Lorentz-violating neutrino oscillations: Difference between revisions

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{{Distinguish|Thirring model}}
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The '''Thirring–Wess model''' or '''Vector Meson model'''
is an exactly solvable
quantum field theory describing the interaction of a  [[fermion field#Dirac fields|Dirac field]] with a vector field in dimension two.
 
==Definition==
The [[Lagrangian density]] is made of three terms:
 
the free vector field <math> A^\mu</math>  is described by
 
:<math>
{(F^{\mu\nu})^2 \over 4}
+{\mu^2\over 2} (A^\mu)^2  
</math>
 
for <math> F^{\mu\nu}= \partial^\mu A^\nu - \partial^\nu A^\mu </math> and the boson mass <math>\mu</math> must be
strictly positive;
the free fermion field <math> \psi </math>
is described by
 
:<math>
\overline{\psi}(i\partial\!\!\!/-m)\psi
</math>
 
where the fermion mass <math>m</math> can be positive or zero.
And the interaction term is
:<math>
qA^\mu(\bar\psi\gamma^\mu\psi)
</math>
 
Although not required to define the massive vector field, there can be  also a gauge-fixing term
:<math>
{\alpha\over 2} (\partial^\mu A^\mu)^2
</math>
for <math> \alpha \ge 0 </math>
 
There is a remarkable difference between the case <math> \alpha > 0 </math> and the case <math> \alpha = 0 </math>:  the latter  requires a [[renormalization|field renormalization]] to absorb divergences of the two point correlation.
 
==History==
This model was introduced by Thirring and Wess as a version  of the [[Schwinger model]] with a vector mass term in the Lagrangian .
 
When the fermion is massless (<math> m= 0 </math>), the model is exactly solvable. One solution was found, for <math> \alpha =1 </math>, by  Thirring and Wess <ref name = tw>{{Cite journal| last = Thirring| first = W| coauthor= Wess J|
authorlink = |
title = Solution of a field theoretical model in one space one time dimensions |
journal = [[Annals Phys.]]|
volume = 27 | issue = | pages = 331–337
| year = 1964| publisher =
| url =
| format =  | id = | accessdate = }}</ref> 
using a method introduced by Johnson for the [[Thirring model]]; and, for <math> \alpha = 0 </math>, two different solutions were given by Brown<ref name = br>{{Cite journal| last = Brown| first = L|
authorlink = |
title = Gauge invariance and Mass in a Two-Dimensional Model|
journal = [[N.Cimento.]]|
volume = 29 | issue = | pages =
| year = 1963| publisher =
| url =
| format = | id = | accessdate = }}</ref> and Sommerfield.<ref name = sm>{{Cite journal| last = Sommerfield | first = C|
authorlink = |
title =  |
journal = [[Annals Phys.]]|
volume = 26 | issue = | pages =
| year = 1964| publisher =
| url =
| format = | id = | accessdate = }}</ref> Subsequently Hagen <ref name = ha>{{Cite journal| last = Hagen| first = C| 
authorlink = |
title =  Current definition and mass renormalization in a Model Field Theory|
journal = [[N. Cimento A]]|
volume = 51 | issue = | pages =
| year = 1967| publisher =  
| url =  
| format = | id = | accessdate = }}</ref> showed (for <math> \alpha = 0 </math>, but it turns out to be true for <math> \alpha \ge 0 </math>) that there is a one parameter family of solutions.
 
==References==
<references/>
 
==External links==
 
{{Quantum field theories}}
 
{{DEFAULTSORT:Thirring-Wess Model}}
[[Category:Quantum field theory]]
[[Category:Exactly solvable models]]
 
 
{{Quantum-stub}}

Latest revision as of 13:10, 5 May 2014

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