Editing Symplectic group

{{Short description|Mathematical group}}{{for| finite groups with all characteristic abelian subgroups cyclic|group of symplectic type}}
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In [[mathematics]], the name '''symplectic group''' can refer to two different, but closely related, collections of mathematical [[Group (mathematics)|groups]], denoted {{math|Sp(2''n'', '''F''')}} and {{math|Sp(''n'')}} for positive integer ''n'' and [[field (mathematics)|field]] '''F''' (usually '''C''' or '''R'''). The latter is called the '''compact symplectic group''' and is also denoted by <math>\mathrm{U
Sp}(n)</math>. Many authors prefer slightly different notations, usually differing by factors of {{math|2}}. The notation used here is consistent with the size of the most common [[Matrix (math)|matrices]] which represent the groups. In [[Élie Cartan|Cartan]]'s classification of the [[simple Lie algebra]]s, the Lie algebra of the complex group {{math|Sp(2''n'', '''C''')}} is denoted {{math|''C<sub>n</sub>''}}, and {{math|Sp(''n'')}} is the [[Real form (Lie theory)#Compact real form|compact real form]] of {{math|Sp(2''n'', '''C''')}}. Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension {{math|''n''}}.

The name "[[Symplectic topology|symplectic]] group" was coined by [[Hermann Weyl]] as a replacement for the previous confusing names ('''line''') '''complex group''' and '''Abelian linear group''', and is the Greek analog of "complex".

The [[metaplectic group]] is a double cover of the symplectic group over '''R'''; it has analogues over other [[local field]]s, [[finite field]]s, and [[adele ring]]s.

=={{math|Sp(2''n'', '''F''')}}==
The symplectic group is a [[classical group]] defined as the set of [[linear transformations]] of a {{math|2''n''}}-dimensional [[vector space]] over the field {{math|'''F'''}} which preserve a [[nondegenerate form|non-degenerate]] [[skew-symmetric matrix|skew-symmetric]] [[bilinear form]]. Such a vector space is called a [[symplectic vector space]], and the symplectic group of an abstract symplectic vector space {{math|''V''}} is denoted {{math|Sp(''V'')}}.  Upon fixing a basis for {{math|''V''}}, the symplectic group becomes the group of {{math|2''n'' × 2''n''}} [[symplectic matrix|symplectic matrices]], with entries in {{math|'''F'''}}, under the operation of [[matrix multiplication]].  This group is denoted either {{math|Sp(2''n'', '''F''')}} or {{math|Sp(''n'', '''F''')}}.  If the bilinear form is represented by the [[nonsingular matrix|nonsingular]] [[skew-symmetric matrix]] Ω, then

:<math>\operatorname{Sp}(2n, F) = \{M \in M_{2n \times 2n}(F) : M^\mathrm{T} \Omega M = \Omega\},</math>

where ''M''<sup>T</sup> is the [[transpose]] of ''M''.  Often Ω is defined to be

:<math>\Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix},</math>

where ''I<sub>n</sub>'' is the identity matrix.  In this case, {{math|Sp(2''n'', '''F''')}} can be expressed as those block matrices <math>(\begin{smallmatrix} A & B \\ C & D \end{smallmatrix})</math>, where <math>A, B, C, D \in M_{n \times n}(F)</math>, satisfying the three equations:

:<math>\begin{align}
-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{align}</math>

Since all symplectic matrices have [[determinant]] {{math|1}}, the symplectic group is a [[subgroup]] of the [[special linear group]] {{math|SL(2''n'', '''F''')}}.  When {{math|1=''n'' = 1}}, the symplectic condition on a matrix is satisfied [[if and only if]] the determinant is one, so that {{math|1=Sp(2, '''F''') = SL(2, '''F''')}}. For {{math|''n'' > 1}}, there are additional conditions, i.e. {{math|Sp(2''n'', '''F''')}} is then a proper subgroup of {{math|SL(2''n'', '''F''')}}.

