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)
Famous quotes containing the words concept of, concept and/or generator:
“To find the length of an object, we have to perform certain
physical operations. The concept of length is therefore fixed when the operations by which length is measured are fixed: that is, the concept of length involves as much as and nothing more than the set of operations by which length is determined.”
—Percy W. Bridgman (1882–1961)
“Behind the concept of woman’s strangeness is the idea that a woman may do anything: she is below society, not bound by its law, unpredictable; an attribute given to every member of the league of the unfortunate.”
—Christina Stead (1902–1983)
“He admired the terrible recreative power of his memory. It was only with the weakening of this generator whose fecundity diminishes with age that he could hope for his torture to be appeased. But it appeared that the power to make him suffer of one of Odette’s statements seemed exhausted, then one of these statements on which Swann’s spirit had until then not dwelled, an almost new word relayed the others and struck him with new vigor.”
—Marcel Proust (1871–1922)