Symplectic group

Inmathematics,the namesymplectic groupcan refer to two different, but closely related, collections of mathematicalgroups,denoted $Sp(2 n, F)$ and $Sp(n)$ for positive integernandfieldF(usuallyCorR). The latter is called thecompact symplectic groupand is also denoted by $\mathrm {USp} (n)$ .Many authors prefer slightly different notations, usually differing by factors of $2$ .The notation used here is consistent with the size of the most commonmatriceswhich represent the groups. InCartan's classification of thesimple Lie algebras,the Lie algebra of the complex group $Sp(2 n, C)$ is denoted $C n$ ,and $Sp(n)$ is thecompact real formof $Sp(2 n, C)$ .Note that when we refer tothe(compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension $n$ .

The name "symplecticgroup "was coined byHermann Weylas a replacement for the previous confusing names (line)complex groupandAbelian linear group,and is the Greek analog of "complex".

Themetaplectic groupis a double cover of the symplectic group overR;it has analogues over otherlocal fields,finite fields,andadele rings.

$Sp(2 n, F)$

The symplectic group is aclassical groupdefined as the set oflinear transformationsof a $2 n$ -dimensionalvector spaceover the field $F$ which preserve anon-degenerate skew-symmetric bilinear form.Such a vector space is called asymplectic vector space,and the symplectic group of an abstract symplectic vector space $V$ is denoted $Sp(V)$ .Upon fixing a basis for $V$ ,the symplectic group becomes the group of $2 n \times 2 n$ symplectic matrices,with entries in $F$ ,under the operation ofmatrix multiplication.This group is denoted either $Sp(2 n, F)$ or $Sp(n, F)$ .If the bilinear form is represented by thenonsingular skew-symmetric matrixΩ, then

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

whereM^Tis thetransposeofM.Often Ω is defined to be

\Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},

whereI_nis the identity matrix. In this case, $Sp(2 n, F)$ can be expressed as those block matrices $({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})$ ,where $A,B,C,D\in M_{n\times n}(F)$ ,satisfying the three equations:

{\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}}

Since all symplectic matrices havedeterminant $1$ ,the symplectic group is asubgroupof thespecial linear group $SL(2 n, F)$ .When $n = 1$ ,the symplectic condition on a matrix is satisfiedif and only ifthe determinant is one, so that $Sp(2, F) = SL(2, F)$ .For $n > 1$ ,there are additional conditions, i.e. $Sp(2 n, F)$ is then a proper subgroup of $SL(2 n, F)$ .

Typically, the field $F$ is the field ofreal numbers $R$ orcomplex numbers $C$ .In these cases $Sp(2 n, F)$ is a real or complexLie groupof real or complex dimension $n (2 n + 1)$ ,respectively. These groups areconnectedbutnon-compact.

Thecenterof $Sp(2 n, F)$ consists of the matrices $I 2 n$ and $- I 2 n$ as long as thecharacteristic of the fieldis not $2$ .^[1]Since the center of $Sp(2 n, F)$ is discrete and its quotient modulo the center is asimple group, $Sp(2 n, F)$ is considered asimple Lie group.

The real rank of the corresponding Lie algebra, and hence of the Lie group $Sp(2 n, F)$ ,is $n$ .

TheLie algebraof $Sp(2 n, F)$ is the set

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

equipped with thecommutatoras its Lie bracket.^[2]For the standard skew-symmetric bilinear form $\Omega =({\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}})$ ,this Lie algebra is the set of all block matrices $({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})$ subject to the conditions

{\begin{aligned}A&=-D^{\mathrm {T} },\\B&=B^{\mathrm {T} },\\C&=C^{\mathrm {T} }.\end{aligned}}

$Sp(2 n, C)$

The symplectic group over the field of complex numbers is anon-compact,simply connected,simple Lie group.

$Sp(2 n, R)$

$Sp(n, C)$ is thecomplexificationof the real group $Sp(2 n, R)$ . $Sp(2 n, R)$ is a real,non-compact,connected,simple Lie group.^[3]It has afundamental group isomorphicto the group ofintegersunder addition. As thereal formof asimple Lie groupits Lie algebra is asplittable Lie algebra.

Some further properties of $Sp(2 n, R)$ :

Theexponential mapfrom theLie algebra $sp (2 n, R)$ to the group $Sp(2 n, R)$ is notsurjective.However, any element of the group can be represented as the product of two exponentials.^[4]In other words,

\forall S\in \operatorname {Sp} (2n,\mathbf {R} )\,\,\exists X,Y\in {\mathfrak {sp}}(2n,\mathbf {R} )\,\,S=e^{X}e^{Y}.

