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In physics, gauge theories are a class of theories based on the idea that symmetry transformations can be performed locally as well as globally. In quantum field theory (QFT the forces between particles are mediated by other particles Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The history of quantum field theory starts with its creation by Dirac when he attempted to quantize the Electromagnetic field in the late 1920s Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these In quantum field theory (QFT the forces between particles are mediated by other particles A global symmetry is a symmetry that holds for all points in the Spacetime under consideration as opposed to a Local symmetry that only holds for an

Extending a global symmetry to a local symmetry transforms a free or non-interacting theory into a theory which models interactions. In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms This extension normally preserves the renormalisability of the original free model. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection This makes gauge theories a useful tool for developing renormalisable unified field theories. In Physics, a unified field theory is a type of Field theory that allows all of the Fundamental forces between Elementary particles to be written

The gauge idea applies to finite-dimensional systems as well (i. e. , systems described by ordinary differential equations).

Contents

History

The earliest field theory having a gauge symmetry was Maxwell's formulation of electrodynamics in 1864. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the invariance of the action under a general coordinate transformation. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation Later Hermann Weyl, in an attempt to unify general relativity and electromagnetism, conjectured (incorrectly, as it turned out) that Eichinvarianz or invariance under the change of scale (or "gauge") might also be a local symmetry of general relativity. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of The concept of scale is applicable if a system is represented proportionally by another system After the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London modified gauge by replacing the scale factor with a complex quantity and turned the scale transformation into a change of phase—a U(1) gauge symmetry). Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Vladimir Aleksandrovich Fock (or Fok, Владимир Александрович Фoк ( December 22 1898 &ndash December 27 1974 Fritz Wolfgang London ( March 7, 1900 &ndash March 30, 1954) was a German -born American theoretical Physicist. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex This explained the electromagnetic field effect on the wave function of a charged quantum mechanical particle. The electromagnetic field is a physical field produced by electrically charged objects. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made This was the first widely recognised gauge theory. It was popularised by Pauli in the 1940s, e. g. R.M. P.13, 203.

In 1954, attempting to resolve some of the great confusion in elementary particle physics, Chen Ning Yang and Robert Mills introduced non-abelian gauge theories as models to understand the strong interaction holding together nucleons in atomic nuclei. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Chen-Ning Franklin Yang ( (born October 1, 1922) is a Chinese -born American Physicist who worked on Statistical mechanics Robert L Mills ( April 15, 1927 - October 27, 1999) was a Physicist, specializing in Quantum field theory, the theory Non-abelian may describe Non-abelian group, in mathematics a group that is not abelian (commutative Non-abelian gauge theory, in physics In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom (Ronald Shaw, working under Abdus Salam, independently introduced the same notion in his doctoral thesis. Abdus Salam ( Urdu: محمد عبد السلام) ( January 29, 1926; Jhang Punjab &ndash November 21, ) Generalizing the gauge invariance of electromagnetism, they attempted to construct a theory based on the action of the (non-abelian) SU(2) symmetry group on the isospin doublet of protons and neutrons. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Physics, and specifically Particle physics, isospin ( isotopic spin, isobaric spin) is a Quantum number related to the The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. This is similar to the action of the U(1) group on the spinor fields of quantum electrodynamics. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the In Physics, a field is a Physical quantity associated to each point of Spacetime. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In particle physics the emphasis was on using quantized gauge theories.

