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 notation used here is consistent with the size of the most common matrices which represent the groups.
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 2n-dimensional vector space over the field F which preserve a non-degenerate skew-symmetric bilinear form.
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.
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.
, satisfying the three 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).
The center of Sp(2n, F) consists of the matrices I2n and −I2n as long as the characteristic of the field is not 2.
[1] 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.
subject to the conditions The symplectic group over the field of complex numbers is a non-compact, simply connected, simple Lie group.
Sp(2n, R) is a real, non-compact, connected, simple Lie group.
[10] 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 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.
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).
Also Sp(1) is the group of quaternions of norm 1, equivalent to SU(2) and topologically a 3-sphere S3.
Rather, it is isomorphic to a subgroup of Sp(2n, 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(2n, C).
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 non-compact symplectic group Sp(2n, 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(2n, 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.
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
[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] 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.
An alternative approach is to consider the evolution of the position and momentum operators under the Heisenberg equation in phase space.
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.