Superspace is the coordinate space of a theory exhibiting supersymmetry.
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.
[1] 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.
Superspace is also commonly used as a synonym for the super vector space.
There are several conventions for constructing a super vector space in use; two of these are described by Rogers[2] and DeWitt.
[3] A third usage of the term "superspace" is as a synonym for a supermanifold: a supersymmetric generalization of a manifold.
A fourth, and completely unrelated meaning saw a brief usage in general relativity; this is discussed in greater detail at the bottom.
The first few assume a definition of superspace as a super vector space.
The four-dimensional examples take superspace to be super Minkowski space.
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.
(For example, the spin groups form a normal part of the study of Riemannian geometry,[4] 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.
Supersymmetric quantum mechanics with N supercharges is often formulated in the superspace R1|2N, which contains one real direction t identified with time and N complex Grassmann directions which are spanned by Θi and Θ*i, where i runs from 1 to N. Consider the special case N = 1.
This superspace is an abelian Lie superalgebra, which means that all of the aforementioned brackets vanish where
One may define functions from this vector space to itself, which are called superfields.
Therefore, superfields may be written as arbitrary functions of t multiplied by the zeroeth and first order terms in the two Grassmann coordinates Superfields, which are representations of the supersymmetry of superspace, generalize the notion of tensors, which are representations of the rotation group of a bosonic space.
One can choose sign conventions such that the derivatives satisfy the anticommutation relations These derivatives may be assembled into supercharges whose anticommutators identify them as the fermionic generators of a supersymmetry algebra where i times the time derivative is the Hamiltonian operator in quantum mechanics.
The supersymmetry variation with supersymmetry parameter ε of a superfield Φ is defined to be We can evaluate this variation using the action of Q on the superfields Similarly one may define covariant derivatives on superspace which anticommute with the supercharges and satisfy a wrong sign supersymmetry algebra The fact that the covariant derivatives anticommute with the supercharges means the supersymmetry transformation of a covariant derivative of a superfield is equal to the covariant derivative of the same supersymmetry transformation of the same superfield.
Thus, generalizing the covariant derivative in bosonic geometry which constructs tensors from tensors, the superspace covariant derivative constructs superfields from superfields.
[5] In supersymmetric quantum field theories one is interested in superspaces which furnish representations of a Lie superalgebra called a supersymmetry algebra.
For this reason, in physical applications one considers an action of the supersymmetry algebra on the four fermionic directions of
The CPT theorem implies that in a unitary, Poincaré invariant theory, which is a theory in which the S-matrix is a unitary matrix and the same Poincaré generators act on the asymptotic in-states as on the asymptotic out-states, the supersymmetry algebra must contain an equal number of left-handed and right-handed Weyl spinors.
Such a theory is said to have extended supersymmetry, and such models have received a lot of attention.
However, in this subsection we are considering the superspace with four fermionic components and so no Weyl spinors are consistent with the CPT theorem.
The word "superspace" is also used in a completely different and unrelated sense, in the book Gravitation by Misner, Thorne and Wheeler.
In modern terms, this particular idea of "superspace" is captured in one of several different formalisms used in solving the Einstein equations in a variety of settings, both theoretical and practical, such as in numerical simulations.