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In physics it seems everything is explained with $\mathbb R$ or $\mathbb C$ typed entitites.

  1. Is there anything in or that would be in future in physics that would need the utility of $p$-adics in an essential way?

  2. Why is everything explainable with only $\mathbb C$ in an essential way even though there are far more numbers of form involving $\mathbb Q_p$ completions $\mathbb C_p$?

Is there a fundamental reference?

Turbo
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  • There are p-adic mathematical physics, actually I have such a book on my shelf that I still haven't found the time to read. – MathematicalPhysicist Sep 28 '19 at 12:58
  • Quaternions appear in Sp(N) (so standard gauge theories, but more importantly in SUSY, with its hyperkäler moduli spaces). $p$-adics appear in the classification of lattices (so String Theory, Condensed Matter Physics, etc). The question is way too broad. – AccidentalFourierTransform Sep 28 '19 at 12:59
  • Related: https://physics.stackexchange.com/q/107290/2451 , https://physics.stackexchange.com/q/361650/2451 https://physics.stackexchange.com/q/15252/2451 and links therein. – Qmechanic Sep 28 '19 at 13:02
  • @MathematicalPhysicist The very fact you have not read it makes it not essential for your working knowledge. – Turbo Sep 28 '19 at 13:49
  • As of your latest edit, it seems like what you meant to ask was why are $\Bbb R$ and $\Bbb C$ more physically helpful than $\Bbb Q_p$ and $\Bbb C_p$? I don't think you should expect every mathematical structure to be physically relevant, only every solution to problems in an already-relevant structure. – J.G. Sep 28 '19 at 14:02
  • @J.G. Numbers are numbers and there are probably essential realities to them. My post is deeper than what you are gathering here. – Turbo Sep 28 '19 at 14:06
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    @T.... If you edit enough to convince those who voted to close (I wasn't one of them) that your question is deep rather than broad, maybe your question will be re-opened, which is the only way it can get another answer. Good luck. – J.G. Sep 28 '19 at 14:14
  • p-adics don’t appear essentially in the description of any observable physics. They could appear in nonobservable physics, but that isn’t surprising, as then both things are just pure math. – knzhou Sep 28 '19 at 17:05
  • @T.... it's in my endless reading material. :-D – MathematicalPhysicist Sep 28 '19 at 19:01
  • It seems that the res. recom. tag does not apply. – Qmechanic Oct 06 '19 at 11:00

1 Answers1

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I think you just need to read more widely to encounter a broad range of number systems being useful in at least theoretical physics. I'll hyperlink to discussions of applications, but just mention the number systems themselves, as some have multiple applications.

There are uses for $p$-adic numbers, split-complex numbers, dual numbers, quaternions, split-quaternions, dual complex numbers, octonions, split-octonions, dual quaternions, Clifford algebras more generally, and Grassmann numbers. (For that last one, I just linked to the overall article because it frequently mentions the relevance to fermions.)

The prevalence of $\Bbb C$ can be explained, or at least half-explained, from many perspectives; Scott Aaronson has looked into why quantum mechanics runs on them in at least three long but good reads. In particular, he looks at why they $\Bbb C$ may be a sweet spot between $\Bbb R$ and $\Bbb H$.

J.G.
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  • The link for p-adics shows no relevancy to essential need for p-adics but gives a purported theory. This posting is only about p-adics and the answerer tells a long irrelevant message and I wonder why it is upvoted. – Turbo Sep 28 '19 at 13:46
  • @T.... I've replaced that link with a better one, but I think you're being unfair about the rest of the answer. The OP's second question was about "far more" number systems. – J.G. Sep 28 '19 at 13:49
  • by number systems I meant $\mathbb Q_p$ completions. – Turbo Sep 28 '19 at 13:50
  • @T.... Then your question should have said so. – J.G. Sep 28 '19 at 13:52
  • You should have read the title carefully and understood context. If unclear you should have asked clarification. – Turbo Sep 28 '19 at 13:52
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    @T... I read the title and question carefully, and so did everyone who put your question on hold for being too broad, because at the time you implied you were interested in just about every number system there is. A few minutes ago, I changed the $p$-adic link. Is it what you wanted now, or do you want more? – J.G. Sep 28 '19 at 13:54
  • How is the p-adic explanation essential when all the experimentation in the world can be explained with $\mathbb C$? You are missing the point. If I needed a few useless pointers I would have asked dumb google search. I would not have searched the database of intelligent humans by posting here. – Turbo Sep 28 '19 at 13:56
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    @T.... The hostility you are exhibiting towards a good-faith answer to your question (and FWIW a good answer to the question as asked) is not helpful. Best policy is to pretend that other users deserve your respect and understanding even when their answers aren't what you are looking for. – J. Murray Sep 28 '19 at 16:08
  • I consider users here intelligent. I am not belittling anyone. – Turbo Sep 28 '19 at 20:44
  • @T.... I think J.Murray was talking more about this. – J.G. Sep 28 '19 at 20:46
  • Ok I have a request can you remove your answer? This post is too messed up. – Turbo Sep 28 '19 at 21:16
  • @T.... Me removing my answer won't help you, since it's unlikely this question will ever be opened. Try asking a fresh question that's much clearer than your current one. I think you'll want to make clear from the opening that you want to know whether anything in physics "requires" $\Bbb Q_p$ or $\Bbb C_p$ in at least the sense certain things currently "require" $\Bbb C$, and if so what, and if not why not? (If that's not how you intended your question, that's fine, but a new one would hopefully make clear what you're asking instead.) – J.G. Sep 28 '19 at 21:38
  • @J.G. I want to delete the post. – Turbo Sep 28 '19 at 22:05