Comment by JPLeRouzic
12 hours ago
> 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?
12 hours ago
> 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?
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.
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.
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 ?
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:
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.
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
2 replies →
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.
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
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.