Is a commutator a scalar?

  • Thread starter Aziza
  • Start date
  • #1
190
1
[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..
 

Answers and Replies

  • #2
1,006
105
Yes, in this case ##i \hbar## is shorthand for "##i \hbar## times the identity operator."
 
  • #3
190
1
Ohhh ok thanks!! I can't believe I went through all of Griffiths and this point was never made clear.
 
  • #4
dextercioby
Science Advisor
Homework Helper
Insights Author
13,024
579
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
The commutator of two scalars is a scalar.
The commutator of two vectors is a matrix. e.g.

[xi,pj]=iħδij

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

Related Threads on Is a commutator a scalar?

Replies
0
Views
1K
Replies
13
Views
1K
Replies
16
Views
2K
Replies
3
Views
4K
  • Last Post
Replies
4
Views
574
Replies
12
Views
2K
Replies
1
Views
1K
Replies
4
Views
1K
Replies
2
Views
2K
Top