# On a Simple Relation between Duffin-Kemmer-Petiau and Tzou Algebras

Authors: Andrzej Okniński
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

A simple link betweenβμ matrices of the Duffin-Kemmer-Petiau theory and ρμ matrices of Tzou representations is constructed. The link consists of a constant unitary transformation of the βμ matrices and a projection onto a lower-dimensional subspace.


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## Body

## 1. Introduction

The Duffin-Kemmer-Petiau (DKP) equations [1–3] have been gaining importance due to their applications to problems in particle and nuclear physics of spin 0 and spin 1 mesons [4–15]. Since spin 1 bosons can be also described by Hagen-Hurley equations [16–18] involving Tzou algebra [19–21], relations between DKP and Tzou algebras deserve a separate study.Recently, we have constructed a similarity transformation that maps the free Duffin-Kemmer-Petiau equations to equations involving the Tzou algebras [22].However, this similarity transformation is rather complicated, is momentum dependent, and transforms the whole operatorβμkμ-m only. In the present paper a simple link between βμ matrices of the DKP theory and ρμ matrices of Tzou representations is constructed. The link consists of a (constant) unitary transformation of the βμ matrices and a projection onto a lower-dimensional subspace.The inspiration has come from our earlier experience in splitting relativistic equations within the spinor formalism. It turns out that the process of splitting the DKP equations leads to subequations involving Tzou algebras; see [19] and references therein. Therefore, our first step to find a simple relation between βμ and ρμ matrices consists in transition to spinor formalism via a unitary transformation.In the next section, Tzou and DKP algebras are described in the setting of relativistic wave equations. In Sections3 and 4 the βμ matrices are converted by application of unitary transformations and projection operators into two sets of Tzou matrices ρμ, one of them having been used in the Hagen-Hurley equations describing spin 1 bosons [16–19]. We discuss our results in the last section. In what follows we use notation and conventions described in [19].


## 2. Tzou and Duffin-Kemmer-Petiau Algebras in Relativistic Wave Equations

Relativistic equations describing elementary particles can be written as(1)ρμpμΨ=mΨ,where pμ=dfi∂/∂xμ and ρμ are some matrices.Equation (1) describes a particle with definite mass on condition that ρμ obey the Tzou commutation relations [20, 21, 23–25]:(2)∑λ,μ,νλρλρμρν=∑λ,μ,νgλμρν,where we sum over all permutations of λ,μ,ν.Important solution of the Tzou conditions (2) was constructed in [1] in form(3)ρμ=12γμ⊗I4×4+I4×4⊗γμ≡βμ.Such βμ obey simpler but more restrictive commutation relations [1, 2]:(4)βλβμβν+βνβμβλ=gλμβν+gνμβλ,for which (1) leads to the Duffin-Kemmer-Petiau (DKP) theory of spin 0 and 1 mesons; see [1–3]. This 16-dimensional representation (3) of βμ matrices (denoted as 16) is reducible and can be decomposed as 16=10⊕5⊕1. Representation 10 (spin 1 case) is realized in terms of 10×10 matrices, while representation 5 (spin 0) involves 5×5 matrices; see [1, 23, 24].In the case of5×5 (spin-0) representation of βμ matrices (1) is equivalent to the following set of equations:(5)pμψ=mψμpνψν=mψ,if we define Ψ in (1) as(6)Ψ=ψμ,ψT=ψ0,ψ1,ψ2,ψ3,ψT,where T denotes transposition of a matrix [19].In the case of10×10 (spin-1) representation of matrices βμ (1) reduces to(7)pμψν-pνψμ=mψμνpμψμν=mψν,with the following definition of Ψ in (1):(8)Ψ=ψμν,ψλT=ψ01,ψ02,ψ03,ψ23,ψ31,ψ12,ψ0,ψ1,ψ2,ψ3T,where ψλ are real and ψμν are purely imaginary [19]. Because of antisymmetry of ψμν we have pνψν=0 and this is spin 1 condition. The set of (7) was first written by Proca [26].It turns out that in the case of the more general equation (2) there are also other representations of ρμ matrices; see [23, 24] for a review. For example, there are two representations 7 for which the corresponding 7×7 matrices ρμ yield the Hagen-Hurley equations for spin 1 bosons [16–18]; see also [19, 23, 24]. There are also two sets of 3×3 matrices ρμ obeying (2); see [27]. In the present work we find a simple relation between DKP and Tzou representations.Several comments concerning equivalence/nonequivalence of formalisms used to describe vector bosons are in order here. The spin1 DKP and Proca theories are equivalent in the free case as well as in the case of minimally coupled interactions. On the other hand, the DKP formalism permits nonstandard couplings via nonminimal interactions, which are not possible in the Proca theory; see [28, 29] and references therein. Finally, the Hagen-Hurley equations are yet another formalism since these equations violate parity and thus their applications are limited to weakly interacting particles [30].


