# De Donder–Weyl theory

In mathematical physics, the De Donder–Weyl theory is a formalism in the calculus of variations over spacetime which treats the space and time coordinates on equal footing. In this framework, a field is represented as a system that varies both in space and in time.

 De Donder–Weyl equations: $\partial p^{i}_a / \partial x^{i} = -\partial H / \partial y^{a}$ $\partial y^{a} / \partial x^{i} = \partial H / \partial p^{i}_a$

## De Donder–Weyl formulation of field theory

The De Donder–Weyl theory is based on a change of variables. Let xi be spacetime coordinates, for i = 1 to n (with n = 4 representing 3 + 1 dimensions of space and time), and ya field variables, for a = 1 to m, and L the Lagrangian density.

$L = L(y^{a},\partial_i y^{a},x^{i})$

With polymomenta pia defined as

$p^{i}_a = \partial L / \partial (\partial_i y^{a})$

and for De Donder–Weyl Hamiltonian function H defined as

$H = p^{i}_a \partial_i y^{a} - L$

the De Donder–Weyl equations are:[1]

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

These canonical equations of motion are covariant. The theory is a formulation of a covariant Hamiltonian field theory and for n = 1 it reduces to Hamiltonian mechanics (see also action principle in the calculus of variations).

The generalization of Poisson brackets to the De Donder–Weyl theory and the representaion of De Donder-Weyl equations in terms of generalized Poisson brackets was found by Kanatchikov in 1993.[2]

## History

The formalism, now known as De Donder–Weyl (DW) theory, was developed by Théophile de Donder[3][4] and Hermann Weyl. Weyl has made his proposal in 1934, inspired by work of Constantin Carathéodory, which in turn was founded on the work of Vito Volterra. The work of De Donder in contrast started from the theory of integral invariants of Élie Cartan.[5] The De Donder–Weyl theory has been known in the calculus of variations since the 1930s and initially found only very rare application in physics. It has recently found increased interest in theoretical physics in the context of quantum field theory[6] and quantum gravity.[7]

In 1970, Jedrzej Śniatycki, author of book on Geometric quantization and quantum mechanics, developed an invariant geometrical formulation of jet bundles building on the work of De Donder and Weyl.[8] Theoretical physicist Igor V. Kanatchikov showed in 1999 that the De Donder–Weyl covariant Hamiltonian field equations can be formulated in terms of Duffin–Kemmer–Petiau matrices.[9]