Duffin–Kemmer–Petiau algebra explained

In mathematical physics, the Duffin–Kemmer–Petiau algebra (DKP algebra), introduced by R.J. Duffin, Nicholas Kemmer and G. Petiau, is the algebra which is generated by the Duffin–Kemmer–Petiau matrices. These matrices form part of the Duffin–Kemmer–Petiau equation that provides a relativistic description of spin-0 and spin-1 particles.

The DKP algebra is also referred to as the meson algebra.^[1]

Defining relations

The Duffin–Kemmer–Petiau matrices have the defining relation^[2]

\beta ^{a} \beta ^{b} \beta ^{c} + \beta ^{c} \beta ^{b} \beta ^{a} = \beta ^{a} η ^{b} + \beta ^{c} η ^{b} where η ^{a} stand for a constant diagonal matrix. The Duffin–Kemmer–Petiau matrices \beta for which η ^{a} consists in diagonal elements (+1,-1,...,-1) form part of the Duffin–Kemmer–Petiau equation. Five-dimensional DKP matrices can be represented as:^{[3] [4]} \beta ^{0} = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} , \beta ^{1} = \begin{pmatrix} 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} , \beta ^{2} = \begin{pmatrix} 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} , \beta ^{3} = \begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{pmatrix} These five-dimensional DKP matrices represent spin-0 particles. The DKP matrices for spin-1 particles are 10-dimensional.^[3] The DKP-algebra can be reduced to a direct sum of irreducible subalgebras for spin‐0 and spin‐1 bosons, the subalgebras being defined by multiplication rules for the linearly independent basis elements.^[5]

Duffin–Kemmer–Petiau equation

The Duffin–Kemmer–Petiau equation (DKP equation, also: Kemmer equation) is a relativistic wave equation which describes spin-0 and spin-1 particles in the description of the standard model. For particles with nonzero mass, the DKP equation is^[2]

(i \hbar \beta ^{a} \partial _{a} - m c) \psi = 0 where \beta ^{a} are Duffin–Kemmer–Petiau matrices, m is the particle's mass, \psi its wavefunction, \hbar the reduced Planck constant, c the speed of light. For massless particles, the term m c is replaced by a singular matrix \gamma that obeys the relations \beta ^{a} \gamma + \gamma \beta ^{a} = \beta ^{a} and \gamma ^{2} = \gamma .

The DKP equation for spin-0 is closely linked to the Klein–Gordon equation^{[4] [6]} and the equation for spin-1 to the Proca equations.^[7] It suffers the same drawback as the Klein–Gordon equation in that it calls for negative probabilities.^[4] Also the De Donder–Weyl covariant Hamiltonian field equations can be formulated in terms of DKP matrices.^[8]

History

The Duffin–Kemmer–Petiau algebra was introduced in the 1930s by R.J. Duffin,^[9] N. Kemmer^[10] and G. Petiau.^[11]

Further reading

• Fernandes . M. C. B. . Vianna . J. D. M. . On the generalized phase space approach to Duffin–Kemmer–Petiau particles. Foundations of Physics . Springer Science and Business Media LLC . 29 . 2 . 1999 . 0015-9018 . 10.1023/a:1018869505031 . 201–219. 118277218 .
• Fernandes . Marco Cezar B. . Vianna . J. David M. . On the Duffin-Kemmer-Petiau algebra and the generalized phase space . Brazilian Journal of Physics . FapUNIFESP (SciELO) . 28 . 4 . 1998 . 0103-9733 . 10.1590/s0103-97331998000400024 . 00. free .
• Book: Sharp . Robert T. . Winternitz . Pavel . Symmetry in physics : in memory of Robert T. Sharp . American Mathematical Society . Providence, R.I. . 2004 . 0-8218-3409-6 . 53953715 . 50 ff. https://books.google.com/books?id=2AqekbVZX4AC&pg=PA50. Bhabha and Duffin–Kemmer–Petiau equations: spin zero and spin one.
• Fainberg . V.Ya. . Pimentel . B.M. . Duffin–Kemmer–Petiau and Klein–Gordon–Fock equations for electromagnetic, Yang–Mills and external gravitational field interactions: proof of equivalence . Physics Letters A . Elsevier BV . 271 . 1–2 . 2000 . 0375-9601 . 10.1016/s0375-9601(00)00330-3 . 16–25. hep-th/0003283. 9595290 .

Notes and References

1. Helmstetter . Jacques . Micali . Artibano . About the Structure of Meson Algebras . Advances in Applied Clifford Algebras . Springer Science and Business Media LLC . 20 . 3–4 . 2010-03-12 . 0188-7009 . 10.1007/s00006-010-0213-0 . 617–629. 122175054 .
2. See introductory section of: Yu V. . Pavlov. gr-qc/0610115v1. Duffin–Kemmer–Petiau equation with nonminimal coupling to curvature. Gravitation & Cosmology. 12. 2006. 2–3. 205–208.
3. See for example Boztosun . I. . Karakoc . M. . Yasuk . F. . Durmus . A. . Asymptotic iteration method solutions to the relativistic Duffin-Kemmer-Petiau equation . Journal of Mathematical Physics . 47 . 6 . 2006 . 0022-2488 . 10.1063/1.2203429 . 062301. math-ph/0604040v1. 119152844 .
4. Book: Capri, Anton Z. . Relativistic quantum mechanics and introduction to quantum field theory . World Scientific . River Edge, NJ . 2002 . 981-238-136-8 . 51850719 . 25.
5. Fischbach . Ephraim . Nieto . Michael Martin . Scott . C. K. . Duffin‐Kemmer‐Petiau subalgebras: Representations and applications . Journal of Mathematical Physics . AIP Publishing . 14 . 12 . 1973 . 0022-2488 . 10.1063/1.1666249 . 1760–1774.
6. Casana . R . Fainberg . V Ya . Lunardi . J T . Pimentel . B M . Teixeira . R G . Massless DKP fields in Riemann–Cartan spacetimes . Classical and Quantum Gravity . 20 . 11 . 2003-05-16 . 0264-9381 . 10.1088/0264-9381/20/11/333 . 2457–2465. gr-qc/0209083v2. 250832154 .
7. Book: Kruglov, Sergey . Symmetry and electromagnetic interaction of fields with multi-spin . Nova Science Publishers . Huntington, N.Y. . 2001 . 1-56072-880-9 . 45202093 . 26.
8. Kanatchikov . Igor V. . On the Duffin-Kemmer-Petiau formulation of the covariant Hamiltonian dynamics in field theory . Reports on Mathematical Physics . 46 . 1–2 . 2000 . 0034-4877 . 10.1016/s0034-4877(01)80013-6 . 107–112. hep-th/9911175v1. 13185162 .
9. Duffin . R. J. . On The Characteristic Matrices of Covariant Systems . Physical Review . American Physical Society (APS) . 54 . 12 . 1938-12-15 . 0031-899X . 10.1103/physrev.54.1114 . 1114.
10. N. Kemmer. The particle aspect of meson theory . Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences . The Royal Society . 173 . 952 . 1939-11-10 . 0080-4630 . 10.1098/rspa.1939.0131 . 91–116. 121843934 .
11. G. Petiau, University of Paris thesis (1936), published in Acad. Roy. de Belg., A. Sci. Mem. Collect.vol. 16, N 2, 1 (1936)