PI talk on Feynman Checkerboard Model and the wave function

[CarlB]

The Perimeter Institute has a talk on the Feynman chessboard / checkerboard model scheduled for a few days from now. This will be recorded and you can play it back, sometime after November 18th. Links:

Speaker(s): Garnet Ord - Ryerson University
Ord's website: http://www.scs.ryerson.ca/~gord/

Abstract: Conventional quantum mechanics answers this question by specifying the required mathematical properties of wavefunctions and invoking the Born postulate. The ontological question remains unanswered. There is one exception to this. A variation of the Feynman chessboard model allows a classical stochastic process to assemble a wavefunction, based solely on the geometry of spacetime paths. A direct comparison of how a related process assembles a Probability Density Function reveals both how and why PDFs and wavefunctions differ from the perspective of an underlying kinetic theory. If the fine-scale motion of a particle through spacetime is continuous and position is a single valued function of time, then we are able to describe ensembles of paths directly by PDFs. However, should paths have time reversed portions so that position is not a single-valued function of time, a simple Bernoulli counting of paths fails, breaking the link to PDF's! Under certain circumstances, correcting the path-counting to accommodate time-reversed sections results in wavefunctions not PDFs. The result is that a single `switch' simultaneously turns on both special relativity and quantum propagation. Physically, fine-scale random motion in space alone yields a diffusive process with PDFs governed by the Telegraph equations. If the fine-scale motion includes both directions in time, the result is a wavefunction satisfying the Dirac equation that also provides a detailed answer to the title question.
PI talk link: http://pirsa.org/08110045

A related arXiv paper by Garnet Ord is:
http://arxiv.org/abs/quant-ph/0411005

A problem with the above paper is that it seems to be in 1+1 dimensions in stead of 3+1.

Other Ord articles:
http://arxiv.org/find/quant-ph/1/au:+Ord_G/0/1/0/all/0/1

Peter Plavchan's paper giving the 3+1 version of Feynman's checkerboard:
http://www.brannenworks.com/plavchan_feynmancheckerboard.pdf

As some of you know, I'm a big fan of the Feynman checkerboard model. It can be generalized to 3 dimensions. I wrote up a blog post with a link to an article showing the generalization earlier this year:
http://carlbrannen.wordpress.com/2008/04/21/481/
It discusses the Lorentz violation that comes with taking the idea from 1+1 dimensions to 3+1.

 

[MTd2]

Lol Carl, I was googling for this stuff right before you posted

I was really looking forward to study these models since you said in your post that it provided a 2 branched solution, one localy Lorentz and the other not. It blew my mind because it provided a way so natural to justify the wick rotation in a so natural way, that it must lead to much deeper insights. For example, this thing works as sounds propagating in a solid, so, I was thinking that if you identified the lorentz invariant slice of this sound wave as the space-like slice that included a neighbohood of an observer inf GR, you maybe one could provide a clear framework to study quantum fields in curved geometries.

I've been thinking about sci fi ideas like identifying the lattice as a spim foam and if you could define holes in the lattice, pictographicaly view as in condensed matter. Except that in a checker board, you can speak in negative probabilities, that is, the probability of not finding something. So, maybe that negative probability, on this kind of crazy lattice, could mean that the negative proability ghosts could be just anocupied states, or negative bounded energy states, in the lattice.

As for the ghost thing, I was thinking that, since it exchages the fermion boson commutative sign, I was thinking of them as living in the lattice, by relating the change of interpretation from sound to space-like surface picture.

 

[CarlB]

Garnet Ord's lecture is up on the PI at the above link. It had me guffawing out loud, if not laughing. The PI audience basically cut him to pieces. But they were very polite about it. The whole thing looked like a not very well put together PhD candidate botching his defense.

He glosses over the 1+1 dimension problem comes up around 45 minutes into the lecture and gets into trouble with the audience who point out that he can never get the whole Dirac equation, but is forever restricted to just two dimensions (i.e. one space and one time). It's like he doesn't see that there is a real problem getting from 1+1 to 3+1. And it seems that the audience is unaware of the rather obscure methods of getting from 1+1 to 3+1 with the Feynman checkerboard model. He talks about the idea being compatible with Bell's inequality and the EPR without seeming to realize that all these examples of quantum oddity require more than 1+1 dimensions in order to be observed. So instead of showing analytically that these things work, he hand waves through it, but his hand waving is incompatible with his equations (which are always 1+1 or even 2+0 dimensions).

