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Sunday, November 25, 2012

Complex-Probability Random Walks and the Emergence of Continuous General-Relativistic Spacetime from Quantum Dynamics


(A post presenting some interesting, but still only half-baked, physics ideas....)

The issue of unifying quantum mechanics and general relativity is perenially bouncing around in the back of my mind.   I don't spend that much time thinking about it, because I decided years ago to focus most of my intellectual energy on AI and understanding the mind, but I can't help now and then revisiting the good old physics problem, and doing occasional relevant background reading....

Of course there are loads of approaches to unified physics out there these days, some of them extremely sophisticated.  Yet I can't help hoping for a conceptually simpler unification.   Here's what I'm thinking today....

I've been enjoying Frank Blume's 2006 paper A Nontemporal Probabilistic Approach to Specialand General Relativity....   It consists of fairly elementary calculations done in pursuit of a philosophical point.  Blume wanted to show that the continuous spacetime assumed in special and general relativity, can be approximated arbitrarily well by discrete random walks.   The subtle point is that these discrete random walks hop around randomly (according to a certain specified probability distribution) not only in space, but also in time.   So Blume's picture has particles hopping back and forth in time, which in his view is in accordance with Julian Barbour's perspective that "physical reality is essentially nontemporal and is best thought of as an ordered sequence of discrete static images" (see Barbour's book  The End of Time).  

I don't feel confident I know how physical reality is "best thought of" ... but I do agree with Barbour and Blume that the view of time as flowing forward from past to future is badly flawed.  This sense of unidirectional time-flow is part of  human psychology, and perhaps part of the dissipative nature of the human mind/body as a macroscopic, thermodynamic system ... but it's not fundamental in the way that people sometimes naively assume.   It's not there in microphysics, either -- at the quantum level the flowing of time from past to future is an alien concept.  If you think this sounds like nonsense, read Barbour's book!

But the philosophy of time is somewhat peripheral to the point I want to make here.   What I've been thinking about is the possibility of replacing Blume's random walk, which is defined in terms of ordinary real-number probabilities, with an analogous random walk defined in terms of complex-number probabilities.   

Saul Youssef, in a series of interesting papers (click here and scroll down to Youssef's name) has shown that if one replaces ordinary real-number probabilities with complex-number probabilities, and adds a few other commonsensical assumptions, then the equations of quantum theory basically pop out.        

This direction of research seems natural once one notes that, according to the basic math of probability theory, there are four options for creating probabilities that obey all the standard probability rules: real-number, complex-number, quaternionic and octonionic probabilities.  Classical physics uses the standard real-number option.  Quantum physics uses the complex-number option.

Ordinary quantum logic uses real-number probabilities, but uses an unusual logic (lattice meet and join on the lattice of subspaces of a complex Hilbert space), which lacks some of the normal rules of Boolean logic, such as distributivity.    Youssef's exotic probability approach retains ordinary Boolean logic rules, but moves to complex number probabilities.   

What I began wondering is: What if you replace Blume's conventional random walk with a random walk in which each movement of a particle is quantified by a certain complex-number probability?

Then a particle may move in various spatiotemporal directions, and there is the possibility for constructive or destructive interference between the different directions.  

And it seems that, in the case where the interference between the different directions cancels out, one would get the same behavior as a real-probability random walk.  

So based on back-of-the-envelope calculations I did the other day, it looks like one can probably get General Relativity to emerge as a statistical approximation to the large-scale behavior of complex-number-probability (quantum) random walks, under conditions of minimal interference.

How far does a perspective like this go, in terms of explaining the particulars of unified physics?  I don't know, and don't seem to have the time to do the rigorous calculations to find out, right now.  But it seems an interesting direction....   If you're a physicist interested in helping work out the details, drop me a line! ...