Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Galilean covariance and the Duffin-Kemmer-Petiau equation

M. de Montigny, F. C. Khanna, Ademir Eugenio de Santana, E. S. Santos, J. D. M. Vianna

Published Jul 28, 2000• 80 citations
Publisher

Abstract

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.
DOI
10.1088/0305-4470/33/31/102

Related papers

On the Galilean Duffin–Kemmer–Petiau equation in arbitrary dimensions

M. de Montigny, E. S. Santos

We obtain the representations of the Galilean covariant Duffin–Kemmer–Petiau equation in an arbitrary number of dimensions. Their purpose is to facilitate the study of nonrelativistic many-body systems with spinless and spin-one fields. A Galilean covariant formalism exploits the tensor structure of relativistic Lorentz-invariant theories by adding one extra spacelike coordinate and working with light-cone coordinates.

International Journal of Modern Physics A•Jun 17, 2020•2 citations

Galilean Duffin Kemmer Petiau algebra and symplectic structure

M.C.B. Fernandes, Ademir Eugenio de Santana, J. D. M. Vianna

We develop the Duffin–Kemmer–Petiau (DKP) approach in the phase-space picture of quantum mechanics by considering DKP algebras in a Galilean covariant context. Specifically, we develop an algebraic calculus based on a tensor algebra defined on a five-dimensional space which plays the role of spacetime background of the non-relativistic DKP equation. The Liouville operator is determined and the Liouville–von Neumann equation is written in two situations: the free particle and a particle in an external electromagnetic field. A comparison between the non-relativistic and the relativistic cases is commented.

Journal of Physics A Mathematical and General•Mar 19, 2003•34 citations

A generalized Duffin-Kemmer-Petiau oscillator

Youcef Nedjadi, Roger C. Barrett

We propose a generic S = 1 relativisitic oscillator model which extends the class of relativistic bosonic oscillators. The Duffin-Kemmer-Petiau (DKP) oscillator we introduced in an earlier work can be recovered as an element of a family of DKP oscillators that can be realized within this generic model. We present the formalism for the exact quantum mechanical treatment of this generic model and, for illustration, compute the eigenvalues of a particular family of relativistic oscillators.

Journal of Physics A Mathematical and General•Aug 7, 1998•42 citations

Galilean Covariance and the Schrödinger Equation

M. Omote, S. Kamefuchi, Yasushi Takahashi +1 more

Galilei transformations are formulated in a 5-dimensional form and on such basis the non-relativistic quantum mechanics is reconstructed. Our approach has the following advantages over others'. (1) The Schrödinger wavefunctions are taken to be vector (not projective) representations of the Galilei group, so that (2) the problem here is considerably simplified. (3) The formulation proceeds in a way quite parallel to that of relativistic quantum mechanics, and what is most important, (4) Galilei covariance is manifest throughout.

Fortschritte der Physik/Progress of Physics•Jan 1, 1989•59 citations

ON THE NONRELATIVISTIC LIMIT OF LINEAR WAVE EQUATIONS FOR ZERO AND UNITY SPIN PARTICLES

P. Yu. Moshin, Jeferson Tomazelli

The nonrelativistic limit of the linear wave equation for zero and unity spin bosons of mass m in the Duffin–Kemmer–Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy–Wouthuysen canonical transformation for a relativistic electron. The interacting case is also analyzed, by considering a power series expansion of the transformed Hamiltonian, thus demonstrating that all features of particle dynamics can be recovered if corrections of order 1/m2 are taken into account through a recursive iteration procedure.

