Exercise [21.09]

deant · 01 Sep 2012, 16:52

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.

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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).

$\blacksquare$