In
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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a supercharge is a generator of
supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
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 half odd-integer ...
into
fermions
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin ( spin , spin , etc.) and obey the Pauli exclusion principle. These particles include all quarks and leptons and ...
, 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
In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
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 of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
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 of a complex vector space. The special case of a 1-dimensional algebra is known a ...
s. 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
Supercharge is described by the
super-Poincaré algebra.
Supercharge commutes with the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
operator:
:
Q , H = 0
So does its adjoint.
See also
*
R-symmetry
References
Supersymmetry
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