The anti-commutator of two elements of a group is defined as [A,B]+ = AB + BA.

A and B are said to anti-commute if [A,B]+ = 0

We can easily show that for any two non-zero complex numbers the anti-commutator is always different from zero. Since the set of complex numbers is a vector space, any two complex numbers will commute and thus ab = ba. So clearly [A,B]+ = 2ab != (does not equal) zero

However, there must obviously be somethings which do possess this anti-commutation property for this to be relevant. An example of this property involves the two complex matricies:

|0  1|
|1  0|


|0 -i|
|i  0|

(where i is the square root of negative one (see imaginary number).
When we apply the anti-commutator to these two matricies we get:

|0  1||0 -i| + |0 -i||0  1| = |i  0| + |-i 0| = |0  0|
|1  0||i  0| + |i  0||1  0| = |0 -i| + | 0 i| = |0  0|

Which is the zero matrix (the analog of zero when we discuss matricies). If you do not follow the matrix multiplication, do not worry, the point is that there do exist mathematical entities which anti-commute.

(If this looks familiar familiar, it is because these matricies correspond to the x and y components of the spin operator in quantum mechanics for a spin one-half particle)

Note that this very similiar to the commutator [A,B] = AB - BA

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