3.1 Definitions and Conventions
15
Then, in the first term we substitute the identity D α D β = −D β D α + {D α , D β }, and in
the second one we substitute the identity D
β D α = −D α D
β
+ {D
β
, D α }. Afterwards,
the anticommutator terms cancel each other, and we rest with
D
β D α D β = 0.
(3.16)
This is a very important identity which role is similar to the properties of the projecting
operators in four-dimensional superfield supersymmetry which will be discussed in
the next chapter. Applying this identity to (3.15) we find another important identity
{D α , D
2
} = 0.
(3.17)
Using the (3.13) and the (3.17) we can derive one more important relation
(D
2
)
2
= .
(3.18)
These properties of the supercovariant derivatives can be used for constructing of the
superfields.
One more very important property of the supercovariant derivatives and the delta
function which can be checked by direct applying the spinor supercovariant derivatives on the delta function is
δ 12 D
2
δ 12 = δ 12 .
(3.19)
It is straightforward to verify as well that
δ 12 D α δ 12 = 0.
(3.20)
The most important superfields used in the known field theory models in the threedimensional superspace are the scalar and the spinor ones. The scalar superfield is
defined in the form of the following θ expansion:
φ(x, θ) = ϕ(x) + θ
α
ψ α (x) − θ
2 F(x).
(3.21)
Its components can be also defined as the projections:
ϕ(x) = φ(x, θ)|;
ψ α (x) = D α φ(x, θ)|;
F(x) = D
2
φ(x, θ)|.
(3.22)
Here and further the symbol | means that the Grassmannian coordinates θ in the corresponding expression are put to zero after taking derivatives. In the expression above,
the ϕ(x) is the usual scalar field, ψ α (x) is the spinor one, and F(x) is the auxiliary
field, whose equations of motion in the usual theory of the scalar superfield without
15
Then, in the first term we substitute the identity D α D β = −D β D α + {D α , D β }, and in
the second one we substitute the identity D
β D α = −D α D
β
+ {D
β
, D α }. Afterwards,
the anticommutator terms cancel each other, and we rest with
D
β D α D β = 0.
(3.16)
This is a very important identity which role is similar to the properties of the projecting
operators in four-dimensional superfield supersymmetry which will be discussed in
the next chapter. Applying this identity to (3.15) we find another important identity
{D α , D
2
} = 0.
(3.17)
Using the (3.13) and the (3.17) we can derive one more important relation
(D
2
)
2
= .
(3.18)
These properties of the supercovariant derivatives can be used for constructing of the
superfields.
One more very important property of the supercovariant derivatives and the delta
function which can be checked by direct applying the spinor supercovariant derivatives on the delta function is
δ 12 D
2
δ 12 = δ 12 .
(3.19)
It is straightforward to verify as well that
δ 12 D α δ 12 = 0.
(3.20)
The most important superfields used in the known field theory models in the threedimensional superspace are the scalar and the spinor ones. The scalar superfield is
defined in the form of the following θ expansion:
φ(x, θ) = ϕ(x) + θ
α
ψ α (x) − θ
2 F(x).
(3.21)
Its components can be also defined as the projections:
ϕ(x) = φ(x, θ)|;
ψ α (x) = D α φ(x, θ)|;
F(x) = D
2
φ(x, θ)|.
(3.22)
Here and further the symbol | means that the Grassmannian coordinates θ in the corresponding expression are put to zero after taking derivatives. In the expression above,
the ϕ(x) is the usual scalar field, ψ α (x) is the spinor one, and F(x) is the auxiliary
field, whose equations of motion in the usual theory of the scalar superfield without