arXiv:0704.3462•Modern Physics Letters A•Jan 20, 2008•7 citations

Spin-1/2 current density in relativistic quantum mechanics

Ole Keller

In the search for a relativistic wave equation for a free particle of rest mass m one must demand that the energy(E)-momentum (p) [dispersion] relation E2 = m2c4 + p2c2 (25.1) is satisfied. By the substitutions E → i~ ∂ ∂t , p→ −i~∇ (25.2) in Eq. (25.1) one obtains an operator, which when acting on a scalar wave function, gives us the Klein-Gordon wave equation for a spinless particle. The Klein-Gordon equation is of second order in the space (r) and time (t) coordinates. If one seeks a covariant first-order differential equation, r and t must play symmetric roles. A natural starting point therefore would be a linear dispersion relation of the form E = βmc2 + cα · p, (25.3) where β and α = (α1, α2, α3) are real and dimensionless. Since Eq. (25.3) must be compatible with Eq. (25.1), squaring of Eq. (25.3) leads to the following relations [already given in Eqs. (10.77)-(10.79)]: β2 = 1, (25.4) αiβ + βαi = 0, (25.5) αiαj + αjαi = 2δij , (25.6) where i, j = 1, 2, 3 (x, y, z). It is clear from Eqs. (25.4)-(25.6) that β and α cannot be numbers, but one can find matrices which satisfy these equations. Since the Hamiltonian appearing upon a substitution of the connections in (25.2) into Eq. (25.3) must be Hermitian, the αi and β matrices also have to be Hermitian, i.e., α†i = αi, i = 1− 3, (25.7) β† = β. (25.8) One can show that the matrices must have even rank and that the rank must be at least 4, see, e.g., [88]. The Dirac equation, which describes the quantum mechanics of a spin1/2 elementary particle, e.g., an electron, relates to a four-dimensional realization of Eqs. (25.4)-(25.8). Representations of a given rank are not unique.

CRC Press eBooks•May 13, 2014•0 citations
We develop the Duffin–Kemmer–Petiau (DKP) approach in the phase-space picture of quantum mechanics by considering DKP algebras in a Galilean covariant context. Specifically, we develop an algebraic calculus based on a tensor algebra defined on a five-dimensional space which plays the role of spacetime background of the non-relativistic DKP equation. The Liouville operator is determined and the Liouville–von Neumann equation is written in two situations: the free particle and a particle in an external electromagnetic field. A comparison between the non-relativistic and the relativistic cases is commented.
Galilei transformations are formulated in a 5-dimensional form and on such basis the non-relativistic quantum mechanics is reconstructed. Our approach has the following advantages over others'. (1) The Schrödinger wavefunctions are taken to be vector (not projective) representations of the Galilei group, so that (2) the problem here is considerably simplified. (3) The formulation proceeds in a way quite parallel to that of relativistic quantum mechanics, and what is most important, (4) Galilei covariance is manifest throughout.
The nonrelativistic limit of the linear wave equation for zero and unity spin bosons of mass m in the Duffin–Kemmer–Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy–Wouthuysen canonical transformation for a relativistic electron. The interacting case is also analyzed, by considering a power series expansion of the transformed Hamiltonian, thus demonstrating that all features of particle dynamics can be recovered if corrections of order 1/m2 are taken into account through a recursive iteration procedure.
In the search for a relativistic wave equation for a free particle of rest mass m one must demand that the energy(E)-momentum (p) [dispersion] relation E2 = m2c4 + p2c2 (25.1) is satisfied. By the substitutions E → i~ ∂ ∂t , p→ −i~∇ (25.2) in Eq. (25.1) one obtains an operator, which when acting on a scalar wave function, gives us the Klein-Gordon wave equation for a spinless particle. The Klein-Gordon equation is of second order in the space (r) and time (t) coordinates. If one seeks a covariant first-order differential equation, r and t must play symmetric roles. A natural starting point therefore would be a linear dispersion relation of the form E = βmc2 + cα · p, (25.3) where β and α = (α1, α2, α3) are real and dimensionless. Since Eq. (25.3) must be compatible with Eq. (25.1), squaring of Eq. (25.3) leads to the following relations [already given in Eqs. (10.77)-(10.79)]: β2 = 1, (25.4) αiβ + βαi = 0, (25.5) αiαj + αjαi = 2δij , (25.6) where i, j = 1, 2, 3 (x, y, z). It is clear from Eqs. (25.4)-(25.6) that β and α cannot be numbers, but one can find matrices which satisfy these equations. Since the Hamiltonian appearing upon a substitution of the connections in (25.2) into Eq. (25.3) must be Hermitian, the αi and β matrices also have to be Hermitian, i.e., α†i = αi, i = 1− 3, (25.7) β† = β. (25.8) One can show that the matrices must have even rank and that the rank must be at least 4, see, e.g., [88]. The Dirac equation, which describes the quantum mechanics of a spin1/2 elementary particle, e.g., an electron, relates to a four-dimensional realization of Eqs. (25.4)-(25.8). Representations of a given rank are not unique.