Expectation value of composite field $T{\bar T}$ in two-dimensional quantum field theory

TL;DR

This work proves that in two-dimensional quantum field theory, the vacuum expectation value of the composite operator $T\bar{T}$ is exactly determined by the expectation values of the energy-momentum tensor components. Using symmetry, OPE analysis, and conformal perturbation theory, Zamolodchikov defines $T\bar{T}$ through a short-distance limit that eliminates singularities, yielding the universal relation $\langle T\bar{T}\rangle = \langle T\rangle\langle \bar{T}\rangle - \langle \Theta\rangle^2$ (on the plane) and its cylinder generalization $\langle T\bar{T}\rangle = \langle T\rangle\langle \bar{T}\rangle - \langle \Theta\rangle^2$. The results hold without integrability and extend to diagonal matrix elements in finite-volume eigenstates, with explicit finite-volume expressions in terms of energy and momentum. The approach provides a nonperturbative link between TTbar and the energy-momentum data, with implications for finite-volume spectra and critical phenomena.

Abstract

I show that the expectation value of the composite field $T{\bar T}$, built from the components of the energy-momentum tensor, is expressed exactly through the expectation value of the energy-momentum tensor itself. The relation is derived in two-dimensional quantum field theory under broad assumptions, and does not require integrability.