Документ взят из кэша поисковой машины. Адрес оригинального документа : http://theory.sinp.msu.ru/comphep_html/tutorial/node100.html
Дата изменения: Tue Aug 15 14:54:18 2000
Дата индексирования: Mon Oct 1 22:51:17 2012
Кодировка:

Local gauge invariance.

It is an invariance of Lagrangian under group transformations $g(x)$ which depend on a point in the space-time manifold. Fields in such theories are divided into two classes: matter fields and gauge fields. The gauge fields $A^{\alpha}_{\mu}(x)$ are vector ones. The number of such fields is equal to the number of group generators. The matter fields $\psi^i(x)$ can have an arbitrary Lorentz structure. Their internal components are transformed according to some representation of the group. Let $D(\omega)$ be an operator which performs an infinitesimal transformation of fields under the local gauge transformation. For the matter fields we have

$$\hat D(\omega)\psi(x)= i\,\omega^{\alpha}(x){\hat \tau}_ {\alpha}\psi(x)\;.$$

For gauge fields the local gauge transformations are defined by

$$(\hat D(\omega) A_{\mu})^{\alpha}(x)= f^{\alpha}_{\beta \gam... ...(x) \omega^{\gamma}(x) + \partial_{\mu} \omega^{\alpha}(x)\;.$$

The following expressions, namely a covariant derivative and a gauge field tension, are used to construct a local invariant Lagrangian:

$$\nabla_{\mu}\psi(x) = \partial_{\mu} \psi(x) - \hat D(A_{\mu}... ...} \psi(x) - iA^{\alpha}_{\mu}(x){\hat \tau}_{\alpha}\psi(x)\;;$$

$$F^{\alpha}_{\mu \nu}(x) = \partial_{\mu}A^{\alpha}_{\nu}(x) ... ...{\alpha}_{\beta\gamma}A^{\beta}_{\mu}(x)A^{\gamma}_{\nu}(x)\;.$$

It can be proved that

$$\hat D(\omega)[\nabla_{\mu}\psi(x)]= \hat \tau_{\alpha}\omega^{\alpha}(x) \nabla_{\mu}\psi(x)\;;$$

$$(\hat D(\omega) F_{\mu \nu})^{\alpha}(x)= f^{\alpha}_{\beta \gamma} F^{\beta}_{\mu \nu}(x)\omega^{\gamma}(x)\;.$$

In these terms the Lagrangian of gauge theory is defined by the following expression [Bjorken&Drell]

$$L = -\, \frac{1}{4g^2} {F^{\alpha}}_{\mu \nu} {F_{\alpha}}^{\mu \nu} + L_m( \nabla_{\mu} \psi, \psi)$$ (1)

where $g$ is the coupling constant and $L_m(\partial_{\mu}\psi,\psi)$ is some Lagrangian of the matter fields which is invariant under the global gauge transformations.