nLab quadratic Hamiltonian

Definition

Given a symplectic vector space $(V,\omega)$, then a function $H \colon V \to \mathbb{R}$ regarded as a Hamiltonian is called quadratic if in terms of linear coordinates it is a degree-2 polynomial. If it is in fact a quadratic form then it is called a homogeneous quadratic Hamiltonian.

Properties

Quantization

Quadratic Hamiltonians enjoy particularly nice properties under quantization. In particular Weyl quantization restricts on them to a homomorphism of Lie algebras from the Poisson bracket to the commutator (e.g. Robbin-Salamon 93).

Relation to metaplectic group and extended affine symplectic group

The quantomorphisms of the symplectic manifold $(V,\omega)$ which come from paths of quadratic Hamiltonians form the extended affine symplectic group $ESp(V,\omega)$. The further subgroup of those coming from homogeneous quadratic Hamiltonians form the metaplectic group (Robbin-Salamon 93)

References

• D. G. Currie, E. J. Saletan, Canonical transformations and quadratic hamiltonians, II Nuovo Cimento B Series 11 , Volume 9, Issue 1, pp 143-153, 1972

• Jean Leray, section 1.1 of Lagrangian analysis and quantum mechanics, MIT press 1981 pdf

• Joel Robbin, Dietmar Salamon, Feynman path integrals on phase space and the metaplectic representation Math. Z. 221 (1996), no. 2, 307–335, (MR98f:58051, doi, pdf), also in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993)

• Monique Combescure, Didier Robert, Quadratic Quantum Hamiltonians revisited (arXiv:math-ph/0509027)