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 Exercise [21.09] 
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Joined: 12 Jul 2010, 07:44
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Post Exercise [21.09]
We first show that the functions e^{i p \chi} are linearly independent for any two distinct values of p.

Consider the fourier transform formulae shown on p.166. The first formula expresses f(\chi) as an arbitrary linear combination of the functions e^{i p \chi}, where \left(2\pi\right)^{-\frac{1}{2}}g(p) gives the "coefficient" or "weight" of each function in the (integral) combination.

If the functions e^{i p \chi} were NOT linearly independent, then there would be two or more distinct hyperfunctions g(p) that would give rise to exactly the same f(\chi). (By the definition of linear independence).

However, each of the fourier equations given is the inverse of the other. So there is a bijection between the space of hyperfunctions f(\chi) and the space of hyperfunctions g(p). Or to put this another way, one can use the 1st formula to compute f(\chi) given an initial g(p), and the 2nd formula to recover the initial g(p) from f(\chi).

This would be impossible if there were more than one distinct g(p) that corresponded to a particular f(\chi). So the functions e^{i p \chi} are indeed linearly independent for any two distinct values of p.


The fourier formulae of p.166 may readily be extended to higher dimensions, and then proof that the functions e^{i p_a x^a} are linearly independent for any two distinct one-forms p_a follows analogously. (We could then take p_a = -\frac{1}{\hbar}P_a so as to correspond exactly to the expressions in the book, if we wished to).


01 Sep 2012, 16:52
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