tag:blogger.com,1999:blog-11295132.post114840609968385443..comments2016-09-24T21:29:58.885-07:00Comments on A Neighborhood of Infinity: Quantum Mechanics and Probability TheoryDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-11295132.post-1148842283878547372006-05-28T11:51:00.000-07:002006-05-28T11:51:00.000-07:00Hmmm...I'm not right. In particle physics we're lo...Hmmm...I'm not right. In particle physics we're looking at representations over C so in fact we're probably looking at a Q⊗C (⊗ over R) valued wavefunction.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1148841632665416722006-05-28T11:40:00.000-07:002006-05-28T11:40:00.000-07:00theo,I'd have thought the |.|² norm would be fine ...theo,<BR/><BR/>I'd have thought the |.|² norm would be fine for reals and quaternions.<BR/><BR/>I think you can get something like quaternion probabilities if you consider physical systems with SU(2) symmetry. Any time you have a particle whose internal symmetry carries the adjoint representation of SU(2) (which is basically the vector space of quaternions x acted on by the unit quaternions via x→qxq^-1) then we have a quaternion valued wave function. (You'd get the adjoint representation if the particle was a bound state of a pair of particles carrying the fundamental representation.) The components of the quaternion tell you about the particle's composition - but if you ignore the particle's inner state and just consider its distribution in space then you can think of it having a quaternion valued 'amplitude' distribution. (I think that's right, I did my particle physics courses nearly 20 years ago...)sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1148796757605552272006-05-27T23:12:00.000-07:002006-05-27T23:12:00.000-07:00I see the main difference as whether the coefficie...I see the main difference as whether the coefficients are complex or real. With, in fact, a little bit more of a twist: you point out both this difference, and that in QM, as opposed to Classical, the coefs are squared when normalizing the states.<BR/><BR/>So, it seems (and I'm thinking out loud more than making any definite or well-defined claims) that we jump all the way from the rig of positive integers to the ring of complex numbers, and over each impose a natural "unit sphere" condition.<BR/><BR/>(In fact, it's there's also something projective going on: we don't, in QM, care about the total phase of our system.)<BR/><BR/>So, one natural question: what is the proper in-between "probability" theory allowing positive and negative, but only real, probabilities? What's the correct norm?<BR/><BR/>Going the other way: what about probability theories with coefficients in the Quaternions?Theohttp://www.blogger.com/profile/10449695704601441213noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1148594095488887922006-05-25T14:54:00.000-07:002006-05-25T14:54:00.000-07:00Some things Oded says are wrong. Like "For example...Some things Oded says are wrong. Like "For example, it says that you cannot make non-Unitary transformations, but this by itself does not mean that you can effect any Unitary transformation that you want." isn't quit correct. There are "universal" quantum logic gates that allow you to get as close as you like to any unitary transformation. However, I still agree with the general tenor of his comments. (Though this is mostly a different topic to what I posted on.)sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1148533208922914152006-05-24T22:00:00.000-07:002006-05-24T22:00:00.000-07:00fair enough. overall I think your intuition is cor...fair enough. overall I think your intuition is correct, in that quantum states, by virtue of being complex states, are more than just higher dimensional classical states. <BR/><BR/>Another discussion you might be interestsed in reading is <BR/><BR/>http://www.wisdom.weizmann.ac.il/~oded/on-qc.html<BR/><BR/>on quantum computing vs classical randomized algorithms; it boils down to similar arguments.Sureshhttp://www.blogger.com/profile/16443460499476270978noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1148492249403981652006-05-24T10:37:00.000-07:002006-05-24T10:37:00.000-07:00In more detail (and it's a long time since I worke...In more detail (and it's a long time since I worked through the proof of Bells' theorem so I may be slightly off) what Bell's theorem shows is that there is no way of interpreting the results of the EPR experiment as coming from a conventional classical setup with some hidden unknown variables described by some (conventional) probability distribution. If you reject anything other than this kind of approach then you are forced to accept some kind of spooky action at a distance.<BR/><BR/>If you look at the discussion <A HREF="http://en.wikipedia.org/wiki/Bell%27s_theorem" REL="nofollow">here</A> you can see how the 'paradox' comes about because we can craft states where the coefficient of a basis element can be negative. What Bell's theorem says is not that classical systems can't be entangled, but that there is a limit to how correlated such systems can be. By exploiting the fact that amplitudes can be complex you can make quantum systems much more correlated than classical ones. And that's why I see this kind of non-convexity as much more interesting than entanglement itself.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1148448456568114412006-05-23T22:27:00.000-07:002006-05-23T22:27:00.000-07:00Well I see entanglement as exactly analogous to jo...Well I see entanglement as exactly analogous to joint probability distributions of non-independent variables. Bell's theorem shows that those joint probabilities can't possibly obey the usual rules of conventional probability. So I don't see it as fundamentally about entanglement.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1148436242411254162006-05-23T19:04:00.000-07:002006-05-23T19:04:00.000-07:00But isn't Bell's theorem really about entanglement...But isn't Bell's theorem really about entanglement ? And since that is really what shows the distinctiveness of QM, isn't entanglement key ?Sureshhttp://www.blogger.com/profile/16443460499476270978noreply@blogger.com