This idea later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory. In quantum field theory (QFT the forces between particles are mediated by other particles The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom. In Physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles such as Quarks, becomes arbitrarily Asymptotic freedom was believed to be an important characteristic of strong interactions. This motivated searching for a strong force gauge theory. This theory, now known as quantum chromodynamics, is a gauge theory with the action of the SU(3) group on the color triplet of quarks. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In Particle physics, color charge is a property of Quarks and Gluons which are related to their Strong interactions in the context of Quantum In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. The Standard Model unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles

In the 1970s, Sir Michael Atiyah began studying the mathematics of solutions to the classical Yang-Mills equations. Sir Michael Francis Atiyah, OM, FRS, FRSE (b April 22, 1929) is a British Mathematician, and one of the Yang-Mills is a Gauge theory of Quantum field theory based on the SU(N group. In 1983, Atiyah's student Simon Donaldson built on this work to show that the differentiable classification of smooth 4-manifolds is very different from their classification up to homeomorphism. Simon Kirwan Donaldson (born August 20 1957 in Cambridge, England) is an English mathematician famous for his work on the Topology of In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose Topological equivalence redirects here see also Topological equivalence (dynamical systems. Michael Freedman used Donaldson's work to exhibit exotic R4s, that is, exotic differentiable structures on Euclidean 4-dimensional space. Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, U In Mathematics, an exotic R 4 is a Differentiable manifold that is Homeomorphic to the Euclidean space R 4 In Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. In 1994, Edward Witten and Nathan Seiberg invented gauge-theoretic techniques based on supersymmetry which enabled the calculation of certain topological invariants. Year 1994 ( MCMXCIV) was a Common year starting on Saturday (link will display full 1994 Gregorian calendar) Edward Witten (born August 26, 1951) is an American Theoretical physicist and Professor at the Institute for Advanced Study Nathan "Nati" Seiberg, born in 1956 is an Israel-born American Theoretical physicist who works on String theory. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of These contributions to mathematics from gauge theory have led to a renewed interest in this area.

An extensive historical discussion can be found in Woit. [1]

Explanation

Many powerful theories in physics are described by Lagrangians which are invariant under certain symmetry transformation groups. The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system In Mathematics and Theoretical physics, an invariant is a property of a system which remains unchanged under some transformation. When they are invariant under a transformation identically performed at every point in the space in which the physical processes occur, they are said to have a global symmetry. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume A global symmetry is a symmetry that holds for all points in the Spacetime under consideration as opposed to a Local symmetry that only holds for an The requirement of local symmetry is much more strict than the requirement of global symmetry. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in space-time. This can be viewed as a generalization of the equivalence principle of general relativity in which each point in spacetime is allowed a choice of local reference (coordinate) frame. The equivalence principle General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 As in that situation, gauge "symmetries" reflect a redundancy in the description of a system. Historically, these ideas were first noticed in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared in relativistic quantum mechanics of electrons (see discussions below). Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.

Sometimes, the term 'gauge symmetry' is used in a more general sense to include any local symmetry, like for example, diffeomorphisms. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable This sense of the term will not be used in this article.

Yang-Mills theories are a particular example of gauge theories with non-abelian symmetry groups specified by the Yang-Mills action (Other gauge theories with a non-abelian gauge symmetry also exist, e. Yang-Mills is a Gauge theory of Quantum field theory based on the SU(N group. Non-abelian may describe Non-abelian group, in mathematics a group that is not abelian (commutative Non-abelian gauge theory, in physics The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Yang-Mills is a Gauge theory of Quantum field theory based on the SU(N group. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation g. , the Chern-Simons model). The Chern-Simons theory is a 3-dimensional Topological quantum field theory of Schwarz type, developed by Shiing-Shen Chern and James Harris Simons

