Duffin–Kemmer–Petiau algebra

In mathematical physics, the Duffin–Kemmer–Petiau (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]

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]

The Duffin–Kemmer–Petiau (DKP) equation, also known as 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]

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

• Fernandes, M. C. B.; Vianna, J. D. M. (1999). "On the generalized phase space approach to Duffin–Kemmer–Petiau particles". Foundations of Physics. 29 (2). Springer Science and Business Media LLC: 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. 28 (4). FapUNIFESP (SciELO): 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. 271 (1–2). Elsevier BV: 16–25. arXiv:hep-th/0003283. doi:10.1016/s0375-9601(00)00330-3. ISSN 0375-9601. S2CID 9595290.