Symplectic group

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For finite groups with all characteristic abelian subgroups cyclic, see group of symplectic type.

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, 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(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, 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 group, and is the Greek analog of "complex".

Sp(2n, F)[edit]

The symplectic group of degree 2n over a field F, denoted Sp(2n, F), is the group of 2n × 2n 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(2n, F).

More abstractly, the symplectic group can be defined as the set of linear transformations of a 2n-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(2n, F) is a real/complex Lie group of real/complex dimension n(2n + 1). These groups are connected but non-compact.

The centre of Sp(2n, F) consists of the matrices I2n and I2n as long as the characteristic of the field is not 2.[1] Here I2n denotes the 2n × 2n identity matrix. The non-triviality of the centre of Sp(2n, 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(2n, 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 AT is the transpose of A and

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

The Lie algebra of Sp(2n, F) is given by the set of 2n × 2n 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(2n, F) is then a proper subgroup of SL(2n, F).

Sp(2n, C)[edit]

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

Sp(2n, R)[edit]

Sp(2n, C) is the complexification of the real group Sp(2n, R). Sp(2n, 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(2n, R):

  • The exponential map from the Lie algebra sp(2n, R) to the group Sp(2n, 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(2n, 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 Zs forms a non-compact subgroup of Sp(2n, 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.

Infinitesimal generators[edit]

The members of the symplectic Lie algebra sp(2n, 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[edit]

For Sp(2,R), the group of 2 × 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[edit]

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(2n, 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)[edit]

The compact symplectic group Sp(n) is often written as USp(2n), indicating the fact that it is isomorphic to the group of unitary symplectic matrices, Sp(n) ≅ U(2n) ∩ Sp(2n, 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 Hn:

\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 S3.

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 Hn (in fact, the only skew-symmetric form is the zero form). Rather, it is isomorphic to a subgroup of Sp(2n, 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(2n, C).

Sp(n) is a real Lie group with (real) dimension n(2n + 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[edit]

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[edit]

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(2n, C) is semisimple and is denoted sp(2n, C). Its split real form is sp(2n, R) and its compact real form is sp(n). These correspond to the Lie groups Sp(2n, R) and Sp(n) respectively.

The algebras, sp(p, np), which are the Lie algebras of Sp(p, np), are the indefinite signature equivalent to the compact form.

Physical significance[edit]

Classical mechanics[edit]

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(2n, R) are canonical transformations on this vector, i.e. they preserve the form of Hamilton's equations.[9]

Quantum mechanics[edit]

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 In is the n × n identity matrix.

Most 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 2n × 2n 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(2n, R), on the initial state.

See also[edit]

Notes[edit]

  1. ^ "Symplectic group", Encyclopedia of Mathematics Retrieved on 13 December 2014.
  2. ^ "Is the symplectic group Sp(2n,R) simple?", Stack Exchange Retrieved on 14 December 2014.
  3. ^ "Is the exponential map for Sp(2n,R) surjective?", Stack Exchange Retrieved on 5 December 2014.
  4. ^ "Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso", Retrieved on 30 January 2015.
  5. ^ "Symplectic Geometry – Arnol'd and Givental", Retrieved on 30 January 2015.
  6. ^ Symplectic Group, (source: Wolfram MathWorld), downloaded February 14, 2012
  7. ^ "Lecture Notes – Lecture 2: Symplectic reduction", Retrieved on 30 January 2015.
  8. ^ Hall 2003 Chapter 1.
  9. ^ Arnold 1989 gives an extensive mathematical overview of classical mechanics. See chapter 8 for symplectic manifolds.

References[edit]