Symplectic group - Wikipedia

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id="toc-Important_subgroups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_between_the_symplectic_groups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_between_the_symplectic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relationship between the symplectic groups</span> </div> </a> <ul id="toc-Relationship_between_the_symplectic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physical_significance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Physical_significance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Physical significance</span> </div> </a> <button aria-controls="toc-Physical_significance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a></th></tr><tr><td class="sidebar-image" style="padding-bottom:0.9em;"><span typeof="mw:File/Frameless"><a href="/wiki/File:E8Petrie.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png" decoding="async" width="180" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/270px-E8Petrie.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/360px-E8Petrie.svg.png 2x" data-file-width="512" data-file-height="514" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Classical_group" title="Classical group">Classical groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li> <li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li> <li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li> <li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li> <li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li> <li><a class="mw-selflink selflink">Symplectic</a> Sp(<i>n</i>)</li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Simple_Lie_group" title="Simple Lie group">Simple Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks hlist" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Classical</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_Lie_group#A_series" title="Simple Lie group">A<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#B_series" title="Simple Lie group">B<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#C_series" title="Simple Lie group">C<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#D_series" title="Simple Lie group">D<sub><i>n</i></sub></a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Exceptional</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">Other Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li> <li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal group</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li> <li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie group–Lie algebra correspondence</a></li> <li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">Exponential map</a></li> <li><a href="/wiki/Adjoint_representation" title="Adjoint representation">Adjoint representation</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Killing_form" title="Killing form">Killing form</a></li><li><a href="/wiki/Index_of_a_Lie_algebra" title="Index of a Lie algebra">Index</a></li></ul></div></li> <li><a href="/wiki/Simple_Lie_algebra" title="Simple Lie algebra">Simple Lie algebra</a></li> <li><a href="/wiki/Loop_algebra" title="Loop algebra">Loop algebra</a></li> <li><a href="/wiki/Affine_Lie_algebra" title="Affine Lie algebra">Affine Lie algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">Semisimple Lie algebra</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagrams</a></li> <li><a href="/wiki/Cartan_subalgebra" title="Cartan subalgebra">Cartan subalgebra</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Root_system" title="Root system">Root system</a></li><li><a href="/wiki/Weyl_group" title="Weyl group">Weyl group</a></li></ul></div></li> <li><div class="hlist"><ul><li><a href="/wiki/Real_form_(Lie_theory)" title="Real form (Lie theory)">Real form</a></li><li><a href="/wiki/Complexification_(Lie_group)" title="Complexification (Lie group)">Complexification</a></li></ul></div></li> <li><a href="/wiki/Split_Lie_algebra" title="Split Lie algebra">Split Lie algebra</a></li> <li><a href="/wiki/Compact_Lie_algebra" title="Compact Lie algebra">Compact Lie algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Representation_of_a_Lie_group" title="Representation of a Lie group">Lie group representation</a></li> <li><a href="/wiki/Lie_algebra_representation" title="Lie algebra representation">Lie algebra representation</a></li> <li><a href="/wiki/Representation_theory_of_semisimple_Lie_algebras" title="Representation theory of semisimple Lie algebras">Representation theory of semisimple Lie algebras</a></li> <li><a href="/wiki/Representations_of_classical_Lie_groups" title="Representations of classical Lie groups">Representations of classical Lie groups</a></li> <li><a href="/wiki/Theorem_of_the_highest_weight" title="Theorem of the highest weight">Theorem of the highest weight</a></li> <li><a href="/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem" title="Borel–Weil–Bott theorem">Borel–Weil–Bott theorem</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Lie groups in <a href="/wiki/Physics" title="Physics">physics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Particle physics and representation theory</a></li> <li><a href="/wiki/Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">Lorentz group representations</a></li> <li><a href="/wiki/Representation_theory_of_the_Poincar%C3%A9_group" title="Representation theory of the Poincaré group">Poincaré group representations</a></li> <li><a href="/wiki/Representation_theory_of_the_Galilean_group" title="Representation theory of the Galilean group">Galilean group representations</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li> <li><a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li> <li><a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a></li> <li><a href="/wiki/Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a></li> <li><a href="/wiki/Armand_Borel" title="Armand Borel">Armand Borel</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below plainlist"> <ul><li><a href="/wiki/Glossary_of_Lie_groups_and_Lie_algebras" title="Glossary of Lie groups and Lie algebras">Glossary</a></li> <li><a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">Table of Lie groups</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" 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href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group"><i>p</i>-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a class="mw-selflink selflink">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the name <b>symplectic group</b> can refer to two different, but closely related, collections of mathematical <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, denoted <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> and <span class="texhtml">Sp(<i>n</i>)</span> for positive integer <i>n</i> and <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <b>F</b> (usually <b>C</b> or <b>R</b>). The latter is called the <b>compact symplectic group</b> and is also denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {USp} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {USp} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ef12766ead658934bf106caf867a483d744d4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.532ex; height:2.843ex;" alt="{\displaystyle \mathrm {USp} (n)}"></span>. Many authors prefer slightly different notations, usually differing by factors of <span class="texhtml">2</span>. The notation used here is consistent with the size of the most common <a href="/wiki/Matrix_(math)" class="mw-redirect" title="Matrix (math)">matrices</a> which represent the groups. In <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a>'s classification