nLab Nakanishi-Lautrup field



Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks (qq)
up-typeup quark (uu)charm quark (cc)top quark (tt)
down-typedown quark (dd)strange quark (ss)bottom quark (bb)
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
ϕ-meson (ss¯s \bar s),
kaon, K*-meson (usu s, dsd s)
eta-meson (uu+dd+ssu u + d d + s s)

charmed heavy mesons:
D-meson (uc u c, dcd c, scs c)
J/ψ-meson (cc¯c \bar c)
bottom heavy mesons:
B-meson (qbq b)
ϒ-meson (bb¯b \bar b)
proton (uud)(u u d)
neutron (udd)(u d d)

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)


in grand unified theory

minimally extended supersymmetric standard model




dark matter candidates


auxiliary fields



In BV-BRST formalism, for gauge fixing Yang-Mills theory (to Lorenz gauge or similar) a contractible chain complex of auxiliary field bundles is introduced for two Lie algebra-valued fields, one in degree zero, called the Nakanishi-Lautrup field, usually denoted “BB” and one in degree -1, called the antighost field, usually denoted C¯\overline{C}. See at quantization of Yang-Mills theory.

Beware that there are also the antifields of the ghost fields, which technically are hence “anti-ghostfields” as opposed to the Nakanishi-Lautrup “antighost-fields”. Whoever is responsible for this bad terminology should be blamed.


Named after Benny Lautrup and Noburo Nakanishi, who is sometimes misspelled as “Takanishi”.

  • Noburo Nakanishi, Covariant quantization of the electromagnetic Field in the Landau Gauge, Prog. Theor. Phys. 35, 1111 (1966)
  • Benny Lautrup, Canonical quantum electrodynamics in covariant gauges, Kong. Dan. Vid. Sel. Mat. Fys. Med. 35, 29 (1967)

Review for the case of electromagnetism and with path integral terminology is in

  • Marc Henneaux, section 9.1 of Lectures on the Antifield-BRST formalism for gauge theories, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 (pdf)

while discussion for general Yang-Mills theory in the context of causal perturbation theory/perturbative algebraic quantum field theory is in

  • Katarzyna Rejzner, section 7.2 of Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)

Last revised on October 20, 2019 at 20:21:49. See the history of this page for a list of all contributions to it.