|
|
| Line 1: |
Line 1: |
| {{See introduction}}
| | They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. This CMS has great flexibility to adapt various extensions and add-ons. If you need a special plugin for your website , there are thousands of plugins that can be used to meet those needs. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility. <br><br> |
|
| |
|
| In [[physics]], a '''gauge theory''' is a type of [[Field theory (physics)|field theory]] in which the [[Lagrangian]] is [[Invariant (physics)|invariant]] under a [[continuous group]] of local transformations.
| | Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. If you are a positive thinker businessman then today you have to put your business online. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. t need to use the back button or the URL to get to your home page. For those who have any concerns relating to wherever as well as how to make use of [http://go1.in/wordpress_backup_plugin_9376764 wordpress dropbox backup], you are able to contact us in our web-page. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer. <br><br>It is very easy to install Word - Press blog or website. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. Enough automated blog posts plus a system keeps you and your clients happy. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. See a product, take a picture, and it gives you an Amazon price for that product, or related products. <br><br>He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. This allows updates to be sent anyone who wants them via an RSS reader or directly to their email. Extra investment in Drupal must be bypassed as value for money is what Drupal provides. Customers within a few seconds after visiting a site form their opinion about the site. |
| | |
| The term ''gauge'' refers to redundant [[degrees of freedom]] in the Lagrangian. The transformations between possible gauges, called ''gauge transformations'', form a [[Lie group]]—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the [[Lie algebra]] of [[group generator]]s. For each group generator there necessarily arises a corresponding [[vector field]] called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the [[Quantum|quanta]] of the gauge fields are called ''[[gauge boson]]s''. If the symmetry group is [[Nonabelian group|non-commutative]], the gauge theory is referred to as ''non-abelian'', the usual example being the [[Yang–Mills theory]].
| |
| | |
| Many powerful theories in physics are described by [[Lagrangian]]s that are [[Invariant (physics)|invariant]] under some symmetry transformation groups. When they are invariant under a transformation identically performed at ''every'' [[Point (geometry)|point]] in the space in which the physical processes occur, they are said to have a [[global symmetry]]. The requirement of [[local symmetry]], the cornerstone of gauge theories, is a stricter constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in space-time.
| |
| | |
| Gauge theories are important as the successful field theories explaining the dynamics of [[elementary particles]]. [[Quantum electrodynamics]] is an [[Abelian group|abelian]] gauge theory with the symmetry group [[Unitary group|U(1)]] and has one gauge field, the [[electromagnetic four-potential]], with the [[photon]] being the gauge boson. The [[Standard Model]] is a non-abelian gauge theory with the symmetry group U(1)×[[Special unitary group#n.3D2|SU(2)]]×[[Special unitary group#n.3D3|SU(3)]] and has a total of twelve gauge bosons: the [[photon]], three [[weak boson]]s and eight [[gluons]].
| |
| | |
| Gauge theories are also important in explaining [[gravitation]] in the theory of [[general relativity]]. Its case is somewhat unique in that the gauge field is a tensor, the [[Lanczos tensor]]. Theories of [[quantum gravity]], beginning with [[gauge gravitation theory]], also postulate the existence of a gauge boson known as the [[graviton]]. Gauge symmetries can be viewed as analogues of the [[principle of general covariance]] of general relativity in which the coordinate system can be chosen freely under arbitrary [[diffeomorphism]]s of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, [[gauge theory gravity]], replaces the principle of general covariance with a true gauge principle with new gauge fields.
| |
| | |
| Historically, these ideas were first stated in the context of [[classical electromagnetism]] and later in [[general relativity]]. However, the modern importance of gauge symmetries appeared first in the [[relativistic quantum mechanics]] of [[electron]]s{{spaced ndash}}[[quantum electrodynamics]], elaborated on below. Today, gauge theories are useful in [[condensed matter physics|condensed matter]], [[nuclear physics|nuclear]] and [[high energy physics]] among other subfields.
