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Say we have a $1D$ lattice with spacing $a$ between two sites. How does one formally map the discrete position basis of the lattice to a continuous one in the limit $a\to 0$.
For instance how does quantities such as $\lvert i\rangle$ with $i=xa$ maps to $\lvert x \rangle $? Or the orthogonality relations $\langle i \lvert j\rangle=\delta_{ij}$ and $\langle x \lvert x'\rangle=\delta(x-x')$?

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It was shown by the German mathematician Cantor that any mapping

$$ f:\mathbb N \rightarrow \mathbb R $$

cannot be surjective, which means that there is no way to map a discrete infinite basis in a Hilbert space into a continuous one. This is rephrased as: a separable Hilbert space cannot be isomorphic to a nonseparable one. This means that there is a negative answer to all your questions.

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  • $\begingroup$ But surely, physicists do take this kind of continuous limits all the time? Does it mean that any procedure of this kind lacks mathematical rigor? $\endgroup$ – Tony Jin Feb 5 '18 at 11:54
  • $\begingroup$ Yes, those limits are mathematically ill-defined. $\endgroup$ – DanielC Feb 5 '18 at 13:28

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