Double-Slit experiment in Quaternion Quantum Mechanics?

[robphy]

I'm not sure if "Quantum Physics" (or "Beyond the Standard Model") is the appropriate place for this question. Please move if necessary.

What does "Quaternion Quantum Mechanics" say about the famous double-slit experiment?

Does it make the same quantitative prediction as standard [Complex] Quantum Mechanics?
Can anyone provide a specific reference?

I just checked out my library's copy of Stephen Adler's (1995) "Quaternionic Quantum Mechanics and Quantum Fields"... looking for a statement about this... but I haven't found it yet. I might start tracing back through the various references... Finkelstein-Jauch (1960s), Birkhoff-vonNeumann (1930s), can you suggest any others?

 

[sweetser]

The double slit experiment

Hello Robphy:

A modern reference not on your list is "Quaternion Quantum Mechanics: Second Quantization and Gauge Fields" by L. P. Horwitz and L. C. Biedenharn, Ann. Phys 157:432, p432, 1984. That does not look like it addresses the double slit experiment, but may be worth your time to scan.

In "Quantum Mechanics of Fundamental Systems 1" edited by Claudio Teitelboim, on p. 5, of Adler's contribution to the book, he makes the point that quantum mechanics over the real field would not have "interesting" quantum mechanics, the stuff of interference that shows up in the two slit experiment. Once you have a complex or quaternion valued wave function, there will be interference.

My guess is there is no reference that addresses quaternions and the double slit experiment as directly as you hope. The rest is my perspective on the topic.

The double slit experiment has far too much focus on the slits, not enough on the source. The slits are a way of collecting the data. If there is only 1 slit, you can still see the complex or quaternion valued wave function through quantum diffusion (basically interference patterns near the edge of the main signal). If there are a thousand slits, you see the interference pattern. The number of slits is irrelevant. It is sad in my opinion how much time is spent discussing the slits instead of the source, the object which is under study.

What is essential is that the source is coherent. Incoherent sources give a Gaussian distribution, so the word coherent cannot be omitted (but often is). Coherent means the source has some order to it. Most often the source is coherent in space, so it passes through a single slit before being shown at the single or multiple slits. The source can also be coherent in time (but I'm not sure how to do that). A source can be coherent in both space and in time, and is known in the lab as a laser. In that coherent source, there will be particular places in time and space where there are zero photons, and other places where there are many photons. The double slit experiment allows us to see these spacetime locations of zero photons (destructive interference) and places with the most photons (constructive interference). The fact that we must know about space and time is what at a minimum means we must use complex numbers - time is not space, but the two are related to each other algebraically - or using quaternions.

A quaternions is three complex numbers that share the same real number. The complex numbers are a subgroup of quaternions. Absolutely anything that can be done with complex numbers is possible to do with quaternions of the form (a, b, 0, 0) because they are the same darn thing. Stephen Adler does not go for "all quaternions, all the time" because quaternion analysis is in an anemic state (I may have made real progress on that topic, but that would require a long digression). There should be quaternion analysis, with complex analysis as a subset.

From my research, my gut instinct says there will be no measurable difference between an approach that uses quaternion quantum mechanics and one that uses complex quaternion mechanics. The folks doing quantum have figured out how to do quaternion math without quaternions, using the subgroup complex number, and patching it up with non-commuting operators as necessary.

doug

 

[robphy]

Thanks for your comments.

 

[Eric Belcastro]

From my research, my gut instinct says there will be no measurable difference between an approach that uses quaternion quantum mechanics and one that uses complex quaternion mechanics. The folks doing quantum have figured out how to do quaternion math without quaternions, using the subgroup complex number, and patching it up with non-commuting operators as necessary.

After beginning studies in the mathematics of quantum mechanics, this is what I am starting to notice. I prefer quaternions, though I noticed that the way non-commutative operators have been used with pairs of complex numbers is basically the same thing, as you said. It would seem to me to be wise to just stubbornly persist in using quaternions (while still understanding the alternatives for the sake of communication), as the general use of quaternions could make certain principles clearer or perhaps even reveal new principles in EM, and relativistic theory. Regardless, I plan on using them in my classes and studies, if only to show that quaternions really are useful, even if my professors don't necessarily "profess" them.

 

[Son Goku]

There should be quaternion analysis, with complex analysis as a subset.

I think the problem is that in quaternionic analysis almost every function besides linear ones are non-analytic. Basically there are 16 Cauchy-Riemann equation analogues that are much more difficult to satisfy. Even the function $$f(q) = q^2, q \in \mathbb{H}$$ is not analytic.


For robphy, see this paper: http://www.scottaaronson.com/papers/island.pdf" , published in Proceedings of the Växjö Conference "Quantum Theory: Reconsideration of Foundations" (A. Khrennikov, ed.), 2004.
The author discusses how essential complex amplitudes are to QM and the problems with real or quaternionic amplitudes.

Caves, Fuchs, and Schack exploited this observation to show that a quantum de Finetti Theorem"
(which justifies Bayesian reasoning) works only if amplitudes are complex.