https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=150.244.37.45formulasearchengine - User contributions [en]2026-07-15T05:24:10ZUser contributionsMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Finite_potential_well&diff=11512Finite potential well2013-10-21T16:44:05Z<p>150.244.37.45: /* Finding wavefunctions for the bound state */ minor edit fixed typo</p>
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<div>In [[theoretical physics|theoretical]] [[condensed matter physics]], '''Bosonization''' is a mathematical procedure by which a system of interacting [[fermions]] in [[dimension|(1+1) dimensions]] can be transformed to a system of massless, non-interacting [[bosons]].<ref name=gogolin>{{cite book|last=Gogolin|first=Alexander O.|title=Bosonization and Strongly Correlated Systems|year=2004|publisher=Cambridge University Press|isbn=0-521-61719-7|url=http://books.google.com/books?id=BZDfFIpCoaAC&dq=Bosonization+and+Strongly+Correlated+Systems&lr=&source=gbs_navlinks_s}}</ref> The method of bosonization was conceived independently by particle physicists [[Sidney Coleman]] and [[Stanley Mandelstam]]; and condensed matter physicists [[Daniel Mattis]] and [[Alan Luther]] in 1975.<ref name=gogolin /> <br />
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The basic physical idea behind bosonization is that [[Dirac sea|particle-hole excitations]] are bosonic in character. However, it was shown by [[Sin-Itiro Tomonaga|Tomonaga]] in 1950 that this principle is only valid in one-dimensional systems.<ref name=senechal>{{cite journal|last=Sénéchal|first=David|title=An Introduction to Bosonization|journal=Theoretical Methods for Strongly Correlated Electrons|year=1999|series=CRM Series in Mathematical Physics|doi=10.1007/0-387-21717-7_4|url=http://www.springerlink.com/content/q8p7094320733111/}}</ref> Bosonization is an [[effective field theory]] that focuses on low-energy excitations.<ref name=glazman>{{cite book|last=Sohn|first=Lydia (ed.)|title=Mesoscopic electron transport|year=1997|publisher=Springer|isbn=0-7923-4737-4|url=http://arxiv.org/pdf/cond-mat/9610037v1.pdf}}</ref> <br />
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Two complex fermions <math>\psi,\bar\psi</math> are written as functions of a boson <math>\phi</math><br />
:<math>\bar\psi_-\psi_+ = :\exp(i\phi):,\qquad \bar\psi_-\psi_+ = :\exp(-i\phi):</math><ref><br />
In actuality, there is a [[cocycle]] prefactor to give correct (anti-)commutation relations with other fields under consideration.</ref><br />
while the inverse map is given by<br />
:<math>\partial\phi=:\bar\psi\psi:</math><br />
All equations are [[normal-order]]ed. The changed statistics arises from [[anomalous dimension]]s of the fields.<br />
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==References==<br />
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[[Category:Quantum field theory]]<br />
[[Category:Condensed matter physics]]<br />
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{{quantum-stub}}</div>150.244.37.45