For all $S$ in $Sp(2 n, R)$ :

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}}.

The matrix

D

ispositive-definiteanddiagonal.The set of such

Z

s forms a non-compact subgroup of

Sp(2 n, R)

whereas

U(n)

forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.^[5]Furthersymplectic matrixproperties can be found on that Wikipedia page.

As aLie group, $Sp(2 n, R)$ has a manifold structure. Themanifoldfor $Sp(2 n, R)$ isdiffeomorphicto theCartesian productof theunitary group $U(n)$ with avector spaceof dimension $n (n +1)$ .^[6]

Infinitesimal generators

The members of the symplectic Lie algebra $sp (2 n, F)$ are theHamiltonian matrices.

These are matrices, $Q$ such that

$Q={\begin{pmatrix}A&B\\C&-A^{\mathrm {T} }\end{pmatrix}}$

where $B$ and $C$ aresymmetric matrices.Seeclassical groupfor a derivation.

Example of symplectic matrices

For $Sp(2, R)$ ,the group of $2 \times 2$ matrices with determinant $1$ ,the three symplectic $(0, 1)$ -matrices are:^[7]

${\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.$

Sp(2n, R)

It turns out that $\operatorname {Sp} (2n,\mathbf {R} )$ can have a fairly explicit description using generators. If we let $\operatorname {Sym} (n)$ denote the symmetric $n\times n$ matrices, then $\operatorname {Sp} (2n,\mathbf {R} )$ is generated by $D(n)\cup N(n)\cup \{\Omega \},$ where

${\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbf {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}$

are subgroups of $\operatorname {Sp} (2n,\mathbf {R} )$ ^[8]^{pg 173}^[9]^{pg 2}.

Relationship with symplectic geometry

Symplectic geometryis the study ofsymplectic manifolds.Thetangent spaceat any point on a symplectic manifold is asymplectic vector space.^[10]As noted earlier, structure preserving transformations of a symplectic vector space form agroupand this group is $Sp(2 n, F)$ ,depending on the dimension of the space and thefieldover which it is defined.

A symplectic vector space is itself a symplectic manifold. A transformation under anactionof the symplectic group is thus, in a sense, a linearised version of asymplectomorphismwhich is a more general structure preserving transformation on a symplectic manifold.

$Sp(n)$

Thecompact symplectic group^[11] $Sp(n)$ is the intersection of $Sp(2 n, C)$ with the $2n\times 2n$ unitary group:

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

It is sometimes written as $USp(2 n)$ .Alternatively, $Sp(n)$ can be described as the subgroup of $GL(n, H)$ (invertiblequaternionicmatrices) that preserves the standardhermitian formon $H n$ :

\langle x,y\rangle ={\bar {x}}_{1}y_{1}+\cdots +{\bar {x}}_{n}y_{n}.

That is, $Sp(n)$ is just thequaternionic unitary group, $U(n, H)$ .^[12]Indeed, it is sometimes called thehyperunitary group.Also Sp(1) is the group of quaternions of norm $1$ ,equivalent to $SU(2)$ and topologically a $3$ -sphere $S 3$ .

Note that $Sp(n)$ isnota symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric $H$ -bilinear form on $H n$ :there is no such form except the zero form. Rather, it is isomorphic to a subgroup of $Sp(2 n, C)$ ,and so does preserve a complexsymplectic formin a vector space of twice the dimension. As explained below, the Lie algebra of $Sp(n)$ is the compactreal formof the complex symplectic Lie algebra $sp (2 n, C)$ .

$Sp(n)$ is a real Lie group with (real) dimension $n (2 n + 1)$ .It iscompactandsimply connected.^[13]

The Lie algebra of $Sp(n)$ is given by the quaternionicskew-Hermitianmatrices, the set of $n -by- n$ quaternionic matrices that satisfy

A+A^{\dagger }=0

where $A †$ is theconjugate transposeof $A$ (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

Important subgroups

Some main subgroups are:

\operatorname {Sp} (n)\supset \operatorname {Sp} (n-1)

\operatorname {Sp} (n)\supset \operatorname {U} (n)

\operatorname {Sp} (2)\supset \operatorname {O} (4)

Conversely it is itself a subgroup of some other groups:

\operatorname {SU} (2n)\supset \operatorname {Sp} (n)

\operatorname {F} _{4}\supset \operatorname {Sp} (4)

\operatorname {G} _{2}\supset \operatorname {Sp} (1)

There are also theisomorphismsof theLie algebras $sp (2) = so (5)$ and $sp (1) = so (3) = su (2)$ .

