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So, the definition of QFT in terms of path integrals is that the partition function is:

$$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$

But does it have any meaning if instead of this $U(1)$ quantum mechanics you replace it with $SU(2)$ of unit quaternions:

$$Z[J] \propto \int e^{iS_1[\phi]+jS_2[\phi]+kS_3[\phi]+J.\phi} D[\phi]$$

Obviously there are three actions $S$ instead of one. So is this kind of thing forbidden? Or is it equivalent to something else? (i.e. could all 3 actions be combined into one?) Is there something special about complex numbers? What is the physical principle or mathematical principle that says that we must only consider complex $U(1)$ phases.

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/137475/2451 , physics.stackexchange.com/q/105278/2451 and link therein. $\endgroup$ – Qmechanic Jun 23 '15 at 22:44
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    $\begingroup$ You mean the physical principles of experimentation and observation? Nothing says that you must use experiments, you can be a philosopher instead or a world builder, it's only the backwards scientists that still try to fit models to experimental data. ;-) $\endgroup$ – CuriousOne Jun 23 '15 at 22:44
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    $\begingroup$ Whatever this "$\mathrm{SU}(2)$ partition function" (this is not the partition function, but the generation functional, btw) is supposed to be, it simply isn't QFT. What would the classical limit (which has only one action) be here? How do we get scattering amplitudes (non-commutativity of quaternions makes expanding the exponential a pain)? You can't just write down something for $Z[J]$ and ask "Why not this?", you have to show that it actually defines a consistent kind of theory. $\endgroup$ – ACuriousMind Jun 23 '15 at 22:49
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    $\begingroup$ That is my question. Does it define a consistent theory. Would it describe some kind of Universe? Or is it forbidden by some mathematical theorem. i.e. Is Quantum Mechanics inevitable or just one of an infinite number of consistent theories? $\endgroup$ – zooby Jun 23 '15 at 22:54
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    $\begingroup$ In model building nothing is forbidden. If you want to know what it describes then you have to add an interpretation to it. Unlike ACuriousMind I do not agree that one needs consistency. Your expression could describe an approximation of a physical system, in which case it would not have to form a consistent theory. We have plenty of useful ad-hoc approximations that don't (aka "every physical theory ever conceived by man"). $\endgroup$ – CuriousOne Jun 23 '15 at 23:00
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Theoretically it can. Adler wrote a book Quaternionic Quantum Mechanics and Quantum Fields, where the details are worked out. See also Arbab's recent paper Quaternionic Quantum Mechanics. However, it is unclear what advantages quaternionic theory offers over the complex one, and analytic issues do not work out very well. Already Hamilton encountered difficulties when trying to develop quaternionic analysis, the theory is quite poor compared to complex one. Adler writes:

"we know that in analogy with complex analyticity, a much more restricted concept of quaternion analyticity has been developed in the mathematical literature... we have not found any context in our development of quaternionic quantum mechanics in which the use of quaternion analyticity seems natural (but there could be one)".

As one commenter put it, "essentially Alder is using complex quantum mechanics with quaternion coefficients only when safe", see more in Google Groups discussion.

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  • $\begingroup$ The Arbab paper is informative, but are we limited to the same equations of motion that we use in the complex case? I am also not sure how many of the physical properties he recovers from these equations actually depend on the algebraic structure of the wave function representation. It seems like a peephole view on a deeper structural level and he has just recovered some special cases. $\endgroup$ – CuriousOne Jun 24 '15 at 3:29
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    $\begingroup$ I don't think this is the same thing as I was talking about. Essentially I think a quaternionic shrodinger equation would involve 3 hamiltonians. e.g. $\frac{\partial\Psi}{\partial t} = (i H_1+ j H_2 + k H_3 ) \Psi $. In that case there would instead of Energy being a scalar it would be a 3-vector. Kind of weird but is it ruled out by mathematics? For example if it required also 3 time co-ordinates that might rule it out. $\endgroup$ – zooby Jun 24 '15 at 11:50
  • $\begingroup$ @zooby I was only answering the title question. As for your specific suggestion, there is nothing mathematically wrong with this evolution equation in a single time variable, it has unique solution for reasonable initial conditions, etc. But as far as physics one would have to interpret and study it, and I do not know if anyone did. $\endgroup$ – Conifold Jun 25 '15 at 0:53
  • $\begingroup$ @CuriousOne I do not think there is such a limitation, but Adler's approach was to take complex equations, replace complex numbers with quaternions, and see what happens. Arbab is more interested in establishing connections between quaternionic and different complex equations, and perhaps recovering usual theories with some additional effects. $\endgroup$ – Conifold Jun 25 '15 at 1:07
  • $\begingroup$ Thanks for the qualification. I simply can't tell which properties of these solutions stem from the structure of the equations and which depend on the choice of complex numbers/quaternions. The complex frequencies are certainly both remarkable and concerning. I don't know what to make of that. $\endgroup$ – CuriousOne Jun 25 '15 at 3:11
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I think Marek Danielewski may just have answered this quesion in this paper from Dec 2020: Foundations of the Quaternion Quantum Mechanics https://www.mdpi.com/1099-4300/22/12/1424#

In summary: quaternions can be viewed as representing compression (the real part) and torsion (the three imaginary parts). They are used in condensed matter physics to model waves in elastic solids or crystals. Marek applies this model to quantum physics, and derives the Schrödinger equation by combining the Cauchy model of the elastic continuum with the Helmholtz decomposition.

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  • $\begingroup$ Hello Chantal Roth, an answer should not only consist of a link, since if the link is no longer working the answer is useless. So i would recommend to, at least briefly, summarize what one would find following the given link. $\endgroup$ – AlmostClueless Dec 24 '20 at 22:48
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    $\begingroup$ Thanks for the hint, makes sense :-). $\endgroup$ – Chantal Roth Dec 31 '20 at 11:29

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