Symplectic group

Template:Short descriptionTemplate:For Template:Lie groups Template:Group theory sidebar In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Template:Math and Template:Math for positive integer n and 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 Template:Math. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Template:Math is denoted Template:Math, and Template:Math is the compact real form of Template:Math. 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 Template:Math.

The name "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 fields, finite fields, and adele rings.

Template:Math

The symplectic group is a classical group defined as the set of linear transformations of a Template:Math-dimensional vector space over the field Template:Math which preserve a non-degenerate skew-symmetric bilinear form. Such a vector space is called a symplectic vector space, and the symplectic group of an abstract symplectic vector space Template:Math is denoted Template:Math. Upon fixing a basis for Template:Math, the symplectic group becomes the group of Template:Math symplectic matrices, with entries in Template:Math, under the operation of matrix multiplication. This group is denoted either Template:Math or Template:Math. If the bilinear form is represented by the 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^T 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_n is the identity matrix. In this case, Template:Math 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 Template:Math, the symplectic group is a subgroup of the special linear group Template:Math. When Template:Math, the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that Template:Math. For Template:Math, there are additional conditions, i.e. Template:Math is then a proper subgroup of Template:Math.

Typically, the field Template:Math is the field of real numbers Template:Math or complex numbers Template:Math. In these cases Template:Math is a real or complex Lie group of real or complex dimension Template:Math, respectively. These groups are connected but non-compact.

The center of Template:Math consists of the matrices Template:Math and Template:Math as long as the characteristic of the field is not Template:Math.<ref>"Symplectic group", Encyclopedia of Mathematics Retrieved on 13 December 2014.</ref> Since the center of Template:Math is discrete and its quotient modulo the center is a simple group, Template:Math is considered a simple Lie group.

The real rank of the corresponding Lie algebra, and hence of the Lie group Template:Math, is Template:Math.

The Lie algebra of Template:Math 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 as its Lie bracket.<ref>Template:Harvnb 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>

Template:Math

The symplectic group over the field of complex numbers is a 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.Template:Sfn

Template:Math

Template:Math is the complexification of the real group Template:Math. Template:Math is a real, non-compact, connected, simple Lie group.<ref>"Is the symplectic group Sp(2n, R) simple?", Stack Exchange Retrieved on 14 December 2014.</ref> 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 Template:Math:

• The exponential map from the Lie algebra Template:Math to the group Template:Math is not surjective. However, any element of the group can be represented as the product of two exponentials.<ref>"Is the exponential map for Sp(2n, 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 Template:Math in Template:Math:

<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 Template:Math is positive-definite and diagonal. The set of such Template:Maths forms a non-compact subgroup of Template:Math whereas Template:Math forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.<ref>"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, Template:Math has a manifold structure. The manifold for Template:Math is diffeomorphic to the Cartesian product of the unitary group Template:Math with a vector space of dimension Template:Math.<ref>"Symplectic Geometry – Arnol'd and Givental", Retrieved on 30 January 2015.</ref>

Infinitesimal generators

The members of the symplectic Lie algebra Template:Math are the Hamiltonian matrices.

These are matrices, <math>Q</math> such that

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

where Template:Math and Template:Math are symmetric matrices. See classical group for a derivation.

Example of symplectic matrices

For Template:Math, the group of Template:Math matrices with determinant Template:Math, the three symplectic Template:Math-matrices are:<ref>Symplectic Group, (source: Wolfram MathWorld), downloaded February 14, 2012</ref>

<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>

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

<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>

are subgroups of <math>\operatorname{Sp}(2n,\mathbf{R})</math><ref>Template:Cite book</ref>^{pg 173}<ref>Template:Cite book</ref>^{pg 2}.

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.<ref>"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 and this group is Template:Math, 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.

Template:Math

The compact symplectic group<ref>Template:Harvnb Section 1.2.8</ref> Template:Math is the intersection of Template:Math 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 Template:Math. Alternatively, Template:Math can be described as the subgroup of Template:Math (invertible quaternionic matrices) that preserves the standard hermitian form on Template:Math:

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

That is, Template:Math is just the quaternionic unitary group, Template:Math.<ref>Template:Harvnb p. 14</ref> Indeed, it is sometimes called the hyperunitary group. Also Sp(1) is the group of quaternions of norm Template:Math, equivalent to Template:Math and topologically a [[3-sphere|Template:Math-sphere]] Template:Math.

Note that Template:Math is not a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric Template:Math-bilinear form on Template:Math: there is no such form except the zero form. Rather, it is isomorphic to a subgroup of Template:Math, and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of Template:Math is the compact real form of the complex symplectic Lie algebra Template:Math.

Template:Math is a real Lie group with (real) dimension Template:Math. It is compact and simply connected.<ref>Template:Harvnb Prop. 13.12</ref>

The Lie algebra of Template:Math is given by the quaternionic skew-Hermitian matrices, the set of Template:Math quaternionic matrices that satisfy

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

where Template:Math is the conjugate transpose of Template:Math (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 isomorphisms of the Lie algebras Template:Math and Template:Math.

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 Template:Math is semisimple and is denoted Template:Math. Its split real form is Template:Math and its compact real form is Template:Math. These correspond to the Lie groups Template:Math and Template:Math respectively.

The algebras, Template:Math, which are the Lie algebras of Template:Math, are the indefinite signature equivalent to the compact form.

Physical significance

Classical mechanics

The non-compact symplectic group Template:Math comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.

Consider a system of Template:Math particles, evolving under 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 Template:Math are, in a certain sense, canonical transformations on this vector, i.e. they preserve the form of Hamilton's equations.<ref>Template:Harvnb gives an extensive mathematical overview of classical mechanics. See chapter 8 for symplectic manifolds.</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 Template:Mvar and all Template:Math in phase space.<ref>Template:Harvnb</ref>

For the special case of a Riemannian manifold, Hamilton's equations describe the geodesics 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 and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London Template:Isbn</ref> In fact, the cotangent bundle of any smooth manifold can be a given a symplectic structure in a canonical way, with the symplectic form defined as the exterior derivative of the tautological one-form.<ref>Template:Cite book</ref>

Quantum mechanics

Template:More citations needed section Consider a system of Template:Math 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,

<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 Template:Math is the Template:Math identity matrix.

Many physical situations only require quadratic Hamiltonians, i.e. Hamiltonians of the form

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

where Template:Math is a Template:Math real, symmetric matrix. This turns out to be a useful restriction and allows us to rewrite the 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 action of [[Symplectic group#Sp.282n.2C R.29|the real symplectic group, Template:Math]], 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

References

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• Template:Cite book
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