Relativistic quantum field

A free real scalar field obeys the Klein-Gordon equation and creates neutral spin-zero particles.

A complex scalar field has distinct particle and antiparticle excitations and a global phase symmetry.

A free Dirac field has Lagrangian density $\mathcal{L}=i\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\psi$ and obeys the Dirac equation.

The Dirac equation is the relativistic spin-one-half field equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$.

A left- or right-handed Weyl spinor transforms in the Lorentz representation $(1/2,0)$ or $(0,1/2)$. Parity exchanges the two chiralities.

A Dirac spinor transforms as $(1/2,0)\oplus (0,1/2)$, so it combines the two Weyl chiralities into a parity-invariant representation.

A gauge field is a connection associated with a local symmetry. An Abelian gauge potential $A_{\mu }$ has field strength $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$.

Yang-Mills theory is the non-Abelian gauge theory with field strengthUnder an infinitesimal gauge transformation it transforms covariantly as $\delta F_{\mu \nu }=-ig[F_{\mu \nu },\alpha ]$.

In four dimensions the Yang-Mills theta term is proportional to $\theta \mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. Its density is a total derivative, but nontrivial gauge-field topology can make its spacetime integral physically relevant in the quantum theory.

The Chern-Simons current satisfies ${\partial }_{\mu }{K}^{\mu }=\mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. It is not gauge invariant even though its divergence is.

A quantum anomaly is the failure of a classical symmetry to survive quantization because the functional measure or regulator cannot preserve it.

A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

The chiral anomaly is the quantum nonconservation of a classically conserved axial current in a gauge-field background.

For massless Dirac fermions, the classical axial current is $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. The chiral anomaly makes its divergence proportional to $F_{\mu \nu }{}^{\star }{F}^{\mu \nu }$.

A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is preserved by renormalization-group flow and constrains possible infrared phases.

't Hooft anomaly matching requires the massless infrared degrees of freedom, topological sector, or symmetry-breaking pattern to reproduce every anomaly of an unbroken global symmetry measured in the ultraviolet theory.

Gauge fixing removes the degeneracy among gauge-equivalent field configurations so that the kinetic operator has an inverse propagator.

Adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\alpha )$ gives a family of Lorentz-covariant gauges; $\alpha =1$ is Feynman gauge.

The Faddeev-Popov determinant is the functional Jacobianthat compensates for the change from integration along a gauge orbit to a gauge-fixing condition $G(A)=0$. It can be represented by a path integral over a Faddeev-Popov ghost field pair.

A Faddeev-Popov ghost field is a Grassmann-valued scalar field whose Gaussian functional integral represents the Faddeev-Popov determinant. Ghosts occur only on internal lines and cancel unphysical gauge-field contributions.

An axial gauge imposes $n^{\mu }{A}_{\mu }=0$ for a fixed vector $n^{\mu }$. Its Faddeev-Popov operator is $n^{\mu }{D}_{\mu }$; on the strict gauge slice its gauge-field-dependent part vanishes, so its ghosts decouple.

BRST symmetry is a nilpotent fermionic symmetry of a gauge-fixed action. Its differential $s$ replaces an infinitesimal gauge parameter by the ghost field and satisfies $s^2=0$.

A gauge-fixing fermion is a Grassmann-odd functional whose BRST transformation $s\Psi$ supplies the gauge-fixing and ghost terms. Nilpotence gives $s(s\Psi )=0$ immediately.

BRST cohomology identifies physical states and observables with BRST-closed objects modulo BRST-exact ones. A change of gauge-fixing fermion changes the action by a BRST-exact term and therefore leaves BRST-cohomology classes unchanged when the measure has no BRST anomaly.