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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">The group U(1)</span>\n> <span id=\"source-block-2052\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<span id=\"concept-2-group-u-1\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-id=\"concept-2-group-u-1\" data-concept-identity=\"urn:uuid:73e79877-a4cc-4221-bbef-694fec4c9a8b\" data-concept-route=\"/concept-identities#concept-73e79877-a4cc-4221-bbef-694fec4c9a8b\" id=\"first-a6eca3bcad136ed9baa5-en\">The group $U(1)$</u>). </span>The elements of the group $U(1)$ are points on the unit circle, which can be labeled by a unit complex number $e^{i\\theta}$, or an angle $\\theta\\in\\mathbf{R}$ with $\\theta$ and $\\theta+N2\\pi$ labeling the same group element for $N\\in\\mathbf{Z}$. Multiplication of group elements is complex multiplication, which by the properties of the exponential satisfies\n>\n> <span id=\"source-block-2053\" class=\"block-anchor\"></span>\n>\n> $$\n> e ^ {i \\theta_ {1}} e ^ {i \\theta_ {2}} = e ^ {i (\\theta_ {1} + \\theta_ {2})}\n> $$\n>\n> <span id=\"source-block-2054\" class=\"block-anchor\"></span>\n>\n> so in terms of angles the group law is addition (mod 2$\\pi$).\n",
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