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Michael Weiss: Dangerous signs of normality

So you replaced the Hamiltonian [...] by its normal-ordered form
So that's what I was doing. Though actually I had already noted dangerous signs of normality in my thought processes, when trying to prove the Baker-Campbell-Hausdorff formula. After all, if we set A = za*/sq2,
B = -z*a/sq2, then eAeB is the normally ordered form of eA+B.

Speaking of Baker-Campbell-Hausdorff: is there a slick proof of their formula, or does one just have to fight it out with Taylor series? I did notice one thing: we can prove pretty easily that eA+Be-Be-A is a number (i.e., a multiple of the identity operator) in the case at hand. First we show that this product of exponentials commutes with A, using the ``derivative = bracket'' trick:

[A, eA+Be-Be-A] = 0

I omit the details (which are not messy), but it's essential here that [A,B] commutes with A and B. Next we appeal to Schur's lemma. The irreducibility hypothesis is satisfied since {|0>, A|0>, ..., An |0>,...} span the whole Hilbert space.

Michael Weiss