3.1 Definitions and Conventions
13
where θ
2
=
1
2
θ
α
θ α . It is clear that if f 0 (x), f 1 (x), f 2 (x) are physical fields, the spin
of f
α
1 (x) differs by 1/2 from the spin of f 0 (x) and of f 2 (x), i.e. f
α
1 (x) is a spinor
field while f 0 (x), f 2 (x) are the scalar ones. In a general case, the f 0 (x), f 1 (x), f 2 (x)
can carry extra indices being tensors of different ranks, the simplest example is the
spinor superfield we discuss further.
The integral over the Grassmannian coordinates is defined as
dθ α θ
β
= δ
β
α ,
(3.4)
which is frequently referred as the conclusion that for Grassmann variables, the
integration is equivalent to the differentiation. We note that actually it implies that
the mass dimensions of θ α and dθ α are not equal but opposite. Actually, for θ α the
dimension is −1/2, and for dθ β or ∂ β , is 1/2, while for ∂ αβ it is 1 as it must be for
the space-time derivative. With using of the definition d
2
θ =
1
2
dθ
α dθ α , we conclude
that the integral (3.4) implies
d
2
θθ
2
= −1.
(3.5)
This result allows to define the Grassmannian delta function
δ
2
(θ ) = −θ
2
,
(3.6)
which satisfies the usual property of the delta function
d
2
θ 1 f (θ 1 )δ
2
(θ 1 − θ 2 ) = f (θ 2 ).
(3.7)
Further, in this chapter we will denote δ
2
(θ 1 − θ 2 ) as δ 12 . We also will employ the
notation δ
5
(z 1 − z 2 ) ≡ δ(z 1 − z 2 ) = δ
3
(x 1 − x 2 )δ 12 . The generators of the supersymmetry transformations, playing the role of translations in the superspace and
also called the supercharges, are defined as
Q α = i∂ α + θ
β
∂ βα .
(3.8)
They evidently commute with the generators of “common” bosonic translations
P αβ = i∂ αβ , while the anticommutator of two supercharges is nontrivial:
{Q α , Q β } = 2P αβ .
(3.9)
We note that the mass dimension of Q α is equal to 1/2. The supersymmetry transformation for the given superfield (x, θ) with the infinitesimal parameter
α is
defined as
δδ(x, θ) = −i
α Q α (x, θ).
(3.10)
13
where θ
2
=
1
2
θ
α
θ α . It is clear that if f 0 (x), f 1 (x), f 2 (x) are physical fields, the spin
of f
α
1 (x) differs by 1/2 from the spin of f 0 (x) and of f 2 (x), i.e. f
α
1 (x) is a spinor
field while f 0 (x), f 2 (x) are the scalar ones. In a general case, the f 0 (x), f 1 (x), f 2 (x)
can carry extra indices being tensors of different ranks, the simplest example is the
spinor superfield we discuss further.
The integral over the Grassmannian coordinates is defined as
dθ α θ
β
= δ
β
α ,
(3.4)
which is frequently referred as the conclusion that for Grassmann variables, the
integration is equivalent to the differentiation. We note that actually it implies that
the mass dimensions of θ α and dθ α are not equal but opposite. Actually, for θ α the
dimension is −1/2, and for dθ β or ∂ β , is 1/2, while for ∂ αβ it is 1 as it must be for
the space-time derivative. With using of the definition d
2
θ =
1
2
dθ
α dθ α , we conclude
that the integral (3.4) implies
d
2
θθ
2
= −1.
(3.5)
This result allows to define the Grassmannian delta function
δ
2
(θ ) = −θ
2
,
(3.6)
which satisfies the usual property of the delta function
d
2
θ 1 f (θ 1 )δ
2
(θ 1 − θ 2 ) = f (θ 2 ).
(3.7)
Further, in this chapter we will denote δ
2
(θ 1 − θ 2 ) as δ 12 . We also will employ the
notation δ
5
(z 1 − z 2 ) ≡ δ(z 1 − z 2 ) = δ
3
(x 1 − x 2 )δ 12 . The generators of the supersymmetry transformations, playing the role of translations in the superspace and
also called the supercharges, are defined as
Q α = i∂ α + θ
β
∂ βα .
(3.8)
They evidently commute with the generators of “common” bosonic translations
P αβ = i∂ αβ , while the anticommutator of two supercharges is nontrivial:
{Q α , Q β } = 2P αβ .
(3.9)
We note that the mass dimension of Q α is equal to 1/2. The supersymmetry transformation for the given superfield (x, θ) with the infinitesimal parameter
α is
defined as
δδ(x, θ) = −i
α Q α (x, θ).
(3.10)