Gauge Symmetry

In Physics , gauge theories are a class of physical theories based on the idea that Symmetry Transformation s can be performed locally as well as globally. '''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.g. the Chern-Simons Model ). Most physical theories are described by Lagrangian s which are Invariant under certain transformations, when the transformations are identically performed at ''every'' Space-time Point —they have Global Symmetries . Gauge theory extends this idea by requiring that the Lagrangians must possess ''local symmetries'' as well—it should be possible to perform these symmetry transformations in a particular region of space-time without affecting what happens in another region. This requirement is a generalized version of the Equivalence Principle of General Relativity .

Gauge "symmetries" reflect a redundancy in the description of a system.

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 . This theory, known as the Standard Model , accurately describes Experiment al 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 Some Formulations of General Relativity , are, in one way or another, gauge theories.

Sometimes, the term gauge symmetry is used in a more general sense to include any local symmetry, like for example, Diffeomorphism s. This sense of the term will ''not '' be used in this article.

A BRIEF HISTORY

The earliest physical theory which had a gauge symmetry was Maxwell 's Electrodynamics . However, the importance of this symmetry remained unnoticed in the earliest formulations. After Einstein 's development of general relativity, Hermann Weyl , in an attempt to unify General Relativity and Electromagnetism , conjectured that ''Eichinvarianz'' or invariance under the change of Scale (or "gauge") might also be a local symmetry of the theory of general relativity. This conjecture was found to lead to some unphysical results. However after the development of Quantum Mechanics , Weyl, Vladimir Fock and Fritz London realized that the idea, with some modifications (replacing the scale factor with a Complex quantity, and turning the scale transformation into a change of Phase —a U(1) gauge symmetry) provided a neat explanation for the effect of an Electromagnetic Field on the Wave Function of a Charge d quantum mechanical Particle . This was the first gauge theory. It was popularised by Pauli in the 1940s, e.g. R.M. P.13, 203 .

In the 1950s, 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 Nucleon s in 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 on the Isospin doublet of Proton s and Neutron s, similar to the action of the U(1) group on the Spinor 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 , that was believed to be an important characteristic of strong interactions—thereby motivating the search for a gauge theory of the strong force. 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 . The Standard Model unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory.

In the Seventies , Sir Michael Atiyah began a program of 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 4- Manifold s is very different from their classification Up To Homeomorphism . Michael Freedman used Donaldson's work to exhibit Fake '''R'''⁴ s, that is, exotic Differentiable Structure s on 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 which enabled the calculation of certain Topological invariants. These contributions to mathematics from gauge theory have led to a renewed interest in this area.

A SIMPLE GAUGE SYMMETRY EXAMPLE FROM ELECTRODYNAMICS

The definition of Electrical Ground in an Electric Circuit is an example of a Gauge Symmetry ; when the Electric Potential s across all points in a circuit are raised by the same amount, the circuit would still operate identically; as the potential differences ( Voltage s) in the circuit are unchanged. A common illustration of this fact is the sight of a bird perched on a high voltage power line without electrocution, as the bird is insulated from ground.

This is called a Global Gauge Symmetry . 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 .

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Gauge Symmetry