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. A gauge symmetry is analogous to how we can describe something within one language through different words (synonyms). δ given a completely fixed choice of gauge, the boundary conditions of an individual configuration are completely described, given a completely fixed gauge and a complete set of boundary conditions, the least action determines a unique mathematical configuration and therefore a unique physical situation consistent with these bounds. This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. This involves a renormalization of the theory. {\displaystyle \Phi } γ Quantization schemes suited to these problems (such as lattice gauge theory) may be called non-perturbative quantization schemes. ∂ ε When they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. μ is the Lie bracket. One can obtain the equations for the gauge theory by: 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. represents the path-ordered operator. 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). {\displaystyle A_{\mu }(x)\rightarrow A'_{\mu }(x)=A_{\mu }(x)+\partial _{\mu }f(x)} The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons. δ Pauli uses the term gauge transformation of the first type to mean the transformation of In many situations, one needs fields which are a list of space-time scalars: (φ1, φ2, ... φN). where * stands for the Hodge dual and the integral is defined as in differential geometry. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom. When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. Any gauge symmetry of the Lagrangian is equivalent to a constraint in the Hamiltonian formalism, i.e. Also, Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields. Historically, the first example of gauge symmetry discovered was classical electromagnetism. Quantization schemes intended to simplify such computations (such as canonical quantization) may be called perturbative quantization schemes. 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. V Other examples are isospin, weak isospin, charm, strangeness and any other flavour symmetry. The formalism of gauge theory carries over to a general setting. V [7], The complete Lagrangian for the gauge theory is now, As a simple application of the formalism developed in the previous sections, consider the case of electrodynamics, with only the electron field. x along dimension but. ] Invariance of this term under gauge transformations is a particular case of a priori classical (geometrical) symmetry. ( μ There are more general nonlinear representations (realizations), but these are extremely complicated. This idea later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory. The Standard Model of particle physics consists of Yang-Mills Theories. 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. is an element of the vector space spanned by the generators {\displaystyle T^{a}} 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. Gauge theories may be quantized by specialization of methods which are applicable to any quantum field theory. Hence the wedge product In physics, a gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups. a ↦ ) A gauge transformation is just a transformation between two such sections. Such quantities can be for example an observable, a tensor or the Lagrangian of a theory. gauge symmetry can be seen as the basis for electromagnetism and conservation of charge. However, to make this interaction physical and not completely arbitrary, the mediator A(x) needs to propagate in space. μ "Localising" this symmetry implies the replacement of θ by θ(x). 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, such that the value of the function and its derivatives at each point represents the action of the gauge transformation on the fiber over that point. A 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. Mathematically, a gauge is just a choice of a (local) section of some principal bundle. ψ In particle physics, the color symmetry of the interaction of quarks is an example of an internal symmetry, that of the strong interaction. In physics, a gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups. ∂ [ A The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the invariance of the action under a general coordinate transformation. does not vanish.). D ( These contributions to mathematics from gauge theory have led to a renewed interest in this area. are the structure constants of the Lie algebra of the generators of the gauge group. {\displaystyle \Phi } V ′ The first gauge theory quantized was quantum electrodynamics (QED). Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the currents.