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Commutators and anticommutators

What is commutation?

What's the difference between 6×7 and 7×6? Nothing!

That's because numbers like 7 and 6 commute - which just means that the order of multiplication doesn't matter.

So if two things multiplied together 'commute', it means that you can swap them around and still get the same answer.
But isn't that always true?

Not always! Even though all numbers commute with each other, you can't always take commutation for granted.
Take matrices, for example. If I have 2 matrices called A and B, and I want to multiply them together, multiplying A by B gives a very different result to multiplying B by A.
Or, if you have two functions, say, f and g, and f contains a derivative operator which takes the derivative of anything written to the right of it, then f·g and g·f could also give very different results.

How can we express the idea of commutation mathematically?
Allow me to introduce the commutator, which is just a mathematical object on paper that looks like this: [ , ]

It's defined as follows:

[A, B]

AB - BA

So to see if A and B commute, and BA is the same as AB, and you subtract one from the other, you'll just get zero.
For example,

[6, 7]	=	6×7 - 7×6
	=	42 - 42
	=	0

So the numbers 6 and 7 commute (just like all other scalar numbers do).

[A, B]

0 if A and B commute.

Anticommutation

If AB = -BA, that's the opposite of commutation, and A and B are said to anticommute.
To go with the commutator, there's also an anticommutator, which is defined as follows:

{A, B}	=	AB + BA
So if {A, B}	=	0, then A and B anti-commute.

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