There is a certain inaccuracy in the way the term symmetry is used in some physics literature, especially in more elementary books about elementary particles and field theory. In (quantum) physics, symmetry is a transformation between physical states that preserves the expectation values of all observables O (in particular the Hamiltonian). S: |φ> → |ψ> = S|φ>; |<ψ|O|ψ>|2=|<φ|O|φ>|2. The usual formulation of physics theories uses fields, which sometimes are not physical quantities. Such are the gauge fields (fiber bundle connections for the mathematicians), which provide a redundant but convenient description of the physical degrees of freedom. The gauge (local) "symmetries" are a reflection of this redundancy. The physical quantities are certain equivalence classes of gauge fields. An analogy can be made with the construction of the real numbers. We can use sequences of rational numbers that have the same limit. Of course, each real number is represented by infinitely many such sequences. We can choose a particular well-defined sequence to represent the real number. This corresponds to the procedure of 'gauge fixing' in gauge theories. The fact that gauge fields are not physical degrees of freedom becomes very clear when we try to quantize them. Then we are forced to work in one way or another with the physical quantities by removing the redundancy (the gauge symmetry). Another important illustration of the problem with the gauge “symmetries” is when we have anomalies. By definition these are symmetries which exist in the classical system, but not in its quantum counterpart. Anomalies are quite usual and also an experimental fact — for example, the axial anomaly in the strong interactions (broken symmetries). However, because gauge symmetries are not symmetries, gauge anomalies are not something that just complicates a proposed quantum theory but something that kills it, i. e. there are no gauge "anomalies", because such theories don't exist. This is why having the exact relation between the number of flavours and quark colours in the Standard model is so important — otherwise there is a gauge anomaly and the theory does not exist. For the same reason, string theories are defined in 10 dimensions. Only then do the anomalies cancel.

Importance

The importance of gauge theories for physics stems from the tremendous success of the mathematical formalism in providing a unified framework to describe the quantum field theories of electromagnetism, the weak force and the strong force. In quantum field theory (QFT the forces between particles are mediated by other particles Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and This theory, known as the Standard Model, accurately describes experimental predictions regarding three of the four fundamental forces of nature, and is a gauge theory with the gauge group SU(3) × SU(2) × U(1). The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms For a basic description see the article on the Standard Model. Modern theories like string theory, as well as some formulations of general relativity, are, in one way or another, gauge theories. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings This page covers applications of the Cartan formalism. For the general concept see Cartan connection. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

A simple gauge symmetry example from electrodynamics

The definition of electrical ground in an electric circuit is an example of a gauge symmetry. In Electrical engineering, the term ground or earth has several meanings depending on the specific application areas An electrical network is an interconnection of Electrical elements such as Resistors Inductors Capacitors Transmission lines Voltage Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations When the electric potentials at all points in a circuit are raised by the same amount, the circuit will still operate identically, as the potential differences (voltages) in the circuit are unchanged. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field Electrical tension (or voltage after its SI unit, the Volt) is the difference of electrical potential between two points of an electrical A common illustration of this fact is the sight of a small bird perched on a high voltage power line without electrocution, because the bird is insulated from the ground (as long as it doesn't complete the circuit by accidentally touching another wire or some grounded structure, or by dropping something).

This is called a global gauge symmetry. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations The absolute value of the potential is immaterial; what matters to circuit operation is the potential differences across the components of the circuit. The definition of the ground point is arbitrary, but once that point is set, then that definition must be followed globally.

In contrast, if some symmetry could be defined arbitrarily from one position to the next, that would be a local gauge symmetry. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

Classical gauge theory

This section requires some familiarity with classical or quantum field theory, and the use of Lagrangians. In quantum field theory (QFT the forces between particles are mediated by other particles The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system

Definitions in this section: gauge group, gauge field, interaction Lagrangian, gauge boson

An example: Scalar O(n) gauge theory

The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between fields which were originally non-interacting. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature

Consider a set of n non-interacting scalar fields, with equal masses m. In Physics, a field is a Physical quantity associated to each point of Spacetime. This system is described by an action which is the sum of the (usual) action for each scalar field φi

\mathcal{S} = \int \, \mathrm{d}^4 x \sum_{i=1}^n \left[ \frac{1}{2} \partial_\mu \varphi_i \partial^\mu \varphi_i - \frac{1}{2}m^2 \varphi_i^2 \right].

