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 y^a field variables, for a = 1 to m, and L the Lagrangian density.

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

With polymomenta pⁱ_a 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 representation 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]

See also

• Hamiltonian field theory
• Covariant Hamiltonian field theory

Further reading

• Cornelius Paufler, Hartmann Römer: De Donder–Weyl equations and multisymplectic geometry, Reports on Mathematical Physics, vol. 49 (2002), no. 2–3, pp. 325–334
• Krzysztof Maurin: The Riemann legacy: Riemannian ideas in mathematics and physics, Part II, Chapter 7.16 Field theories for calculus of variation for multiple integrals, Kluwer Academic Publishers, ISBN 0-7923-4636-X, 1997, p. 482 ff.

References

1. Igor V. Kanatchikov: Towards the Born–Weyl quantization of fields, arXiv:quant-ph/9712058v1 (submitted on 31 December 1997)
2. Igor V. Kanatchikov: On the Canonical Structure of the De Donder-Weyl Covariant Hamiltonian Formulation of Field Theory I. Graded Poisson brackets and equations of motion, arXiv:hep-th/9312162v1 (submitted on 20 Dec 1993)
3. Théophile de Donder, "Théorie invariantive du calcul des variations," Gauthier-Villars, 1930.
4. Frédéric Hélein: Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory In Haïm Brézis, Felix E. Browder, Abbas Bahri, Sergiu Klainerman, Michael Vogelius (ads.): Noncompact problems at the intersection of geometry, analysis, and topology, American Mathematical Society, 2004, pp. 127–148, p. 131, ISBN 0-8218-3635-8,
5. Roger Bielawski, Kevin Houston, Martin Speight: Variational Problems in Differential Geometry, London Mathematical Society Lecure Notes Series, no. 394, University of Leeds, 2009, ISBN 978-0-521-28274-1, p. 104 f.
6. Igor V. Kanatchikov: De Donder–Weyl theory and a hypercomplex extension of quantum mechanics to field theory, arXiv:hep-th/9810165v1 (submitted on 21 October 1998)
7. I.V. Kanatchikov: Precanonical Quantum Gravity: quantization without the space-time decomposition, arXiv:gr-qc/0012074 (submitted on 20 December 2000)
8. Jedrzej Śniatycki, 1970. Cited after: Yvette Kosmann-Schwarzbach: The Noether Theorems: Invariance and Conservation Laws in the 20th Century, Springer, 2011, ISBN 978-0-387-87867-6, p. 111
9. Igor V. Kanatchikov: On the Duffin–Kemmer–Petiau formulation of the covariant Hamiltonian dynamics in field theory, arXiv:hep-th/9911/9911175v1 (submitted on 23 November 1999)