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- Gauge Field Theories
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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 by introducing additional fields the Faddeev—Popov ghosts 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.
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. 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.
Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential A , with. 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. 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 scalar fields , with equal masses m.
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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-structures. Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the currents.
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. 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. This new "derivative" is called a gauge covariant derivative and takes the form. 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. The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian. This term introduces interactions 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. In the quantized version of the obtained classical field theory , the quanta of the gauge field A x are called gauge bosons.
The interpretation of the interaction Lagrangian in quantum field theory is of scalar bosons interacting by the exchange of these gauge bosons. 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.
This formulation of the Lagrangian is called a Yang—Mills action. Other gauge invariant actions also exist e. 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. 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.
An appropriate covariant derivative is then. Identifying the "charge" e not to be confused with the mathematical constant e in the symmetry description 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. The gauge principle is therefore seen to naturally introduce the so-called minimal coupling of the electromagnetic field to the electron field.
Gauge theories are usually discussed in the language of differential geometry. Mathematically, a gauge is just a choice of a local 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 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 fact, a result in general gauge theory shows that affine representations i. 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 representations realizations , but these are extremely complicated. Still, nonlinear sigma models transform nonlinearly, so there are applications. If there is 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.
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 1-form , which is called the gauge potential in physics.
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This is evidently not an intrinsic but a frame-dependent quantity. The curvature form F , a Lie algebra-valued 2-form that is an intrinsic quantity, is constructed from a connection form by. Under such an infinitesimal gauge transformation,. Not all gauge transformations can be generated by infinitesimal gauge transformations in general.
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. See instanton for an example. A quantity which is gauge-invariant i. The formalism of gauge theory carries over to a general setting. For example, it is sufficient to ask that a vector bundle have a metric connection ; when one does so, one finds that the metric connection satisfies the Yang-Mills equations of motion.
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.
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.
Gauge Field Theories
The main point to quantization is to be able to compute 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 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.
Gauge Field Theories: An Introduction with Applications - M. W. Guidry, Mike Guidry - Google книги
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. Some of the symmetries of the classical theory are then seen not to hold in the quantum theory; a phenomenon called an anomaly. Among the most well known are:. A pure gauge is the set of field configurations obtained by a gauge transformation on the null-field configuration, i.
So it is a particular "gauge orbit" in the field configuration's space. From Wikipedia, the free encyclopedia. Physical theory with fields invariant under the action of local "gauge" Lie groups. For a more accessible and less technical introduction to this topic, see Introduction to gauge theory. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations.
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Please help to improve this article by introducing more precise citations. September Learn how and when to remove this template message. Feynman diagram. Standard Model. Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism. Incomplete theories. Anderson P. Main article: Yang—Mills theory.
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See also: Dirac equation , Maxwell's equations , and Quantum electrodynamics. Main article: Quantum gauge theory. Gauge principle Aharonov—Bohm effect Coulomb gauge Electroweak theory Gauge covariant derivative Gauge fixing Gauge gravitation theory Gauge group mathematics Kaluza—Klein theory Lorenz gauge 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 PRL symmetry breaking papers.
Bibcode : RvMP Bibcode : PhRv Constructing Quarks. University of Chicago Press. Welke opties voor jouw bestelling beschikbaar zijn, zie je bij het afronden van de bestelling. Schrijf een review. E-mail deze pagina. Auteur: Nicola Cabibbo Luciano Maiani. Co-auteur: Omar Benhar.
Samenvatting Written by world-leading experts in particle physics, this new book from Luciano Maiani and Omar Benhar, with contributions from the late Nicola Cabibbo, is based on Feynman's path integrals. Key elements of gauge theories are described-Feynman diagrams, gauge-fixing, Faddeev-Popov ghosts-as well as renormalization in Quantum Electrodynamics. Quarks and QCD interactions are introduced. Renormalization group and high momentum behaviour of the coupling constants is discussed in QED and QCD, with asymptotic freedom derived at one-loop.
These concepts are related to the Higgs boson and models of grand unification. It will be an excellent book for the serious student and a good reference for the professional practitioner. Let me add that, scattered through the pages, we can find occasional traces of Nicola Cabibbo's style. The volume ends with an illuminating description of the expectation generated by the recent discovery of the Higgs boson, combined with the lack of evidence for super-symmetric particles in the mass range 0.
The reader is masterfully guided through the subtleties of the quantum field theory and elementary particle physics from simple examples in Quantum Mechanics to salient details of modern theory. Toon meer Toon minder. Recensie s This book on gauge theories completes a trilogy on the foundations of the Standard Theory of the fundamental interactions, but is fully self-contained. It combines the depth and conceptual accuracy expected from key contributors to the field with an effective pedagogical approach, tested over several decades of courses.
The mathematical formalism is never overwhelming, and the reader is guided through the concepts and techniques of quantum gauge theories along a path always driven by physics, including recent developments on the Higgs boson. This book is a great legacy of the Rome School' for students, teachers and researchers in particle physics, not only theorists but also curious experimentalists.
A must-have in any physics library, this book is highly original, centring on deep problems of quantum field theory, and consistently well-motivated by experimental phenomena. The symmetries of nature are the guiding principle, illuminating the presentation of the concept of renormalisation and its applications in gauge theories, from a thorough and pedagogical analysis of Quantum Electrodynamics, to the exploration of Quantum Chromodynamics and the SM, including the role of the Higgs field, and beyond.
Everybody can find some food for thought here: from the graduate student to the most hardened researcher. This book is full of physical insights and is highly recommended. The book covers a vast amount of material, including the basics of quantum field theory, gauge theories and their applications to the Standard Model or Theory of particle physics, in addition to an exploration of its potential extensions. Following the discovery of the Higgs particle, it is welcome to have a complete discussion on the basic theoretical aspects of the most fundamental theory we know so far.
Based on a set of lectures given by the authors over the years, this book is very pedagogic and well structured.
The path integral formalism is used throughout the book, and its simple presentation and excellent balance between concepts and calculational tools makes it an ideal textbook for an upper division undergraduate courses, though it will also prove useful as an excellent text for PhD students and professional physicists working in a variety of different fields.
I was right. This volume, the third in the series, is as clearly and beautifully written as the previous ones. It is an excellent introduction to the quantum theory of gauge fields and their applications to particle physics.