Supersymmetric quantum mechanics (SUSY QM)

Supersymmetric quantum mechanics factors partner Hamiltonians through first-order operators. Its odd supercharges anticommute to the Hamiltonian; on a Riemannian manifold, the Hilbert space can be realized by square-integrable differential forms with supercharges given by a differential and its adjoint.

The fermionic Fock space of $n$ modes is the exterior algebra ${\Lambda }^{\bullet }{\mathbb{C}}^n$. Creation acts by exterior multiplication and annihilation by contraction.

A fermionic operator is odd under fermion parity: if $P=(-1{)}^N$ is the parity operator, then $PAP=-A$. A fermionic creation operator or fermionic annihilation operator is an example. Interchanging odd insertions in a time-ordered product contributes the corresponding fermionic sign. Odd parity alone does not imply that every pair of such operators has zero anticommutator.

A Clifford vacuum is a state annihilated by all the chosen fermionic annihilation operators. In a massless supermultiplet it labels the starting helicity, and applying fermionic raising and lowering operators constructs the remaining states.

Fermionic creation and annihilation operators satisfy $\{a_i,a_j^{†}\}={\delta }_{ij}$ and vanishing equal-type anticommutators. These relations make each mode either empty or singly occupied.

The fermion number operator counts occupied fermionic modes. Its parity $(-1{)}^F$ is positive on even exterior degree and negative on odd exterior degree.

A supertrace is the trace weighted by fermion parity. Supersymmetry makes paired positive-energy states cancel in a supertrace.

An equivariant supertrace inserts a symmetry $f$ as well as fermion parity, $\mathrm{Tr}((-1{)}^{F}fA)$. In a path integral, $f$ produces boundary conditions twisted by its action.

A fermionic path integral integrates Grassmann-valued histories. Its Gaussian integral is a determinant, in contrast with the inverse determinant produced by a bosonic Gaussian integral.

The Berezin integral is differentiation with respect to a Grassmann variable: $\int d\theta 1=0$ and $\int d\theta \theta =1$. A finite-dimensional fermionic Gaussian integral gives a determinant or Pfaffian.

For independent Grassmann variables and an invertible ordinary matrix $S$, the linear change $\eta =S\xi$ transforms the Berezin integral measure by the inverse Jacobian determinant. Indeed, the highest-degree monomial transforms by $\det S$, and coefficient extraction must undo that factor. For paired variables $\eta =S\xi$ and $\bar{\eta }=\bar{\xi }{S}^{-1}$ the two factors cancel. The order of odd variables and differentials fixes the overall sign and must be kept consistent.

The short-time limit rescales a Euclidean time circle toward zero size. Supersymmetry cancels paired nonzero-mode determinants, leaving an integral over constant or symmetry-fixed zero modes.

For a real function $h$, the twisted de Rham differential is $d_h=d+dh\wedge =e^{-h}d{e}^h$. Its adjoint and $d_h$ define a supersymmetric Hamiltonian, and its square-integrable cohomology describes zero-energy states.

Partner Hamiltonians reverse the order of two first-order factors and share corresponding positive-energy states.

A Pöschl-Teller potential is a solvable one-dimensional potential built from ${\mathrm{sech}}^{2}x$. The attractive memberis reflectionless when $n$ is a positive integer.

LetThenThus the one-soliton Schrodinger operator is paired with the free Schrodinger operator. Its normalizable state annihilated by $A$ is proportional to $\mathrm{sech}(\chi x)$.