Relationship between the symplectic groups

Every complex,semisimple Lie algebrahas asplit real formand acompact real form;the former is called acomplexificationof the latter two.

The Lie algebra of $Sp(2 n, C)$ issemisimpleand is denoted $sp (2 n, C)$ .Itssplit real formis $sp (2 n, R)$ and itscompact real formis $sp (n)$ .These correspond to the Lie groups $Sp(2 n, R)$ and $Sp(n)$ respectively.

The algebras, $sp (p, n - p)$ ,which are the Lie algebras of $Sp(p, n - p)$ ,are theindefinite signatureequivalent to the compact form.

Physical significance

Classical mechanics

The non-compact symplectic group $Sp(2 n, R)$ comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.

Consider a system of $n$ particles, evolving underHamilton's equationswhose position inphase spaceat a given time is denoted by the vector ofcanonical coordinates,

\mathbf {z} =(q^{1},\ldots,q^{n},p_{1},\ldots,p_{n})^{\mathrm {T} }.

The elements of the group $Sp(2 n, R)$ are, in a certain sense,canonical transformationson this vector, i.e. they preserve the form ofHamilton's equations.^[14]^[15]If

\mathbf {Z} =\mathbf {Z} (\mathbf {z},t)=(Q^{1},\ldots,Q^{n},P_{1},\ldots,P_{n})^{\mathrm {T} }

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

{\dot {\mathbf {Z} }}=M({\mathbf {z} },t){\dot {\mathbf {z} }},

where

M(\mathbf {z},t)\in \operatorname {Sp} (2n,\mathbf {R} )

for all $t$ and all $z$ in phase space.^[16]

For the special case of aRiemannian manifold,Hamilton's equations describe thegeodesicson that manifold. The coordinates $q^{i}$ live on the underlying manifold, and the momenta $p_{i}$ live in thecotangent 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 $H={\tfrac {1}{2}}g^{ij}(q)p_{i}p_{j}$ where $g^{ij}$ is the inverse of themetric tensor $g_{ij}$ on the Riemannian manifold.^[17]^[15]In fact, the cotangent bundle ofanysmooth manifold can be a given asymplectic structurein a canonical way, with the symplectic form defined as theexterior derivativeof thetautological one-form.^[18]

Quantum mechanics

Consider a system of $n$ particles whosequantum stateencodes its position and momentum. These coordinates are continuous variables and hence theHilbert 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 theHeisenberg equationinphase space.

Construct a vector ofcanonical coordinates,

\mathbf {\hat {z}} =({\hat {q}}^{1},\ldots,{\hat {q}}^{n},{\hat {p}}_{1},\ldots,{\hat {p}}_{n})^{\mathrm {T} }.

Thecanonical commutation relationcan be expressed simply as

[\mathbf {\hat {z}},\mathbf {\hat {z}} ^{\mathrm {T} }]=i\hbar \Omega

where

\Omega ={\begin{pmatrix}\mathbf {0} &I_{n}\\-I_{n}&\mathbf {0} \end{pmatrix}}

and $I n$ is the $n \times n$ identity matrix.

Many physical situations only require quadraticHamiltonians,i.e.Hamiltoniansof the form

{\hat {H}}={\frac {1}{2}}\mathbf {\hat {z}} ^{\mathrm {T} }K\mathbf {\hat {z}}

where $K$ is a $2 n \times 2 n$ real,symmetric matrix.This turns out to be a useful restriction and allows us to rewrite theHeisenberg equationas

{\frac {d\mathbf {\hat {z}} }{dt}}=\Omega K\mathbf {\hat {z}}

The solution to this equation must preserve thecanonical commutation relation.It can be shown that the time evolution of this system is equivalent to anactionofthe real symplectic group, $Sp(2 n, R)$ ,on the phase space.

