Editing Operator (physics) (section)

==Generators==
If the transformation is [[infinitesimal]], the operator action should be of the form

: <math> I + \epsilon A, </math>

where <math>I</math> is the identity operator, <math>\epsilon</math> is a parameter with a small value, and <math>A</math> will depend on the transformation at hand, and is called a [[Generator (mathematics)#Differential equations|generator of the group]]. Again, as a simple example, we will derive the generator of the space translations on 1D functions.

As it was stated, <math>T_a f(x)=f(x-a)</math>. If <math>a=\epsilon</math> is infinitesimal, then we may write

: <math>T_\epsilon f(x)=f(x-\epsilon)\approx f(x) - \epsilon f'(x).</math>

This formula may be rewritten as

: <math>T_\epsilon f(x) = (I-\epsilon D) f(x)</math>

where <math>D</math> is the generator of the translation group, which in this case happens to be the ''derivative'' operator. Thus, it is said that the generator of translations is the derivative.