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In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold for superfields. It is acted on by the super Poincaré algebra. ==Definition== The underlying supermanifold of super Minkowski space is isomorphic to a super vector space given by the direct sum of ordinary Minkowski spacetime in ''d'' dimensions (often taken to be 4) and a number ''N'' of real spinor representations of the Lorentz algebra. (When ''d'' is 2 mod 4 this is slightly ambiguous because there are 2 different real spin representations, so one needs to replace ''N'' by a pair of integers ''N''=''N''1+''N''2, though some authors use a different convention and take ''N'' copies of both spin representations.) However this construction is misleading for two reasons: first, super Minkowski space is really an affine space over a group rather than a group, or in other words it has no distinguished "origin", and second, the underlying supergroup of translations is not a super vector space but a nilpotent supergroup of nilpotent length 2. This supergroup has the following Lie algebra. Suppose that ''M'' is Minkowski space, and ''S'' is a finite sum of irreducible real spinor representations. Then there is an invariant symmetric bilinear map () from ''S''×''S'' to ''M'' that is positive definite in the sense that the image of ''s''×''s'' is in the closed positive cone of ''M'', and is nonzero if ''s'' is nonzero. This bilinear map is unique up to isomorphism. The Lie superalgebra has ''M'' as its even part, ''S'' as its odd or fermionic part, and the Lie bracket is given by () (and the Lie bracket of anything in ''M'' with anything is zero). The dimensions of the irreducible real spinor representations for various dimensions ''d'' of spacetime are given by the following table: The table repeats whenever the dimension increases by 8, except that the dimensions of the spin representations are multiplied by 16. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Super Minkowski space」の詳細全文を読む スポンサード リンク
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