Superfield
Encyclopedia
In theoretical physics
, one often analyzes theories with supersymmetry
in which superfields play a very important role. In four dimensions, the simplest example - namely the minimal N=1 supersymmetry - may be written using a superspace
with four extra fermionic coordinates
, transforming as a two-component spinor
and its conjugate. More generally there are 4N extra fermionic coordinates .
Superfields were introduced by Abdus Salam
and J. A. Strathdee in their 1974 article Supergauge Transformations.
A more coordinate-free description of the superspace is that it's the quotient space
of the super-Poincaré group divided by the Lorentz group
.
Every superfield, i.e. a field that depends on all coordinates of the superspace (or in other words, an element of a module
of the algebra of functions over superspace), may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called chiral superfields, that, in the chiral representation of supersymmetry, depend only on the variables
but not their conjugates. The last term in the corresponding expansion, namely 
, is called the F-term. Other superfields include vector superfields.
There also exist superfields in theories with larger supersymmetry.
Manifestly supersymmetric Lagrangian
s may also be written as integral
s over the whole superspace. Some special terms, such as the superpotential
, may be written as integrals over
s only. They are also referred to as F-terms, much like the terms in the ordinary potential that arise from these terms of the supersymmetric Lagrangian.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, one often analyzes theories with supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
in which superfields play a very important role. In four dimensions, the simplest example - namely the minimal N=1 supersymmetry - may be written using a superspace
Superspace
"Superspace" has had two meanings in physics. The word was first used by John Wheeler to describe the configuration space of general relativity; for example, this usage may be seen in his famous 1973 textbook Gravitation....
with four extra fermionic coordinates
, transforming as a two-component spinorSpinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
and its conjugate. More generally there are 4N extra fermionic coordinates .
Superfields were introduced by Abdus Salam
Abdus Salam
Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk (Urdu: محمد عبد السلام, pronounced , (January 29, 1926– November 21, 1996) was a Pakistani theoretical physicist and Nobel laureate in Physics for his work on the electroweak unification of the...
and J. A. Strathdee in their 1974 article Supergauge Transformations.
A more coordinate-free description of the superspace is that it's the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the super-Poincaré group divided by the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
.
Every superfield, i.e. a field that depends on all coordinates of the superspace (or in other words, an element of a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
of the algebra of functions over superspace), may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called chiral superfields, that, in the chiral representation of supersymmetry, depend only on the variables
but not their conjugates. The last term in the corresponding expansion, namely , is called the F-term. Other superfields include vector superfields.There also exist superfields in theories with larger supersymmetry.
Manifestly supersymmetric Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
s may also be written as integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s over the whole superspace. Some special terms, such as the superpotential
Superpotential
Superpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
, may be written as integrals over
s only. They are also referred to as F-terms, much like the terms in the ordinary potential that arise from these terms of the supersymmetric Lagrangian.