of the <a href="/wiki/Simple_Lie_algebra" title="Simple Lie algebra">simple Lie algebras</a>, the Lie algebra of the complex group <span class="texhtml">Sp(2<i>n</i>, <b>C</b>)</span> is denoted <span class="texhtml"><i>C<sub>n</sub></i></span>, and <span class="texhtml">Sp(<i>n</i>)</span> is the <a href="/wiki/Real_form_(Lie_theory)#Compact_real_form" title="Real form (Lie theory)">compact real form</a> of <span class="texhtml">Sp(2<i>n</i>, <b>C</b>)</span>. Note that when we refer to <i>the</i> (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension <span class="texhtml"><i>n</i></span>. </p><p>The name "<a href="/wiki/Symplectic_topology" class="mw-redirect" title="Symplectic topology">symplectic</a> group" was coined by <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> as a replacement for the previous confusing names (<b>line</b>) <b>complex group</b> and <b>Abelian linear group</b>, and is the Greek analog of "complex". </p><p>The <a href="/wiki/Metaplectic_group" title="Metaplectic group">metaplectic group</a> is a double cover of the symplectic group over <b>R</b>; it has analogues over other <a href="/wiki/Local_field" title="Local field">local fields</a>, <a href="/wiki/Finite_field" title="Finite field">finite fields</a>, and <a href="/wiki/Adele_ring" title="Adele ring">adele rings</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sp(2n,_F)"><span id="Sp.282n.2C_F.29"></span><span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=1" title="Edit section: Sp(2n, F)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The symplectic group is a <a href="/wiki/Classical_group" title="Classical group">classical group</a> defined as the set of <a href="/wiki/Linear_transformations" class="mw-redirect" title="Linear transformations">linear transformations</a> of a <span class="texhtml">2<i>n</i></span>-dimensional <a href="/wiki/Vector_space" title="Vector space">vector space</a> over the field <span class="texhtml"><b>F</b></span> which preserve a <a href="/wiki/Nondegenerate_form" class="mw-redirect" title="Nondegenerate form">non-degenerate</a> <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric</a> <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a>. Such a vector space is called a <a href="/wiki/Symplectic_vector_space" title="Symplectic vector space">symplectic vector space</a>, and the symplectic group of an abstract symplectic vector space <span class="texhtml"><i>V</i></span> is denoted <span class="texhtml">Sp(<i>V</i>)</span>. Upon fixing a basis for <span class="texhtml"><i>V</i></span>, the symplectic group becomes the group of <span class="texhtml">2<i>n</i> × 2<i>n</i></span> <a href="/wiki/Symplectic_matrix" title="Symplectic matrix">symplectic matrices</a>, with entries in <span class="texhtml"><b>F</b></span>, under the operation of <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. This group is denoted either <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> or <span class="texhtml">Sp(<i>n</i>, <b>F</b>)</span>. If the bilinear form is represented by the <a href="/wiki/Nonsingular_matrix" class="mw-redirect" title="Nonsingular matrix">nonsingular</a> <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrix</a> Ω, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>M</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>×</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi mathvariant="normal">Ω</mi> <mi>M</mi> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb8668fee7c231253ef21c9ad894b230bae7d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.762ex; height:3.176ex;" alt="{\displaystyle \operatorname {Sp} (2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \},}"></span></dd></dl> <p>where <i>M</i><sup>T</sup> is the <a href="/wiki/Transpose" title="Transpose">transpose</a> of <i>M</i>. Often Ω is defined to be </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0c04aaf32bd0a98a332f176c9ba0499801ea01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.21ex; height:6.176ex;" alt="{\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},}"></span></dd></dl> <p>where <i>I<sub>n</sub></i> is the identity matrix. In this case, <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> can be expressed as those block matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3fca5853838886a4f58c35e55be79aa87871ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.01ex; margin-bottom: -0.328ex; width:5.944ex; height:3.343ex;" alt="{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,D\in M_{n\times n}(F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,D\in M_{n\times n}(F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe2778040259eb78eea9c7a1f6fa769684dedad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.428ex; height:2.843ex;" alt="{\displaystyle A,B,C,D\in M_{n\times n}(F)}"></span>, satisfying the three equations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}-C^{\mathrm {T} }A+A^{\mathrm {T} }C&=0,\\-C^{\mathrm {T} }B+A^{\mathrm {T} }D&=I_{n},\\-D^{\mathrm {T} }B+B^{\mathrm {T} }D&=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>−</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>A</mi> <mo>+</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>C</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>B</mi> <mo>+</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>D</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>B</mi> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>D</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-C^{\mathrm {T} }A+A^{\mathrm {T} }C&=0,\\-C^{\mathrm {T} }B+A^{\mathrm {T} }D&=I_{n},\\-D^{\mathrm {T} }B+B^{\mathrm {T} }D&=0.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/118fa9cad377b15a4e912285aa6e3b66f895c43d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:21.601ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}-C^{\mathrm {T} }A+A^{\mathrm {T} }C&=0,\\-C^{\mathrm {T} }B+A^{\mathrm {T} }D&=I_{n},\\-D^{\mathrm {T} }B+B^{\mathrm {T} }D&=0.\end{aligned}}}"></span></dd></dl> <p>Since all symplectic matrices have <a href="/wiki/Determinant" title="Determinant">determinant</a> <span class="texhtml">1</span>, the symplectic group is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> <span class="texhtml">SL(2<i>n</i>, <b>F</b>)</span>. When <span class="texhtml"><i>n</i> = 1</span>, the symplectic condition on a matrix is satisfied <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the determinant is one, so that <span class="texhtml">Sp(2, <b>F</b>) = SL(2, <b>F</b>)</span>. For <span class="texhtml"><i>n</i> > 1</span>, there are additional conditions, i.e. <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> is then a proper subgroup of <span class="texhtml">SL(2<i>n</i>, <b>F</b>)</span>. </p><p>Typically, the field <span class="texhtml"><b>F</b></span> is the field of <a href="/wiki/Real_number" title="Real number">real numbers</a> <span class="texhtml"><b>R</b></span> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <span class="texhtml"><b>C</b></span>. In these cases <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> is a real or complex <a href="/wiki/Lie_group" title="Lie group">Lie group</a> of real or complex dimension <span class="texhtml"><i>n</i>(2<i>n</i> + 1)</span>, respectively. These groups are <a href="/wiki/Connected_space" title="Connected space">connected</a> but <a href="/wiki/Compact_group" title="Compact group">non-compact</a>. </p><p>The <a href="/wiki/Center_(group_theory)" title="Center (group theory)">center</a> of <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> consists of the matrices <span class="texhtml"><i>I</i><sub>2<i>n</i></sub></span> and <span class="texhtml">−<i>I</i><sub>2<i>n</i></sub></span> as long as the <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic of the field</a> is not <span class="texhtml">2</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Since the center of <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> is discrete and its quotient modulo the center is a <a href="/wiki/Simple_group" title="Simple group">simple group</a>, <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> is considered a <a href="/wiki/Simple_Lie_group#Comments_on_the_definition" title="Simple Lie group">simple Lie group</a>. </p><p>The real rank of the corresponding Lie algebra, and hence of the Lie group <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span>, is <span class="texhtml"><i>n</i></span>. </p><p>The <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span> is the set </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sp}}(2n,F)=\{X\in M_{2n\times 2n}(F):\Omega X+X^{\mathrm {T} }\Omega =0\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>×</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mi>X</mi> <mo>+</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sp}}(2n,F)=\{X\in M_{2n\times 2n}(F):\Omega X+X^{\mathrm {T} }\Omega =0\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe802a19b3a89324a375752788730ceee62636d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:48.987ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {sp}}(2n,F)=\{X\in M_{2n\times 2n}(F):\Omega X+X^{\mathrm {T} }\Omega =0\},}"></span></dd></dl> <p>equipped with the <a href="/wiki/Commutator#Ring_theory" title="Commutator">commutator</a> as its Lie bracket.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For the standard skew-symmetric bilinear form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =({\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =({\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc29fb1eb253a663bd18b229dad012f34946880e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.032ex; margin-bottom: -0.306ex; width:11.046ex; height:3.509ex;" alt="{\displaystyle \Omega =({\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}})}"></span>, this Lie algebra is the set of all block matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3fca5853838886a4f58c35e55be79aa87871ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.01ex; margin-bottom: -0.328ex; width:5.944ex; height:3.343ex;" alt="{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}"></span> subject to the conditions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A&=-D^{\mathrm {T} },\\B&=B^{\mathrm {T} },\\C&=C^{\mathrm {T} }.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=-D^{\mathrm {T} },\\B&=B^{\mathrm {T} },\\C&=C^{\mathrm {T} }.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a82af045b61764889f4645aee3084e5d43997a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:11.414ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}A&=-D^{\mathrm {T} },\\B&=B^{\mathrm {T} },\\C&=C^{\mathrm {T} }.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Sp(2n,_C)"><span id="Sp.282n.2C_C.29"></span><span class="texhtml">Sp(2<i>n</i>, <b>C</b>)</span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=2" title="Edit section: Sp(2n, C)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The symplectic group over the field of complex numbers is a <a href="/wiki/Compact_group" title="Compact group">non-compact</a>, <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>, <a href="/wiki/Simple_Lie_group" title="Simple Lie group">simple Lie group</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Sp(2n,_R)"><span id="Sp.282n.2C_R.29"></span><span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=3" title="Edit section: Sp(2n, R)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="texhtml">Sp(<i>n</i>, <b>C</b>)</span> is the <a href="/wiki/Complexification_(Lie_group)" title="Complexification (Lie group)">complexification</a> of the real group <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span>. <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> is a real, <a href="/wiki/Compact_group" title="Compact group">non-compact</a>, <a href="/wiki/Connected_space" title="Connected space">connected</a>, <a href="/wiki/Simple_Lie_group" title="Simple Lie group">simple Lie group</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It has a <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to the group of <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> under addition. As the <a href="/wiki/Real_form" class="mw-redirect" title="Real form">real form</a> of a <a href="/wiki/Simple_Lie_group" title="Simple Lie group">simple Lie group</a> its Lie algebra is a <a href="/wiki/Split_Lie_algebra" title="Split Lie algebra">splittable Lie algebra</a>. </p><p>Some further properties of <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span>: </p> <ul><li>The <a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a> from the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> <span class="texhtml"><b>sp</b>(2<i>n</i>, <b>R</b>)</span> to the group <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> is not <a href="/wiki/Surjective_function" title="Surjective function">surjective</a>. However, any element of the group can be represented as the product of two exponentials.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In other words,</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall S\in \operatorname {Sp} (2n,\mathbf {R} )\,\,\exists X,Y\in {\mathfrak {sp}}(2n,\mathbf {R} )\,\,S=e^{X}e^{Y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>S</mi> <mo>∈</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>S</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall S\in \operatorname {Sp} (2n,\mathbf {R} )\,\,\exists X,Y\in {\mathfrak {sp}}(2n,\mathbf {R} )\,\,S=e^{X}e^{Y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08ded0aa1591bc17452c2d7fb018d3562239cd30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.216ex; height:3.176ex;" alt="{\displaystyle \forall S\in \operatorname {Sp} (2n,\mathbf {R} )\,\,\exists X,Y\in {\mathfrak {sp}}(2n,\mathbf {R} )\,\,S=e^{X}e^{Y}.}"></span></dd></dl></dd></dl> <ul><li>For all <span class="texhtml"><i>S</i></span> in <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span>:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=OZO'\quad {\text{such that}}\quad O,O'\in \operatorname {Sp} (2n,\mathbf {R} )\cap \operatorname {SO} (2n)\cong U(n)\quad {\text{and}}\quad Z={\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>O</mi> <mi>Z</mi> <msup> <mi>O</mi> <mo>′</mo> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>such that</mtext> </mrow> <mspace width="1em" /> <mi>O</mi> <mo>,</mo> <msup> <mi>O</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>D</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=OZO'\quad {\text{such that}}\quad O,O'\in \operatorname {Sp} (2n,\mathbf {R} )\cap \operatorname {SO} (2n)\cong U(n)\quad {\text{and}}\quad Z={\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c0040b4cf5e3353adf42e5e971052eead0ce7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:87.257ex; height:6.176ex;" alt="{\displaystyle S=OZO'\quad {\text{such that}}\quad O,O'\in \operatorname {Sp} (2n,\mathbf {R} )\cap \operatorname {SO} (2n)\cong U(n)\quad {\text{and}}\quad Z={\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}.}"></span></dd></dl></dd> <dd>The matrix <span class="texhtml"><i>D</i></span> is <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite</a> and <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal</a>. The set of such <span class="texhtml"><i>Z</i></span>s forms a non-compact subgroup of <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> whereas <span class="texhtml">U(<i>n</i>)</span> forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Further <a href="/wiki/Symplectic_matrix" title="Symplectic matrix">symplectic matrix</a> properties can be found on that Wikipedia page.