Michael Weiss: Vacuum energy,  Schmotons  John Baez: Messing around

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 e^Ae^B is the normally ordered form of e^A+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 e^A+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, e^A+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.
 

3/10/1998