The Lagrangian (density) can be compactly written as

\ \mathcal{L} = \frac{1}{2} (\partial_\mu \Phi)^T \partial^\mu \Phi - \frac{1}{2}m^2 \Phi^T \Phi

by introducing a vector of fields

\ \Phi = ( \varphi_1, \varphi_2,\ldots, \varphi_n)^T .

The term \partial_\mu is sometimes confusing to those who have not seen it before. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation It is Einstein notation for the partial derivative of Φ in each of the four dimensions. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant It is now transparent that the Lagrangian is invariant under the transformation

\Phi \mapsto G \Phi

whenever G is a constant matrix belonging to the n-by-n orthogonal group O(n). In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n This is the global symmetry of this particular Lagrangian, and the symmetry group is often called the gauge group; the mathematical term is structure group, especially in the theory of G-structures. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. In Differential geometry, a G -structure on an n - Manifold M, for a given Structure group G, is a G Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the current

\ J^{a}_{\mu} = i\partial_\mu \Phi^T T^{a} \Phi

where the Ta matrices are generators of the SO(n) group. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the There is one conserved current for every generator.

Now, demanding that this Lagrangian should have local O(n)-invariance requires that the G matrices (which were earlier constant) should be allowed to become functions of the space-time coordinates x. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point

Unfortunately, the G matrices do not "pass through" the derivatives. When G = G(x),

\ \partial_\mu (G \Phi)^T \partial^\mu (G \Phi) \neq \partial_\mu \Phi^T \partial^\mu \Phi.

This suggests defining a "derivative" D with the property

\ D_\mu (G(x) \Phi(x)) = G(x) D_\mu \Phi.

It can be checked that such a "derivative" (called a covariant derivative) is

\ D_\mu = \partial_\mu + g A_\mu(x)

where the gauge field A(x) is defined to have the transformation law

\ A_{\mu}(x) \mapsto G(x)A_{\mu}(x)G^{-1}(x) - \frac{1}{g} \partial_\mu G(x) G^{-1}(x)

and g is the coupling constant - a quantity defining the strength of an interaction. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold.

The gauge field is an element of the Lie algebra, and can therefore be expanded as

\ A_{\mu}(x)= \sum_a A_{\mu}^a (x) T^a

There are therefore as many gauge fields as there are generators of the Lie algebra.

Finally, we now have a locally gauge invariant Lagrangian

\ \mathcal{L}_\mathrm{loc} = \frac{1}{2} (D_\mu \Phi)^T D^\mu \Phi -\frac{1}{2}m^2 \Phi^T \Phi.

Pauli calls gauge transformation of the first type to the one applied to fields as Φ, while the compensating transformation in A is said to be a gauge transformation of the second type.

Feynman diagram of scalar bosons interacting via a gauge boson
Feynman diagram of scalar bosons interacting via a gauge boson

The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian

\ \mathcal{L}_\mathrm{int} = \frac{g}{2} \Phi^T A_{\mu}^T \partial^\mu \Phi + \frac{g}{2} (\partial_\mu \Phi)^T A^{\mu} \Phi + \frac{g^2}{2} (A_\mu \Phi)^T A^\mu \Phi.

This term introduces interactions between the n scalar fields just as a consequence of the demand for local gauge invariance. Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described Interaction is a kind of action that occurs as two or more objects have an Effect upon one another In the quantized version of this classical field theory, the quanta of the gauge field A(x) are called gauge bosons. In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature The interpretation of the interaction Lagrangian in quantum field theory is of scalar bosons interacting by the exchange of these gauge bosons. In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein

The Yang-Mills Lagrangian for the gauge field

Main article: Yang-Mills

Our picture of classical gauge theory is almost complete except for the fact that to define the covariant derivatives D, one needs to know the value of the gauge field A(x) at all space-time points. Yang-Mills is a Gauge theory of Quantum field theory based on the SU(N group. Yang-Mills is a Gauge theory of Quantum field theory based on the SU(N group. Instead of manually specifying the values of this field, it can be given as the solution to a field equation. Further requiring that the Lagrangian which generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is (conventionally) written as

