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How to find the generator of translation?

  1. Nov 14, 2004 #1
    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:
    [tex][J^i, K^j] = i \epsilon^{ijk}K^k[/tex]

    Can one work out the generator for space translation, [tex]P[/tex]? so that one can show explicitly that:

    [tex][K^i, P^j] = 0[/tex]

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

    OR

    there is no matrix form for these two generators?
     
  2. jcsd
  3. Nov 16, 2004 #2

    pervect

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    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.
     
  4. Nov 16, 2004 #3
    Isn't the generator for space translations [tex]e^{i \hat{p}a/\hbar}[/tex]? Or is a generator something different?
     
  5. Nov 23, 2004 #4
    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 [tex]x_5[/tex] is always 1.
     
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