# Is a commutator a scalar?

1. Dec 15, 2014

### Aziza

[A,B] = AB-BA, so the commutator should be a matrix in general, but yet
[x,p]=i*hbar...which is just a scalar. Unless by this commutator, we mean i*hbar*(identity matrix) ?

I am asking because I see in a paper the following:

tr[A,B]

Which I interpret to mean the trace of the commutator [A,B]. But if [A,B] is just a scalar, then trace of a scalar should always just be the scalar..

2. Dec 15, 2014

### The_Duck

Yes, in this case $i \hbar$ is shorthand for "$i \hbar$ times the identity operator."

3. Dec 15, 2014

### Aziza

Ohhh ok thanks!! I can't believe I went through all of Griffiths and this point was never made clear.

4. Dec 16, 2014

### dextercioby

Since Griffiths doesn't get into the mathematics of the CCRs, nor specifically treats the old matrix mechanics, it's quite understandable that he leaves out the unit operator/matrix.

5. Dec 16, 2014