# nLab auxiliary field

Contents

### Context

#### Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## Idea

In quantum field theory an “auxiliary field” is a field type that is introduced on top of the fields in the field theory genuinely of interest, in order to bring that field theory of actual interest into another form – usually while not actually changing it up to equivalence – which lends itself better to certain purposes.

## Examples

For example under suitable conditions a field theory with constraints may be formulated equivalently as another field theory without explicit constraints, but with an auxiliary Lagrange multiplier field which does induce that constraint after all, but indirectly so via its equation of motion.

In BV-BRST formalism, a powerful generalization of this idea of Lagrange multiplier is the usage of auxiliary fields that allow to define “gauge fixing fermions” to implement gauge fixing via their equations of motion.

For instance in the quantization of Yang-Mills theory the Nakanishi-Lautrup field (and its “antighost field”) are auxiliary fields that are introduced in order to indirectly induce gauge fixing to Lorentz gauge. The BV-BRST formalism also makes precise how exactly this introduction of auxiliary fields does not change the field theory, up to equivalence: the auxiliary fields change the BV-BRST complex (which effectivley defines the field theory) only up to a quasi-isomorphism, i.e. the relevant notion of equivalence in this context.

In fact even before it comes to gauge fixing, BV-BRST formalism introduces a variety of auxiliary fields: the ghosts, the ghosts-of-ghosts, etc., and the antifields.

Last revised on August 2, 2018 at 03:10:09. See the history of this page for a list of all contributions to it.