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:
“Terror is as much a part of the concept of truth as runniness is of the concept of jam. We wouldn’t like jam if it didn’t, by its very nature, ooze. We wouldn’t like truth if it wasn’t sticky, if, from time to time, it didn’t ooze blood.”
—Jean Baudrillard (b. 1929)
“The heritage of the American Revolution is forgotten, and the American government, for better and for worse, has entered into the heritage of Europe as though it were its patrimony—unaware, alas, of the fact that Europe’s declining power was preceded and accompanied by political bankruptcy, the bankruptcy of the nation-state and its concept of sovereignty.”
—Hannah Arendt (1906–1975)
“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)