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) for positive integer n and field F (usually C or R). The latter is called the compact symplectic group. 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(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 as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex".
The symplectic group is a classical group defined as the set of linear transformations of a 2n-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 2n × 2n symplectic matrices, with entries in F, under the operation of matrix multiplication. This group is denoted either Sp(2n, F) or Sp(n, F). If the bilinear form is represented by the nonsingular skew-symmetric matrix Ω, then
where MT is the transpose of M. Often Ω is defined to be
where In is the identity matrix. In this case, Sp(2n, F) can be expressed as those block matrices , where , satisfying the equations:
Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, 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(2n, F) is then a proper subgroup of SL(2n, F).
Typically, the field F is the field of real numbers R or complex numbers C. In these cases Sp(2n, F) is a real/complex Lie group of real/complex dimension n(2n + 1). These groups are connected but non-compact.
The center of Sp(2n, F) consists of the matrices I2n and −I2n as long as the characteristic of the field is not 2. Since the center of Sp(2n, F) is discrete and its quotient modulo the center is a simple group, Sp(2n, F) is considered a simple Lie group.
The real rank of the corresponding Lie algebra, and hence of the Lie group Sp(2n, F), is n.
The Lie algebra of Sp(2n, F) is the set
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. 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. In other words,
- For all S in Sp(2n, R):
- 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. Further symplectic matrix properties can be found on that Wikipedia page.
- As a Lie group, Sp(2n, R) has a manifold structure. The manifold for Sp(2n, R) is diffeomorphic to the Cartesian product of the unitary group U(n) with a vector space of dimension n(n+1).
The members of the symplectic Lie algebra sp(2n, F) are the Hamiltonian matrices.
These are matrices, such that
Example of symplectic matrices
For Sp(2, R), the group of 2 × 2 matrices with determinant 1, the three symplectic (0, 1)-matrices are:
It turns out that can have a fairly explicit description using generators. If we let denote the symmetric matrices, then is generated by where
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. 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.
The compact symplectic group Sp(n) is the intersection of Sp(2n, C) with the unitary group:
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: there is no such form except 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 the compact real form of the complex symplectic Lie algebra sp(2n, C).
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:
Relationship between the symplectic groups
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, n − p), which are the Lie algebras of Sp(p, n − p), are the indefinite signature equivalent to the compact form.
The compact symplectic group Sp(n) comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.
are new canonical coordinates, then, with a dot denoting time derivative,
for all t and all z in phase space.
For the special case of a Riemannian manifold, Hamilton's equations describe the geodesics on that manifold. The coordinates live in the tangent bundle to the 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. The cotangent bundle of any Riemanninan manifold is a special case of a symplectic manifold.
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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
and In is the n × n identity matrix.
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 phase space.
- Orthogonal group
- Unitary group
- Projective unitary group
- Symplectic manifold, Symplectic matrix, Symplectic vector space, Symplectic representation
- Representations of classical Lie groups
- Hamiltonian mechanics
- Metaplectic group
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