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| "'''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 number]]s rather than real numbers. The ordinary space dimensions correspond to [[boson]]ic degrees of freedom, the anticommuting dimensions to [[fermionic]] degrees of freedom.
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| See also [[supermanifold]] (although the definition of a superspace as a supermanifold here does not agree with the definition used in that article).
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| '''R'''<sup>''m''|''n''</sup> is the '''Z'''<sub>2</sub>-[[graded vector space]] with '''R'''<sup>''m''</sup> as the even subspace and '''R'''<sup>''n''</sup> as the odd subspace. The same definition applies to '''C'''<sup>m|n</sup>.
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| The word "superspace" was first used by [[John Archibald Wheeler|John Wheeler]] in an unrelated sense to describe the [[configuration space]] of [[general relativity]]; for example, this usage may be seen in his 1973 textbook ''[[Gravitation (book)|Gravitation]]''.
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| ==Examples==
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| ===Trivial examples===
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| The smallest superspace is a point which contains neither bosonic nor fermionic directions. Other trivial examples include the ''n''-dimensional real plane '''R'''<sup>n</sup>, which is a [[vector space]] extending in ''n'' real, bosonic directions and no fermionic directions. The vector space '''R'''<sup>0|n</sup>, which is the ''n''-dimensional real [[Grassmann algebra]]. The space '''R'''<sup>1|1</sup> of one even and one odd direction is known as the space of [[dual number]]s, introduced by [[William Kingdon Clifford|William Clifford]] in 1873.
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| ===The superspace of supersymmetric quantum mechanics===
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| [[Supersymmetric quantum mechanics]] with ''N'' [[supercharge]]s is often formulated in the superspace '''R'''<sup>1|2''N''</sup>, which contains one real direction ''t'' identified with [[time]] and ''N'' complex Grassmann directions which are spanned by Θ<sub>''i''</sub> and Θ<sup>*</sup><sub>''i''</sub>, where ''i'' runs for 1 to ''N''.
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| Consider the special case ''N'' = 1. The superspace '''R'''<sup>1|2</sup> is a 3-dimensional vector space. A given coordinate therefore may be written as a triple (''t'', Θ, Θ<sup>*</sup>). The coordinates form a [[Lie superalgebra]], in which the gradation degree of ''t'' is even and that of Θ and Θ<sup>*</sup> 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]] on two even coordinates and on one even and one odd coordinate while it is an [[anticommutator]] on two odd coordinates. This superspace is an abelian Lie superalgebra, which means that all of the forementioned brackets vanish
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| :::<math>\left[ t,t\right]=\left[ t, \theta\right]=\left[ t, \theta^*\right]=\left\{\theta, \theta\right\}=\left\{ \theta, \theta^*\right\} =\left\{ \theta^*, \theta^*\right\}=0</math>
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| where <math>[a,b]</math> is the commutator of ''a'' and ''b'' and <math>\{a,b\}</math> is the anticommutator of ''a'' and ''b''.
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| One may define functions from this vectorspace to itself, which are called [[superfield]]s. The above algebraic relations imply that, if we expand our superfield as a [[power series]] in Θ and Θ<sup>*</sup> then we will only find terms at the zeroeth and first orders, because Θ<sup>2</sup> = Θ<sup>*2</sup> = 0. Therefore superfields may be written as arbitrary functions of ''t'' multiplied by the zeroeth and first order terms in the two Grassmann coordinates
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| :::<math>\Phi \left(t,\Theta,\Theta^* \right)=\phi(t)+\Theta\Psi(t)-\Theta^*\Phi^*(t)+\Theta\Theta^* F(t)</math>
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| Superfields, which are representations of the [[supersymmetry]] of superspace, generalize the notion of [[tensor]]s, which are representations of the rotation group of a bosonic space.
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| One may then define derivatives in the Grassmann directions, which take the first order term in the expansion of a superfield to the zeroeth order term and annihilate the zeroeth order term. One can choose sign conventions such that the derivatives satisfy the anticommutation relations
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| :::<math>\left\{\frac{\partial}{\partial \theta}\,,\Theta\right\}=\left\{\frac{\partial}{\partial \theta^*}\,,\Theta^*\right\}=1</math>
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| These derivatives may be assembled into [[supercharge]]s
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| :::<math>Q=\frac{\partial}{\partial \theta}-i\Theta^*\frac{\partial}{\partial t}\quad \text{and} \quad Q^\dagger=\frac{\partial}{\partial \theta^*}+i\Theta\frac{\partial}{\partial t}</math>
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| whose anticommutators identify them as the fermionic generators of a [[supersymmetry]] algebra
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| :::<math>\left\{ Q,Q^\dagger\,\right\}=2i\frac{\partial}{\partial t}</math>
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| where ''i'' times the time derivative is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator in [[quantum mechanics]]. Both ''Q'' and its adjoint anticommute with themselves. The supersymmetry variation with supersymmetry parameter ε of a superfield Φ is defined to be
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| :::<math>\delta_\epsilon\Phi=(\epsilon^* Q+\epsilon Q^\dagger)\Phi.</math>
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| We can evaluate this variation using the action of ''Q'' on the superfields
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| :::<math>\left[Q,\Phi \right]=\left(\frac{\partial}{\partial \theta}\,+i\theta^*\frac{\partial}{\partial t}\right)\Phi=\psi+\theta^*\left(F+i\dot{\phi}\right)-i\theta\theta^*\dot{\psi}.</math>
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| Similarly one may define [[covariant derivative]]s on superspace
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| :::<math>D=\frac{\partial}{\partial \theta}-i\theta\frac{\partial}{\partial t}\quad \text{and} \quad D^\dagger=\frac{\partial}{\partial \theta^*}-i\theta\frac{\partial}{\partial t}</math>
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| which anticommute with the supercharges and satisfy a wrong sign supersymmetry algebra
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| :::<math>\left\{D,D^\dagger\right\}=-2i\frac{\partial}{\partial t}</math>.
