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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: 0.414s | 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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ColinHayhurst ◴[21 Nov 24 16:47 UTC] No.42206146[source]▶
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

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ValentinA23 ◴[21 Nov 24 16:52 UTC] No.42206198[source]▶
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...

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1. ColinHayhurst ◴[21 Nov 24 17:14 UTC] No.42206406[source]▶
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.