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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 *e*^{A}e^{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}^{+B}*e*^{-B}*e*^{-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}^{+B}*e*^{-B}*e*^{-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>*, ..., A*^{n} |0>*,.*..} span the whole Hilbert
space.

*Michael Weiss*

*3/10/1998*