Chapter 1
Introduction
We prove, once and for all, that people who don’t use
superspace are really out of it.
“Stuperspace”
The idea of supersymmetry is now considered as one of the basic concepts of theoretical high energy physics (see e.g. [1]). The supersymmetry, being a fundamental symmetry allowing to relate bosons and fermions, provides possibilities to construct theories with much better renormalization properties since some bosonic and fermionic
divergent contributions in supersymmetric theories cancel each other. Moreover,
there are essentially finite four-dimensional supersymmetry theories without higher
derivatives, e.g. N = 4 super-Yang-Mills (SYM) theory (the detailed discussion of
the finiteness of this theory is presented in [2]), and, probably, N = 8 supergravity
[3]. In the three-dimensional case, N = 6 and N = 8 supersymmetric Chern-Simons
theories are known to be finite [4]. Due to this essential improvement of renormalization properties, there is a common expectation that the expected unified theory
of all fundamental interactions must be supersymmetric (see e.g. [5] and references
therein).
The concept of supersymmetry was introduced in known papers by Volkov and
Akulov [6] and Golfand and Lichtman [7] in early 70s. It got a further advance in
[8, 9] (the history of arising and development of the concept of the supersymmetry,
including biographical aspects, can be found in the book [10]). The essential breakthrough in supersymmetric field theory was achieved with introducing the idea of
a superfield [11] (see also [12, 13]). The reason for it consists in the fact that the
superfield formulation, first, allows to maintain manifest supersymmetric covariance
at all steps of calculations, with all calculations are performed in a very compact
manner (indeed, any supergraph corresponds to a set of the Feynman diagrams for
component fields), second, automatically takes into account the famous “miraculous
cancellations” of ultraviolet divergences responsible for an essential improvement
of the renormalization behavior of supersymmetric field theories [14]. Moreover, in
the context of the noncommutative field theory it turns out to be that the supersym© 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_1
1
Introduction
We prove, once and for all, that people who don’t use
superspace are really out of it.
“Stuperspace”
The idea of supersymmetry is now considered as one of the basic concepts of theoretical high energy physics (see e.g. [1]). The supersymmetry, being a fundamental symmetry allowing to relate bosons and fermions, provides possibilities to construct theories with much better renormalization properties since some bosonic and fermionic
divergent contributions in supersymmetric theories cancel each other. Moreover,
there are essentially finite four-dimensional supersymmetry theories without higher
derivatives, e.g. N = 4 super-Yang-Mills (SYM) theory (the detailed discussion of
the finiteness of this theory is presented in [2]), and, probably, N = 8 supergravity
[3]. In the three-dimensional case, N = 6 and N = 8 supersymmetric Chern-Simons
theories are known to be finite [4]. Due to this essential improvement of renormalization properties, there is a common expectation that the expected unified theory
of all fundamental interactions must be supersymmetric (see e.g. [5] and references
therein).
The concept of supersymmetry was introduced in known papers by Volkov and
Akulov [6] and Golfand and Lichtman [7] in early 70s. It got a further advance in
[8, 9] (the history of arising and development of the concept of the supersymmetry,
including biographical aspects, can be found in the book [10]). The essential breakthrough in supersymmetric field theory was achieved with introducing the idea of
a superfield [11] (see also [12, 13]). The reason for it consists in the fact that the
superfield formulation, first, allows to maintain manifest supersymmetric covariance
at all steps of calculations, with all calculations are performed in a very compact
manner (indeed, any supergraph corresponds to a set of the Feynman diagrams for
component fields), second, automatically takes into account the famous “miraculous
cancellations” of ultraviolet divergences responsible for an essential improvement
of the renormalization behavior of supersymmetric field theories [14]. Moreover, in
the context of the noncommutative field theory it turns out to be that the supersym© 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_1
1