why I can't define a commutator operator in this way？

Question

I am trying to define some operators I used frequently.For example, I defined a KroneckerProduct operator as $ \otimes $ as follows:

a_$ \otimes $b_:=a.b-b.a

and this works fine.

Then I want to define a commutator used in quantum mechanics act like follows: $${\left[ {a,b} \right]_ - } = a.b - b.a$$ $${\left[ {a,b} \right]_ + } = a.b + b.a$$ where a and b are matrix.

So I naively write the definition in matheamtica as follows:

$${\left[ {a{\rm{\_}},b{\rm{\_}}} \right]_ - }: = a.b - b.a$$ $${\left[ {a{\rm{\_}},b{\rm{\_}}} \right]_ + }: = a.b + b.a$$

But this will not work.

1.So how to define commutator in this form (appearance)?

2.and why the definition for a_$\otimes$b_ hold, the similar definition for commutator [a_,b_]not hold?

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There is another question which actually should create another post. But I choose just to add it here.

If I defined the KroneckerProduct as the operator [CircleTimes] in the way I did, then I found that in parallel computing, I can't simply distribute such a definition of [CircleTimes] like:

DistributeDefinitions[$\otimes$]

Mathematica doesn't allow me to do that. Of course the 'Parallelize' will not distribute the $\otimes$ definition automatically either. So if an expression contains a $\otimes$, Simply 'Parallelize' it, the computation will get stuck and no answer will come out.

So my second question is

how to distribute such a operator definition in parallel computing ?

Answer 1

Maybe it's this that you are looking for.

Using the notation Package you can write:

I used a image to show these yellow box. To create it, use the notation pallet that Popup when you call the Package. This is the pallet:

Use the second button to do your new notation.

More information about this in Wolfram help in this link.

Answer 2

Too long for a comment but not a complete answer. From the end

-I don't know how (or if it is possible) to distribute the operation in parallel computing.

-See this, and this question for defining a commutator.

In particular there is advice in the second one by Szabolics on how to use better notation if you don't like the one below.

-An anticommutator would be sufficiently defined by a rule I think, like so:

HoldAll[(l m + m l + l^2 + m m l )] /. {(a_ b_ + b_ a_) -> 
Anitcom[a, b], a_ a_ -> 0, a_^_->0}

Where you can define Anticom on a case by case basis.