A mathematical operator (typically a binary operator, represented by *) is anticommutativeif and only if it is true that This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P...
x * y = −(y * x)
for all x and y on the operator's valid domain (e.g. R for subtraction, and R3vectors for cross products). In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but this...
Examples of the use of anticommutative operators include:
In mathematics, subtraction is one of the four basic arithmetic operations. ... In mathematics, the cross product is a binary operation on vectors in vector space. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
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.
where [a,b] is the commutator of a and b and {a,b} is the anticommutator of a and b.
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.
The Maxwell equations are formed from a combinations of commutators and anticommutators of the differential operator and the electric and magnetic fields E and B respectively (for isolated charges in a vacuum.
The electric field E is the vector part of the anticommutator of the conjugates of the differential operator and the 4-potential.
The homogeneous terms are formed from the sum of both orders of the commutator and anticommutator.
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