Quantum Field Theory
... no spooky action at a distance (Einstein)
Early Results
Relativistic Quantum Mechanics Klein-Gordon Equation
Dirac Equation
The Dawn of QFT Spinors
Feynman Slash Notation
Klein-Gordon Field
Dirac Field
Grassman Variable
Conformal Field Theory
Countdown to the Standard Model
From a framework to a model Yang-Mills Theory
Quantum Electrodynamics
Quantum Chromodynamics
Electroweak Theory
Higgs Mechanism
Standard Model
Semi-Classical Gravity and the Dark Age Hawking Radiation
Chandrashekhar Limit
Problems with the Standard Model
Beyond the Standard Model Beyond the Standard Model
Quantum Gravity
Theory of Everything
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 FormulationEdit this section

Starting PointEdit this section

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. [1]\

De Donder-Weyl EquationsEdit this section

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} $

ReferencesEdit this section

  1. Paufler, Cornelius; Romer, Hartmann. (2002). "De Donder–Weyl equations and multisymplectic geometry". Reports on Mathematical Physics 49 (2 - 3): 325 - 334.