Generators

Generators are a very important thing in group theory and therefore theoretical physics. They are not hard to understand, though. You might have heard phrases like the following:

The quantum mechanical momentum generates translations.

Or perhaps rather this:

The quantum mechanical angular momentum generates rotations.

What does that mean?

I will start with a Taylor series. Say you have a simple function f \colon
\mathbb R \to \mathbb R. Its Taylor series around the point x can then be expanded like so:

f(x_0 + \alpha) = f(x_0) + f'(x_0) \alpha + \frac12 f''(x_0) \alpha^2 +
\mathrm O(\alpha^3) \,.

The whole series can be written with a summation sign. Then it looks like this:

f(x_0 + \alpha) = \sum_{n = 0}^\infty \frac{1}{n!}
\frac{\mathrm d^n f}{\mathrm d x^n} (x_0) \alpha^n \,.

I hope this is all fine up to this point. Now I will rewrite the same expression by just slightly reordering the terms.

f(x_0 + \alpha) = \sum_{n = 0}^\infty \frac{\alpha^n}{n!}
\left[ \frac{\mathrm d^n}{\mathrm d x^n} f(x) \right]_{x = x_0}

One can even factor out the function from the sum. This means that the derivative operator \mathrm d/\mathrm dx has to act outside of the square bracket. So be it.

f(x_0 + \alpha) = \left[ \sum_{n = 0}^\infty \frac{\alpha^n}{n!}
\left.\frac{\mathrm d^n}{\mathrm d x^n}\right|_{x = x_0} \right] f(x)

The sum is extracted from the function f and the differential operator acts outside of square bracket already. It is not too far fetched to write the sum as an exponential function.

f(x_0 + \alpha) = \exp\left( \alpha
\left.\frac{\mathrm d}{\mathrm d x}\right|_{x = x_0} \right) f(x)

One now calls \mathrm d / \mathrm d x the generator, sometimes denoted by T or so. The parameter \alpha is the amount of the transformation desired.

All that was in the notation of mathematics. The notation used in physics has a couple imaginary units in it. One does the following:

\exp(\alpha T) \mapsto \exp\left(\mathrm i \alpha [- \mathrm i T] \right) \,.

Then physicists still call \alpha the parameter but the generator now is -
\mathrm i T. When you now look at the momentum operator in quantum mechanics, \hat p_x = - \mathrm i \hbar \partial_x, you will see that this generates translation in the x-direction with the Taylor series in mind.