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Fischer Decomposition for Massless Fields of Spin 1 in Dimension 4

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Abstract

The massless field equations for lower integer and half-integer values of spin in Minkowski space are fundamental equations in mathematical physics. Their counterpart in Euclidean spacetime is a system of elliptic equations, which was already studied from the viewpoint of function theory in the framework of so-called Hodge systems for differential forms of various degrees. In dimension 4 it is possible to substitute spinor calculus for the usual tensor notation. In the present paper we concentrate on the case of the massless field equation for spin 1 in dimension 4, and we treat, in a spinor formalism, a fundamental concept of its function theory: the Fischer decomposition of polynomial spinor fields, for which we give simple and independent proofs.

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Acknowledgements

Lukáš Krump and Vladimír Souček gratefully acknowledge support by the Czech Grant GA CR 17-01171S.

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Authors and Affiliations

  1. Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium

    F. Brackx & H. De Schepper

  2. Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha, Czech Republic

    L. Krump & V. Souček

Authors
  1. F. Brackx
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  2. H. De Schepper
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  3. L. Krump
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  4. V. Souček
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Correspondence to H. De Schepper.

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Communicated by Irene Sabadini, Michael Shapiro, Daniele Struppa.

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Brackx, F., De Schepper, H., Krump, L. et al. Fischer Decomposition for Massless Fields of Spin 1 in Dimension 4. Complex Anal. Oper. Theory 12, 439–456 (2018). https://doi.org/10.1007/s11785-017-0697-x

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  • DOI: https://doi.org/10.1007/s11785-017-0697-x

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