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Zbl 1111.81112
Nikolov, Nikolay M.; Todorov, Ivan T.
Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT.
(English)
[J] Rev. Math. Phys. 17, No. 6, 613-667 (2005). ISSN 0129-055X

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.
MSC 2000:
*81T05 Axiomatic quantum field theory
81R10 Repres. of infinite-dim. groups and algebras from quantum theory
81T40 Two-dimensional field theories, etc.
11Z05 Miscellaneous appl. of number theory

Keywords: 4-dimensional conformal field theory; thermal correlation functions; elliptic functions; modular forms

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