Comment 4
: The formulations with
λ
and
ˆ
λ
are thus equivalent up to field redefinitions (rescalings).
Also the other ambiguities in the Noether method we encountered before can be shown to lead to equivalent
formulations up to field redefinitions. We leave this as an exercise.
Comment 5
: The local superspace approach will produce a polynomial supergravity action which begins
with
1
2
(1

2
h
)[
.
ϕ
2
+
iλ
.
λ
]. We can obtain the complete action and transformation rules by rescaling the
results in (
13.5
) as follows
λ
=
√
1

2
h
ˆ
λ
;
ψ
=
ˆ
ψ/
√
1

2
h
;
= ˆ
/
√
1

2
h
(14.16)
15
Of course any rescaling will produce an action which is still susy invariant and Einstein invariant, but
most rescalings will lead to nonpolynomial actions and/or nonpolynomial transformation laws. However,
the rescaling in (
14.16
) leads to both a polynomial action and polynomial transformation laws. There is
nothing wrong with nonpolynomial actions and/or nonpolynomial transformation rules, but polynomial
expresssions are easier to deal with.
The result reads
L
=
1
2
(1

2
h
)[
.
ϕ
2
+
i
ˆ
λ
ˆ
.
λ
]

i
ˆ
ψ
.
ϕ
ˆ
λ
δϕ
=
i
ˆ
ˆ
λ
+
ξ
.
ϕ
;
δ
ˆ
λ
=

.
ϕ
ˆ+
ξ
ˆ
.
λ
+
1
2
.
ξ
ˆ
λ
δ
ˆ
ψ
= (1

2
h
)
ˆ
.
+
.
h
ˆ+
ξ
ˆ
.
ψ

1
2
.
ξ
ˆ
ψ
;
δh
=

i
ˆ
ˆ
ψ
+
ξ
.
h
+
1
2
(1

2
h
)
.
ξ
(14.17)
15
Rigid Superspace
Superspace was introduced by Salam and Strathdee who constructed it as an application of a general
grouptheoretical approach called the group manifold approach. We shall discuss this approach, but first
we want to give a more direct way why we shall use anticommuting as well as commuting coordinates. The
superalgebra which followed from our QM model contained the crucial anticommutators
{
Q, Q
}
= 2
H
.
In higher dimensions this generalizes to
{
Q
α
, Q
β
} ∼
γ
μ
αβ
P
μ
. It seems interesting to treat
Q
α
and
P
μ
(in
our case
H
and
Q
) on equal footing. Then, since
P
μ
is a translation generator, also
Q
should correspond
to a translation generator. The coordinates which
P
μ
translates are spacetime coordinates
x
μ
. Then the
coordinates which
Q
α
translates must be a new kind of anticommuting coordinates
θ
α
.
(
θ
α
should be
anticommuting because
Q
α
are anticommuting.) The natural choice (not the only choice, there are other
choices) for anticommuting
θ
α
is as Grassmann variables; if one has two of them, say
θ
1
and
θ
2
, they satisfy
θ
1
θ
2
=

θ
2
θ
1
. In particular
θθ
= 0. So for our QM model, we are led to consider a twodimensional space
with coordinates
t
and
θ
, and fields (called superfields) depending on
φ
(
t, θ
). An expression in terms of
θ
has then only two terms
φ
(
t, θ
) =
ϕ
1
(
t
) +
θϕ
2
(
t
)
(15.1)
In the group manifold approach (and, more generally, in the coset manifold approach) one associates
with each generator of a (super)algebra a coordinate. The superalgebra of our QM model is
{
Q, Q
}
= 2
H
,
[
Q, H
] = 0, [
H, H
] = 0 (of course). It f
t
is associated with H, there has to be an anticommuting coordinate
θ
which is associated with Q. It satisfies
θθ
= 0. The we can construct group elements as follows
g
=
e
itH
+
θQ
;
g
†
=
g

1
(15.2)
16
The coordinates
t
and
θ
parametrize points in superspace. Since H, Q and
t