Comment by JPLeRouzic
7 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?
7 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 ?
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
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.
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.