✎ Duality [The Physics Travel Guide]

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====== Duality ======

<tabbox Why is it interesting?> 
Oftentimes in physics, we can only calculate things in terms of a [[advanced_notions:quantum_field_theory:perturbation_theory|perturbation series]]. Especially, in quantum field theory, this is almost always the case. The perturbation series in [[theories:quantum_field_theory|quantum field theory]] is usually a series in the small coupling constant of the interaction in question. However, when this interaction constant becomes too large, such a perturbative approach no longer yields a good approximation. This is especially problematic for quantum chromodynamics at low energies, where the strong coupling constant becomes too large.

A duality can be an enormously powerful tool in such a situation. A dual theory describes the same physics, but in a different way. Especially the coupling constant of a dual theory is inversely proportional to the original coupling constant. Hence, whenever the coupling constant becomes too large the calculation can be done in the dual theory instead, where we can work with a small coupling constant.

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<blockquote>The idea of duality has been at the centre of many important developments in the theoretical physics of the last 50 years. In fundamental physics, the notion of duality has been applied to very different kinds of theories. First, there is the dual resonance model of the late sixties, from which early string theory originated. Successively, one of the most important developments of this idea was the generalization, proposed by Claus Montonen and David Olive in 1977, of electromagnetic duality in the framework of quantum field theory. This was later extended to the context of string theory, where dualities have also spawned recent developments in fundamental physics, offering a window into non-perturbative physics, and motivating both the M theory conjecture and gauge-gravity duality<cite>https://arxiv.org/pdf/1803.09443.pdf<\cite></blockquote>
<tabbox Layman> 
A duality is a formal or theoretical equivalence between two theories.

<blockquote>The crux of many problems in physics is the correct choice of variables with which to label
the degrees of freedom. Often the best choice is very different from the obvious choice; a
name for this phenomenon is ‘duality’. We will study many examples of it (Kramers-Wannier,
Jordan-Wigner, bosonization, Wegner, particle-vortex, perhaps others). This word is dangerous (it is one of the forbidden words on my blackboard) because it is about ambiguities
in our (physics) language. I would like to reclaim it.<cite>[[https://mcgreevy.physics.ucsd.edu/s14/239a-lectures.pdf|Where do quantum field theories come from?]] by McGreevy</cite></blockquote>
  
<tabbox Student> 
This situation is what people call a duality:

<diagram>
| Lagrangian 1 | | Lagrangian 2|
| |!| | | | !| |
| |`@2| a QFT|  '@8| |
</diagram>
 
However, take note that also if there are three Lagrangians that yield the same QFT, or even n-Lagrangians, people usually don't speak of a triality, or n-ality, but instead call all such situations dualities.

So the thing is that we have two (or more) descriptions - here defined in terms of Lagrangians - that yield the same physical observable results. Hence they describe the same physical situation, but the perspective is changed.

Especially the role of the fundamental degrees of freedom gets switched. The topological excitations of theory 1 become the elementary particles in theory 2 and vice versa.

For example, in [[models:classical_electrodynamics|Maxwell's theory]] supplemented with magnetic monopoles in the usual description, our elementary particles are electrons and the magnetic monopoles are [[advanced_notions:quantum_field_theory:solitons|topological solitons]]. However, in the dual description, the magnetic monopoles are the elementary particles and the electrons are topological solitons.

--> Source#

<blockquote>
In general, the dual theory will describe the interactions of the topological excitations of the original theory. In the case of U(1), these excitations are monopoles in d=3, and monopole loops in d=4.

<cite>[[https://books.google.de/books?id=aKiPXUdkKNAC&lpg=PA18&ots=vMRFe-Girj&dq=duality%20U(1)%20gauge%20theory%20and%20xy%20model&hl=de&pg=PA18#v=onepage&q=duality%20U(1)%20gauge%20theory%20and%20xy%20model&f=false|page 18 in Gauge Field Theories Theoretical Studies and Computer Simulations
by Giarczynski]]</cite>
</blockquote>
<--

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**Examples**

--> Duality of Maxwell's Equations#

The Maxwell equations without electrically charged particles, i.e. only with the electromagnetic field do not change their form under the exchange of the electric and magnetic fields. Thus it is said that the electric and the magnetic field are dual to each other.

However, with electrically charged particles this duality does no longer exist. It was realised in 1931 by Dirac that the duality would still exist with electrically charged particles if there exist **magnetic monopoles**. One way to think about magnetic monopoles is as topological non-trivial configurations of the electromagnetic field for which the electromagnetic field is infinitely large at one point.

While the coupling strength of the electrically charged particles to the electromagnetic field is characterized by the fine structure constant $ \alpha = 1/137$, the duality requires that magnetic charges coupling with  $1/\alpha = 137$ to the electromagnetic field. Therefore, they should be easy to be observed, but never were observed.

The hope is that if one can find a similar duality for general Yang-Mills theories such as QCD one could finally understand what goes on in the strong coupling regime by considering the dual theory, which in turn would be very weakly coupled.

(Source: Not Even Wrong by P. Woit page 138)

<blockquote>To quantize the system we must make a choice. It turns out that consistency requires
the coupling strengths of the two types of currents to be inversely proportional to
each other.<cite>Modern Quantum Mechanics by Banks</cite></blockquote>
 
<--

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**Recommended Resources:**

* [[https://arxiv.org/abs/1603.08334|Comparing Dualities and Gauge Symmetries]] by Sebastian De Haro, Nicholas Teh, Jeremy N. Butterfield
<tabbox Researcher> 
<blockquote>
While a [[basic_tools:symmetry|symmetry]] is a relation within a single theory (e.g. an automorphism of the space state and-or the set of quantities of the theory), a duality is a relation between different theories. Or between different descriptions of the same theory in case the duality is a self-duality. In such a case, the duality is a symmetry in the usual sense of model theory on the semantic conception of theories. ...

Physicists tend to construe duality as an isomorphism of theories: the rough idea is that there is a duality if two theories make the same predictions for all the physical quantities that one can write down in the theory. ...

More precisely, we define a duality as an isomorphism between two models of a single theory, where ‘model’ is here understood not in the usual sense, but as a mathematical representation of the theory (i.e. as a homomorphism from the theory to a mathematical structure that does the representing). What we here call the ‘single theory’ is a bare theory, and the originally-given two dual theories are now called ‘models’. 
  
<cite>https://arxiv.org/pdf/1803.09443.pdf<\cite></blockquote>

</tabbox>

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