Commuting, non-commuting, anti-commuting

  • Thread starter Lapidus
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We have

[A,B] equals zero, commuting

[A,B] not equals zero, not commuting

[A,B] = - [B,A] , anti-commuting


So then we can say

[X,P] anticommutes, since [X,P]= -[P,X] , and

[X,P] does not commute, since [X,P] = ih

I find that confusing. Is there something I missed? (Is this on the one hand math language for the Lie algebra, which needs to be anti-commuting, and on the other hand physics language for commuting and non-commuting observables?)

Also, for femions there is the anti-commuting relations {A,B}. Here A,B anticommute if {A,B} is zero. But X and P for bosons anticommute, why are we here not using the anticommutator.

Can someone unconfuse me?
thanks
 

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  • #2
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Oooops,

just noticed that [A,B] = -[B,A] means antisymmetric!
 
  • #3
PeterDonis
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(Is this on the one hand math language for the Lie algebra, which needs to be anti-commuting, and on the other hand physics language for commuting and non-commuting observables?)
They're the same underlying mathematics, just applied in different ways.

Also, for femions there is the anti-commuting relations {A,B}. Here A,B anticommute if {A,B} is zero.
Yes, but *what* do these relations apply to for fermions? It's not operators like X and P; those do not commute for *any* quantum object, whether it's a boson or a fermion, as you note. For fermions, the actual *states* anti-commute, in the sense that, for example, if we take a two-fermion state and swap the fermions, the state flips sign. This is just the mathematical realization of the Pauli exclusion principle.

X and P for bosons anticommute, why are we here not using the anticommutator.
Because the Pauli exclusion principle doesn't apply to bosons; boson states do not flip sign if two bosons are swapped. Mathematically, this means that boson states can be described by ordinary numbers, which commute with each other under all algebraic operations. Fermion states, on the other hand, can't be described by ordinary numbers, because fermion states have to anti-commute under certain operations (like swapping two fermions).

As for X and P, those are operators, not states, and the key physical question for operators is whether they commute, i.e., whether they give the same joint result regardless of the order in which they are applied. That's why you only see commutators evaluated for operators; the anti-commutator for operators doesn't mean anything, physically.
 
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[A,B] = - [B,A] , anti-commuting
No. [A, B] = -[B, A] is a general property of the commutator (or Lie brackets more generally), true for any operators A and B:

(AB - BA) = -(BA - AB)

We say that A and B anticommute only if {A,B} = 0, that is AB + BA = 0. X and P do not anticommute.
 
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