The 2(2S + 1)-formalism and its connection with other descriptions

Abstract

In the framework of the Joos-Weinberg 2(2S + 1)-theory for massless particles, the dynamical invariants have been derived from the Lagrangian density which is considered to be a 4-vector. A la Majorana interpretation of the 6-component "spinors'", the field operators of S = 1 particles, as the left-and right-circularly polarized radiation, leads us to the conserved quantities which are analogous to those obtained by Lipkin and Sudbery. The scalar Lagrangian of the Joos-Weinberg theory is shown to be equivalent to the Lagrangian of a free massless field, introduced by Hayashi. As a consequence of a new "gauge" invariance this skew-symmetric field describes physical particles with the longitudinal components only. The interaction of the spinor field with the Weinberg's 2(2S + 1)-component massless field is considered. New interpretation of the Weinberg field function is proposed.

On Some Geometrical Aspects of Space-Time Description and Relativity

“…Here, due to the celebration of Valeriy's anniversary and his great work on rep theories and equations of motion, we do not want to carry coals to Newcastle, so we skip most rep details here, but relate later (see sec. IV B) few of our aspects and identifications to Valeriy's overviews [27], [28], [29], [30], [31] on different reps and formalisms, and especially the 2(2S+1)-Formalism, or (S, 0) ⊕ (0, S)-reps, with S = 1.…”

“…For now, we want to mention (as the first and as an early example) only the relation to generating lines of the one-sheeted hyperboloid 27 . Each generating family of lines of the hyperboloid can be derived from three Complexe, so by Lie transfer [20] or by the differential rep discussed in appendix E, we may relate as well the two operator sets L i and K j , each comprising three generators 28 .…”

“…10. Gilmore's detailed approach by coset spaces as well as Helgason's marvelous work [38], [39] can be seen as a mechanism and framework to comprise, generalize and supersede Wigner's approach by boosts [85] which has been used by Weinberg's formalism ( [81], [82], [83], and see also [27], [28], [29], [30], and [31]). We have summarized some aspects in II F later in order to identify the operators and the reductive algebra structure.…”

“…Especially Valeriy has published a lot of great work on various aspects of the different rep theories, so we feel free to skip most of the discussion here. Instead, we focus in this subsection on the overlapping conceptual aspects which are relevant for our later discussion here, and as such, feeling guided by Valeriy's papaers [27], [28], [29], and [30], we want to refer here only to Weinberg's presentations [81], [82], and [83] on spin reps. Why?…”

“…For our considerations later, the second aspect of Weinberg's 'idea' has already been re-presented in Valeriy's overviews appropriately 58 ([27], [28], [29], and [30]) by rearranging the field components E and B into a 'sixspinor' Ψ according to…”