Comment by Ono-Sendai
16 hours ago
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
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
https://deferentialgeometry.org/papers/Doran,%20Lasenby%20-%...
page 28, equation 2.36. Thanks a lot I'll take a dive into this
Note: my inquiry was motivated by this:
https://blog.revolutionanalytics.com/2014/01/the-fourier-tra...
1 reply →
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.
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.
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