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en>Michael Hardy at 17:58, 9 November 2012
2012-11-09T17:58:24Z
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<p><b>New page</b></p><div>{{more footnotes|date=December 2012}}<br />
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The '''Nielsen-Ninomiya theorem''' is a [[no-go theorem]] in physics, in particular in [[lattice gauge theory]], concerning the possibility of defining a theory of [[Chirality (physics)|chiral]] [[fermions]] on a [[Lattice model (physics)|lattice]] in even dimensions. The theorem can be stated as follows: let <math>S[\psi]</math> be the (Euclidean) action describing fermions <math>\psi</math> on a regular lattice of even dimensions with periodic boundary conditions, and suppose that ''S'' is local, [[hermitian]] and [[translation invariant]]; then the theory describes as many left-handed as right-handed states. Equivalently, the theorem implies that there are as many states of chirality +1 as of chirality -1. The proof of the theorem relies on the [[Poincaré-Hopf theorem]] or on similar results in [[algebraic topology]]. <br />
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Since the [[Standard Model (mathematical formulation)|Standard Model]] is chiral (left- and right-handed fermions are treated differently by weak interactions, for example), the Nielsen-Ninomiya theorem implies that for simulating some Standard Model phenomena at least one of the assumptions of the theorem needs to be violated.<br />
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==References==<br />
*{{Citation|last1=Nielsen|first1=H.B.|last2=Ninomiya|first2=M.|title=A no-go theorem for regularizing chiral fermions|journal=[[Physics Letters|Phys. Lett.]]|volume=B105|pages=219|year=1981}}<br />
*{{Citation|last1=Nielsen|first1=H.B.|last2=Ninomiya|first2=M.|title=Absence of neutrinos on a lattice: (I). Proof by homotopy theory|journal=[[Nuclear Physics (journal)|Nucl. Phys.]]|volume=B185|pages=20|year=1981}}<br />
*{{Citation|last1=Nielsen|first1=H.B.|last2=Ninomiya|first2=M.|title=Absence of neutrinos on a lattice: (II). Intuitive topological proof|journal=[[Nuclear Physics (journal)|Nucl. Phys.]]|volume=B193|pages=173|year=1981}}<br />
* I. Montvay and G. Münster, ''Quantum Fields on a Lattice'', Cambridge University Press 1997<br />
*{{Citation | last1=Itzykson | first1=Claude | last2=Drouffe | first2=Jean-Michel | title=Statistical field theory. Vol. 1 | publisher=[[Cambridge University Press]] | series=Cambridge Monographs on Mathematical Physics | isbn=978-0-521-34058-8 | mr=1175176 | year=1989}}<br />
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[[Category:Quantum_field_theory]]<br />
[[Category:Lattice_models]]</div>
en>Michael Hardy