## 3. From Representation 5 of the DKP Algebra to Representations 3 of the Tzou Algebra

As explained in the Introduction, we first carry out a unitary transformationC(9)C=121001001i00000020-1i00100-10which transforms the wave-function (6) into the spinor form:(10)Ψ→CΨ=12ψ0+ψ3,ψ1+iψ2,2ψ,-ψ1+iψ2,ψ0-ψ3T,i.e., involving spinor components ψAB˙ rather than vector ones ψμ; see [27] for the definition of ψAB˙.Then we apply two projection operators,P=diag1,1,1,0,0 and Q=diag0,0,1,1,1, projecting onto three-dimensional subspaces, to define new matrices:(11a)ρ^μ=12PCβμC-1P,(11b)ρˇμ=12QCβμC-1Q.The matrices ρ^μ which can be written in explicit compact form as(12)
ρ
^
μ
p
μ
=
0
0
p
0
+
p
3
0
0
0
0
p
1
+
i
p
2
0
0
p
0
-
p
3
-
p
1
+
i
p
2
0
0
0
0
0
0
0
0
0
0
0
0
0
,fulfilling the Tzou commutation relations (2). Matrices ρ^μ can be immediately read off from (12) as they are proportional to pμ’s. This result was obtained in [22] but for the whole operator ρ^μkμ only and not for individual matrices ρ^μ.While these matrices belong to reducible representation of the Tzou algebra (2), they induce the irreducible representation 3:(13)ρμpμ=00p0+p300p1+ip2p0-p3-p1+ip20.The matricesρˇμ lead to another irreducible representation 3 of the Tzou algebra. These two representations 3 were described in [27].


## 4. From Representation 10 of the DKP Algebra to Representations 7 of the Tzou Algebra

We start carrying out a unitary transformationC(14)C=12100i0000000100i0000000100i00000000002000000000020000000000200000000002100-i0000000100-i0000000100-i0000,which transforms the wave-function (8), Ψ→CΨ, into the form involving self-dual and antiself-dual tensors, ψSμν and ψAμν. More exactly, ψSμν=ψμν+iψ^μν and ψAμν=ψμν-iψ^μν, where a tensor ψ^μν dual to a given antisymmetric tensor ψμν is defined as ψ^μν=dfi/2ϵμνκλψκλ and ψ^Sμν=ψSμν, ψ^Aμν=-ψAμν. Self-dual and antiself-dual tensors ψSμν and ψAμν are directly related to symmetric spinors ξAB and ηC˙D˙ [27, 31, 32].Next, we apply projection operators,P=diag⁡1,1,1,1,1,1,1,0,0,0 and Q=diag⁡0,0,0,1,1,1,1,1,1,1, projecting onto seven-dimensional subspaces, to define new matrices:(15a)ρ^μ=12PCβμC-1P,(15b)ρˇμ=12QCβμC-1Q.The 10×10 matrices ρ^μand ρˇμ are reducible representations of Tzou algebra (2) and induce irreducible representations 7. For example, the matrices ρμ induced by ρ^μ can be written in explicit compact form as(16)ρμpμ=000p1p0ip3-ip2000p2-ip3p0ip1000p3ip2-ip1p0-p1-p2-p30000p0-ip3ip20000ip3p0-ip10000-ip2ip1p00000.EquationρμpμΨ=mΨ is unitarily equivalent to the Hagen-Hurley equation [19].


## 5. Summary

We have constructed transformations of the Duffin-Kemmer-Petiau matrices, corresponding to representations5 and 10, inducing matrices of representations 3 and 7 of the Tzou algebra, respectively. These transformations consist of a (renormalized) constant unitary transformation and a projection operator onto lower-dimensional subspace; see (11a), (11b), (15a), and (15b). The computations have been carried out in a particular representation of the corresponding matrices. On the other hand, all the results are valid in all other representations which can be obtained via a unitary or similarity transformation of these matrices.These Tzou representations arise in the process of splitting relativistic equations, leading to subsolutions of these equations. For example, representation7 appearing in the Hagen-Hurley theory of spin 1 bosons [16–18] arises in the process of splitting the DKP equation written for representation 10 of βμ matrices [19].These simple formulae can be contrasted with complicated, momentum dependent, and similarity transformations described in [22]. Moreover, those similarity transformations transform correctly the whole DKP operator βμkμ-m only; see (12) in [22].



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