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 $\langle x \lvert x'\rangle=\delta(x-x')$?
$\begingroup$
$\endgroup$
1
-
$\begingroup$ Related: physics.stackexchange.com/q/89958/2451 , physics.stackexchange.com/q/273423/2451 , physics.stackexchange.com/q/330416/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Feb 2, 2018 at 18:16
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
2
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.
-
$\begingroup$ 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? $\endgroup$ Commented Feb 5, 2018 at 11:54
-
$\begingroup$ Yes, those limits are mathematically ill-defined. $\endgroup$– DanielCCommented Feb 5, 2018 at 13:28
$\begingroup$
$\endgroup$
Actually I think we can try to formalize it? Maybe embed $L^2(\mathbb N)$ into $L^2(\mathbb R)$ by having a constant value on each interval $[n \epsilon, (n+1)\epsilon)$. Then let $\epsilon \rightarrow 0$.
We get a sequence of subspaces. A vector should be approximated increasingly well by its projection to each subspace as $\epsilon \rightarrow 0$.
