ing a detector operator built out of the stress tensor The integral above is the detector working time integral, a period of time during which the detector is collecting energy. The limit appears because we place the detector at infinity far away from the “collision” region. Thanks to the insertion point being $r \vec n$ the calorimeter collects only the radiation propagating in the $\vec n$ direction.

In this note we define a detector operator in a slightly different manner (see also ) where $n=(1, \vec n)$, $\bar n=(1,-\vec n)$ so that $x .n = t-r$, $x . \bar n = t+r$ and $\vec 0_{\perp}$ stands for coordinates perpendicular to $n$ and $\bar n$.

Notice that in general two definitions are not equivalent. Indeed, the definition   corresponds to the detector that collects all sorts of excitations, massive (propagating to the future and from the past time-like infinity $i^{\pm}$) as well as massless (propagating to and from the null infinity ${\cal I}^{\pm}$). A usual physical setup corresponds to the choice of the state $| \Psi \ra$ with a compact support, such that the detector triggers only the outgoing particles propagating to $i^+$ and ${\cal I}^{+}$. At the very end of the calculation we can take the limit in which the wave function can become a simple plane wave as we consider below. If we instead start from the wave function which is nowhere zero we will detect the ingoing particles that hit the detector from the outside. In CFTs for the simple states created by the primary operator with momentum $q$ it is easy to check that energy flux at $i^+$ is zero.

The definition   is designed to collect only radiation at ${\cal I}^+$ from the very beginning. We first send the detector to the future null infinity and then integrate over the null working time. We can use simple plane wave states since it is impossible to hit the detector inserted along the null line from the outside. Another advantage of   is that it is manifestly Lorentz invariant. Thus, qualitatively we see that in CFTs the two definitions 