### 2.1
*N* = 1, 2, 4 supersymmetric quantum mechanics

Let us analyze a quantum system, defined by a Hamiltonian \(\hat{H}\) (Hermitian in nature) acting on some Hilbert space which is constructed in terms of N self-adjoint operators \(\hat{M}_{i} = \hat{M}_{i}^{\dag }\). Such quantum system is called supersymmetric provided the following anti-commutation relation is valid.

$$\left\{ {\hat{M}_{i} ,\hat{M}_{j} } \right\} = \hat{H}\delta_{ij} ,\quad \left( {i, \, j = 0,1,2,3,4, \ldots \ldots ,N} \right)$$

(1)

where curly bracket represents anticommuting relation. The self-adjoint operators \(\hat{M}_{i} \;\& \;\hat{M}_{j}\) are called supercharges, and the Hamiltonian \(\hat{H}\) is said to be SUSY Hamiltonian.

### 2.2
*N* = 1 quaternionic supersymmetric quantum mechanics

*N* = 1 SQM is defined in terms of only one supercharge called a generator of *N* = 1 SUSY. One-dimensional supersymmetric quantum mechanics is then described by the graded algebra and can be expressed as

$$\left\langle {\psi \left| {\hat{H}} \right|\psi } \right\rangle = \left\langle {\psi \left| {\hat{M}^{\dag } \hat{M}} \right|\psi } \right\rangle + \left\langle {\psi \left| {\hat{M}\hat{M}^{\dag } } \right|\psi } \right\rangle = \left| {\hat{M}} \right|\psi > |^{2} + \left| {\hat{M}^{\dag } } \right|\psi > |^{2} \ge 0$$

(2)

Here \(\left| {\hat{H}} \right|\psi >\) is the corresponding eigenstate. We may extend *N* = 1 quaternionic SUSY to the relativistic quantum mechanics where a system [22] is defined by a Pauli Hamiltonian for a spin 1/2 particle in an external magnetic field. Let us consider two quaternion gauge potentials \(\vec{A}_{\mu } \left( {x,t} \right)\) and \(\vec{B}_{\mu } \left( {x,t} \right)\). The two external quaternion gauge magnetic fields are given as

$$C = \vec{\nabla } \times \vec{A}_{\mu } \left( {x,t} \right)\;{\text{and}}\;C^{\prime } = \vec{\nabla } \times \vec{B}_{\mu } \left( {x,t} \right)$$

(3)

where \(\vec{A}_{\mu } \left( {x,t} \right)\) and \(\vec{B}_{\mu } \left( {x,t} \right)\) are quaternion potentials (μ = 0, 1, 2, 3) and defined as

$$\vec{A}_{\mu } \left( {x,t} \right) = A_{0} + \mathop \sum \limits_{l = 1}^{3} e_{l} \vec{A}_{l} \;{\text{and}}\;\vec{B}_{\mu } \left( {x,t} \right) = B_{0} + \mathop \sum \limits_{l = 1}^{3} e_{l} \vec{B}_{l}$$

(4)

Here \(A_{0}\) and \(B_{0}\) are the scalar part of electric and magnetic field, respectively, and \(\vec{A}_{l} , \vec{B}_{l}\) are vector part of the corresponding electric and magnetic field. We may introduce self-adjoint supercharge (\(\hat{M}_{D}\)) in electromagnetic field system in terms of momentum (\(\vec{p}_{l}\)), electric (\(\vec{A}_{l} )\), and magnetic field (\(\vec{B}_{l}\)), i.e.,

$$\hat{M}_{D} = ie_{l} \left( {\vec{p}_{l} - ie_{l} \vec{A}_{l} + ie_{l} \vec{B}_{l} } \right) = \hat{M}_{D}^{\dag }$$

(5)

in the above equation, if we substitute the scalar part of quaternion i. e. \(e_{l} \to e_{0} \to 1\), we get only one supercharge which is the generator of *N* = 1 SUSY; henceforth, we get Pauli Hamiltonian [21]

$$\hat{H}_{P} = 2\hat{M}_{D}^{2} = 2\left\{ {ie_{l} \left( {\vec{p}_{l} - ie_{l} \vec{A}_{l} + ie_{l} \vec{B}_{l} } \right)} \right\}^{2}$$

