Bhabha first-order wave equations. III. Poincaré generators
About: This article is published in Physical Review D.The article was published on 1975-03-15. It has received 19 citations till now. The article focuses on the topics: Spin-½ & Wave equation.
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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 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
TL;DR: In this paper, the authors derived a set of general inverse and ST operators for arbitrary-spin Bhabha fields and decouple the "particle components" from the "subsidiary components" in the Hamiltonian equations for integer spin (where, as was the case for DKP, they find that their solution is an identity in terms of the particle-components solution).
Abstract: Beginning with the Bhabha first-order wave equation of maximum spin 1 [the Duffin-Kemmer-Petiau (DKP) equation], where Sakata and Taketani (ST) separated out the "particle components" from the built in "subsidiary components," we derive for the first time the Hamiltonian equation for the "subsidiary components," and show that its solution is an identity in terms of the particle-components solution. We then derive a set of general inverse and ST operators for arbitrary-spin Bhabha fields. With these generalized operators we can discuss and understand the mass and spin composition of a general Bhabha so(5) field, it being a particular sum of [$2\ifmmode\times\else\texttimes\fi{}(2S+1)$] components for each particular mass and spin ($S$) state, as well as built in "subsidiary components" for integer spin. We then can use these general inverse and ST operators to (a) derive the general Bhabha Hamiltonian for arbitrary spin, (b) decouple the "particle components" from the "subsidiary components" in the Hamiltonian equations for integer spin (where, as was the case for DKP, we find that the Hamiltonian "subsidiary components" solution is an identity in terms of the particle-components solution), and (c) decouple the $\mathcal{S}+\frac{1}{2} (\mathcal{S})$ different mass states for half-integer (integer) spin. We discuss the physical implications of this observation and other aspects of our results.
31 citations
TL;DR: In this paper, it was shown that the arbitrary-spin Bhabha fields with minimal electromagnetic coupling are causal in both the c-number and q-number theories, and the KG divisors in closed form were obtained in terms of the elementary symmetric functions.
Abstract: It is demonstrated that the arbitrary-spin Bhabha fields with minimal electromagnetic coupling are causal in both the c-number and q-number theories. The Klein-Gordon (KG) divisors in closed form are first obtained in terms of the elementary symmetric functions. c-number causality is easily demonstrated for half-integer spin with the Velo-Zwanziger method and integer spin by using Wightman's suggestion involving the KG divisors. For the q-number demonstration, an indefinite-metric second-quantized formalism is set up, and the above KG divisors are used to show causality in closed form for arbitrary spin. In both the c-number and q-number theories a special handling of the integer-spin subsidiary components is necessary. Discussion focuses on the Bhabha indefinite metric and on the connection between the number of derivatives in a theory and the occurrence or nonoccurence of causality. (AIP)
25 citations
TL;DR: In this article, the Lagrange multiplier was used to separate the subsidiary conditions from the equations of motion in a consistent way by means of a Lagrange multiplicative multiplier, which may be the solution to various pathologies afflicting high-spin field theories.
Abstract: It is proposed that the method of keeping the subsidiary conditions separate from the equations of motion in a consistent way by means of a Lagrange multiplier may be the solution to the various pathologies afflicting high-spin field theories. The distinct features of this approach are discussed with reference to a massive spin-3/2 field. It is shown that in this formulation there is no causality violation for propagation in an external field and the energy spectrum in a homogeneous magnetic field is real. Quantization is carried out in a Hilbert space with indefinite metric. For minimal electromagnetic interaction a unitary S matrix is constructed by introducing a fictitious particle and an additional vertex.
21 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, a canonical transformation on the Dirac Hamiltonian for a free particle is obtained in which positive and negative energy states are separately represented by two-component wave functions.
Abstract: By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and $c$. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
1,715 citations
919 citations
TL;DR: In this article, the picture of an elementary relativistic quantum-mechanical particle can now be roughly outlined as follows: the "exact theory" is one of quantized waves, the particle characteristics appearing as consequences of the noncommutation of the wave amplitudes.
Abstract: 1. Introduction For many years a central problem of theoretical physics has been to set up a satisfactory relativistic theory of elementary particles. This problem is yet far from solution, the notorious occurrence of infinite self-energies and similar divergencies having hitherto frustrated all attempts at complete formulation. Nevertheless, definite advances towards the understanding of the general problem have recently been made, not so much by improvement of the theory as by a more detailed study of all its possible types and variants and the resulting clarification of the essential underlying principles. The picture of an elementary relativistic quantum-mechanical “particle ” can now be roughly outlined as follows: The “exact theory” is one of quantized waves, the particle characteristics appearing as consequences of the non-commutation of the wave amplitudes. There exist possible theories with any given integral or half-integral value s of the “ spin” of the particle, the number of independent states of polarization of the corresponding waves then being 2 s + 1 = N (Fierz 1939; Fierz and Pauli 1939).
482 citations
433 citations