</dd></dl> <ul><li>As a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> has a manifold structure. The <a href="/wiki/Manifold" title="Manifold">manifold</a> for <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> is <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphic</a> to the <a href="/wiki/Manifold#Cartesian_products" title="Manifold">Cartesian product</a> of the <a href="/wiki/Unitary_group" title="Unitary group">unitary group</a> <span class="texhtml">U(<i>n</i>)</span> with a <a href="/wiki/Vector_space" title="Vector space">vector space</a> of dimension <span class="texhtml"><i>n</i>(<i>n</i>+1)</span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Infinitesimal_generators">Infinitesimal generators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=4" title="Edit section: Infinitesimal generators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The members of the symplectic Lie algebra <span class="texhtml"><b>sp</b>(2<i>n</i>, <b>F</b>)</span> are the <a href="/wiki/Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian matrices</a>. </p><p> These are matrices, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> such that</p><blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\begin{pmatrix}A&B\\C&-A^{\mathrm {T} }\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mo>−</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\begin{pmatrix}A&B\\C&-A^{\mathrm {T} }\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1cc598d8c62f195ee6ab3e382c0c0e4091934d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.168ex; height:6.176ex;" alt="{\displaystyle Q={\begin{pmatrix}A&B\\C&-A^{\mathrm {T} }\end{pmatrix}}}"></span></p></blockquote><p>where <span class="texhtml"><i>B</i></span> and <span class="texhtml"><i>C</i></span> are <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a>. See <a href="/wiki/Classical_group" title="Classical group">classical group</a> for a derivation. </p><div class="mw-heading mw-heading3"><h3 id="Example_of_symplectic_matrices">Example of symplectic matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=5" title="Edit section: Example of symplectic matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> For <span class="texhtml">Sp(2, <b>R</b>)</span>, the group of <span class="texhtml">2 × 2</span> matrices with determinant <span class="texhtml">1</span>, the three symplectic <span class="texhtml">(0, 1)</span>-matrices are:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></p><blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78eef0a778694186d0a7cec9a7ef88777b6b0fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.856ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.}"></span></p></blockquote> <div class="mw-heading mw-heading4"><h4 id="Sp(2n,_R)_2"><span id="Sp.282n.2C_R.29_2"></span>Sp(2n, R)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=6" title="Edit section: Sp(2n, R)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> It turns out that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68730f9f62424bdc5ad61fac08400a7303f6d3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.989ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}"></span> can have a fairly explicit description using generators. If we let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sym</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b53f7a393ab1e376574e61af57e6cd8591b0d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.66ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (n)}"></span> denote the symmetric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matrices, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68730f9f62424bdc5ad61fac08400a7303f6d3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.989ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}"></span> is generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(n)\cup N(n)\cup \{\Omega \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">Ω</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(n)\cup N(n)\cup \{\Omega \},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240fe0c850eacaac14aaf60c2c0062d198cf8c7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.211ex; height:2.843ex;" alt="{\displaystyle D(n)\cup N(n)\cup \{\Omega \},}"></span> where</p><blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbf {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>D</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>A</mi> <mo>∈</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>N</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>B</mi> <mo>∈</mo> <mi>Sym</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbf {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f58cbafa84f09e68c60189d79369de3b9eb2bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:43.061ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbf {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}}"></span></p></blockquote><p>are subgroups of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68730f9f62424bdc5ad61fac08400a7303f6d3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.989ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}"></span><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup>pg 173</sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup>pg 2</sup>. </p><div class="mw-heading mw-heading3"><h3 id="Relationship_with_symplectic_geometry">Relationship with symplectic geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=7" title="Edit section: Relationship with symplectic geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic geometry</a> is the study of <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifolds</a>. The <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> at any point on a symplectic manifold is a <a href="/wiki/Symplectic_vector_space" title="Symplectic vector space">symplectic vector space</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> As noted earlier, structure preserving transformations of a symplectic vector space form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> and this group is <span class="texhtml">Sp(2<i>n</i>, <b>F</b>)</span>, depending on the dimension of the space and the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> over which it is defined. </p><p>A symplectic vector space is itself a symplectic manifold. A transformation under an <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of the symplectic group is thus, in a sense, a linearised version of a <a href="/wiki/Symplectomorphism" title="Symplectomorphism">symplectomorphism</a> which is a more general structure preserving transformation on a symplectic manifold. </p> <div class="mw-heading mw-heading2"><h2 id="Sp(n)"><span id="Sp.28n.29"></span><span class="texhtml">Sp(<i>n</i>)</span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=8" title="Edit section: Sp(n)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>compact symplectic group</b><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <span class="texhtml">Sp(<i>n</i>)</span> is the intersection of <span class="texhtml">Sp(2<i>n</i>, <b>C</b>)</span> with the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n\times 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mo>×</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n\times 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f264bcf2cf920cb907b8c9a50120da5d34a27a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.955ex; height:2.176ex;" alt="{\displaystyle 2n\times 2n}"></span> unitary group: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (n):=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {U} (2n)=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {SU} (2n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩</mo> <mi mathvariant="normal">U</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>SU</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (n):=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {U} (2n)=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {SU} (2n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe870de8b0ede75e6f113e470040e64a89861c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.79ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (n):=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {U} (2n)=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {SU} (2n).