(6)

which is described as the Pauli Hamiltonian for a spin − 1/2 particles. Accordingly, we may write Dirac Hamiltonian \({\widehat{H}}_{D}\) as

$$\hat{H}_{D} = \mathop \sum \limits_{l = 1}^{3} \alpha_{l} \left( {\vec{p}_{l} - ie_{l} \vec{A}_{l} + ie_{l} \vec{B}_{l} } \right) + \beta m = \left[ {\begin{array}{*{20}l} m \hfill & {ie_{l} \left( {\vec{p}_{l} - ie_{l} \vec{A}_{l} + ie_{l} \vec{B}_{l} } \right)} \hfill \\ {ie_{l} \left( {\vec{p}_{l} - ie_{l} \vec{A}_{l} + ie_{l} \vec{B}_{l} } \right)} \hfill & { - m} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} m & {\hat{M}_{D}^{\dag } } \\ {\hat{M}_{D} } & { - m} \\ \end{array} } \right]$$

(7)

; similarly Hamiltonian for *N* = 1 SUSY is obtained by substituting \(e_{l} \to e_{0} \to 1\); here we can replace Dirac matrices (\(\alpha_{l} , \beta\)) with quaternions (\(e_{l}\)) as

$$\alpha_{l} = \left[ {\begin{array}{*{20}l} 0 \hfill & {ie_{l} } \hfill \\ {ie_{l} } \hfill & 0 \hfill \\ \end{array} } \right]\;{\text{and}}\;\beta = \left[ {\begin{array}{*{20}l} I \hfill & 0 \hfill \\ 0 \hfill & { - I} \hfill \\ \end{array} } \right]$$

(8)

Squaring the Dirac Hamiltonian, we get

$$\hat{H}_{D}^{2} = \left[ {\begin{array}{*{20}l} {\hat{M}_{D} \hat{M}_{D}^{\dag } + m^{2} } \hfill & 0 \hfill \\ 0 \hfill & {\hat{M}_{D} \hat{M}_{D}^{\dag } + m^{2} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\hat{M}_{D}^{2} + m^{2} } \hfill & 0 \hfill \\ 0 \hfill & {\hat{M}_{D}^{2} + m^{2} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\frac{{\hat{H}_{P} }}{2} + m^{2} } \hfill & 0 \hfill \\ 0 \hfill & {\frac{{\hat{H}_{P} }}{2} + m^{2} } \hfill \\ \end{array} } \right]$$

(9)

where \(\hat{H}_{P}\) is Pauli's Hamiltonian; we also get the relationship for Dirac and Pauli Hamiltonian

$$\left[ {\hat{M}_{D} , \hat{H}_{D} } \right] = 0 ,\;\; \left[ {\hat{M}_{D} , \hat{H}_{P} } \right] = 0 ,\;\;\left[ {\hat{M}_{D} , \hat{M}_{D} } \right] = \hat{H}_{P}$$

(10)

### 2.3
*N* = 2 quaternionic supersymmetric quantum mechanics

*N* = 2 SUSY was discussed by Witten [16] as a simple model, which consists of an additional spin − 1/2 degree of freedom, and accordingly, we may write two supercharges \(\hat{M}_{1} \& \hat{M}_{2}\) in terms of quaternion units \(e_{1}\) and \(e_{2}\) and potential ∅(x) as follows

$$\hat{M}_{1} = \frac{1}{\sqrt 2 }\left( {p \times ie_{1 } + \emptyset \left( x \right) \times ie_{2} } \right)\;\& \;\hat{M}_{2} = \frac{1}{\sqrt 2 }\left( {p \times ie_{2 } - \emptyset \left( x \right) \times ie_{1} } \right)$$

(11)

Then the Hamiltonian is given by

$$\hat{H} = 2\hat{M}_{1}^{2} = 2\hat{M}_{2}^{2} = p^{2} + \emptyset^{2} - 2e_{3} p\emptyset = p^{2} + \emptyset^{2} - 2i\sigma_{3} p\left( {{\raise0.7ex\hbox{${ - {\text{d}}}$} \!\mathord{\left/ {\vphantom {{ - {\text{d}}} {{\text{d}}x}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{\text{d}}x}$}}} \right)\emptyset = p^{2} + \emptyset^{2} - 2\sigma_{3} \emptyset^{\prime }$$

