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I came across the notion of duality recently to explain a physical concept.

  • What is duality?

  • Why does it occur?

  • How do I know if two things are dual?

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    $\begingroup$ Duality is an old and confused concept from the early stages of quantum mechanics when people noticed that quantum objects had properties that could be assigned to both particles and waves. To be honest, one shouldn't use it any longer. It was never very useful and there are much better ways to think about quantum phenomena. $\endgroup$ – CuriousOne Feb 16 '16 at 12:20
  • $\begingroup$ @CuriousOne I am interested to find out why you think such ideas should not be used any more? $\endgroup$ – Farcher Feb 16 '16 at 12:27
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    $\begingroup$ @Farcher: Because it hasn't worked in 1925 and it still doesn't work. Quantum phenomena aren't dual, they are fundamentally different. Duality is the application of the "law of the excluded middle" on something that was never a binary problem. You can't make any physical predictions with "duality". Not one. Worse, still, you can't even realize either of the corner cases. The real behavior of a quantum system is not "particle" (which would require infinite momentum and energy) or "wave" (which would require an infinite detector) but it is "neither". $\endgroup$ – CuriousOne Feb 16 '16 at 12:34
  • $\begingroup$ @CuriousOne Then i think that you missed the point which is that initially one has to be economical with the truth to make progress. I have been extremely impressed with the sophistication of your explanations, analysis and knowledge but I would guess that has come after you accepting things in the past which you now know not to be entirely correct? $\endgroup$ – Farcher Feb 16 '16 at 12:47
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    $\begingroup$ @Farcher: One does not have to be economical with the correct explanation after the correct explanation has been found. All one has to do is to teach it rather than to fall back on incorrect explanation attempts that predate it. Are you telling people that it seems natural that moving things are always coming to a rest but that the world just doesn't seem to work that way and that we have to keep looking for an explanation why they don't? No, of course not. You are simply teaching Newton. So where does this fascination for the wrong explanation come from in QM? $\endgroup$ – CuriousOne Feb 16 '16 at 13:03
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Duality between two different theories means that these two theories when applied to a problem yield the same answers. The dualities can be approximate or exact (ex: the gauge/gravity duality, string dualities etc.) depending on whether the calculations performed using the two different theories agree exactly or only up to a certain limit.

Dualities are usually a characteristic feature of the underlying mathematical logic and structure. As a vague example, just because you can explain effectively say the structure of stars using hydrodynamics and quantum mechanics separately, does not mean that hydrodynamics is dual to quantum mechanics. The duality occurs because the underlying mathematical structure between the two objects end up calculating the exact same thing or reduce to the same quantity in certain limits. Of course, there are some dualities which are well understood mathematically like mirror symmetry while some others which are still well accepted conjectures (from a mathematical framepoint) like the holographic duality. But there are dualities in physics and given their mathematical origin, they must also exist in mathematics, and they do.

Knowing two things are dual requires you to understand how to construct and calculate quantities precisely using the theories between which you want to describe a duality. However, proving two things are dual is much more mathematical.

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Duality arises in physics when two different aspects of nature are apparently related to each other by a common underlying theory. There are various examples of dualities in physics. eg Wave particle duality, electric field/ magnetic field duality, gauge/ gravity duality. Usually it appears when the underlying theory is viewed from two different limits.

In order to verify if the two or more aspects of nature are dual or not, you need to know about the basic underlying theory, and see whether the aspects manifest themselves as the limits of the theory / or are sub-parts of the full theory.

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protected by Qmechanic Feb 16 '16 at 12:45

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