbation can be inserted at any time moment
$t^{\prime}$ between 0 and $t$, and we obtain

$\displaystyle U\ =\
e^{i\hat{H}_{0}t}+i\int_{0}^{t}dt^{\prime}e^{i(t-t^{\prime})\hat{H}_{0}}\hat{V}e^{it^{\prime}\hat{H}_{0}}+\ldots\,.$
(A.2)

Choosing $\hat{H}_{0}=\theta X^{j}t_{j},\ \hat{V}=\theta\,\delta X^{j}t_{j},\
t=1$, renaming $t^{\prime}\to\tau$ and recalling the notation
$\omega=e^{R}=e^{i\theta X^{j}t_{j}}$, we derive

$\displaystyle\omega(X+\delta X)=\omega(X)+\delta
X^{j}\partial_{j}\omega(X)+\ldots\ =\
\omega(X)\left[1+i\theta\int_{0}^{1}d\tau\,e^{-\tau R}\delta X^{j}t_{j}e^{\tau
R}+\ldots\right],$ (A.3)

from which (2.16) follows.

As a consequence, we derive for the product in (2.7) that $\epsilon^{a}$ is
equal to $E^{a}_{j}\,\delta X^{j}$, with the vielbein $E^{a}_{j}$ having the
form (2.17). Then $\delta X^{j}$ is expressed via $\epsilon^{a}$ with the
vielbein $E_{a}^{j}$ written in (2.8).

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