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The question whether surreal or hyperreal numbers (that both contain the reals, even if they have the same cardinality) could be useful to provide a more satisfactory theory of QM is maybe more interesting. -yuggib

Background

I indirectly ended up on this stack, much to my surprise. Little by little I have been working to piece the puzzle together.

I have been pondering this question for a while now, but was not able to formulate it so succinctly until I saw the above quote.

I still don't feel able to properly convey my intuition around why this is a good approach..

Abstract

Star

is not zero, but neither positive nor negative, and is therefore said to be fuzzy and confused with (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0

Fuzzy logic

is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1

Qubit

—usually taken to have the value “0” and “1”, like a bit. However, whereas the state of a bit can only be either 0 or 1, the general state of a qubit according to quantum mechanics can be a superposition of both.

Motivation

"It is likely to be a fresh research question, and you the person one most interested in the entire world in finding an answer --- in which case it is most probably up to you (and your opportunity!) to obtain the answer first." -Niel

While the motivation to wanting to make this work may not be apparent at first, a big piece of what I'm wanting to accomplish is creating a quantum algorithm based on surreal constructions for a quantum intelligence that can use game theory for computation.

I have placed a couple different bounties in an attempt to push along this research.

Question

How can surreal maths be used in quantum computing?

Sanchayan Dutta
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user820789
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    I'm voting to close this question as off-topic because this appears to ask about speculative models of physics. (This is not to say it is not an interesting question in principle, but it is not appropriate for this site at this point --- at least not without more development to present how it could be used. Before jumping all the way to "is this useful for QM", it is worth considering whether it makes any sense in relation to the framework of quantum computation; and before you do that, you should consider what a "surreal-valued-amplitude" model of computation might look like.) – Niel de Beaudrap Jul 13 '18 at 09:53
  • Can you tell us what it is about quantum theory that you find unsatisfactory? What aspect can be made more satisfactory? – user1271772 No more free time Jul 13 '18 at 12:46
  • @NieldeBeaudrap "Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science" – user820789 Jul 13 '18 at 14:22
  • @meowzz: But not arbitrary paradigms of mathematics and computer science. One cannot generate a topic in categorical quantum mechanics by considering some subject in mathematics or computer science, and then trying to use it some way or another to describe or elaborate on quantum mechanics. For instance, considering Red-Black trees and then applying it to describe complex-valued polynomials in order to describe quantum amplitude functions, would not be an example of CQM (even though that could just conceivably be interesting as a way to describe quantum states on some other grounds). – Niel de Beaudrap Jul 13 '18 at 14:26
  • @user1271772 It's not that I particularly have any qualms w/ QM. I'm more specifically interested in how surreal maths can be used. – user820789 Jul 13 '18 at 14:27
  • @NieldeBeaudrap re: "surreal-valued-amplitude model of computation" = yes – user820789 Jul 13 '18 at 14:29
  • @meowzz: I'm more specifically interested in how surreal maths can be used --- that's asking us to speculate on fundamental physics, which is not the purpose of this forum. // "surreal-valued-amplitude model of computation" = yes --- that's a matter of models of computation, plausibly counting complexity; while this has connections to quantum computation, it is not an example of it. It is a reasonably interesting idea to think of exploring, but even so, that would not really be appropriate to ask on any SE unless you have reason to believe that someone has already explored the question. – Niel de Beaudrap Jul 13 '18 at 14:31
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    @meowzz: If you are interested in exploring the idea --- and there is at least some potential in it --- I recommend that you read / research a little about counting complexity, and see how surreal numbers might be incorporated into that line of thought. This won't necessarily end up coinciding with quantum computation, but it is the right way (in my opinion) to explore how such ideas might relate to quantum computation. – Niel de Beaudrap Jul 13 '18 at 14:34
  • @meowzz: sorry --- my vote to close stands. An interesting idea to explore possibly, but too speculative inasmuch as it is meant to relate to quantum computing. (I would do the same for a question about wormholes or physics over p-adic integers.) – Niel de Beaudrap Jul 13 '18 at 14:38
  • As for the question about asking it on any SE: it is conceivably on-topic for the Theoretical Computer Science SE, but if I understand the landscape of research there well enough, it is unlikely that you will get a satisfactory response even if you present it entirely in terms of counting complexity (as opp. quantum computing). It is likely to be a fresh research question, and you the person one most interested in the entire world in finding an answer --- in which case it is most probably up to you (and your opportunity!) to obtain the answer first. – Niel de Beaudrap Jul 13 '18 at 14:41
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    The reason I did not vote to close is because it is about quantum theory, and while this SE is called "quantum computing" (so perhaps this is more appropriate for the physics SE), I don't have a problem if it gets asked here. "Quantum information", "quantum communication", "quantum metrology" and "quantum foundations" are also fine in my opinion. The IQC in Waterloo has quantum foundations people that don't do "quantum computing" per say, but it's still called "IQC". This is a "quantum foundations" question as far as I see. – user1271772 No more free time Jul 13 '18 at 14:44
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    I have cleaned up the comments on this question. Please remember comments are for clarification of the question. If you want to discuss this question, please also take it to chat. See also this meta post about this question. – auden Jul 13 '18 at 15:29
  • @meowzz This question does not seem to be related to "categorical-quantum-mechanics". Could you kindly remove that tag? – Sanchayan Dutta Jul 13 '18 at 15:40
  • @blue Could you kindly explain (in chat?) why it is not an appropriate tag? Based on both the tag description & the Wikipedia page, it's seems a good fit to me. – user820789 Jul 13 '18 at 15:45
  • @meowzz Categorical quantum mechanics is a specific approach to quantum mechanics with concepts from certain areas of mathematics like category theory, logic and type theory. Your question basically asks whether surreal numbers are helpful in quantum mechanics and not about whether it is relevant to the formulation of a specific approach to quantum mechanics. People will only get confused seeing that tag on that question and you'll probably get irrelevant answers. – Sanchayan Dutta Jul 13 '18 at 15:48
  • @Blue I am looking for a surreal approach to QM. Games (surreal pseudo-numbers) interest me more than surreal numbers themselves. – user820789 Jul 13 '18 at 15:55
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    @meowzz Categorical quantum mechanics is not a surreal approach to QM. In case you're interested about CQM in particular, ask about it in a new question. Making that change to your question now, would render the existing answer invalid, which is unfair. – Sanchayan Dutta Jul 13 '18 at 16:00
  • @NieldeBeaudrap hopefully this is an ok format. – user820789 Jul 13 '18 at 16:25
  • I'm having trouble narrowing in on the question. I mean, to me, the surreal system is just one approach to constructing numbers; seems like the easiest way to use them in quantum theory is to just use them, which seems like far too trivial an answer. So are you asking about if there are problems in quantum theory for which the canonical approach to handling them invokes surreal construction, or...? – Nat Jul 13 '18 at 18:59
  • @Nat I think "if they are invoked" or "where they might naturally be invoked" is along the lines of what I'm after – user820789 Jul 13 '18 at 19:16
  • @heather I would like to answer my own question. – user820789 Jul 19 '18 at 02:06

1 Answers1

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In quantum field theory there are Feynman path integrals that diverge and for this there is a concept of "renormalization". At least one approach to this uses surreal numbers but it is not very mainstream.

user1271772 No more free time
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