}"></span></dd></dl> <p>It is sometimes written as <span class="texhtml">USp(2<i>n</i>)</span>. Alternatively, <span class="texhtml">Sp(<i>n</i>)</span> can be described as the subgroup of <span class="texhtml">GL(<i>n</i>, <b>H</b>)</span> (invertible <a href="/wiki/Quaternion" title="Quaternion">quaternionic</a> matrices) that preserves the standard <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">hermitian form</a> on <span class="texhtml"><b>H</b><sup><i>n</i></sup></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle ={\bar {x}}_{1}y_{1}+\cdots +{\bar {x}}_{n}y_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle ={\bar {x}}_{1}y_{1}+\cdots +{\bar {x}}_{n}y_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e53bfeb3f40876341ed1e7ca24e6e5f3ba103e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.961ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle ={\bar {x}}_{1}y_{1}+\cdots +{\bar {x}}_{n}y_{n}.}"></span></dd></dl> <p>That is, <span class="texhtml">Sp(<i>n</i>)</span> is just the <a href="/wiki/Classical_group#Sp(p,_q)_–_the_quaternionic_unitary_group" title="Classical group">quaternionic unitary group</a>, <span class="texhtml">U(<i>n</i>, <b>H</b>)</span>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Indeed, it is sometimes called the <b>hyperunitary group</b>. Also Sp(1) is the group of quaternions of norm <span class="texhtml">1</span>, equivalent to <span class="texhtml"><a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a></span> and topologically a <a href="/wiki/3-sphere" title="3-sphere"><span class="texhtml">3</span>-sphere</a> <span class="texhtml">S<sup>3</sup></span>. </p><p>Note that <span class="texhtml">Sp(<i>n</i>)</span> is <i>not</i> a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric <span class="texhtml"><b>H</b></span>-bilinear form on <span class="texhtml"><b>H</b><sup><i>n</i></sup></span>: there is no such form except the zero form. Rather, it is isomorphic to a subgroup of <span class="texhtml">Sp(2<i>n</i>, <b>C</b>)</span>, and so does preserve a complex <a href="/wiki/Symplectic_manifold" title="Symplectic manifold"> symplectic form</a> in a vector space of twice the dimension. As explained below, the Lie algebra of <span class="texhtml">Sp(<i>n</i>)</span> is the compact <a href="/wiki/Real_form" class="mw-redirect" title="Real form">real form</a> of the complex symplectic Lie algebra <span class="texhtml"><b>sp</b>(2<i>n</i>, <b>C</b>)</span>. </p><p><span class="texhtml">Sp(<i>n</i>)</span> is a real Lie group with (real) dimension <span class="texhtml"><i>n</i>(2<i>n</i> + 1)</span>. It is <a href="/wiki/Compact_space" title="Compact space">compact</a> and <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>The Lie algebra of <span class="texhtml">Sp(<i>n</i>)</span> is given by the quaternionic <a href="/wiki/Skew-Hermitian" class="mw-redirect" title="Skew-Hermitian">skew-Hermitian</a> matrices, the set of <span class="texhtml"><i>n</i>-by-<i>n</i></span> quaternionic matrices that satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+A^{\dagger }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+A^{\dagger }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e70e2a25a48dc5399c72f60d072b75b28953de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.55ex; height:2.843ex;" alt="{\displaystyle A+A^{\dagger }=0}"></span></dd></dl> <p>where <span class="texhtml">A<sup>†</sup></span> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <span class="texhtml">A</span> (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator. </p> <div class="mw-heading mw-heading3"><h3 id="Important_subgroups">Important subgroups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=9" title="Edit section: Important subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some main subgroups are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (n)\supset \operatorname {Sp} (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⊃</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (n)\supset \operatorname {Sp} (n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f53e3634cd72f26cc68f863f85a95b229baea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.679ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (n)\supset \operatorname {Sp} (n-1)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (n)\supset \operatorname {U} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⊃</mo> <mi mathvariant="normal">U</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (n)\supset \operatorname {U} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6926d15af0902483823355dd7eb6b86731a38be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.835ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (n)\supset \operatorname {U} (n)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (2)\supset \operatorname {O} (4)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⊃</mo> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2)\supset \operatorname {O} (4)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b42728a991ef77a7b431f96df9bc978830650c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.435ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (2)\supset \operatorname {O} (4)}"></span></dd></dl> <p>Conversely it is itself a subgroup of some other groups: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SU} (2n)\supset \operatorname {Sp} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SU</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⊃</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SU} (2n)\supset \operatorname {Sp} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e50c4ae0843a994d6d21fb3390d5f5af65f7456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.29ex; height:2.843ex;" alt="{\displaystyle \operatorname {SU} (2n)\supset \operatorname {Sp} (n)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {F} _{4}\supset \operatorname {Sp} (4)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⊃</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {F} _{4}\supset \operatorname {Sp} (4)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a5f735ddc54b43970f6509204163cbde7a76a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.227ex; height:2.843ex;" alt="{\displaystyle \operatorname {F} _{4}\supset \operatorname {Sp} (4)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {G} _{2}\supset \operatorname {Sp} (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊃</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {G} _{2}\supset \operatorname {Sp} (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e735b83cf9bb4928ef0d4ab66ab63fc2338d8a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.534ex; height:2.843ex;" alt="{\displaystyle \operatorname {G} _{2}\supset \operatorname {Sp} (1)}"></span></dd></dl> <p>There are also the <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a> of the <a href="/wiki/Lie_algebras" class="mw-redirect" title="Lie algebras">Lie algebras</a> <span class="texhtml"><b>sp</b>(2) = <b>so</b>(5)</span> and <span