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CAF123
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An infinitesimal transformation of position coordinates in a d dimensional Minkowski space may be written as $$x^{'\mu} = x^{\mu} + \omega_a \frac{\delta x^{\mu}}{\delta \omega_a}$$ The corresponding change in some field defined over the space is $$\Phi '(x') = \Phi(x) + \omega_a \frac{\delta F}{\delta \omega_a},$$ where ##\Phi '(x') := F(\Phi(x))##.
The ##\left\{\omega_a\right\}## are a set of infinitesimal parameters. The generator ##G_a## of a symmetry transformation is defined by the following infinitesimal transformation at a same point $$\delta_{\omega} \Phi(x) = \Phi '(x) - \Phi(x) = -i\omega_a G_a \Phi(x)$$
Could someone explain what this equation means? I have been trying to connect to what I worked on in Lie Algebras and the fact that combinations of the generators and the parameters from the identity span the Lie algebra.
Many thanks.
The ##\left\{\omega_a\right\}## are a set of infinitesimal parameters. The generator ##G_a## of a symmetry transformation is defined by the following infinitesimal transformation at a same point $$\delta_{\omega} \Phi(x) = \Phi '(x) - \Phi(x) = -i\omega_a G_a \Phi(x)$$
Could someone explain what this equation means? I have been trying to connect to what I worked on in Lie Algebras and the fact that combinations of the generators and the parameters from the identity span the Lie algebra.
Many thanks.