Intertwining operator in thermal CFTd

Abstract

It has long been known that two-point functions of conformal field theory (CFT) are nothing but the integral kernels of intertwining operators for two equivalent representations of conformal algebra. Such intertwining operators are known to fulfill some operator identities -the intertwining relations -in the representation space of conformal algebra. Meanwhile, it has been known that the S-matrix operator in scattering theory is nothing but the intertwining operator between the Hilbert spaces of in-and out-particles. Inspired by this algebraic resemblance, in this paper, we develop a simple Lie-algebraic approach to momentum-space two-point functions of thermal CFT living on the hyperbolic space-time H 1 × H d−1 by exploiting the idea of Kerimov's intertwining operator approach to exact S-matrix. We show that in thermal CFT on H 1 × H d−1 , the intertwining relations reduce to certain linear recurrence relations for two-point functions in the complex momentum space. By solving these recurrence relations, we obtain the momentum-space representations of advanced and retarded two-point functions as well as positive-and negative-frequency two-point Wightman functions for a scalar primary operator in arbitrary space-time dimension d ≥ 3.

Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs

“…C.2 Derivation of G a,b ∆ (ω E , k) We provide here some details on the Fourier transform leading to (5.18). Essentially, the calculation is the same as a Euclidean version of Appendix A.2 of [61] with the additional replacement of Bessel functions I ω E (u) → I ω E (au) and K ik (u) → b −(d−2)/2 K ik (bu) to take the propagator off-shell. We now recall some of the essential steps.…”

Conformal Ward–Takahashi Identity at Finite Temperature

“…where O(t) is an arbitrary Heisenberg operator and H is assumed to satisfy H|Ω = 0. Once we have the identity (15), we can prove that the Wightman functions with respect to the state |Ω satisfy (13). Indeed, by using the inner product notation ( * , * ) we have (see also Chapter 5 of [15])…”

“…For a full account of the KMS condition we refer to [9,10]. Now, let us take a closer look at the boundary conditions (13). These conditions are best understood in statistical mechanics for finite degrees of freedom in a finite box.…”

“…The above discussion is based on the expectation value with respect to the density matrix ρ = e −βH /Z. However, the boundary conditions (13) themselves can be formulated without recourse to the density matrix. Suppose that there exist a state |Ω and an antiunitary operator J such that the following identity holds:…”

“…Contrary to this popular belief, however, finite temperature and conformal invariance can in fact be compatible with each other: If conformal field theory (CFT) is thermalized via the Unruh effect, conformal symmetry remains intact even at finite temperature. The purpose of this paper is to report our recent work on this subject [13] and to see how the conformal symmetry determines finite-temperature twopoint functions in momentum space. The key is the intertwining relations of conformal algebra so(2, d) [6,8,12,16], which follow from the conformal Ward-Takahashi identities for two-point functions.…”