\ \mathcal{L}_\mathrm{gf} = - \frac{1}{2} \operatorname{Tr}(F^{\mu \nu} F_{\mu \nu})

with

\ F_{\mu \nu} = \frac{1}{ig}[D_\mu, D_\nu]

and the trace being taken over the vector space of the fields. In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added This is called the Yang-Mills action. Other gauge invariant actions also exist (e. g. nonlinear electrodynamics, Born-Infeld action, Chern-Simons model, theta term etc. Nonlinear optics (NLO is the branch of Optics that describes the behaviour of Light in nonlinear media, that is media in which the dielectric polarization In physics the Born-Infeld theory is a Nonlinear Generalization of Electromagnetism (see Nonlinear electrodynamics) The Chern-Simons theory is a 3-dimensional Topological quantum field theory of Schwarz type, developed by Shiing-Shen Chern and James Harris Simons In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics ).

Note that in this Lagrangian there is not a field Φ whose transformation counterweights the one of A. Invariance of this term under gauge transformations is a particular case of a prior classical (or geometrical, if you prefer) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated gauge fixing, but even after restriction, gauge transformations are possible (see Sakurai, Advanced Quantum Mechanics, sect 1-4). In the Physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees

The complete Lagrangian for the O(n) gauge theory is now

\ \mathcal{L} = \mathcal{L}_\mathrm{loc} + \mathcal{L}_\mathrm{gf} = \mathcal{L}_\mathrm{global} + \mathcal{L}_\mathrm{int} + \mathcal{L}_\mathrm{gf}

A simple example: Electrodynamics

As a simple application of the formalism developed in the previous sections, consider the case of electrodynamics, with only the electron field. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The bare-bones action which generates the electron field's Dirac equation is

\mathcal{S} = \int \bar\psi(i \hbar c \, \gamma^\mu \partial_\mu - m c^2 ) \psi \, \mathrm{d}^4x.

The global symmetry for this system is

\ \psi \mapsto e^{i \theta} \psi.

The gauge group here is U(1), just the phase angle of the field, with a constant θ. In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

"Local"ising this symmetry implies the replacement of θ by θ(x).

An appropriate covariant derivative is then

\ D_\mu = \partial_\mu - i \frac{e}{\hbar} A_\mu.

Identifying the "charge" e with the usual electric charge (this is the origin of the usage of the term in gauge theories), and the gauge field A(x) with the four-vector potential of electromagnetic field results in an interaction Lagrangian

\ \mathcal{L}_\mathrm{int} = \frac{e}{\hbar}\bar\psi(x) \gamma^\mu \psi(x) A_{\mu}(x) = J^{\mu}(x) A_{\mu}(x).

where Jμ(x) is the usual four vector electric current density. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Vector calculus, a vector potential is a Vector field whose curl is a given vector field The electromagnetic field is a physical field produced by electrically charged objects. In relativity, a four-vector is a vector in a four-dimensional real Vector space, called Minkowski space. The gauge principle is therefore seen to naturally introduce the so-called minimal coupling of the electromagnetic field to the electron field.

Adding a Lagrangian for the gauge field Aμ(x) in terms of the field strength tensor exactly as in electrodynamics, one obtains the Lagrangian which is used as the starting point in quantum electrodynamics. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics.

\ \mathcal{L}_{\mathrm{QED}} = \bar\psi(i\hbar c \, \gamma^\mu D_\mu - m c^2 )\psi - \frac{1}{4 \mu_0}F_{\mu\nu}F^{\mu\nu}.

See also: Dirac equation, Maxwell's equations, Quantum electrodynamics

Mathematical formalism

Gauge theories are usually discussed in the language of differential geometry. In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Mathematically, a gauge is just a choice of a (local) section of some principal bundle. In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G A gauge transformation is just a transformation between two such sections.

