In the scalar case, the text is probably just considering the (time-only version of) a massless scalar field (setting $m^2=0$), because the goal is to illustrate SUSY in the simplest possible way. That's probably also why they didn't include a potential. We don't stop calling it Klein-Gordon just because we're considering a limiting case ($m^2\to 0$ and $V\to 0$).

The Dirac side is more interesting. The Dirac equation can be formulated in any number of dimensions, and the case of one time and no space dimensions might be easier to appreciate if we consider the generalization first. In $1+D$ dimensions for any $D\in\{0,1,2,...\}$, the massless version of the Dirac equation is $$ \gamma^a\partial_a\psi=0 \tag{1} $$ where the $\gamma$s are the smallest possible set of linearly independent matrices (with complex components) satisfying the Clifford algebra relation $$ \gamma^a\gamma^b+\gamma^b\gamma^a = 2\eta^{ab}, \tag{2} $$ where $\eta^{ab}$ is the Minkowski metric. How small can these matrices be? This is an exercise in linear algebra (cf Dimension of Dirac $\gamma$ matrices). The matrices need to have size $2^n\times 2^n$ where $n$ is the integer part of $(1+D)/2$. Here's a table: $$ \begin{matrix} 1+D & n & 2^n \\ \hline 1 & 0 & 1 \\ 2 & 1 & 2 \\ 3 & 1 & 2 \\ 4 & 2 & 4 \\ 5 & 2 & 4 \\ 6 & 3 & 8 \\ 7 & 3 & 8 \\ \end{matrix} $$ The pattern should be clear. The case described in the question has $1+D=1$, so the $\gamma$-matrices have size $1\times 1$, and therefore the spinor $\psi$ only needs one component.

For some values of $1+D$, we can have a Majorana representation of the Clifford algebra, in which they have only real components but still have the same size $2^n\times 2^n$. We can also consider pseudo-Majorana representations (often also called Majorana representations), in which the components are purely imaginary. In either case, we can take $\psi$ to satisfy a reality condition of the form $\psi^*=C\psi$ for some matrix $C$. This is possible, in particular, when $1+D=1$. Based on the notation in the question, the text is considering a Majorana spinor, so its one-and-only component is self-adjoint (the Grassmann version of "real").

In the scalar case, the text is probably just considering the (time-only version of) a massless scalar field (setting $m^2=0$), because the goal is to illustrate SUSY in the simplest possible way. That's probably also why they didn't include a potential. We don't stop calling it Klein-Gordon just because we're considering a limiting case ($m^2\to 0$ and $V\to 0$).

The Dirac side is more interesting. The Dirac equation can be formulated in any number of dimensions, and the case of one time and no space dimensions might be easier to appreciate if we consider the generalization first. In $1+D$ dimensions for any $D\in\{0,1,2,...\}$, the massless version of the Dirac equation is $$ \gamma^a\partial_a\psi=0 \tag{1} $$ where the $\gamma$s are the smallest possible set of matrices (with complex components) satisfying the Clifford algebra relation $$ \gamma^a\gamma^b+\gamma^b\gamma^a = 2\eta^{ab}, \tag{2} $$ where $\eta^{ab}$ is the Minkowski metric. How small can these matrices be? This is an exercise in linear algebra (cf Dimension of Dirac $\gamma$ matrices). The matrices need to have size $2^n\times 2^n$ where $n$ is the integer part of $(1+D)/2$. Here's a table: $$ \begin{matrix} 1+D & n & 2^n \\ \hline 1 & 0 & 1 \\ 2 & 1 & 2 \\ 3 & 1 & 2 \\ 4 & 2 & 4 \\ 5 & 2 & 4 \\ 6 & 3 & 8 \\ 7 & 3 & 8 \\ \end{matrix} $$ The pattern should be clear. The case described in the question has $1+D=1$, so the $\gamma$-matrices have size $1\times 1$, and therefore the spinor $\psi$ only needs one component.

For some values of $1+D$, we can have a Majorana representation of the Clifford algebra, in which they have only real components but still have the same size $2^n\times 2^n$. We can also consider pseudo-Majorana representations (often also called Majorana representations), in which the components are purely imaginary. In either case, we can take $\psi$ to satisfy a reality condition of the form $\psi^*=C\psi$ for some matrix $C$. This is possible, in particular, when $1+D=1$. Based on the notation in the question, the text is considering a Majorana spinor, so its one-and-only component is self-adjoint (the Grassmann version of "real").

