Supersymmetric Quantum Mechanics - Example

Example

Let's look at the example of a one-dimensional nonrelativistic particle with a 2D (i.e., two states) internal degree of freedom called "spin" (it's not really spin because "real" spin is a property of 3D particles). Let b be an operator which transforms a "spin up" particle into a "spin down" particle. Its adjoint b† then transforms a spin down particle into a spin up particle; the operators are normalized such that the anticommutator {b,b†}=1. And of course, b2=0. Let p be the momentum of the particle and x be its position with =i. Let W (the "superpotential") be an arbitrary complex analytic function of x and define the supersymmetric operators

Note that Q₁ and Q₂ are self-adjoint. Let the Hamiltonian

where W' is the derivative of W. Also note that {Q₁,Q₂}=0. This is nothing other than N = 2 supersymmetry. Note that acts like an electromagnetic vector potential.

Let's also call the spin down state "bosonic" and the spin up state "fermionic". This is only in analogy to quantum field theory and should not be taken literally. Then, Q₁ and Q₂ maps "bosonic" states into "fermionic" states and vice versa.

Let's reformulate this a bit:

Define

and of course,

and

An operator is "bosonic" if it maps "bosonic" states to "bosonic" states and "fermionic" states to "fermionic" states. An operator is "fermionic" if it maps "bosonic" states to "fermionic" states and vice versa. Any operator can be expressed uniquely as the sum of a bosonic operator and a fermionic operator. Define the supercommutator [,} as follows: Between two bosonic operators or a bosonic and a fermionic operator, it is none other than the commutator but between two fermionic operators, it is an anticommutator.

Then, x and p are bosonic operators and b, Q and are fermionic operators.

Let's work in the Heisenberg picture where x, b and are functions of time.

Then,

This is nonlinear in general: i.e., x(t), b(t) and do not form a linear SUSY representation because isn't necessarily linear in x. To avoid this problem, define the self-adjoint operator . Then,

and we see that we have a linear SUSY representation.

Now let's introduce two "formal" quantities, ; and with the latter being the adjoint of the former such that

and both of them commute with bosonic operators but anticommute with fermionic ones.

Next, we define a construct called a superfield:

f is self-adjoint, of course. Then,

Incidentally, there's also a U(1)_R symmetry, with p and x and W having zero R-charges and having an R-charge of 1 and b having an R-charge of -1.