# Algebraic Structure of Dirac Hamiltonians in Non-Commutative Phase Space

Horacio Falomir1 1Instituto de Física La Plata (CONICET and Universidad
Nacional de La Plata) CC 67 (1900) La Plata, Argentina
<EMAIL_ADDRESS>,  Joaquin Liniado1<EMAIL_ADDRESS>and
Pablo Pisani1<EMAIL_ADDRESS>
###### Abstract.

In this article we study two-dimensional Dirac Hamiltonians with non-
commutativity both in coordinates and momenta from an algebraic perspective.
In order to do so, we consider the graded Lie algebra $\mathfrak{sl}(2|1)$
generated by Hermitian bilinear forms in the non-commutative dynamical
variables and the Dirac matrices in $2+1$ dimensions. By further defining a
total angular momentum operator, we are able to express a class of Dirac
Hamiltonians completely in terms of these operators. In this way, we analyze
the energy spectrum of some simple models by constructing and studying the
representation spaces of the unitary irreducible representations of the graded
Lie algebra $\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$. As application of our
results, we consider the Landau model and a fermion in a finite cylindrical
well.

###### Contents

  1. 1 Introduction
  2. 2 The Non-Commutative Two-Dimensional Phase Space
  3. 3 Algebraic Structure of Dirac Hamiltonians
  4. 4 The Irreducible Representations of $\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$
    1. 4.1 Quadratic Casimir Operator of $\mathfrak{sl}(2|1)$
    2. 4.2 The $\mathfrak{so}(2)$ factor
  5. 5 The Landau Problem
  6. 6 The Cylindrical Well
  7. 7 Conclusions
  8. A Adjoint Representation of $\mathfrak{sl}(2|1)$
  9. B Action of the Odd Generators
    1. B.1 Case $\sigma=2k$
    2. B.2 Case $\sigma=-2k$
  10. C Unitary Transformations in $\mathfrak{sl}(2,\mathbb{R})$

##  1\. Introduction

Non-commutativity (NC) in physics is an area of research that has provided
different models with peculiar properties, ever since its appearance in the
1930s’ in the seminal paper by Peierls [1] (see also [2, 3]), describing non-
relativistic electrons subject to a strong magnetic field. Non-commutative
geometries [4] have been considered in a wide range of settings, originally in
the context of regularization schemes in quantum field theory [5, 6, 7], as
well as in string theory, in the description of low energy effective theories
of $D-$branes in background magnetic fields [8]. From a slightly different
perspective, non-commutativity has also stimulated interest in the study of
quantum mechanical systems with deformed algebra of commutators [9, 14, 15,
16, 17, 18, 19, 20, 21, 10, 11, 12, 13], since these present notable
properties either because of their phenomenological consequences or as an
arena for the use of alternative techniques (_cf._ [22] for a detailed
historical review and a list of references).

Non-commutativity has been implemented through the Moyal product of ordinary
functions [15, 19, 21] or, alternatively, by explicit realizations of
dynamical variables as operators on the Hilbert space satisfying a deformed
Heisenberg algebra of the form [14, 15, 16]

(1.1)
$\left[\hat{X}_{i},\hat{X}_{j}\right]=i\theta\epsilon_{ij}\,,\quad\left[\hat{X}_{i},\hat{P}_{j}\right]=i\delta_{ij}\,,\quad\left[\hat{P}_{i},\hat{P}_{j}\right]=i\kappa\epsilon_{ij}\,,$

where $\kappa,\theta$ are real parameters describing phase-space NC, and we
take $\hbar=1$. Several authors have pointed out that non-relativistic quantum
mechanical models on the non-commutative phase space present two quantum
phases [14, 16, 20] –according to the value of the dimensionless parameter
$\kappa\theta$– separated by a critical point where the effective dimension of
the system is reduced [14, 16, 20, 18, 10, 11]. In particular, for the case of
Dirac Hamiltonians see [12, 13].

In this context a class of non-relativistic two-dimensional rotationally
invariant Hamiltonians have been studied from an algebraic point of view [22].
Making use of the commutation relations (1.1), it has been shown that the
rotationally invariant _Hermitian quadratic forms_ in
$\hat{X}_{i},\hat{P}_{i}$, $i=1,2$ generate, in particular, a non-abelian
three-dimensional Lie algebra corresponding to $\mathfrak{sl}(2,\mathbb{R})$
or $\mathfrak{su}(2)$ according to $\kappa\theta<1$ or $\kappa\theta>1$. By
further constructing the generator of rotations $\hat{L}$ in the non-
commutative phase space, it was established that the most general rotationally
invariant non-relativistic Hamiltonian can be expressed as a function of the
generators of that Lie algebra and $\hat{L}$ itself, so that the
characteristic subspaces of these Hamiltonians are representation spaces of
the unitary irreducible representations (irreps) of the Lie algebras
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$ or
$\mathfrak{su}(2)\oplus\mathfrak{so}(2)$.

The aim of this article is to extend this algebraic study to two-dimensional
_fermionic_ systems: we study Dirac Hamiltonians in the non-commutative phase
space. We show that due to the spinorial structure of Dirac equation, the
natural generalization is to consider a graded extension of
$\mathfrak{sl}(2,\mathbb{R})$, the Lie algebra appearing111For simplicity, we
restrict to the parameter region $\kappa\theta<1$ for it is the one which can
be connected with the ordinary (commutative) case. As mentioned above, the Lie
algebra corresponding to this region is precisely
$\mathfrak{sl}(2,\mathbb{R})$. in the Schrödinger case [22]. In fact, by
constructing a total angular momentum operator $\mathcal{L}$, we manage to
express some Dirac-type Hamiltonians in terms of $\mathcal{L}$ and the
generators of the graded Lie algebra $\mathfrak{sl}(2|1)$, which we then
employ to compute, in two examples, the energy spectrum by studying, as in the
previous case, the representation spaces of the irreducible representations of
the graded Lie algebra. Specifically, we discuss the non-commutative version
of an electron subject to a uniform magnetic field (the NC Landau problem),
and separately, the bound states of a particle in the background of a finite
cylindrical well.

The article is organized as follows: in section 2 we introduce the non-
commutative dynamical variables and define the generator of rotations in the
non-commutative phase space. We then establish the conventions regarding the
Dirac Hamiltonians in two dimensions we will consider throughout the article.
In section 3 we present a detailed construction of the generators of the
graded Lie algebra $\mathfrak{sl}(2|1)$ in terms of the non-commutative phase-
space variables, and the Dirac matrices in $2+1$ dimensions. Furthermore, we
construct the total angular momentum operator which will generate an
additional one dimensional Lie algebra $\mathfrak{so}(2)$ required to fully
express the Hamiltonians under consideration. In section 4 we provide a
complete characterization of the unitary irreducible representations of
$\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$ which we then employ in sections 5
and 6 to solve the problems of a particle in a uniform magnetic field and in a
finite cylindrical well respectively. Finally, three appendices are included
to clarify some intermediary calculations.

##  2\. The Non-Commutative Two-Dimensional Phase Space

Non-commutativity on a two-dimensional phase space can be implemented by the
following commutation relations between the Hermitian operators $\hat{X}_{i}$
and $\hat{P}_{i}$

(2.1)
$\left[\hat{X}_{i},\hat{X}_{j}\right]=i\theta\epsilon_{ij}\,,\quad\left[\hat{X}_{i},\hat{P}_{j}\right]=i\delta_{ij}\,,\quad\left[\hat{P}_{i},\hat{P}_{j}\right]=i\kappa\epsilon_{ij}\,,\quad
i=1,2$

where $\theta$ and $\kappa$ are the (real) non-commutativity parameters (we
choose a unit system in which $\hbar=c=1$). With no loss of generality, we can
take $\theta\geq 0$. These relations reduce to the Heisenberg algebra for
ordinary dynamical variables in the $\theta,\kappa\to 0$ limit,

(2.2)
$\left[\hat{x}_{i},\hat{x}_{j}\right]=0\,,\quad\left[\hat{x}_{i},\hat{p}_{j}\right]=i\delta_{ij}\,,\quad\left[\hat{p}_{i},\hat{p}_{j}\right]=0\,.$

From equation (2.1), it follows that the generator of rotations on the NC
phase-space [16] is given by

(2.3)
$\hat{L}:=\frac{1}{\left(1-{\kappa\theta}\right)}\left\\{\left(\hat{X}_{1}\hat{P}_{2}-\hat{X}_{2}\hat{P}_{1}\right)+{{\frac{\theta}{2}\left({\hat{P}_{1}}^{2}+{\hat{P}_{2}}^{2}\right)}+{\frac{\kappa}{2}\left({\hat{X}_{1}}^{2}+{\hat{X}_{2}}^{2}\right)}}\right\\}\,,$

for $\kappa\theta\neq 1$. Indeed, it can be seen that $\hat{\mathbf{X}}$ and
$\hat{\mathbf{P}}$ transform as vectors under rotations on the plane

(2.4)
$\left[\hat{L},\hat{X}_{i}\right]=i\,\epsilon_{ij}\hat{X}_{j}\,,\quad\left[\hat{L},\hat{P}_{i}\right]=i\,\epsilon_{ij}\hat{P}_{j}\,.$

Thus,

(2.5)
$\left[\hat{L},\hat{\mathbf{X}}^{2}\right]=0\,,\quad\left[\hat{L},\hat{\mathbf{P}}^{2}\right]=0\,.$

Furthermore, for the critical value $\kappa_{c}=1/\theta$ (where $\hat{L}$ in
equation (2.3) is ill-defined), the commutator algebra in equation (2.1)
reduces to

(2.6)
$\left[\hat{X}_{i},\hat{X}_{j}\right]=i\epsilon_{ij}\theta\,,\quad\left[\hat{X}_{i},\hat{P}_{j}\right]=i\delta_{ij}\,,\quad\left[\hat{P}_{i},\hat{P}_{j}\right]=i\epsilon_{ij}\theta^{-1}\,,$

which can be satisfied by just one pair of dynamical variables
$\hat{X}_{1}=-\theta\hat{P}_{2}$ and $\hat{P}_{1}=\frac{\hat{X}_{2}}{\theta}$.
Therefore, as a consequence of the degeneracy of the symplectic form in
classical phase space for this particular value of $\kappa$, there occurs a
_dimensional reduction_ [16, 20]. Indeed, two elements in the deformed
Heisenberg algebra defined by equation (2.1),

(2.7)
$\hat{P}=\hat{P}_{2}+\kappa\hat{X}_{1},\quad\hat{Q}=\hat{X}_{2}-\theta\hat{P}_{1},$

which are proportional to the _magnetic translation generators_ (See [22],
equations (2.8-9), for example), commute with $\hat{X}_{1}$ and $\hat{P}_{1}$
and satisfy

(2.8) $\left[\hat{Q},\hat{P}\right]=i(1-\kappa\theta).$

Thus, they are well-defined and belong to the center of the algebra when
$\kappa=\kappa_{c}$, where the system becomes effectively one-dimensional.

