Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted $Sp(2 n, F)$ and $Sp(n)$ . The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually differing by factors of $2$ . The notation used here is consistent with the size of the matrices used to represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group $Sp(2 n, C)$ is denoted $C n$ , and $Sp(n)$ is the compact real form of $Sp(2 n, C)$ . 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 $n$ .

The name "symplectic group" is due to Hermann Weyl (details) as a replacement for the previous confusing names of (line) complex group and Abelian linear group, and is the Greek analog of "complex".

$Sp(2 n, F)$

The symplectic group of degree $2 n$ over a field $F$ , denoted $Sp(2 n, F)$ , is the group of $2 n \times 2 n$ symplectic matrices with entries in $F$ , and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant $1$ , the symplectic group is a subgroup of the special linear group $SL(2 n, F)$ .

More abstractly, the symplectic group can be defined as the set of linear transformations of a $2 n$ -dimensional vector space over $F$ that preserve a non-degenerate, skew-symmetric, bilinear form, see classical group for this definition. Such a vector space is called a symplectic vector space. The symplectic group of an abstract symplectic vector space $V$ is also denoted $Sp(V)$ .

Typically, the field $F$ is the field of real numbers, $R$ , or complex numbers, $C$ . In this case $Sp(2 n, F)$ is a real/complex Lie group of real/complex dimension $n (2 n + 1)$ . These groups are connected but non-compact.

The centre of $Sp(2 n, F)$ consists of the matrices $I 2 n$ and $- I 2 n$ as long as the characteristic of the field is not $2$ .^[1] Here $I 2n$ denotes the $2 n \times 2 n$ identity matrix. The non-triviality of the centre of $Sp(2 n, F)$ and its relation to the simplicity of the group is discussed here.

The real rank of the Lie Algebra, and hence, the Lie Group for $Sp(2 n, F)$ is $n$ .

The condition that a symplectic matrix preserves the symplectic form can be written as

S \in \operatorname{Sp}(2n,F) \quad \text{iff} \quad S^\text{T}\Omega S = \Omega

where A^T is the transpose of A and

\Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix}.

The Lie algebra of $Sp(2 n, F)$ is given by the set of $2 n \times 2 n$ matrices A (with entries in F) that satisfy

\Omega A + A^\mathrm{T} \Omega = 0.

When $n = 1$ , the symplectic condition on a matrix is satisfied if and only if the 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 $SL(2 n, F)$ .

$Sp(2 n, C)$

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

$Sp(2 n, R)$

$Sp(2 n, C)$ is the complexification of the real group $Sp(2 n, R)$ . $Sp(2 n, R)$ is a real, non-compact, connected, simple Lie group.^[2] It has a fundamental group isomorphic to the group of integers under addition. As the real form of a simple Lie group its Lie algebra is a splittable Lie algebra.

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

The exponential map from the Lie algebra $sp (2 n, R)$ to the group $Sp(2 n, R)$ is not surjective. However, any element of the group may be generated by the group multiplication of two elements.^[3] In other words

\forall \; S \in \operatorname{Sp}(2n,\mathbb{R}) \; \exists \; X,Y \in \mathfrak{sp}(2n,\mathbb{R}) \text{ such that } S = e^Xe^Y.

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

S = OZO' \quad \text{such that} \quad O, O' \in \operatorname{Sp}(2n,\mathbb{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

is positive-definite and diagonal. 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.^[4] Further symplectic matrix properties can be found on that Wikipedia page.

As a Lie group, $Sp(2 n, R)$ has a manifold structure. The manifold for $Sp(2 n, R)$ is diffeomorphic to the Cartesian product of the unitary group $U(n)$ with a vector space of dimension $n (n +1)$ .^[5]

Infinitesimal generators

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

These are matrices, $Q$ such that

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

where $B$ and $C$ are symmetric matrices. See classical group for 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:^[6]

\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}.

Relationship with symplectic geometry

Symplectic geometry is the study of symplectic manifolds. The tangent space at any point on a symplectic manifold is a symplectic vector space.^[7] As noted earlier, structure preserving transformations of a symplectic vector space form a group and this group is $Sp(2 n, F)$ , depending on the dimension of the space and the field over which it is defined.

A symplectic vector space is itself a symplectic manifold. A transformation under an 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.