Typically, the field {{math|'''F'''}} is the field of [[real number]]s {{math|'''R'''}} or [[complex number]]s {{math|'''C'''}}. In these cases {{math|Sp(2''n'', '''F''')}} is a real or complex [[Lie group]] of real or complex dimension {{math|''n''(2''n'' + 1)}}, respectively. These groups are [[connected space|connected]] but [[Compact group|non-compact]].

The [[Center (group theory)|center]] of {{math|Sp(2''n'', '''F''')}} consists of the matrices {{math|''I''<sub>2''n''</sub>}} and {{math|−''I''<sub>2''n''</sub>}} as long as the [[Characteristic (algebra)|characteristic of the field]] is not {{math|2}}.<ref>[http://www.encyclopediaofmath.org/index.php/Symplectic_group "Symplectic group"], ''[[Encyclopedia of Mathematics]]'' Retrieved on 13 December 2014.</ref> Since the center of {{math|Sp(2''n'', '''F''')}} is discrete and its quotient modulo the center is a [[simple group]], {{math|Sp(2''n'', '''F''')}} is considered a [[Simple Lie group#Comments on the definition|simple Lie group]].

The real rank of the corresponding Lie algebra, and hence of the Lie group {{math|Sp(2''n'', '''F''')}}, is {{math|''n''}}.

The [[Lie algebra]] of {{math|Sp(2''n'', '''F''')}} is the set

:<math>\mathfrak{sp}(2n,F) = \{X \in M_{2n \times 2n}(F) : \Omega X + X^\mathrm{T} \Omega = 0\},</math>

equipped with the [[Commutator#Ring theory|commutator]] as its Lie bracket.<ref>{{harvnb|Hall|2015}} Prop. 3.25</ref>  For the standard skew-symmetric bilinear form <math>\Omega = (\begin{smallmatrix} 0 & I \\ -I & 0 \end{smallmatrix})</math>, this Lie algebra is the set of all block matrices <math>(\begin{smallmatrix} A & B \\ C & D \end{smallmatrix})</math> subject to the conditions

:<math>\begin{align}
A &= -D^\mathrm{T}, \\
B &= B^\mathrm{T}, \\
C &= C^\mathrm{T}.
\end{align}</math>

==={{math|Sp(2''n'', '''C''')}}===
The symplectic group over the field of complex numbers is a [[Compact group|non-compact]], [[simply connected]], [[simple Lie group]]. The definition of this group includes '''no''' conjugates (contrary to what one might naively expect) but instead it is exactly the same as the definition bar the field change.{{sfn|Hall|2015|p=10}}

==={{math|Sp(2''n'', '''R''')}}===
{{math|Sp(''n'', '''C''')}} is the [[Complexification (Lie group)|complexification]] of the real group {{math|Sp(2''n'', '''R''')}}. {{math|Sp(2''n'', '''R''')}} is a real, [[Compact group|non-compact]], [[Connected space|connected]], [[simple Lie group]].<ref>[https://math.stackexchange.com/q/1051400 "Is the symplectic group Sp(2''n'', '''R''') simple?"], ''[[Stack Exchange]]'' Retrieved on 14 December 2014.</ref> It has a [[fundamental group]] [[Group isomorphism|isomorphic]] to the group of [[integers]] under addition. As the [[real form]] of a [[simple Lie group]] its Lie algebra is a [[Split Lie algebra|splittable Lie algebra]].

Some further properties of {{math|Sp(2''n'', '''R''')}}:

* The [[exponential map (Lie theory)|exponential map]] from the [[Lie algebra]] {{math|'''sp'''(2''n'', '''R''')}} to the group {{math|Sp(2''n'', '''R''')}} is not [[Surjective function|surjective]]. However, any element of the group can be represented as the product of two exponentials.<ref>[https://math.stackexchange.com/q/1051255 "Is the exponential map for Sp(2''n'', '''R''') surjective?"], ''[[Stack Exchange]]'' Retrieved on 5 December 2014.</ref> In other words,
::<math>\forall S \in \operatorname{Sp}(2n,\mathbf{R})\,\, \exists X,Y \in \mathfrak{sp}(2n,\mathbf{R}) \,\, S = e^Xe^Y. </math>

* For all {{math|''S''}} in {{math|Sp(2''n'', '''R''')}}:
::<math>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}.</math>
:The matrix {{math|''D''}} is [[Positive-definite matrix|positive-definite]] and [[Diagonal matrix|diagonal]]. The set of such {{math|''Z''}}s forms a non-compact subgroup of {{math|Sp(2''n'', '''R''')}} whereas {{math|U(''n'')}} forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.<ref>[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"], Retrieved on 30 January 2015.</ref> Further [[symplectic matrix]] properties can be found on that Wikipedia page.

