Lectures on quantum field theory

Lecture 30 on 30 April 2015: An introduction to supersymmetry: Supersymmetric quantum mechanics. In the simplest model, there are two supersymmetric charges, and the hamiltonian is twice the square of each. The supersymmetry is exact if these charges annihilate the vacuum, in which case the ground-state energy is zero, and one can solve for the wave-function of the ground state as long as the superpotential W(x) --> ± ∞ as x --> ± ∞ or W(x) --> ∓ ∞ as x --> ± ∞. In this case supersymmetry is exact, and the excited states occur in pairs of equal energy. But when the superpotential W(x) --> ∞ as x --> ± ∞ or when W(x) --> − ∞ as x --> ± ∞, then the zero-energy wave function is not normalizable and no state of zero energy exists. In this case supersymmetry is dynamically broken, and the excited states do not occur in pairs of equal energy. Supersymmetric quantum-mechanical systems are easier to analyze than similar systems without supersymmetry. All these features have analogs in quantum field theory. A quantum field theory with exact supersymmetry has a ground state of zero energy. The density of dark energy is not zero, but it is small compared to the energy density ( ∞ )^4 of a theory of free bosons and to that − ( ∞ )^4 of a theory of free fermions. And the energy densities of theories with broken supersymmetry diverge no worse than ± ( ∞ )^2. My notes on supersymmetry (theory.phys.unm.edu/cahill/523-14/ss.pdf) use dotted and undotted indicies, which may be more trouble than they're worth. The change in the action density of a susy theory is a total divergence of a current K. The equations of motion imply that the change in the action density also is the divergence of the susy Noether current J. So the difference of these two total divergences vanishes, and the difference of these two currents is the conserved susy current S = J - K. The space integral of the time component of S gives us the susy charges. Their anticommutators are linear combinations of the Pauli matrices and the 4-momentum operators of the quantum field theory. A superfield is a function both of a point in spacetime and of some anticommuting variables. Equivalently, a superfield is a function of a point in superspace. Superfields in superspace provide a simple way of making actions that are supersymmetric. The word super is used too much in physics.