The Exponential Map
The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations the idea works like this.
The translation for a finite value of may be obtained by repeated application of the infinitesimal translation:
with the standing for the application times. If is large, each of the factors may be considered to be infinitesimal:
But this limit may be rewritten as an exponential:
To be convinced of the validity of this formal expression, we may expand the exponential in a power series:
The right-hand side may be rewritten as
which is just the Taylor expansion of, which was our original value for .
The mathematical properties of physical operators are a topic of great importance in itself. For further information, see C*-algebra and Gelfand-Naimark theorem.
Read more about this topic: Operator (physics)
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