| |
| | |
| ==History and importance==
| |
| | |
| The earliest field theory having a gauge symmetry was [[James Clerk Maxwell|Maxwell]]'s formulation of [[classical electrodynamics|electrodynamics]] in 1864. The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, [[David Hilbert|Hilbert]] had derived the [[Einstein field equations]] by postulating the invariance of the [[Action (physics)|action]] under a general coordinate transformation. Later [[Hermann Weyl]], in an attempt to unify [[general relativity]] and [[electromagnetism]], conjectured that ''Eichinvarianz'' or invariance under the change of [[scale (measurement)|scale]] (or "gauge") might also be a local symmetry of general relativity. After the development of [[quantum mechanics]], Weyl, [[Vladimir Fock]] and [[Fritz London]] modified gauge by replacing the scale factor with a [[complex number|complex]] quantity and turned the scale transformation into a change of [[phase (waves)|phase]], which is a U(1) gauge symmetry. This explained the [[electromagnetic field]] effect on the [[wave function]] of a [[electric charge|charge]]d quantum mechanical [[Elementary particle|particle]]. This was the first widely recognised gauge theory, popularised by [[Wolfgang Pauli|Pauli]] in the 1940s.<ref>[[Wolfgang Pauli]] (1941) "[http://prola.aps.org/abstract/RMP/v13/i3/p203_1 Relativistic Field Theories of Elementary Particles,]" ''Rev. Mod. Phys.'' '''13''': 203–32.</ref>
| |
| | |
| In 1954, attempting to resolve some of the great confusion in [[elementary particle physics]], [[Chen Ning Yang]] and [[Robert Mills (physicist)|Robert Mills]] introduced '''non-abelian gauge theories''' as models to understand the [[strong interaction]] holding together [[nucleon]]s in [[atomic nucleus|atomic nuclei]]. (Ronald Shaw, working under [[Abdus Salam]], independently introduced the same notion in his doctoral thesis.) Generalizing the gauge invariance of electromagnetism, they attempted to construct a theory based on the action of the (non-abelian) SU(2) symmetry [[group (mathematics)|group]] on the [[isospin]] doublet of [[proton]]s and [[neutron]]s. This is similar to the action of the [[U(1)]] group on the [[spinor]] [[field (physics)|field]]s of [[quantum 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. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called [[asymptotic freedom]]. 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 charge|color]] triplet of [[quarks]]. The [[Standard Model]] unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory.
| |
| | |
| In the 1970s, Sir [[Michael Atiyah]] began studying the mathematics of solutions to the classical [[Yang–Mills]] equations. In 1983, Atiyah's student [[Simon Donaldson]] built on this work to show that the [[differentiable]] classification of [[smooth function|smooth]] 4-[[manifold]]s is very different from their classification [[up to]] [[homeomorphism]]. [[Michael Freedman]] used Donaldson's work to exhibit [[exotic R4|exotic '''R'''<sup>4</sup>]]s, that is, exotic [[differentiable structure]]s on [[Euclidean space|Euclidean]] 4-dimensional space. 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]] that enabled the calculation of certain [[topology|topological]] invariants (the [[Seiberg–Witten invariant]]s). These contributions to mathematics from gauge theory have led to a renewed interest in this area.
| |
| | |
| The importance of gauge theories in physics is exemplified in the tremendous success of the mathematical formalism in providing a unified framework to describe the [[quantum field theory|quantum field theories]] of [[electromagnetism]], the [[weak force]] and the [[strong force]]. This theory, known as the [[Standard model|Standard Model]], accurately describes experimental predictions regarding three of the four [[fundamental force]]s of nature, and is a gauge theory with the gauge group [[SU(3) × SU(2) × U(1)]]. Modern theories like [[string theory]], as well as [[general relativity]], are, in one way or another, gauge theories.
| |
| | |
| :''See Pickering<ref name=Pickering>
| |
| {{cite book
| |
| |last=Pickering |first=A.
| |
| |title=Constructing Quarks
| |
| |url=http://www.amazon.com/Constructing-Quarks-Sociological-History-Particle/dp/0226667995/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1235837296&sr=8-1
| |
| |publisher=[[University of Chicago Press]]
| |
| |year=1984
| |
| |isbn=0-226-66799-5
| |
| }}</ref> for more about the history of gauge and quantum field theories.''
| |
| | |
| == Description ==
| |
| | |
| ===Global and local symmetries===
| |
| In [[physics]], the mathematical description of any physical situation usually contains excess [[Degrees of freedom (physics and chemistry)|degrees of freedom]]; the same physical situation is equally well described by many equivalent mathematical configurations. For instance, in [[Newtonian dynamics]], if two configurations are related by a [[Galilean transformation]] (an [[inertial]] change of reference frame) they represent the same physical situation. These transformations form a [[group (mathematics)|group]] of "[[symmetry in physics|symmetries]]" of the theory, and a physical situation corresponds not to an individual mathematical configuration but to a class of configurations related to one another by this symmetry group.
| |
| | |
| This idea can be generalized to include local as well as global symmetries, analogous to much more abstract "changes of coordinates" in a situation where there is no preferred "[[inertial]]" coordinate system that covers the entire physical system. A gauge theory is a mathematical model that has symmetries of this kind, together with a set of techniques for making physical predictions consistent with the symmetries of the model.
| |
| | |
| ===Example of global symmetry===
| |
| When a quantity occurring in the mathematical configuration is not just a number but has some geometrical significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation. For instance, if one description of a pattern of fluid flow states that the fluid velocity in the neighborhood of (''x''=1, ''y''=0) is 1 m/s in the positive ''x'' direction, then a description of the same situation in which the coordinate system has been rotated clockwise by 90 degrees states that the fluid velocity in the neighborhood of (''x''=0, ''y''=1) is 1 m/s in the positive ''y'' direction. The coordinate transformation has affected both the coordinate system used to identify the ''location'' of the measurement and the basis in which its ''value'' is expressed. As long as this transformation is performed globally (affecting the coordinate basis in the same way at every point), the effect on values that represent the ''rate of change'' of some quantity along some path in space and time as it passes through point ''P'' is the same as the effect on values that are truly local to ''P''.
