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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.

Qmechanic
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2 Answers2

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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.

Conifold
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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.

Chantal Roth
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