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Natural Science Forum / Physics / Research / October 2004Does decoherence explain phase space emergence from Hilbert spaces?
Hi, I am thinking about classical-quantum correspondence and the decoherence theory. I think that any theory that shows how classical emerges from the quantum must explain the following: 1) Why are objects around us not in a superposition of states? 2) How phase space emerges from infinite dimentional Hilbert spaces 3) How to derive classical Hamilton's equations from Schrodinger's equation. It seems to me that decoherence only tries to solve the first problem. Or has there been work in decoherence theory about 2) and 3) above? What are the theories that explain 2) and 3) and what is their relationship with decoherence theory? regards, Kanwar > Hi, > [quoted text clipped - 16 lines] > regards, > Kanwar First of all, I don't think the idea is to have classical phase space emerge from Hilbert space. In a sense, the notions of classical phase space are already present in QM if you use Wigner's phase space distribution approach. Due to the commutivity rules of QM, however, this can be somewhat different from the standard classical theory. For instance, we cannot specify actual points in phase space, but only finite closed regions dictated by the uncertainty principle. Thus, QM phase spaces are fundamental examples of what is referred to as non-commutative geometry. But interestingly, most of the fundamental ideas are still there in one form or another. If you are not familiar with Wigner's ideas, you would definitely benefit from looking them up. Googling on "Wigner" and "phase space" would be good place to begin. >>I am thinking about classical-quantum correspondence and the decoherence >>theory. I think that any theory that shows how classical emerges from [quoted text clipped - 15 lines] > distribution approach. Due to the commutivity rules of QM, however, > this can be somewhat different from the standard classical theory. In the classical limit hbar to 0, this difference disappears. Arnold Neumaier > Hi, > [quoted text clipped - 13 lines] > What are the theories that explain 2) and 3) and what is their > relationship with decoherence theory? 2): Coherent states. 3): Nelson's stochastic mechanics In many cases, arbitrary states decohere into coherent states. Arnold Neumaier Hi, Thanks to everyone who answered my question. It made me think more. Wigner distribution seems to be an attempt to study quantum mechanics in terms of phase space which is already assumed to exist. And coherent states seems to be the closest quantum approximation of a body with a definite position and momentum. My point here is that using these we cannot hope to get back phase space. This is because phase space was the input to both these concepts and getting them back is no surprise. My question is : if we start with a purely quantum universe with only Hilbert spaces then how does position and momentum become the basis of our existance. Are these the preferred basis and does this basis emerge after selection by the envirioment ? Would an imaginary "animal" with atomic dimentions know anything about position and momentum? If yes then would these concepts be a part of a large number of others with no special status. And if not then how do we emerge with an obsession with these concepts? regards Kanwar Arnold Neumaier wrote: > Igor wrote: > > Kanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<4149914C.FAC80FAA@cadence.com>... [quoted text clipped - 22 lines] > > Arnold Neumaier > Hi, > [quoted text clipped - 10 lines] > our existance. Are these the preferred basis and does this basis emerge > after selection by the envirioment ? My own take on this is that this is an arbitrary basis that we've only chosen since it goes over rather nicely to quantities that we can measure with no real problem as h approaches zero. Frankly, I'm not really sure what other basis would be available to us, but since the wave function evolves in a completely deterministic way, the elements of the Hilbert space as well as their time derivatives may be a better choice, although these are technically nonobservable. Something along the line of the Bohm analysis may be in order here. > Would an imaginary "animal" with atomic dimentions know anything about > position and momentum? If yes then would these concepts be a part of a > large number of others with no special status. And if not then how do we > emerge with an obsession with these concepts? I doubt that such a creature would have any understanding of our concepts of mechanics at all. I do wonder whether one would be able to "see" the wave function at that level. If so, that would probably be the basis of their own mechanical universe. If not, there may not be any real answer to that question. Again, I think the only reason we insist on talking about position and momentum is because they are fundamental quantities of classical mechanics and we prefer to understand the quantum world in terms of things we already know. But this may not be the best approach, however. >>Would an imaginary "animal" with atomic dimentions know anything about >>position and momentum? If yes then would these concepts be a part of a [quoted text clipped - 4 lines] > concepts of mechanics at all. I do wonder whether one would be able > to "see" the wave function at that level. Someone competent wondered already in 1940, and turned it into a bunch of nice stories: Mr Tompkins in Paperback : Comprising 'Mr Tompkins in Wonderland' and 'Mr Tompkins Explores the Atom' by George Gamow http://www.amazon.com/exec/obidos/tg/detail/-/0521447712/ Happy reading! Arnold Neumaier > Thanks to everyone who answered my question. It made me think more. > Wigner distribution seems to be an attempt to study quantum mechanics in > terms of phase space which is already assumed to exist. No. No classical input is needed. One can get Wigner distributions from quantum theories which never mention the classical theory. Wigner distributions don't even relate directly to phase space since they are not probability distributions on phase space. Once you have canonical commutation relations, you get from it a symplectic structure and associated Wigner distributions. A classical phase space picture only arises in the limit hbar to 0. > My question is : if we start with a purely quantum universe with only > Hilbert spaces then how does position and momentum become the basis of > our existance. Are these the preferred basis and does this basis emerge > after selection by the envirioment ? Position and momentum are not a 'preferred basis' in any technical sense, in particular not in the sense of decoherence theory. The form of the interaction with the environment decides upon whether there is a preferred set of states; sometimes it is momentum states, sometimes it is position states, sometimes coherent states. Arnold Neumaier > Thanks to everyone who answered my question. It made me think more. > Wigner distribution seems to be an attempt to study quantum mechanics in > terms of phase space which is already assumed to exist. And coherent > states seems to be the closest quantum approximation of a body with a > definite position and momentum. It's worth noting that there are other sorts of distribution along the lines of the Wigner -- notably the (Glauber-Sudarshan) P and (Husimi) Q ones. Further, there are versions of these with expanded domains, as e.g. the complex-P or positive-P distributions. The P distributions are very useful in optics -- they use a coherent state basis, so a coherent state is represented (in all its intrinsic quantum uncertainty) by a delta function; the positive-P is nice since it is guaranteed to be positive definite, so you can treat it like a probablility distribution (unlike e.g. the Wigner or P), and generate stochastic equations for numerical solutions. You can also go even further, adding "gauges" to optimise a model according to what you are tring to calculate. Peter Drummond at UQ has done a lot of this stuff.  Signature---------------------------------+--------------------------------- Dr. Paul Kinsler Blackett Laboratory (QOLS) (ph) +44-20-759-47520 (fax) 47714 Imperial College London, Dr.Paul.Kinsler@physics.org SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/ |
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