| |
| | |
| ===Use of fiber bundles to describe local symmetries===
| |
| In order to adequately describe physical situations in more complex theories, it is often necessary to introduce a "coordinate basis" for some of the objects of the theory that do not have this simple relationship to the coordinates used to label points in space and time. (In mathematical terms, the theory involves a [[fiber bundle]] in which the fiber at each point of the base space consists of possible coordinate bases for use when describing the values of objects at that point.) In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point (a ''local section'' of the fiber bundle) and express the values of the objects of the theory (usually "[[field theory (physics)|fields]]" in the physicist's sense) using this basis. Two such mathematical configurations are equivalent (describe the same physical situation) if they are related by a transformation of this abstract coordinate basis (a change of local section, or ''gauge transformation'').
| |
| | |
| In most gauge theories, the set of possible transformations of the abstract gauge basis at an individual point in space and time is a finite-dimensional [[Lie group]]. The simplest such group is [[U(1)]], which appears in the modern formulation of [[quantum electrodynamics#Mathematics|quantum electrodynamics (QED)]] via its use of [[complex number]]s. QED is generally regarded as the first, and simplest, physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the ''gauge group'' of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the (finite-dimensional) Lie group, whose value at each point represents the action of the gauge transformation on the fiber over that point.
| |
| | |
| A gauge transformation with constant parameter at every point in space and time is analogous to a rigid rotation of the geometric coordinate system; it represents a [[global symmetry]] of the gauge representation. As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity. A gauge transformation whose parameter is ''not'' a constant function is referred to as a [[local symmetry]]; its effect on expressions that involve a [[derivative]] is qualitatively different from that on expressions that don't. (This is analogous to a non-inertial change of reference frame, which can produce a [[Coriolis effect]].)
| |
| | |
| ===Gauge fields===
| |
| The "gauge covariant" version of a gauge theory accounts for this effect by introducing a [[gauge field]] (in mathematical language, an [[Ehresmann connection]]) and formulating all rates of change in terms of the [[covariant derivative]] with respect to this connection. The gauge field becomes an essential part of the description of a mathematical configuration. A configuration in which the gauge field can be eliminated by a gauge transformation has the property that its [[field strength]] (in mathematical language, its [[curvature]]) is zero everywhere; a gauge theory is ''not'' limited to these configurations. In other words, the distinguishing characteristic of a gauge theory is that the gauge field does not merely compensate for a poor choice of coordinate system; there is generally no gauge transformation that makes the gauge field vanish.
| |
| | |
| When analyzing the [[Dynamics (physics)|dynamics]] of a gauge theory, the gauge field must be treated as a dynamical variable, similarly to other objects in the description of a physical situation. In addition to its [[Fundamental interaction|interaction]] with other objects via the covariant derivative, the gauge field typically contributes [[energy]] in the form of a "self-energy" term. One can obtain the equations for the gauge theory by:
| |
| * starting from a naïve [[ansatz]] without the gauge field (in which the derivatives appear in a "bare" form);
| |
| * listing those global symmetries of the theory that can be characterized by a continuous parameter (generally an abstract equivalent of a rotation angle);
| |
| * computing the correction terms that result from allowing the symmetry parameter to vary from place to place; and
| |
| * reinterpreting these correction terms as couplings to one or more gauge fields, and giving these fields appropriate self-energy terms and dynamical behavior.
| |
| This is the sense in which a gauge theory "extends" a global symmetry to a local symmetry, and closely resembles the historical development of the gauge theory of gravity known as [[general relativity]].
| |
| | |
| ===Physical experiments===
| |
| Gauge theories are used to model the results of physical experiments, essentially by:
| |
| * limiting the universe of possible configurations to those consistent with the information used to set up the experiment, and then
| |
| * computing the probability distribution of the possible outcomes that the experiment is designed to measure.
| |
| The mathematical descriptions of the "setup information" and the "possible measurement outcomes" (loosely speaking, the "boundary conditions" of the experiment) are generally not expressible without reference to a particular coordinate system, including a choice of gauge. (If nothing else, one assumes that the experiment has been adequately isolated from "external" influence, which is itself a gauge-dependent statement.) Mishandling gauge dependence in boundary conditions is a frequent source of [[anomaly (physics)|anomalies]] in gauge theory calculations, and gauge theories can be broadly classified by their approaches to anomaly avoidance.
| |
| | |
| ===Continuum theories===
| |
| The two gauge theories mentioned above (continuum electrodynamics and general relativity) are examples of continuum field theories. The techniques of calculation in a continuum theory implicitly assume that:
| |
| * given a completely fixed choice of gauge, the boundary conditions of an individual configuration can in principle be completely described;
| |
| * given a completely fixed gauge and a complete set of boundary conditions, the principle of least action determines a unique mathematical configuration (and therefore a unique physical situation) consistent with these bounds;
| |
| * the likelihood of possible measurement outcomes can be determined by:
| |
| ** establishing a probability distribution over all physical situations determined by boundary conditions that are consistent with the setup information,
| |
| ** establishing a probability distribution of measurement outcomes for each possible physical situation, and
| |
| ** convolving these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information; and
| |
| * fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.
