and is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra. Nilpotency in physics Any ladder operator in a finite dimensional space is nilpotent. They represent creation and annihilation operators, which transform from one state to another, for example the raising and lowering Pauli matrices . An operand that satisfies is nilpotent. Grassmann numbers which allow a path integral representation for Fermionic fields are nilpotents since their squares vanish. The BRST charge is an important example in physics. As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition. More generally, in view of the above definitions, an operator is nilpotent if there is such that (the zero function). Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with ). Both are linked, also through supersymmetry and Morse theory, as shown by Edward Witten in a celebrated article. The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space. More generally, the technique of microadditivity (which can used to derive theorems in physics) makes use of nilpotent or nilsquare infinitesimals and is part smooth infinitesimal analysis. Algebraic nilpotents The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions , and complex octonions . If a nilpotent infinitesimal is a variable tending to zero, it can be shown that any sum of terms for which it is the subject is an indefinitely small proportion of the first order term. See also Idempotent element (ring theory) Unipotent Reduced ring Nil ideal References Ring theory 0 (number) Algebraic properties of elements