Symplectic group

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

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.[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(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.[4] In other words,

${\displaystyle \forall S\in \operatorname {Sp} (2n,\mathbf {R} )\,\,\exists X,Y\in {\mathfrak {sp}}(2n,\mathbf {R} )\,\,S=e^{X}e^{Y}.}$

• For all S in Sp(2n, R):

${\displaystyle S=OZO'\quad {\text{such that}}\quad O,O'\in \operatorname {Sp} (2n,\mathbf {R} )\cap \operatorname {SO} (2n)\cong U(n)\quad {\text{and}}\quad Z={\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}.}$

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.[5] 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).[6]

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

These are matrices, ${\displaystyle Q}$ such that

${\displaystyle Q={\begin{pmatrix}A&B\\C&-A^{\mathrm {T} }\end{pmatrix}}}$

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]

${\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.}$

It turns out that ${\displaystyle \operatorname {Sp} (n,\mathbf {R} )}$ can have a fairly explicit description using generators. If we let ${\displaystyle \operatorname {Sym} (n)}$ denote the symmetric ${\displaystyle n\times n}$ matrices, then ${\displaystyle \operatorname {Sp} (n,\mathbf {R} )}$ is generated by ${\displaystyle D(n)\cup N(n)\cup \{\Omega \},}$ where

${\displaystyle {\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbf {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}}$

are subgroups of ${\displaystyle \operatorname {Sp} (n,\mathbf {R} )}$[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(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.

Some main subgroups are:

${\displaystyle \operatorname {Sp} (n)\supset \operatorname {Sp} (n-1)}$

${\displaystyle \operatorname {Sp} (n)\supset \operatorname {U} (n)}$

${\displaystyle \operatorname {Sp} (2)\supset \operatorname {O} (4)}$

Conversely it is itself a subgroup of some other groups:

${\displaystyle \operatorname {SU} (2n)\supset \operatorname {Sp} (n)}$

${\displaystyle \operatorname {F} _{4}\supset \operatorname {Sp} (4)}$

${\displaystyle \operatorname {G} _{2}\supset \operatorname {Sp} (1)}$

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

The compact symplectic group Sp(n) 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,

${\displaystyle \mathbf {z} =(q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})^{\mathrm {T} }.}$

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

${\displaystyle \mathbf {Z} =\mathbf {Z} (\mathbf {z} ,t)=(Q^{1},\ldots ,Q^{n},P_{1},\ldots ,P_{n})^{\mathrm {T} }}$

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

${\displaystyle {\dot {\mathbf {Z} }}=M({\mathbf {z} },t){\dot {\mathbf {z} }},}$

where

${\displaystyle M(\mathbf {z} ,t)\in \operatorname {Sp} (2n,\mathbf {R} )}$

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 ${\displaystyle q^{i}}$ live in the tangent bundle to the manifold, and the momenta ${\displaystyle p_{i}}$ 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 ${\displaystyle H={\tfrac {1}{2}}g^{ij}(q)p_{i}p_{j}}$ where ${\displaystyle g^{ij}}$ is the inverse of the metric tensor ${\displaystyle g_{ij}}$ on the Riemannian manifold.[17][15] 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.

Construct a vector of canonical coordinates,

${\displaystyle \mathbf {\hat {z}} =({\hat {q}}^{1},\ldots ,{\hat {q}}^{n},{\hat {p}}_{1},\ldots ,{\hat {p}}_{n})^{\mathrm {T} }.}$

The canonical commutation relation can be expressed simply as

${\displaystyle [\mathbf {\hat {z}} ,\mathbf {\hat {z}} ^{\mathrm {T} }]=i\hbar \Omega }$

where

${\displaystyle \Omega ={\begin{pmatrix}\mathbf {0} &I_{n}\\-I_{n}&\mathbf {0} \end{pmatrix}}}$

and I_n is the n × n identity matrix.

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

${\displaystyle {\hat {H}}={\frac {1}{2}}\mathbf {\hat {z}} ^{\mathrm {T} }K\mathbf {\hat {z}} }$

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

${\displaystyle {\frac {d\mathbf {\hat {z}} }{dt}}=\Omega K\mathbf {\hat {z}} }$

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.