| |
| These assumptions are close enough to be valid across a wide range of energy scales and experimental conditions, to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life, from light, heat, and electricity to eclipses and spaceflight. They fail only at the smallest and largest scales (due to omissions in the theories themselves) and when the mathematical techniques themselves break down (most notably in the case of [[turbulence]] and other [[chaos (physics)|chaotic]] phenomena).
| |
| | |
| ===Quantum field theories===
| |
| Other than these "classical" continuum field theories, the most widely known gauge theories are [[quantum field theories]], including [[quantum electrodynamics]] and the [[Standard Model]] of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariant [[action integral]] that characterizes "allowable" physical situations according to the [[principle of least action]]. However, continuum and quantum theories differ significantly in how they handle the excess degrees of freedom represented by gauge transformations. Continuum theories, and most pedagogical treatments of the simplest quantum field theories, use a [[gauge fixing]] prescription to reduce the orbit of mathematical configurations that represent a given physical situation to a smaller orbit related by a smaller gauge group (the global symmetry group, or perhaps even the trivial group).
| |
| | |
| More sophisticated quantum field theories, in particular those that involve a [[non-abelian]] gauge group, break the gauge symmetry within the techniques of [[Perturbation theory (quantum mechanics)|perturbation theory]] by introducing additional fields (the [[Faddeev–Popov ghost]]s) and counterterms motivated by [[anomaly cancellation]], in an approach known as [[BRST quantization]]. While these concerns are in one sense highly technical, they are also closely related to the nature of measurement, the limits on knowledge of a physical situation, and the interactions between incompletely specified experimental conditions and incompletely understood physical theory. The mathematical techniques that have been developed in order to make gauge theories tractable have found many other applications, from [[solid-state physics]] and [[crystallography]] to [[low-dimensional topology]].
| |
| | |
| == Classical gauge theory ==
| |
| | |
| ===Classical electromagnetism===
| |
| Historically, the first example of gauge symmetry discovered was classical [[electromagnetism]]. In [[electrostatics]], one can either discuss the electric field, '''E''', or its corresponding [[electric potential]], ''V''. Knowledge of one makes it possible to find the other, except that potentials differing by a constant, <math>V \rightarrow V+C</math>, correspond to the same electric field. This is because the electric field relates to ''changes'' in the potential from one point in space to another, and the constant ''C'' would cancel out when subtracting to find the change in potential. In terms of [[vector calculus]], the electric field is the [[gradient]] of the potential, <math>\mathbf{E} = -\nabla V</math>. Generalizing from static electricity to electromagnetism, we have a second potential, the [[magnetic vector potential|vector potential]] '''A''', with
| |
| :<math>\begin{align}
| |
| \mathbf{E} &= -\nabla V - \frac{\partial \mathbf{A}}{\partial t}\\
| |
| \mathbf{B} &= \nabla \times \mathbf{A}
| |
| \end{align}</math>
| |
| | |
| The general gauge transformations now become not just <math>V \rightarrow V+C</math> but
| |
| :<math>\begin{align}
| |
| \mathbf{A} &\rightarrow \mathbf{A} + \nabla f\\
| |
| V &\rightarrow V - \frac{\partial f}{\partial t}
| |
| \end{align}</math>
| |
| | |
| where ''f'' is any function that depends on position and time. The fields remain the same under the gauge transformation, and therefore [[Maxwell's equations]] are still satisfied. That is, Maxwell's equations have a gauge symmetry.
| |
| | |
| === An example: Scalar O(''n'') gauge theory ===
| |
| :''The remainder of this section requires some familiarity with classical or [[quantum field theory]], and the use of [[Lagrangian]]s.''
| |
| | |
| :''Definitions in this section: [[gauge group]], [[gauge field]], [[interaction Lagrangian]], [[gauge boson]].''
| |
| | |
| The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between originally non-interacting fields.
| |
| | |
| Consider a set of ''n'' non-interacting real [[field (physics)|scalar field]]s, with equal masses ''m''. This system is described by an [[action (physics)|action]] that is the sum of the (usual) action for each scalar field <math>\varphi_i</math>
| |
| | |
| :<math> \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]</math>
| |
| | |
| The Lagrangian (density) can be compactly written as
| |
| | |
| :<math>\ \mathcal{L} = \frac{1}{2} (\partial_\mu \Phi)^T \partial^\mu \Phi - \frac{1}{2}m^2 \Phi^T \Phi </math>
| |
| | |
| by introducing a [[vector (geometry)|vector]] of fields
| |
| :<math>\ \Phi = ( \varphi_1, \varphi_2,\ldots, \varphi_n)^T</math>
| |
| | |
| The term <math>\partial_\mu</math> is [[Einstein notation]] for the [[partial derivative]] of <math>\Phi</math> in each of the four dimensions. It is now transparent that the Lagrangian is invariant under the transformation
| |
| | |
| :<math>\ \Phi \mapsto \Phi' = G \Phi </math>
| |
| | |
| whenever ''G'' is a ''constant'' [[matrix (mathematics)|matrix]] belonging to the ''n''-by-''n'' [[orthogonal group]] O(''n''). This is seen to preserve the Lagrangian, since the derivative of <math>\Phi</math> transforms identically to <math>\Phi</math> and both quantities appear inside dot products in the Lagrangian (orthogonal transformations preserve the dot product).
