# Does the existence of dualities imply a more fundamental structure?

I was wondering if the existence of some kind of duality in physics always implies the existence of some underlying more fundamental structure/concept?

Let me give a few example from history:

1. Wave-particle duality$~\Rightarrow~$ Existence of quantum particle.

2. Heisenberg's Matrix Mechanics $\&$ Schrodinger's wave formulation of QM $~\Rightarrow~$ Existence of Dirac formulation of QM.

3. Magnetic field $\&$ electric field $~\Rightarrow~$ Existence of an electromagnetic theory.

Similarly, can one conclude that

1. for example, from AdS/CFT correspondence,

2. or more generally, because there is an holographic equivalence between quantum gravity in $D+1$ dimensions, and QFT in $D$ dimensions,

then there must be a more fundamental underlying structure that incorporate both sides of the correspondence?

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Can you please clarify your abbreviations to me? – Bernhard Feb 12 '12 at 13:32

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Qm=quantum mechanics. Qft= Quantum field theory. Cft=Conformal field theory. Ads/cft= anti de Sitter/CFT – Manishearth♦ Feb 12 '12 at 13:38

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If we have two different descriptions of the same phenomenon, then yeah, we always hope that there is a "unified" picture that tells us why. Logically there's no reason that this must be the case though. Maybe we have a philosopher here who can argue that a thing's thingness depends on it having a unique description. – wsc Feb 12 '12 at 22:46

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@Qmechanic why is this closed? – Dilaton Dec 29 '12 at 17:13

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Hi @Qmechanic, maybe it is somewhat broad but I thought it nevertheless contains an interesting ordered line of thought which could be adressed by someone (else) knowledgabel enough. I would be interested in seing such an answer too. Maybe Lumo (or Ron :-( ...) could do it. – Dilaton Dec 29 '12 at 17:49
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## 2 Answers

I think AdS/CFT went the other way. People knew about the unifying concept (string theory) first and "derived" AdS/CFT from worldsheet duality in string theory. But I guess it could've gone the other way in an alternate history.

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I asked Lumo if he has an answer to this question. He did not like the question too much... ;-) Nevertheless he gave some nice clarifying comments and explained what is wrong with it and how he thinks about the issues mentioned. I think his comments make a very decent answer here anyway (and hope he does not mind that I post them here). So here we go:

\begin{quote}

Dualities are obviously important and unify several seemingly different descriptions. This is by definition of dualities. In this most general sense, they are analogous to the wave-particle dualism and unification of pictures in quantum mechanics and perhaps other things (unification of electricity and magnetism is substantially different).

The quantum particle is the same thing as the object displaying both wave and particle properties, so the "two" concepts related by the arrow on that line are really the same concept, and the whole relationship claim is vacuous or tautological.

In the same way, the matrix and wave mechanics may be unified but the unification is nothing else than the Dirac formalism for quantum mechanism so the two parts of the relationship are - assuming that the relationship between the pictures is found - a priori equivalent, too. We already have this description for dualities in string theory, sort of, too. One may discuss physics in the description-invariant way. The problem is that we don't have a universal definition of the "Hamiltonian" or "action" but we may still write the general equations with a Hamiltonian or an action that is duality-invariant. This situation differs from the simplest models of quantum mechanics where the Hamiltonian could have been written down "exactly". In string theory, the expressions for the "Hamiltonian" or whatever defines the dynamics depends on the description and it is often incomplete, so the dualities can't be formulated as a sharp mathematical claim at this moment. They're still perfectly true according to all the evidence and tests we may do and assuming it is indeed the case, and it seems to be the case beyond any reasonable doubt, the equivalence is the same equivalence as the equivalence between pictures (Heis/Schr) in quantum mechanics or representations (position/momentum) in quantum mechanics.

Electromagnetism is a bit different because the electromagnetic field contains both the electric vector and the magnetic vector as independent degrees of freedom, so electromagnetism isn't about 2 views on the same 1 thing. It is about 2 things that naturally collaborate and are linked by symmetries and transform into each other under the Lorentz transformations. It's a different relationship than the equivalence in dualities.

\end{quote}

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I made this answer CW because it contains what Lumo said; so everybody who agrees with his comments can freely upvote and I dont need to have a bad concience for any upvotes that may occur :-) – Dilaton Mar 20 at 21:54
Hi @Dilaton: Do you have an Internet link to the above quote by Lubos Motl? – Qmechanic♦ Mar 20 at 22:19
@Qmechanic not directly, it is in the comments below a TRF article and at these comments I can not link directly ... I can at most link to the corresponding TRF post. – Dilaton Mar 20 at 22:22
– Dilaton Mar 20 at 22:27
Ok. Link good enough for now. – Qmechanic♦ Mar 20 at 22:30

## protected by Qmechanic♦Mar 20 at 21:25

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