class="texhtml"><b>sp</b>(1) = <b>so</b>(3) = <b>su</b>(2)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_between_the_symplectic_groups">Relationship between the symplectic groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=10" title="Edit section: Relationship between the symplectic groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every complex, <a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple Lie algebra</a> has a <a href="/wiki/Real_form_(Lie_theory)#Split_real_form" title="Real form (Lie theory)">split real form</a> and a <a href="/wiki/Real_form_(Lie_theory)#Compact_real_form" title="Real form (Lie theory)">compact real form</a>; the former is called a <a href="/wiki/Complexification" title="Complexification">complexification</a> of the latter two. </p><p>The Lie algebra of <span class="texhtml">Sp(2<i>n</i>, <b>C</b>)</span> is <a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple</a> and is denoted <span class="texhtml"><b>sp</b>(2<i>n</i>, <b>C</b>)</span>. Its <a href="/wiki/Real_form_(Lie_theory)#Split_real_form" title="Real form (Lie theory)">split real form</a> is <span class="texhtml"><b>sp</b>(2<i>n</i>, <b>R</b>)</span> and its <a href="/wiki/Real_form_(Lie_theory)#Compact_real_form" title="Real form (Lie theory)">compact real form</a> is <span class="texhtml"><b>sp</b>(<i>n</i>)</span>. These correspond to the Lie groups <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> and <span class="texhtml">Sp(<i>n</i>)</span> respectively. </p><p>The algebras, <span class="texhtml"><b>sp</b>(<i>p</i>, <i>n</i> − <i>p</i>)</span>, which are the Lie algebras of <span class="texhtml">Sp(<i>p</i>, <i>n</i> − <i>p</i>)</span>, are the <a href="/wiki/Metric_signature" title="Metric signature">indefinite signature</a> equivalent to the compact form. </p> <div class="mw-heading mw-heading2"><h2 id="Physical_significance">Physical significance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=11" title="Edit section: Physical significance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Classical_mechanics">Classical mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=12" title="Edit section: Classical mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The non-compact symplectic group <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket. </p><p>Consider a system of <span class="texhtml"><i>n</i></span> particles, evolving under <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamilton's equations</a> whose position in <a href="/wiki/Phase_space" title="Phase space">phase space</a> at a given time is denoted by the vector of <a href="/wiki/Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} =(q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})^{\mathrm {T} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {z} =(q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})^{\mathrm {T} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a373eae2160c2aa44c8177aa268a13f4ac3da822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.595ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} =(q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})^{\mathrm {T} }.}"></span></dd></dl> <p>The elements of the group <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span> are, in a certain sense, <a href="/wiki/Canonical_transformations" class="mw-redirect" title="Canonical transformations">canonical transformations</a> on this vector, i.e. they preserve the form of <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamilton's equations</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-A&M_15-0" class="reference"><a href="#cite_note-A&M-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> If </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} =\mathbf {Z} (\mathbf {z} ,t)=(Q^{1},\ldots ,Q^{n},P_{1},\ldots ,P_{n})^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} =\mathbf {Z} (\mathbf {z} ,t)=(Q^{1},\ldots ,Q^{n},P_{1},\ldots ,P_{n})^{\mathrm {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b336d826bd82107278d788f6316a3679229c556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.161ex; height:3.176ex;" alt="{\displaystyle \mathbf {Z} =\mathbf {Z} (\mathbf {z} ,t)=(Q^{1},\ldots ,Q^{n},P_{1},\ldots ,P_{n})^{\mathrm {T} }}"></span></dd></dl> <p>are new canonical coordinates, then, with a dot denoting time derivative, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {Z} }}=M({\mathbf {z} },t){\dot {\mathbf {z} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {Z} }}=M({\mathbf {z} },t){\dot {\mathbf {z} }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d46669fc15daec063a86d9a60c94eaa7446ad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.88ex; height:3.176ex;" alt="{\displaystyle {\dot {\mathbf {Z} }}=M({\mathbf {z} },t){\dot {\mathbf {z} }},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(\mathbf {z} ,t)\in \operatorname {Sp} (2n,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(\mathbf {z} ,t)\in \operatorname {Sp} (2n,\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaef9ad084ecbdb59e3ff1d9914288f78faaf8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.142ex; height:2.843ex;" alt="{\displaystyle M(\mathbf {z} ,t)\in \operatorname {Sp} (2n,\mathbf {R} )}"></span></dd></dl> <p>for all <span class="texhtml mvar" style="font-style:italic;">t</span> and all <span class="texhtml"><b>z</b></span> in phase space.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>For the special case of a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, Hamilton's equations describe the <a href="/wiki/Geodesic" title="Geodesic">geodesics</a> on that manifold. The coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3b6e58595ab90741577c4f9f63875fba9b7c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.879ex; height:3.009ex;" alt="{\displaystyle q^{i}}"></span> live on the underlying manifold, and the momenta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> live in the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a>. This is the reason why these are conventionally written with upper and lower indexes; it is to distinguish their locations. The corresponding Hamiltonian consists purely of the kinetic energy: it is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\tfrac {1}{2}}g^{ij}(q)p_{i}p_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\tfrac {1}{2}}g^{ij}(q)p_{i}p_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5658a2b727113d2dff10d1241a94732b571600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.343ex; height:3.509ex;" alt="{\displaystyle H={\tfrac {1}{2}}g^{ij}(q)p_{i}p_{j}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b14a4aa3b277a89268fd9026b8f16a749199cb10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.595ex; height:3.009ex;" alt="{\displaystyle g^{ij}}"></span> is the inverse of the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c1130c3dec178129b287a3672c72f88e773832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}"></span> on the Riemannian manifold.