January 1928: Dirac equation unifies quantum mechanics and special relativity

> Quaternions

I know nothing of physics, but it seems to me that rotation fingerprints are everywhere in physics. Is this just me or is there something more tangible in this remark?

It's not just you. Dirac fields are constantly rotating. In fact the solutions are called spinors. (e.g. things that spin). There are a lot of rotations at the quantum level. It's also why complex numbers show up a lot in q.m.

Of course. The solutions of the Dirac equations live in space and space has rotation symmetry. These solution have to transform in some way under rotations.

I've been trying to get an intuitive understanding of why multiplying by e^ix leads to a rotation in the complex plane, without going into Taylor series (too algebraic, not enough geometric). I tried to find a way to calculate the value of e in a rotational setting, maybe there is a way to reinterpret compound interests as compound rotation. Any insight ?

IANAM but I'd go with "it's implicit in how complex numbers are defined". Complex numbers are a thing made up by humans (as are negative numbers), and we got to define i as "up the y-axis". Once you do that, and note that a rotation is therefore cos angle plus i sin angle, add in that e^something is an eigenfuncion of differentiation, and you're pretty much there.

Fwiw I think it's Maclaurin series for this.

Edit: obviously should be j not i.

Rotations and spin are deeply tied into the geometrical nature of a space. It's not just you. It's core to understanding the nature of matter itself.

Cartan had only just invented spinors as an object in themselves (ignoring clifford) so a lot of the physics stuff was done in parallel or even without the knowledge the mathematicians had.

One possibility: take the unit circle, and a vertical line tangent to the circle at (1,0). Then e^ix takes that line and wraps it around the circle. This

Complex numbers and (Pauli/Dirac) matrices not required if you use Geometric Algebra. I highly recommend the book by Doran and Lasenby [0], or you can get the details from their papers, notably [1].

[0] Geometric Algebra for Physicists, CUP, 2003

[1] https://arxiv.org/abs/quant-ph/0509178

p281 for Dirac equation. But I suggest you start at least from the beginning of Chapter 8. Earlier, obviously if you don't know Geometric Algebra. It's worth it; many examples but one is that the four Maxwell equations are expressed as one compact equation with geometric intuition.

Euler's formula is a specific case of the exponential map from Lie theory. This means e^x can be used with all sorts of interesting x types, and it often has surprisingly intuitive behavior! When x is a real number you get continuous growth. When x is a purely imaginary number you get continuous rotation. When x is complex you get continuous growth and rotation. When x is a matrix you get a continuous linear transformation (growth, rotation, and shear). What's the similarity here? Euler's formula treats it's argument as a transformation which gets continuously applied in infinitesimal amounts. This also explains the formula for calculating the value of e:

    e = lim (1 + 1 / n) ^ n where (x -> infinity)

https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)

https://en.wikipedia.org/wiki/Matrix_exponential

https://www.youtube.com/watch?v=O85OWBJ2ayo

My favorite intuitive explanation was actually written by science fiction author, Greg Egan. It takes the exact approach you're asking for, reinterpreting compound interest in a 2d rotational context on the complex plane, and doesn't use more than high school math:

https://gregegan.net/FOUNDATIONS/04/found04.html#s2

Fig. 7 is the money shot.

The first thing to understand is that multiplying a complex number by i rotates the complex number by 90 degrees counter-clockwise around the origin. For example, 1 * i = i (e.g. 1 + 0i is mapped to 0 + 1i). And i*i = -1 (e.g. 0 + 1i is mapped to (-1 + 0i) and so on. e^ix is a continuous generalisation of this discrete rotation, as I understand it.

There's a lot of rotations and stuff in QM, yes. That you should do it with quaternions is mostly an internet thing. I have a PhD in physics, and I never encountered quaternions in any course I took, and from seeing curricula in other universities I haven't seen it there either.