Editing Symplectic group (section)

===Quantum mechanics===
{{More citations needed section|date=October 2019}}
Consider a system of {{math|''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 picture|Heisenberg equation]] in [[phase space]].

Construct a vector of [[canonical coordinates]],

:<math>\mathbf{\hat{z}} = (\hat{q}^1, \ldots , \hat{q}^n, \hat{p}_1, \ldots , \hat{p}_n)^\mathrm{T}. </math>

The [[canonical commutation relation]] can be expressed simply as

:<math> [\mathbf{\hat{z}},\mathbf{\hat{z}}^\mathrm{T}] = i\hbar\Omega </math>

where

:<math> \Omega = \begin{pmatrix} \mathbf{0} & I_n \\ -I_n & \mathbf{0}\end{pmatrix} </math>

and {{math|''I''<sub>''n''</sub>}} is the {{math|''n'' × ''n''}} identity matrix.

Many physical situations only require quadratic [[Hamiltonian (quantum mechanics)|Hamiltonians]], i.e. [[Hamiltonian (quantum mechanics)|Hamiltonians]] of the form

:<math>\hat{H} = \frac{1}{2}\mathbf{\hat{z}}^\mathrm{T} K\mathbf{\hat{z}}</math>

where {{math|''K''}} is a {{math|2''n'' × 2''n''}} real, [[symmetric matrix]]. This turns out to be a useful restriction and allows us to rewrite the [[Heisenberg picture|Heisenberg equation]] as

:<math>\frac{d\mathbf{\hat{z}}}{dt} = \Omega K \mathbf{\hat{z}}</math>

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 [[Group action (mathematics)|action]] of [[Symplectic group#Sp.282n.2C R.29|the real symplectic group, {{math|Sp(2''n'', '''R''')}}]], on the phase space.