Symplectic basis explained

In linear algebra, a standard symplectic basis is a basis

{ e} _{i,} { f} _{i} of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form \omega , such that \omega({ e} _{i,} { e} _{j)} = 0 = \omega({ f} _{i,} { f} _{j),} \omega({ e} _{i,} { f} _{j)} = \delta _{ij} . A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the Gram–Schmidt process.^[1] The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite.

See also

• Darboux theorem
• Symplectic frame bundle
• Symplectic spinor bundle
• Symplectic vector space

References

• da Silva, A.C., Lectures on Symplectic Geometry, Springer (2001). .
• Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel .

Notes and References

1. Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006), p.7 and pp. 12–13