Supercharge: Difference between revisions
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In [[theoretical physics]], a '''supercharge''' is a generator of [[supersymmetry]] transformations. It is an example of the general notion of a [[charge (physics)|charge]] in physics. |
In [[theoretical physics]], a '''supercharge''' is a generator of [[supersymmetry]] transformations. It is an example of the general notion of a [[charge (physics)|charge]] in physics. |
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Supercharge, denoted by the symbol Q, is an operator which transforms [[bosons]] into [[fermions]], and vice versa. Since the supercharge operator changes a particle with [[Spin (physics)|spin]] one-half to a particle with [[Spin (physics)|spin]] one or zero, the supercharge itself is a [[spinor]] that carries one half unit of spin.<ref name="ss to the rescue">{{cite web |title=Supersymmetry to the rescue? |url=http://www.superstringtheory.com/experm/exper4a.html |website=The Official String Theory Web Site |url-status=dead |archive-url=https://web.archive.org/web/20181102164833/http://www.superstringtheory.com/experm/exper4a.html |archive-date=2018-11-02}}</ref><ref>{{cite web |last1=von Hippel |first1=Matthew |title=Supersymmetry, to the Rescue! |url=https://4gravitons.com/supersymmetry-to-the-rescue/ |website=4 Gravitons The trials and tribulations of four gravitons and a postdoc |access-date=18 April 2021}}</ref> |
Supercharge, denoted by the symbol Q, is an operator which transforms [[bosons]] into [[fermions]], and vice versa. Since the supercharge operator changes a particle with [[Spin (physics)|spin]] one-half to a particle with [[Spin (physics)|spin]] one or zero, the supercharge itself is a [[spinor]] that carries one half unit of spin.<ref name="ss to the rescue">{{cite web |title=Supersymmetry to the rescue? |url=http://www.superstringtheory.com/experm/exper4a.html |website=The Official String Theory Web Site |url-status=dead |archive-url=https://web.archive.org/web/20181102164833/http://www.superstringtheory.com/experm/exper4a.html |archive-date=2018-11-02}}</ref><ref>{{cite web |last1=von Hippel |first1=Matthew |title=Supersymmetry, to the Rescue! |url=https://4gravitons.com/supersymmetry-to-the-rescue/ |website=4 Gravitons The trials and tribulations of four gravitons and a postdoc |access-date=18 April 2021}}</ref> |
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Depending on the context, supercharges may also be called ''Grassmann variables'' or ''Grassmann directions''; they are generators of the [[exterior algebra]] of anti-commuting numbers, the [[Grassmann number]]s. All these various usages are essentially synonymous; they refer to the <math>\mathbb{Z}_2</math> [[ |
Depending on the context, supercharges may also be called ''Grassmann variables'' or ''Grassmann directions''; they are generators of the [[exterior algebra]] of anti-commuting numbers, the [[Grassmann number]]s. All these various usages are essentially synonymous; they refer to the <math>\mathbb{Z}_2</math> [[signed set|grading]] between bosons and fermions, or equivalently, the grading between ''[[c-number]]s'' and ''a-numbers''. Calling it a charge emphasizes the notion of a symmetry at work. |
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==Commutation== |
==Commutation== |
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Supercharge is described by the [[ |
Supercharge is described by the [[super-Poincaré algebra]]. |
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Supercharge commutes with the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator: |
Supercharge commutes with the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator: |
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:[ Q , H ] = 0 |
:[ Q , H ] = 0 |
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So does its adjoint |
So does its adjoint. |
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== See also == |
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:<math> [ Q^* , H ] = 0 |
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* [[R-symmetry]] |
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==References== |
==References== |
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[[Category:Supersymmetry]] |
[[Category:Supersymmetry]] |
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Latest revision as of 11:48, 13 November 2023
In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge in physics.
Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and vice versa. Since the supercharge operator changes a particle with spin one-half to a particle with spin one or zero, the supercharge itself is a spinor that carries one half unit of spin.[1][2]
Depending on the context, supercharges may also be called Grassmann variables or Grassmann directions; they are generators of the exterior algebra of anti-commuting numbers, the Grassmann numbers. All these various usages are essentially synonymous; they refer to the grading between bosons and fermions, or equivalently, the grading between c-numbers and a-numbers. Calling it a charge emphasizes the notion of a symmetry at work.
Commutation[edit]
Supercharge is described by the super-Poincaré algebra.
Supercharge commutes with the Hamiltonian operator:
- [ Q , H ] = 0
So does its adjoint.
See also[edit]
References[edit]
- ^ "Supersymmetry to the rescue?". The Official String Theory Web Site. Archived from the original on 2018-11-02.
- ^ von Hippel, Matthew. "Supersymmetry, to the Rescue!". 4 Gravitons The trials and tribulations of four gravitons and a postdoc. Retrieved 18 April 2021.
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