(12)

here \(\sigma_{3}\) is the Pauli spin matrix related to the quaternion unit as \(\sigma_{3} = ie_{3}\). Here we can see that spin is the natural outcome of using quaternion units. Now Hamiltonian can be written in terms of super partner Hamiltonians \(\hat{H}_{ + }\) and \(\hat{H}_{ - }\) as

$$\hat{H} = \left[ {\begin{array}{*{20}l} {\hat{H}_{ + } } \hfill & 0 \hfill \\ 0 \hfill & {\hat{H}_{ - } } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {p^{2} + \emptyset^{2} + 2\emptyset^{\prime } } \hfill & 0 \hfill \\ 0 \hfill & {p^{2} + \emptyset^{2} - 2\emptyset^{\prime } } \hfill \\ \end{array} } \right]$$

(13)

For *N* = 2 SUSY, two complex supercharges \(\hat{M}_{1} \;\& \;\hat{M}_{2}\) are formed by substituting quaternion units by complex unit—*i* and Hamiltonian \(\hat{H}\) satisfying the following relations

$$\hat{M}_{1} \hat{M}_{2} = - \hat{M}_{2} \hat{M}_{1} = - \frac{1}{2}\left( {ip^{2} + i\emptyset^{2} + p\emptyset } \right)\;\& \;\hat{H} = 2\hat{M}_{1}^{2} = 2\hat{M}_{2}^{2} = p^{2} + \emptyset^{2} - 2ip\emptyset = p^{2} + \emptyset^{2} - 2\emptyset^{\prime }$$

(14)

Let the two complex supercharges are given by

$$\hat{M} = \frac{1}{\sqrt 2 }\left( {\hat{M}_{1 } + i\hat{M}_{2} } \right)\;\& \;\hat{M}^{\dag } = \frac{1}{\sqrt 2 }\left( {\hat{M}_{1 } - i\hat{M}_{2} } \right)$$

(15)

where *i* is a complex quantity and belongs to c (1, *i*) space. These supercharge \(\hat{M}, \hat{M}^{\dag }\) and Hamiltonian \(\hat{H}\) should satisfy the SUSY algebra,

$$\hat{M}^{2} = \hat{M}^{\dag } = 0\;\& \;\hat{H} = \left\{ {\hat{M}, \hat{M}^{\dag } } \right\}$$

Now to satisfy the relation \(\hat{M}^{2} = \hat{M}^{\dag 2} = 0\), the complex supercharges are given by nilpotent matrices, which are defined as follows

$$\hat{M} = \left[ {\begin{array}{*{20}l} 0 \hfill & a \hfill \\ 0 \hfill & 0 \hfill \\ \end{array} } \right]\;\& \;\hat{M}^{\dag } = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill \\ {a^{\dag } } \hfill & 0 \hfill \\ \end{array} } \right]$$

(16)

Hence the Hamiltonian becomes

$$\hat{H} = \left\{ {\hat{M},\hat{M}^{\dag } } \right\} = \left[ {\begin{array}{*{20}l} {aa^{\dag } } \hfill & 0 \hfill \\ 0 \hfill & {a^{\dag } a} \hfill \\ \end{array} } \right]$$

(17)

where \(a \&\) \(a^{\dag }\) are annihilation and creation operators defined as

$$a = \frac{1}{{\sqrt {2\omega } }}\left( { - ip + \omega q} \right) \;\& \;a^{\dag } = \frac{1}{{\sqrt {2\omega } }}\left( {ip + \omega q} \right)$$

(18)

Substituting \(2\omega = 1\) and \(\omega q = U\left( x \right), \;U\left( x \right)\) is real super potential; we can write \(a\) and \(a^{\dag }\) in the following manner

$$a = \left( { - ip + U\left( x \right)} \right) = - \frac{{\text{d}}}{{{\text{d}}x}} + U\;{\text{and}}\;a^{\dag } = \left( {ip + U\left( x \right)} \right) = \frac{{\text{d}}}{{{\text{d}}x}} + U$$

(19)

