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

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.

LaTex. 56 pages. The concept of global conformal invariance set in a historical perspective (new Sect. 1.1 in the Introduction), references added; minor corrections in the rest of the paper

References in corpus (9)

• Vertex Algebras and Algebraic Curves
• Classification of Two-dimensional Local Conformal Nets with c<1 and 2-cohomology Vanishing for Tensor Categories
• Rationality of conformally invariant local correlation functions on compactified Minkowski space
• Vertex Algebras in Higher Dimensions and Globally Conformal Invariant Quantum Field Theory
• Four Dimensional CFT Models with Rational Correlation Functions
• Globally conformal invariant gauge field theory with rational correlation functions
• On hot bangs and the arrow of time in relativistic quantum field theory
• Elliptic functions and temperature inversion symmetry on spheres
• Braided Structure in 4-dimensional conformal Quantum Field Theory

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