Chiral symmetry
Encyclopedia
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.
(QCD) with two massless quarks u and d. The Lagrangian is
In terms of left-handed and right-handed spinors it becomes
(Hereby i is the imaginary unit and
the well-known Dirac operator
.)
Defining
it can be written as
The Lagrangian is unchanged under a rotation of
by any 2 x 2 unitary matrix L, and 
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.
The axial symmetry
acts as
and it does not correspond to a conserved quantity because it is violated due to quantum anomaly
.
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 differing masses of the quarks,
is only an approximate symmetry to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone boson
s.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, chiral symmetry is a possible symmetry
Symmetry in physics
In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...
of the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
under which the left-handed
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
and right-handed
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
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.
Example: u and d quarks in QCD
Consider quantum chromodynamicsQuantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(QCD) with two massless quarks u and d. The Lagrangian is
In terms of left-handed and right-handed spinors it becomes
(Hereby i is the imaginary unit and
the well-known Dirac operatorDirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
.)
Defining
it can be written as
The Lagrangian is unchanged under a rotation of
by any 2 x 2 unitary matrix L, and 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 intoThe vector symmetry
acts asand corresponds to baryon number conservation.
The axial symmetry
acts asand it does not correspond to a conserved quantity because it is violated due to quantum anomaly
Anomaly (physics)
In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...
.
The remaining chiral symmetry
turns out to be spontaneously brokenSpontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
by quark condensate into the vector subgroup
, known as isospinIsospin
In physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...
. The Goldstone bosons corresponding to the three broken generators are the pions. In real world, because of the differing masses of the quarks,
is only an approximate symmetry to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosonPseudo-Goldstone boson
Pseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken , and would thus engender massless Goldstone bosons. The additional explicit symmetry breaking...
s.