# The lure of conformal symmetry

## TL;DR

This paper explores the role of conformal symmetry, Clifford algebra, and modular forms in understanding massless particles, fields, and their thermodynamic properties, connecting mathematical structures to physical phenomena.

## Contribution

It links conformal symmetry and Clifford algebra to the representation theory of the conformal group and extends vertex operator algebra concepts to higher dimensions.

## Key findings

- Unitary positive energy ladder representations elucidate massless particles.
- Thermal correlation functions are elliptic functions with modular properties.
- Partition functions reproduce black body radiation law.

## Abstract

The Clifford algebra, generated by the real (Majorana) gamma-matrices and by a hermitian gamma_5, gives room to the reductive Lie algebra u(2,2) of the conformal group extended by the u(1) helicity operator. Its unitary positive energy ladder representations, constructed by Gerhard Mack and the author 50 years ago, opened the way to a better understanding of zero-mass particles and fields and their relation to the space of bound states of the hydrogen atom. They became a prototypical example of a minimal representation of a non-compact reductive group introduced during the subsequent decade by Joseph. By the mid 1980's I have developed an analytic approach to compactified Minkowski space, suited for extending the notion of vertex operator algebras to higher dimensions. Another 20 years later Nikolay Nikolov and I realized that thermal correlation functions of massless fields are doubly periodic elliptic functions while the logarithmic derivative of the corresponding partition function is a modular form reproducing Planck's black body radiation law.