Knowlpedia

Symplectic Form

A nondegenerate skew-symmetric bilinear form B on a real vector space

A symplectic form on a real vector space KK is a bilinear form B:K×K→RB:K\times K\to\mathbb R with B(x,y)=−B(y,x)B(x,y)=-B(y,x) and nondegeneracy: B(x,⋅)=0⇒x=0B(x,\cdot)=0\Rightarrow x=0.

Remarks

Key properties (used here):

  • Defines the Weyl phase e−iB(z1,z2)/2e^{-iB(z_1,z_2)/2} in the .
  • Symplectic operators satisfy B(Tx,Ty)=B(x,y)B(Tx,Ty)=B(x,y).
Examples
  • On R2n\mathbb R^{2n}, B((p,q),(p′,q′))=p⋅q′−q⋅p′B((p,q),(p',q'))=p\cdot q'-q\cdot p'.