Duffin–Kemmer–Petiau algebra

{{Short description|Algebra generated by the Duffin-Kemmer-Petiau matrices}}

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.{{cite journal | last1=Helmstetter | first1=Jacques | last2=Micali | first2=Artibano | title=About the Structure of Meson Algebras | journal=Advances in Applied Clifford Algebras | publisher=Springer Science and Business Media LLC | volume=20 | issue=3–4 | date=2010-03-12 | issn=0188-7009 | doi=10.1007/s00006-010-0213-0 | pages=617–629| s2cid=122175054 }}

Defining relations

The Duffin–Kemmer–Petiau matrices have the defining relationSee introductory section of: {{cite journal|first=Yu V. |last=Pavlov|arxiv=gr-qc/0610115v1|title=Duffin–Kemmer–Petiau equation with nonminimal coupling to curvature|journal=Gravitation & Cosmology|volume=12|year=2006|issue=2–3|pages=205–208}}

: $\beta^{a} \beta^{b} \beta^{c} + \beta^{c} \beta^{b} \beta^{a} = \beta^{a} \eta^{b c} + \beta^{c} \eta^{b a}$

where $\eta^{a b}$ stand for a constant diagonal matrix. The Duffin–Kemmer–Petiau matrices $\beta$ for which $\eta^{a b}$ consists in diagonal elements (+1,-1,...,-1) form part of the Duffin–Kemmer–Petiau equation. Five-dimensional DKP matrices can be represented as:See for example {{cite journal | last1=Boztosun | first1=I. | last2=Karakoc | first2=M. | last3=Yasuk | first3=F. | last4=Durmus | first4=A. | title=Asymptotic iteration method solutions to the relativistic Duffin-Kemmer-Petiau equation | journal=Journal of Mathematical Physics | volume=47 | issue=6 | year=2006 | issn=0022-2488 | doi=10.1063/1.2203429 | page=062301|arxiv=math-ph/0604040v1| s2cid=119152844 }}{{cite book | last=Capri | first=Anton Z. | title=Relativistic quantum mechanics and introduction to quantum field theory | publisher=World Scientific | publication-place=River Edge, NJ | date=2002 | isbn=981-238-136-8 | oclc=51850719 | page=25|url=https://books.google.com/books?id=tTJHB5hepQUC&pg=PA25}}

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

These five-dimensional DKP matrices represent spin-0 particles. The DKP matrices for spin-1 particles are 10-dimensional. 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.{{cite journal | last1=Fischbach | first1=Ephraim | last2=Nieto | first2=Michael Martin | last3=Scott | first3=C. K. | title=Duffin‐Kemmer‐Petiau subalgebras: Representations and applications | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=14 | issue=12 | year=1973 | issn=0022-2488 | doi=10.1063/1.1666249 | pages=1760–1774}}

Duffin–Kemmer–Petiau equation

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

: $(i \hbar \beta^{a} \partial_a - m c) \psi = 0$

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{{cite journal | last1=Casana | first1=R | last2=Fainberg | first2=V Ya | last3=Lunardi | first3=J T | last4=Pimentel | first4=B M | last5=Teixeira | first5=R G | title=Massless DKP fields in Riemann–Cartan spacetimes | journal=Classical and Quantum Gravity | volume=20 | issue=11 | date=2003-05-16 | issn=0264-9381 | doi=10.1088/0264-9381/20/11/333 | pages=2457–2465|arxiv=gr-qc/0209083v2| s2cid=250832154 }} and the equation for spin-1 to the Proca equations.{{cite book | last=Kruglov | first=Sergey | title=Symmetry and electromagnetic interaction of fields with multi-spin | publisher=Nova Science Publishers | publication-place=Huntington, N.Y. | date=2001 | isbn=1-56072-880-9 | oclc=45202093 | page=26|url=https://books.google.com/books?id=E6Elkxs9PaIC&pg=PA26}} 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.{{cite journal | last=Kanatchikov | first=Igor V. | title=On the Duffin-Kemmer-Petiau formulation of the covariant Hamiltonian dynamics in field theory | journal=Reports on Mathematical Physics | volume=46 | issue=1–2 | year=2000 | issn=0034-4877 | doi=10.1016/s0034-4877(01)80013-6 | pages=107–112|arxiv=hep-th/9911175v1| s2cid=13185162 }}

History

The Duffin–Kemmer–Petiau algebra was introduced in the 1930s by R.J. Duffin,{{cite journal | last=Duffin | first=R. J. | title=On The Characteristic Matrices of Covariant Systems | journal=Physical Review | publisher=American Physical Society (APS) | volume=54 | issue=12 | date=1938-12-15 | issn=0031-899X | doi=10.1103/physrev.54.1114 | pages=1114}} N. Kemmer{{cite journal |author=N. Kemmer| title=The particle aspect of meson theory | journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences | publisher=The Royal Society | volume=173 | issue=952 | date=1939-11-10 | issn=0080-4630 | doi=10.1098/rspa.1939.0131 | pages=91–116| s2cid=121843934 |url=http://rspa.royalsocietypublishing.org/content/173/952/91.full.pdf+html}} and G. Petiau.G. Petiau, University of Paris thesis (1936), published in Acad. Roy. de Belg., A. Sci. Mem. Collect.vol. 16, N 2, 1 (1936)

References

Category:Algebras

Duffin–Kemmer–Petiau algebra

Defining relations

Duffin–Kemmer–Petiau equation

History

Further reading

References