Symplectomorphism

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(Redirected from Canonical transformation)

In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. Specifically, let (M1, ω1) and (M2, ω2) by symplectic manifolds. A map f : M1M2 is a symplectomorphism if it is a diffeomorphism and the pullback of ω2 under f is equal to ω1:

<math>f^{*}\omega_2 = \omega_1\,<math>

Symplectomorphisms are usually called canonical transformations by physicists.

The flow of a symplectic vector field on a symplectic manifold is a symplectomorphism. This follows from the closedness of the symplectic form and Cartan's formula for the Lie derivative in terms of the exterior derivative. As a direct consequence we have Liouville's theorem: the symplectic volume is invariant under a Hamiltionan flow. Since

{H,H} = XH(H) = 0

the flow of a Hamiltonian vector field also preserves H. In physics this is interpreted as the law of conservation of energy. Liouville's theorem is interpreted as the conservation of phase volume in Hamiltonian systems, which is the basis for classical statistical mechanics.

We have shown that there is a one-to-one correspondence between infinitesimal symplectomorphisms and closed one-forms on a symplectic manifold. If the first Betti number of the manifold is zero, and it is connected, the latter set is the same as the space of smooth functions modulo addition of constants.

Unlike Riemannian manifolds, symplectic manifolds are extremely non-rigid: they have many symplectomorphisms coming from Hamiltonian vector fields. The fundamental difference between Riemannian and symplectic geometry is that a symplectic manifold has no local invariants: according to Darboux's theorem for every point x in a symplectic manifold there is a local coordinate system with coordinates, called the canonical coordinates,

p1,...,pn, q1,...,qn,

such that

<math>\omega=\sum_n dq^i \wedge dp_i<math>

Finite-dimensional subgroups of the group of symplectomorphisms are Lie groups. Representations of these Lie groups (after <math>\hbar<math>-deformations, in general) on Hilbert spaces are called quantizations. When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization; this is a more common way of looking at it in physics.

Locally, symplectomorphisms can be generated by a generating function over a (local) Darboux coordinates. See Hamilton-Jacobi equation.

References

  • Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0-198-50451-9.
  • Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.
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