28 Feb



GAUGE in measurement :

Gauge Theory :

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.

The term gauge refers to redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group which is referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. 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 quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, the gauge theory is referred to as non-abelian, the usual example being the Yang–Mills theory.

Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic field, with the photon being the gauge boson. 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.

Many powerful theories in physics are described by Lagrangians which are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every 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 symmetries can be viewed as analogues of the equivalence principle of general relativity in which each point in spacetime is allowed a choice of local reference (coordinate) frame. Both symmetries reflect a redundancy in the description of a system.

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 electronsquantum electrodynamics, elaborated on below. Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.

Classical electromagnetism

Historically, the first example of gauge symmetry to be discovered was classical electromagnetism. In static electricity, 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, V \rightarrow V+C, 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, \mathbf{E}=-\nabla V. Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential A, with

\mathbf{E} = - \nabla V - \frac{\partial \mathbf{A}}{\partial t}
\mathbf{B} =  \nabla \times \mathbf{A}\ .

The general gauge transformations now become not just V \rightarrow V+C but

\mathbf{A} \rightarrow \mathbf{A}+\nabla f
V \rightarrow V-\frac{\partial f}{\partial t}\ ,

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 Lagrangians.
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 fields which were originally non-interacting.

Consider a set of n non-interacting scalar fields, with equal masses m. This system is described by an action which is the sum of the (usual) action for each scalar field \varphi_i

 \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].

The Lagrangian (density) can be compactly written as

\ \mathcal{L} = \frac{1}{2} (\partial_\mu \Phi)^T \partial^\mu \Phi - \frac{1}{2}m^2 \Phi^T \Phi

by introducing a vector of fields

\ \Phi = ( \varphi_1, \varphi_2,\ldots, \varphi_n)^T .

The term \partial_\mu is Einstein notation for the partial derivative of Φ in each of the four dimensions. It is now transparent that the Lagrangian is invariant under the transformation

\ \Phi \mapsto \Phi^' = G \Phi

whenever G is a constant matrix belonging to the n-by-n orthogonal group O(n). This is seen to preserve the Lagrangian since the derivative of Φ will transform identically to Φ and both quantities appear inside dot products in the Lagrangian (orthogonal transformations preserve the dot product).

\ (\partial_\mu \Phi) \mapsto (\partial_\mu \Phi)^' = G \partial_\mu \Phi

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 current

\ J^{a}_{\mu} = i\partial_\mu \Phi^T T^{a} \Phi

where the Ta matrices are generators 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 coordinates x.

Unfortunately, the G matrices do not “pass through” the derivatives, when G = G(x),

\ \partial_\mu (G \Phi) \neq G (\partial_\mu \Phi)

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 Φ will again transform identically with Φ

\ (D_\mu \Phi)^' = G D_\mu \Phi.

This new “derivative” is called a covariant derivative and takes the form

\ D_\mu = \partial_\mu + g A_\mu

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

\ A^'_\mu = G A_\mu G^{-1} - \frac{1}{g} (\partial_\mu G)G^{-1}

The gauge field is an element of the Lie algebra, and can therefore be expanded as

\ A_{\mu} =  \sum_a A_{\mu}^a T^a

There are therefore as many gauge fields as there are generators of the Lie algebra.

Finally, we now have a locally gauge invariant Lagrangian

\ \mathcal{L}_\mathrm{loc} = \frac{1}{2} (D_\mu \Phi)^T D^\mu \Phi -\frac{1}{2}m^2 \Phi^T \Phi.

Pauli calls gauge transformation of the first type to the one applied to fields as Φ, while the compensating transformation in A is said to be a gauge transformation of the second type.

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

\ \mathcal{L}_\mathrm{int} = \frac{g}{2} \Phi^T A_{\mu}^T \partial^\mu \Phi + \frac{g}{2}  (\partial_\mu \Phi)^T A^{\mu} \Phi + \frac{g^2}{2} (A_\mu \Phi)^T A^\mu \Phi.

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. That is dealt with in the next section by adding yet another term, \mathcal{L}_{\mathrm{gf}}, to the Lagrangian. 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.

The Yang–Mills Lagrangian for the gauge field

Main article: 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 A(x) 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 which generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is (conventionally) written as

\ \mathcal{L}_\mathrm{gf} = - \frac{1}{2} \operatorname{Tr}(F^{\mu \nu} F_{\mu \nu})


\ F_{\mu \nu} = \frac{1}{ig}[D_\mu, D_\nu]

and the 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, theta term etc.).

Note that in this Lagrangian term there is no field whose transformation counterweighs the one of A. 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.[3]

The complete Lagrangian for the gauge theory is now

\ \mathcal{L} = \mathcal{L}_\mathrm{loc} + \mathcal{L}_\mathrm{gf} = \mathcal{L}_\mathrm{global} + \mathcal{L}_\mathrm{int} + \mathcal{L}_\mathrm{gf}

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 which generates the electron field’s Dirac equation is

 \mathcal{S} = \int \bar\psi(i \hbar c \, \gamma^\mu \partial_\mu - m c^2 ) \psi \, \mathrm{d}^4x.

The global symmetry for this system is

\ \psi \mapsto e^{i \theta} \psi.

The gauge group here is U(1), just the 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

\ D_\mu = \partial_\mu - i \frac{e}{\hbar} A_\mu.

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

\ \mathcal{L}_\mathrm{int} = \frac{e}{\hbar}\bar\psi(x) \gamma^\mu \psi(x) A_{\mu}(x) = J^{\mu}(x) A_{\mu}(x).

where Jμ(x) 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 Aμ(x) in terms of the field strength tensor exactly as in electrodynamics, one obtains the Lagrangian which is used as the starting point in quantum electrodynamics.\ \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}.

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Ditulis oleh pada Februari 28, 2011 in Bagaimana Dengan Fisika ?


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