Although gauge theory is dominated by the study of connections (primarily because it's mainly studied by high-energy physicists), the idea of a connection is not central to gauge theory in general. In Mathematics, and specifically Differential geometry, a connection form is a manner of organizing the data of a connection using the language of Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In fact, a result in general gauge theory shows that affine representations (i. An affine representation of a topological ( Lie) group G on an Affine space A is a continuous ( smooth) Group e. affine modules) of the gauge transformations can be classified as sections of a jet bundle satisfying certain properties. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Differential geometry, the jet bundle is a certain construction which makes a new smooth Fiber bundle out of a given smooth fiber bundle There are representations which transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations which transform as a connection form (called by physicists gauge transformations of the second kind) (this is an affine representation) and other more general representations, such as the B field in BF theory. In Mathematics, and specifically Differential geometry, a connection form is a manner of organizing the data of a connection using the language of The BF model is a topological field theory which when quantized, becomes a Topological quantum field theory. And of course, we can consider more general nonlinear representations (realizations), but that is extremely complicated. But still, nonlinear sigma models transform nonlinearly, so there are applications. In Quantum field theory, a nonlinear &sigma model describes a Scalar field &Sigma which takes on values in a nonlinear manifold called the Target manifold

If we have a principal bundle P whose base space is space or spacetime and structure group is a Lie group, then the sections of P form a principal homogeneous space of the group of gauge transformations. In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a principal homogeneous space, or torsor, for a group G is a set X on which G acts freely and

We can define a connection (gauge connection) on this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle. In Mathematics, and specifically Differential geometry, a connection form is a manner of organizing the data of a connection using the language of In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. In Mathematics, the theory of Fiber bundles with a Structure group G (a Topological group) allows an operation of creating an associated If we choose a local frame (a local basis of sections) then we can represent this covariant derivative by the connection form A, a Lie algebra-valued 1-form which is called the gauge potential in physics and which is evidently not an intrinsic but a frame-dependent quantity. In Mathematics, and specifically Differential geometry, a connection form is a manner of organizing the data of a connection using the language of In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. From this connection form we can construct the curvature form F, a Lie algebra-valued 2-form which is an intrinsic quantity, by

\bold{F}=\mathrm{d}\bold{A}+\bold{A}\wedge\bold{A}

where d stands for the exterior derivative and \wedge stands for the wedge product. In Differential geometry, the curvature form describes Curvature of a connection on a Principal bundle. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms (\bold{A} is an element of the vector space spanned by the generators Ta, and so the components of \bold{A} do not commute with one another. Hence the wedge product \bold{A}\wedge\bold{A} does not vanish. )

Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie algebra valued scalar, ε. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication Under such an infinitesimal gauge transformation,

\delta_\varepsilon \bold{A}=[\varepsilon,\bold{A}]-\mathrm{d}\epsilon

where [\cdot,\cdot] is the Lie bracket. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

One nice thing is that if \delta_\varepsilon X=\varepsilon X, then \delta_\varepsilon DX=\varepsilon DX where D is the covariant derivative

DX\ \stackrel{\mathrm{def}}{=}\ \mathrm{d}X+\bold{A}X.

Also, \delta_\varepsilon \bold{F}=\varepsilon \bold{F}, which means F transforms covariantly.

Not all gauge transformations can be generated by infinitesimal gauge transformations in general. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have An example is when the base manifold is a compact manifold without boundary such that the homotopy class of mappings from that manifold to the Lie group is nontrivial. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be For a different notion of boundary related to Manifolds see that article In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be See instanton for an example. An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics.

