Abstract
A simple link between matrices of the Duffin-Kemmer-Petiau theory and matrices of Tzou representations is constructed. The link consists of a constant unitary transformation of the matrices and a projection onto a lower-dimensional subspace.
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 and spin mesons [4–15]. Since spin 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 operator only. In the present paper a simple link between matrices of the DKP theory and matrices of Tzou representations is constructed. The link consists of a (constant) unitary transformation of the 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 and 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 Sections 3 and 4 the matrices are converted by application of unitary transformations and projection operators into two sets of Tzou matrices , one of them having been used in the Hagen-Hurley equations describing spin 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 aswhere and are some matrices.
Equation (1) describes a particle with definite mass on condition that obey the Tzou commutation relations [20, 21, 23–25]:where we sum over all permutations of .
Important solution of the Tzou conditions (2) was constructed in [1] in formSuch obey simpler but more restrictive commutation relations [1, 2]:for which (1) leads to the Duffin-Kemmer-Petiau (DKP) theory of spin and mesons; see [1–3]. This -dimensional representation (3) of matrices (denoted as ) is reducible and can be decomposed as . Representation (spin case) is realized in terms of matrices, while representation (spin ) involves matrices; see [1, 23, 24].
In the case of (spin-) representation of matrices (1) is equivalent to the following set of equations:if we define in (1) aswhere denotes transposition of a matrix [19].
In the case of (spin-) representation of matrices (1) reduces towith the following definition of in (1):where are real and are purely imaginary [19]. Because of antisymmetry of we have and this is spin 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 matrices; see [23, 24] for a review. For example, there are two representations for which the corresponding matrices yield the Hagen-Hurley equations for spin bosons [16–18]; see also [19, 23, 24]. There are also two sets of matrices 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 spin 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 transformation which transforms the wave-function (6) into the spinor form:i.e., involving spinor components rather than vector ones ; see [27] for the definition of .
Then we apply two projection operators, and , projecting onto three-dimensional subspaces, to define new matrices:The matrices which can be written in explicit compact form asfulfilling the Tzou commutation relations (2). Matrices can be immediately read off from (12) as they are proportional to ’s. This result was obtained in [22] but for the whole operator only and not for individual matrices .
While these matrices belong to reducible representation of the Tzou algebra (2), they induce the irreducible representation :
The matrices lead to another irreducible representation of the Tzou algebra. These two representations 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 transformation which transforms the wave-function (8), , into the form involving self-dual and antiself-dual tensors, and . More exactly, and , where a tensor dual to a given antisymmetric tensor is defined as and , . Self-dual and antiself-dual tensors and are directly related to symmetric spinors and [27, 31, 32].
Next, we apply projection operators, and , projecting onto seven-dimensional subspaces, to define new matrices:The matrices and are reducible representations of Tzou algebra (2) and induce irreducible representations . For example, the matrices induced by can be written in explicit compact form as
Equation is unitarily equivalent to the Hagen-Hurley equation [19].
5. Summary
We have constructed transformations of the Duffin-Kemmer-Petiau matrices, corresponding to representations and , inducing matrices of representations and 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, representation appearing in the Hagen-Hurley theory of spin bosons [16–18] arises in the process of splitting the DKP equation written for representation of 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 only; see in [22].
Data Availability
The manuscript is self-contained, no additional data have been necessary.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.