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. Defining relations 
The Duffin–Kemmer–Petiau matrices have the defining relation

where  stand for a constant diagonal matrix. The Duffin–Kemmer–Petiau matrices  for which  consists in diagonal elements (+1,-1,...,-1) form part of the Duffin–Kemmer–Petiau equation. Five-dimensional DKP matrices can be represented as:
, , , 
These five-dimensional DKP matrices represent spin-0 particles. The DKP matrices for spin-1 particles are 10-dimensional. 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. 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

where  are Duffin–Kemmer–Petiau matrices,  is the particle's mass,  its wavefunction,  the reduced Planck constant,  the speed of light. For massless particles, the term  is replaced by a singular matrix  that obeys the relations  and . The DKP equation for spin-0 is closely linked to the Klein–Gordon equation and the equation for spin-1 to the Proca equations. It suffers the same drawback as the Klein–Gordon equation in that it calls for negative probabilities. Also the De Donder–Weyl covariant Hamiltonian field equations can be formulated in terms of DKP matrices. History 
The Duffin–Kemmer–Petiau algebra was introduced in the 1930s by R.J. Duffin, N. Kemmer and G. Petiau. Further reading 
 
 
 
 
 Kruglov, S. I. (2011). "Field theory of massive and massless vector particles in the Duffin-Kemmer-Petiau formalism". Int.J.Mod.Phys.A 26,  2487-2501 [arXiv:1009.4895]. Kruglov, S. I. (2010) "Kalb-Ramond fields in the Petiau-Duffin-Kemmer formalism and scale invariance " Mod.Phys.Lett.A 25, 2745-2751 [arXiv:1002.3574]. References 

Algebras