In
theoretical physics, a supercharge is a generator of
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
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
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
one-half to a particle with
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
one or zero, the supercharge itself is a
spinor that carries one half unit of spin.
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-number
The term Number C (or C number) is an old nomenclature used by Paul Dirac which refers to real and complex numbers. It is used to distinguish from operators (q-numbers or quantum numbers) in quantum mechanics.
Although c-numbers are commuting ...
s'' and ''a-numbers''. Calling it a charge emphasizes the notion of a symmetry at work.
Commutation
Supercharge is described by the
Super-Poincaré algebra.
Supercharge commutes with the
Hamiltonian operator:
:
Q , H
Q, or q, is the seventeenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is pronounced , most commonly spelled ''cue'', but ...
= 0
So does its adjoint.
References
Supersymmetry
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