From: Amara Graps (amara@amara.com)
Date: Fri Sep 22 2000 - 19:16:12 MDT
scecir:
>> Otherwise it is not easy to normalize, that is to compute the entropy
>> involved in a quantum measurement.
David Blenkinsop :
>Try looking up the the "complete" version of what entropy is all about
>in Chapter 4 of _Nanosystems_,
I suggest that Drexler might want expand his definition of entropy a
little bit.
In 1998, I heard a talk by Ariel Caticha [1] describing a different
way to treat state vectors. He calls the idea "array entropy", which
was originally introduced by Edwin Jaynes [2], but Jaynes thought
that it was inadequate for the entropy of a quantum system. Caticha,
however, expanded on Jaynes' idea in a nice way. He assigned
amplitudes and wave functions, not just to the system, but to *the
whole experimental setup*.
[In the following, I synthesize a bit from Caticha's article.
For the full article, get the conference proceedings [1] or look at
the xxx server, where he he has two slightly earlier articles available:
http://xxx.lanl.gov/abs/quant-ph/9803086
"Consistency and Linearity in Quantum Theory"
http://xxx.lanl.gov/abs/quant-ph/9804012
"Consistency, Amplitudes and Probabilities in Quantum Theory"]
One of the objectives of quantum theory is to predict the outcomes
of experiments. Amplitudes are used to predict the outcomes of
experiments, where, mathematically, the amplitudes are calculated
via "Hilbert norms" (These norms, or inner products, are the means
to measure the distance between wave functions). Interpreting the
amplitudes can then be used to prove the "Born postulate" provided
that amplitudes are assigned to the experimental setup. Caticha
calls this extra assignment a "constraint" that the amplitudes be
assigned consistently.
NOTE: Caticha says in his appendix that Born's postulate is a
theorem that has been independently discovered several times: A.M.
Gleason, J. Rat. Mech. Anal. 6, 885 (1957); D. Finkelstein, Trans.
NY Acad. Sci. 25, 621 (1963); J.B. Hartle, Am. J. Phys. 36, 704
(1968); N. Graham, in "The Many-Worlds Interpretation of Quantum
Mechanics" ed. by B.S. DeWitt and N. Graham (Princeton, 1973). The
limit N->infinity where N is the number of replicas of the system is
further discussed in E. Farhi, J. Goldstone, and S. Gutman, Ann.
Phys. 192, 368 (1989).
His approach, to consistently assign amplitudes to the experimental
setup, is to assign a single complex number to each setup in a way
that relations among the experimental parts translate into relations
among the corresponding complex numbers. The consistency comes in by
the requirement that if there are two different ways to compute this
complex number, then the two answers must agree.
Caticha uses a path integral approach [3] to cover all possible
combinations of starting points and interactions prior to the
observing time. These lead to the same numerical value for the
amplitude, that is, the outcome of the experiment. These amplitudes
can then be inserted in the quantum mechanical equation of motion-
the Schroedinger equation.
How to get from the Schroedinger equation to the Born postulate? The
Born postulate states that the probability of N independent
particles is the product of the wavefunction (amplitude-squared) for
each of the particles, and the sum of each those is unity. [Note:
here is the connection to the Many Worlds Interpretation.] The
Schroedinger equation is an equation of motion. Caticha shows that
the connection can be made by using a Hilbert norm.
After Caticha shows the connection, he wants to evolve the system in
time, and he does that with an entropy array. Once he applies the
entropy array, then he can measure any observable in the system, and
he says that the time evolution of states is "linear" and "unitary".
(I'm still pondering the significance of those last two things)
EXTRA NOTE: Yes, this is a kind of Bayesian approach. If one adopts
a Bayesian approach to probability, then the Schroedinger wave
equation simply becomes a posterior probability describing
our incomplete information about the quantum system,
rather than wave functions that collapse in reality upon our
observation. It could clear up a lot of confusion. I posted
something to the extropians list about Bayesian approaches last
year, and you can find that message here:
http://www.amara.com/ftpstuff/bayesian.txt
Amara
References
[1] Caticha, Ariel, _Probability and Entropy in Quantum Theory_ in
Maximum Entropy and Bayesian Methods_ (conf proceedings from
MaxEnt'98 conference, Garching, Germany, July 1998), Kluwer
Academic Publ., 1999.
(BTW: This 9page mathematical paper is surprisingly readable..!)
[2] Jaynes, E.T. "Jaynes: Papers on Probability, Statistics and
Stastical Physics," edited by R.D. Rosenkrantz, Reidel, Dordrecht,
1983.
[3] R.P. Feynman, Rev. Mod. Phys. 20, 267 (1948); R.P. Feynman and
A.R. Hibbs, "Quantum Mechanics and Path Integrals," (McGraw-Hill,
1965).
********************************************************************
Amara Graps email: amara@amara.com
Computational Physics vita: finger agraps@shell5.ba.best.com
Multiplex Answers URL: http://www.amara.com/
********************************************************************
"Sometimes I think I understand everything. Then I regain
consciousness." --Ashleigh Brilliant
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