A note on the Duffin-Kemmer-Petiau equation in (1+1) space-time dimensions

Abstract

In the last years several papers addressed the supposed spin-1 sector of the massive Duffin-Kemmer-Petiau (DKP) equation restricted to (1+1) space-time dimensions. In this note we show explicitly that this is a misleading approach, since the DKP algebra in (1+1) dimensions admits only a spin-0 representation. Our result also is useful to understand why several recent papers found coincident results for both spin-0 and spin-1 sectors of the DKP theory in (3+1) dimensions when the dynamics is restricted to one space dimension.The Duffin-Kemmer-Petiau (DKP) equation is a first order wave equation similar to the Dirac one, which in its original formulation in (3+1) space-time dimensions describes spin-0 and spin-1 fields or particles 1-5 . In recent years some papers addressed the DKP equation in strict (1+1) space-time dimensions in situations involving interactions and addressed the supposed spin-1 sector of the theory 6-9 . In this note we show explicitly that such an approach is misleading; by using the (1+1)-dimensional analogs of the original DKP spin-0 and spin-1 projection operators we show that the supposed "spin-1" sector of the theory restricted to (1+1) dimensions actually is unitarily equivalent to its spin-0 sector, which describes a (pseudo)scalar field. We illustrate this equivalence by explicitly building the lowest dimensional (irreducible) representation of the theory. At the end we comment how our result explain why several authors in recent years found identical results for both the spin-0 and spin-1 sectors of the (3+1) dimensional DKP equation when the dynamics is restricted to only one space dimension 10-20 .DKP equation in (3+1) dimensions. We start by recalling some basic results about the free DKP equation in (3+1) space-time dimensions. The equation is given by 1-5 (we use natural units = c = 1)where m is the particle's mass, ψ is the DKP wave function and β µ are matrices satisfying the DKP algebrawhere g µν is the Minkowski metric tensor in (3+1) dimensions with signature (+, −, −, −).It is well known that there are only three irreducible representations (irrep's) of DKP algebra in (3+1) dimensions: one is trivial, having dimension 1, and the other two are nontrivial, having dimensions 5 and 10, corresponding respectively to scalar (spin-0) and vector (spin-1) fields 4,21,22 . Under infinitesimal Lorentz transformations x ′µ = Λ µ ν x ν , with Λ µν = g µν + ω µν , ω µν = −ω νµ , the DKP spinor ψ transforms as ψ → U ψ, where 5 U = 1 + 1 2 ω µν S µν , S µν = [β µ , β ν ] .

Duffin-Kemmer-Petiau oscillator in topologically charged Ellis-Bronnikov-type wormhole

“…In addition, spin-0 system of the DKP equation in the presence of Gödel-type space-time [16] and in a topologically trivial space-time background [17], and the DKP oscillator in a Gödel-type space-time [18] were studied. The equivalency of the spin-one and spin-zero representation of the DKP equation in one dimension was discussed by Lunardi in [19]. Some inaccurate discussion in the literature based on the DKP equations in one dimension was pointed out by Chargui in [20].…”

Investigation of the DKP Equation for A Two-Dimensional Black Hole

“…An alternative way to overcome this problem is to evaluate the DKP equation in the lower dimensions in order to obtain the behavior of semi-classical and quantum gravity. In the literature, we find many studies related to the general relativity in (1 + 1)-dimensions [12][13][14][15].…”

The explicit relation between the DKP equation and the Klein-Gordon equation

“…Boutabia-Chéraitia et al [11] presented a calculation of the Green's function of the DKP equation in the case of scalar and vectorial particles interacting with a square barrier potential and its relation to the KG equation. Recently, Lunardi [12] has shown that the supposed spin-1 sector of the theory restricted to (1 + 1) space-time dimensions actually is unitarily equivalent to its spin-0 sectors. This work is organized as follows: In Section 2, we give the DKP equation for the spin-0 with 5 × 5 beta matrices and 10 × 10 beta matrices for the spin-1.…”