The free Hamiltonian we will consider is of the standard Dirac-type, which in
terms of the non-commutative momenta, may be expressed as

(2.9) $\hat{H}=\boldsymbol{\alpha}\cdot\hat{\mathbf{P}}-\beta m$

where $\beta=\gamma^{0}$ and $\alpha_{i}=\beta\gamma^{i}$ with $\gamma^{\mu}$,
$(\mu=0,1,2$) the Dirac matrices in $2+1$ space-time dimensions. These satisfy
the Clifford algebra222We choose the signature
$\eta^{\mu\nu}=\rm{diag}(1,-1,-1)$.
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$ which can be realised by
choosing

(2.10)
$\gamma^{0}=\sigma_{3},\quad\gamma^{1}=i\sigma_{2},\quad\gamma^{2}=-i\sigma_{1}\,.$

Thus, we have

(2.11) $\hat{H}=\hat{\mathbf{P}}\cdot\boldsymbol{\sigma}-m\sigma_{3}.$

which is the two dimensional non-commutative Dirac Hamiltonian of a free
massive particle. Later on, we will also consider the Hamiltonian of the non-
commutative version of an electron subject to a uniform magnetic field, and of
an electron confined within a finite cylindrical well.

##  3\. Algebraic Structure of Dirac Hamiltonians

In the previous section, we noticed that the non-singularity of the symplectic
form establishes a critical value $\kappa_{c}=1/\theta$, that defines two
different regions, namely $\kappa<\kappa_{c}$ and $\kappa>\kappa_{c}$.
However, only the first one can be connected with the ordinary (commutative)
case, and in the following we shall therefore restrict ourselves to this
region of parameters. Bearing this in mind, we show that the characteristic
subspaces of the Hamiltonians we will consider are characterized by unitary
irreducible representations of the graded Lie algebra
$\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$.

Taking $0<\kappa<\kappa_{c}$333As a matter of fact, $\kappa<0$ is a perfectly
valid choice. However, the corresponding algebraic construction is very
similar (_cf._ [22]) and we will therefore omit it. we may define the
generators $\mathcal{J}_{0},\mathcal{J}_{\pm}$ as

(3.1)
$\begin{array}[]{c}\displaystyle\mathcal{J}_{0}=\frac{\sqrt{{\kappa\theta}}}{4}\left(\frac{\mathbf{{{\hat{P}}}}^{2}}{\kappa}+\frac{\mathbf{{{\hat{X}}}}^{2}}{\theta}+{2\hat{L}}\right)={\mathcal{J}_{0}}^{\dagger}\,,\\\
\\\
\displaystyle\mathcal{J}_{\pm}=\frac{\sqrt{{\kappa\theta}}}{4\sqrt{1-{\theta\kappa}}}\left\\{\frac{\mathbf{{{\hat{X}}}}^{2}}{\theta}-\frac{\mathbf{{{\hat{P}}}}^{2}}{\kappa}\mp\frac{i}{\sqrt{\kappa\theta}}\left(\mathbf{\hat{P}}\cdot\mathbf{\hat{X}}+\mathbf{\hat{X}}\cdot\mathbf{\hat{P}}\right)\right\\}={\mathcal{J}_{\mp}}^{\dagger}\,,\end{array}$

that satisfy the commutator algebra of $\mathfrak{sl}(2,\mathbb{R})$

(3.2)
$\left[\mathcal{J}_{0},\mathcal{J}_{\pm}\right]=\pm\mathcal{J}_{\pm}\,,\quad\left[\mathcal{J}_{+},\mathcal{J}_{-}\right]=-2\mathcal{J}_{0}\,,$

where $\mathcal{J}_{\pm}=\mathcal{J}_{1}\pm i\mathcal{J}_{2}$, with
$\mathcal{J}_{i},\,\,(i=0,1,2)$, Hermitian operators. Moreover, the quadratic
Casimir operator ${\mathcal{J}}^{2}=\mathcal{J}_{0}(\mathcal{J}_{0}\mp
1)-\mathcal{J}_{\pm}\mathcal{J}_{\mp}=\mathcal{J}_{0}^{2}-\mathcal{J}_{1}^{2}-\mathcal{J}_{2}^{2}$
is given by

(3.3)
$\begin{array}[]{c}\displaystyle{\mathcal{J}}^{2}=\frac{{\kappa\theta}}{16\left(1-{\kappa\theta}\right)}\left\\{\left(1-{\kappa\theta}\right)\left(\frac{\mathbf{\hat{X}}^{2}}{\theta}+\frac{\mathbf{\hat{P}}^{2}}{\kappa}+2{\hat{L}}\right)^{2}-\right.\\\
\\\
\displaystyle\left.-\left(\frac{\mathbf{\hat{X}}^{2}}{\theta}-\frac{\mathbf{\hat{P}}^{2}}{\kappa}\right)^{2}-\frac{\left(\mathbf{\hat{P}}\cdot\mathbf{\hat{X}}+\mathbf{\hat{X}}\cdot\mathbf{\hat{P}}\right)^{2}}{\theta\kappa}\right\\}\,,\end{array}$

which notably, is not a positive definite operator [23]. Furthermore,
equations (2.4)-(2.5) imply

(3.4)
$\left[\hat{L},\mathcal{J}_{0}\right]=0\,,\quad\left[\hat{L},\mathcal{J}_{\pm}\right]=0\quad\Longrightarrow\quad\left[\hat{L},{\mathcal{J}}^{2}\right]=0\,.$

In fact, one can also verify that [22]

(3.5)
$\mathcal{J}^{2}=\frac{1}{4}\left\\{\hat{L}^{2}-1\right\\}\geq-\frac{1}{4}\,,$

which imposes a constraint on the irreducible representations of the Lie
algebra $\mathfrak{sl}(2,\mathbb{R})$. In particular, this relation implies
that only the discrete classes of irreducible unitary representations of
$\mathfrak{sl}(2,\mathbb{R})$ will be of interest for our purposes [23, 24].
The construction we have described above has been done for the bosonic case in
[22].

From equation (2.11), we note that the Dirac Hamiltonian in $2+1$ dimensions
contains a kinetic term, $\hat{\mathbf{P}}\cdot\boldsymbol{\sigma}$, which is
linear in momentum. Furthermore, in the next sections, we will discuss
problems with operators of the form
$\hat{\mathbf{X}}\times\boldsymbol{\sigma}=\epsilon^{ij}\hat{X}_{i}\sigma_{j}$,
that is, linear in the NC coordinate operators. In analogy to what is usually
done in supersymmetric theories, we consider the $\mathbb{Z}_{2}-$graded
extension of $\mathfrak{sl}(2,\mathbb{R})$ in order to incorporate fermionic-
type operators. We start by defining

(3.6) $\Sigma:=\hat{L}+\sigma_{3}$

which, together with $\mathcal{J}^{2}$ and $\mathcal{J}_{\pm}$ generate the
_even_ subspace $\mathfrak{u}(1)\oplus\mathfrak{sl}(2,\mathbb{R})$ of the
graded Lie algebra. To construct the _odd_ generators, we define first

(3.7)
$\begin{array}[]{c}\mathcal{Q}_{\pm}:=\displaystyle{\frac{1}{2}\sqrt[4]{\frac{\kappa\theta}{1-\kappa\theta}}}\left\\{\left(\frac{\hat{X}_{1}}{\sqrt{\theta}}\mp
i\,\frac{\hat{P}_{1}}{\sqrt{\kappa}}\right)\sigma_{1}+\left(\frac{\hat{X}_{2}}{\sqrt{\theta}}\mp
i\,\frac{\hat{P}_{2}}{\sqrt{\kappa}}\right)\sigma_{2}\right\\}\,,\\\ \\\
\tilde{\mathcal{Q}}_{\pm}:=\sigma_{3}\mathcal{Q}_{\pm}\,.\end{array}$

In terms of $\mathcal{Q}_{\pm}$ and $\tilde{\mathcal{Q}}_{\pm}$ we define a
pair of fermionic multiplets $(\mathcal{R}_{+},\mathcal{R}_{-})$ and
$(\mathcal{S}_{+},\mathcal{S}_{-})$ given by

(3.8)
$\begin{split}&\mathcal{R}_{+}:=\frac{1}{2}\sqrt[4]{\frac{1-\sqrt{\kappa\theta}}{1+\sqrt{\kappa\theta}}}\left(\mathcal{Q}_{+}+\widetilde{\mathcal{Q}}_{+}\right)\,,\quad\mathcal{R}_{-}:=\frac{1}{2}\sqrt[4]{\frac{1+\sqrt{\kappa\theta}}{1-\sqrt{\kappa\theta}}}\left(\mathcal{Q}_{-}+\widetilde{\mathcal{Q}}_{-}\right)\\\
&\mathcal{S}_{+}:=\frac{1}{2}\sqrt[4]{\frac{1+\sqrt{\kappa\theta}}{1-\sqrt{\kappa\theta}}}\left(\mathcal{Q}_{+}-\widetilde{\mathcal{Q}}_{+}\right)\,,\quad\mathcal{S}_{-}:=\frac{1}{2}\sqrt[4]{\frac{1-\sqrt{\kappa\theta}}{1+\sqrt{\kappa\theta}}}\left(\mathcal{Q}_{-}-\widetilde{\mathcal{Q}}_{-}\right)\,.\end{split}$

As expected, the fermionic elements are nilpotent,

(3.9) $\mathcal{R}_{\pm}^{2}=\mathcal{S}_{\pm}^{2}=0\,,$

and they satisfy

(3.10)
$\mathcal{R}_{+}\mathcal{R}_{-}=\mathcal{S}_{+}\mathcal{S}_{-}=0\,,\quad\mathcal{R}_{+}^{\dagger}=\mathcal{S}_{-}\,,\quad\mathcal{R}_{-}^{\dagger}=\mathcal{S}_{+}\,.$

Thus, the set of eight generators
$\left\\{\mathcal{J}_{0},\mathcal{J}_{\pm},\Sigma,\mathcal{R}_{\pm},\mathcal{S}_{\pm}\right\\}$
span the graded Lie algebra $\mathfrak{sl}(2|1)$, whose (anti)commutation
relations are given by444We have chosen different conventions than, for
example, in [25].

