Chapter 33

Supersymmetry, Some

Simple Examples

If one considers fermionic and bosonic quantum systems that each separately
have operators coming from Lie algebra or superalgebra representations on their
state spaces, when one combines the systems by taking the tensor product, these
operators will continue to act on the combined system. In certain special cases
new operators with remarkable properties will appear that mix the fermionic
and bosonic systems and commute with the Hamiltonian (these operators are
often given by some sort of “square root” of the Hamiltonian). These are gener-
ically known as “supersymmetries” and provide new information about energy
eigenspaces. In this chapter we’ll examine in detail some of the simplest such
quantum systems, examples of “supersymmetric quantum mechanics”.

33.1 The supersymmetric oscillator

In the previous chapters we discussed in detail

• The bosonic harmonic oscillator inddegrees of freedom, with state space
  Fdgenerated by applyingdcreation operatorsaB†jan arbitrary number
  of times to a lowest energy state| 0 〉B. The Hamiltonian is

H=

1

2

~ω

∑d

j=1

(aB†jaBj+aBjaB†j) =

∑d

j=1

(

NBj+

1

2

)

~ω

whereNBjis the number operator for thej’th degree of freedom, with

eigenvaluesnBj= 0, 1 , 2 ,···.

• The fermionic oscillator inddegrees of freedom, with state spaceFd+
  generated by applyingdcreation operatorsaFjto a lowest energy state