Editing Operator (physics) (section)

==The exponential map==
The whole group may be recovered, under normal circumstances, from the generators, via the [[exponential map (Lie theory)|exponential map]]. In the case of the translations the idea works like this.

The translation for a finite value of <math>a</math> may be obtained by repeated application of the infinitesimal translation:

: <math>T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)</math>

with the <math>\cdots</math> standing for the application <math>N</math> times. If <math>N</math> is large, each of the factors may be considered to be infinitesimal:

: <math>T_a f(x) = \lim_{N\to\infty} \left(I - \frac{a}{N} D\right)^N f(x).</math>

But this limit may be rewritten as an exponential:

: <math>T_a f(x) = \exp(-aD) f(x).</math>

To be convinced of the validity of this formal expression, we may expand the exponential in a [[power series]]:

: <math>T_a f(x) = \left( I - aD + {a^2 D^2 \over 2!} - {a^3 D^3 \over 3!} + \cdots \right) f(x).</math>

The right-hand side may be rewritten as

: <math>f(x) - af'(x) + \frac{a^2}{2!} f''(x) - \frac{a^3}{3!} f^{(3)}(x) + \cdots</math>

which is just the Taylor expansion of <math>f(x-a)</math>, which was our original value for <math>T_a f(x)</math>.

The mathematical properties of physical operators are a topic of great importance in itself. For further information, see [[C*-algebra]] and [[Gelfand–Naimark theorem]].