The supercharges for this case are given by

$$\hat{M} = \left[ {\begin{array}{*{20}l} 0 \hfill & { - \frac{d}{dx} + U} \hfill \\ 0 \hfill & 0 \hfill \\ \end{array} } \right]\;\& \;\hat{M}^{\dag } = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill \\ {\frac{{\text{d}}}{{{\text{d}}x}} + U} \hfill & 0 \hfill \\ \end{array} } \right]$$

(20)

and the Hamiltonian is given by

$$\hat{H} = \left[ {\begin{array}{*{20}l} {\hat{H}_{ + } } \hfill & 0 \hfill \\ 0 \hfill & {\hat{H}_{ - } } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} { - \frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }} - U^{\prime } + U^{2} } \hfill & 0 \hfill \\ 0 \hfill & { - \frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }} + U^{\prime } + U^{2} } \hfill \\ \end{array} } \right]$$

(21)

Here \(\hat{H}\)_{,} \(\hat{M}\), and \(\hat{M}^{\dag }\) satisfy the SUSY algebra given by

$$\left[ {\hat{H}, \hat{M}} \right] = \left[ {\hat{H}, \hat{M}^{\dag } } \right] = 0,\;\;\widehat{{\{ M}} ,\hat{M}\} = \left\{ {\hat{M}^{\dag } , \hat{M}^{\dag } } \right\} = 0\;\& \;\hat{H} = \left\{ {\hat{M}, \hat{M}^{\dag } } \right\}$$

(22)

where square bracket represents commutation relation while curly bracket represents anticommutation relation. We consider \(\psi\) as a two-component (\(\psi_{a}\) & \(\psi_{b}\)) spinor given by

$$\psi = \left[ {\begin{array}{*{20}c} {\psi_{a} } \\ {\psi_{b} } \\ \end{array} } \right]$$

(23)

and necessary condition for SUSY to be a good supersymmetry that supercharges annihilate the ground state

$$\hat{M}\left| {\psi > = \hat{M}^{\dag } } \right|\psi > = 0$$

(24)

In terms of energy, the condition for SUSY to be a good SUSY is that the ground state energy should be zero. Using Eqs. (20) and (23), we get

$$- \psi_{b}^{\prime } + U\psi_{b} = 0\;{\text{and}}\;\psi_{a}^{\prime } + U\psi_{a} = 0$$

(25)

where *U* is given by

$$U = \frac{{\psi_{b}^{\prime } }}{{\psi_{b} }} = - \frac{{\psi_{a}^{\prime } }}{{\psi_{a} }};$$

So that

$$\psi_{b} = e^{{ - \mathop \smallint \limits_{{x_{0} }}^{x} \vec{U}\left( s \right).{\text{d}}\vec{S}}} \;{\text{and}}\;\psi_{a} = e^{{ - \mathop \smallint \limits_{{x_{0} }}^{x} \vec{U}\left( s \right).{\text{d}}\vec{S}}} .$$

(26)

Let us define the quaternion wave function as two-component complex spinors

$$\psi = \left[ {\begin{array}{*{20}c} {\psi_{a} } \\ {\psi_{b} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\psi^{ + } } \\ {\psi^{ - } } \\ \end{array} } \right]$$

(27)

### 2.4
*N* = 4 supersymmetric quantum mechanics

According to Hull [22], *N* = 4 SUSY QM can be formed from *N* = 2 SUSY QM by extending the complex number i to three imaginary units, described as quaternion units. Thus *N* = 4 SUSY QM can be obtained by replacing *i* by \(\vec{e}\) in Eqs. (18) and (19). Then we get

$$a = \left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)\;{\text{and}}\;a^{\dag } = \left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)$$

(28)

Here we consider *U* as the quaternionic super potential by taking only the imaginary part and leaving the real part and is defined by

$$U = \mathop \sum \limits_{l = 1}^{3} e_{l} w_{l} = e_{1} w_{1} + e_{2} w_{2} + e_{3} w_{3}$$

(29)

$$U^{\dag } = - \mathop \sum \limits_{l = 1}^{3} e_{l} w_{l} = - e_{1} w_{1} - e_{2} w_{2} - e_{3} w_{3}$$

(30)

