In mathematical physics, Yang–Mills theory is a
gauge theory based on a
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
SU(''N''), or more generally any
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
,
reductive Lie algebra
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: \mathfr ...
. Yang–Mills theory seeks to describe the behavior of elementary particles using these
non-abelian Lie groups and is at the core of the unification of the
electromagnetic force
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and
weak force
Weak may refer to:
Songs
* Weak (AJR song), "Weak" (AJR song), 2016
* Weak (Melanie C song), "Weak" (Melanie C song), 2011
* Weak (SWV song), "Weak" (SWV song), 1993
* Weak (Skunk Anansie song), "Weak" (Skunk Anansie song), 1995
* "Weak", a song ...
s (i.e. U(1) × SU(2)) as well as
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
, the theory of the
strong force
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the n ...
(based on SU(3)). Thus it forms the basis of our understanding of the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
of particle physics.
History and theoretical description
In 1953, in a private correspondence,
Wolfgang Pauli
Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
formulated a six-dimensional theory of
Einstein's field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, extending the five-dimensional theory of
Kaluza, Klein,
Fock and others to a higher-dimensional internal space.
However, there is no evidence that Pauli developed the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of a
gauge field
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles", he refrained from publishing his results formally.
Although Pauli did not publish his six-dimensional theory, he gave two talks about it in Zürich. Recent research shows that an extended Kaluza–Klein theory is in general not equivalent to Yang–Mills theory, as the former contains additional terms.
Chen Ning Yang long considered the idea of non-abelian gauge theories. Only after meeting
Robert Mills did he introduce the junior scientist to the idea and lay the key hypothesis that Mills would use to assist in creating a new theory. This eventually became the Yang–Mills theory, as Mills himself discussed,
"During the academic year 1953-1954, Yang was a visitor to Brookhaven National Laboratory
Brookhaven National Laboratory (BNL) is a United States Department of Energy national laboratory located in Upton, Long Island, and was formally established in 1947 at the site of Camp Upton, a former U.S. Army base and Japanese internment c ...
...I was at Brookhaven also...and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length...I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's."
In early 1954, Yang and Mills
extended the concept of gauge theory for
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s, e.g.
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
, to non-abelian groups to provide an explanation for strong interactions. The idea by Yang–Mills was criticized by Pauli, as the
quanta of the Yang–Mills field must be massless in order to maintain
gauge invariance
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
. The idea was set aside until 1960, when the concept of particles acquiring mass through
symmetry breaking
In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observe ...
in massless theories was put forward, initially by
Jeffrey Goldstone
Jeffrey Goldstone (born 3 September 1933) is a British theoretical physicist and an ''emeritus'' physics faculty member at the MIT Center for Theoretical Physics.
He worked at the University of Cambridge until 1977. He is famous for the discove ...
,
Yoichiro Nambu
was a Japanese-American physicist and professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded half of the Nobel Prize in Physics in 2008 for the discovery in 1960 of the mechanism ...
, and
Giovanni Jona-Lasinio
Giovanni Jona-Lasinio (born 1932), sometimes called Gianni Jona, is an Italian theoretical physicist, best known for his works on quantum field theory and statistical mechanics. He pioneered research concerning spontaneous symmetry breaking, and ...
.
Yang–Mills theory was independently discovered by Ronald Shaw in January 1954, a graduate student at the
University of Cambridge
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
. Since the resulting massless particles did not seem to be found in nature at the time, Shaw and his supervisor
Abdus Salam
Mohammad Abdus Salam Salam adopted the forename "Mohammad" in 1974 in response to the anti-Ahmadiyya decrees in Pakistan, similarly he grew his beard. (; ; 29 January 192621 November 1996) was a Punjabi Pakistani theoretical physicist and a ...
chose not to publish the results. Shortly after Yang and Mills published their paper, Salam encouraged Shaw to publish his work to mark his contribution, however Shaw declined and instead it only forms a chapter in his PhD thesis published in 1956.
This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of both
electroweak unification and quantum chromodynamics (QCD). The electroweak interaction is described by the gauge group SU(2) × U(1), while QCD is an
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
Yang–Mills theory. The massless gauge bosons of the electroweak SU(2) × U(1) mix after
spontaneous symmetry breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the ...
to produce the 3 massive weak bosons (, , and ) as well as the still-massless
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
field. The dynamics of the photon field and its interactions with matter are, in turn, governed by the U(1) gauge theory of quantum electrodynamics. The
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
combines the
strong interaction
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the n ...
with the unified electroweak interaction (unifying the
weak and
electromagnetic interaction
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
) through the symmetry group SU(3) × SU(2) × U(1). In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed
running of the coupling constants it is believed they all converge to a single value at very high energies.
Phenomenology
Phenomenology may refer to:
Art
* Phenomenology (architecture), based on the experience of building materials and their sensory properties
Philosophy
* Phenomenology (philosophy), a branch of philosophy which studies subjective experiences and a ...
at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why
confinement
Confinement may refer to
* With respect to humans:
** An old-fashioned or archaic synonym for childbirth
** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods
** Civil confi ...
has not been theoretically proven, though it is a consistent experimental observation. This shows why QCD confinement at low energy is a mathematical problem of great relevance, and why the
Yang–Mills existence and mass gap
The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 f ...
problem is a
Millennium Prize Problem.
Mathematical overview
Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
:
with the generators
of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, indexed by , corresponding to the ''F''-quantities (the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
or field-strength form) satisfying
:
Here, the are
structure constant
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting pr ...
s of the Lie algebra (totally antisymmetric if the generators of the Lie algebra are normalised such that
is proportional with
), the
covariant derivative is defined as
:
is the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
(matching the size of the generators),
is the
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
potential, and ''g'' is the
coupling constant. In four dimensions, the coupling constant ''g'' is a pure number and for a SU(''N'') group one has
The relation
:
can be derived by the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
:
The field has the property of being self-interacting and the equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only by
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
with small nonlinearities.
Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for ''a'' indices (e.g.
), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature,
.
From the given Lagrangian one can derive the equations of motion given by
:
Putting
, these can be rewritten as
:
A
Bianchi identity In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie alge ...
holds
:
which is equivalent to the
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
:
since
. Define the Hodge star operator, dual strength tensor
, then the Bianchi identity can be rewritten as
:
A source
enters into the equations of motion as
:
Note that the currents must properly change under gauge group transformations.
We give here some comments about the physical dimensions of the coupling. In ''D'' dimensions, the field scales as