Superspace is the coordinate space of a theory exhibiting
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in
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 ...
s rather than real numbers. The ordinary space dimensions correspond to
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
ic degrees of freedom, the anticommuting dimensions to
fermionic
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
degrees of freedom.
The word "superspace" was first used by
John Wheeler in an unrelated sense to describe the
configuration space of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
; for example, this usage may be seen in his 1973 textbook ''
Gravitation
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
''.
Informal discussion
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 mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super Poincaré alg ...
. In this case, one takes ordinary
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, and extends it with anti-commuting fermionic degrees of freedom, taken to be anti-commuting
Weyl spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
s from the
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
associated to the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
. Equivalently, the super Minkowski space can be understood as the quotient of the
super Poincaré algebra
Super may refer to:
Computing
* SUPER (computer program), or Simplified Universal Player Encoder & Renderer, a video converter / player
* Super (computer science), a keyword in object-oriented programming languages
* Super key (keyboard butto ...
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
In mathematics, a super vector space is a \mathbb Z_2- graded vector space, that is, a vector space over a field \mathbb K with a given decomposition of subspaces of grade 0 and grade 1. The study of super vector spaces and their generalization ...
. This is taken to be an ordinary
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, together with additional coordinates taken from the
Grassmann algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
, i.e. coordinate directions that are
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 ...
s. There are several conventions for constructing a super vector space in use; two of these are described by Rogers
Alice Rogers
Frances Alice Rogers is a British mathematician and mathematical physicist. She is an emeritus professor of mathematics at King's College London.
Research
Rogers' research concerns mathematical physics and more particularly supermanifolds, gene ...
, ''Supermanifolds: Theory and Applications'', World Scientific (2007) . and DeWitt.
Bryce DeWitt
Bryce Seligman DeWitt (January 8, 1923 – September 23, 2004), was an American theoretical physicist noted for his work in gravitation and quantum field theory.
Life
He was born Carl Bryce Seligman, but he and his three brothers, including ...
, ''Supermanifolds'', Cambridge University Press (1984) .
A third usage of the term "superspace" is as a synonym for a
supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
Informal definition
An informal definition is com ...
: a supersymmetric generalization of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. 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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
; this is discussed in greater detail at the bottom.
Examples
Several examples are given below. The first few assume a definition of superspace as a
super vector space
In mathematics, a super vector space is a \mathbb Z_2- graded vector space, that is, a vector space over a field \mathbb K with a given decomposition of subspaces of grade 0 and grade 1. The study of super vector spaces and their generalization ...
. This is denoted as R
''m'', ''n'', the Z
2-
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
with R
''m'' as the even subspace and R
''n'' as the odd subspace. The same definition applies to C
m, n.
The four-dimensional examples take superspace to be
super Minkowski space
In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super Poincaré alg ...
. Although similar to a vector space, this has many important differences: First of all, it is an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, having no special point denoting the origin. Next, the fermionic coordinates are taken to be anti-commuting
Weyl spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
s from the
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
, rather than being
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 ...
s. 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
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
, and the Clifford algebra has an isomorphism to the exterior algebra, but its relation to the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
and the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
, used to construct the
spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
s, give it a deep geometric significance. (For example, the spin groups form a normal part of the study of
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
, quite outside the ordinary bounds and concerns of physics.)
Trivial examples
The smallest superspace is a point which contains neither bosonic nor fermionic directions. Other trivial examples include the ''n''-dimensional real plane R
n, which is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
extending in ''n'' real, bosonic directions and no fermionic directions. The vector space R
0, n, which is the ''n''-dimensional real
Grassmann algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
. The space R
1, 1 of one even and one odd direction is known as the space of
dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
s, introduced by
William Clifford in 1873.
The superspace of supersymmetric quantum mechanics
Supersymmetric quantum mechanics
In theoretical physics, supersymmetric quantum mechanics is an area of research where supersymmetry are applied to the simpler setting of plain quantum mechanics, rather than quantum field theory. Supersymmetric quantum mechanics has found ap ...
with ''N''
supercharge
In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge in physics.
Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and v ...
s is often formulated in the superspace R
1, 2''N'', which contains one real direction ''t'' identified with
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
and ''N'' complex
Grassmann directions which are spanned by Θ
''i'' and Θ
*''i'', where ''i'' runs from 1 to ''N''.
Consider the special case ''N'' = 1. The superspace R
1, 2 is a 3-dimensional vector space. A given coordinate therefore may be written as a triple (''t'', Θ, Θ
*). The coordinates form a
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...
, in which the gradation degree of ''t'' is even and that of Θ and Θ
* is odd. This means that a bracket may be defined between any two elements of this vector space, and that this bracket reduces to the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
on two even coordinates and on one even and one odd coordinate while it is an
anticommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
on two odd coordinates. This superspace is an abelian Lie superalgebra, which means that all of the aforementioned brackets vanish
:::
where