* As a [[Lie group]], {{math|Sp(2''n'', '''R''')}} has a manifold structure. The [[manifold]] for {{math|Sp(2''n'', '''R''')}} is [[Diffeomorphism|diffeomorphic]] to the [[Manifold#Cartesian products|Cartesian product]] of the [[unitary group]] {{math|U(''n'')}} with a [[vector space]] of dimension {{math|''n''(''n''+1)}}.<ref>[http://www.maths.ed.ac.uk/~aar/papers/arnogive.pdf "Symplectic Geometry – Arnol'd and Givental"], Retrieved on 30 January 2015.</ref>

===Infinitesimal generators===
The members of the symplectic Lie algebra {{math|'''sp'''(2''n'', '''F''')}}  are the [[Hamiltonian matrix|Hamiltonian matrices]].

These are matrices, <math>Q</math> such that<blockquote><math>Q = \begin{pmatrix} A & B \\ C & -A^\mathrm{T} \end{pmatrix}</math></blockquote>where {{math|''B''}} and {{math|''C''}} are [[Symmetric matrix|symmetric matrices]]. See [[classical group]] for a derivation.

===Example of symplectic matrices===
For {{math|Sp(2, '''R''')}}, the group of {{math|2 × 2}} matrices with determinant {{math|1}}, the three symplectic {{math|(0, 1)}}-matrices are:<ref>[http://mathworld.wolfram.com/SymplecticGroup.html Symplectic Group], (source: [[Wolfram MathWorld]]), downloaded February 14, 2012</ref><blockquote><math>\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}. </math></blockquote>

==== Sp(2n, R) ====
It turns out that <math>\operatorname{Sp}(2n,\mathbf{R})</math> can have a fairly explicit description using generators. If we let <math>\operatorname{Sym}(n)</math> denote the symmetric <math>n\times n</math> matrices, then <math>\operatorname{Sp}(2n,\mathbf{R})</math> is generated by <math>D(n)\cup N(n) \cup \{\Omega\},</math> where<blockquote><math>\begin{align}
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{align}</math></blockquote>are subgroups of <math>\operatorname{Sp}(2n,\mathbf{R})</math><ref>{{Cite book|last=Gerald B. Folland.|url=https://www.worldcat.org/oclc/945482850|title=Harmonic analysis in phase space|date=2016|publisher=Princeton Univ Press|isbn=978-1-4008-8242-7|location=Princeton|page=173|oclc=945482850}}</ref><sup>pg 173</sup><ref>{{Cite book|last=Habermann, Katharina, 1966-|url=http://worldcat.org/oclc/262692314|title=Introduction to symplectic Dirac operators|date=2006|publisher=Springer|isbn=978-3-540-33421-7|oclc=262692314}}</ref><sup>pg 2</sup>.

===Relationship with symplectic geometry===
[[Symplectic geometry]] is the study of [[symplectic manifold]]s. The [[tangent space]] at any point on a symplectic manifold is a [[symplectic vector space]].<ref>[https://empg.maths.ed.ac.uk/Activities/BRST/ "Lecture Notes – Lecture 2: Symplectic reduction"], Retrieved on 30 January 2015.</ref> As noted earlier, structure preserving transformations of a symplectic vector space form a [[Group (mathematics)|group]] and this group is {{math|Sp(2''n'', '''F''')}}, depending on the dimension of the space and the [[Field (mathematics)|field]] over which it is defined.