| |
| | |
| :<math>\ (\partial_\mu \Phi) \mapsto (\partial_\mu \Phi)' = G \partial_\mu \Phi </math>
| |
| | |
| This characterizes 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-structure]]s. Incidentally, [[Noether's theorem]] implies that invariance under this group of transformations leads to the conservation of the ''currents''
| |
| :<math>\ J^{a}_{\mu} = i\partial_\mu \Phi^T T^{a} \Phi</math>
| |
| | |
| where the ''T<sup>a</sup>'' matrices are [[generating set of a group|generator]]s of the SO(''n'') group. 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]] [[coordinate]]s ''x''.
| |
| | |
| Unfortunately, the ''G'' matrices do not "pass through" the derivatives, when ''G'' = ''G''(''x''),
| |
| | |
| :<math>\ \partial_\mu (G \Phi) \neq G (\partial_\mu \Phi) </math>
| |
| | |
| The failure of the derivative to commute with "G" introduces an additional term (in keeping with the product rule), which spoils the invariance of the Lagrangian. In order to rectify this we define a new derivative operator such that the derivative of <math>\Phi</math> again transforms identically with <math>\Phi</math>
| |
| | |
| :<math>\ (D_\mu \Phi)' = G D_\mu \Phi</math>
| |
| | |
| This new "derivative" is called a [[covariant derivative]] and takes the form
| |
| | |
| :<math>\ D_\mu = \partial_\mu + i g A_\mu </math>
| |
| | |
| Where ''g'' is called the coupling constant; a quantity defining the strength of an interaction.
| |
| After a simple calculation we can see that the '''gauge field''' ''A''(''x'') must transform as follows
| |
| | |
| :<math>\ A'_\mu = G A_\mu G^{-1} + \frac{i}{g} (\partial_\mu G)G^{-1}</math>
| |
| | |
| The gauge field is an element of the Lie algebra, and can therefore be expanded as
| |
| | |
| :<math>\ A_{\mu} = \sum_a A_{\mu}^a T^a </math>
| |
| | |
| There are therefore as many gauge fields as there are generators of the Lie algebra.
| |
| | |
| Finally, we now have a ''locally gauge invariant'' Lagrangian
| |
| | |
| :<math>\ \mathcal{L}_\mathrm{loc} = \frac{1}{2} (D_\mu \Phi)^T D^\mu \Phi -\frac{1}{2}m^2 \Phi^T \Phi</math>
| |
| | |
| Pauli calls ''gauge transformation of the first type'' to the one applied to fields as <math>\Phi</math>, while the compensating transformation in <math>A</math> is said to be a ''gauge transformation of the second type''.
| |
| | |
| [[Image:Feynman-Diagram.svg|thumb|right|200px|[[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'''
| |
| | |
| :<math>\ \mathcal{L}_\mathrm{int} = i\frac{g}{2} \Phi^T A_{\mu}^T \partial^\mu \Phi + i\frac{g}{2} (\partial_\mu \Phi)^T A^{\mu} \Phi - \frac{g^2}{2} (A_\mu \Phi)^T A^\mu \Phi</math>
| |
| | |
| This term introduces [[interaction]]s between the ''n'' scalar fields just as a consequence of the demand for local gauge invariance. However, to make this interaction physical and not completely arbitrary, the mediator ''A''(''x'') needs to propagate in space. That is dealt with in the next section by adding yet another term, <math>\mathcal{L}_{\mathrm{gf}}</math>, to the Lagrangian. In the [[Quantization (physics)|quantized]] version of the obtained [[classical field theory]], the [[quantum|quanta]] of the gauge field ''A''(''x'') are called [[gauge boson]]s. The interpretation of the interaction Lagrangian in quantum field theory is of [[scalar (physics)|scalar]] [[boson]]s interacting by the exchange of these gauge bosons.
