How to find the generator of translation?

[kakarukeys]

The Galilei group contains rotations, Galilean transformations, space translation and time translation.

It is easy to work out generators for rotations and Galilean transfromations in matrix form.

And they obey:
$$[J^i, K^j] = i \epsilon^{ijk}K^k$$

Can one work out the generator for space translation, $$P$$? so that one can show explicitly that:

$$[K^i, P^j] = 0$$

and same for time translation.
$$[K^i, H] = i P^i$$

OR

there is no matrix form for these two generators?

 

[pervect]

Nobody else has taken a shot at this, so I'll put my $.02 in, though I'm afraid I can't give you as definite an answer as you'd like (well, for that matter, as I'd like).

Usual matrix notation is linear, so you can write down (x,y) -> ax + by in matrix form. However, you want to write a transform of the form x -> x+a. This isn't a linear tranform. It appears to me that you can do this by just defining a "variable" that's equal to a constant. To avoid winding up with non-square matrices, you'll have to add a dummy line, that describes how a constant transforms. Well, a constant is always equal to itself, it doesn't depend on the other variables, so the matrix entry for how a constant transforms will have to say that it's equal to 1 x itself, no other variable affects it.

 

[da_willem]

Isn't the generator for space translations $$e^{i \hat{p}a/\hbar}$$? Or is a generator something different?

 

[kakarukeys]

I do not quite understand what you said.
But I found the answer.
The matrix can be found by expanding the 4D space to 5D, provided that $$x_5$$ is always 1.