(3.11)
$\begin{array}[]{c}\left[\Sigma,{\mathcal{R}}_{\pm}\right]={\mathcal{R}}_{\pm}\,,\quad\quad\left[\Sigma,{\mathcal{S}}_{\pm}\right]=-{\mathcal{S}}_{\pm}\,,\\\
\\\
\left[\mathcal{J}_{0},{\mathcal{R}}_{\pm}\right]=\pm\frac{1}{2}{\mathcal{R}}_{\pm}\,,\quad\quad\left[\mathcal{J}_{0},{{\mathcal{S}}}_{\pm}\right]=\pm\frac{1}{2}{{\mathcal{S}}}_{\pm}\,,\\\
\\\
\left[\mathcal{J}_{\pm},{\mathcal{R}}_{\mp}\right]=\mp{\mathcal{R}}_{\pm}\,,\quad\quad\left[\mathcal{J}_{\pm},{\mathcal{S}}_{\mp}\right]=\mp{\mathcal{S}}_{\pm}\,,\\\
\\\
\left[\mathcal{J}_{0},\mathcal{J}_{\pm}\right]=\pm\mathcal{J}_{\pm}\,,\quad\quad\left[\mathcal{J}_{+},\mathcal{J}_{-}\right]=-2\mathcal{J}_{0}\,,\\\
\\\
\left\\{{\mathcal{R}}_{\pm},{\mathcal{S}}_{\pm}\right\\}=\mathcal{J}_{\pm}\,,\quad\quad\left\\{{\mathcal{R}}_{\pm},{\mathcal{S}}_{\mp}\right\\}=\mathcal{J}_{0}\mp\frac{\Sigma}{2}\,,\\\
\end{array}$

where unspecified (anti)commutators vanish. Next, we define a _total_ angular
momentum operator

(3.12) $\mathcal{L}:=\hat{L}+\frac{\sigma_{3}}{2}\,$

which commutes with all the generators of $\mathfrak{sl}(2|1)$ and that will
be required to express the Hamiltonians we will consider in what follows. As a
matter of fact, these will be completely expressible in terms of
$\hat{\mathbf{P}}\cdot\boldsymbol{\sigma}$,
$\hat{\mathbf{X}}\times\boldsymbol{\sigma}$, $\hat{\mathbf{X}}^{2}$, $\Sigma$
and $\mathcal{L}$ itself. We thus write for future use,

(3.13)
$\begin{array}[]{c}\displaystyle\hat{\mathbf{P}}\cdot\boldsymbol{\sigma}=\hat{P}_{i}\,\sigma_{i}=i\sqrt[4]{\frac{\kappa}{\theta}}\left[\sqrt{1+\sqrt{\kappa\theta}}\left(\mathcal{R}_{+}-\mathcal{S}_{-}\right)-\sqrt{1-\sqrt{\kappa\theta}}\left(\mathcal{R}_{-}-\mathcal{S}_{+}\right)\right]\,,\\\
\\\
\displaystyle\hat{\mathbf{X}}\times\boldsymbol{\sigma}=\epsilon^{ij}\hat{X}_{i}\,\sigma_{j}=-i\sqrt[4]{\frac{\theta}{\kappa}}\left[\sqrt{1+\sqrt{\kappa\theta}}\left(\mathcal{R}_{+}-\mathcal{S}_{-}\right)+\sqrt{1-\sqrt{\kappa\theta}}\left(\mathcal{R}_{-}-\mathcal{S}_{+}\right)\right]\,.\end{array}$

In this way, each characteristic subspace of the Hamiltonians is a
representation space of a unitary irreducible representation of the direct sum
of Lie algebras $\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$, where the second
factor is generated by $\mathcal{L}$.

##  4\. The Irreducible Representations of
$\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$

In this section we present a detailed construction of the irreducible
representations of the direct sum of Lie algebras
$\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$. We start with the unitary
irreducible representations in the discrete classes of the Lie algebra
$\mathfrak{sl}(2,\mathbb{R})$, restriction due to the constraint on
$\mathcal{J}^{2}$ being bounded below (_cf._ equation (3.5) and [24, 23]).
These representations can be characterized by the eigenvalue of the Casimir
operator $\mathcal{J}^{2}$ which takes values $\lambda=k(k-1)$ with
$k=\tfrac{1}{2},1,\tfrac{3}{2},\dots$ . For a given $k$ the representation
space is generated by the simultaneous eigenvectors of $\mathcal{J}^{2}$ and
$\mathcal{J}_{0}$,

(4.1) $\begin{array}[]{c}\mathcal{J}^{2}|k,m\rangle=k(k-1)|k,m\rangle\\\ \\\
\mathcal{J}_{0}|k,m\rangle=m|k,m\rangle\,,\quad\text{with}\quad
m=k,k+1,\dots\text{ or }m=-k,-(k+1),\dots\,.\\\ \end{array}$

We recall that the even subspace of the graded extension has an additional
$\mathfrak{u}(1)$ factor generated by $\Sigma$ and thus we consider the
vectors which are also eigenvectors of this operator. We denote these by
$|k,m,\sigma\rangle$, such that

(4.2)
$\begin{array}[]{c}\Sigma|k,m,\sigma\rangle=\sigma|k,m,\sigma\rangle\,,\quad\text{with
}\sigma\in\mathbb{R}\,.\\\ \end{array}$

We denote $\langle k,\sigma\rangle={\rm span}(\\{|k,m,\sigma\rangle\\})$.

Next we characterize the irreducible representation spaces of the graded Lie
algebra, which are the direct sum of a finite number of irreducible
representation spaces of the even subalgebra,

(4.3) $\langle k_{0},\sigma_{0}\rangle\oplus\langle
k_{1},\sigma_{1}\rangle\oplus\dots\oplus\langle
k_{n},\sigma_{n}\rangle\quad\text{with}\quad k_{0}\leq k_{1}\leq\dots\leq
k_{n}\,.$

To determine which values of $k$ appear in this direct sum decomposition we
use the fermionic generators. At this point, it is worth stressing that the
vectors $|k_{i},m_{i},\sigma_{i}\rangle\in\langle k_{i},\sigma_{i}\rangle$
will have values of $m_{i}=k_{i},k_{i}+1,\dots$ or
$m_{i}=-k_{i},-(k_{i}+1),\dots$ but not both. In particular, without loss of
generality, the construction we present assumes that
$m_{0}=k_{0},k_{0}+1,\dots$ , given that the negative $m_{0}$ case follows in
exactly the same way. Notably, in §4.2, we will argue that for the physical
systems we are considering, we require the value of $m$ to be bounded below
and thus, in practice, we will only make use of the representations we
construct explicitly in this section.

So let us consider the _minimum_ weight vector
$|k_{0},k_{0},\sigma_{0}\rangle\in\langle k_{0},\sigma_{0}\rangle$. Using the
commutation relations (3.11), it is straightforward to see that

(4.4)
$\begin{array}[]{c}\mathcal{J}_{0}\mathcal{R}_{-}|k_{0},k_{0},\sigma_{0}\rangle=\left(k_{0}-\tfrac{1}{2}\right)\mathcal{R}_{-}|k_{0},k_{0},\sigma_{0}\rangle\,,\end{array}$

from where we conclude that $\mathcal{R}_{-}|k_{0},k_{0},\sigma_{0}\rangle=0$.
With a similar argument, we obtain
$\mathcal{S}_{-}|k_{0},k_{0},\sigma_{0}\rangle=0$. We now proceed with
$\mathcal{R}_{+}$. By considering the action of the product of operators
$\mathcal{J}_{0}\mathcal{R}_{+}$, $\Sigma\mathcal{R}_{+}$ and using the
commutation relations (3.11) it is easy to verify that

(4.5)
$\mathcal{R}_{+}|k_{0},k_{0},\sigma_{0}\rangle=\sqrt{k_{0}-\tfrac{\sigma_{0}}{2}}|k_{0}+\tfrac{1}{2},k_{0}+\tfrac{1}{2},\sigma_{0}+1\rangle\in\langle
k_{0}+\tfrac{1}{2},\sigma_{0}+1\rangle\,,$

which is nonvanishing as long as $\sigma_{0}\neq 2k_{0}$. Moreover, by
successively applying $\mathcal{J}_{+}$, we construct a complete basis for the
subspace $\langle k_{0}+\tfrac{1}{2},\sigma_{0}+1\rangle$. With similar
arguments we have

(4.6)
$\mathcal{S}_{+}|k_{0},k_{0},\sigma_{0}\rangle=\sqrt{k_{0}+\tfrac{\sigma_{0}}{2}}|k_{0}+\tfrac{1}{2},k_{0}+\tfrac{1}{2},\sigma_{0}-1\rangle\in\langle
k_{0}+\tfrac{1}{2},\sigma_{0}-1\rangle\,,$

which is nonvanishing as long as $\sigma_{0}\neq-2k_{0}$. Again, with the
successive application of $\mathcal{J}_{+}$ we construct the complete basis
for the subspace $\langle k_{0}+\tfrac{1}{2},\sigma_{0}-1\rangle$. We further
recall that the odd generators are nilpotent and that their crossed products
are identically vanishing (equations (3.9) and (3.10)) and thus

(4.7)
$\mathcal{R}_{\pm}|k_{0}+\tfrac{1}{2},k_{0}+\tfrac{1}{2},\sigma_{0}+1\rangle=0=\mathcal{S}_{\pm}|k_{0}+\tfrac{1}{2},k_{0}+\tfrac{1}{2},\sigma_{0}-1\rangle\,.$

But, from equation (3.11), we have

(4.8)
$\begin{array}[]{c}\mathcal{R}_{-}|k_{0}+\tfrac{1}{2},k_{0}+\tfrac{1}{2},\sigma_{0}-1\rangle=\sqrt{k_{0}+\tfrac{\sigma_{0}}{2}}|k_{0},k_{0},\sigma_{0}\rangle\\\
\\\
\mathcal{S}_{-}|k_{0}+\tfrac{1}{2},k_{0}+\tfrac{1}{2},\sigma_{0}+1\rangle=\sqrt{k_{0}-\tfrac{\sigma_{0}}{2}}|k_{0},k_{0},\sigma_{0}\rangle\end{array}$

and therefore, the action of $\mathcal{S}_{-}$ or $\mathcal{R}_{-}$ on vectors
in $\langle k_{0}+\tfrac{1}{2},\sigma_{0}+1\rangle$ and in $\langle
k_{0}+\tfrac{1}{2},\sigma_{0}-1\rangle$ respectively, takes us back to
$\langle k_{0},\sigma_{0}\rangle$.

Finally, we are left with the analysis of
$\mathcal{R}_{+}\mathcal{S}_{+}|k_{0},k_{0},\sigma\rangle$ and
$\mathcal{S}_{+}\mathcal{R}_{+}|k_{0},k_{0},\sigma\rangle$. In contrast with
the previous cases, these vectors are not orthogonal to the subspace $\langle
k_{0},\sigma_{0}\rangle$; for instance, we can take the linear combination

(4.9)
$\\{\mathcal{R}_{+},\mathcal{S}_{+}\\}|k_{0},k_{0},\sigma_{0}\rangle=\mathcal{J}_{+}|k_{0},k_{0},\sigma_{0}\rangle=\sqrt{2k_{0}}|k_{0},k_{0}+1,\sigma_{0}\rangle\in\langle
k_{0},\sigma_{0}\rangle\,.$

Indeed, to construct a vector orthogonal to the minimal weight subspace we
look for the value of $a$ such that

(4.10) $\begin{split}0&=\langle
k_{0},k_{0}+1,\sigma_{0}|\left(\mathcal{S}_{+}\mathcal{R}_{+}|k_{0},k_{0},\sigma_{0}\rangle-a|k_{0},k_{0}+1,\sigma_{0}\rangle\right)\\\
&=\frac{1}{\sqrt{2k_{0}}}\langle
k_{0},k_{0},\sigma_{0}|\mathcal{J}_{+}^{\dagger}\mathcal{S}_{+}\mathcal{R}_{+}|k_{0},k_{0},\sigma_{0}\rangle-a\\\
&=\frac{1}{\sqrt{2k_{0}}}\|\mathcal{R}_{+}|k_{0},k_{0},\sigma_{0}\rangle\|^{2}-a\\\
&=\frac{1}{\sqrt{2k_{0}}}(k_{0}-\tfrac{\sigma_{0}}{2})-a\,,\end{split}$

where we have used (4.9) and (3.11). Then,
$a=\frac{1}{\sqrt{2k_{0}}}\left(k_{0}-\tfrac{\sigma_{0}}{2}\right)$ is the
adequate choice. Moreover, by considering the action of the operators
$\mathcal{J}_{0}$ and $\Sigma$ on this vector, and taking its norm, we
conclude that

(4.11)
$\begin{split}\left(\mathcal{S}_{+}\mathcal{R}_{+}-\tfrac{k_{0}-\sigma_{0}/2}{2k_{0}}\mathcal{J}_{+}\right)&|k_{0},k_{0},\sigma_{0}\rangle\\\
&=\sqrt{\tfrac{(2k_{0}+1)}{8k_{0}}\left(4k_{0}^{2}-\sigma_{0}^{2}\right)}|k_{0}+1,k_{0}+1,\sigma_{0}\rangle\in\langle
k_{0}+1,\sigma_{0}\rangle\,.\end{split}$

Notice that if $\sigma_{0}=\pm 2k_{0}$, then
$\left(\mathcal{S}_{+}\mathcal{R}_{+}-\frac{k_{0}-\sigma_{0}/2}{2k_{0}}\mathcal{J}_{+}\right)|k_{0},k_{0},\sigma_{0}\rangle=0$.
Once again, with the successive application of $\mathcal{J}_{+}$ we construct
a complete basis for the subspace $\langle k_{0}+1,\sigma_{0}\rangle$. With
equation (4.11) we have exhausted all of the possibilities, given that any
additional combination of products of fermionic operators will vanish, due to
equations (3.9) and (3.10).

