Part of the London Black Hole Information Paradox Journal Club (for Zoom link, please subscribe here: https://forms.gle/wLYi4A72tEUwuzGK6)

The Hamiltonian formulation of mechanics has many advantages, but its standard presentation destroys manifest covariance. This can be avoided by using the "covariant phase formalism" of Iyer and Wald, but until recently this formalism has suffered from several ambiguities related to boundary terms and total derivatives. In this talk I will present a new version of the formalism which incorporates boundary effects from the beginning. This eliminates all ambiguities, and leads to an algorithmic procedure for covariantly constructing the phase space and Hamiltonian of any Lagrangian field theory. It also allows us to confirm that the Poisson bracket in covariant phase space is indeed equivalent to an old proposal of Peierls for computing Poisson brackets covariantly. Along the way I'll illustrate the formalism using various examples. Based on work with Jie-qiang Wu.