Bhabha first-order wave equations: I. C, P, and T
TL;DR: In this paper, the first-order Bhabha transformation matrices for the Dirac field were derived in various representations, including the transformation matrix for the first order wave equations for arbitrary spin, of which the DKP and Duffin-Kemmer-Petiau (DKP) are special examples.
Abstract: We discuss properties of Bhabha first-order wave equations for arbitrary spin, of which the Dirac and Duffin-Kemmer-Petiau (DKP) equations are special examples. The $C$, $P$, and $T$ transformation matrices for the Dirac field are reviewed in various representations, and the $C$, $P$, and $T$ transformation matrices for the DKP and general Bhabha cases are then derived. The Bhabha transformation matrices are polynomials of order $2\mathcal{S}$ in the algebra matrices, where $\mathcal{S}$ is the maximum spin of a particular Bhabha algebra. For the cases $\mathcal{S}=1 \mathrm{and} \frac{1}{2}$ they reduce to the DKP and Dirac transformation matrices. We also discuss $C$, $P$, and $T$ for the Sakata-Taketani (ST) reduction of the DKP equation, and explicitly exhibit the "subsidiary component" ST Hamiltonian equation, as well as the known "particle component" ST equation. Throughout we emphasize that physical insight which can be gained from the use of the first-order Bhabha formalism, including a possible connection between meson nonconservation and $\mathrm{CP}$ violation.
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TL;DR: In this article, the authors consider the use of the Duffin-Kemmer-Petiau (DKP) relativistic equation for practical calculations and derive the system of first-order coupled differential radial equations which enable the energy eigenvalues as well as the full wavefunctions to be evaluated.
Abstract: In view of recent interest in the use of the Duffin-Kemmer-Petiau (DKP) relativistic equation, the authors consider some of its properties which are needed for practical calculations. They also address the unresolved problem of the spinless DKP boson in a central field and derive the system of first-order coupled differential radial equations which enable the energy eigenvalues as well as the full wavefunctions to be evaluated. As an example they calculate the free DKP spherical waves and then solve the problem of a pionic atom with a point Coulomb interaction only.
94 citations
TL;DR: In this article, scalar bosons are described by the Duffin-Kemmer-Petiau (DKP) formalism and the effects of topological defect in the equation of motion, energy spectrum, and DKP spinor are analyzed and discussed in detail.
Abstract: The quantum dynamics of scalar bosons embedded in the background of a cosmic string is considered. In this work, scalar bosons are described by the Duffin–Kemmer–Petiau (DKP) formalism. In particular, the effects of this topological defect in the equation of motion, energy spectrum, and DKP spinor are analyzed and discussed in detail. The exact solutions for the DKP oscillator in this background are presented in closed form.
58 citations
TL;DR: In this paper, a generic S = 1 relativisitic oscillator model is proposed, which extends the class of relativistic bosonic oscillators and can be realized within this generic model.
Abstract: 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.
49 citations
TL;DR: In this article, the authors simplify and clarify the mathematical language used to express quaternionic quantum mechanics (QQM) and obtain a rapid QM counterpart of standard quantum mechanical results.
Abstract: This paper is an attempt to simplify and clarify the mathematical language used to express quaternionic quantum mechanics (QQM). In our quaternionic approach the choice of “complex” geometries allows an appropriate definition of momentum operator and gives the possibility to obtain consistent formulations of standard theories. Barred operators represent the key to realizing a set of translation rules between quaternionic and complex quantum mechanics (QM). These translations enable us to obtain a rapid quaternionic counterpart of standard quantum mechanical results.
39 citations
TL;DR: In this article, the authors 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 equations (for spinless and spin 1 particles).
Abstract: 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 non-relativistic limits of the corresponding equations are obtained directly, without taking any low-velocity approximation. As a simple illustration, we discuss the harmonic oscillator.
39 citations
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TL;DR: In this article, it was shown that the incompleteness of the previous theories lying in their disagreement with relativity or, alternatetically, with the general transformation theory of quantum mechanics leads to an explanation of all duplexity phenomena.
Abstract: The new quantum mechanics, when applied to the problem of the structure of the atom with point-charge electrons, does not give results in agreement with experiment. The discrepancies consist of “duplexity ” phenomena, the observed number of stationary states for an electron in an atom being twice the number given by the theory. To meet the difficulty, Goudsmit and Uhlenbeck have introduced the idea of an electron with a spin angular momentum of half a quantum and a magnetic moment of one Bohr magneton. This model for the electron has been fitted into the new mechanics by Pauli,* and Darwin,† working with an equivalent theory, has shown that it gives results in agreement with experiment for hydrogen-like spectra to the first order of accuracy. The question remains as to why Nature should have chosen this particular model for the electron instead of being satisfied with the point-charge. One would like to find some incompleteness in the previous methods of applying quantum mechanics to the point-charge electron such that, when removed, the whole of the duplexity phenomena follow without arbitrary assumptions. In the present paper it is shown that this is the case, the incompleteness of the previous theories lying in their disagreement with relativity, or, alternatetively, with the general transformation theory of quantum mechanics. It appears that the simplest Hamiltonian for a point-charge electron satisfying the requirements of both relativity and the general transformation theory leads to an explanation of all duplexity phenomena without further assumption. All the same there is a great deal of truth in the spinning electron model, at least as a first approximation. The most important failure of the model seems to be that the magnitude of the resultant orbital angular momentum of an electron moving in an orbit in a central field of force is not a constant, as the model leads one to expect.
