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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*) for positive integer *n* and field **F** (usually **C** or **R**). The latter is called the **compact symplectic group** and is also denoted by . Many authors prefer slightly different notations, usually differing by factors of 2. 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 Sp(2*n*, **C**) is denoted *C _{n}*, and Sp(

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.

The symplectic group is a classical group defined as the set of linear transformations of a 2*n*-dimensional vector space over the field **F** 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 *V* is denoted Sp(*V*). Upon fixing a basis for *V*, the symplectic group becomes the group of 2*n* × 2*n* symplectic matrices, with entries in **F**, under the operation of matrix multiplication. This group is denoted either Sp(2*n*, **F**) or Sp(*n*, **F**). If the bilinear form is represented by the nonsingular skew-symmetric matrix Ω, then

where *M*^{T} is the transpose of *M*. Often Ω is defined to be

where *I _{n}* is the identity matrix. In this case, Sp(2

Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2*n*, **F**). 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**).

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

The center of Sp(2*n*, **F**) consists of the matrices *I*_{2n} and −*I*_{2n} as long as the characteristic of the field is not 2.^{[1]} Since the center of Sp(2*n*, **F**) is discrete and its quotient modulo the center is a simple group, Sp(2*n*, **F**) is considered a simple Lie group.

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

The Lie algebra of Sp(2*n*, **F**) is the set

equipped with the commutator as its Lie bracket.^{[2]} For the standard skew-symmetric bilinear form , this Lie algebra is the set of all block matrices subject to the conditions

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

Sp(*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.^{[3]} 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 can be represented as the product of two exponentials.^{[4]}In other words,

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

- 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.^{[5]}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).^{[6]}

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

These are matrices, such that

where *B* and *C* are symmetric matrices. See classical group for a derivation.

For Sp(2, **R**), the group of 2 × 2 matrices with determinant 1, the three symplectic (0, 1)-matrices are:^{[7]}

It turns out that can have a fairly explicit description using generators. If we let denote the symmetric matrices, then is generated by where

are subgroups of ^{[8]}^{pg 173}^{[9]}^{pg 2}.

Symplectic geometry is the study of symplectic manifolds. The tangent space at any point on a symplectic manifold is a symplectic vector space.^{[10]} 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.

The **compact symplectic group**^{[11]} Sp(*n*) is the intersection of Sp(2*n*, **C**) with the unitary group:

It is sometimes written as USp(2*n*). Alternatively, Sp(*n*) can be described as the subgroup of GL(*n*, **H**) (invertible quaternionic matrices) that preserves the standard hermitian form on **H**^{n}:

That is, Sp(*n*) is just the quaternionic unitary group, U(*n*, **H**).^{[12]} 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}: 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 complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of Sp(*n*) is the compact 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 and simply connected.^{[13]}

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

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

Some main subgroups are:

Conversely it is itself a subgroup of some other groups:

There are also the isomorphisms of the Lie algebras **sp**(2) = **so**(5) and **sp**(1) = **so**(3) = **su**(2).

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.

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 under Hamilton's equations whose position in phase space at a given time is denoted by the vector of canonical coordinates,

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.^{[14]}^{[15]} If

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

where

for all t and all **z** in phase space.^{[16]}

For the special case of a Riemannian manifold, Hamilton's equations describe the geodesics on that manifold. The coordinates live on the underlying manifold, and the momenta 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 where is the inverse of the metric tensor on the Riemannian manifold.^{[17]}^{[15]} 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.^{[18]}

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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,

The canonical commutation relation can be expressed simply as

where

and *I*_{n} is the *n* × *n* identity matrix.

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

where *K* is a 2*n* × 2*n* real, symmetric matrix. This turns out to be a useful restriction and allows us to rewrite the Heisenberg equation as

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.