In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, super Minkowski space or Minkowski
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 numb ...
is a
supersymmetric
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 ...
extension of
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 ...
, sometimes used as the base
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 ...
(or rather,
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 ...
) for
superfield
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra.
Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply ...
s. It is acted on by 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 ...
.
Construction
Abstract construction
Abstractly, super Minkowski space is the space of (right)
cosets
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ha ...
within the
Super Poincaré group of
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 ...
, that is,
:
.
This is analogous to the way ordinary
Minkowski spacetime
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 inert ...
can be identified with the (right) cosets within the
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
of the Lorentz group, that is,
:
.
The coset space is naturally
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
, and the nilpotent, anti-commuting behavior of the fermionic directions arises naturally 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 with the Lorentz group.
Direct sum construction
For this section, the dimension of the Minkowski space under consideration is
.
Super Minkowski space can be concretely realized as the direct sum of Minkowski space, which has coordinates
, with 'spin space'. The dimension of 'spin space' depends on the number
of supercharges in the associated
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 ...
to the super Minkowski space under consideration. In the simplest case,
, the 'spin space' has 'spin coordinates'
with
, where each component is a
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 ...
. In total this forms 4 spin coordinates.
The notation for
super Minkowski space is then
.
There are theories which admit
supercharges. Such cases have
extended supersymmetry
In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4).
Extende ...
. For such theories, super Minkowski space is labelled
, with coordinates
with
.
Definition
The underlying
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 ...
of super Minkowski space is isomorphic to 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 ...
given by the direct sum of ordinary Minkowski spacetime in ''d'' dimensions (often taken to be 4) and a number
of real spinor representations of the Lorentz algebra. (When
this is slightly ambiguous because there are 2 different real spin representations, so one needs to replace
by a pair of integers
, though some authors use a different convention and take
copies of both spin representations.)
However this construction is misleading for two reasons: first, super Minkowski space is really 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 ...
over a group rather than a group, or in other words it has no distinguished "origin", and second, the underlying
supergroup Supergroup or super group may refer to:
* Supergroup (music), a music group formed by artists who are already notable or respected in their fields
* Supergroup (physics), a generalization of groups, used in the study of supersymmetry
* Supergroup ...
of translations is not a super vector space but a nilpotent supergroup of nilpotent length 2.
This supergroup has the following
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 ...
. Suppose that
is Minkowski space (of dimension
), and
is a finite sum of irreducible real
spinor representation
In mathematics, the spin representations are particular projective representations of the orthogonal group, orthogonal or special orthogonal groups in arbitrary dimension and metric signature, signature (i.e., including indefinite orthogonal groups ...
s for
-dimensional Minkowski space.
Then there is an invariant, symmetric bilinear map
. It is positive definite in the sense that, for any
, the element