It's a Hilbert space, $\mu_H$ stands for the Haar measure on $U(1)$, but what does $S^1$ mean? I found it in one of my quantum mechanics books which approaches from a very 'mathematical' way.

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    $\begingroup$ Can you give a reference to the book? $\endgroup$
    – rob
    Jan 26 '16 at 8:29
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    $\begingroup$ $S^1$ is a circle, a "1-sphere". So this is the space of square-integrable functions over a periodic variable. $\endgroup$ Jan 26 '16 at 8:35
  • $\begingroup$ Thank you! Now it makes sense. @rob: I can't give a reference to the book, it's not written in english and it is not available online. But my question has been answered, thanks! $\endgroup$
    – user32109
    Jan 26 '16 at 8:50
  • $\begingroup$ @user32109 Dead-tree references are also okay. $\endgroup$
    – rob
    Jan 26 '16 at 10:00
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    $\begingroup$ $U(1)\cong S^1$. $\endgroup$
    – Qmechanic
    Jan 26 '16 at 14:21

The (Lie-)group $U(1)$ is the topological space $S^1$ (what we call a circle together with its standard open subsets) together with a rule how to multiply its points. In its representation as numbers in ${\mathbb C}$ with absolute value $1$, we have ${\mathrm e}^{{\mathrm i}\alpha}\bullet{\mathrm e}^{{\mathrm i}\beta}:={\mathrm e}^{{\mathrm i}\left(\alpha+\beta\right)}$.


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