What's your opinions on the Axiom of Choice?

[Hurkyl]

Now, to try an answer to your question I would say that the eigenvectors of the position operator in the position representation are:

\hat{x} = x \rightarrow eigenVectors(\hat{x}) = \{\phi_{x'}:\mathbb{R} \rightarrow \mathbb{C} | \phi_{x'}(x) \mapsto \delta(x - x') , x'\in \mathbb{R}\}

\hat{p} = i \frac{\partial}{\partial x} \rightarrow eigenVectors(\hat{p}) = \{\phi_{k}:\mathbb{C} \rightarrow \mathbb{R} | \phi_{k}(x) \mapsto e^{i k x} , k\in \mathbb{R}\}

In physics we say that either of these sets of eigenvectors are an uncountable basis!

The linear combinations become definite integrals covering the range of the eigenvalues:

f(x) = \int_{-\infty} ^{\infty} c_{x&#039;} \phi_{x&#039;}(x) dx&#039; [/itex]<br /> <br /> where the coefficients are given by Fourier&#039;s trick:<br /> <br /> c_{x&amp;#039;} = \int_{-\infty} ^{\infty} f(x) \phi_{x&amp;#039;}(x) dx [/itex]&lt;br /&gt; &lt;br /&gt; I&amp;#039;m interested to learn more about what&amp;#039;s really going on here.

&lt;br /&gt; Well, the thing here is that neither of these &amp;quot;bases&amp;quot; contain a single element of your Hilbert space! &lt;img src=&quot;https://www.physicsforums.com/styles/physicsforums/xenforo/smilies/oldschool/bugeye.gif&quot; class=&quot;smilie&quot; loading=&quot;lazy&quot; alt=&quot;:bugeye:&quot; title=&quot;Bug Eye :bugeye:&quot; data-shortname=&quot;:bugeye:&quot; /&gt; The position and momentum operators, acting on our Hilbert space, actually do not have any nonzero eigenvectors.&lt;br /&gt; &lt;br /&gt; There are at least two ways to make sense of this. One is via a &amp;quot;rigged Hilbert space&amp;quot;, where in addition to our Hilbert space, we consider a smaller subspace of &amp;quot;test functions&amp;quot;, its dual space of &amp;quot;generalized functions&amp;quot;. The position and momentum operators have eigenvectors in this larger space of generalized functions. (distribution is another buzzword to look for)&lt;br /&gt; &lt;br /&gt; Another is through a &amp;quot;direct integral&amp;quot; of Hilbert spaces, which generalizes the (finite) direct sum of Hilbert spaces in a way similar to how the ordinary integral can be viewed as generalizing finite sums of numbers. We can write our Hilbert space as a direct integral of a copy of &lt;b&gt;C&lt;/b&gt; at each point in the real line. Now, your &amp;quot;basis vectors&amp;quot; are not elements of our Hilbert space of interest, but instead a choice of basis vectors for these individual Hilbert spaces.

 

[Hurkyl]

now i haven't read the whole thread but why again are we discussing the truth value of an axiom?

Because people aren't used to separating the notion of a mathematical theory and their favorite interpretation of that theory.