The Yang-Mills action is now given by

\frac{1}{4g^2}\int \operatorname{Tr}[*F\wedge F]

where * stands for the Hodge dual and the integral is defined as in differential geometry. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is

A quantity which is gauge-invariant i. e. invariant under gauge transformations is the Wilson loop, which is defined over any closed path, γ, as follows:

\chi^{(\rho)}\left(\mathcal{P}\left\{e^{\int_\gamma A}\right\}\right)

where χ is the character of a complex representation ρ and \mathcal{P} represents the path-ordered operator. In Gauge theory, a Wilson loop (named after Kenneth Wilson) is a gauge-invariant observable obtained from the Holonomy of the Gauge connection In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

Quantization of gauge theories

Gauge theories may be quantized by specialization of methods which are applicable to any quantum field theory. In quantum field theory (QFT the forces between particles are mediated by other particles However, because of the subtleties imposed by the gauge constraints (see section on Mathematical formalism, above) there are many technical problems to be solved which do not arise in other field theories. At the same time, the richer structure of gauge theories allow simplification of some computations: for example Ward identities connect different renormalization constants. In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection

Methods and aims

The first gauge theory to be quantized was quantum electrodynamics (QED). Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. The first methods developed for this involved gauge fixing and then applying canonical quantization. In Physics, canonical quantization is one of many procedures for quantizing a Classical theory. The Gupta-Bleuler method was also developed to handle this problem. In Quantum field theory, the Gupta-Bleuler formalism is a way of quantizing the Electromagnetic field. Non-abelian gauge theories are now handled by a variety of means. Methods for quantization are covered in the article on quantization. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory.

The main point to quantization is to be able to compute quantum amplitudes for various processes allowed by the theory. In Quantum mechanics, a probability amplitude is a complex -valued function that describes an uncertain or unknown quantity Technically, they reduce to the computations of certain correlation functions in the vacuum state. Correlation functions contain information about the distribution of points or events or things across some space/time In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy. This involves a renormalization of the theory. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection

When the running coupling of the theory is small enough, then all required quantities may be computed in perturbation theory. In Physics, a coupling constant, usually denoted g, is a number that determines the strength of an Interaction. This article describes perturbation theory as a general mathematical method Quantization schemes intended to simplify such computations (such as canonical quantization) may be called perturbative quantization schemes. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. At present some of these methods lead to the most precise experimental tests of gauge theories.

However, in most gauge theories, there are many interesting questions which are non-perturbative. Quantization schemes suited to these problems (such as lattice gauge theory) may be called non-perturbative quantization schemes. In Physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized onto a lattice. Precise computations in such schemes often require supercomputing, and are therefore less well-developed currently than other schemes. A supercomputer is a Computer that is at the frontline of processing capacity particularly speed of calculation (at the time of its introduction

Anomalies

Some of the symmetries of the classical theory are then seen not to hold in the quantum theory — a phenomenon called an anomaly. 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 Among the most well known are:

References

  1. ^ Peter Woit (2006). Not even wrong: the failure of string theory and the search for unity in physical law. New York: Basic Books. ISBN 0465092756.  

See also

External links


The Clay Mathematics Institute has offered the prize of USD 1000000 for each of 7 great problems in mathematics The Aharonov-Bohm effect, sometimes called the Ehrenberg-Siday-Aharonov-Bohm effect is a quantum mechanical phenomenon by which a charged particle is affected by Bell's theorem is a theorem that shows that the predictions of Quantum mechanics (QM are not intuitive and touches upon fundamental philosophical issues that relate to modern In the Physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the For a basic description see the article on the Standard Model. The gauge covariant derivative (ˌgeɪdʒ koʊˌvɛəriənt dɪˈrɪvətɪv is like a generalization of the Covariant derivative used in General relativity. In Physics, Kaluza–Klein theory (or KK theory, for short is a model that seeks to unify the two fundamental forces of Gravitation and In Electromagnetism, the Lorenz gauge or Lorenz gauge condition are common misnomers for a particular choice of the electromagnetic Four-potential A^a Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the See Gauge theory for the classical preliminaries In order to quantize a gauge theory like for example Yang-Mills theory Chern-Simons or BF model Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these In the Physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees
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