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| The fact that the covariant derivatives anticommute with the supercharges means the supersymmetry transformation of a covariant derivative of a superfield is equal to the covariant derivative of the same supersymmetry transformation of the same superfield. Thus, generalizing the covariant derivative in bosonic geometry which constructs tensors from tensors, the superspace covariant derivative constructs superfields from superfields.
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| ===Four-dimensional ''N'' = 1 superspace===
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| Perhaps the most popular superspace in [[physics]] is ''d''=4 ''N''=1 [[super Minkowski space]] '''R'''<sup>4|4</sup>, which is the [[direct sum of modules|direct sum]] of four real bosonic dimensions and four real Grassmann dimensions. In [[supersymmetric]] [[quantum field theory|quantum field theories]] one is interested in superspaces which furnish [[group representation|representations]] of a [[Lie superalgebra]] called a [[supersymmetry algebra]]. The bosonic part of the supersymmetry algebra is the [[Poincaré algebra]], while the fermionic part is constructed using [[spinor]]s of Grassmann numbers.
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| For this reason, in physical applications one considers an action of the supersymmetry algebra on the four fermionic directions of '''R'''<sup>4|4</sup> such that they transform as a [[spinor]] under the Poincaré subalgebra. In four dimensions there are three distinct irreducible 4-component spinors. There is the [[Majorana spinor]], the left-handed [[Weyl spinor]] and the right-handed Weyl spinor. The [[CPT theorem]] implies that in a [[unitary]]{{Disambiguation needed|date=January 2012}}, Poincaré invariant theory, which is a theory in which the [[S-matrix]] is a [[unitary matrix]] and the same Poincaré generators act on the asymptotic in-states as on the asymptotic out-states, the supersymmetry algebra must contain an equal number of left-handed and right-handed Weyl spinors. However, since each Weyl spinor has four components, this means that if one includes any Weyl spinors one must have 8 fermionic directions. Such a theory is said to have [[extended supersymmetry]], and such models have received a lot of attention. For example, supersymmetric gauge theories with eight supercharges and fundamental matter have been solved by [[Nathan Seiberg]] and [[Edward Witten]], see [[Seiberg–Witten gauge theory]]. However in this subsection we are considering the superspace with four fermionic components and so no Weyl spinors are consistent with the CPT theorem.
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| ''Note'': There are many [[sign convention]]s in use and this is only one of them.
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| This leaves us with one possibility, the four fermionic directions transform as a Majorana spinor θ<sub>α</sub>. We can also form a conjugate spinor
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| :::<math>\overline{\theta}\ \stackrel{\mathrm{def}}{=}\ i\theta^\dagger\gamma^0=-\theta^\perp C</math>
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| where ''C'' is the charge conjugation matrix, which is defined by the property that when it conjugates a [[gamma matrix]], the gamma matrix is negated and transposed. The first equality is the definition of {{Overline|θ}} while the second is a consequence of the Majorana spinor condition θ<sup>*</sup> = iγ<sub>0</sub>Cθ. The conjugate spinor plays a role similar to that of θ<sup>*</sup> in the superspace '''R'''<sup>1|2</sup>, except that the Majorana condition, as manifested in the above equation, imposes that θ and θ<sup>*</sup> are not independent.
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| In particular we may construct the supercharges
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| :::<math>Q=-\frac{\partial}{\partial\overline{\theta}}+\gamma^\mu\theta\partial_\mu</math>
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| which satisfy the supersymmetry algebra
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| :::<math>\left\{Q,Q\right\}=\left\{\overline{Q},Q\right\}C=2\gamma^\mu\partial_\mu C=-2i\gamma^\mu P_\mu C</math>
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| where <math>P=i\partial_\mu</math> is the 4-[[momentum]] operator. Again the covariant derivative is defined like the supercharge but with the second term negated and it anticommutes with the supercharges. Thus the covariant derivative of a supermultiplet is another supermultiplet.
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| ==See also==
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| ===Spaces===
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| * [[Chiral superspace]]
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| * [[Harmonic superspace]]
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| * [[Projective superspace]]
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| ===Formalisms===
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| * [[ADM formalism]]
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| * [[Hamilton–Jacobi–Einstein equation]]
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| * [[Wheeler–DeWitt equation]]
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| ==References==
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| *{{Citation | editor1-last=[[:uk:Дуплій Степан Анатолійович|Duplij]] | editor1-first=Steven | editor2-last=[[Warren Siegel|Siegel]]| editor2-first=Warren| editor3-last=Bagger | editor3-first=Jonathan | title=Concise Encyclopedia of Supersymmetry And Noncommutative Structures in Mathematics and Physics | publisher=[[Springer Publishing|Springer]] | location=Berlin, New York | isbn=978-1-4020-1338-6 | year=2005}} (Second printing)
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| [[Category:Geometry]]
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| [[Category:Supersymmetry]]
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| [[Category:General relativity]]
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| [[hu:Szupertér]]
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