All in all, we observe that the irreducible representations of the graded Lie
algebra $\mathfrak{sl}(2|1)$ may be characterized by
$k=\tfrac{1}{2},1,\tfrac{3}{2},\dots$, the integer or half-integer appearing
in the expression of the eigenvalue of the quadratic Casimir operator of the
even subalgebra $\mathcal{J}^{2}$. For a given value of $k$, if $\sigma\neq\pm
2k$, the representation space is the direct sum of four different
representation spaces of $\mathfrak{u}(1)\oplus\mathfrak{sl}(2,\mathbb{R})$

(4.12) $\langle k,\sigma\rangle\oplus\langle
k+\tfrac{1}{2},\sigma+1\rangle\oplus\langle
k+\tfrac{1}{2},\sigma-1\rangle\oplus\langle k+1,\sigma\rangle\,,$

whereas if $\sigma=\pm 2k$, the graded representation space is the direct sum
of only two of these representation spaces

(4.13) $\begin{array}[]{c}\langle k,\sigma\rangle\oplus\langle
k+\tfrac{1}{2},\sigma-1\rangle\,\quad\quad\text{if}\quad\quad\sigma=2k\\\ \\\
\langle k,\sigma\rangle\oplus\langle
k+\tfrac{1}{2},\sigma+1\rangle\,\quad\quad\text{if}\quad\quad\sigma=-2k\end{array}$

In appendix B we compute the action of the fermionic generators in a general
vector of each representation space for $\sigma=\pm 2k$.

###  4.1. Quadratic Casimir Operator of $\mathfrak{sl}(2|1)$

Having constructed the irreducible representations of $\mathfrak{sl}(2|1)$, we
compute the quadratic Casimir operator $\mathcal{C}^{2}$ of the graded Lie
algebra. This can be done using the _Killing form_ $\mathbf{K}$ of the graded
Lie algebra, whose components are [26]

(4.14) $K_{ab}:={\rm Str}[T_{a}T_{b}]={\rm Tr}[T_{a}T_{b}\mathcal{I}]$

where $T_{a},T_{b}$ are the generators of $\mathfrak{sl}(2|1)$ in the adjoint
representation, with $a,b=1,\dots 4$ corresponding to the even subalgebra,
$a,b=5,\dots,8$ to the odd generators, and the matrix $\mathcal{I}$ is defined
as

(4.15) $\mathcal{I}:={\rm diag}(1,1,1,1,-1,-1,-1,-1)\,.$

In Appendix A we write down the generators in the adjoint representation as
well as the Killing form. This, in turn, can be used to calculate the
quadratic Casimir operator as

(4.16)
$\begin{split}\mathcal{C}^{2}&=\left(\mathcal{J}_{0},\mathcal{J}_{+},\mathcal{J}_{-},\Sigma,\mathcal{R}_{+},\mathcal{R}_{-},\mathcal{S}_{+},\mathcal{S}_{-}\right)\mathbf{K}^{-1}\left(\mathcal{J}_{0},\mathcal{J}_{+},\mathcal{J}_{-},\Sigma,\mathcal{R}_{+},\mathcal{R}_{-},\mathcal{S}_{+},\mathcal{S}_{-}\right)^{t}\\\
&=\mathcal{J}_{0}^{2}-\tfrac{1}{2}\left(\mathcal{J}_{+}\mathcal{J}_{-}+\mathcal{J}_{-}\mathcal{J}_{+}\right)+\mathcal{J}_{0}-\left(\mathcal{R}_{+}\mathcal{S}_{-}+\mathcal{S}_{+}\mathcal{R}_{-}\right)-\tfrac{1}{4}\Sigma^{2}\,.\end{split}$

Now, if we act with $\mathcal{C}^{2}$ on the minimum weight vector555Note that
given that $\mathcal{C}^{2}$ commutes with every generator of
$\mathfrak{sl}(2|1)$, the eigenvalue of the minimum weight vector of $\langle
k,\sigma\rangle$ characterizes the whole representation space. of the
representation space $\langle k,\sigma\rangle$ we get (_cf._ appendix B for
the action of the odd generators)

(4.17)
$\mathcal{C}^{2}|k,k,\sigma\rangle=\left(k^{2}-\frac{\sigma^{2}}{4}\right)|k,k,\sigma\rangle\,.$

On the other hand, using the commutation relations in (3.11) and the defining
relations of the generators (equation (3.8)), it is straightforward to verify
that, in terms of the dynamical variables in (2.1), the quadratic Casimir
operator can be written as

(4.18)
$\mathcal{C}^{2}=\mathcal{J}^{2}-\tfrac{1}{4}\left(\hat{L}^{2}-1\right)=0\,,$

where the last equality follows from equation (3.5). Thus, from equations
(4.17) and (4.18) we conclude that the allowed values of $\sigma$ are
precisely $\sigma=\pm 2k$. This implies that, taking into account our previous
construction of the representation space of $\mathfrak{sl}(2|1)$, the
representation space of the _irreps_ we will be interested in are restricted
to

(4.19) $\begin{cases}\langle k,\sigma\rangle\oplus\langle
k+\tfrac{1}{2},\sigma-1\rangle\quad\text{if}\quad\sigma=2k\\\ \\\ \langle
k,\sigma\rangle\oplus\langle
k+\tfrac{1}{2},\sigma+1\rangle\quad\text{if}\quad\sigma=-2k\,.\end{cases}$

###  4.2. The $\mathfrak{so}(2)$ factor

As we discussed in §3, in order to express the Hamiltonians we will consider
in this article, we require an additional operator $\mathcal{L}$. This
operator commutes with all of the generators of $\mathfrak{sl}(2|1)$ so that
we take it to be the generator of an additional one dimensional Lie algebra
$\mathfrak{so}(2)$. The irreducible representations of this Lie algebra are
characterized by a real number666Below we show that in our setting, $j$ will
actually take half-integer values. $j\in\mathbb{R}$ such that, for a given
$j$, the representation space is the one dimensional vector space generated by
$|j\rangle$ with $\mathcal{L}|j\rangle=j|j\rangle$.

Thus, the characteristic subspaces of the Hamiltonians we will study are
representation spaces of unitary irreducible representations of the direct sum
of Lie algebras $\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$, which we denote by
$\langle k,\sigma,j\rangle$, whose vectors we write in the form
$|k,m,\sigma,j\rangle$. In particular, taking into account the discussion of
the representations selected by the quadratic Casimir operator in §4.1, we
have just to consider

(4.20) $\begin{cases}\langle k,\sigma,j\rangle\oplus\langle
k+\tfrac{1}{2},\sigma-1,j\rangle\quad\text{if}\quad\sigma=2k\\\ \\\ \langle
k,\sigma,j\rangle\oplus\langle
k+\tfrac{1}{2},\sigma+1,j\rangle\quad\text{if}\quad\sigma=-2k\,.\end{cases}$

Notably, given that the vectors of the above representation spaces are
eigenvectors of both $\Sigma$ and $\mathcal{L}$, they are also eigenvectors of
the linear combinations $\Sigma_{3}:=2(\Sigma-\mathcal{L})$ and
$L:=2\mathcal{L}-\Sigma$. In fact, for $\sigma=2k$ we have,

(4.21)
$\begin{array}[]{c}\Sigma_{3}|k,m,2k,j\rangle=2(2k-j)|k,m,2k,j\rangle\\\ \\\
\Sigma_{3}|k+\tfrac{1}{2},m,2k-1,j\rangle=2(2k-1-j)|k+\tfrac{1}{2},m,2k-1,j\rangle\,.\\\
\end{array}$

In particular, for the applications we are interested in, $\Sigma_{3}$ is
represented by the third Pauli matrix $\sigma_{3}$ (_cf._ equations (3.6) and
(3.12)), whose eigenvalues are $\pm 1$. Thus, we conclude that in our setting,
$2(2k-j)=1$ while $2(2k-1-j)=-1$, which gives us an expression for $j$ in
terms of $k$, given by the half-integer $j=2k-\tfrac{1}{2}$. Similarly, we may
act on the vectors of the representation space corresponding to $\sigma=2k$
with $L$, to get

(4.22)
$\begin{array}[]{c}L|k,m,2k,2k-\tfrac{1}{2}\rangle=(2k-1)|k,m,2k,2k-\tfrac{1}{2}\rangle\\\
\\\
L|k+\tfrac{1}{2},m,2k-1,2k-\tfrac{1}{2}\rangle=2k|k+\tfrac{1}{2},m,2k-1,2k-\tfrac{1}{2}\rangle\,.\end{array}$

In the representation defined by the dynamical variables (2.3), $L$ is
represented by $\hat{L}$ (_cf._ equations (3.6) and (3.12)), so the eigenvalue
$\ell$ of $\hat{L}$ is related to the eigenvalue of $\mathcal{J}^{2}$ by
$\ell=2k-1\geq 0$, consistent with equation (3.5). On the other hand, if
$\sigma=-2k$, with a completely analogous calculation, we get
$j=\tfrac{1}{2}-2k$ and $\ell=-(2k-1)<0$, which again, is consistent with
equation (3.5).

Taking this into account, and in order to simplify notation for the examples
we will consider in the next sections, we relabel our vectors using a
different set of quantum numbers, namely, $k$, $\ell$ and $\pm$, the latter,
corresponding to the eigenvalue of $\sigma_{3}$. Explicitly,

(4.23) $|k,m,\sigma,j\rangle\longrightarrow|k,m,\ell,\pm\rangle\,.$

Note that in terms of these quantum numbers, $\sigma=2k$ and $\sigma=-2k$
correspond to positive and negative angular momentum respectively, so that
equation (4.20) becomes

(4.24) $\begin{cases}\langle k,\ell,+\rangle\oplus\langle
k+\tfrac{1}{2},\ell+1,-\rangle\quad\text{if}\quad\ell=2k-1\geq 0\\\ \\\
\langle k,\ell,-\rangle\oplus\langle
k+\tfrac{1}{2},\ell-1,+\rangle\quad\text{if}\quad\ell=-2k+1\leq
0\,.\end{cases}$

Having included the $\mathfrak{so}(2)$ factor, we may now argue why the
eigenvalues of $\mathcal{J}_{0}$ have to be positive. From equation (3.1) we
may solve for

(4.25)
$\frac{\hat{\mathbf{X}}^{2}}{\theta}=\frac{2}{\sqrt{\kappa\theta}}\mathcal{J}_{0}+\sqrt{\frac{1-\kappa\theta}{\kappa\theta}}(\mathcal{J}_{+}+\mathcal{J}_{-})-\hat{L}\,,$

and taking into account that this is the square of a Hermitian operator, we
have

(4.26) $0\leq\langle
k,m,\ell,\pm|\hat{\mathbf{X}}^{2}|k,m,\ell,\pm\rangle=\frac{2m}{\sqrt{\kappa\theta}}-\ell\,,$

from where we conclude that, for any given $\ell$ (positive or negative)
$2m\geq\ell\sqrt{\kappa\theta}$. This implies that $m$ is bounded from below
and thus $m=k,k+1,\dots$.

