Is the commutator of two operators always a scalar?

In summary, the commutator [A,B] is a matrix in general, but in the case of [x,p]=i*hbar it is just a scalar. However, in some cases, such as in the trace of the commutator, [A,B] is shorthand for "i*hbar times the identity operator." This point is often not explicitly stated in textbooks. The commutator of two scalars is a scalar, while the commutator of two vectors is a matrix, with the unity matrix being an example.
  • #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:


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


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

1. What is a commutator?

A commutator is a mathematical operation that calculates the difference between two quantities by subtracting them in reverse order. It is often denoted by the symbol [A,B] where A and B are the quantities being operated on.

2. Is a commutator a scalar or a vector?

A commutator can be either a scalar or a vector, depending on the quantities being operated on. If the quantities are both scalars, then the commutator will also be a scalar. If at least one of the quantities is a vector, then the commutator will be a vector.

3. What are some examples of commutators?

Some common examples of commutators include the cross product in vector calculus, the commutator in quantum mechanics, and the Poisson bracket in classical mechanics. These operations involve subtracting quantities in reverse order to calculate a difference.

4. What is the significance of a commutator in mathematics?

The commutator is a fundamental operation in mathematics that helps to define and understand the properties of different mathematical structures. It is particularly important in fields such as quantum mechanics and group theory, where it is used to study the properties of operators and groups, respectively.

5. How is a commutator different from other types of mathematical operations?

The main difference between a commutator and other mathematical operations is that it involves subtracting quantities in reverse order. This means that the result of a commutator can change depending on the order in which the quantities are subtracted, whereas most other operations are commutative and the order of the operands does not affect the result.

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