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 algebra.
Construction
Abstract construction Abstractly, super Minkowski space is the space of (right) cosets within the Super Poincaré group of Lorentz group, that is,
Super Minkowski space ≅ Super Poincaré group Lorentz group {\displaystyle {\text{Super Minkowski space}}\cong {\frac {\text{Super Poincaré group}}{\text{Lorentz group}}}} . This is analogous to the way ordinary Minkowski spacetime can be identified with the (right) cosets within the Poincaré group of the Lorentz group, that is,
Minkowski space ≅ Poincaré group Lorentz group {\displaystyle {\text{Minkowski space}}\cong {\frac {\text{Poincaré group}}{\text{Lorentz group}}}} . The coset space is naturally affine, and the nilpotent, anti-commuting behavior of the fermionic directions arises naturally from the Clifford algebra associated with the Lorentz group.
Direct sum construction For this section, the dimension of the Minkowski space under consideration is d = 4 {\displaystyle d=4} . Super Minkowski space can be concretely realized as the direct sum of Minkowski space, which has coordinates x μ {\displaystyle x^{\mu }} , with 'spin space'. The dimension of 'spin space' depends on the number N {\displaystyle {\mathcal {N}}} of supercharges in the associated super Poincaré algebra to the super Minkowski space under consideration. In the simplest case, N = 1 {\displaystyle {\mathcal {N}}=1} , the 'spin space' has 'spin coordinates' ( θ α , θ ¯ α ˙ ) {\displaystyle (\theta _{\alpha },{\bar {\theta }}^{\dot {\alpha }})} with α , α ˙ = 1 , 2 {\displaystyle \alpha ,{\dot {\alpha }}=1,2} , where each component is a Grassmann number. In total this forms 4 spin coordinates. The notation for N = 1 {\displaystyle {\mathcal {N}}=1} super Minkowski space is then R 4 | 4 {\displaystyle \mathbb {R} ^{4|4}} . There are theories which admit N {\displaystyle {\mathcal {N}}} supercharges. Such cases have extended supersymmetry. For such theories, super Minkowski space is labelled R 4 | 4 N {\displaystyle \mathbb {R} ^{4|4{\mathcal {N}}}} , with coordinates ( θ α I , θ ¯ J α ˙ ) {\displaystyle (\theta _{\alpha }^{I},{\bar {\theta }}^{J{\dot {\alpha }}})} with I , J = 1 , ⋯ , N {\displaystyle I,J=1,\cdots ,{\mathcal {N}}} .
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