De Donder-Weyl Theory

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Related	De Donder-Weyl Theory

De Donder-Weyl Theory is a formulation of Quantum Field Theory which claims tthat the De Donder-Weyl Equations are "more" explicitly Lorentz-Invariant than the standard Quantum Field Theory.

However, in reality, all this does is introduce a more funny Hamiltonian using a "polymomenta" instead of the standard momenta, which make the Hamiltonian manifestly Lorentz-Invariant.

Mathematical Formulation

Starting Point

The Hamiltonian in Physics is generally strongly tied to the idea of time, as opposed to space, since it is trivial that the Hamiltonian describes the Time-Evolution. This can be seen in, for example, Noether's Theorem and Schrodinger's Equation.

For similar reasons, in Quantum Field Theory,

$\mathcal{H} = \pi \partial_0 {\phi} - \mathcal{L}$

We thus see, that this is a rather irritating statement for someone who is comfortable with the world of manifestly Lorentz-Invariant creatures.

Therefore, we may instead define the so - called "Polymomenta", as De-Donder and Weyl called it, so that

$\mathcal{H} = \pi_{\mu}\partial^{\mu} \phi - \mathcal{L}$

Where $\pi^\mu$ is the "Polymomenta". This is known as the De Donder-Weyl Equation. \

De Donder-Weyl Equations

The result of this is rather trivially the "De Donder-Weyl Equations":

$\frac{\partial p^{i}_a}{\partial x^{i}} = -\frac{\partial H}{\partial y^{a}}$ $\frac{\partial y^{a}}{\partial x^{i}} = \frac{\partial H}{\partial p^{i}_a}$