# Symmetry 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.