From Duffin-Kemmer-Petiau to Tzou algebras in relativistic wave equations
aa r X i v : . [ phy s i c s . g e n - ph ] J a n From Duffin-Kemmer-Petiau to Tzou algebras inrelativistic wave equations
Andrzej Okni´nski ∗ Chair of Mathematics and Physics, Politechnika ´Swi¸etokrzyska,Al. 1000-lecia PP 7, 25-314 Kielce, PolandOctober 14, 2018
Abstract
We study relation between the Duffin-Kemmer-Petiau algebras andsome representations of Tzou algebras. Working in the setting of rela-tivistic wave equations we reduce, via a similarity transformation, fiveand ten dimensional Duffin-Kemmer-Petiau algebras to three and sevendimensional Tzou algebras, respectively.
The Duffin-Kemmer-Petiau (DKP) equations [1–3] have been becoming increas-ingly popular due to their applications to problems in particle and nuclearphysics involving spin 0 and spin 1 mesons [4–15].The DKP equations involve 5 × ×
10 matrices, corresponding torepresentations denoted as and , respectively, and obeying commutationrelations defining the DKP algebra [1–3]. However, there are several attemptsto describe spin 0 and spin 1 mesons using the so-called Tzou algebras [16, 17],see [18,19] as well as [20] and references therein. Since the spin 0 as well as spin1 bosons have been described within approaches using different algebras it seemsuseful to investigate possible relations between DKP and Tzou formalisms.In the next Section Tzou and DKP algebras are described shortly in thecontext of relativistic wave equations. In Section 3 the free DKP equationsin momentum representation are converted by a similarity transformation toequations involving the Tzou algebra, one of them being the Hagen-Hurleyequation. Our results are summarized in the last Section. In what followswe use notation and conventions described in [21]. ∗ Email: fi[email protected] Relativistic wave equations: Tzou and Duffin-Kemmer-Petiau algebras
Equations describing spin 0 and 1 bosons, can be written as: ρ µ p µ Ψ = m Ψ , (1)where p µ df = i ∂∂x µ and ρ µ are matrices with properties described below.Eq. (1) describes a particle with definite mass if ρ µ obey the Tzou commu-tation relations [16–19, 22]: X λλ,µ,ν ρ λ ρ µ ρ ν = X λ,µ,ν g λµ ρ ν , (2)where we sum over all permutations of λ, µ, ν .A special solution of the Tzou relations (2) was constructed in Ref. [1] inform: ρ µ = 12 ( γ µ ⊗ I × + I × ⊗ γ µ ) ≡ β µ . (3)Such β µ obey simpler but more restrictive commutation relations [1, 2]: β λ β µ β ν + β ν β µ β λ = g λµ β ν + g νµ β λ , (4)for which Eq. (1) leads to the Duffin-Kemmer-Petiau (DKP) theory of spin 0and 1 mesons, see [1–3]. This reducible 16-dimensional representation (3) of β µ matrices (denoted as ) can be decomposed as = ⊕ ⊕ . Representation (spin 1 case) is realized in terms of 10 ×
10 matrices, while representation (spin 0) involves 5 × β µ (theone-dimensional representation is trivial, i.e. all β µ = 0).It turns out that in the case of more general Eqs. (2) there are also otherrepresentations of ρ µ matrices, see [18, 19] for a review. For example, there aretwo representations for which the corresponding 7 × ρ µ yield theHagen-Hurley equations for spin 1 bosons [23–25], see also [18,19] and Subsection2.2 in [21]. There are also two sets of 3 × ρ µ obeying (2), see [26]. Theaim of this work is to find possible links between DKP and Tzou representations. Representations and of the DKP algebra (4) are irreducible. However, ifwe relax these conditions demanding only Tzou conditions (2) then these repre-sentations can be decomposed. We shall work in the momentum representation,Ψ ( x ) = ψ k e − ik · x , ρ µ k µ ψ k = mψ k . 2 .1 Explicit decomposition of representation 5 The β µ k µ matrix for irreducible DKP representation of matrices β µ reads: β µ k µ = k k k k k − k − k − k , (5)while the matrix ˆ ρ µ k µ , corresponding to reducible Tzou representation ⊕ ⊕ (note that ˆ ρ µ fulfill conditions (2)), is:ˆ ρ µ k µ = k + k k + ik k − k − k + ik . (6)Both matrices have the same characteristic polynomials, hence there exists asimilarity transformation V converting β µ k µ to ˆ ρ µ k µ , i.e. V β µ k µ V − = ˆ ρ µ k µ .To find V we first diagonalize both matrices S − β µ k µ S = diag ( κ, − κ, , ,
0) = T − ˆ ρ µ k µ T , κ = p k − k − k − k , where columns of S and T are eigenvectorsof β µ k µ and ˆ ρ µ k µ , respectively: S = k κ − k κ k κ − k κ k κ − k κ k κ − k κ k k − k k − k k , (7) T = k + k k + ik k + k k + ik k − ik k − k κ k − ik k + k κ − k + ik k + k . (8)Finally, we have V = T S − . We have assumed that denominators in Eqs. (7),(8) do not vanish. However, in special cases when some (or all) denominatorsdo vanish (i.e. if k = 0, or k = k = 0, or k = k = k = 0), transformations S , T , V still exist.Note that V depends on k µ and thus V β µ V − = ˆ ρ µ . Accordingly, whilematrices β µ belong to representation of the DKP algebra (they are also arepresentation of the Tzou algebra), matrices ˆ ρ µ are representation of the Tzoualgebra only. It follows from Eq. (6) that the irreducible representation of3he Tzou algebra is: ρ µ k µ = k + k k + ik k − k − k + ik , (9)see also Eqs. (14), (16) in [26]. Matrices β µ of the DKP representation can be found in [1, 18, 19]. Thematrix β µ k µ reads: β µ k µ = k k k k
00 0 0 0 0 0 k k k − k − k k k − k − k − k − k k k − k k − k k k k − k (10)and the matrix ˆ ρ µ k µ , with matrices ˆ ρ µ corresponding to reducible Tzou repre-sentation ⊕ ⊕ ⊕ (note again that ˆ ρ µ fulfill conditions (2)), is:ˆ ρ µ k µ = − ik − ik − k k − ik k − ik − k − ik − k k − ik − ik − ik − ik ik − k k k ik − k − k k ik (11)Proceeding exactly as in Subsection 3.1 we construct matrix V of the simi-larity transformation V β µ k µ V − = ˆ ρ µ k µ . Finally, the matrix ρ µ k µ is the 7 × as ⊕ ⊕ ⊕ . Equation ρ µ p µ Ψ = m Ψ is one of the Hagen-Hurleyequations describing spin 1 bosons, cf. Eq. (16) and the text below in [21].4
Summary
We have demonstrated that the free Duffin-Kemmer-Petiau equations in the mo-mentum representation can be converted by application of a similarity transfor-mation to equations involving Tzou algebra. More exactly, there exists a matrix V , dependent on k µ , such that: V ( β µ k µ − m ) V − = ˆ ρ µ k µ − m, (12a) V β µ V − = ˆ ρ µ , (12b)where β µ fulfill the Duffin-Kemmer-Petiau relations (4), while ˆ ρ µ belong to the(reducible) Tzou algebra (2). Therefore, in the case of spin 1 bosons we haveconstructed a link (a similarity transformation) between the Duffin-Kemmer-Petiau equation with 10 ×
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