theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, 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 ...

transformations. It is an example of the general notion of a

charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...

in physics. Supercharge, denoted by the symbol Q, is an operator which transforms

bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...

into

fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...

, 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 ...

one-half to a particle with

one or zero, the supercharge itself is a

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

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 In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...

of anti-commuting numbers, the

Grassmann number In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as ...

s. All these various usages are essentially synonymous; they refer to the

\mathbb_2

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, th ...

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 = 0 So does its adjoint.

References

Supersymmetry {{quantum-stub