Chapter 4
Four-Dimensional Superfield
Supersymmetry
Basing on our experience with studies of three-dimensional supersymmetric field
theory models, now we turn to our main aim—study of a superfield description for
four-dimensional supersymmetric field theories. We discuss their superfield formulations and manners to calculate quantum corrections for these theories.
4.1 General Properties of the Four-Dimensional
Superspace
Now, after we have described in details the superfield approach for the supersymmetric theories formulated in the three-dimensional space-time, let us go to the usual,
four-dimensional case.
The essential difference of the four-dimensional space-time from the threedimensional one consists in the fact that the Lorentz group SO(1, 3) characterizing the rotational symmetry of the space-time, possesses two mutually conjugated
and linearly independent spinor representations, denoted by undotted and dotted
indices respectively. Each of these representations is realized by a group of unimodular complex 2 × 2 matrices, that is, SL(2, C), and two linear spaces where
these representations are acting are the spaces of undotted and dotted spinors ψ
α , ¯
ψ
˙
α
respectively. It is easy to see that the invariant tensors of these two representations are
the Levi-Civita symbols
αβ ,
˙
α ˙
β . These tensors play the role of metrics for spinors
and will be used to form the invariant scalar products of any spinors:
ψ
α
=
αβ
ψ β ; ψ α = ψ
β
βα ;
¯
ψ ˙
α = ˙
α ˙
β
¯
ψ
˙
β
; ¯
ψ
˙
α
= ¯
ψ ˙
β
˙
β ˙
α
.
(4.1)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Petrov, Quantum Superfield Supersymmetry, Fundamental Theories of Physics 202,
https://doi.org/10.1007/978-3-030-68136-4_4
49
Four-Dimensional Superfield
Supersymmetry
Basing on our experience with studies of three-dimensional supersymmetric field
theory models, now we turn to our main aim—study of a superfield description for
four-dimensional supersymmetric field theories. We discuss their superfield formulations and manners to calculate quantum corrections for these theories.
4.1 General Properties of the Four-Dimensional
Superspace
Now, after we have described in details the superfield approach for the supersymmetric theories formulated in the three-dimensional space-time, let us go to the usual,
four-dimensional case.
The essential difference of the four-dimensional space-time from the threedimensional one consists in the fact that the Lorentz group SO(1, 3) characterizing the rotational symmetry of the space-time, possesses two mutually conjugated
and linearly independent spinor representations, denoted by undotted and dotted
indices respectively. Each of these representations is realized by a group of unimodular complex 2 × 2 matrices, that is, SL(2, C), and two linear spaces where
these representations are acting are the spaces of undotted and dotted spinors ψ
α , ¯
ψ
˙
α
respectively. It is easy to see that the invariant tensors of these two representations are
the Levi-Civita symbols
αβ ,
˙
α ˙
β . These tensors play the role of metrics for spinors
and will be used to form the invariant scalar products of any spinors:
ψ
α
=
αβ
ψ β ; ψ α = ψ
β
βα ;
¯
ψ ˙
α = ˙
α ˙
β
¯
ψ
˙
β
; ¯
ψ
˙
α
= ¯
ψ ˙
β
˙
β ˙
α
.
(4.1)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Petrov, Quantum Superfield Supersymmetry, Fundamental Theories of Physics 202,
https://doi.org/10.1007/978-3-030-68136-4_4
49