In the following sections we use this algebraic construction to solve two
different problems: A charged particle in a uniform magnetic field, and a
particle in a finite cylindrical well.

##  5\. The Landau Problem

Let us consider the problem of a charged massive particle of spin $1/2$ in a
uniform magnetic field perpendicular to the NC plane. We consider the
Hamiltonian given in (2.11) with a minimal coupling to the electromagnetic
field,

(5.1) $\hat{P_{i}}\to\hat{P}_{i}-q\hat{A}_{i}\,,$

where we choose

(5.2) $\hat{A}_{i}=-\frac{\mathcal{B}}{2}\epsilon_{ij}\hat{X}_{j}\,,$

for some real constant $\mathcal{B}$. Note that, according to the covariant
definition of the field tensors, this gauge field produces a uniform magnetic
field,

(5.3)
$B=\frac{1}{iq}\,[P_{1}-qA_{1},P_{2}-qA_{2}]=\mathcal{B}+\frac{\kappa}{q}+\frac{\theta
q\mathcal{B}^{2}}{4}\,.$

The Landau Hamiltonian reads

(5.4)
$\hat{H}=\hat{\mathbf{P}}\cdot\boldsymbol{\sigma}-\omega\hat{\mathbf{X}}\times\boldsymbol{\sigma}-m\sigma_{3}\,,\quad\text{with
}\omega=\frac{q\mathcal{B}}{2}\,.$

For the moment we do not make any assumption on whether $q\mathcal{B}$ is
positive or negative.

Using Eq. (3.13) we may express the Hamiltonian as

(5.5) $\begin{array}[]{c}\hat{H}=\displaystyle
i\sqrt[4]{\frac{\kappa}{\theta}}\left(\sqrt{1+\sqrt{\kappa\theta}}\left(\mathcal{R}_{+}-\mathcal{S}_{-}\right)-\sqrt{1-\sqrt{\kappa\theta}}\left(\mathcal{R}_{-}-\mathcal{S}_{+}\right)\right)\\\
\displaystyle+i\omega\sqrt[4]{\frac{\theta}{\kappa}}\left(\sqrt{1+\sqrt{\kappa\theta}}\left(\mathcal{R}_{+}-\mathcal{S}_{-}\right)+\sqrt{1-\sqrt{\kappa\theta}}\left(\mathcal{R}_{-}-\mathcal{S}_{+}\right)\right)-m\sigma_{3}\,.\\\
\end{array}$

We now apply a unitary transformation in order to simplify the expression to
ultimately be able to diagonalize the Hamiltonian,

(5.6)
$\begin{split}\hat{H}_{r}&=e^{i\alpha\mathcal{J}_{2}}\hat{H}e^{-i\alpha\mathcal{J}_{2}}\\\
&=e^{i\alpha\mathcal{J}_{2}}\hat{H}_{m=0}e^{-i\alpha\mathcal{J}_{2}}-m\sigma_{3}\quad\text{since}\quad[\mathcal{J}_{2},\sigma_{3}]=0\,.\\\
\end{split}$

Here $\hat{H}_{r}$ is the transformed Hamiltonian, $\hat{H}_{m=0}$ is the
Hamiltonian $\hat{H}$ with $m$ set equal to zero and $\alpha\in\mathbb{R}$ to
guarantee the unitarity of the transformation. Recalling that
$\mathcal{J}_{2}=\frac{\mathcal{J}_{+}-\mathcal{J}_{-}}{2i}$, using the
commutation relations in (3.11) and the Hausdorff-Baker-Campbell formula, we
get

(5.7)
$\hat{H}_{r}=\frac{iM}{\sqrt[4]{\kappa\theta}}\left(\mathcal{R}_{+}-\mathcal{S}_{-}\right)-\frac{iK}{\sqrt[4]{\kappa\theta}}\left(\mathcal{R}_{-}-\mathcal{S}_{+}\right)-m\sigma_{3}\,,$

where we have defined

(5.8)
$\begin{array}[]{c}M=\sqrt{1+\sqrt{\kappa\theta}}\left(\sqrt{\kappa}+\omega\sqrt{\theta}\right)\cosh\left(\frac{\alpha}{2}\right)+\sqrt{1-\sqrt{\kappa\theta}}\left(\sqrt{\kappa}-\omega\sqrt{\theta}\right)\sinh\left(\frac{\alpha}{2}\right)\\\
\\\
K=\sqrt{1-\sqrt{\kappa\theta}}\left(\sqrt{\kappa}-\omega\sqrt{\theta}\right)\cosh\left(\frac{\alpha}{2}\right)+\sqrt{1+\sqrt{\kappa\theta}}\left(\sqrt{\kappa}+\omega\sqrt{\theta}\right)\sinh\left(\frac{\alpha}{2}\right)\,.\end{array}$

Next, we look for the values of $\alpha$ such that either $M=0$ or $K=0$, in
order to simplify this expression for $\hat{H}_{r}$ given in equation (5.7).
Explicitly, we have

(5.9) $\begin{array}[]{c}M=0\quad\text{ if and only if
}\quad\displaystyle\tanh{\left(\frac{\alpha}{2}\right)}=-\frac{\sqrt{1+\sqrt{\kappa\theta}}}{\sqrt{1-\sqrt{\kappa\theta}}}\cdot\frac{\sqrt{\kappa}+\omega\sqrt{\theta}}{\sqrt{\kappa}-\omega\sqrt{\theta}}\\\
\\\ K=0\quad\text{ if and only if
}\quad\displaystyle\tanh\left(\frac{\alpha}{2}\right)=-\frac{\sqrt{1-\sqrt{\kappa\theta}}}{\sqrt{1+\sqrt{\kappa\theta}}}\cdot\frac{\sqrt{\kappa}-\omega\sqrt{\theta}}{\sqrt{\kappa}+\omega\sqrt{\theta}}\,.\end{array}$

In particular, for $\alpha$ to be real, it is necessary and sufficient that

(5.10) $\left|\tanh\left(\frac{\alpha}{2}\right)\right|<1\,.$

Thus, combining equations (5.9) and (5.10) we get the following conditions

(5.11) $\begin{array}[]{c}K=0\,\text{ and }\,\alpha\in\mathbb{R}\quad\text{if
and only if}\quad\omega^{2}\theta+2\omega+\kappa>0\\\ \\\ M=0\,\text{ and
}\,\alpha\in\mathbb{R}\quad\text{if and only
if}\quad\omega^{2}\theta+2\omega+\kappa<0\,.\end{array}$

Note the magnetic field reappears in the expression
$\omega^{2}\theta+2\omega+\kappa=qB$.

As we show below, the Landau problem in the NC phase-space gives the same
result as the commutative one as long as one uses the physical magnetic field
$B$. If we take $q>0$, the case $K=0$ corresponds to $B>0$, whereas $M=0$
corresponds to $B<0$. Although the Hamiltonian in each of these cases takes a
different form, the procedure for solving the spectrum is exactly the same;
thus, we only discuss the case $B>0$.

Taking $\alpha$ such that $K=0$, from equation (5.8) we may solve for $M$ to
get777In fact, when solving for $M$ we get two possibilities for the
expression of the Hamiltonian that differ by a minus sign in the first term,
depending on whether $\alpha>0$ or $\alpha<0$. We take $\alpha>0$ given that
in both cases the spectrum is the same.

(5.12)
$\hat{H}_{r}=i\sqrt{2qB}\left(\mathcal{R}_{+}-\mathcal{S}_{-}\right)-m\sigma_{3}\,.$

We recall that, depending on the value of $\ell=\pm(2k-1)$, the
representations of $\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$ that intervene
are different. Hence we consider, for $\ell=2k-1\geq 0$, in light of the
action of the generators $\mathcal{R}_{+}$ and $\mathcal{S}_{-}$ (_cf._
Appendix B), an eigenvector as a linear combination of

(5.13)
$|k,k+n+1,\ell,+\rangle\quad\text{and}\quad\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\ell+1,-\right\rangle\,.$

The eigenvalue equation for the Hamiltonian (5.12), leads to a system of two
linear equations whose solutions are the energy spectrum

(5.14) $E_{\ell\geq 0}^{\pm}=\pm\sqrt{2qB(n+1)+m^{2}}\quad\text{with}\quad
n=0,1,2,\dots\,,$

which is independent of $\ell$. Note that the set of vectors
$|k,k,\ell,+\rangle$ with $\ell\geq 0$ are eigenvectors of the Hamiltonian
associated to the ground-state energy level $E=-m$, which is therefore
infinitely degenerate. The fact that there are no states with energy $E=m$ is
characteristic to the two dimensional setting, as discussed in [27].
Similarly, for every $n\geq 0$, we have an infinite number of states with the
same energy with a different value of $\ell=0,1,2\dots$. On the other hand, if
$\ell=-2k+1\leq 0$, we take an eigenvector as a linear combination of

(5.15)
$|k,k+n,\ell,-\rangle\quad\text{and}\quad\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\ell-1,+\right\rangle\,.$

Again, the eigenvalue equation for $\hat{H}_{r}$ leads to a system of two
linear equations whose solutions are the energy spectrum for negative angular
momentum

(5.16) $E^{\pm}_{\ell\leq
0}=\pm\sqrt{2qB(n-\ell+1)+m^{2}}\quad\text{with}\quad n=0,1,2,\dots\text{ and
}\ell=0,-1,-2\dots\,.$

We note, in contrast to the positive angular momentum case, that the
degeneracy is now finite. In fact, we may put together equations (5.14) and
(5.16) to write the spectrum for arbitrary $\ell$ as

(5.17)
${E^{+}=\sqrt{2qB\left(n-\tfrac{1}{2}(\ell-|\ell|)+1\right)+m^{2}}\,,\quad
E^{-}=-\sqrt{2qB\left(n-\tfrac{1}{2}\left(\ell-|\ell|\right)\right)+m^{2}}}$

with $n=0,1,2\dots$ and $\ell=0,\pm 1,\pm 2,\dots$ . This expression for the
spectrum agrees with the well known result in the commutative plane [27] if
written in terms of the physical magnetic field $B$. As expected, in the limit
$\kappa,\theta\to 0$ the spectrum converges smoothly to the commutative one.
Furthermore, our result agrees with that of [28], where the Landau problem in
the NC phase-space has also been considered. In this case, coordinate NC is
achieved via a Moyal-Weyl product of ordinary functions, whereas a Bopp’s
shift is used to realize momentum NC.

