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In Dirac's Principles of Quantum Mechanics (pg. 86 eq. 5), the quantum commutator is motivated by looking for a bracket that satisfies the same properties of the Poisson bracket. When deriving the commutator, he first calculates

$$ \{u_1 u_2 , v\}_{P.B.} $$ however, ensuring that the ordering of $u_1$ and $u_2$ is preserved. My confusion is that he allows $u_1$ and $u_2$ to commute with $v$ in his derivation. Is this because he is viewing the Poisson bracket in the abstract sense i.e.

$$ F\times F \rightarrow F $$ where $F$ is the set of all functions on phase space?

Replacing the coordinates with ones that are Grassmann-odd valued, would I pick up a minus sign whenever $u_1$ or $u_2$ is commuted with $v$, or would I let the commute exactly like the derivation of Dirac's? I'm assuming I would get the anticommutator but I feel like I'm missing something important.

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1 Answers1

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  1. Ref. 1 is only discussing bosonic/Grassmann-even variables, so the classical variables trivially commute, but Dirac has secretly the corresponding operator identity for the commutator in mind when he writes down the Poisson property/Leibniz rule (5) for the Poisson bracket, and then the order of operators is of course crucial.

  2. The generalization to supernumber-valued (operators) reads $$ [u_1u_2,v]~=~[u_1,v]u_2(-1)^{|v||u_2|}+u_1[u_2,v],\tag{5}$$ where $|\cdot|$ denotes the Grassmann-parity. See also e.g. Wikipedia and this related Phys.SE.

References:

  1. P.A.M. Dirac, Principles of QM, 4th ed, 1958; $\S21$ p. 86.
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