Harmonic Oscillator and Lie Superalgebra
The harmonic oscillator provides the simplest setting where operator factorization leads naturally to supersymmetric structure. The Hamiltonian is written as
\[H^{(\mathrm{LHO})}=p^2+q^2\]with canonical commutation relation
\[[q,p]=i\]where $\hbar=1$. Define first-order operators
\[A=q+ip,\qquad B=q-ip\]and compute
\[AB=(q+ip)(q-ip)=q^2+p^2+i(pq-qp)\]using $[q,p]=i\Rightarrow pq-qp=-i$ gives
\[AB=q^2+p^2+1=H^{(\mathrm{LHO})}+1\]similarly
\[BA=(q-ip)(q+ip)=q^2+p^2+i(qp-pq)=H^{(\mathrm{LHO})}-1\]hence
\[H^{(\mathrm{LHO})}=AB-1=BA+1\]which shows two distinct factorizations differing by a constant shift. Define partner Hamiltonians
\[H^{(L)}=BA=H^{(\mathrm{LHO})}-1,\qquad H^{(R)}=AB=H^{(\mathrm{LHO})}+1\]and combine them into a matrix super-Hamiltonian
\[\mathcal H=\begin{pmatrix}H^{(L)}&0\\0&H^{(R)}\end{pmatrix}\]acting on two-component states
\[\Psi(q)=\begin{pmatrix}\psi_L(q)\\\psi_R(q)\end{pmatrix}\]introducing two sectors. Define supercharges
\[Q=\begin{pmatrix}0&0\\A&0\end{pmatrix},\qquad \tilde Q=\begin{pmatrix}0&B\\0&0\end{pmatrix}\]which map $\Psi$ as
\[Q\Psi=\begin{pmatrix}0\\A\psi_L\end{pmatrix},\qquad \tilde Q\Psi=\begin{pmatrix}B\psi_R\\0\end{pmatrix}\]exchanging sectors. The algebra uses commutator $[X,Y]=XY-YX$ and anticommutator $\{X,Y\}=XY+YX$. The generators $\mathcal H$ (even) and $Q,\tilde Q$ (odd) form a Lie superalgebra. Compute nilpotency
\[Q^2=0,\qquad \tilde Q^2=0\]hence
\[\{Q,Q\}=0,\qquad \{\tilde Q,\tilde Q\}=0\]and evaluate
\[Q\tilde Q=\begin{pmatrix}0&0\\0&AB\end{pmatrix},\qquad \tilde Q Q=\begin{pmatrix}BA&0\\0&0\end{pmatrix}\]giving
\[\{Q,\tilde Q\}=\begin{pmatrix}BA&0\\0&AB\end{pmatrix}=\mathcal H\]and commutation
\[[\mathcal H,Q]=0,\qquad [\mathcal H,\tilde Q]=0\]since
\[H^{(R)}A=(AB)A=A(BA)=AH^{(L)},\qquad H^{(L)}B=(BA)B=B(AB)=BH^{(R)}\]establishing conserved supercharges. Introduce fermionic operators
\[F^\dagger=\begin{pmatrix}0&0\\1&0\end{pmatrix},\qquad F=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad N_F=\begin{pmatrix}0&0\\0&1\end{pmatrix}\]satisfying
\[F^2=0,\qquad (F^\dagger)^2=0,\qquad F^\dagger F=N_F,\qquad FF^\dagger=\begin{pmatrix}1&0\\0&0\end{pmatrix}\]encoding two-state fermionic structure. The supercharges factorize as
\[Q\sim a\,F^\dagger,\qquad \tilde Q\sim a^\dagger F\]combining bosonic ladder operators with fermionic operators. The vacuum is
\[\langle q|0\rangle=\begin{pmatrix}e^{-q^2/2}/\sqrt{\pi}\\0\end{pmatrix}\]satisfying
\[Q|0\rangle=0,\qquad \tilde Q|0\rangle=0\]indicating unbroken supersymmetry. With oscillator energies
\[E_n=2n+1\]the partner spectra become
\[E_n^{(L)}=2n,\qquad E_n^{(R)}=2n+2\]showing paired excited states and a single unpaired zero-energy ground state. This construction establishes the full supersymmetric structure via factorization, matrix formulation, supercharges, and Lie superalgebra relations.