Finally, it is worth noting that the spectrum described in (5.17) shows a
smooth limit for $\kappa\rightarrow\kappa_{c}^{-}$, with
$\kappa_{c}=1/\theta$, as in the non-relativistic case [22]. Indeed, even
though the algebraic structure of the Hamiltonian is very different for
$\kappa>\kappa_{c}$ (whose description following the lines of the article can
be performed considering the graded extension of $\mathfrak{su}(2)$), it can
be seen that the spectrum of the Landau model presents the same form as in
(5.17) (also with infinite degeneracy), having the same
$\kappa\rightarrow\kappa_{c}^{+}$ limit. Even more, one can straightforwardly
show that the dimensional reduction taking place at the critical point, as
mentioned in Section 2, reduces the Hamiltonian in (5.4) to that of a harmonic
oscillator whose spectrum coincides precisely with both lateral limits for
$\kappa\rightarrow\kappa_{c}$, also with infinite degeneracy this time due to
the center of the deformed Heisenberg algebra. Thus, the spectrum of the
Landau model in this two-dimensional non-commutative phase space is continuous
at $\kappa\theta=1$.

##  6\. The Cylindrical Well

In this section we study a Dirac particle in a finite cylindrical well of
radius $A$ and depth $V_{0}$ in the non-commutative plane. In principle, we
could add to the Hamiltonian a position dependent step potential
$V_{0}\Theta\left(\hat{\mathbf{X}}^{2}-A^{2}\right)$, which corresponds to the
timelike component of a vector potential. However, this procedure leads to the
Klein paradox, in which a large variation of the potential in a small region
of space induces a solution which can only be interpreted in terms of particle
creation; as a consequence, one does not obtain the wave function of a single
particle confined within the interior of the well [29]. A way out of this
problem is to consider a Lorentz scalar potential, instead of a vector
potential. This can be done by adding a space-dependent mass term888Note that
a mass term –contrary to a vector potential– acts in the same way for
particles and antiparticles, so no pair creation is to be expected.

(6.1)
$\hat{H}=\hat{\mathbf{P}}\cdot\boldsymbol{\sigma}-\left(m+V_{0}\Theta\left(\hat{\mathbf{X}}^{2}-A^{2}\right)\right)\sigma_{3}$

where $V_{0}$ is a constant with mass units.

In a similar fashion as with the Landau Hamiltonian, we look for a unitary
transformation to simplify our problem. From equations (3.1) and (3.3) we may
express $\hat{\mathbf{X}}^{2}$ in terms of the generators of the graded
algebra, and consider the transformation

(6.2)
$e^{i\alpha\mathcal{J}_{2}}\left(\frac{\hat{\mathbf{X}}^{2}}{\theta}\right)e^{-i\alpha\mathcal{J}_{2}}=2\mathcal{J}_{0}-\hat{L}\,.$

with $\tanh(\alpha)=\sqrt{1-\kappa\theta}$, as shown in Appendix C (_cf._
equation (C.4)). For this choice of $\tanh(\alpha)$ the kinetic term
transforms as in the previous section. Taking into account that
$[\sigma_{3},\mathcal{J}_{2}]=0$, the transformed Hamiltonian is given by

(6.3)
$\begin{array}[]{c}\hat{H}_{r}=e^{i\alpha\mathcal{J}_{2}}\hat{H}e^{-i\alpha\mathcal{J}_{2}}=i\sqrt{\frac{2}{\theta}}\left[(\mathcal{R}_{+}-\mathcal{S}_{-})-\sqrt{1-\kappa\theta}(\mathcal{R}_{-}-\mathcal{S}_{+})\right]\\\
\\\
-\left(m+V_{0}\Theta\left[\theta\left\\{2\mathcal{J}_{0}-\hat{L}\right\\}-A^{2}\right]\right)\sigma_{3}\,.\end{array}$

We now carry out the algebraic solution of the eigenvalue problem. As before,
representations will differ depending on the value of $\ell=\pm(2k-1)$, so we
start with $\ell=2k-1$. The representation space for the graded Lie algebra
for positive $\ell$ is given by $\langle k,\ell,+\rangle\oplus\left\langle
k+\tfrac{1}{2},\ell+1,-\right\rangle$ (with the notation in (4.23) - (4.24)).
Hence, we propose a general eigenvector of the form

(6.4) $|\psi\rangle_{\ell\geq
0}=\sum_{n=0}^{\infty}\alpha_{n}|k,k+n,\ell,+\rangle+\sum_{n=0}^{\infty}\beta_{n}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\ell+1,-\right\rangle\,.$

Considering the eigenvalue equation $\hat{H}_{r}|\psi\rangle=E|\psi\rangle$
and taking into account that these vectors are linearly independent, we get a
system of coupled three-term recurrence relations for the coefficients
$\alpha_{n}$ and $\beta_{n}$ given by

(6.5)
$\begin{array}[]{c}\left[V_{0}\Theta\left(\theta\\{2n+1\\}-A^{2}\right)+m+E\right]\alpha_{n}-i\frac{\sqrt{2}}{\sqrt{\theta}}\sqrt{n}\,\beta_{n-1}+\\\
\\\ +i\sqrt{\frac{2}{\theta}}\sqrt{(2k+n)(1-\kappa\theta)}\,\beta_{n}=0\\\ \\\
\left[V_{0}\Theta\left(\theta\\{2n+1\\}-A^{2}\right)+m-E\right]\beta_{n}-i\sqrt{\frac{2}{\theta}}\sqrt{n+1}\,\alpha_{n+1}+\\\
\\\ +i\sqrt{\frac{2}{\theta}}\sqrt{(2k+n)(1-\kappa\theta)}\,\alpha_{n}=0\,.\\\
\end{array}$

Due to the presence of the Heaviside function, we define the matching integer

(6.6)
$n_{0}^{(+)}=\left[\frac{A^{2}}{2\theta}-\frac{1}{2}\right]\in\mathbb{N}_{0}\,,$

with $[\,\cdot\,]:\mathbb{R}\to\mathbb{N}_{0}$ the floor function, so that

(6.7)
$\begin{cases}\begin{array}[]{c}\Theta(\theta(2n+1)-A^{2})=0\quad\text{if}\quad
n\leq n_{0}^{(+)}\\\ \Theta(\theta(2n+1)-A^{2})=1\quad\text{if}\quad n\geq
n_{0}^{(+)}+1\,.\end{array}\end{cases}$

We start by simplifying the recurrence relations in equation (6.5) with the
definitions

(6.8) $C_{n}=\frac{\sqrt{(2k+n-1)!}}{\sqrt{n!}}\alpha_{n}\,,\quad
D_{n}=\frac{\sqrt{(2k+n)!}}{\sqrt{n!}\sqrt{\theta}}\beta_{n}\,\quad\text{and}\quad
b=\sqrt{1-\kappa\theta}\,,$

so that the resulting recurrence relations are given by

(6.9) $\begin{array}[]{c}(E+m)C_{n}-i\sqrt{2}(D_{n-1}+bD_{n})=0\\\ \\\
(E-m)\theta D_{n}-i\sqrt{2}b(2k+n)C_{n}+i\sqrt{2}(n+1)C_{n+1}=0\\\
\end{array}$

if $n\leq n_{0}^{(+)}$, while if $n\geq n_{0}^{(+)}+1$, we get

(6.10) $\begin{array}[]{c}(E+m+V_{0})C_{n}-i\sqrt{2}(D_{n-1}+bD_{n})=0\\\ \\\
(E-m-V_{0})\theta D_{n}-i\sqrt{2}b(2k+n)C_{n}+i\sqrt{2}(n+1)C_{n+1}=0\,.\\\
\end{array}$

Equations (6.9) may be used to solve for the coefficients $C_{n}$ with $n\leq
n_{0}^{(+)}+1$ and $D_{n}$ with $n\leq n_{0}^{(+)}$, whereas equations (6.10)
may be used to solve for the coefficients $C_{n}$ with $n\geq n_{0}^{(+)}+1$
and $D_{n}$ with $n\geq n_{0}^{(+)}$. We note that the coefficients
$C_{n_{0}^{(+)}+1}$ and $D_{n_{0}^{(+)}}$ may be computed using either pair of
equations, a matching condition which will be used later on to determine the
spectrum.

Let us start by considering equation (6.9) for $n=0$. This will set the
initial condition

(6.11) $(E+m)C_{0}-i\sqrt{2}bD_{0}=0\,,$

which allows us to write $D_{0}$ in terms of $C_{0}$, and thus, by replacing
in the second equation of (6.9), solve for $C_{1}$. In principle, this
procedure could be employed iteratively to solve for the coefficients $C_{n}$
and $D_{n}$ up to $n_{0}^{(+)}+1$ and $n_{0}^{(+)}$ respectively. Notice that
in iteratively solving (6.10) for $n\geq n_{0}^{(+)}+1$ we should not impose
an _initial condition_ as in (6.11), but rather the normalizability of the
eigenvector, which demands the coefficient to vanish sufficiently fast.

However, in order to get closed expressions for the coefficients, we will
proceed differently. Let us define the generating functions

(6.12) $F(x):=\sum_{n=0}^{\infty}C_{n}x^{n}\quad\text{and}\quad
G(x):=\sum_{n=0}^{\infty}D_{n}x^{n}\,.$

We multiply the first equation of (6.9) and (6.10) by $x^{n}$ and take the sum
from $n=1$. Similarly, we multiply the second equation of (6.9) and (6.10) by
$x^{n+1}$ and sum from $n=0$. Equations (6.9) and (6.10) thus turn into the
set of coupled differential equations

(6.13) $\begin{array}[]{c}(E+m)F(x)-i\sqrt{2}(x-b)G(x)=0\\\ \\\
(E-m)\,x\,\theta\,G(x)-i2\sqrt{2}\,k\,b\,x\,F(x)+ix\sqrt{2}(1-bx)F^{\prime}(x)=0\end{array}$

and

(6.14) $\begin{array}[]{c}(E+m+V_{0})F(x)-i\sqrt{2}(x-b)G(x)=C_{0}V_{0}\\\ \\\
(E-m-
V_{0})\,x\,\theta\,G(x)-i2\sqrt{2}\,k\,b\,x\,F(x)+ix\sqrt{2}(1-bx)F^{\prime}(x)=0\,,\end{array}$

where we have used equation (6.11) in (6.13). We note that the first equation
of (6.14) is –besides the different constant coefficient multiplying the first
term– the non-homogeneous version of the first equation of (6.13). The
solutions for equation (6.13) denoted by $F_{L}$ and $G_{L}$, are given by

(6.15)
$\begin{array}[]{c}F_{L}(x)=(x-b)^{\frac{\theta(m^{2}-E^{2})}{2(b^{2}-1)}}(1-bx)^{\frac{\theta(E^{2}-m^{2})}{2(b^{2}-1)}-2k}\\\
\\\
G_{L}(x)=-{\displaystyle\frac{i(E+m)}{\sqrt{2}}}(x-b)^{-1+\frac{\theta(m^{2}-E^{2})}{2(b^{2}-1)}}(1-bx)^{\frac{\theta(E^{2}-m^{2})}{2(b^{2}-1)}-2k}\end{array}$

while those for equation (6.14), denoted by $F_{H}$ and $G_{H}$ are

(6.16)
$\begin{array}[]{c}F_{H}(x)=\displaystyle\frac{C_{0}V_{0}}{E+m+V_{0}}\,_{2}F_{1}\left(1,2k,1+\frac{\theta(E^{2}-(m+V_{0})^{2})}{2(b^{2}-1)},\frac{b(b-x)}{b^{2}-1}\right)\\\
\\\
G_{H}(x)=\displaystyle\frac{iC_{0}V_{0}}{\sqrt{2}(b-x)}\left[\,{}_{2}F_{1}\left(1,2k,1+\frac{\theta(E^{2}-(m+V_{0})^{2})}{2(b^{2}-1)},\frac{b(b-x)}{b^{2}-1}\right)-1\right]\end{array}$

