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January 1928: Dirac equation unifies quantum mechanics and special relativity

(www.aps.org)
109 points thunderbong | 1 comments | 21 Nov 24 02:17 UTC | HN request time: 0s | source
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JPLeRouzic ◴[21 Nov 24 08:58 UTC] No.42202394[source]▶
> 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?

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Ono-Sendai ◴[21 Nov 24 11:11 UTC] No.42203111[source]▶
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.
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ValentinA23 ◴[21 Nov 24 14:28 UTC] No.42204619[source]▶
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 ?
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1. itishappy ◴[21 Nov 24 17:28 UTC] No.42206559[source]▶
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