# 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 . Its Taylor series around the point can then be expanded like so:

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

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

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

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

One now calls the generator, sometimes denoted by or so. The parameter 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:

Then physicists still call the parameter but the generator now is . When you now look at the momentum operator in quantum mechanics, , you will see that this generates translation in the -direction with the Taylor series in mind.