Re: operator satisfying relations [H,T] = i hbar

mingyuming@xxxxxxxxx wrote:
Hi,

I am reading John Baez's writing on energy-time uncertainty and have a
question about his claim of the non-existance of any observable
varaible T satisfying relations [H,T] = i hbar. In fact, for a
classical harmonic oscillator, H=P if we use action-angle variables P
and Q, and {P,Q}=1 as usual, i.e {H,Q}=1. So the Q would satisfy [H,Q]
= i hbar after quantization. But this seems to create a paradox due to
his following argument. I would appreciate any comments or refence
material. Thanks a lot.

Your construction of the supposed paradox is not clear to me, but I
suspect that any apparent contradiction between your example and John
Baez's assertion may be because your H may not be bounded below. His
assertion was that for any H which *is* bounded below, an Hermition Q
satisfying

HQ - QH = i

cannot exist.

There is a sketch of an argument justifying the assertion in a
review of "Quantum Paradoxes" by Y. Aharonov and D. Rohrlich at
www.math.umb.edu/~sp/paradoxes.pdf . It is basically a formal
calculation showing that if Q exists, then the spectral measure (hence
the spectrum) of H must be invariant under translation (hence cannot be
bounded below). The calculation is rigorous for bounded, Hermitian H,
and Q. It is algebraically correct (e.g., satisfies the usual standard
of mathematical rigor of physics texts) for any Hermition H, Q. I have
the impression that the argument may be originally due to Wigner, but
I'm not sure.

Aharonov and Rohrlich seem to assume that for any H, Q will exist,
which the argument shows is false. But that's just my interpretation.
The book is so vaguely written that it's hard to tell what they are
assuming.

I would be interested to hear what others think of this book. There
are two reviews of it on amazon.com, mine and one of Michael Steiner,
who is a serious researcher. He liked the book, giving it a a five-star
rating (the highest). I didn't, and gave it two stars (one step above
the lowest).

.

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