computing the probability distribution of the possible outcomes that the experiment is designed to measure. , and so the components of 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. ( {\displaystyle T^{a}} where * stands for the Hodge dual and the integral is defined as in differential geometry. When analyzing the dynamics of a gauge theory, the gauge field must be treated as a dynamical variable, similar to other objects in the description of a physical situation. {\displaystyle \delta _{\varepsilon }DX=\varepsilon DX} [2], The symmetry group of Supergravity is a local symmetry, whereas supersymmetry is a global symmetry. "Localising" this symmetry implies the replacement of θ by θ(x). QED is generally regarded as the first, and simplest, physical gauge theory. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. At the same time, the richer structure of gauge theories allows simplification of some computations: for example Ward identities connect different renormalization constants. 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. ( Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom. μ ( ′ {\displaystyle [\cdot ,\cdot ]} 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, where the A {\displaystyle J^{\mu }(x)={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)} a μ transforms identically to γ 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. where a ) However, continuum and quantum theories differ significantly in how they handle the excess degrees of freedom represented by gauge transformations. a Such quantities can be for example an observable, a tensor or the Lagrangian of a theory. Other gauge invariant actions also exist (e.g., nonlinear electrodynamics, Born–Infeld action, Chern–Simons model, theta term, etc.). ( X One assumes an adequate experiment isolated from "external" influence that is itself a gauge-dependent statement. ¯ A That is, Maxwell's equations have a gauge symmetry. and both quantities appear inside dot products in the Lagrangian (orthogonal transformations preserve the dot product). 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. → is an element of the vector space spanned by the generators , ⋅ ε (This is analogous to a non-inertial change of reference frame, which can produce a Coriolis effect.). ψ ⋅ This is seen to preserve the Lagrangian, since the derivative of ) There are therefore as many gauge fields as there are generators of the Lie algebra. {\displaystyle \Phi } {\displaystyle \wedge } The first gauge theory quantized was quantum electrodynamics (QED). ε 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. Thus, in the abelian case, where μ 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. 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, F There is one conserved current for every generator. ) {\displaystyle \partial _{\mu }\Phi } However, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below. A gauge symmetry is analogous to how we can describe something within one language through different words (synonyms). where There are many global symmetries (such as SU(2) of isospin symmetry) and local symmetries (like SU(2) of weak interactions) in particle physics. V x Methods for quantization are covered in the article on quantization. μ where D is the covariant derivative. For example, in weather prediction these may be temperature, pressure, humidity, etc. where the Ta matrices are generators of the SO(n) group. {\displaystyle \delta _{\varepsilon }X=\varepsilon X} establishing a probability distribution over all physical situations determined by boundary conditions consistent with the setup information, establishing a probability distribution of measurement outcomes for each possible physical situation, This page was last edited on 26 November 2020, at 08:16.
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