| |
| | |
| === The Yang–Mills Lagrangian for the gauge field ===
| |
| {{Main|Yang–Mills theory}}
| |
| The picture of a classical gauge theory developed in the previous section is almost complete, except for the fact that to define the covariant derivatives ''D'', one needs to know the value of the gauge field <math>A(x)</math> at all space-time points. 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 that generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is (conventionally) written as
| |
| | |
| :<math>\ \mathcal{L}_\mathrm{gf} = - \frac{1}{2} \operatorname{Tr}(F^{\mu \nu} F_{\mu \nu}) </math>
| |
| | |
| with
| |
| | |
| :<math>\ F_{\mu \nu} = \frac{1}{ig}[D_\mu, D_\nu] </math>
| |
| | |
| and the [[trace (linear algebra)|trace]] being taken over the [[vector space]] of the fields. This is called the '''Yang–Mills action'''. Other gauge invariant actions also exist (e.g., [[nonlinear electrodynamics]], [[Born–Infeld action]], [[Chern–Simons model]], [[strong CP problem|theta term]], etc.).
| |
| | |
| Note that in this Lagrangian term there is no field whose transformation counterweighs the one of <math>A</math>. Invariance of this term under gauge transformations is a particular case of ''a priori'' classical (geometrical) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated [[gauge fixing]], but even after restriction, gauge transformations may be possible.<ref>Sakurai, ''Advanced Quantum Mechanics'', sect 1–4</ref>
| |
| | |
| The complete Lagrangian for the gauge theory is now
| |
| :<math>\ \mathcal{L} = \mathcal{L}_\mathrm{loc} + \mathcal{L}_\mathrm{gf} = \mathcal{L}_\mathrm{global} + \mathcal{L}_\mathrm{int} + \mathcal{L}_\mathrm{gf} </math>
| |
| | |
| === An example: Electrodynamics ===
| |
| As a simple application of the formalism developed in the previous sections, consider the case of [[electrodynamics]], with only the [[electron]] field. The bare-bones action that generates the electron field's [[Dirac equation]] is
| |
| | |
| :<math> \mathcal{S} = \int \bar\psi(i \hbar c \, \gamma^\mu \partial_\mu - m c^2 ) \psi \, \mathrm{d}^4x</math>
| |
| | |
| The global symmetry for this system is
| |
| | |
| :<math> \psi \mapsto e^{i \theta} \psi</math>
| |
| | |
| The gauge group here is [[U(1)]], just the [[complex number|phase angle]] of the field, with a constant ''θ''.
| |
| | |
| "Local"ising this symmetry implies the replacement of θ by θ(''x''). An appropriate covariant derivative is then
| |
| | |
| :<math>\ D_\mu = \partial_\mu - i \frac{e}{\hbar} A_\mu</math>
| |
| | |
| 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
| |
| | |
| :<math>\ \mathcal{L}_\mathrm{int} = \frac{e}{\hbar}\bar\psi(x) \gamma^\mu \psi(x) A_{\mu}(x) = J^{\mu}(x)A_{\mu}(x)</math>
| |
| | |
| where <math>J^{\mu}(x)</math> is the usual [[four vector]] electric current density. 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 <math>A_{\mu}(x)</math> in terms of the [[Maxwell's equations|field strength tensor]] exactly as in electrodynamics, one obtains the Lagrangian used as the starting point in [[quantum electrodynamics]].
| |
| | |
| :<math>\ \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}</math>
| |
| | |
| :''See also: [[Dirac equation]], [[Maxwell's equations]], [[Quantum electrodynamics]]''
| |
| | |
| == Mathematical formalism ==
| |
| Gauge theories are usually discussed in the language of [[differential geometry]]. Mathematically, a ''gauge'' is just a choice of a (local) [[section (fiber bundle)|section]] of some [[principal bundle]]. A '''gauge transformation''' is just a transformation between two such sections.
| |
| | |
| Although gauge theory is dominated by the study of [[connection form|connections]] (primarily because it's mainly studied by [[high-energy physics|high-energy physicists]]), the idea of a connection is not central to gauge theory in general. In fact, a result in general gauge theory shows that [[affine representation]]s (i.e., affine [[module (mathematics)|modules]]) of the gauge transformations can be classified as sections of a [[jet bundle]] satisfying certain properties. There are representations that transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations that transform as a [[connection form]] (called by physicists gauge transformations of the second kind, an affine representation)—and other more general representations, such as the B field in [[BF theory]]. There are more general [[nonlinear realization|nonlinear representations]] (realizations), but are extremely complicated. Still, [[nonlinear sigma model]]s transform nonlinearly, so there are applications.
| |
| | |
| If there is a [[principal bundle]] ''P'' whose [[Fibre bundle|base space]] is [[space]] or [[spacetime]] and [[Fibre bundle|structure group]] is a [[Lie group]], then the sections of ''P'' form a [[principal homogeneous space]] of the group of gauge transformations.
