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--- advanced_notions:duality [2018/04/01 13:34]
jakobadmin [Researcher]
+++ advanced_notions:duality [2020/04/12 15:04] (current)
jakobadmin
@@ Line 1: / Line 1: @@
 ====== 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.
+<tabbox Intuitive>
+A duality is like a translation scheme between languages. The same thing, say a chair, can be described either in English through the word "chair", but equivalently, for example, in German through the word "Stuhl". For certain purposes, one language is better suited than another. For example, the relatively simple English word "pollution" is translated into German the horribly long word "Umweltverschmutzung". So if you're dealing with an issue surrounding pollution, the English language is better suited.
-----
+A bit more precisely a duality is a formal or theoretical equivalence between two theories. Often physicists call the definition of the duality transformation that takes us from one language to another the ‘dictionary’.
+Take note that instead of different descriptions of the same thing in different languages, we can have many synonyms for one and the same thing //within// one language. In the physical context this goes under the name [[advanced_tools:gauge_symmetry|gauge symmetry]].
-----
-<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>
+<tabbox Concrete>
 This situation is what people call a duality:
@@ Line 46: / Line 40: @@
 by Giarczynski]]</cite>
 </blockquote>
+<blockquote>There should be two "dual equivalent" field formulations of the same theory in which electric (Noether) and magnetic (topological) quantum numbers exchange roles.<cite>[[http://www.indiana.edu/~jpac/QCDRef/1970s/MAGNETIC%20MONOPOLES%20AS%20GAUGE%20PARTICLES%20-%20Montonen%20-%201977.pdf|Magnetic monopoles as gauge particles?]] by C. Montonen and D. Olive</cite></blockquote>
 <--
-**Recommended Resources:**
-  * [[https://arxiv.org/abs/1603.08334|Comparing Dualities and Gauge Symmetries]] by Sebastian De Haro, Nicholas Teh, Jeremy N. Butterfield
+----
-<tabbox Researcher>
+**Examples**
-<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.
-<cite>https://arxiv.org/pdf/1803.09443.pdf<\cite></blockquote>
-<tabbox Examples>
---> Duality of Maxwell's Equations#
+--> Electric-magnetic duality in classical electromagnetism#
@@ Line 78: / Line 68: @@
 <--
+For a nice discussion of other examples see https://arxiv.org/pdf/1803.09443.pdf
+----
-<tabbox FAQ>
+**Recommended Resources:**
+  * [[https://arxiv.org/abs/1603.08334|Comparing Dualities and Gauge Symmetries]] by Sebastian De Haro, Nicholas Teh, Jeremy N. Butterfield
+----
+<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. <cite>[[https://mcgreevy.physics.ucsd.edu/s14/239a-lectures.pdf|Where do quantum field theories come from?]] by McGreevy</cite></blockquote>
+<blockquote>
+There are many theories, which have more than one Lagrangian. So that's the opposite. Either we have no Lagrangian at all or we have more than one Lagrangian.
+<cite>Duality and emergent gauge symmetry - Nathan Seiberg</cite>
+</blockquote>
+<tabbox Abstract>
+<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’.
-<tabbox History>
+<cite>https://arxiv.org/pdf/1803.09443.pdf</cite></blockquote>
-</tabbox>
+<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:canonical|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.
+----
+<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>
+<blockquote>Duality symmetries: mapping two theories or two different descriptions of a theory one-to-one to
+the other. The prime example is self-duality for the free Maxwell theory: you
+can mutually exchange the electric and the magnetic field, and this replacement
+leaves Maxwell’s equations invariant. Another example is quantum physics either
+described by Schrödinger’s wave equation or with Heisenberg’s matrix mechanics.
+Today, dualities are experiencing a boom, triggered by so called string theory.
+The five families of string models exhibit astoundingly many dualities, if one
+introduces branes; for a non-expert this is nicely described in [433]. Another very
+influential development was the discovery of a duality between anti-de Sitter grav-
+ity and a conformal field theory, sometimes named after its inventor as Maldacena
+conjecture. <cite>Symmetries in Fundamental Physics by Sundermeyer</cite></blockquote>
+</tabbox>

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