3,034 citations
TL;DR: In this paper, the possibilita di pervenire a piena simmetrizzazione formale della teoria quantistica dell’elettrone e del positrone facendo uso di un nuovo processo di quantizzazione.
Abstract: Si dimostra la possibilita di pervenire a una piena simmetrizzazione formale della teoria quantistica dell’elettrone e del positrone facendo uso di un nuovo processo di quantizzazione. Il significato delle equazioni di
Dirac
ne risulta alquanto modificato e non vi e piu luogo a parlare di stati di energia negativa; ne a presumere per ogni altro tipo di particelle, partieolarmente neutre, l’esistenza di « antiparticelle » corrispondenti ai « vuoti » cut energia negativa.
1,444 citations
TL;DR: In this paper, the authors show that the Gleichungen der elektrischen Teilchen nehmen hierbei auch in elektromagnetisehen Feldern the Gestalt von Gleichingsen geodischen Linien an.
Abstract: Auf den folgenden Seiten mSchte ich auf einen einfachen Zusammenhang hin- weisen zwischen der yon Kaluza 1) vorgeschlagencn Theorie filr den Zusammen- hang zwischen Elektromagnetismus und Gravitation cinerseits und der von de Broglic 2) und SchrSdinger 3) angegebenen Methodc zur Behandlung der Quantenprobleme andererseits. Die Theorie yon Kaluza geht darauf hinaus, die zehn Einsteinsehen Gravitationspotentiale gik und die vicr elektromagnetisehen Potentiale ~i in Zusammenhang zu bringen mit den Koeffizienten Yi~ eines Linienelementes von einem Riemannschen Raum~ der atdler den vier gewOhn- lichen Dimensionen noch eine ffinfte Dimension enth~ilt. Die Bewegungsgleichungen der elektrischen Teilchen nehmen hierbei auch in elektromagnetisehen Feldern die Gestalt von Gleichungen geod~tiseher Linien an. Wenn dieselben als Strahlen- gleichungen gedeutet werden, indem die Materie als eine Art Wellenausbreitun~ betraehtet wird, kommt man fast von selbst zu einer partiellen Differential- gleichung zweiter Ordnung, die als eine Verallgemeinerung der gewShnlichen Wellengleiehung angesehen werden kann. Werden nun solche LSsungen dieser Gleichung betrachtct, bei denen die f[infte Dimension rein harmonisch auftritt mit einer bestimmten mit der Planekschen Konstante zusammenh~ngenden Periode, so kommt man eben zu den obenerw~hnten quantentheoretischen ~ethoden.
1,337 citations
TL;DR: In this article, the authors show that the Gleichungen der elektrischen Teilchen nehmen hierbei auch in elektromagnetisehen Feldern the Gestalt von Gleichingsen geodischen Linien an.
Abstract: Auf den folgenden Seiten mSchte ich auf einen einfachen Zusammenhang hin- weisen zwischen der yon Kaluza 1) vorgeschlagencn Theorie filr den Zusammen- hang zwischen Elektromagnetismus und Gravitation cinerseits und der von de Broglic 2) und SchrSdinger 3) angegebenen Methodc zur Behandlung der Quantenprobleme andererseits. Die Theorie yon Kaluza geht darauf hinaus, die zehn Einsteinsehen Gravitationspotentiale gik und die vicr elektromagnetisehen Potentiale ~i in Zusammenhang zu bringen mit den Koeffizienten Yi~ eines Linienelementes von einem Riemannschen Raum~ der atdler den vier gewOhn- lichen Dimensionen noch eine ffinfte Dimension enth~ilt. Die Bewegungsgleichungen der elektrischen Teilchen nehmen hierbei auch in elektromagnetisehen Feldern die Gestalt von Gleichungen geod~tiseher Linien an. Wenn dieselben als Strahlen- gleichungen gedeutet werden, indem die Materie als eine Art Wellenausbreitun~ betraehtet wird, kommt man fast von selbst zu einer partiellen Differential- gleichung zweiter Ordnung, die als eine Verallgemeinerung der gewShnlichen Wellengleiehung angesehen werden kann. Werden nun solche LSsungen dieser Gleichung betrachtct, bei denen die f[infte Dimension rein harmonisch auftritt mit einer bestimmten mit der Planekschen Konstante zusammenh~ngenden Periode, so kommt man eben zu den obenerw~hnten quantentheoretischen ~ethoden.
1,188 citations
919 citations