Anticommutative property

In mathematics, pairs of elements a and b in a ring or algebra may have the anticommutative property: ab = − ba. The property is true of the binary operation of subtraction.

As ab – ba = 0 expresses commutativity, so ab + ba = 0 expresses anti-commutativity.

Instances

For example, matrix multiplication of these 2x2 matrices, with entries from a ring with −1, anticommute:

(0110)×(100−1)=(0−110)
(100−1)×(0110)=(01−10)

Here the reversed order of factors results in the additive inverse matrix.

In multilinear algebra over a field, the wedge product has the anticommutative property.

Sophus Lie used the commutator on a continuous group to study the structure with Lie algebra. The commutator of two group elements g and h is [g,h]=gh−hg, which is anticommutative.

W. R. Hamilton initiated the study of quaternion algebras which exhibit anticommutativity of some elements.

When an algebra has all basis elements anticommute with each other, and have squares equal to one, then the square of a general element equals the sum of the squares of the coordinates since anticommutativity implies cross products sum to zero:

If ei2=1,eiej+ejei=0,i≠j ,then(∑i=1nxiei)2=∑i=1nxi2.

Quaternion algebras

Given a field F , the vector space (F 4, + ) is made into a quaternion algebra Q by defining some vector products according to the anti-commutative rule. One dimension of Q commutes with all the others, it is the scalar field F. The other three dimensions of Q are denoted i, j, and k, so that {1, i, j, k} is a basis of Q. The multiplication x in Q = (F 4, + , x ) is anticommutative when different i, j, or k are factors. The product x may be as in the quaternion group or as in the dihedral group of order eight.

Fermions

In quantum mechanics the permutation of factors is accomplished with the exchange operator. In 1926 P. A. M. Dirac wrote "the variables that describe a dynamical system do not obey the commutative law of multiplication but satisfy instead certain quantum conditions." The resulting Fermi–Dirac statistics uses commutativity and anticommutativity to distinguish bosons from fermions. Dirac used the term antisymmetrical: "An antisymmetrical eigenfunction vanishes identically when two of the electrons are in the same orbit. This means that in the solution of the problem with antisymmetrical eigenfunctions there can be no stationary states with two or more electrons in the same orbit, which is just Pauli's exclusion principle." One the other hand, David Hestenes uses antisymmetric on page 8 of his book Spacetime Algebra (1966) but later uses the term anticommutative.

References

  1. ↑ Hermann Grassmann (2000)[1862] Lloyd Kannenberg (translator), Extension Theory, American Mathematical Society ISBN 0-8126-9275-6, ISBN 0-8126-9276-4
  2. ↑ Robert Gilmore (1974) Lie Groups, Lie Algebras and Some of Their Applications, page 94, John Wiley & Sons ISBN 0-471-30179-5
  3. ↑ Marcel Riesz (1958) Clifford Numbers and Spinors, pages 5 & 6, University of Maryland via Internet Archive
  4. ↑ Voight, John (2021). Quaternion Algebras. Graduate Texts in Mathematics. Vol. 288. Springer Nature. p. 23. doi:10.1007/978-3-030-56694-4. ISBN 978-3-030-56692-0.
  5. 1 2 P. A. M. Dirac (1926) "On the theory of quantum mechanics", Proceedings of the Royal Society #762 doi:10.1098/rspa.1926.0133
  6. ↑ David Hestenes (1966) Spacetime Algebra, compare page 8 to pages 33, 34, 41, 54, and 65