# On the Wave Equation of Meson

01 Jan 1955-Progress of Theoretical Physics Supplement (Oxford University Press)-Vol. 1, Iss: 1, pp 84-97

About: This article is published in Progress of Theoretical Physics Supplement.The article was published on 1955-01-01 and is currently open access. It has received 10 citations till now. The article focuses on the topics: Wave packet & Wave equation.

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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.

45 citations

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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

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TL;DR: In this paper, the most general physically admissible positive-definite inner product on the space of Proca fields was constructed, up to a trivial scaling, which defines a five-parameter family of Lorentz invariant inner products that were used to construct a genuine Hilbert space for the quantum mechanics of the Proca field.

Abstract: We construct the most general physically admissible positive-definite inner product on the space of Proca fields. Up to a trivial scaling this defines a five-parameter family of Lorentz invariant inner products that we use to construct a genuine Hilbert space for the quantum mechanics of Proca fields. If we identify the generator of time translations with the Hamiltonian, we obtain a unitary quantum system that describes first-quantized Proca fields and does not involve the conventional restriction to the positive-frequency fields. We provide a rather comprehensive analysis of this system. In particular, we examine the conserved current density responsible for the conservation of the probabilities, explore the global gauge symmetry underlying the conservation of the probabilities, obtain a probability current density, construct position, momentum, helicity, spin, and angular momentum operators, and determine the localized Proca fields. We also compute the generalized parity (P), generalized time-reversal ...

24 citations

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TL;DR: In this article, the most general physically admissible positive-definite inner product on the space of Proca fields was constructed, and a five-parameter family of Lorentz invariant inner products were used to construct a genuine Hilbert space for the quantum mechanics of the Proca field.

Abstract: We construct the most general physically admissible positive-definite inner product on the space of Proca fields. Up to a trivial scaling this defines a five-parameter family of Lorentz invariant inner products that we use to construct a genuine Hilbert space for the quantum mechanics of Proca fields. If we identify the generator of time-translations with the Hamiltonian, we obtain a unitary quantum system that describes first-quantized Proca fields and does not involve the conventional restriction to the positive-frequency fields. We provide a rather comprehensive analysis of this system. In particular, we examine the conserved current density responsible for the conservation of the probabilities, explore the global gauge symmetry underlying the conservation of the probabilities, obtain a probability current density, construct position, momentum, helicity, spin, and angular momentum operators, and determine the localized Proca fields. We also compute the generalized parity ($\cP$), generalized time-reversal ($\cT$), and generalized charge or chirality ($\cC$) operators for this system and offer a physical interpretation for its $\cP\cT$-, $\cC$-, and $\cC\cP\cT$-symmetries.

24 citations

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19 citations