Chiral symmetry

The Chiral Symmetry (from Greek χέρι hand) is a possible symmetry of the Lagrangian in quantum field theory, which in many cases - is given and then plays an important role, eg for the pion - at least approximately.

This left-handed and right-handed portion of the Fermionic fields are independently transformed. The chiral symmetry transformation can be divided into a component that is left-handed and right-handed fraction treated equally (vector symmetry), and a component that they " opposed " treated ( Axial symmetry). The latter portion dissipates by quark condensation in the former phase.

Example: u-and d - quarks in QCD

Consider the quantum chromodynamics ( QCD) with two massless quarks u and d is the Lagrangian

The i here means the imaginary unit and the Dirac operator in the Feynman slash notation. U and d are the usual sizes of the Dirac theory, with four components.

After the quantum mesons composed of one quark and one antiquark are composed, eg from one and one. That does not, however, change the following derivation or just " cum grano salis ".

In the representation of left-handed and right-handed spinors so initially obtained

It is defined

It thus follows

The Lagrangian is invariant under rotation of each with unitary 2x2 matrices L and in rotation with the unitary 2x2 matrices R. This symmetry of the Lagrangian function is called flavor symmetry or chiral symmetry and as quoted. They can be broken down into the following symmetries

The vector symmetry is

And corresponds to the baryon number conservation.

The corresponding axial operation is

It does not correspond to conserved quantity, since it is broken by a quantum anomaly.

It turns out that the remaining chiral symmetry for vector subgroup ( the isospin group) is spontaneously broken. The symmetry breaking is manifested by an appropriate case, complete quark condensate.

The Goldstone bosons, corresponding to the three broken generators of the transformation, the pions. Since the masses of the quarks are not equal, is only approximate symmetry of the system. The pions are therefore not "real", massless Goldstone bosons, but so-called pseudo - Goldstone bosons.

Chiral limit

To be distinguished from the " chiral symmetry " is the so-called " chiral limit " ( ) of a single Dirac equation. This limit is best for neutrinos and their antiparticles with their well-defined chirality realized ( " left-handed helix " or " right-handed screw " with respect to spin and momentum for neutrinos and anti- neutrinos ), as well as in the solid state in graphene.