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Register a new userFamily of Boundary Poisson Brackets
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Authors
K. Bering
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We find a new d-parameter family of ultra-local boundary Poisson brackets that satisfy the Jacobi identity. The two already known cases (hep-th/9305133, hep-th/9806249 and hep-th/9901112) of ultra-local boundary Poisson brackets are included in this new continuous family as special cases.
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We discuss how the integrators used for the Hybrid Monte Carlo (HMC) algorithm not only approximately conserve some Hamiltonian $H$ but exactly conserve a nearby shadow Hamiltonian (tilde H), and how the difference $Delta H equiv tilde H - H $ may be expressed as an expansion in Poisson brackets. By measuring average values of these Poisson brackets over the equilibrium distribution $propto e^{-H}$ generated by HMC we can find the optimal integrator parameters from a single simulation. We show that a good way of doing this in practice is to minimize the variance of $Delta H$ rather than its magnitude, as has been previously suggested. Some details of how to compute Poisson brackets for gauge and fermion fields, and for nested and force gradient integrators are also presented.
Numerical lattice gauge theory computations to generate gauge field configurations including the effects of dynamical fermions are usually carried out using algorithms that require the molecular dynamics evolution of gauge fields using symplectic integrators. Sophisticated integrators are in common use but are hard to optimise, and force-gradient integrators show promise especially for large lattice volumes. We explain why symplectic integrators lead to very efficient Monte Carlo algorithms because they exactly conserve a shadow Hamiltonian. The shadow Hamiltonian may be expanded in terms of Poisson brackets, and can be used to optimize the integrators. We show how this may be done for gauge theories by extending the formulation of Hamiltonian mechanics on Lie groups to include Poisson brackets and shadows, and by giving a general method for the practical computation of forces, force-gradients, and Poisson brackets for gauge theories.
We introduce new invariants associated to collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
This paper is a fusion of a survey and a research article. We focus on certain rigidity phenomena in function spaces associated to a symplectic manifold. Our starting point is a lower bound obtained in an earlier paper with Zapolsky for the uniform norm of the Poisson bracket of a pair of functions in terms of symplectic quasi-states. After a short review of the theory of symplectic quasi-states, we extend this bound to the case of iterated Poisson brackets. A new technical ingredient is the use of symplectic integrators. In addition, we discuss some applications to symplectic approximation theory and present a number of open problems.
We introduce the notion of a multiplicative Poisson $lambda$-bracket, which plays the same role in the theory of Hamiltonian differential-difference equations as the usual Poisson $lambda$-bracket plays in the theory of Hamiltonian PDE. We classify multiplicative Poisson $lambda$-brackets in one difference variable up to order 5. Applying the Lenard-Magri scheme to a compatible pair of multiplicative Poisson $lambda$-brackets of order 1 and 2, we establish integrability of some differential-difference equations, generalizing the Volterra chain.
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