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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 fieldF (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".
where MT is the transpose of M. Often ? 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 determinant1, the symplectic group is a subgroup of the special linear groupSL(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).
The exponential map from the Lie algebrasp(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 popflock.com resource page.
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 groupSp(n) is the intersection of Sp(2n, C) with the unitary group:
It is sometimes written as USp(2n). Alternatively, Sp(n) can be described as the subgroup of GL(n, H) (invertible quaternionic matrices) that preserves the standard hermitian form on Hn:
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-sphereS3.
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).
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.
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.