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-A&M_15-1" class="reference"><a href="#cite_note-A&M-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> In fact, the cotangent bundle of <i>any</i> smooth manifold can be a given a <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic structure</a> in a canonical way, with the symplectic form defined as the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> of the <a href="/wiki/Tautological_one-form" title="Tautological one-form">tautological one-form</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=13" title="Edit section: Quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Symplectic_group" title="Special:EditPage/Symplectic group">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">October 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Consider a system of <span class="texhtml"><i>n</i></span> particles whose <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a> encodes its position and momentum. These coordinates are continuous variables and hence the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, in which the state lives, is infinite-dimensional. This often makes the analysis of this situation tricky. An alternative approach is to consider the evolution of the position and momentum operators under the <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg equation</a> in <a href="/wiki/Phase_space" title="Phase space">phase space</a>. </p><p>Construct a vector of <a href="/wiki/Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {z}} =({\hat {q}}^{1},\ldots ,{\hat {q}}^{n},{\hat {p}}_{1},\ldots ,{\hat {p}}_{n})^{\mathrm {T} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {z}} =({\hat {q}}^{1},\ldots ,{\hat {q}}^{n},{\hat {p}}_{1},\ldots ,{\hat {p}}_{n})^{\mathrm {T} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd6363149699bebc59edf6674ab4cb23d41a14d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.719ex; height:3.176ex;" alt="{\displaystyle \mathbf {\hat {z}} =({\hat {q}}^{1},\ldots ,{\hat {q}}^{n},{\hat {p}}_{1},\ldots ,{\hat {p}}_{n})^{\mathrm {T} }.}"></span></dd></dl> <p>The <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relation</a> can be expressed simply as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {\hat {z}} ,\mathbf {\hat {z}} ^{\mathrm {T} }]=i\hbar \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbf {\hat {z}} ,\mathbf {\hat {z}} ^{\mathrm {T} }]=i\hbar \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76c61fc8dde04d402ffce61a81ef17605edcc36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.305ex; height:3.343ex;" alt="{\displaystyle [\mathbf {\hat {z}} ,\mathbf {\hat {z}} ^{\mathrm {T} }]=i\hbar \Omega }"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ={\begin{pmatrix}\mathbf {0} &I_{n}\\-I_{n}&\mathbf {0} \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ={\begin{pmatrix}\mathbf {0} &I_{n}\\-I_{n}&\mathbf {0} \end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8b481cd8de5efbb30ca93ac125fef1eaee921b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.563ex; height:6.176ex;" alt="{\displaystyle \Omega ={\begin{pmatrix}\mathbf {0} &I_{n}\\-I_{n}&\mathbf {0} \end{pmatrix}}}"></span></dd></dl> <p>and <span class="texhtml"><i>I</i><sub><i>n</i></sub></span> is the <span class="texhtml"><i>n</i> × <i>n</i></span> identity matrix. </p><p>Many physical situations only require quadratic <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonians</a>, i.e. <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonians</a> of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}={\frac {1}{2}}\mathbf {\hat {z}} ^{\mathrm {T} }K\mathbf {\hat {z}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}={\frac {1}{2}}\mathbf {\hat {z}} ^{\mathrm {T} }K\mathbf {\hat {z}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3a435183e4ff57a79cdd77ede4cddc9d38bb94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.319ex; height:5.176ex;" alt="{\displaystyle {\hat {H}}={\frac {1}{2}}\mathbf {\hat {z}} ^{\mathrm {T} }K\mathbf {\hat {z}} }"></span></dd></dl> <p>where <span class="texhtml"><i>K</i></span> is a <span class="texhtml">2<i>n</i> × 2<i>n</i></span> real, <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a>. This turns out to be a useful restriction and allows us to rewrite the <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg equation</a> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {\hat {z}} }{dt}}=\Omega K\mathbf {\hat {z}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {\hat {z}} }{dt}}=\Omega K\mathbf {\hat {z}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab6f9236166b411ab9a483729607f3eba82595d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.568ex; height:5.509ex;" alt="{\displaystyle {\frac {d\mathbf {\hat {z}} }{dt}}=\Omega K\mathbf {\hat {z}} }"></span></dd></dl> <p>The solution to this equation must preserve the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relation</a>. It can be shown that the time evolution of this system is equivalent to an <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of <a class="mw-selflink-fragment" href="#Sp.282n.2C_R.29">the real symplectic group, <span class="texhtml">Sp(2<i>n</i>, <b>R</b>)</span></a>, on the phase space. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li> <li><a href="/wiki/Metaplectic_group" title="Metaplectic group">Metaplectic group</a></li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal group</a></li> <li><a href="/wiki/Paramodular_group" title="Paramodular group">Paramodular group</a></li> <li><a href="/wiki/Projective_unitary_group" title="Projective unitary group">Projective unitary group</a></li> <li><a href="/wiki/Representations_of_classical_Lie_groups" title="Representations of classical Lie groups">Representations of classical Lie groups</a></li> <li><a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic manifold</a>, <a href="/wiki/Symplectic_matrix" title="Symplectic matrix">Symplectic matrix</a>, <a href="/wiki/Symplectic_vector_space" title="Symplectic vector space">Symplectic vector space</a>, <a href="/wiki/Symplectic_representation" title="Symplectic representation">Symplectic representation</a></li> <li><a href="/wiki/Unitary_group" title="Unitary group">Unitary group</a></li> <li><a href="/wiki/%CE%9810" title="Θ10">Θ10</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=15" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Symplectic_group">"Symplectic group"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i> Retrieved on 13 December 2014.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Prop. 3.25</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/1051400">"Is the symplectic group Sp(2<i>n</i>, <b>R</b>) simple?"</a>, <i><a href="/wiki/Stack_Exchange" title="Stack Exchange">Stack Exchange</a></i> Retrieved on 14 December 2014.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/1051255">"Is the exponential map for Sp(2<i>n</i>, <b>R</b>) surjective?"</a>, <i><a href="/wiki/Stack_Exchange" title="Stack Exchange">Stack Exchange</a></i> Retrieved on 5 December 2014.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.maths.nottingham.ac.uk/personal/ga/papers/2602.pdf">"Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso"</a>, Retrieved on 30 January 2015.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.maths.ed.ac.uk/~aar/papers/arnogive.pdf">"Symplectic Geometry – Arnol'd and Givental"</a>, Retrieved on 30 January 2015.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/SymplecticGroup.html">Symplectic Group</a>, (source: <a href="/wiki/Wolfram_MathWorld" class="mw-redirect" title="Wolfram MathWorld">Wolfram MathWorld</a>), downloaded February 14, 2012</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGerald_B._Folland.2016" class="citation book cs1">Gerald B. Folland. (2016). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/945482850"><i>Harmonic analysis in phase space</i></a>. Princeton: Princeton Univ Press. p. 173. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-8242-7" title="Special:BookSources/978-1-4008-8242-7"><bdi>978-1-4008-8242-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/945482850">945482850</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Harmonic+analysis+in+phase+space&rft.place=Princeton&rft.pages=173&rft.pub=Princeton+Univ+Press&rft.date=2016&rft_id=info%3Aoclcnum%2F945482850&rft.isbn=978-1-4008-8242-7&rft.au=Gerald+B.+Folland.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F945482850&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHabermann,_Katharina,_1966-2006" class="citation book cs1">Habermann, Katharina, 1966- (2006). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/262692314"><i>Introduction to symplectic Dirac operators</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-33421-7" title="Special:BookSources/978-3-540-33421-7"><bdi>978-3-540-33421-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/262692314">262692314</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+symplectic+Dirac+operators&rft.pub=Springer&rft.date=2006&rft_id=info%3Aoclcnum%2F262692314&rft.isbn=978-3-540-33421-7&rft.au=Habermann%2C+Katharina%2C+1966-&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F262692314&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>) CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://empg.maths.ed.ac.uk/Activities/BRST/">"Lecture Notes – Lecture 2: Symplectic reduction"</a>, Retrieved on 30 January 2015.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Section 1.2.8</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> p. 14</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Prop. 13.12</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFArnold1989">Arnold 1989</a> gives an extensive mathematical overview of classical mechanics. See chapter 8 for <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifolds</a>.</span> </li> <li id="cite_note-A&M-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-A&M_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-A&M_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Ralph Abraham</a> and <a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Jerrold E. Marsden</a>, <i>Foundations of Mechanics</i>, (1978) Benjamin-Cummings, London <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-0102-X" title="Special:BookSources/0-8053-0102-X">0-8053-0102-X</a></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoldstein1980">Goldstein 1980</a>, Section 9.3</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Jurgen Jost, (1992) <i>Riemannian Geometry and Geometric Analysis</i>, Springer.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFda_Silva2008" class="citation book cs1">da Silva, Ana Cannas (2008). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/978-3-540-45330-7"><i>Lectures on Symplectic Geometry</i></a>. Lecture Notes in Mathematics. Vol. 1764. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 9. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-45330-7">10.1007/978-3-540-45330-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-42195-5" title="Special:BookSources/978-3-540-42195-5"><bdi>978-3-540-42195-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Symplectic+Geometry&rft.place=Berlin%2C+Heidelberg&rft.series=Lecture+Notes+in+Mathematics&rft.pages=9&rft.pub=Springer+Berlin+Heidelberg&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-3-540-45330-7&rft.isbn=978-3-540-42195-5&rft.aulast=da+Silva&rft.aufirst=Ana+Cannas&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2F978-3-540-45330-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_group&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnold1989" class="citation cs2"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold, V. I.</a> (1989), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalmeth0000arno"><i>Mathematical Methods of Classical Mechanics</i></a></span>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 60 (second ed.), <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96890-3" title="Special:BookSources/0-387-96890-3"><bdi>0-387-96890-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+of+Classical+Mechanics&rft.series=Graduate+Texts+in+Mathematics&rft.edition=second&rft.pub=Springer-Verlag&rft.date=1989&rft.isbn=0-387-96890-3&rft.aulast=Arnold&rft.aufirst=V.+I.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalmeth0000arno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2015" class="citation cs2">Hall, Brian C. (2015), <i>Lie groups, Lie algebras, and representations: An elementary introduction</i>, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319134666" title="Special:BookSources/978-3319134666"><bdi>978-3319134666</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+groups%2C+Lie+algebras%2C+and+representations%3A+An+elementary+introduction&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2015&rft.isbn=978-3319134666&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFultonHarris1991" class="citation cs2"><a href="/wiki/William_Fulton_(mathematician)" title="William Fulton (mathematician)">Fulton, W.</a>; <a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, J.</a> (1991), <i>Representation Theory, A first Course</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 129, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97495-8" title="Special:BookSources/978-0-387-97495-8"><bdi>978-0-387-97495-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Representation+Theory%2C+A+first+Course&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1991&rft.isbn=978-0-387-97495-8&rft.aulast=Fulton&rft.aufirst=W.&rft.au=Harris%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstein1980" class="citation book cs1"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, H.</a> (1980) [1950]. "Chapter 7". <i>Classical Mechanics</i> (2nd ed.). Reading MA: <a href="/wiki/Addison-Wesley_Publishing_Company" class="mw-redirect" title="Addison-Wesley Publishing Company">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-02918-9" title="Special:BookSources/0-201-02918-9"><bdi>0-201-02918-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+7&rft.btitle=Classical+Mechanics&rft.place=Reading+MA&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1980&rft.isbn=0-201-02918-9&rft.aulast=Goldstein&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2003" class="citation cs2">Lee, J. M. (2003), <i>Introduction to Smooth manifolds</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 218, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95448-1" title="Special:BookSources/0-387-95448-1"><bdi>0-387-95448-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Smooth+manifolds&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=2003&rft.isbn=0-387-95448-1&rft.aulast=Lee&rft.aufirst=J.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRossmann2002" class="citation cs2">Rossmann, Wulf (2002), <i>Lie Groups – An Introduction Through Linear Groups</i>, Oxford Graduate Texts in Mathematics, Oxford Science Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-859683-9" title="Special:BookSources/0-19-859683-9"><bdi>0-19-859683-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups+%E2%80%93+An+Introduction+Through+Linear+Groups&rft.series=Oxford+Graduate+Texts+in+Mathematics&rft.pub=Oxford+Science+Publications&rft.date=2002&rft.isbn=0-19-859683-9&rft.aulast=Rossmann&rft.aufirst=Wulf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerraroOlivaresParis2005" class="citation arxiv cs2">Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (March 2005), "Gaussian states in continuous variable quantum information", <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0503237">quant-ph/0503237</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Gaussian+states+in+continuous+variable+quantum+information&rft.date=2005-03&rft_id=info%3Aarxiv%2Fquant-ph%2F0503237&rft.aulast=Ferraro&rft.aufirst=Alessandro&rft.au=Olivares%2C+Stefano&rft.au=Paris%2C+Matteo+G.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+group" class="Z3988"></span>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q936434#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00571095">Japan</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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Symplectic group - Wikipedia