A symplectic vector space is itself a symplectic manifold. A transformation under an [[Group action (mathematics)|action]] of the symplectic group is thus, in a sense, a linearised version of a [[symplectomorphism]] which is a more general structure preserving transformation on a symplectic manifold.

=={{math|Sp(''n'')}}==

The '''compact symplectic group'''<ref>{{harvnb|Hall|2015}} Section 1.2.8</ref> {{math|Sp(''n'')}} is the intersection of {{math|Sp(2''n'', '''C''')}} with the <math>2n\times 2n</math> unitary group:

:<math>\operatorname{Sp}(n):=\operatorname{Sp}(2n;\mathbf C)\cap\operatorname{U}(2n)=\operatorname{Sp}(2n;\mathbf C)\cap\operatorname {SU} (2n).</math>

It is sometimes written as {{math|USp(2''n'')}}. Alternatively, {{math|Sp(''n'')}} can be described as the subgroup of {{math|GL(''n'', '''H''')}} (invertible [[quaternion]]ic matrices) that preserves the standard [[hermitian form]] on {{math|'''H'''<sup>''n''</sup>}}:

:<math>\langle x, y\rangle = \bar x_1 y_1 + \cdots + \bar x_n y_n.</math>

That is, {{math|Sp(''n'')}} is just the [[Classical group#Sp(p, q) – the quaternionic unitary group|quaternionic unitary group]], {{math|U(''n'', '''H''')}}.<ref>{{harvnb|Hall|2015}} p. 14</ref> Indeed, it is sometimes called the '''hyperunitary group'''. Also Sp(1) is the group of quaternions of norm {{math|1}}, equivalent to {{math|[[SU(2)]]}} and topologically a [[3-sphere|{{math|3}}-sphere]] {{math|S<sup>3</sup>}}.

Note that {{math|Sp(''n'')}} is ''not'' a symplectic group in the sense of the previous section&mdash;it does not preserve a non-degenerate skew-symmetric {{math|'''H'''}}-bilinear form on {{math|'''H'''<sup>''n''</sup>}}: there is no such form except the zero form. Rather, it is isomorphic to a subgroup of {{math|Sp(2''n'', '''C''')}}, and so does preserve a complex [[symplectic manifold | symplectic form]] in a vector space of twice the dimension. As explained below, the Lie algebra of {{math|Sp(''n'')}} is the compact [[real form]] of the complex symplectic Lie algebra {{math|'''sp'''(2''n'', '''C''')}}.

{{math|Sp(''n'')}} is a real Lie group with (real) dimension {{math|''n''(2''n'' + 1)}}. It is [[Compact space|compact]] and [[simply connected]].<ref>{{harvnb|Hall|2015}} Prop. 13.12</ref>

The Lie algebra of {{math|Sp(''n'')}} is given by the quaternionic [[skew-Hermitian]] matrices, the set of {{math|''n''-by-''n''}} quaternionic matrices that satisfy

:<math>A+A^{\dagger} = 0</math>

where {{math|A<sup>†</sup>}} is the [[conjugate transpose]] of {{math|A}} (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

===Important subgroups===
Some main subgroups are:

: <math>\operatorname{Sp}(n) \supset \operatorname{Sp}(n-1)</math>
: <math>\operatorname{Sp}(n) \supset \operatorname{U}(n) </math>
: <math>\operatorname{Sp}(2) \supset \operatorname{O}(4)</math>

Conversely it is itself a subgroup of some other groups:

: <math>\operatorname{SU}(2n) \supset \operatorname{Sp}(n)</math>
: <math>\operatorname{F}_4 \supset \operatorname{Sp}(4)</math>
: <math>\operatorname{G}_2 \supset \operatorname{Sp}(1)</math>

There are also the [[isomorphism]]s of the [[Lie algebras]] {{math|1='''sp'''(2) = '''so'''(5)}} and {{math|1='''sp'''(1) = '''so'''(3) = '''su'''(2)}}.

==Relationship between the symplectic groups==
Every complex, [[semisimple Lie algebra]] has a [[Real form (Lie theory)#Split real form|split real form]] and a [[Real form (Lie theory)#Compact real form|compact real form]]; the former is called a [[complexification]] of the latter two.