Notes

^"Symplectic group",Encyclopedia of MathematicsRetrieved on 13 December 2014.
^Hall 2015Prop. 3.25
^"Is the symplectic group Sp(2n,R) simple? ",Stack ExchangeRetrieved on 14 December 2014.
^"Is the exponential map for Sp(2n,R) surjective? ",Stack ExchangeRetrieved on 5 December 2014.
^"Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso",Retrieved on 30 January 2015.
^"Symplectic Geometry – Arnol'd and Givental",Retrieved on 30 January 2015.
^Symplectic Group,(source:Wolfram MathWorld), downloaded February 14, 2012
^Gerald B. Folland. (2016).Harmonic analysis in phase space.Princeton: Princeton Univ Press. p. 173.ISBN 978-1-4008-8242-7.OCLC 945482850.
^Habermann, Katharina, 1966- (2006).Introduction to symplectic Dirac operators.Springer.ISBN 978-3-540-33421-7.OCLC 262692314.{{cite book}}:CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^"Lecture Notes – Lecture 2: Symplectic reduction",Retrieved on 30 January 2015.
^Hall 2015Section 1.2.8
^Hall 2015p. 14
^Hall 2015Prop. 13.12
^Arnold 1989gives an extensive mathematical overview of classical mechanics. See chapter 8 forsymplectic manifolds.
^^a ^bRalph AbrahamandJerrold E. Marsden,Foundations of Mechanics,(1978) Benjamin-Cummings, LondonISBN 0-8053-0102-X
^Goldstein 1980,Section 9.3
^Jurgen Jost, (1992)Riemannian Geometry and Geometric Analysis,Springer.
^da Silva, Ana Cannas (2008).Lectures on Symplectic Geometry.Lecture Notes in Mathematics. Vol. 1764. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 9.doi:10.1007/978-3-540-45330-7.ISBN 978-3-540-42195-5.

References

Arnold, V. I.(1989),Mathematical Methods of Classical Mechanics,Graduate Texts in Mathematics,vol. 60 (second ed.),Springer-Verlag,ISBN 0-387-96890-3
Hall, Brian C. (2015),Lie groups, Lie algebras, and representations: An elementary introduction,Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,ISBN 978-3319134666
Fulton, W.;Harris, J.(1991),Representation Theory, A first Course,Graduate Texts in Mathematics,vol. 129,Springer-Verlag,ISBN 978-0-387-97495-8.
Goldstein, H.(1980) [1950]. "Chapter 7".Classical Mechanics(2nd ed.). Reading MA:Addison-Wesley.ISBN 0-201-02918-9.
Lee, J. M. (2003),Introduction to Smooth manifolds,Graduate Texts in Mathematics,vol. 218,Springer-Verlag,ISBN 0-387-95448-1
Rossmann, Wulf (2002),Lie Groups – An Introduction Through Linear Groups,Oxford Graduate Texts in Mathematics, Oxford Science Publications,ISBN 0-19-859683-9
Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (March 2005), "Gaussian states in continuous variable quantum information",arXiv:quant-ph/0503237.

[1] "Symplectic group",Encyclopedia of MathematicsRetrieved on 13 December 2014.

[2] Hall 2015Prop. 3.25

[3] "Is the symplectic group Sp(2n,R) simple? ",Stack ExchangeRetrieved on 14 December 2014.

[4] "Is the exponential map for Sp(2n,R) surjective? ",Stack ExchangeRetrieved on 5 December 2014.

[5] "Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso",Retrieved on 30 January 2015.

[6] "Symplectic Geometry – Arnol'd and Givental",Retrieved on 30 January 2015.

[7] Symplectic Group,(source:Wolfram MathWorld), downloaded February 14, 2012

[8] Gerald B. Folland. (2016).Harmonic analysis in phase space.Princeton: Princeton Univ Press. p. 173.ISBN 978-1-4008-8242-7.OCLC 945482850.

[9] Habermann, Katharina, 1966- (2006).Introduction to symplectic Dirac operators.Springer.ISBN 978-3-540-33421-7.OCLC 262692314.{{cite book}}:CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)

[10] "Lecture Notes – Lecture 2: Symplectic reduction",Retrieved on 30 January 2015.

[11] Hall 2015Section 1.2.8

[12] Hall 2015p. 14

[13] Hall 2015Prop. 13.12

[14] Arnold 1989gives an extensive mathematical overview of classical mechanics. See chapter 8 forsymplectic manifolds.

[A&M-15] Ralph AbrahamandJerrold E. Marsden,Foundations of Mechanics,(1978) Benjamin-Cummings, LondonISBN 0-8053-0102-X

[16] Goldstein 1980,Section 9.3

[17] Jurgen Jost, (1992)Riemannian Geometry and Geometric Analysis,Springer.

[18] Silva, Ana Cannas (2008).Lectures on Symplectic Geometry.Lecture Notes in Mathematics. Vol. 1764. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 9.doi:10.1007/978-3-540-45330-7.ISBN 978-3-540-42195-5.

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Symplectic group

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