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We use a five-dimensional approach to Galilean covariance to investigate the non-relativistic Duffin-Kemmer-Petiau first-order wave equations for spinless particles. The corresponding representation is generated by five 6 × 6 matrices. We consider the harmonic oscillator as an example.

We analyse the Poincare gauge structure proposed by D. Cangemi and R. Jackiw (CJ) (Ann. Phys. (N.Y.) 225, 229), in association with general Lie algebras and, in particular, with Galilean symmetries. In this context, the CJ method is formulated as an embedding scheme of metric spaces, and aspects of Galilean covariance are then used to analyse: (i) the nonrelativistic limits of the electromagnetic field; (ii) a Galilean counterpart of the CJ theory; (iii) geometrical common structures of Lorentzian and Galilean physics; and (iv) a covariant formalism for classical mechanics. Academic Press

We apply a five-dimensional formulation of Galilean covariance to construct non-relativistic Bhabha first-order wave equations which, depending on the representation, correspond either to the well known Dirac equation (for particles with spin 1/2) or the Duffin-Kemmer-Petiau equation (for spinless and spin 1 particles). Here the irreducible representations belong to the Lie algebra of the 'de Sitter group' in 4 + 1 dimensions, SO(5, 1). Using this approach, the nonrelativistic limits of the corresponding equations are obtained directly, without taking any low-velocity approximation. As a simple illustration, we discuss the harmonic oscillator.

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