The Lie algebra of {{math|Sp(2''n'', '''C''')}} is [[Semisimple Lie algebra|semisimple]] and is denoted {{math|'''sp'''(2''n'', '''C''')}}. Its [[Real form (Lie theory)#Split real form|split real form]] is {{math|'''sp'''(2''n'', '''R''')}} and its [[Real form (Lie theory)#Compact real form|compact real form]] is {{math|'''sp'''(''n'')}}. These correspond to the Lie groups {{math|Sp(2''n'', '''R''')}} and {{math|Sp(''n'')}} respectively.

The algebras, {{math|'''sp'''(''p'', ''n'' − ''p'')}}, which are the Lie algebras of {{math|Sp(''p'', ''n'' − ''p'')}}, are the [[Metric signature|indefinite signature]] equivalent to the compact form.

==Physical significance==
===Classical mechanics===
The non-compact symplectic group  {{math|Sp(2''n'', '''R''')}} comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.

Consider a system of {{math|''n''}} particles, evolving under [[Hamiltonian mechanics|Hamilton's equations]] whose position in [[phase space]] at a given time is denoted by the vector of [[canonical coordinates]],

:<math>\mathbf{z} = (q^1, \ldots , q^n, p_1, \ldots , p_n)^\mathrm{T}.</math>

The elements of the group {{math|Sp(2''n'', '''R''')}} are, in a certain sense, [[canonical transformations]] on this vector, i.e. they preserve the form of [[Hamiltonian mechanics|Hamilton's equations]].<ref>{{harvnb|Arnold|1989}} gives an extensive mathematical overview of classical mechanics. See chapter 8 for [[symplectic manifold]]s.</ref><ref name="A&M" /> If

:<math>\mathbf{Z} = \mathbf Z(\mathbf z, t) = (Q^1, \ldots , Q^n, P_1, \ldots , P_n)^\mathrm{T}</math>

are new canonical coordinates, then, with a dot denoting time derivative,

:<math>\dot {\mathbf Z} = M({\mathbf z}, t) \dot {\mathbf z},</math>

where

:<math>M(\mathbf z, t) \in \operatorname{Sp}(2n, \mathbf R)</math>

for all {{mvar|t}} and all {{math|'''z'''}} in phase space.<ref>{{harvnb|Goldstein|1980|loc=Section 9.3}}</ref>

For the special case of a [[Riemannian manifold]], Hamilton's equations describe the [[geodesic]]s on that manifold. The coordinates <math>q^i</math> live on the underlying manifold, and the momenta <math>p_i</math> live in the [[cotangent bundle]]. 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 <math>H=\tfrac{1}{2}g^{ij}(q)p_ip_j</math> where <math>g^{ij}</math> is the inverse of the [[metric tensor]] <math>g_{ij}</math> on the Riemannian manifold.<ref>Jurgen Jost, (1992) ''Riemannian Geometry and Geometric Analysis'', Springer.</ref><ref name="A&M">[[Ralph Abraham (mathematician)|Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London {{isbn|0-8053-0102-X}}</ref> In fact, the cotangent bundle of ''any'' smooth manifold can be a given a [[symplectic manifold|symplectic structure]] in a canonical way, with the symplectic form defined as the [[exterior derivative]] of the [[tautological one-form]].<ref>{{Cite book|last=da Silva|first=Ana Cannas|url=http://link.springer.com/10.1007/978-3-540-45330-7|title=Lectures on Symplectic Geometry|date=2008|publisher=Springer Berlin Heidelberg|isbn=978-3-540-42195-5|series=Lecture Notes in Mathematics|volume=1764|location=Berlin, Heidelberg|pages=9|doi=10.1007/978-3-540-45330-7}}</ref>

===Quantum mechanics===
{{More citations needed section|date=October 2019}}
Consider a system of {{math|''n''}} particles whose [[quantum state]] encodes its position and momentum. These coordinates are continuous variables and hence the [[Hilbert space]], 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 [[Heisenberg picture|Heisenberg equation]] in [[phase space]].