$Sp(n)$

The compact symplectic group $Sp(n)$ is often written as $USp(2 n)$ , indicating the fact that it is isomorphic to the group of unitary symplectic matrices, $Sp(n) ≅ U(2 n) \cap Sp(2 n, C)$ .^[8] Although the $Sp(n)$ notation is more common, and hence used here, it can be confusing in that the general idea of the symplectic group – including the compact, real and complex forms – can be represented as $Sp(n)$ . For example this is used in the sidebar at the top of this page in the list of classical groups.

$Sp(n)$ is the subgroup of $GL(n, H)$ (invertible quaternionic matrices) that preserves the standard hermitian form on $H n$ :

\langle x, y\rangle = \bar x_1 y_1 + \cdots + \bar x_n y_n

That is, $Sp(n)$ is just the quaternionic unitary group, $U(n, H)$ . Indeed, it is sometimes called the hyperunitary 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 $Sp(n)$ is not a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric ( $H$ -bilinear) form on $H n$ (in fact, the only skew-symmetric form is the zero form). Rather, it is isomorphic to a subgroup of $Sp(2 n, C)$ , and so does preserve a complex symplectic form in a vector space of dimension twice as high. As explained below, the Lie algebra of $Sp(n)$ is a real form of the complex symplectic Lie algebra $sp (2 n, C)$ .

$Sp(n)$ is a real Lie group with (real) dimension $n (2 n + 1)$ . It is compact, connected, and simply connected.

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

A+A^{\dagger} = 0

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

Important subgroups

The compact symplectic group $Sp(n)$ comes up in quantum physics as a symmetry on Poisson brackets so it is important to understand its 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 the isomorphisms of the Lie algebras $sp (2) = so (5)$ and $sp (1) = so (3) = su (2)$ .

Relationship between the symplectic groups

Every complex, semisimple Lie algebra has a split real form and a compact real form; the former is called a complexification of the latter two.

The Lie algebra of $Sp(2 n, C)$ is semisimple and is denoted $sp (2 n, C)$ . Its split real form is $sp (2 n, R)$ and its compact real form is $sp (n)$ . 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 the indefinite signature equivalent to the compact form.

Physical significance

Classical mechanics

Consider a system of $n$ particles, evolving under Hamilton's equations whose position in phase space at a given time is denoted by the vector of canonical coordinates,

\mathbf{z} = (q_1, \ldots , q_n, p_1, \ldots , p_n)^T.

The elements of the group $Sp(2 n, R)$ are, in a certain sense, canonical transformations on this vector, i.e. they preserve the form of Hamilton's equations.^[9] If

\mathbf{Z} = \mathbf Z(\mathbf z, t) = (Q_1, \ldots , Q_n, P_1, \ldots , P_n)^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 \mathrm{Sp}(2n, \mathbb R)

for all $t$ and all $z$ in phase space.^[10]

Quantum mechanics

Consider a system of $n$ 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 equation in phase space.

Construct a vector of canonical coordinates,

\mathbf{\hat{z}} = (\hat{q}_1, \ldots , \hat{q}_n, \hat{p}_1, \ldots , \hat{p}_n)^T.

The canonical commutation relation can be expressed simply as

[\mathbf{\hat{z}},\mathbf{\hat{z}}^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 quadratic Hamiltonians, i.e. Hamiltonians of the form

\hat{H} = \frac{1}{2}\mathbf{\hat{z}}^TK\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 the Heisenberg equation as

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

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 action of the real symplectic group, $Sp(2 n, R)$ , on the phase space.

Notes

↑ "Symplectic group", Encyclopedia of Mathematics Retrieved on 13 December 2014.
↑ "Is the symplectic group Sp(2n,R) simple?", Stack Exchange Retrieved on 14 December 2014.
↑ "Is the exponential map for Sp(2n,R) surjective?", Stack Exchange Retrieved 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
↑ "Lecture Notes – Lecture 2: Symplectic reduction", Retrieved on 30 January 2015.
↑ Hall 2003 Chapter 1.
↑ Arnold 1989 gives an extensive mathematical overview of classical mechanics. See chapter 8 for symplectic manifolds.
↑ Goldstein 1980, Section 9.3

References

Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60 (second ed.), Springer-Verlag, ISBN 0-387-96890-3
Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics 222, Springer-Verlag, ISBN 0-387-40122-9

Fulton, W.; Harris, J. (1991), Representation Theory, A first Course, Graduate Texts in Mathematics 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 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.org .

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