52
4 Four-Dimensional Superfield Supersymmetry
4 fermionic ones θ
α
, ¯
θ
˙
α so it is 8-dimensional and will be denoted henceforth as
R
4|4 . We will refer to it as to the four-dimensional superspace. It is natural to assume
that the superfields of the forms like (4.3), in the new supersymmetric field theory,
will be dynamical variables playing the role of fields in the superspace. The task
of this chapter consists in development of the quantum theory for superfields in the
four-dimensional superspace basing on principles of the standard methodology of
quantum field theory.
We introduce derivatives on the superspace in a manner similar to the threedimensional case, that is, we use the same definition of the (left) Grassmannian
derivative with respect to θ
α (3.1), and the derivative with respect to ¯
θ ˙
α is defined
analogously.
Then, to introduce the integral we start with the usual Grassmannian definition
dθθ = 1, which we generalize to
dθ α θ
β
= δ
β
α .
It is a convention. Just as in three-dimensional theories, the θ and dθ have different
(and opposite) mass dimensions—again, the dimension of θ is equal to −
1
2
, and of
dθ—to
1
2
, and variation δθ never should be mixed with differential dθ since they
have different dimensions. Then, an integral from a constant is zero,
dθ · 1 = 0,
this identity is caused by suggestion of the translation invariance due to which the
relation
dθ(θ + λ) =
dθθ for constant λ must be satisfied, hence λ
dθ = 0.
We introduce the following scalar measures for Grassmann integration:
d
2
θ = −
1
4
dθ
α dθ α , d
2 ¯
θ = −
1
4
¯
dθ ˙
α
¯
dθ
˙
α , d
4
θ = d
2
θd
2 ¯
θ.
(4.8)
These measures satisfy the relations
d
2
θθ
2
=
d
2 ¯
θ ¯
θ
2
=
d
4
θθ
2 ¯
θ
2
= 1.
(4.9)
Since
∂θ
α
∂θ β ≡ ∂ α θ
β
= δ
α
β as well as
dθ α θ
β
= δ
β
α (similarly, ∂
˙
α ¯
θ ˙
β =
d ¯
θ
˙
α ¯
θ ˙
β =
δ
˙
α
˙
β
) we conclude that integration and differentiation in Grassmannian space described
by (θ
α
, ¯
θ ˙
α ) are equivalent, similarly to the three-dimensional case. In particular, we
see that
4 Four-Dimensional Superfield Supersymmetry
4 fermionic ones θ
α
, ¯
θ
˙
α so it is 8-dimensional and will be denoted henceforth as
R
4|4 . We will refer to it as to the four-dimensional superspace. It is natural to assume
that the superfields of the forms like (4.3), in the new supersymmetric field theory,
will be dynamical variables playing the role of fields in the superspace. The task
of this chapter consists in development of the quantum theory for superfields in the
four-dimensional superspace basing on principles of the standard methodology of
quantum field theory.
We introduce derivatives on the superspace in a manner similar to the threedimensional case, that is, we use the same definition of the (left) Grassmannian
derivative with respect to θ
α (3.1), and the derivative with respect to ¯
θ ˙
α is defined
analogously.
Then, to introduce the integral we start with the usual Grassmannian definition
dθθ = 1, which we generalize to
dθ α θ
β
= δ
β
α .
It is a convention. Just as in three-dimensional theories, the θ and dθ have different
(and opposite) mass dimensions—again, the dimension of θ is equal to −
1
2
, and of
dθ—to
1
2
, and variation δθ never should be mixed with differential dθ since they
have different dimensions. Then, an integral from a constant is zero,
dθ · 1 = 0,
this identity is caused by suggestion of the translation invariance due to which the
relation
dθ(θ + λ) =
dθθ for constant λ must be satisfied, hence λ
dθ = 0.
We introduce the following scalar measures for Grassmann integration:
d
2
θ = −
1
4
dθ
α dθ α , d
2 ¯
θ = −
1
4
¯
dθ ˙
α
¯
dθ
˙
α , d
4
θ = d
2
θd
2 ¯
θ.
(4.8)
These measures satisfy the relations
d
2
θθ
2
=
d
2 ¯
θ ¯
θ
2
=
d
4
θθ
2 ¯
θ
2
= 1.
(4.9)
Since
∂θ
α
∂θ β ≡ ∂ α θ
β
= δ
α
β as well as
dθ α θ
β
= δ
β
α (similarly, ∂
˙
α ¯
θ ˙
β =
d ¯
θ
˙
α ¯
θ ˙
β =
δ
˙
α
˙
β
) we conclude that integration and differentiation in Grassmannian space described
by (θ
α
, ¯
θ ˙
α ) are equivalent, similarly to the three-dimensional case. In particular, we
see that