Elliptic Thermal Correlation Functions and Modular Forms in a Globally Conformal Invariant QFT

TL;DR

This paper demonstrates that in globally conformal invariant quantum field theories, thermal correlation functions can be expressed as elliptic functions, revealing deep connections between conformal symmetry, elliptic functions, and thermodynamics.

Contribution

It proves that thermal expectation values in GCI QFT can be represented as elliptic functions and explores their modular properties and thermodynamic limits.

Findings

Thermal correlation functions are elliptic functions under convergence.

Explicit computation of thermal 2-point functions for free fields.

Analysis of modular transformation properties of Gibbs energy.

Abstract

Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space-time dimensions implies the Huygens' principle, and hence, rationality of correlation functions of observable fields (see Commun. Math. Phys. 218 (2001) 417-436; hep-th/0009004). The conformal Hamiltonian $H$ has discrete spectrum assumed here to be finitely degenerate. We then prove that thermal expectation values of field products on compactified Minkowski space can be represented as finite linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and $\tau$) whose coefficients are, in general, formal power series in $q^{1/2}=e^{i\pi\tau}$ involving spherical functions of the "space-like" fields' arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2-point functions of free fields are computed and shown to display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature $\tau$ ($Im(\tau)=\beta/(2\pi)>0$). The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.