Specifically, if in a collision between n incoming particles from which m outgoing particles arise, the outcome of the collision depends on a certain S matrix, by adding one or more gravitons to the n + m particles, the resulting S matrix (let it be S') differs from the initial S only by a factor that does not depend in any way, except for the momentum, on the type of particles to which the gravitons couple.[2]
The theorem also holds by putting photons in place of gravitons, thus obtaining a corresponding soft photon theorem.
The theorem is used in the context of attempts to formulate a theory of quantum gravity in the form of a perturbative quantum theory, that is, as an approximation of a possible, as yet unknown, exact theory of quantum gravity.[3]
Given particles whose interaction is described by a certain initial S matrix, by adding a soft graviton (i.e., whose energy is negligible compared to the energy of the other particles) that couples to one of the incoming or outgoing particles, the resulting S' matrix is, leaving off some kinematic factors,
where p is the momentum of the particle interacting with the graviton, ϵμν is the graviton polarization, pG is the momentum of the graviton, ε is an infinitesimal real quantity which helps to shape the integration contour, and the factor η is equal to 1 for outgoing particles and -1 for incoming particles.
The formula comes from a power series and the last term with the big O indicates that terms of higher order are not considered. Although the series differs depending on the spin of the particle coupling to the graviton, the lowest-order term shown above is the same for all spins.[1]
In the case of multiple soft gravitons involved, the factor in front of S is the sum of the factors due to each individual graviton.
If a soft photon (whose energy is negligible compared to the energy of the other particles) is added instead of the graviton, the resulting matrix S' is