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> Yet the technique employed to make the theory useful — renormalization — repulsed Dirac because he found it mathematically ugly.

Perhaps if he had used quaternion the solution will not be mathematically ugly or can even be beautiful [1].

[1] A quaternion formulation of the Dirac equation:

https://mauritssilvis.nl/research/publications/silvis-rug10....

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elashri 5 days ago | parent | next [-]

Dirac was not working in vaccum . Klein-Jordan equation was the simplest and the most obvious extension of Schrodinger equation in relativistic manner.

So historically, Dirac was focused on correcting the Klein-Gordon equation, which had issues with negative probabilities and describing electron behavior. His goal was to find a relativistic equation that resolved these problems while maintaining consistency with his own matrix mechanics formulation of quantum mechanics.

By extending his matrix mechanics formalism, Dirac derived an equation that not only addressed the issues with the Klein-Gordon equation but also predicted the existence of antimatter. I would argue that Dirac's approach was consistent with his established framework, and while he found renormalization mathematically unsatisfactory, it does not diminish the validity of his method in deriving the Dirac equation. I doubt he focused on any elegant solutions, he was actually quite happy working with matrix mechanics framework.

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jesuslop 5 days ago | parent [-]

Bohr was a big shot, Nobel prized establishment authority. In Weimberg QFT book he recalls a fragment of Dirac's memoirs:

"I remember once when I was in Copenhagen, that Bohr asked me what I was working on and I told him I was trying to get a satisfactory relativistic theory of the electron, and Bohr said 'But Klein and Gordon have already done that!' That answer first rather disturbed me. Bohr seemed quite satisfied by Klein's solution, but I was not because of the negative probabilities that it led to. I just kept on with it, worrying about getting a theory which would have only positive probabilities."

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phkahler 5 days ago | parent [-]

Is there a relationship between the negative probabilities of Klein and the negative energy of Dirac? Did his formulation just move the problem? If so, does it imply anything? Like are probability and energy related?

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elashri 5 days ago | parent [-]

Klein-Jordan equation does have both problems, negative probabilities and energies. Dirac equation solved negative probabilities and now predicts positive probabilities for both positive and negative energy states. But the negative energies problem still exists and Dirac used different interpenetration to explain them and did not get rid of them (which we knew later that this was the correct things to do). So he came with the famous negative energy solutions interpreted as antiparticles.

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superposeur 5 days ago | parent [-]

It’s worth mentioning that, brilliant as Dirac’s “sea of filled negative energy states” picture was, no one believes that interpretation now. The Dirac equation is better seen as the classical equation of motion for the Grassmann-valued electron field (just as Maxwell’s equations are the classical eom for photon field). There are only positive-energy states (=quantized excitations of the field). I do think popular accounts should begin mentioning this in order not to keep reinforcing the old Dirac sea interpretation.

	▲	codethief 5 days ago \| parent [-]
		> no one believes that interpretation now I know of at least one (tenured) person that does, at least to some degree: Felix Fenster at Regensburg University. When I met him years ago, he said taking the Dirac Sea interpretation seriously was what caused him to come up with his own program for a theory of quantum gravity, called Causal Fermion Systems[0]. I haven't looked into his theory in detail but I did find a reference to the Dirac sea[1]: > In order to obtain a causal fermion system, we first have to choose a Hilbert space. The space of negative-energy solutions of the Dirac equation (i.e. the Dirac sea) turns out to be a good choice. […] As a side remark, it is worth noting that the Dirac sea vacuum is to be seen as an effective model describing a particular minimizing causal fermion system. It is one particular physical system that we can describe as a minimizing causal fermion system. But we should really only think of it as an effective description, in the sense that it describes only the macroscopic structure of spacetime, whereas its microscopic structure on the Planck scale is essentially unknown. […] The idea of the Dirac Sea did, however, play an important role in the conception of the causal fermion systems framework, and most of the existing literature is written with that point of view in mind. A more detailed motivation for why it is a natural starting point can be found here[2]. [0]: https://causal-fermion-system.com/ [1]: https://causal-fermion-system.com/intro-phys/ [2]: https://causal-fermion-system.com/theory/physics/why-dirac-s...

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cornel_io 5 days ago | parent | prev | next [-]

That reformulation doesn't let you avoid renormalization, does it?

	▲	terminalbraid 5 days ago \| parent [-]
		No, it doesn't.

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kelseyfrog 5 days ago | parent | prev [-]

Thank you for posing the quaternion formulation. It inspired me to search for a geometric algebra version of the same equation and was happy to find that it also exists[1].

1. https://fondationlouisdebroglie.org/AFLB-342/aflb342m679.pdf