Dear Rob,
> For instance, famously, you can perfectly describe the momentum or the
> position of a particle, but not both at the same time. This is
> Heisenburg's
> Uncertainty Principle.
It is, and the Uncertainty Principle is perhaps the clearest formal
expression we have of the homely truth "you can't know everything at
once". A more classical version of the the same argument would follow
from the observation that, if you had a machine that tried to run a
Laplacian / deterministic model of the universe, its physical size and
the speed of light would limit the amount of information it could
synchronously process.
> So it is not so much the fact of going from a continuous quality to a
> discrete
> quality which is interesting, it is the necessary incompleteness of
> description in terms of discrete qualities abstracted from a
> distribution
> which is where I think we should be focusing, in analogy with the
> Uncertainty
> Principle of physics.
Is this similar to asking whether all such quantization is a "lossy"
transformation? Is this what you mean by incompleteness?
> Dominic, I have only read the publically available chapter of your
> book. You
> mention a "vector model" for quantum mechanics. Do you have anything
> on the
> Web which talks about that? I can only recall ever having met
> descriptions of
> QM in terms of functions.
The article at http://plato.stanford.edu/entries/qm/ looks like a good
place to begin for QM and vectors.
In broad strokes, the history of vectors and functional analysis became
very closely linked in the 1840s and 1850s, partly through Hamilton's
work on quaternions and the theory of analytic functions on 4-space.
Functions over the real numbers form a vector space - you can add two
functions together, and multiply any function by a scalar. As a result,
mathematicians came to realize that Fourier analysis could be described
in vectors - each of the functions sin(nx) and cos(nx) (for x a real
number, n an integer) is a basis vector, and any piecewise smooth
function can be expanded (uniquely) as a vector, using these functions
as a basis. The Fourier series coefficients are thus interpreted as the
coordinates of a vector in this basis. This vector space is clearly
infinite-dimensional, because a Fourier series expansion can be
infinitely long. (Note again that this means you will never work with
complete information once you've quantized your functions.) Make your
functions complex-valued, and introduce a metric based on
complex-conjugation, and you've got Hilbert spaces, around 1900 I
think.
In the 1930's, Paul Dirac, John von Neumann, and others, used this
formulation of functional analysis as the basis for formal quantum
theory, much of which boils down to the analysis of self-adjoint
operators on Hilbert space. Each function is a state-vectors, can be
normalized and operated on. The resulting operator algebra (group of
linear transformations under composition) is non-commutative, and this
how the formal theory accounts for the Uncertainty Principle - the
lower bound on the uncertainty of two observables is given by the
magnitude of the commutator of their self-adjoint matrices.
Clear as mud? ;)
> I agree completely with your message, but would only add that while
> quantum
> analogies can be very informative for lexis, where I think it really
> gets
> interesting is in syntax, which responds very nicely to a kind of
> "quantum"
> analysis in terms of generating new quantum qualities (particles?), a
> new one
> for each new sentence.
This may be partly because composition is so far modelled more robustly
in syntax that it is in (parts of) semantics? Just trying to figure out
what compositional rules apply a list of noun-noun compounds extracted
from a corpus is very hard - and this is just combining 2 "particles"!
Some of the most interesting structures that arise in QM involve
entanglement, and I dare say that some of the structures in syntax are
as rich in this "multiply composed, new particles / systems arise"
property. I don't have the expertise to do any proper analysis here,
though.
Best wishes,
Dominic
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