are known as supercharges. In the above expression are the generators of translation and are the Pauli matrices. The index runs over the values A dot is used over the index to remind that this index transforms according to the inequivalent conjugate spinor representation; one must never accidentally contract these two types of indexes. The Pauli matrices can be considered to be a direct manifestation of the Littlewood–Richardson rule mentioned before: they indicate how the tensor product of the two spinors can be re-expressed as a vector. The index of course ranges over the space-time dimensions It is convenient to work with Dirac spinors instead of Weyl spinors; a Dirac spinor can be thought of as an element of ; it has four components. The Dirac matrices are thus also four-dimensional, and can be expressed as direct sums of the Pauli matrices. The tensor product then gives an algebraic relation to the Minkowski metric which is expressed as: and This then gives the full algebra which are to be combined with the normal Poincaré algebra. It is a closed algebra, since all Jacobi identities are satisfied and can have since explicit matrix representations. Following this line of reasoning will lead to supergravity. Extended supersymmetry It is possible to add more supercharges. That is, we fix a number which by convention is labelled , and define supercharges with These can be thought of as many copies of the original supercharges, and hence satisfy and but can also satisfy and where is the central charge. Super-Poincaré group and superspace Just as the Poincaré algebra generates the Poincaré group of isometries of Minkowski space, the super-Poincaré algebra, an example of a Lie super-algebra, generates what is known as a supergroup. This can be used to define superspace with supercharges: these are the right cosets of the Lorentz group within the super-Poincaré group. Just as has the interpretation as being the generator of spacetime translations, the charges , with , have the interpretation as generators of superspace translations in the 'spin coordinates' of superspace. That is, we can view superspace as the direct sum of Minkowski space with 'spin dimensions' labelled by coordinates . The supercharge generates translations in the direction labelled by the coordinate By counting, there are spin dimensions. Notation for superspace The superspace consisting of Minkowski space with supercharges is therefore labelled or sometimes simply . SUSY in 3 + 1