Nielsen-Ninomiya theorem

The Nielsen-Ninomiya theorem from lattice theory says that when a field theory is discretized with fermions, the number of right- and left-handed particles is the same. The only prerequisite are some general assumptions about the Hamilton operator . The theorem is named after Holger Bech Nielsen and Masao Ninomiya and was established in 1981.

overview

One tries a simple discretization of a field theory with fermions by looking at the effect

${\displaystyle S={\overline {\psi }}\,(\gamma ^{\mu }\partial _{\mu }+m)\,\psi }$

If the derivative is replaced by differential quotients , it can be seen that this leads to a doubling of the particles for every existing dimension. The number of left- and right-handed particles is the same.

formulation

Nielsen and Ninomiya have shown that the phenomenon of doubling occurs under very general conditions: For every quantum number there is an equal number of left- and right-handed particles if the Hamiltonian is translation-invariant , local and Hermitian .

Fermions on the grid

There are different approaches for simulating fermions on the grid .

Wilson's idea was to provide all “Dopplers” with an additional mass that goes towards infinity in the limes continuum and thus decouple the Dopplers.

Susskind pursued a different approach : Here the Dopplers are treated as an additional flavor (quantum number). These particles are called staggered fermions .

However, generation of chiral fermions on the lattice - for example for a simulation of a minimal standard model - is possible despite the Nielsen-Ninomiya theorem; for this only one requirement of the theorem has to be violated. One variant is the breaking of the translation invariance by introducing an additional dimension, as happens with the domain wall fermions .

literature

• HB Nielsen and M. Ninomiya, Absence of neutrinos on a lattice. 1. Proof by homotopy theory Nucl. Phys. B185 (1981) 20.
• HB Nielsen and M. Ninomiya, Absence of neutrinos on a lattice. 2. Intuitive topological proof Nucl. Phys. B193 (1981) 173.
• HB Nielsen and M. Ninomiya, No Go Theorem for Regularizing Chiral Fermions Phys. Lett. B105 (1981) 219.
• D. Friedan, A proof of the Nielsen-Ninomiya theorem Comm. Math. Phys. 85 (1982) 481.

Individual evidence

1. ^ KG Wilson, Quarks and strings on a lattice New Phenomena in Subnuclear Physics Part A (1975) 69.
2. ^ J. Kogut and L. Susskind, Hamilon formulation of Wilson's lattice gauge theories Phys. Rev. D 11 (1975) 395.
3. ↑ T.Banks, J. Kogut and L. Susskind, Strong coupling calculations of lattice gauge theories: (1 + 1) dimensional exercises Phys. Rev. D 13 (1976) 1043.