Duffin–Kemmer–Petiau algebra

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]

${\displaystyle \beta ^{a}\beta ^{b}\beta ^{c}+\beta ^{c}\beta ^{b}\beta ^{a}=\beta ^{a}\eta ^{bc}+\beta ^{c}\eta ^{ba}}$ 

where ${\displaystyle \eta ^{ab}}$  stand for a constant diagonal matrix. The Duffin–Kemmer–Petiau matrices ${\displaystyle \beta }$  for which ${\displaystyle \eta ^{ab}}$  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]

${\displaystyle \beta ^{0}={\begin{pmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{pmatrix}}}$ , ${\displaystyle \quad \beta ^{1}={\begin{pmatrix}0&0&-1&0&0\\0&0&0&0&0\\1&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{pmatrix}}}$ , ${\displaystyle \quad \beta ^{2}={\begin{pmatrix}0&0&0&-1&0\\0&0&0&0&0\\0&0&0&0&0\\1&0&0&0&0\\0&0&0&0&0\end{pmatrix}}}$ , ${\displaystyle \quad \beta ^{3}={\begin{pmatrix}0&0&0&0&-1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\1&0&0&0&0\end{pmatrix}}}$ 

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]

${\displaystyle (i\hbar \beta ^{a}\partial _{a}-mc)\psi =0}$ 

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

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. (1999). "On the generalized phase space approach to Duffin–Kemmer–Petiau particles". Foundations of Physics. Springer Science and Business Media LLC. 29 (2): 201–219. doi:10.1023/a:1018869505031. ISSN 0015-9018. S2CID 118277218.
• Fernandes, Marco Cezar B.; Vianna, J. David M. (1998). "On the Duffin-Kemmer-Petiau algebra and the generalized phase space". Brazilian Journal of Physics. FapUNIFESP (SciELO). 28 (4): 00. doi:10.1590/s0103-97331998000400024. ISSN 0103-9733.
• Sharp, Robert T.; Winternitz, Pavel (2004). "Bhabha and Duffin–Kemmer–Petiau equations: spin zero and spin one". Symmetry in physics : in memory of Robert T. Sharp. Providence, R.I.: American Mathematical Society. p. 50 ff. ISBN 0-8218-3409-6. OCLC 53953715.
• Fainberg, V.Ya.; Pimentel, B.M. (2000). "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): 16–25. arXiv:hep-th/0003283. doi:10.1016/s0375-9601(00)00330-3. ISSN 0375-9601. S2CID 9595290.

References

1. Helmstetter, Jacques; Micali, Artibano (2010-03-12). "About the Structure of Meson Algebras". Advances in Applied Clifford Algebras. Springer Science and Business Media LLC. 20 (3–4): 617–629. doi:10.1007/s00006-010-0213-0. ISSN 0188-7009. S2CID 122175054.
2. See introductory section of: Pavlov, Yu V. (2006). "Duffin–Kemmer–Petiau equation with nonminimal coupling to curvature". Gravitation & Cosmology. 12 (2–3): 205–208. arXiv:gr-qc/0610115v1.
3. See for example Boztosun, I.; Karakoc, M.; Yasuk, F.; Durmus, A. (2006). "Asymptotic iteration method solutions to the relativistic Duffin-Kemmer-Petiau equation". Journal of Mathematical Physics. 47 (6): 062301. arXiv:math-ph/0604040v1. doi:10.1063/1.2203429. ISSN 0022-2488. S2CID 119152844.
4. Capri, Anton Z. (2002). Relativistic quantum mechanics and introduction to quantum field theory. River Edge, NJ: World Scientific. p. 25. ISBN 981-238-136-8. OCLC 51850719.
5. Fischbach, Ephraim; Nieto, Michael Martin; Scott, C. K. (1973). "Duffin‐Kemmer‐Petiau subalgebras: Representations and applications". Journal of Mathematical Physics. AIP Publishing. 14 (12): 1760–1774. doi:10.1063/1.1666249. ISSN 0022-2488.
6. Casana, R; Fainberg, V Ya; Lunardi, J T; Pimentel, B M; Teixeira, R G (2003-05-16). "Massless DKP fields in Riemann–Cartan spacetimes". Classical and Quantum Gravity. 20 (11): 2457–2465. arXiv:gr-qc/0209083v2. doi:10.1088/0264-9381/20/11/333. ISSN 0264-9381. S2CID 250832154.
7. Kruglov, Sergey (2001). Symmetry and electromagnetic interaction of fields with multi-spin. Huntington, N.Y.: Nova Science Publishers. p. 26. ISBN 1-56072-880-9. OCLC 45202093.
8. Kanatchikov, Igor V. (2000). "On the Duffin-Kemmer-Petiau formulation of the covariant Hamiltonian dynamics in field theory". Reports on Mathematical Physics. 46 (1–2): 107–112. arXiv:hep-th/9911175v1. doi:10.1016/s0034-4877(01)80013-6. ISSN 0034-4877. S2CID 13185162.
9. Duffin, R. J. (1938-12-15). "On The Characteristic Matrices of Covariant Systems". Physical Review. American Physical Society (APS). 54 (12): 1114. doi:10.1103/physrev.54.1114. ISSN 0031-899X.
10. N. Kemmer (1939-11-10). "The particle aspect of meson theory". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 173 (952): 91–116. doi:10.1098/rspa.1939.0131. ISSN 0080-4630. S2CID 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)