nLab
quadratic Hamiltonian

Context

Physics

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Surveys, textbooks and lecture notes

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    • Axiomatizations

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    • Types of quantum field thories

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Geometric quantization

: Lagrangians and Action functionals + Geometric Quantization

Prerequisites

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Prequantum field theory

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Geometric quantization

Applications

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Contents

Definition

Given a symplectic vector space (V,ω)(V,\omega), then a function H:VH \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,ω)(V,\omega) which come from paths of quadratic Hamiltonians form the extended affine symplectic group ESp(V,ω)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)

Created on January 1, 2015 at 22:45:31. See the history of this page for a list of all contributions to it.