Relativistic Spin-One Boson Plasma

Abstract

We derive the polarization tensor for a system of charged spin-one bosons and antibosons in the case of no external magnetic field. This requires a thorough exposition of relativistic spin-one quantum mechanics, and thus we initially focus upon the Sakata-Taketani equation and its free-field solutions. We employ these results to evaluate the matrix elements required for the calculation of the polarization tensor, which itself is derived via the self-consistent random-phase approximation (RP A) method. It is from this tensor that we obtain the longitudinal and transverse dielectric response functions for this plasma. We evaluate these response functions at zero temperature, and exhibit the characteristic modes of oscillation. Finally, we discuss possible generalizations of this work, in particular to a finite-temperature plasma, and to one with an external magnetic field. In this paper, we present a study of the relativistic spin-one particle-antiparticle plasma, in the presence of no external magnetic field. The course of the investigation begins with a review of single particle theories of spin-one (vector) bosons, which we present in § 2 of this work. In § 3 of the paper, we present a detailed treatment of the six-component formalism for spin-one bosons, first developed by Sakata and Taketani/) which we then employ in our linear response theory calculations for the plasma. To our knowledge, this particular formalism is under-represented in the literature pertinent to spin-one bosons, and hence our exposition is an attempt to clarify its general features, and underscore its particular utility in work of the nature we have undertaken. The latter part of the paper (§§ 4 and 5) contains the linear response calculations proper. We set about deriving the polarization four-tensor, employing a method first· proposed by Harris,z) this being the self-consistent random-phase approximation (RPA) method. We then obtain the characteristic modes of oscillation of the plasma. At present, we concern ourselves solely with the plasma properties of the polari­ zation tensor, leaving a complete study of the vacuum modes of oscillation and their renormalization to a later paper, in which we also propose to introduce the presence of an external magnetic field. Employing our plane-wave solutions of the Sakata­ Taketani equation for the case of no external fields, we proceed to calculate the longitudinal and transverse dielectric response functions, which are obtained via the employment of the relationship between the covariant polarization four-tensor and the dielectric three-tensor. In the case of the longitudinal response function, we present the formal result which is valid for all temperatures, and we then evaluate the longitudinal and transverse response functions at zero temperature, from which we obtain the modes of oscillation. Our work follows on from that of Kowalenko, Frankel and Hines (KFH),3) who studied the spin-zero pair plasma by employing a self-consistent field method to find