Hence the supercharges for this case are deduced as

$$\hat{M} = \left[ {\begin{array}{*{20}l} 0 \hfill & {\left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)} \hfill \\ 0 \hfill & 0 \hfill \\ \end{array} } \right]\;\& \;\hat{M}^{\dag } = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill \\ {\left( { - \vec{e}_{l} .\vec{p}_{l} + U^{\dag } } \right)} \hfill & 0 \hfill \\ \end{array} } \right]$$

(31)

if we replace \(\vec{e}_{l} \to \vec{e}_{0} ,\vec{e}_{1} ,\vec{e}_{2} ,\vec{e}_{3}\) we obtain 4 supercharges for *N* = 4 SQM called generators of *N* = 4 SQM.

and the Hamiltonian becomes

$$\hat{H} = \left[ {\begin{array}{*{20}l} {\left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)\left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)} \hfill & 0 \hfill \\ 0 \hfill & {\left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)\left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)} \hfill \\ \end{array} } \right]$$

(32)

Similarly, we get four super-partner Hamiltonians for *N* = 4 SQM by replacing \(\vec{e}_{l} \to \vec{e}_{0} ,\vec{e}_{1} ,\vec{e}_{2} ,\vec{e}_{3}\) and satisfy the usual SUSY algebra given by Eq. (22). Equation (32) reduces to the following expression of supercharges on using the value of U from Eqs. (30) and (31), i.e.,

$$\hat{M} = \left[ {\begin{array}{*{20}l} 0 \hfill & {\vec{e}_{l} \left( { - \vec{p}_{l} + w_{l} } \right)} \hfill \\ 0 \hfill & 0 \hfill \\ \end{array} } \right]\;\& \;\hat{M}^{\dag } = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill \\ { - \vec{e}_{l} \left( {\vec{p}_{l} - w_{l} } \right)} \hfill & 0 \hfill \\ \end{array} } \right]$$

(33)

Corresponding wave function is given by

$$\psi = \psi_{0} + e_{1} \psi_{1} + e_{2} \psi_{2} + e_{3} \psi_{3}$$

Or

$$\psi = \left[ {\begin{array}{*{20}c} {\psi_{a} } \\ {\psi_{b} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\psi^{ + } } \\ {\psi^{ - } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\psi_{0} } \\ {\psi_{1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\psi_{2} } \\ {\psi_{3} } \\ \end{array} } \\ \end{array} } \right]$$

where \(\psi^{ + }\) and \(\psi^{ - }\) are again two-component spinors corresponding to an upper and lower component of Dirac spinor in the following manner

$$\psi^{ + } = \psi_{0} + e_{1} \psi_{1} = \left[ {\begin{array}{*{20}l} {\phi^{ + } } \hfill \\ 0 \hfill \\ \end{array} } \right]\;{\text{and}}\;\psi^{ - } = \psi_{2} - e_{1} \psi_{3} = \left[ {\begin{array}{*{20}l} 0 \hfill \\ {\phi^{ - } } \hfill \\ \end{array} } \right]$$

(34)

where we have used \(a = \left( { - \vec{e}_{l} .\vec{p}_{l} + U} \right)\). If we substitute quaternion basis elements \(e_{l}\) by \(e_{l} = - i\sigma_{l}\), so that \(a = - i\sigma_{l} \left( { - \vec{p}_{l} + w_{l} } \right)\), here \(\sigma_{l}\) is the Pauli spin matrix. Showing that by using quaternionic algebra spin automatically appears in the structure. Hence spin naturally occurs in quaternionic quantum mechanics, which is not possible in *N* = 2 supersymmetric quantum mechanics. Replacing \(e_{l} = - i\sigma_{l}\), we get the following representation for Dirac matrices as follows

$$\gamma_{l} = \left[ {\begin{array}{*{20}l} {e_{l} } \hfill & 0 \hfill \\ 0 \hfill & {e_{l} } \hfill \\ \end{array} } \right]$$

(35)

where

$$\gamma_{l}^{\dag } = - \gamma_{l} ,\;\;tr\gamma_{l} = 0 \;\& \;\gamma_{l} \gamma_{k} + \gamma_{k} \gamma_{l} = - 2\delta_{lk}$$

(36)

These are the matrices, which have been used by Rotelli [20] in formulating the quaternionic Dirac equation.