The coefficients, are finally given by

(6.17)
$C_{n}=\begin{cases}\begin{array}[]{c}\frac{1}{n!}F_{L}^{(n)}(0)\quad\text{for
}n\leq n_{0}^{(+)}+1\\\ \\\ \frac{1}{n!}F_{H}^{(n)}(0)\quad\text{for }n\geq
n_{0}^{(+)}+1\end{array}\end{cases}\quad
D_{n}=\begin{cases}\begin{array}[]{c}\frac{1}{n!}G_{L}^{(n)}(0)\quad\text{for
}n\leq n_{0}^{(+)}\\\ \\\ \frac{1}{n!}G_{H}^{(n)}(0)\quad\text{for }n\geq
n_{0}^{(+)}\end{array}\end{cases}\,.$

Now, given the linearity of the recurrence relations, the coefficients
obtained by differentiation of the generating functions, are defined up to an
$n$-independent scale $R$. So, for the matching conditions, we have

(6.18)
$\frac{R}{(n_{0}^{(+)}+1)!}F_{L}^{(n_{0}^{(+)}+1)}(0)=\frac{1}{(n_{0}^{(+)}+1)!}F_{H}^{(n_{0}^{(+)}+1)}(0)\,.$

and

(6.19)
$\frac{R}{n_{0}^{(+)}!}G_{L}^{(n_{0}^{(+)})}(0)=\frac{1}{n_{0}^{(+)}!}G_{H}^{(n_{0}^{(+)})}(0)\,.$

By eliminating $R$ we get an implicit expression for the energy spectrum,

(6.20)
$G_{L}^{(n_{0}^{(+)})}(0)F_{H}^{(n_{0}^{(+)}+1)}(0)=F_{L}^{(n_{0}^{(+)}+1)}(0)G_{H}^{(n_{0}^{(+)})}(0)\,,$

with $F_{L}$ and $G_{L}$ given in equation (6.15), and $F_{H}$ and $G_{H}$ in
(6.16).

We now turn our attention to the negative angular momentum case. The
representation space of the graded Lie algebra for negative $\ell$ is given by
$\langle k,\ell,-\rangle\oplus\left\langle
k+\tfrac{1}{2},\ell-1,+\right\rangle$ so we take a general linear combination
of the form

(6.21)
$|\psi\rangle_{\ell<0}=\sum_{n=0}^{\infty}\alpha_{n}|k,k+n,\ell,-\rangle+\sum_{n=0}^{\infty}\beta_{n}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\ell-1,+\right\rangle\,.$

From the eigenvalue equation $\hat{H}|\psi\rangle_{\ell\leq
0}=E|\psi\rangle_{\ell\leq 0}$, we get a new set of coupled recurrence
relations given by

(6.22)
$\begin{array}[]{c}\left[V_{0}\Theta\left(\theta\\{4k+2n-1\\}-A^{2}\right)+m-E\right]\alpha_{n}-i\sqrt{\frac{2}{\theta}}\sqrt{2k+n}\,\beta_{n}+\mbox{}\\\
\\\
\mbox{}+i\sqrt{\frac{2}{\theta}}\sqrt{n}\sqrt{1-\kappa\theta}\,\beta_{n-1}=0\\\
\\\
\left[V_{0}\Theta\left(\theta\\{4k+2n+1\\}-A^{2}\right)+m+E\right]\beta_{n}-i\sqrt{\frac{2}{\theta}}\sqrt{2k+n}\,\alpha_{n}+\mbox{}\\\
\\\
\mbox{}+i\sqrt{\frac{2}{\theta}}\sqrt{(n+1)(1-\kappa\theta)}\,\alpha_{n+1}=0\,.\\\
\end{array}$

Notice that the arguments of the Heaviside step functions is different in this
case. We define a new matching integer

(6.23) $n_{0}^{(-)}=\left[\frac{A^{2}}{2\theta}-\frac{1}{2}\right]-2k\,,$

so that, together with the definitions given in equation (6.8), we may write
simplified expressions for our recurrence relations (6.22),

(6.24) $\begin{array}[]{c}(E-m)C_{n}-i\sqrt{2}(bD_{n-1}+D_{n})=0\\\ \\\
(E+m)\theta D_{n}-i\sqrt{2}(2k+n)C_{n}+i\sqrt{2}b(n+1)C_{n+1}=0\\\
\end{array}$

if $n\leq n_{0}^{(-)}$, while if $n=n_{0}^{(-)}+1$ we get

(6.25)
$\begin{array}[]{c}(E-m)C_{n_{0}^{(-)}+1}-i\sqrt{2}(bD_{n_{0}^{(-)}}+D_{n_{0}^{(-)}+1})=0\\\
\\\ (E+m+V_{0})\theta
D_{n_{0}^{(-)}+1}-i\sqrt{2}(2k+n_{0}^{(-)}+1)C_{n_{0}^{(-)}+1}+i\sqrt{2}b(n_{0}^{(-)}+2)C_{n_{0}^{(-)}+2}=0\,,\\\
\end{array}$

and finally for $n\geq n_{0}^{(-)}+2$

(6.26) $\begin{array}[]{c}(E-m-V_{0})C_{n}-i\sqrt{2}(bD_{n-1}+D_{n})=0\\\ \\\
(E+m+V_{0})\theta D_{n}-i\sqrt{2}(2k+n)C_{n}+i\sqrt{2}b(n+1)C_{n+1}=0\,.\\\
\end{array}$

Next, we proceed similarly to the previous case, by defining generating
functions as in (6.12), and rewriting our recurrence relations in terms of
$F(x)$ and $G(x)$. Equations (6.24) and (6.26) thus become respectively

(6.27) $\begin{array}[]{c}(E-m)F(x)-i\sqrt{2}(bx-1)G(x)=0\\\ \\\
(E+m)\,x\,\theta\,G(x)-i2\sqrt{2}\,k\,x\,F(x)+ix\sqrt{2}(b-x)F^{\prime}(x)=0\end{array}$

and

(6.28) $\begin{array}[]{c}(E+m+V_{0})F(x)-i\sqrt{2}(x-b)G(x)=C_{0}V_{0}\\\ \\\
(E-m-
V_{0})\,x\,\theta\,G(x)-i2\sqrt{2}\,k\,b\,x\,F(x)+ix\sqrt{2}(1-bx)F^{\prime}(x)=0\,,\end{array}$

whose solutions are

(6.29)
$\begin{array}[]{c}F_{L}(x)=(x-b)^{-2k+\frac{\theta(m^{2}-E^{2})}{2(b^{2}-1)}}(1-bx)^{\frac{\theta(E^{2}-m^{2})}{2(b^{2}-1)}}\\\
\\\
G_{L}(x)={\displaystyle\frac{i(E-m)}{\sqrt{2}}}(x-b)^{-2k+\frac{\theta(m^{2}-E^{2})}{2(b^{2}-1)}}(1-bx)^{\frac{\theta(E^{2}-m^{2})}{2(b^{2}-1)}-1}\end{array}$

for equation (6.27), and

(6.30) $\begin{array}[]{c}F_{H}(x)=\displaystyle\frac{C_{0}V_{0}}{E-m-
V_{0}}\,_{2}F_{1}\left(1,2k,1+\frac{\theta(E^{2}-(m+V_{0})^{2})}{2(b^{2}-1)},\frac{bx-1}{b^{2}-1}\right)\\\
\\\
G_{H}(x)=-\displaystyle\frac{iC_{0}V_{0}}{\sqrt{2}(bx-1)}\left[\,{}_{2}F_{1}\left(1,2k,1+\frac{\theta(E^{2}-(m+V_{0})^{2})}{2(b^{2}-1)},\frac{bx-1}{b^{2}-1}\right)-1\right]\end{array}$

for equation (6.28). The coefficients may be computed as in (6.17), again up
to a global scale $R$. Notice that we have two different ways to compute the
coefficient $D_{n_{0}^{(-)}+1}$: either with the first equation of (6.25) in
terms of $D_{n_{0}^{(-)}}$ and $C_{n_{0}^{(-)}+1}$ or with the first equation
of (6.26) in terms of $C_{n_{0}^{(-)}+2}$ and $D_{n_{0}^{(-)}+2}$. Similarly,
we have two ways of computing $C_{n_{0}^{(-)}+1}$; either with the second
equation of (6.24) or with the second equation of (6.25). Concretely, we have

(6.31)
$Rb\left[(E-m)\frac{F_{L}^{(n_{0}^{(-)}+1)}}{{(n_{0}^{(-)}+1})!}-i\sqrt{2}b\frac{G_{L}^{(n_{0}^{(-)})}}{{n_{0}^{(-)}}!}\right]=(E-m-
V_{0})\frac{F_{H}^{(n_{0}^{(-)}+2)}}{{(n_{0}^{(-)}+2})!}-i\sqrt{2}\frac{G_{H}^{(n_{0}^{(-)}+2)}}{{(n_{0}^{(-)}+2})!}$

(6.32) $\frac{R(2k+n_{0}^{(-)}+1)}{b}\left[(E+m)\theta
G_{L}^{(n_{0}^{(-)})}-i\sqrt{2}(2k+n_{0}^{(-)})F_{L}^{(n_{0}^{(-)})}\right]\\\
=-\left[(E+m+V_{0})\theta
G_{H}^{(n_{0}^{(-)}+1)}+i\sqrt{2}bF_{H}^{(n_{0}^{(-)}+2)}\right]$

Once again, by eliminating $R$, we get an implicit expression for the energy
spectrum in terms of derivatives of the generating functions as defined in
equations (6.29) and (6.30).

##  7\. Conclusions

In the framework of quantum mechanics in the non-commutative phase space, we
have studied the algebraic structure of fermionic systems described by Dirac
Hamiltonians in $(2+1)-$dimensions.

In so doing, we have considered non-vanishing commutators not only for
coordinates but also for momenta, as in Eq. (2.6). In this non-commutative
phase space we have constructed the generator of rotations, $\hat{L}$, that
reduces to its ordinary counterpart in the $\kappa,\theta\to 0$ limit, but is
singular at the critical value of the momentum non-commutativity parameter,
$\kappa_{c}=1/\theta$. At this point, a dimensional reduction takes place,
since the system can be described in terms of only one pair of conjugate
dynamical variables. This fact requires separate descriptions for the regions
$\kappa>\kappa_{c}$ and $\kappa<\kappa_{c}$ and thus, these systems present
two quantum phases, where only the latter can be connected with the ordinary
(commutative) case. For this reason, we have restricted ourselves to the
region $\kappa<\kappa_{c}$.

In [22], where non-relativistic systems were studied from this perspective, it
was found that the rotationally invariant Hermitian quadratic forms in the
non-commutative dynamical variables generate, in particular, a three-
dimensional Lie algebra which corresponds to the non-compact
$\mathfrak{sl}(2,\mathbb{R})$. With the goal of extending this method to
Dirac-type Hamiltonians, we noticed that the adequate structure to incorporate
in the formalism is the graded Lie algebra $\mathfrak{sl}(2|1)$. Indeed, we
constructed the set of eight generators of $\mathfrak{sl}(2|1)$ in terms of
the non-commutative dynamical variables $\hat{X}_{i}$, $\hat{P}_{i}$, $i=1,2$,
the Pauli matrices, $\sigma_{i}$, $i=0,1,2$ and the generator of rotations
$\hat{L}$.

