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Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions x, y, z, ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers. The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom. The word "superspace" was first used by John Wheeler in an unrelated sense to describe the configuration space of general relativity; for example, this usage may be seen in his 1973 textbook Gravitation. There are several similar, but not equivalent, definitions of superspace that have been used, and continue to be used in the mathematical and physics literature. One such usage is as a synonym for super Minkowski space. In this case, one takes ordinary Minkowski space, and extends it with anti-commuting fermionic degrees of freedom, taken to be anti-commuting Weyl spinors from the Clifford algebra associated to the Lorentz group. Equivalently, the super Minkowski space can be understood as the quotient of the super Poincaré algebra modulo the algebra of the Lorentz group. A typical notation for the coordinates on such a space is with the overline being the give-away that super Minkowski space is the intended space. Superspace is also commonly used as a synonym for the super vector space. This is taken to be an ordinary vector space, together with additional coordinates taken from the Grassmann algebra, i.e. coordinate directions that are Grassmann numbers. There are several conventions for constructing a super vector space in use; two of these are described by Rogers and DeWitt. A third usage of the term "superspace" is as a synonym for a supermanifold: a supersymmetric generalization of a manifold. Note that both super Minkowski spaces and super vector spaces can be taken as special cases of supermanifolds. A fourth, and completely unrelated meaning saw a brief usage in general relativity; this is discussed in greater detail at the bottom.

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Superspace

Grassmann number

In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed.

Supergroup (physics)

The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them. However the functions may have even and odd parts. Moreover, a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.