4.2 Field Theory Models in the Four-Dimensional Superspace
61
1/2 + 0+ + 0− = 1;
0+ 0− = 0± 1/2 = 0− 0+ = 1/2 0± = 0,
(4.48)
i.e. they form the complete and orthogonal set of the projecting operators. It implies
that any superfield can be represented as a linear combination of chiral, antichiral
and linear ones.
Now let us obtain the component structure of the action (4.46). In principle, it can
be done via a straightforward use of the expression (4.46) and the projections (4.28).
However, such a way is very cumbersome. Therefore we use another manner: since
the chiral and complete measures are related by the rule
d
8 z =
d
6 z(−
¯
D
2
4
), we
can rewrite the action (4.46) in an equivalent form (cf. [66]):
S =
1
64
d
6 zW
α W α ,
(4.49)
where
W α = − ¯
D
2 D α V
(4.50)
is an Abelian superfield strength. First, it is clear that W α is chiral. Second, one can
obtain its component expansion (in the chiral representation):
W α = 4[λ α + θ
β f βα + θ α D −
i
2
θ
2
∂ α ˙
β
¯
λ
˙
β
].
(4.51)
or, in the form of projections,
W α | = 4λ α ;
D (β W α) | = 8 f αβ ;
D
2 W α | = 8i∂ α ˙
β
¯
λ
˙
β
;
−
1
2
D
α W α | = 4D.
(4.52)
The f αβ = ∂ β ˙
β A
˙
β
α + ∂ α ˙
β A
˙
β
β here is the (symmetric) bispinor form of the usual
stress tensor F ab , i.e. f αβ =
1
2
(σ
ab
) αβ F ab . In this case, the action (4.49) is reduced
to
S =
1
4
d
4 x(−
1
2
f
αβ f αβ − i ¯
λ
˙
α
∂ ˙
αβ λ
β
+
1
2
D
2
),
(4.53)
that is, the action of the free supersymmetric electrodynamics.
The key property of the action (4.53) is that it involves only higher components
of the superfield V (4.24). From the formal viewpoint, it means that the lower components of this superfield, that is, C(x), χ α (x), ¯
χ
˙
α
(x), M(x), ¯
M(x) can be removed