In the scalar case, the text is probably just considering the (time-only version of) a massless scalar field (setting $m^2=0$), because the goal is to illustrate SUSY in the simplest possible way. That's probably also why they didn't include a potential. We don't stop calling it Klein-Gordon just because we're considering a limiting case ($m^2\to 0$ and $V\to 0$).

The Dirac side is more interesting. The Dirac equation can be formulated in any number of dimensions, and the case of one time and no space dimensions might be easier to appreciate if we consider the generalization first. In $1+D$ dimensions for any $D\in\{0,1,2,...\}$, the massless version of the Dirac equation is $$ \gamma^a\partial_a\psi=0 \tag{1} $$ where the $\gamma$s are the smallest possible set of linearly independent matrices (with complex components) satisfying $$ \gamma^a\gamma^b+\gamma^b\gamma^a = 2\eta^{ab}, \tag{2} $$ where $\eta^{ab}$ is the Minkowski metric. How small can these matrices be? This is an exercise in linear algebra. The matrices need to have size $2^n\times 2^n$ where $n$ is the integer part of $(1+D)/2$. Here's a table: $$ \begin{matrix} 1+D & n & 2^n \\ \hline 1 & 0 & 1 \\ 2 & 1 & 2 \\ 3 & 1 & 2 \\ 4 & 2 & 4 \\ 5 & 2 & 4 \\ 6 & 3 & 8 \\ 7 & 3 & 8 \\ \end{matrix} $$ The pattern should be clear. The case described in the question has $1+D=1$, so the $\gamma$-matrices have size $1\times 1$, and therefore the spinor $\psi$ only needs one component.

For some values of $1+D$, we can have a Majorana representation of the $\gamma$-matrices, in which they have only real components but still have the same size $2^n\times 2^n$. We can also consider pseudo-Majorana representations (often also called Majorana representations), in which the components are purely imaginary. In either case, we can take $\psi$ to satisfy a reality condition of the form $\psi^*=C\psi$ for some matrix $C$. This is possible, in particular, when $1+D=1$. Based on the notation in the question, the text is considering a Majorana spinor, so its one-and-only component is self-adjoint (the Grassmann version of "real").

In the scalar case, the text is probably just considering the (time-only version of) a massless scalar field (setting $m^2=0$), because the goal is to illustrate SUSY in the simplest possible way. That's probably also why they didn't include a potential. We don't stop calling it Klein-Gordon just because we're considering a limiting case ($m^2\to 0$ and $V\to 0$).

The Dirac side is more interesting. The Dirac equation can be formulated in any number of dimensions, and the case of one time and no space dimensions might be easier to appreciate if we consider the generalization first. In $1+D$ dimensions for any $D\in\{0,1,2,...\}$, the massless version of the Dirac equation is $$ \gamma^a\partial_a\psi=0 \tag{1} $$ where the $\gamma$s are the smallest possible set of linearly independent matrices (with complex components) satisfying the Clifford algebra relation $$ \gamma^a\gamma^b+\gamma^b\gamma^a = 2\eta^{ab}, \tag{2} $$ where $\eta^{ab}$ is the Minkowski metric. How small can these matrices be? This is an exercise in linear algebra (cf Dimension of Dirac $\gamma$ matrices). The matrices need to have size $2^n\times 2^n$ where $n$ is the integer part of $(1+D)/2$. Here's a table: $$ \begin{matrix} 1+D & n & 2^n \\ \hline 1 & 0 & 1 \\ 2 & 1 & 2 \\ 3 & 1 & 2 \\ 4 & 2 & 4 \\ 5 & 2 & 4 \\ 6 & 3 & 8 \\ 7 & 3 & 8 \\ \end{matrix} $$ The pattern should be clear. The case described in the question has $1+D=1$, so the $\gamma$-matrices have size $1\times 1$, and therefore the spinor $\psi$ only needs one component.

For some values of $1+D$, we can have a Majorana representation of the Clifford algebra, in which they have only real components but still have the same size $2^n\times 2^n$. We can also consider pseudo-Majorana representations (often also called Majorana representations), in which the components are purely imaginary. In either case, we can take $\psi$ to satisfy a reality condition of the form $\psi^*=C\psi$ for some matrix $C$. This is possible, in particular, when $1+D=1$. Based on the notation in the question, the text is considering a Majorana spinor, so its one-and-only component is self-adjoint (the Grassmann version of "real").