In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently. The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as vector symmetry, and a component that actually treats them differently, known as axial symmetry. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... This article or section does not cite its references or sources. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ... In physics, a Dirac field is a fermionic field (usually a quantized field, as usual in quantum field theory) associated with spin 1/2 fermions such as the electron or muon. ...

Example: u and d quarks in QCD

Consider quantum chromodynamics (QCD) with two massless quarks u and d. The Lagrangian is Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... For other uses of this term, see: Quark (disambiguation) 1974 discovery photograph of a possible charmed baryon, now identified as the Σc++ In particle physics, the quarks are subatomic particles thought to be elemental and indivisible. ...

In terms of left-handed and right-handed spinors it becomes

Defining

it can be written as

The Lagrangian is unchanged under a rotation of q_L by any 2 x 2 unitary matrix L, and q_R by any 2 x 2 unitary matrix R. This symmetry of the Lagrangian is called flavor symmetry or chiral symmetry, and denoted as . It can be decomposed into

The vector symmetry acts as

and corresponds to baryon number conservation. In particle physics, the baryon number is an approximate conserved quantum number. ...

The axial symmetry acts as

and it does not correspond to a conserved quantity because it is violated due to quantum anomaly. In physics, an anomaly is a classical symmetry — a symmetry of the Lagrangian — that is broken in quantum field theories. ...

The remaining chiral symmetry turns out to be spontaneously broken by quark condensate into the vector subgroup , known as isospin. The Goldstone bosons corresponding to the three broken generators are the pions. In real world, because of the masses of the quarks and electromagnetism, is only an approximate symmetry to begin with, therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons. Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... This article may be too technical for most readers to understand. ... Isospin (isotopic spin, isobaric spin) is a physical quantity which is mathematically analogous to spin. ... In particle and condensed matter physics, Goldstone bosons are bosons that appear in models with spontaneously broken symmetry. ... In particle physics, pion (short for the Greek pi meson = P middle) is the collective name for three subatomic particles discovered in 1947: π0, π+ and π−. Pions are the lightest mesons. ... Pseudo-Goldstone bosons arise in a quantum field theory with an approximate symmetry such that if the symmetry were exact, then there would be spontaneous symmetry breaking (SSB) and the consequent formation of Goldstone bosons. ...