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In this paper, Wald presents a quite general construction of a sympletic form for classical fields. If I understood (which I might have not, and in that case corrections are highly appreciated), the procedure is somewhat along these lines:

  1. Let a spacetime $(M,g)$ be given and let $\mathcal{L}$ be a Lagrangian density specifying a field theory in $(M,g)$. The action is $$S[\phi]=\int_M \mathcal{L}[\phi](x) \sqrt{g} \ d^4x.$$

  2. The equations of motion of the theory follow from $\delta S =0$. It turns out that when we compute $\delta S$ we get a boundary term. So the full variation is $$\delta S = \int \bigg[\bigg(\dfrac{\partial \mathcal{L}}{\partial \phi}+\nabla_a \dfrac{\partial\mathcal{L}}{\partial (\nabla_a\phi)} \delta\phi\bigg)+\nabla_a\bigg(\dfrac{\partial \mathcal{L}}{\partial(\nabla_a \phi)}\delta \phi\bigg) \bigg]\sqrt{g} d^4x $$

    The boundary term is usually discarded when computing the equations of motion out of the argument that everything vanishes at infinity.

  3. Nonetheless, Wald calls this thing the presympletic current potential density $$\Theta^a= \dfrac{\partial \mathcal{L}}{\partial (\nabla_a \phi)}\delta\phi$$

  4. The phase space $\Gamma$ of the theory is the manifold of solutions to the equations of motion. A tangent vector is a variation $\delta \phi$ of a solution $\phi$, which is itself a function.

    In the above, $\Theta^a$ is a map defined at each $\phi\in \Gamma$ and is a linear function of variations $\delta \phi \in T_\phi \Gamma$. It takes values on vector fields defined on spacetime.

  5. One can then define the presympletic current to be a map $\omega^a$ defined on $T_\phi\Gamma\times T_\phi\Gamma$ taking values on vector fields on spacetime by $$\omega^aa[\phi,\delta_1 \phi,\delta_2\phi]=\delta_1 \Theta^a[\phi,\delta_2 \phi]-\delta_2\Theta^a[\phi,\delta_1\phi]$$

    What we mean is: compute $\Theta^a[\phi,\delta_2\phi]$ and then vary this considering that the field varies by $\delta \phi_1$, and the same for the other term.

  6. Define the sympletic form by integrating the pre-sympletic current over a Cauchy surface $$\Omega_\phi(\delta_1 \phi,\delta_2\phi)=\int_{\Sigma} \omega^a[\phi,\delta_1\phi,\delta_2\phi] d\Sigma_a$$ By construction this is, at each $\phi$ a bilinear function of two variations, which is skew symmetric and takes values on $\mathbb{R}$. Thus it is a two-form on $\Gamma$ as a sympletic form would be.

As a recipe this is fine. Indeed this yields the expected sympletic forms.

But how on Earth would anyone think of doing this? I mean, the main reasons I'm unconfortable with it, is that I look at it trying to see how anyone would think of doing all this to get a sympletic form for a classical field theory and I simply can't get it.

So what is the intuition behind this construction? What is the motivation to construct the sympletic form for classical field theories like that? How would one get the idea of doing this?

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Well, this is the covariant Hamiltonian formulation, see e.g. Ref. 1 and this Phys.SE post. In contrast, the standard Hamiltonian formulation of Lagrangian field theories in $n$ spacetime dimensions singles out a temporal direction when defining the canonical momentum and performing the standard Legendre transformation, so that manifest covariance is broken.

It is therefore natural to introduce $n$ de-Donder-Weyl polymomenta. But then we have lost a one-to-one correspondence between positions and momenta, which is vital to the symplectic structure.

This is recovered in the covariant construction via a choice of Cauchy hypersurface $\Sigma$. Similarly, the rest of the covariant Hamiltonian formulation is natural covariant generalizations of the standard Hamiltonian construction.

References:

  1. 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.
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  • $\begingroup$ Let's see if I got it. We wish to generalize $\omega=\sum_i p_i dq^i$ from which the sympletic form follows as $d\omega$. Given a Cauchy surface $\Sigma$ with normal $n^\mu$ the label $i$ becomes the points $x\in \Sigma$ and the sum becomes one integral. Furthermore, $dq^i$ becomes $x^\ast$ defined so that given a variation $\delta \phi$ we have $x^\ast(\delta \phi)=\delta\phi(x)$. So we define the polymomenta $p_a^\mu$ and construct $$\int_\Sigma n_\mu(x) p_a^\mu(x) x^\ast d\Sigma$$ which is a one-form on covariant phase space. We then take $\delta$ of that to get $\Omega$. Is that the idea? $\endgroup$ – user1620696 Mar 12 at 2:29

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