| |
| | |
| [[Connection form|Connection]]s (gauge connection) define this principal bundle, yielding a [[covariant derivative]] ∇ in each [[associated vector bundle]]. If a local frame is chosen (a local basis of sections), then this covariant derivative is represented by the [[connection form]] ''A'', a [[Lie algebra]]-valued [[Differential form|1-form]], which is called the '''gauge potential''' in [[physics]]. This is evidently not an intrinsic but a frame-dependent quantity. The [[curvature form]] ''F'' is constructed from a connection form, a [[Lie algebra]]-valued [[Differential form|2-form]] that is an intrinsic quantity, by
| |
| | |
| :<math>\bold{F}=\mathrm{d}\bold{A}+\bold{A}\wedge\bold{A}</math>
| |
| | |
| where d stands for the [[exterior derivative]] and <math>\wedge</math> stands for the [[wedge product]]. (<math>\bold{A}</math> is an element of the vector space spanned by the generators <math>T^{a}</math>, and so the components of <math>\bold{A}</math> do not commute with one another. Hence the wedge product <math>\bold{A}\wedge\bold{A}</math> does not vanish.)
| |
| | |
| Infinitesimal gauge transformations form a [[Lie algebra]], which is characterized by a smooth [[Lie algebra]] valued [[scalar (mathematics)|scalar]], ε. Under such an [[infinitesimal]] gauge transformation,
| |
| | |
| :<math>\delta_\varepsilon \bold{A}=[\varepsilon,\bold{A}]-\mathrm{d}\varepsilon</math>
| |
| | |
| where <math>[\cdot,\cdot]</math> is the [[Lie algebra|Lie bracket]].
| |
| | |
| One nice thing is that if <math>\delta_\varepsilon X=\varepsilon X</math>, then <math>\delta_\varepsilon DX=\varepsilon DX</math> where D is the covariant derivative
| |
| | |
| :<math>DX\ \stackrel{\mathrm{def}}{=}\ \mathrm{d}X + \bold{A}X</math>
| |
| | |
| Also, <math>\delta_\varepsilon \bold{F} = \varepsilon \bold{F}</math>, which means <math>\bold{F}</math> transforms covariantly.
| |
| | |
| Not all gauge transformations can be generated by [[infinitesimal]] gauge transformations in general. An example is when the [[Fibre bundle|base manifold]] is a [[Compact space|compact]] [[manifold]] without [[Boundary (topology)|boundary]] such that the [[homotopy]] class of mappings from that [[manifold]] to the Lie group is nontrivial. See [[instanton]] for an example.
| |
| | |
| The ''Yang–Mills action'' is now given by
| |
| | |
| :<math>\frac{1}{4g^2}\int \operatorname{Tr}[*F\wedge F]</math>
| |
| | |
| where * stands for the [[Hodge dual]] and the integral is defined as in [[differential form|differential geometry]].
| |
| | |
| A quantity which is '''gauge-invariant''' (i.e., [[Invariant (physics)|invariant]] under gauge transformations) is the [[Wilson loop]], which is defined over any closed path, γ, as follows:
| |
| | |
| :<math>\chi^{(\rho)}\left(\mathcal{P}\left\{e^{\int_\gamma A}\right\}\right)</math>
| |
| | |
| where χ is the [[group representation|character]] of a complex [[group representation|representation]] ρ and <math>\mathcal{P}</math> represents the path-ordered operator.
| |
| | |
| ==Quantization of gauge theories==
| |
| {{main|Quantum gauge theory}}
| |
| | |
| Gauge theories may be quantized by specialization of methods which are applicable to any [[quantum field theory]]. 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 allows simplification of some computations: for example [[Ward identities]] connect different [[renormalization]] constants.
| |
| | |
| ===Methods and aims===
| |
| | |
| The first gauge theory quantized was [[quantum electrodynamics]] (QED). The first methods developed for this involved gauge fixing and then applying [[canonical quantization]]. The [[Gupta–Bleuler]] method was also developed to handle this problem. Non-abelian gauge theories are now handled by a variety of means. Methods for quantization are covered in the article on [[quantization (physics)|quantization]].
| |
| | |
| The main point to quantization is to be able to compute [[probability amplitude|quantum amplitudes]] for various processes allowed by the theory. Technically, they reduce to the computations of certain [[correlation functions]] in the [[vacuum state]]. This involves a [[renormalization]] of the theory.
| |
| | |
| When the [[running coupling]] of the theory is small enough, then all required quantities may be computed in [[perturbation theory]]. Quantization schemes intended to simplify such computations (such as [[quantization (physics)#Canonical quantization|canonical quantization]]) may be called '''perturbative quantization schemes'''. 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'''. Precise computations in such schemes often require [[supercomputing]], and are therefore less well-developed currently than other schemes.
| |
| | |
| ===Anomalies===
| |
| | |
| Some of the symmetries of the classical theory are then seen not to hold in the quantum theory; a phenomenon called an '''[[anomaly (physics)|anomaly]]'''. Among the most well known are:
| |
| *The [[scale anomaly]], which gives rise to a ''running coupling constant''. In QED this gives rise to the phenomenon of the [[Landau pole]]. In [[Quantum Chromodynamics]] (QCD) this leads to [[asymptotic freedom]].
| |
| *The [[chiral anomaly]] in either chiral or vector field theories with fermions. This has close connection with [[topology]] through the notion of [[instanton]]s. In QCD this anomaly causes the decay of a [[pion]] to two [[photon]]s.
