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Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT. (English) Zbl 1111.81112
Summary: 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. 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 $q^{\frac12} = e^{i\pi\tau}$ 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 $\tau(\text{lm}\tau = \frac{\beta}{2\pi} > 0)$. The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.

81T05Axiomatic quantum field theory; operator algebras
81R10Infinite-dimensional groups and algebras motivated by physics
81T40Two-dimensional field theories, conformal field theories, etc.
11Z05Miscellaneous applications of number theory
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