Non-linear sigma model: Difference between revisions

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{{Short description|Class of quantum field theory models}}

In [[quantum field theory]], a '''nonlinear ''σ'' model''' describes a [[scalar field]] {{mvar|Σ}} whichthat takes on values in a nonlinear manifold called the '''target manifold''' &nbsp;''T''. The non-linear ''σ''-model was introduced by {{harvtxt|Gell-Mann|Lévy|1960|loc=section §6}}, who named it after a field corresponding to a spinless[[spin (physics)|sp]] [[meson]] called ''σ'' in their model.<ref>{{Citation | last2=Lévy | first1=M. | last1=Gell-Mann | first2=M. | s2cid=122945049 | title=The axial vector current in beta decay | publisher=Italian Physical Society | doi=10.1007/BF02859738 | year=1960 | journal=Il Nuovo Cimento | issn=1827-6121 | volume=16 | issue=4 | pages=705–726| bibcode=1960NCim...16..705G }}</ref> This article deals primarily with the quantization of the non-linear sigma model; please refer to the base article on the [[sigma model]] for general definitions and classical (non-quantum) formulations and results.

==Description==

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This represents a [[Ricci flow]], obeying [[Einstein field equations]] for the target manifold as a fixed point. The existence of such a fixed point is relevant, as it grants, at this order of perturbation theory, that [[conformal field theory|conformal invariance]] is not lost due to quantum corrections, so that the [[quantum field theory]] of this model is sensible (renormalizable).

Further adding nonlinear interactions representing flavor-chiral anomalies results in the [[Wess–Zumino–Witten model]],<ref>{{cite journal |first=E. |last=Witten |s2cid=122018499 |title=Non-abelian bosonization in two dimensions |journal=[[Communications in Mathematical Physics]] |volume= 92| issue= 4 |year=1984 | pages= 455–472 | doi= 10.1007/BF01215276|bibcode = 1984CMaPh..92..455W |url=http://projecteuclid.org/euclid.cmp/1103940923 }}</ref> which

augments the geometry of the flow to include [[Torsion tensor|torsion]], preserving renormalizability and leading to an [[infrared fixed point]] as well, on account of [[teleparallelism]] ("geometrostasis").<ref>{{Cite journal | last1 = Braaten | first1 = E. | last2 = Curtright | first2 = T. L. | last3 = Zachos | first3 = C. K. | doi = 10.1016/0550-3213(85)90053-7 | title = Torsion and geometrostasis in nonlinear sigma models | journal = Nuclear Physics B | volume = 260 | issue = 3–4 | pages = 630 | year = 1985 |bibcode = 1985NuPhB.260..630B }}</ref>

{{further|Ricci flow}}

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[[Category:Quantum field theory]]

[[Category:Mathematical physics]]

[[Category:Murray Gell-Mann]]