A Complex numbers in QM

That doesn't follow. The argument for complex numbers in SR is in principle completely seperate from the argument for complex numbers in QM.
Then why were you telling me about "Lorentzian structure" and "Wick rotation" in the context of QM?

Auto-Didact said:
More importantly, I see that you are focussing alot on #158; I would like to paraphrase Bohr by saying "Every sentence I utter must be understood not as an affirmation, but as a question."
Sorry for taking your post seriously.
Auto-Didact said:
This thread is about complex numbers in QM in general; I think therefore that the arguments I give in other posts in favor of the fundamental place of complex numbers in QT cannot be dismissed so lightly.

I read your post #158 and commented on it. As far as I remember, I did not read your earlier posts, so I could not have dismissed them, neither lightly nor heavily. I don't think I have to read the entire long thread to post anything.

Auto-Didact said:
It is obvious to me that we have very opposing views of how we basically regard what a physical theory is; the difference between our views are what historically was called the applied mathematics view versus the pure mathematics view of physics.

The derivation of spin from first principles by Dirac is to me clearly a result of the mathematical existence of spinors, whether that is/was acknowledged or not by physicists at the time or even today. The existence of spin as a mathematical object can be demonstrated to be a consequence of the existence of spinors, with the gamma matrices operators which act on spinors. In this sense, gamma matrices are non-commuting elements of the Clifford algebra, giving spinors more degrees of freedom than scalar wavefunctions. This is what my post in #158 was alluding to.

In contrast, from what I understand from your point of view, it seems you would claim that spin is just a physical quantity following from any mathematical model capable of describing aspects of the physics, whether or not these descriptions can on the face of it immediately be shown (through trivial efforts/arguments) to be equivalent to some other purely mathematical model of spin we already have; indeed, such a pragmatic view is referred to as an applied mathematics view of physics.
OK, I got it. I have a bad habit to read texts exactly as they are written. You have explained that your post #158 means something different from what is written there. I give up.
Then why were you telling me about "Lorentzian structure" and "Wick rotation" in the context of QM?
Lorentzian structure: because I was talking about QFT, seeing that the discussion was focusing on the Dirac equation. There is an argument to make that QM is always a limiting case of QFT; if one argues this way, whether or not they believe in the argument, then the argument about the complex nature of SR is extremely relevant.

Wick rotation: as I said, convenience techniques, such as e.g. the Wick rotation, which make the mathematics more tractable tend to give results of limited validity. By analogy, I was making the case that your algebraic technique and gauge transformation, resulting in a real fourth order version of the Dirac equation, may have similar limitations. This is why I asked if the resulting equation carries over correctly to QFT in curved spacetime.
Sorry for taking your post seriously.
You misunderstand my intent for clarification; we are speaking about what is fundamental in physical theory, i.e. not just contemporary physics but what should or will likely be continued to be regarded as fundamental in future physics yet to be discovered. As I remarked before this requires a certain view of physics.

In fact, the usual attitude of certainty learned during training is almost never warranted in such discussions, as opposed to discussions about non-fundamental (textbook level) physics, where the level of certainty is rather high because it can almost always trivially be empirically (through experiment) or mathematically justified (e.g. even by a smart undergraduate student).

In other words, discussing foundations necessarily brings with it more uncertainty than in the rest of physics, exactly as Bohr remarked. The only guide theoretical physics has ever known in this intrinsically uncertain endeavor is to rely on pure mathematics not yet appreciated to be relevant to physics; this is often called 'relying on mathematical beauty' and is a concept deeply misunderstood by many physicists today (e.g. Hossenfelder).

Relying on mathematical beauty is truly an art, which cannot be reduced to brainless application of simple techniques and equation churning; it requires development of correct esthetic senses from pure mathematics and therefore requires a finer touch. Importantly, using this methodology in fundamental physics to generate hypotheses often results in highly mathematically elaborate models which stretch across all of physics making them extremely specific and therefore extremely open to falsification.

This is in stark contrast to more specialized topics in physics, which are better served by the applied mathematics view in conjunction with experimental reasoning. I am a strong advocate that thinking about foundational issues in the more pure mathematics manner, and therefore doing fundamental physics in such a way, is the most productive theoretical methodology available and the only tried and true one with precedent of success; the results of all the biggest names in physics attest to this.

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