Supersymmetric quantum mechanics and the Witten index

Let $A=(\partial_x+u(x))/\sqrt2$ on square-integrable wavefunctions on the real line, where u is a real smooth function. With a common supersymmetry-preserving domain and vanishing integration-by-parts boundary terms, its adjoint is $A^\dagger=(-\partial_x+u)/\sqrt2$. Define Q to map the upper, even-parity sector into the lower, odd-parity sector through A. Set $H=\{Q,Q^\dagger\}$ in this lesson. This normalization has no extra factor of two and fixes the oscillator energy spacings quoted below.

Multiplication gives partner potentials involving u squared and opposite signs of its derivative. Both Hamiltonians are nonnegative because they are of the forms A dagger A and A A dagger. If an upper-sector eigenstate has positive energy E, applying A produces a lower-sector state of the same energy; its norm squared is E times the original norm. Division by square root E normalizes the partner. At zero energy this construction can vanish, allowing an unpaired state rather than forcing equal numbers of zero modes.

An even zero mode obeys $A\psi_B=0$, giving a wavefunction proportional to the exponential of minus the integral of u. An odd zero mode obeys A dagger psi F equals zero and has the opposite exponential. Normalization requires the integral of the squared magnitude over the entire real line to converge. A solution decaying at one end but growing at the other is not a ground state. Boundary conditions on a finite interval would also have to preserve both the adjoint relation and the supersymmetry action.

For u equal to x, the even solution is Gaussian and normalizable, while the odd solution grows. The even Hamiltonian has energies zero, one, two, and so on, and the odd Hamiltonian starts at one. Each positive energy is paired and the ground state is unpaired. Reversing the sign of u interchanges which parity contains the Gaussian. These examples show that the sign of an index is a grading convention and zero-mode orientation, rather than a negative number of physical states.

A nonzero well-defined index proves at least one supersymmetric zero-energy state exists. A zero index does not prove supersymmetry breaking because equal numbers of even and odd zero modes can cancel. To conclude breaking one must show the actual absence of normalizable zero-energy states and the existence of a suitable ground state. An infimum reached only at infinity is not a normalizable vacuum. These logical distinctions reappear in field theories with flat directions, runaway potentials, and continuous spectra.

The usual deformation-invariance argument cancels positive-energy pairs while varying parameters smoothly. It assumes states cannot escape to infinity, the relevant domains remain compatible, and the trace or spectral regulator behaves properly. Noncompact directions or a continuum reaching zero can violate those assumptions. For example, changing the asymptotic growth of u can send a normalizable Gaussian into a nonnormalizable limit. Declaring the index topological without examining this boundary behavior would hide the very mechanism by which its apparent value can change.