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bryant_j_j@yahoo.com
science forum beginner

Joined: 22 Mar 2006
Posts: 3

Posted: Mon Jun 19, 2006 8:48 pm    Post subject: CCR and unitary evolution

Hi all,

Suppose we have a closed quantum mechanical system which evolves (in
the Heisenberg picture) such that all the required canonical
commutation relations (e.g., [p(t),q(t)]=i\hbar ) are satisfied for all
t >=0. Is the associated evolution necessarily unitary (we don't know
the exact evolution, but only that the commutation relation holds for
all time)? If so, which result states this, and if not are there
counter examples to this?

TIA.

JJ
science forum beginner

Joined: 02 May 2005
Posts: 21

Posted: Fri Jun 23, 2006 10:05 pm    Post subject: Re: CCR and unitary evolution

On Mon, 19 Jun 2006 20:48:29 +0000 (UTC), bryant_j_j@yahoo.com wrote:

 Quote: Hi all, Suppose we have a closed quantum mechanical system which evolves (in the Heisenberg picture) such that all the required canonical commutation relations (e.g., [p(t),q(t)]=i\hbar ) are satisfied for all t >=0. Is the associated evolution necessarily unitary (we don't know the exact evolution, but only that the commutation relation holds for all time)? If so, which result states this, and if not are there counter examples to this? TIA. JJ The devil is inthe details. p and q are unbounded operators, and there

are known pathological examples of p,q pairs formally satisfying
CCR but not unitarily equivalent to the Schrodinger pair.

If you replace CCR by its Weyl's form - then the answer is yes, by the
von Neumann uniqueness theorem. But if you are desperate and ready to
deal with pathological domains - then you can find counterexamples.

More information in Gerard G Emch, "Algebraic methods in statistical
mechanics and quantum filed theory", Ch. 3 and C. R. Putnam,
"Commutation properies of Hilbert space operators and related topics",
Ch. 4.

ark
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