# Dirac equation from random walk

• CarlB
In summary, the Dirac equation can be obtained as the result of a random walk. Ord's papers provide a description of the procedure. The relationship between statistical mechanics and quantum field theory is explored by analogy to field theory and stat mech. But the analogy ends when one is forced to go beyond the Wiener measure. Feynman diagrams provide a more simple and elegant approach to combining massless propagators into a massive propagator. The method for doing so is similar to that which is obtained if one assumes a Higgsless mass mechanism. This mechanism will violate unitarity.f

#### CarlB

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I was unaware that one could obtain the Dirac equation as a result of a random walk. I believe that this has been done by other researchers, but I found it by Ord's papers:
http://arxiv.org/find/quant-ph/1/au:+Ord_G/0/1/0/all/0/1

Anyone else find these interesting?

My interest is due to the relationship between statistical mechanics and quantum field theory. I'd love to be able to interpret QM as a classical statistical theory for an unusual variety of space-time, as suggested by the author in the above papers.

Carl

It is effectively interesting. I don't own the necessary scientific package to help you, but visit for example http://www.calphysics.org You will find a lot of articles in the direction of your research.

The analogy between field theory and stat mech ends as soon as you are compelled to go beyond the Wiener measure into the murky waters of functional measure spaces.

Haelfix said:
The analogy between field theory and stat mech ends as soon as you are compelled to go beyond the Wiener measure into the murky waters of functional measure spaces.

If one concludes that time is more complicated than the usual single dimension, the analogy can survive.

In the example of the random walks adding up to the Dirac propagator, the problem with statistical mechanics is that it doesn't allow a single element of a statistical ensemble to specify the sum of a collection of random walks. But that's a problem with our usual understanding of time (and therefore what an element of an ensemble that evolves in time must be), not our usual definition of statistical mechanics.

Ensembles of ensembles. Sort of makes one think of the old second quantization.

Carl

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I found Tony Smith's generalization of the Feynman chessboard model to 4D to be interesting:
http://www.arxiv.org/abs/quant-ph/9503015

There is also a connection to the "Zitterbewegung" model of David Hestenes:
http://modelingnts.la.asu.edu/html/Impl_QM.html [Broken]

But I believe that all these things can be accomplished more simply by Feynman diagrams. One begins with a massless Dirac propagator. This is easy to derive, it's the simplest wave equation that one can write in Hestenes' geometric algebra:

$$\nabla \psi = 0$$

where

$$\nabla = \hat{x}\partial_x + \hat{y}\partial_y + \hat{z}\partial_z + \hat{t}\partial_t$$

and $$\hat{x}, \hat{y}$$ etc. are the canonical basis vectors that generate the Clifford algebra (which Hestenes calls the "Spacetime Algebra" or STA).

To get the massive Dirac propagator, one simply assumes an interaction between the left and right handed particles. There are two Feynman diagrams; both are trivial. The first is a left handed particle changes into a right handed particle. The second is a right handed particle changes into a left handed one. In both cases, the momentum is reversed but the spin direction is preserved. The vertex is given a value equal to the mass of the complete particle. For example, the mass of the electron.

This method for combining the massless propagators into a massive propagator is similar to what you would obtain if you assumed a Higgs mechanism and then deleted all the Higgs propagators. Note that this is not the usual "Higgsless" mass mechanism as illustrated in Fig 1 of:
http://arxiv.org/PS_cache/hep-ph/pdf/0508/0508185.pdf [Broken]

But similar to the Higgsless mass mechanisms, this mechanism will violate unitarity.

Carl

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