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Epistemology - m a t h 4 w i s d o m - g m a i l
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The elements of a symplectic group/manifold preserve a (symplectic) 2-form {$s(x,y)$} which is bilinear and antisymmetric: {$s(x,y)=-s(x,y)$}. When we write this out in terms of a matrix: {$s(x,y) = {x^a}{s_{ab}}{y^b} = {x^T}Sy$} then {$s_{ba} = -s_{ab}$}. In particular, {$s_{aa}=0$}. The matrix S is invertible iff {$det(S)\neq 0$}. {$det(S)= \sum_{\sigma \in S_n} sgn(\sigma)a_{1\sigma(1)}\dots a_{n\sigma(n)} $} In this sum, note that we get zero terms whenever {$i = \sigma(i)$}. For each remaining term, we have an involution which matches: {$a_{1\sigma(1)}\dots a_{n\sigma(n)}$} and {$a_{\sigma(1)1}\dots a_{\sigma(n) n} = -a_{1\sigma(1)}\dots -a_{n\sigma(n)} = (-1)^n a_{1\sigma(1)}\dots a_{n\sigma(n)} $} The terms add to zero if {$(-1)^n = -1 $}, which is the case when {$n$} is odd. Thus {$n$} is even. |

This page was last changed on January 19, 2019, at 06:20 PM