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

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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?

This post imported from StackExchange Physics at 2014-03-17 04:17 (UCT), posted by SE-user Revo
asked Feb 12, 2012
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Can you please clarify your abbreviations to me?

This post imported from StackExchange Physics at 2014-03-17 04:17 (UCT), posted by SE-user Bernhard
Qm=quantum mechanics. Qft= Quantum field theory. Cft=Conformal field theory. Ads/cft= anti de Sitter/CFT

This post imported from StackExchange Physics at 2014-03-17 04:17 (UCT), posted by SE-user Manishearth
@wsc I think the answer should be given by a physicist, as Weinberg has said once that philosophy has lots its way long time ago.

This post imported from StackExchange Physics at 2014-03-17 04:17 (UCT), posted by SE-user Revo
@Qmechanic why is this closed?

This post imported from StackExchange Physics at 2014-03-17 04:17 (UCT), posted by SE-user Dilaton
@Dilaton: I closed the question(v1) as not constructive, as it seems formulated too broad to be true in every case. Did you want to add an answer?

This post imported from StackExchange Physics at 2014-03-17 04:17 (UCT), posted by SE-user Qmechanic
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@Qmechanic could you, as we agreed upon, reopen this for a while such that I can put down what Lumo said about this issue on TRF as I asked him in the form of a CW answer? He did not have an outline of an underlying structure, but he explained what he thinks about it.

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Dilaton
... thanks @Qmechanic :-)

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Dilaton

## 2 Answers

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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.

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user josh
answered Feb 12, 2012 by (195 points)
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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}

Here you can read Lumo's original nice comment.

answered Mar 20, 2013 by (4,305 points)
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 :-)

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Dilaton
Hi @Dilaton: Do you have an Internet link to the above quote by Lubos Motl?

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Qmechanic
@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.

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Dilaton
... ok, it is the second comment below this TRF article.

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Dilaton
Ok. Link good enough for now.

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Qmechanic
@Dilaton: You can click on the timestamp on the link to the comment: e.g. motls.blogspot.com/2012/12/…

This post imported from StackExchange Physics at 2014-03-17 04:18 (UCT), posted by SE-user Dimensio1n0

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