Concept of Generator
If the transformation is infinitesimal, the operator action should be of the form
where is the identity operator, is a small parameter, and will depend on the transformation at hand, and is called a 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, . If is infinitesimal, then we may write
This formula may be rewritten as
where 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.
Read more about this topic: Operator (physics)
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