| |
| *The [[gauge anomaly]], which must cancel in any consistent physical theory. In the [[electroweak theory]] this cancellation requires an equal number of [[quark]]s and [[lepton]]s.
| |
| | |
| ==Pure gauge==
| |
| A pure gauge is the set of field configurations obtained by a [[gauge transformation]] on the null field configuration. So it is a particular "gauge orbit" in the field configuration's space.
| |
| | |
| In the abelian case, where <math>A_\mu (x) \rightarrow A'_\mu(x) = A_\mu(x)+ \partial_\mu f(x)</math>, the pure gauge is the set of field configurations <math>A'_\mu(x) = \partial_\mu f(x)</math> for all ''f''(''x'').
| |
| | |
| ==See also==
| |
| {{multicol begin}}
| |
| *[[Gauge principle]]
| |
| *[[Aharonov–Bohm effect]]
| |
| *[[Coulomb gauge]]
| |
| *[[Electroweak theory]]
| |
| *[[Gauge covariant derivative]]
| |
| *[[Gauge fixing]]
| |
| *[[Gauge gravitation theory]]
| |
| *[[Gauge group (mathematics)]]
| |
| *[[Kaluza–Klein theory]]
| |
| *[[Lie algebra]]
| |
| *[[Lie group]]
| |
| *[[Lorenz gauge]]
| |
| {{multicol-break}}
| |
| *[[Quantum chromodynamics]]
| |
| * [[Gluon field]]
| |
| * [[Gluon field strength tensor]]
| |
| *[[Quantum electrodynamics]]
| |
| * [[Electromagnetic four-potential]]
| |
| * [[Electromagnetic tensor]]
| |
| *[[Quantum field theory]]
| |
| *[[Quantum gauge theory]]
| |
| *[[Standard Model]]
| |
| *[[Standard Model (mathematical formulation)]]
| |
| *[[Symmetry breaking]]
| |
| *[[Symmetry in physics]]
| |
| *[[Symmetry in quantum mechanics]]
| |
| *[[Ward identities]]
| |
| *[[Yang–Mills theory]]
| |
| *[[Yang–Mills existence and mass gap]]
| |
| *[[1964 PRL symmetry breaking papers]]
| |
| {{multicol-end}}
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| == Bibliography ==
| |
| ;General readers:
| |
| * Schumm, Bruce (2004) ''Deep Down Things''. Johns Hopkins University Press. Esp. chpt. 8. A serious attempt by a physicist to explain gauge theory and the [[Standard Model]] with little formal mathematics.
| |
| | |
| ;Texts:
| |
| *{{cite book |last=Bromley |first=D.A. |title=Gauge Theory of Weak Interactions |publisher=[[Springer Science+Business Media|Springer]] |year=2000 |isbn=3-540-67672-4}}
| |
| *{{cite book |first1=T.-P. |last1=Cheng |first2=L.-F. |last2=Li |title=Gauge Theory of Elementary Particle Physics |publisher=[[Oxford University Press]] |year=1983 |isbn=0-19-851961-3 }}
| |
| *{{cite book |last=Frampton |first=P. |authorlink=Paul Frampton |title=Gauge Field Theories |edition=3rd |publisher=[[Wiley-VCH]] |year=2008 |isbn=}}
| |
| *{{cite book |last=Kane |first=G.L. |title=Modern Elementary Particle Physics |publisher=Perseus Books |year=1987 |isbn=0-201-11749-5}}
| |
| | |
| ;Articles:
| |
| *{{cite journal |year=1997 |arxiv=hep-ph/9705211 |title= Introduction to Gauge Theories |last1=Becchi | first1=C.|bibcode = 1997hep.ph....5211B |page=5211 }}
| |
| *{{cite web |last=Gross |first=D. |authorlink=David Gross |year=1992 |url=http://psroc.phys.ntu.edu.tw/cjp/v30/955.pdf |title=Gauge theory – Past, Present and Future|accessdate=2009-04-23}}
| |
| *{{cite journal |doi=10.1119/1.1491265 |arxiv=physics/0204034 |title=From Lorenz to Coulomb and other explicit gauge transformations |last1=Jackson | first1=J.D. |year=2002 |journal=Am.J.Phys |volume=70 |pages= 917–928 |bibcode = 2002AmJPh..70..917J |issue=9 }}
| |
| *{{cite journal |year=1999 |arxiv=math-ph/9902027 |title=Preparation for Gauge Theory |last1=Svetlichny |first1=George|bibcode = 1999math.ph...2027S |pages=2027 }}
| |
| | |
| ==External links==
| |
| * {{springer|title=Gauge transformation|id=p/g043400}}
| |
| * [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Yang-Mills Yang–Mills equations on DispersiveWiki]
| |
| * [http://www.scholarpedia.org/article/Gauge_theories Gauge theories on Scholarpedia]
| |
| | |
| {{DEFAULTSORT:Gauge theory}}
| |
| [[Category:Gauge theories]]
| |
| [[Category:Theoretical physics]]
| |