Worse was around 64 minutes where he gets into trouble with configuration space for multiple particles. This is a really big problem for ideas like this.

Of course my solution to the configuration space problem is to work in density matrix formalism instead of wave function formalism. And for me, time is a much stranger object than the usual linear object. This is hinted at around 77 minutes, when they start talking about proper time.

 

[Maaneli]

Dear Carl,

Sorry to bump this up so late but I couldn't resist responding after finding your posts.

He glosses over the 1+1 dimension problem comes up around 45 minutes into the lecture and gets into trouble with the audience who point out that he can never get the whole Dirac equation, but is forever restricted to just two dimensions (i.e. one space and one time). It's like he doesn't see that there is a real problem getting from 1+1 to 3+1.

Actually, Ord and his collaborates have thought quit extensively about how to go to 3+1 dimensions, and he has in fact (I know this from private communication with him) recently completed the 3+1 dimensional entwined path model, but it is not published yet. Here is an earlier preliminary paper that made some progress in that direction:

On the Dirac Equation in 3 + 1 Dimensions
Ord G. N. and Mckeon D. G. C.
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WB1-45R7GBB-2J&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=b4910a8a1968d90c84a5a931f0c2979e

Your comment may have also been influenced by a comment made by one of the audience members, Rob Spekkens, who argued, with his example using the Bloch sphere, that Ord cannot not obtain both the real and imaginary parts of the total solution to the 1+1 dimensional Dirac equation, but only the real or imaginary parts. The problem with Rob's comment is that it seems he was failing to appreciate that Ord's chessboard model is in 1+1 dimensions, where, as is well known, there is no spin. Consequently, the wavefunction solution that one can get can be real or complex, depending on the formulation and it is a two component wavefunction. When Rob mentioned the Bloch sphere, he was probably thinking of spin with the expectation that the solutions of the Dirac equation should give you the whole of the sphere. This is true in 3D but not in 1D. The 1D version misses the 'x' and 'y' directions that would give you the sphere essentially as a surface of rotation.

He talks about the idea being compatible with Bell's inequality and the EPR without seeming to realize that all these examples of quantum oddity require more than 1+1 dimensions in order to be observed.

I think it is important to realize that entanglement nonlocality is certainly possible for wavefunction solutions to the 1+1 Dirac and Schroedinger equation. EPR entanglement is in fact merely a consequence of the Heisenberg uncertainty relations between x and p coordinates for wave packet solutions to the Schroedinger equation. And Ord's model can, in principle, describe these phenomena. For Bell inequality violations, he would have to extend to 2+1 and 3+1 dimensions. But in fact he and his collaborators already have a 2+1 dimensional extension of their model for the Schroedinger equation, and the 3+1 version is a straightforward generalization:

Random walks and Schrödinger's equation in dimensions
G N Ord et al
http://www.iop.org/EJ/abstract/0305-4470/30/3/008

And, as I already mentioned, they already have a 3+1 dimensional extension of the model for the Dirac equation. It's just a matter now of pushing his model to 2 particle situations.

Worse was around 64 minutes where he gets into trouble with configuration space for multiple particles. This is a really big problem for ideas like this.

I don't know why you think this, but I don't see any "big problems" with multiple particles for ideas like these. For example, one can certainly construct a 2-particle diffusion equation from Einstein's random walk model applied to two particles on the same spacetime with the same initial conditions. The resultant transition probability will be a function on a 2D dimensional space, where D is the number of spatial dimensions. And for N particles, it would just be a function on N*D dimensional space. The extension to Ord's entwined path model would be straightforward, where one just allows the random walks of each particle in the N particle system to 'double back' on themselves, so that the resultant wave equation is an N particle Dirac equation instead of an N particle Telegraphers equation (in the relativistic case).

 

[CarlB]

Dear Maaneli,

Thanks for bringing this up. Too bad his paper isn't on arXiv or apparently available anywhere else on the web. Next time I'm at a university, I'll download the paper and read it.

I'm a believer in Feynman's checkerboard model so I know that there are solutions to the problem of going to 3+1 dimensions. I was looking forward to his lecture. My complaint with Ord's lecture at PI was that he botched it, not that the subject is wrong. I doubt that anyone decided to further explore the subject after watching that lecture.

I believe that the 3+1 Feynman checkerboard is related to a paper I'm trying to get published on Foundations of Physics proposing that spin-1/2 arises from a higher energy version of spin that acts more like position. The paper is discussed on Tommaso's blog:
http://www.scientificblogging.com/q...rannen_position_spin_and_particle_generations

Carl