Construct a vector of [[canonical coordinates]],

:<math>\mathbf{\hat{z}} = (\hat{q}^1, \ldots , \hat{q}^n, \hat{p}_1, \ldots , \hat{p}_n)^\mathrm{T}. </math>

The [[canonical commutation relation]] can be expressed simply as

:<math> [\mathbf{\hat{z}},\mathbf{\hat{z}}^\mathrm{T}] = i\hbar\Omega </math>

where

:<math> \Omega = \begin{pmatrix} \mathbf{0} & I_n \\ -I_n & \mathbf{0}\end{pmatrix} </math>

and {{math|''I''<sub>''n''</sub>}} is the {{math|''n'' × ''n''}} identity matrix.

Many physical situations only require quadratic [[Hamiltonian (quantum mechanics)|Hamiltonians]], i.e. [[Hamiltonian (quantum mechanics)|Hamiltonians]] of the form

:<math>\hat{H} = \frac{1}{2}\mathbf{\hat{z}}^\mathrm{T} K\mathbf{\hat{z}}</math>

where {{math|''K''}} is a {{math|2''n'' × 2''n''}} real, [[symmetric matrix]]. This turns out to be a useful restriction and allows us to rewrite the [[Heisenberg picture|Heisenberg equation]] as

:<math>\frac{d\mathbf{\hat{z}}}{dt} = \Omega K \mathbf{\hat{z}}</math>

The solution to this equation must preserve the [[canonical commutation relation]]. It can be shown that the time evolution of this system is equivalent to an [[Group action (mathematics)|action]] of [[Symplectic group#Sp.282n.2C R.29|the real symplectic group, {{math|Sp(2''n'', '''R''')}}]], on the phase space.

==See also==
* [[Hamiltonian mechanics]]
* [[Metaplectic group]]
* [[Orthogonal group]]
* [[Paramodular group]]
* [[Projective unitary group]]
* [[Representations of classical Lie groups]]
* [[Symplectic manifold]], [[Symplectic matrix]], [[Symplectic vector space]], [[Symplectic representation]] 
* [[Unitary group]]
* [[Θ10]]

==Notes==
{{reflist}}

==References==
*{{citation|last=Arnold|first=V. I.|title=Mathematical Methods of Classical Mechanics|year=1989|edition=second|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics]]|volume=60|isbn=0-387-96890-3|author-link=Vladimir Arnold|url-access=registration|url=https://archive.org/details/mathematicalmeth0000arno}} 
*{{Citation| last=Hall|first=Brian C.|title=Lie groups, Lie algebras, and representations: An elementary introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
*{{citation|last1=Fulton|first1=W.|last2=Harris|first2=J.|author-link=William Fulton (mathematician)|author-link2=Joe Harris (mathematician)|title=Representation Theory, A first Course|year=1991|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics]]|volume=129|isbn=978-0-387-97495-8}}.
*{{cite book|last=Goldstein|first=H.|author-link=Herbert Goldstein|title=Classical Mechanics|edition=2nd|publisher=[[Addison-Wesley Publishing Company|Addison-Wesley]]|location=Reading MA|isbn=0-201-02918-9|year=1980|orig-year=1950|chapter=Chapter 7}}
*{{citation|last = Lee|first = J. M.|title=Introduction to Smooth manifolds|year=2003|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics]]|isbn=0-387-95448-1|volume=218}}
*{{citation|last=Rossmann|first= Wulf|title=Lie Groups – An Introduction Through Linear Groups|publisher=Oxford Science Publications|year=2002|series=Oxford Graduate Texts in Mathematics|isbn=0-19-859683-9}}
*{{cite arXiv|last1=Ferraro|first1=Alessandro|last2=Olivares|first2=Stefano|last3=Paris|first3=Matteo G. A.|title=Gaussian states in continuous variable quantum information| date=March 2005 |eprint=quant-ph/0503237|mode=cs2}}.

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[[Category:Lie groups]]
[[Category:Symplectic geometry]]