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 [29]. 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 τ ) whose coefficients are, in general, formal power series in q 1 2 = e iπτ 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 τ (Im τ = β 2π > 0). The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.

Partition functions, the Bekenstein bound and temperature inversion in anti-de Sitter space and its conformal boundary

“…Thus one expects elliptic functions and modular behaviour to arise. This has been seen to happen, for free fields at least, on R × S n−2 [25,26] and on AdS 4 [5]. In what follows, we shall discuss the behaviour of the energies under temperature inversion in certain R × S n−2 and AdS n examples.…”

“…(See, for example, [26] for a recent discussion, with references to earlier work.) Thermal correlators are always periodic or antiperiodic in imaginary time, and for conformally-invariant fields on R × S n−2 , or on AdS n , they are typically also periodic or antiperiodic in real time.…”

Addendum to ‘Area density of localization entropy: I. The case of wedge localization’

“…In a very interesting recent paper [32] it was shown that such a "relativistic box" interpolation is always possible for conformal theories in arbitrary spacetime dimensions and that it is deeply related to Irving Segal's attempt to use the Dirac-Weyl compactification of Minkowski spacetime for cosmological purposes. In the n-dimensional case H is the zero component of the energy-momentum operator and K the zero component of its conformal reflected counterpart.…”

Minimal Representations and Reductive Dual Pairs in Conformal Field Theory

“…There is a natural expansion of the free massless field ϕ into discrete modes since ϕ can be viewed (as any GCI field) as defined on the conformal compactificationM of Minkowski space M . To this end it is handy to use the complex variable parametrization 10) introduced in [T86] (and later explored in [N] and [NT05]). M is embedded in M by setting…”