Is the commutator of two operators always a scalar?

[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..

 

[The_Duck]

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

 

[Aziza]

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

 

[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.

 

[nuclearhead]

The commutator of two scalars is a scalar.
The commutator of two vectors is a matrix. e.g.

[x_i,p_j]=iħδ_ij

in this case it is the unity matrix. The trace of this in 4 dimensions would be 4.