Superspace

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. 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 ( x , θ , θ ¯ ) {displaystyle (x, heta ,{ar { heta }})} 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. Several examples are given below. The first few assume a definition of superspace as a super vector space. This is denoted as Rm|n, the Z2-graded vector space with Rm as the even subspace and Rn as the odd subspace. The same definition applies to Cm|n. The four-dimensional examples take superspace to be super Minkowski space. Although similar to a vector space, this has many important differences: First of all, it is an affine space, having no special point denoting the origin. Next, the fermionic coordinates are taken to be anti-commuting Weyl spinors from the Clifford algebra, rather than being Grassmann numbers. The difference here is that the Clifford algebra has a considerably richer and more subtle structure than the Grassmann numbers. So, the Grassmann numbers are elements of the exterior algebra, and the Clifford algebra has an isomorphism to the exterior algebra, but its relation to the orthogonal group and the spin group, used to construct the spin representations, give it a deep geometric significance. (For example, the spin groups form a normal part of the study of Riemannian geometry, quite outside the ordinary bounds and concerns of physics.) The smallest superspace is a point which contains neither bosonic nor fermionic directions. Other trivial examples include the n-dimensional real plane Rn, which is a vector space extending in n real, bosonic directions and no fermionic directions. The vector space R0|n, which is the n-dimensional real Grassmann algebra. The space R1|1 of one even and one odd direction is known as the space of dual numbers, introduced by William Clifford in 1873.