Quantum Mechanics for Mathematicians

for the same reason as in the oscillator case: repeated factors ofaFora†Fvanish.
Taking as the Hamiltonian the same square as before, we find

H=(Q++Q−)^2

=

1

2

(W′(Q) +iP)(W′(Q)−iP)a†FaF+

1

2

(W′(Q)−iP)(W′(Q) +iP)aFa†F

=

1

2

(W′(Q)^2 +P^2 )(a†FaF+aFa†F) +

1

2

(i[P,W′(Q)])(a†FaF−aFa†F)

=

1

2

(W′(Q)^2 +P^2 ) +

1

2

(i[P,W′(Q)])σ 3

ButiPis the operator corresponding to infinitesimal translations inQ, so we
have
i[P,W′(Q)] =W′′(Q)

and

H=

1

2

(W′(Q)^2 +P^2 ) +

1

2

W′′(Q)σ 3

For different choices ofWthis gives a large class of quantum systems that can
be used as toy models to investigate properties of ground states. All have the
same state space
H=HB⊗F 1 +=L^2 (R)⊗C^2

(using the Schr ̈odinger representation for the bosonic factor). The energy eigen-
values will be non-negative, and energy eigenvectors with positive energy will
occur in pairs
|ψ〉, (Q++Q−)|ψ〉
For any quantum system, an important question is that of whether it has
a unique lowest energy state. If the lowest energy state is not unique, and a
symmetry group acts non-trivially on the space of lowest energy states, the sym-
metry is said to be “spontaneously broken”, a situation that will be discussed
in section 39.4. In supersymmetric quantum mechanics systems, thinking in
terms of Lie superalgebras, one callsQ 1 the generator of the action of a su-
persymmetry, withHinvariant under the supersymmetry in the sense that the
commutator ofQ 1 andHis zero. The question of how the supersymmetry acts
on the lowest energy state depends on whether or not solutions can be found to
the equation
(Q++Q−)| 0 〉=Q 1 | 0 〉= 0

which will be a lowest energy state with zero energy. If such a solution does exist,
one describes the ground state| 0 〉as “invariant under the supersymmetry”. If
no such solution exists,Q 1 will take a lowest energy state to another, different,
lowest energy state, in which case one says that one has “spontaneously broken
supersymmetry”. The question of whether a given supersymmetric theory has
its supersymmetry spontaneously broken or not is one that has become of great
interest in the case of much more sophisticated supersymmetric quantum field
theories. There, hopes (so far unrealized) of making contact with the real world
rely on finding theories where the supersymmetry is spontaneously broken.