Constraints on Massless Particles from Symmetries and Quantum Mechanics
Two big ideas that emerged in the early 20
th
century are: 1) forces arise from the exchange of
particles, and 2) Every particle is labelled by just two quantum numbers: Mass and Spin. Though
remarkable in their own right, the combination of these two ideas has gifted us a profound
understanding of the physics of our Universe: The fundamental forces of Nature are
characterised by the mass and spin of the particle being exchanged.
As we shall explore in this project, this is a beautiful consequence of symmetries and Quantum
Mechanics. We will study how elementary particles are identified with unitarity irreducible
representations of the Poincaré group, the symmetry group of Special Relativity. Using this
relation, we will show how to derive the corresponding relativistic linear wave equations. We
will then move on to interactions, exploring how the interactions of massless particles of
non-zero spin (which mediate long range forces) are non-trivially constrained by Poincaré
symmetry.
These ideas culminate in the groundbreaking result of Nobel Laureate Steven Weinberg, who in
the 60s showed that the fundamental features of long-range forces arise purely from the
assumptions of Poincaré symmetry and Unitarity. In particular, Weinberg proved that if
massless spin-1 particles are responsible for a long-range force then charge must be conserved
-- recovering precisely what we observe for the electric charge in our universe from basic
mathematical consistency!
The strength of such arguments lies in their universality, with the same principles applying to
massless particles of any spin: In exactly the same way, Weinberg also showed that long-range
interactions mediated by a massless spin-2 particle must give rise to Einstein’s famous
equivalence principle! And we don’t not have to stop at spin-2. Applied to higher-spins, the
same reasoning explains why we do not see any other long-range forces in our universe:
Massless particles of spin-3 and higher do not interact consistently to give a long-range force!
This project is a great opportunity to acquire a deeper understanding of concepts that will be
introduced in the AQT course, while at the same time covering important material that a first
course on QFT will not have time to discuss.
Prerequisite: Quantum Mechanics
Co-requisite: Advanced Quantum Theory
Resources:
Steven Weinberg, The Quantum Theory of Fields, Volume 1: Foundations
Mikhail Vasiliev, Sections 2-6 of notes: Introduction into Higher-Spin Gauge Theory.
(do not be discouraged by later sections of these notes, which are far beyond what is required for
the project)
Xavier Bekaert and Nicolas Boulanger, The unitary representations of the Poincaré group
in any spacetime dimension.
