Originally Posted by studiot With the greatest respect it is not incorrect. 
I still disagree, of course. In this post I'll explain why. oint. I'll take
the most well known quantum mechanics texts, which I have, and post the definition of
commutator as these texts use them. I'll then post the exact definition as given by that text and post the reference to the text.
First I'll start with Science World  Wolfram
First take a look at the definition and symbol for inner product at:
http://mathworld.wolfram.com/InnerProduct.html
The symbol for it is <a, b>. It's a generalization of the dot product.
Next we'll look at commutators:
http://scienceworlScienceWorld  Wolfram
http://scienceworld.wolfram.com/phys...mmutators.html
In quantum mechanics, two quantities can be measured to any degree of precision only if their operators commute, in which case they have simultaneous eigenfunctions

Now we'll look at Wolfram  Mathworld
https://en.wikibooks.org/wiki/Quantu...nd_Commutators
Note the definition from the Contents under
Mathematical Definition of Commutator which is located at the following link:
https://en.wikibooks.org/wiki/Quantu..._of_Commutator
It says (I can't do carets in ascii so I'm using boldface instead).
From
Commutators http://quantummechanics.ucsd.edu/ph1...s/node109.html
The following are from my collection of quantum mechanics textbooks;
Quantum Mechanics  3rd Ed. by Eugen Merzbacher. Commutator is defined on page 37 as [
F,
H] =
FH 
HF (Eq. 3.45).
Introductory Quantum Mechanics  3rd Ed. by Richard L. Liboff. Commutator is defined on page 134 as [
A,
B] =
AB 
BA (Eq. 5.49).
Quantum Mechanics in a Nutshell by Gerald D. Mahan. Commutator is defined on page 37 as [
A,
B] =
AB  [B]B[b]
A (Eq. 2.167).
Quantum Mechanics: The Theoretical Minimum; What you need to know to start doing physics by Leonard Susskind & Art Friedman. Commutator is defined on page 111 as [
L,
M] =
LM
ML.
Introduction to Quantum Mechanics  2nd Ed. by David J. Griffiths. Commutator is defined on page 43 as [
A,
B] =
AB 
BA (Eq. 2.48).
Lectures in Quantum Mechanics by Steven Weinberg. Commutator is defined on page 25 as [
A,
B] =
AB 
BA
Originally Posted by studiot I have seen it used for the inner product. 
When using a term in a particular field, like here where we're talking about quantum mechanics, the definition is as defined in that field if there is no other way that it's used as is the case in QM. Therefore since the symbol [A, B] never has any other meaning other than [A, B] = AB  BA there is no need to define it unless its used when discussing QM.
Originally Posted by studiot Kreider, page 257 equation 2 for example. 
If that is a text what is the name of it?
Originally Posted by studiot I have also seen other notations used, there are many unfortunately. 
I've never seen it used any other way. Please provide a few examples.
Originally Posted by studiot That is why it is encumbent upon the promoter to specify what he means by his notation.
You did not. 
I disagree. The commutator is never used in any other way in QM so there is no need to do that since the meaning is implied. I didn't do so because there was no reason for it.
This entire thread is on QM. In particular its about the uncertainty principle so its meaning is implied. In fact if you look up the uncertainty principle in Wikipedia you would have seen this. That's what I do almost all the time. I recommend that you do the same thing too.
Originally Posted by studiot I am sorry I misuderstood but notation does not detract from the correctness of my assertions. 
What assertions are you referring to?
Originally Posted by studiot Also unfortunately another popular inner product notation < > is also used for Bra and Ket in Qunatum Mechanics so is best avoided. 
In almost all quantum mechanics texts both the commutator [A, B] and the inner product <a, b> are used. As I showed you using the Wolfram dictionaries they have one and only one meaning. And that's what I assumed to be true to the best of my knowledge. If its used any other way then I've never seen it and that would be the source of your disagreement with me.
Originally Posted by studiot Oh and by the way, does not AB  BA conform to the mathematical definition of an inner product for the vectors A and B? 
The former applies only to the operators while the later only applies to dot product.
Originally Posted by studiot Oh and by the by the way bra and ket is also a form of inner product. 
Yup. I'm quite aware of that, than you.
Originally Posted by studiot They get everywhere don't they? 
Yeah. They sure do!
Originally Posted by studiot The use of square brackets in relation to commutators derives from group theory
[a, b] = a^1b^1ab for ab in the group. 
My abstract algebra text does not use the symbol [ , ]. And the commutator used in abstract algebra is entirely different then the commutator in general. In fact it only refers to the "commutator of the group."