in terms of basis elements ofV∗, the coordinate functionsvj. There is an anal-
ogous theorem in symplectic geometry (for a proof, see for instance Proposition
1.1 of [8]), which says that a basis of a symplectic vector spaceVcan always be
found so that the dual basis coordinate functions come in pairsqj,pj, with the
symplectic formωthe same one we have found based on the Poisson bracket,
that given by equation 14.7. Note that one difference between Euclidean and
symplectic geometry is that a symplectic vector space will always be even di-
mensional.
Digression. For those familiar with differential manifolds, vector fields and
differential forms, the notion of a symplectic vector space can be extended to:
Definition(Symplectic manifold).A symplectic manifoldMis a manifold with
a differential two-formω(·,·)(called a symplectic two-form) satisfying the con-
ditions
- ωis non-degenerate (i.e., for a nowhere zero vector fieldX,ω(X,·)is a
nowhere zero one-form). - dω= 0, in which caseωis said to be closed.
The cotangent bundleT∗N of a manifoldN (i.e., the space of pairs of a
point onNtogether with a linear function on the tangent space at that point)
provides one class of symplectic manifolds, generalizing the linear caseN=Rd,
and corresponding physically to a particle moving onN. A simple example that
is neither linear nor a cotangent bundle is the sphereM=S^2 , withωthe area
two-form. The Darboux theorem says that, by an appropriate choice of local
coordinatesqj,pjonM, symplectic two-formsωcan always be written in such
local coordinates as
ω=∑dj=1dqj∧dpjUnlike the linear case though, there will in general be no global choice of coor-
dinates for which this true. Later on, our discussion of quantization will rely
crucially on having a linear structure on phase space, so will not apply to general
symplectic manifolds.
Note that there is no assumption here thatMhas a metric (i.e., it may
not be a Riemannian manifold). A symplectic two-formωis a structure on a
manifold analogous to a metric but with opposite symmetry properties. Whereas
a metric is a symmetric non-degenerate bilinear form on the tangent space at
each point, a symplectic form is an antisymmetric non-degenerate bilinear form
on the tangent space.
14.4 For further reading
Some good sources for discussions of symplectic geometry and the geometrical
formulation of Hamiltonian mechanics are [2], [8] and [13].