Canonical transformations in the covariant phase space formalism

As the title says, I'm looking for an explanation on how to apply canonical transformations when using the covariant phase space formalism. I'm familiar with the topic, but I haven't found a good explanation for canonical transformations. The only one I could find is in the book by DeWitt, but it is explained on page 700 on the second volume and I can't even get past the first volume, it's a bit heavy on the notation and the generalizations to include Grassmann variables and all that.

My Attempt:

In the covariant phase space formalism we start with an action principle

$$$$S=\int _M L\tag{1}$$$$ that integrates a $$D$$-form in spacetime over a manifold $$M$$. A variation of the action respect to the dynamical degrees of freedom, collectively denoted by $$\phi^a$$ is

$$$$\delta S =\int _M E_a [\phi]\delta \phi^a+\int_{\partial M}\theta[\phi,\delta\phi]\tag{2}$$$$

where we used $$\delta L=E_a\delta\phi^a+d\theta$$. $$\theta$$ is the pre-symplectic potential and $$E_a$$ are the Euler-Lagrange equations. The equations of motion are obtained by demanding $$\delta S=0$$ to get

$$$$E_a =0.\tag{3}$$$$

The presymplectic potential needs to satisfy boundary conditions for the theory to be well defined, so we get $$$$\theta |_{\Gamma}=dC+F\tag{4}$$$$

where $$C$$ is an allowed corner term and $$F$$ is a flux term that exists for null boundaries and represents the failure of the symplectic form to be zero at the boundaries.

With the presymplectic potential we can build a presymplectic current as

$$$$\omega[\delta\phi_1,\delta\phi_2]=\delta\phi_1 \theta[\phi,\delta\phi_2]-\delta\phi_2\theta[\phi,\delta\phi_1].\tag{5}$$$$

Integrating this over a Cauchy surface $$\Sigma$$ we get the presymplectic form $$\tilde{\Omega}$$

$$$$\tilde{\Omega}=\int_\Sigma \omega.\tag{6}$$$$

My idea would be to first obtain a generalized version of Hamilton's equations. That would look like

$$$$\delta H_s =\tilde{\Omega}\big[\delta \phi,\delta_s \phi\big]\tag{7}$$$$

which means that the conserved charge $$H_s$$ for the symmetry transformation $$s$$ generates the change $$\delta_s \phi$$. If we take $$s$$ to be time translations, this is exactly Hamilton's equations. Then, the idea would be to find transformations on the fields $$\phi^a$$ that keep this equation unchanged.

Alternatively, $$H_s$$ might also be thought of as the infinitesimal generator of the canonical transformation $$s$$, although I'm not sure if that generally makes sense. I'm stuck.

• Is the flux and corner terms terminology from some reference? Commented Nov 21, 2023 at 11:22
• It is from the modern understanding of the covariant phase space formalism when applied to boundary symmetries in general relativity. The review by Harlow (arxiv.org/abs/1906.08616) includes the corner term and there's a bunch of work on corner terms recently. The flux is also an important part of the Wald-Zoupas construction for asymptotic conserved charges. Commented Nov 21, 2023 at 20:01

1. Ref. 1 is talking about the Peierls bracket in the Lagrangian formalism. It is equivalent to the presymplectic structure in the covariant Hamiltonian formalism, cf. e.g. section 4.1 in Ref. 3.

2. Ref. 1 defines canonical transformations (CT) as Poisson morphisms wrt. the Peierls bracket. This corresponds to pre-symplectomorphisms in the covariant Hamiltonian formalism.

3. Concerning OP's last paragraphs (v3): The Hamiltonian is not defined per se in the covariant Hamiltonian formalism, cf. e.g. Ref. 2 and this & this Phys.SE posts. Instead the EOMs are given by the EL equations.

References:

1. B.S. DeWitt, The Global Approach to QFT, Vol. 2, 2003; chapter 33.

2. C. Crnkovic & E. Witten, Covariant description of canonical formalism in geometrical theories. Published in Three hundred years of gravitation (Eds. S. W. Hawking and W. Israel), (1987) 676.

3. D. Harlow & J.-q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146, arXiv:1906.08616.

• Thanks for your answer! As I said in my question, I know the DeWitt book but I was hoping for a slightly shorter than 700 pages answer. I'll check the links to other posts and the Witten paper! Commented Nov 21, 2023 at 20:07