tag:blogger.com,1999:blog-11295132.post8906909326265082919..comments2015-07-21T17:35:21.901-07:00Comments on A Neighborhood of Infinity: Monads, Vector Spaces and Quantum Mechanics pt. IIDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-11295132.post-38627611820463885182011-10-26T08:05:23.487-07:002011-10-26T08:05:23.487-07:00Quantum mechanics can already be seen as a theory ...Quantum mechanics can already be seen as a theory of projection-valued measures (ie. a theory where probability measures take "non-commuting values"). This point of view has been well established since Von Neumann.<br /><br />Answers to questions "is my spin up" or "is the particle within the Borel set A" are answered with a closed subspace (equivalently with the projection onto this closed space), such that familiar laws of measure theory hold : P(\cup_i A_i) = \sum_i P(A_i) for disjoint A_i, P(never) = the zero projector, P(always) = identity.<br /><br />Such projection valued measures naturally yield self adjoint operators (and conversely, by the Spectral Theorem) which are the "observables" that we know about.<br /><br />Heck, one could even speak of toposes here, with the projections of some fixed Hilbert space as the subobject classifier and Hilbert tensor products as products (objects, ie. state spaces are Hilbert spaces), particular subspaces as equalizers, C as an initial object, etc. <br /><br />Projection valued measures form some sort of monad: a measure is a device which integrates functions, its type is something like (a -> r) -> r (r fixed, here to a vector space of projectors), and the "monad of measures" is therefore a kind of Cont monad...<br /><br />This should perhaps make sense in Haskell. Thanks again for this blog.kebabnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-17673320138805977802010-06-28T04:19:34.730-07:002010-06-28T04:19:34.730-07:00Really nice review! Unboxing is the best part of g...Really nice review! Unboxing is the best part of getting something! thanks for a nice post.infinity downlinehttp://infinitymakesmoney.com/overview.phpnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-65956062695917664312007-03-06T15:26:00.000-08:002007-03-06T15:26:00.000-08:00I've met Youssef's stuff before though I've not re...I've met Youssef's stuff before though I've not read it properly. But I do agree very strongly with his opening sentence<BR/><BR/>"If it werenâ€™t for the weight of history, it would seem natural to take quantum mechanical phenomena as an indication that something has gone wrong with probability theory and to attempt to explain such phenomena by modifying probability theory itself, rather than by invoking quantum mechanics."<BR/><BR/>Except that it seems to me that quantum mechanics <EM>is</EM> a modified probability theory. Skimming ahead I did notice that like me, he eliminates mixed states. I did it more out of computational convenience that anything - but on thinking further I suspect I'm doing the same thing as Youssef. To be honest, I wasn't trying to do anything non-standard, I just wanted to do some textbook standard QM and show it it formally looks just like probability theory to the point where you can share code between a probability and quantum monad.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-61326136415913097992007-03-06T14:59:00.000-08:002007-03-06T14:59:00.000-08:00This reminds me of Youssef's theory that bayesian ...This reminds me of Youssef's theory that bayesian probabilities should actually really be complex numbers, and that quatum phyisics makes more sense if they are. I've never quite got my head round it, though. You can find it via Youseff's web page:<BR/>http://physics.bu.edu/~youssef/Alexnoreply@blogger.com