Our construction also required the definition of a total angular momentum
operator $\mathcal{L}$, that together with the generators of the graded Lie
algebra were sufficient to express completely the Hamiltonians considered in
this article. Thus, the problem associated to computing the spectrum of these
Dirac-like Hamiltonians, was translated into the study of the unitary
irreducible representations of the graded Lie algebra
$\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$, where the second factor is
generated by $\mathcal{L}$. For completeness, we have included a detailed
construction of these _irreps_ in a general setting, specifying those which
are relevant to our problem.

From this algebraic point of view, we have analyzed two different problems,
namely, the non-commutative extensions of the Landau model and of a
cylindrical well. Indeed, taking into account that the characteristic
subspaces of these Hamiltonians are unitary irreducible representations of the
graded Lie algebra $\mathfrak{sl}(2|1)\oplus\mathfrak{so}(2)$, we obtained in
each case an expression for the energy spectrum.

For the charged particle subject to a uniform magnetic field perpendicular to
the NC plane, we found the well-known Landau levels with a constant gap
between them proportional to the (covariant) magnetic field.

In the second case, we considered a cylindrical well potential. Since the
coordinates are represented by Hermitian operators on a Hilbert space, the
boundary of the well is implemented by means of an orthogonal projector onto a
subspace of the representation space defined by a condition on the mean square
radius. This allows to reduce the eigenvalue equation into a pair of coupled
linear three-term recursion for the expansion coefficients of the solution,
which includes a potential term that changes when going from the interior to
the exterior of the well. This occurs for a particular eigenvalue of
$\mathcal{J}_{0}$, and the equations involving the corresponding coefficient
imposes a set of two matching conditions which appear as an effective boundary
condition and provide an implicit expression for the energy spectrum.

##  Appendix A Adjoint Representation of $\mathfrak{sl}(2|1)$

In this appendix we include the matrices corresponding to the generators of
$\mathfrak{sl}(2|1)$ in the adjoint representation, the Killing form
$\mathbf{K}$ and its inverse $\mathbf{K}^{-1}$. We recall that the graded
algebra is spanned by the set of eight generators
$\\{\mathcal{J}_{0},\mathcal{J}_{\pm},\Sigma,\mathcal{R}_{\pm},\mathcal{S}_{\pm}\\}$.
Thus,

(A.1) $\begin{array}[]{c}\mathcal{J}^{\rm
ad}_{-}=\left(\begin{array}[]{cccccccc}0&2&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0\\\
\end{array}\right)\,,\par\,\,\mathcal{J}^{\rm
ad}_{+}=-\left(\begin{array}[]{cccccccc}0&0&2&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&0&0\\\ \end{array}\right)\,,\\\ \\\
\mathcal{J}^{\rm ad}_{0}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{2}&0&0&0\\\ 0&0&0&0&0&-\frac{1}{2}&0&0\\\
0&0&0&0&0&0&\frac{1}{2}&0\\\ 0&0&0&0&0&0&0&-\frac{1}{2}\\\
\end{array}\right)\,,\,\,\Sigma^{\rm
ad}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&-1&0\\\ 0&0&0&0&0&0&0&-1\\\ \end{array}\right)\,,\par\\\ \\\
\mathcal{R}^{\rm ad}_{+}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-\frac{1}{2}\\\
-\frac{1}{2}&0&0&-1&0&0&0&0\\\ 0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ \end{array}\right)\,,\,\,\mathcal{R}^{\rm
ad}_{-}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&\frac{1}{2}&0\\\ 0&1&0&0&0&0&0&0\\\
\frac{1}{2}&0&0&-1&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\right)\,,\\\ \\\ \mathcal{S}^{\rm
ad}_{+}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&1&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ -\frac{1}{2}&0&0&1&0&0&0&0\\\ 0&0&-1&0&0&0&0&0\\\
\end{array}\right)\,,\,\,\mathcal{S}^{\rm
ad}_{-}=\left(\begin{array}[]{cccccccc}0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&0&-\frac{1}{2}&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ \frac{1}{2}&0&0&1&0&0&0&0\\\
\end{array}\right)\,,\\\ \\\
\mathbf{K}=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&0&-2&0&0&0&0&0\\\ 0&-2&0&0&0&0&0&0\\\ 0&0&0&-4&0&0&0&0\\\ 0&0&0&0&0&0&0&2\\\
0&0&0&0&0&0&-2&0\\\ 0&0&0&0&0&2&0&0\\\ 0&0&0&0&-2&0&0&0\\\
\end{array}\right)\,,\,\,\mathbf{K}^{-1}=-\left(\begin{array}[]{cccccccc}-1&0&0&0&0&0&0&0\\\
0&0&\frac{1}{2}&0&0&0&0&0\\\ 0&\frac{1}{2}&0&0&0&0&0&0\\\
0&0&0&\frac{1}{4}&0&0&0&0\\\ 0&0&0&0&0&0&0&\frac{1}{2}\\\
0&0&0&0&0&0&-\frac{1}{2}&0\\\ 0&0&0&0&0&\frac{1}{2}&0&0\\\
0&0&0&0&-\frac{1}{2}&0&0&0\\\ \end{array}\right)\,,\par\end{array}$

##  Appendix B Action of the Odd Generators

Here we summarize the action of the odd generators on vectors of the
representation space for $\sigma=\pm 2k.$

###  B.1. Case $\sigma=2k$

(B.1) $\begin{array}[]{c}\mathcal{R}_{+}|k,k+n,\sigma\rangle=0\\\ \\\
\mathcal{R}_{-}|k,k+n,\sigma\rangle=0\end{array}$

(B.2)
$\begin{array}[]{c}\mathcal{S}_{+}|k,k+n,\sigma\rangle=\sqrt{2k+n}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma-1\right\rangle\\\
\\\
\mathcal{S}_{-}|k,k+n,\sigma\rangle=\sqrt{n}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+(n-1),\sigma-1\right\rangle\\\
\\\ \end{array}$

(B.3)
$\begin{array}[]{c}\mathcal{R}_{+}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma-1\right\rangle=\sqrt{n+1}|k,k+n+1,\sigma\rangle\\\
\\\
\mathcal{R}_{-}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma-1\right\rangle=\sqrt{2k+n}|k,k+n,\sigma\rangle\end{array}$
(B.4)
$\begin{array}[]{c}\mathcal{S}_{+}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma-1\right\rangle=0\\\
\\\
\mathcal{S}_{-}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma-1\right\rangle=0\\\
\\\ \end{array}$

###  B.2. Case $\sigma=-2k$

(B.5)
$\begin{array}[]{c}\mathcal{R}_{+}|k,k+n,\sigma\rangle=\sqrt{2k+n}\left|k+\tfrac{1}{2},k+n+\tfrac{1}{2},\sigma+1\right\rangle\\\
\\\
\mathcal{R}_{-}|k,k+n,\sigma\rangle=\sqrt{n}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+(n-1),\sigma+1\right\rangle\end{array}$
(B.6) $\begin{array}[]{c}\mathcal{S}_{+}|k,k+n,\sigma\rangle=0\\\ \\\
\mathcal{S}_{-}|k,k+n,\sigma\rangle=0\\\ \\\ \end{array}$

(B.7)
$\begin{array}[]{c}\mathcal{R}_{+}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma+1\right\rangle=0\\\
\\\
\mathcal{R}_{-}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma+1\right\rangle=0\end{array}$
(B.8)
$\begin{array}[]{c}\mathcal{S}_{+}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma+1\right\rangle=\sqrt{n+1}\left|k,k+n+1,\sigma\right\rangle\\\
\\\
\mathcal{S}_{-}\left|k+\tfrac{1}{2},k+\tfrac{1}{2}+n,\sigma+1\right\rangle=\sqrt{2k+n}\left|k,k+n,\sigma\right\rangle\\\
\\\ \end{array}$

##  Appendix C Unitary Transformations in $\mathfrak{sl}(2,\mathbb{R})$

The generators in the fundamental (nonunitary irreducible) representation of
$\mathfrak{sl}(2,\mathbb{R})$ can be chosen as [30]

(C.1) $X_{0}=-\frac{1}{2}\sigma_{2}\,\quad X_{1}=\frac{i}{2}\sigma_{1}\,\quad
X_{2}=\frac{i}{2}\sigma_{3}\,,$

where $\sigma_{i}$, $i=1,2,3$ are the Pauli matrices. These generators satisfy
the commutation relations

(C.2) $[X_{\mu},X_{\nu}]=-i\epsilon_{\mu\nu\lambda}X^{\lambda}\,$

where $X^{\mu}=\eta^{\mu\nu}X_{\nu}$ with $\eta^{\mu\nu}={\rm diag}(1,-1,-1)$.
It is straightforward to verify by direct computation that

(C.3) $e^{i\alpha X_{2}}\left(AX_{0}+BX_{1}\right)e^{-i\alpha
X_{2}}=A\sqrt{1-\frac{B^{2}}{A^{2}}}X_{0}$

for $A,B\in\mathbb{R}$ with $|A|>|B|$, if we take $\tanh(\alpha)=B/A$. Then,
in any unitary representation of $\mathfrak{sl}(2,\mathbb{R})$ (with Hermitian
generators $\mathcal{J}_{i}$) we also have

(C.4)
$e^{i\alpha\mathcal{J}_{2}}\left(A\mathcal{J}_{0}+B\mathcal{J}_{1}\right)e^{-i\alpha\mathcal{J}_{2}}=A\sqrt{1-\frac{B^{2}}{A^{2}}}\mathcal{J}_{0}\,.$

Indeed, the left hand side of Eq. (C.4) can be formally written as

(C.5)
$\begin{array}[]{c}\displaystyle\sum_{k=0}^{\infty}\frac{(i\alpha)^{k}}{k!}\left(\left[\mathcal{J}_{2},\cdot\,\right]^{k}\left(A\mathcal{J}_{0}+B\mathcal{J}_{1}\right)\right)\\\
\\\
=(A\cosh\alpha-B\sinh\alpha)\mathcal{J}_{0}+(B\cosh\alpha-A\sinh\alpha)\mathcal{J}_{1}\end{array}$

The coefficient of $\mathcal{J}_{1}$ on the right hand side of this equation
vanishes if we choose $\tanh(\alpha)=B/A$ (for $|B/A|<1$), in which case one
gets the right hand side of Eq. (C.4).

Similarly,

(C.6)
$\begin{array}[]{c}e^{i\beta\mathcal{J}_{1}}\left(A\mathcal{J}_{0}+B\mathcal{J}_{2}\right)e^{-i\beta\mathcal{J}_{1}}=\\\
\\\
=(A\cosh\beta+B\sinh\beta)\mathcal{J}_{0}+(A\sinh\beta+B\cosh\beta)\mathcal{J}_{2}\,.\end{array}$

Therefore, by choosing $\tanh(\beta)=-B/A$, for real $A,B$ with $|A|>|B|$, one
gets

(C.7)
$e^{i\beta\mathcal{J}_{1}}\left(A\mathcal{J}_{0}+B\mathcal{J}_{2}\right)e^{-i\beta\mathcal{J}_{1}}=A\sqrt{1-\frac{B^{2}}{A^{2}}}\mathcal{J}_{0}\,.$

Acknowledgments:

This work was supported by CONICET, and UNLP under project X909.

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