In this simple quantum mechanical system, one can try and explicitly solve
the equationQ 1 |ψ〉= 0. States can be written as two-component complex
functions
|ψ〉=(
ψ+(q)
ψ−(q))
and the equation to be solved is
(Q++Q−)|ψ〉=1
√
2
((W′(Q) +iP)a†F+ (W′(Q)−iP)aF)(
ψ+(q)
ψ−(q))
=
1
√
2
((
W′(Q) +
d
dq)(
0 1
0 0
)
+
(
W′(Q)−
d
dq)(
0 0
1 0
))(
ψ+(q)
ψ−(q))
=
1
√
2
(
W′(Q)
(
0 1
1 0
)
+
d
dq(
0 1
−1 0
))(
ψ+(q)
ψ−(q))
=
1
√
2
(
0 1
−1 0
)(
d
dq
−W′(Q)σ 3)(
ψ+(q)
ψ−(q))
= 0
which has general solution
(
ψ+(q)
ψ−(q))
=eW(q)σ^3(
c+
c−)
=
(
c+eW(q)
c−e−W(q))
for complex constantsc+,c−. Such solutions can only be normalizable if
c+= 0, lim
q→±∞
W(q) = +∞or
c−= 0, lim
q→±∞
W(q) =−∞
If, for example,W(q) is an odd polynomial, one will not be able to satisfy either
of these conditions, so there will be no solution, and the supersymmetry will be
spontaneously broken.
33.3 Supersymmetric quantum mechanics and differential forms
If one considers supersymmetric quantum mechanics in the case ofddegrees of
freedom and in the Schr ̈odinger representation, one has
H=L^2 (Rd)⊗Λ∗(Cd)the tensor product of complex-valued functions onRd(acted on by the Weyl
algebra Weyl(2d,C)) and anticommuting functions onCd (acted on by the
Clifford algebra Cliff(2d,C)). There are two operatorsQ+andQ−, adjoints of
each other and of square zero. If one has studied differential forms, this should