Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Friday, March 23, 2007

Bohmian Bloom

Carl Brannen appears to have posted nibbles all over the place regarding a new way of looking at the delta parameter in the 3x3 mass matrices, which should bring a smile to the face of the Bohmian mechanics. The nice thing about a good theory is that everybody is happy.

It involves exchanging a classical function $\psi$ for an exponentiated (quantum) version. For us, as always, this is about the profound interaction of addition and multiplication thought of as monads. (This came up recently in our discussion of the Riemann hypothesis).

And for some fun, here is my stylised version of Kuperberg's generalised 6j symbol, from the spider paper. The mushy rectangles are Jones-Wenzl idempotents.


Blogger CarlBrannen said...

Bohmian mechanics take the \psi of the Schroedinger equation and rewrites it as \psi = R e^{iS} where R is the square root of probability density.

Let E be energy, then statistical mechanics, probabilities are proportional to exp(-E/kT). Put that back into the Bohmian prescription and you've got

\psi = e^{ -E/kt + iS }.

March 23, 2007 6:23 PM  
Blogger Mahndisa S. Rigmaiden said...

03 23 07

Hidden degrees of freedom bother me, which is why I wonder...But who knows!

March 23, 2007 7:34 PM  
Blogger Doug said...

1 - Hi Kea, your image of 'Kuperberg spiders' reminds me very much of Petri Nets. I suspect a relationship.

There are many images of Petri Nets on Google Images. Here is one example, with the basics.

Mechatronics, Version 1.0, August 31, 2001, Copyright, Hugh Jack 1993-2001 [Grand Valley State U]

2 - Hi Mahndisa,
Degrees of freedom [dimensions] are often treated as strategies in various forms of game theory such as Math Plus Algebra.

March 24, 2007 6:05 AM  
Blogger Mahndisa S. Rigmaiden said...

03 23 07

Doug I know that! I was referring to the hidden degrees of freedom that Bohmian mechanics uses as a backbone. I don't think it is sensible and have problems with the interpretation of quantum theory. I figure simplicity is the way to go but hidden degrees of freedom are not simple enough, I fear. See this link from Stanford Encyclopedia of Philosophy with general overview of the theory plus objections.

Lastly, degrees of freedom are a bit more than dimensions. From fractal standpoint, the degree of freedom as a dimensional parameter makes little sense, which is why I prefer to discuss Hausdorff dimensions rather than box dimensionality.

March 24, 2007 6:28 AM  
Blogger Kea said...

Hi all. Doug, it would not be surprising to find concurrency ideas in these categorified knot systematics. And I agree with Mahndisa that conventional 'hidden degrees of freedom' are yucky. Fortunately, as I understand it, this is not at all what Carl is talking about. The usual cheap idea still imagines an objective space on which matter is placed. But here the underlying classical reality is much more subtle: it emerges from the world of measurement as a shared reality between observers. The classical functions that appear must be very, very a zeta function is special.

March 24, 2007 11:03 AM  
Blogger L. Riofrio said...

Another neat post. The diagrams ar particularly fun once again. Something like this coud lead to better theiories of atomic structure.

March 24, 2007 12:05 PM  
Blogger Kea said...

Indeed, Louise. It is difficult to model some of the wonderful new materials being made in the lab with a rough-and-ready standard model. It is time we had something better.

March 24, 2007 12:28 PM  

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