anticommute

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anticommute

[‚an·tē·kə′myüt]
(mathematics)
Two operators anticommute if their anticommutator is equal to zero.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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These symmetry transformations have two innate properties: nilpotency of order two and absolute anticommutativity. First property elaborates the fermionic nature of the (anti-)BRST symmetries whereas latter one insures that BRST and anti-BRST transformations are linearly independent of each other.
The latter is parametrized by the superspace coordinates [Z.sup.M] = ([x.sup.[mu]], [eta], [bar.[eta]]) where [x.sup.[mu]] ([mu] = 0,1,..., D - 1) are the bosonic coordinates and ([eta], [bar.[eta]]) are a pair of Grassmannian variables obeying nilpotency and anticommutativity properties (i.e., [[eta].sup.2] = [bar.[eta]] = 0, [eta][bar.[eta]] + [bar.[eta]][eta] = 0).
In our Section 6, we show the nilpotency and anticommutativity properties of the (anti-)BRST and (anti-)co-BRST transformations (and corresponding generators) in terms of the translational generators along the directions of Grassmannian variables.
be the well-known Pauli-matrices which has these properties: [[sigma].sup.2.sub.i] = I, (I is 2 x 2 identity matrix) [[sigma].sup.*.sub.i] = [[sigma].sub.i] (self-adjointness) i = 1, 2, 3 and for i [not equal to] j, [[sigma].sub.i][[sigma].sub.j] = -[[sigma].sub.j][[sigma].sub.i] (anticommutativity).
As an algebraic point of departure and theoretical physics point of view, the Hochschild cohomology H[H.sup.*](A) of an associative algebra has natural product with a Lie type bracket of degree -1, satisfying Jacobi identity and graded anticommutativity such that both natural product and Lie type bracket are compatible to make H[H.sup.*] (A) a Gerstenhaber algebra.
The minus sign is a consequence of the anticommutativity of the spin operators, which requires the use of quantum logic, rather than classical logic.
Now we dwell a bit on the geometrical meaning of the absolute anticommutativity of the (anti-)co-BRST charges [Q.sub.ad] in the language of the translational generators ([[partial derivative].sub.[theta]] and [[partial derivative].sub.[bar.[theta]]]) along the Grassmannian directions of the supermanifold.
However, if we take the definition of the generator for the transformation [s.sub.ad], then, [s.sub.ad][Q.sub.d] = i{[Q.sub.d],[Q.sub.ad]} = 0 due to the nilpotency ([s.sup.2.sub.ad] = 0) of [s.sub.ad] which in turn implies the absolute anticommutativity of the (anti-)co-BRST charges [Q.sub.(a)d].
In the language of theoretical physics, the nilpotency property ensures the fermionic (supersymmetric-type) nature of the (anti-)BRST symmetries and the linear independence of BRST and anti-BRST symmetries is encoded in the property of absolute anticommutativity of the above (anti-)BRST symmetries.
The superfield approach to BRST formalism [1-8] provides the geometrical basis for the properties of nilpotency and absolute anticommutativity which are associated with the (anti-)BRST symmetries.
We capture the nilpotency and absolute anticommutativity properties of the (anti-)BRST and (anti-)co-BRST symmetries and their generators in our present formalism.
In Section 6, we discuss the properties of nilpotency and absolute anticommutativity of the (anti-)BRST and (anti-)co-BRST transformations and corresponding charges.