Paul BaldufFor Sunday, my #paperOfTheDay is "On spin three self interactions" from 1984. It had been known since Fierz and Pauli in the 1930s what the Lagrangian density for a massless free field of arbitrary integer spin s looks like: The field variable is a rank-s tensor, and its kinetic term involves certain second derivatives and tensor contractions.
The present paper is a very clear and instructive exposition of how this fact restricts the form of possible interactions: 1. From the known free-field Lagrangian, one concludes that not all entries of the field tensor are independent, but instead, they satisfy certain conditions, expressible by a differential operator (think spin 1: del_mu A^mu=0). 2. These identities imply that the field has a gauge invariance. 3. Expand an interaction in powers of the field. The lowest order involves three fields. Gauge invariance completely fixes its functional form, up to redefinitions of the field variable. In particular, a 3-valent interaction vertex for spin-s fields contains s derivatives. 4. This can be iterated for interaction terms with more than 3 field variables. 5. Additionally, one obtains a commutator relation for the gauge transformation, i.e. its Lie algebra structure. In principle, this determines what the gauge symmetry is, macroscopically, if one can recognize the commutator as the Lie bracket of some known group.
The authors apply this programme to the spin-3 field. The resulting expressions are somewhat large, and it's not entirely clear how to interpret them intuitively, but still I like the clarity of their exposition. https://link.springer.com/article/10.1007/BF01410362
The present paper is a very clear and instructive exposition of how this fact restricts the form of possible interactions: 1. From the known free-field Lagrangian, one concludes that not all entries of the field tensor are independent, but instead, they satisfy certain conditions, expressible by a differential operator (think spin 1: del_mu A^mu=0). 2. These identities imply that the field has a gauge invariance. 3. Expand an interaction in powers of the field. The lowest order involves three fields. Gauge invariance completely fixes its functional form, up to redefinitions of the field variable. In particular, a 3-valent interaction vertex for spin-s fields contains s derivatives. 4. This can be iterated for interaction terms with more than 3 field variables. 5. Additionally, one obtains a commutator relation for the gauge transformation, i.e. its Lie algebra structure. In principle, this determines what the gauge symmetry is, macroscopically, if one can recognize the commutator as the Lie bracket of some known group.
The authors apply this programme to the spin-3 field. The resulting expressions are somewhat large, and it's not entirely clear how to interpret them intuitively, but still I like the clarity of their exposition. https://link.springer.com/article/10.1007/BF01410362
On spin three self interactions - Zeitschrift für Physik C Particles and Fields
In this paper we start the construction of a self interacting massless spin three field theory. The first order self interaction is constructed. The free field gauge invariance is modified by a term of first order in the coupling constant. Finally the abelian gauge transformation of the free Lagrangian is shown to become non-abelian.