Posts tagged with systems theory

Both direct sum and tensor product are standard ways of putting together little Hilbert spaces to form big ones. They are used for different purposes. Suppose we have two physical systems…. Roughly speaking, if … a physical system’s … states are either of A OR of B, its Hilbert space will be [a] direct sum…. If we have a system whose states are states of A AND states of B, its Hilbert space will be [a] tensor product….
MEASURE SPACE   disjoint union  Cartesian product
HILBERT SPACE   direct sum      tensor product

John Baez

(Source: math.ucr.edu)

I had judged The Emperor’s New Mind by the negative reviews but never actually picked it up. It has a lot of great stuff, almost like an “early draft” of The Road to Reality.

All I knew about Emperor’s New Mind before was that it invokes quantum mechanics to explain free will, which was perceived as “icky” by people who study the brain. (Despite that, like quantum nonsense, the “greats” of QM—Bohr, Schrödinger—also weighed in with QM/free-will speculations (do you hear me, Conrad&Kochen? Quantum communication folks?) — because, let’s be real here, free will is a millennia-old conundrum and I think we’d all appreciate it if the people who understand compositions of Hilbert spaces weighed in on whether and what the latest “master theory” (bringer of semiconductors = transistors, LCD’s, lasers, MRI/PET and certain polymers/piezoelectrics/other materials) would say about the age-old question)

I got a bit more of the debate whilst reading about pi-1 sentences, which is a computability/knowability/logic dealio. But again, this was the level of “What’s RP’s argument in a nutshell?” rather than “Is here anything worth reading in the 400 pages?”. It’s a lot of good.

Complex systems are ones with a large effective number of strongly-interdependent variables.

This excludes both low-dimensional systems, and high-dimensional ones where the variables are either independent, or so strongly coupled that only a few variables effectively determine all the rest.
Cosma Rohilla Shalizi

(Source: stat.cmu.edu)