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--- advanced_notions:quantum_field_theory:solitons [2018/03/17 13:49]
jakobadmin [Layman]
+++ advanced_notions:quantum_field_theory:solitons [2018/05/05 12:38] (current)
jakobadmin ↷ Links adapted because of a move operation
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 <tabbox Why is it interesting?>
-[[advanced_tools:feynman_diagrams|Feynman diagrams]] do not describe everything that can happen in a [[theories:quantum_theory:quantum_field_theory|quantum field theory]]. There can be classical solutions of the field equations that describe larger lumps of field excitations that aren't describable by Feynman diagrams.
+[[advanced_tools:feynman_diagrams|Feynman diagrams]] do not describe everything that can happen in a [[theories:quantum_field_theory:canonical|quantum field theory]]. There can be classical solutions of the field equations that describe larger lumps of field excitations that aren't describable by Feynman diagrams.
 Classical solutions of the field equations with finite energy are called solitons.
-Such solutions are important to describe, for example, the vacuum of a theory. A famous example is the [[advanced_notions:quantum_field_theory:cd_vacuum|QCD vacuum]] which can only understand with the help of [[advanced_notions:quantum_field_theory:instantons|instantons]].
+Such solutions are important to describe, for example, the vacuum of a theory. A famous example is the [[advanced_notions:quantum_field_theory:qcd_vacuum|QCD vacuum]] which can only understand with the help of [[advanced_notions:quantum_field_theory:instantons|instantons]].
 In addition, there is an [[http://www.pbs.org/wgbh/nova/blogs/physics/2011/12/beautiful-losers-kelvins-vortex-atoms/|old dream]] that all elementary particles could be explained as topological solitons. (There are lots of problems with this idea, but at least, [[https://plus.google.com/+UrsSchreiber/posts/Z2LfHsyxgR8|instantons come somewhat close]].)
+While solitons are rare in particle physics, they are found frequently in condensed matter physics.
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 <tabbox Student>
+A soliton is a wave-packet that keeps a stable shape while propagating. The defining features of a soliton are:
+  - A soliton is of permanent form, which basically means that they are static (=time independent) solutions of the field equations
+  - A soliton is localized within a finite region. In other words, its energy and spatial size is finite.
+  - When a soliton interacts with another soliton it emerges from the collision in exactly the same shape it had before. Only a phase shift is possible.
+----
 {{ :advanced_notions:quantum_field_theory:dissipationvsnonlinearitycropped.png?nolink&600|}}
 Solitons are stable through the interplay of dissipation and non-linearity of the underlying wave equations.
@@ Line 75: / Line 86: @@
   * Non-linearity of the wave equations can result in waves that get __steeper__ over time. A good example are the waves that can be observed at a beach.
+----
+**Recommended Textbooks**
+  * Solitons and Instantons by Ramamurti Rajaraman - is the best introductory book on solitons and related topics
+  * Topological Solitons by Manton and Sutcliff - is the second-best introductory book on solitons
+  * [[http://scipp.ucsc.edu/~haber/ph218/classicallumpsreview_Infanger.pdf|Classical lumps and their quantum descendants]] by Sidney Coleman - a "must read" lecture for anyone interested in solitons
+  * Classical Solutions in Quantum Field Theory: Solitons and Instantons by Erick Weinberg - contains several helpful chapters
+  * Classical Theory of Gauge Fields by Rubakov - is great to dive deeper and contains many alternative perspectives that can't be found anywhere else.
+  * Quarks, Leptons & Gauge Fields by Kerson Huang - contains several extremely helpful chapters regarding solitons etc.
+  * Quantum Field Theory by Lewis H. Ryder - contains, like Huang's book - a particular nice chapter on solitons and instantons
 <tabbox Researcher>
-<note tip>
+<blockquote>Many field theoretical models at the heart of our present understanding of elementary
-The motto in this section is: //the higher the level of abstraction, the better//.
+particles and their interactions have topological defects between the solutions of their
-</note>
+classical counterparts. Because hadrons, particles interacting via strong subnuclear forces, are of
+two types -heavy (baryons), and light (mesons)- it was tempting to think of them respectively as
+quantum solitons and light quanta. This point of view was pioneered by Skyrme and Finkelstein
+as early as the sixties. The first author even proposed a variation that encompasses solitons
+on the (at that time fashionable) Gell-Mann/Levy sigma model of strong interactions . In the
+Skyrme model, the solitons, usually referred to as Skyrmions, would describe the classical limit
+of baryons whereas mesons were associated with light quanta.<cite>https://arxiv.org/pdf/hep-th/0611180.pdf</cite></blockquote>
+----
+  * Geometry of Yang-Mills Fields by M. F. ATIYAH plus chapters in
+  * Geometry of Physics by Frankel and
+  * Topology and Geometry for Physicists by Nash and Sen
 <tabbox Examples>
@@ Line 96: / Line 134: @@
 <tabbox FAQ>
+-->Do we really only care about finite energy solutions?#
+At first, it seems completely reasonable to consider finite energy field configurations. However, take note that usually in QFT we only deal with field configurations with infinite field energy. The "normal" wave solutions of our equations of motion etc. that we use to describe elementary particles are classical infinite energy solutions. Source: page 56 in Quarks, Leptons & Gauge Fields by K. Huang
+See also Coleman, The Use of Instantons page 284 in Aspects of Symmetry: "In fact, it is configurations of finite energy that are unimportant; to be precise, they form a set of measure zero in function space. [...] The only reason we are interested in configurations of finite action is that we are interested in doing semiclassical approximations, and a configuration of infinite action does indeed give zero if it is used as the center point of a Gaussian integral." )
+<--
 <tabbox History>
 Solitons were first described in 1834 by John Scott Russell:

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