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Is a commutator a scalar?

  1. Dec 15, 2014 #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..
     
  2. jcsd
  3. Dec 15, 2014 #2
    Yes, in this case ##i \hbar## is shorthand for "##i \hbar## times the identity operator."
     
  4. Dec 15, 2014 #3
    Ohhh ok thanks!! I can't believe I went through all of Griffiths and this point was never made clear.
     
  5. Dec 16, 2014 #4

    dextercioby

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    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.
     
  6. Dec 16, 2014 #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.
     
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