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        "quote": "\n\n> [!definition] 定义 · 受限辛群与受限正交群\n>\n> [<u class=\"concept-term\" data-concept-reference=\"concepts/symplectic-geometry/restricted-symplectic-group\" data-concept-identity=\"urn:uuid:32fecc2c-a1e4-4f9b-b57c-68d2bed32ce5\" data-concept-route=\"/concept-identities#concept-32fecc2c-a1e4-4f9b-b57c-68d2bed32ce5\" id=\"first-3ad5554d4db186cd1885-zh\">受限辛群(restricted symplectic group)</u>](../../../../../concepts/symplectic-geometry/restricted-symplectic-group) $Sp_{\\mathrm{res}}$ 由保持无限维向量空间上的辛结构、并满足下述条件的线性变换 $g$ 组成：\n>\n> $$\n> \\operatorname{tr}(A^\\dagger A)<\\infty,\\qquad A=[J_g,J_0].\n> $$\n>\n> [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/restricted-orthogonal-group\" data-concept-identity=\"urn:uuid:be89d96d-f2fe-4fd0-b474-8739def4a433\" data-concept-route=\"/concept-identities#concept-be89d96d-f2fe-4fd0-b474-8739def4a433\" id=\"first-8f341fbb416145e08efb-zh\">受限正交群(restricted orthogonal group)</u>](../../../../../concepts/linear-algebra/restricted-orthogonal-group) $SO_{\\mathrm{res}}$ 则由保持无限维空间上的内积、且对 $[J_g,J_0]$ 满足同一条件的线性变换组成。\n",
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        "quote": "\n\n> [!definition] Definition · Restricted symplectic and orthogonal groups\n>\n> The [<u class=\"concept-term\" data-concept-reference=\"concepts/symplectic-geometry/restricted-symplectic-group\" data-concept-identity=\"urn:uuid:32fecc2c-a1e4-4f9b-b57c-68d2bed32ce5\" data-concept-route=\"/concept-identities#concept-32fecc2c-a1e4-4f9b-b57c-68d2bed32ce5\" id=\"first-3ad5554d4db186cd1885-en\">restricted symplectic group</u>](../../../../../concepts/symplectic-geometry/restricted-symplectic-group) $Sp_{\\mathrm{res}}$ consists of symplectic linear transformations $g$ on the infinite-dimensional symplectic space satisfying\n>\n> $$\n> \\operatorname{tr}(A^\\dagger A)<\\infty,\\qquad A=[J_g,J_0].\n> $$\n>\n> The [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/restricted-orthogonal-group\" data-concept-identity=\"urn:uuid:be89d96d-f2fe-4fd0-b474-8739def4a433\" data-concept-route=\"/concept-identities#concept-be89d96d-f2fe-4fd0-b474-8739def4a433\" id=\"first-8f341fbb416145e08efb-en\">restricted orthogonal group</u>](../../../../../concepts/linear-algebra/restricted-orthogonal-group) $SO_{\\mathrm{res}}$ consists of inner-product-preserving linear transformations satisfying the same condition on $[J_g,J_0]$.\n",
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