What is a soft photon? I accidentally came across the words "soft photon" today after reading a few blogs. There was some discussion of special situations involving gauge redundancies and a theorem by Weinberg. 


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*What is a soft photon?

*What is the simplest clearest possible description of Weinberg's soft photon theorem is? 

*How is it motivated and then derived?
 A: A soft photon is a photon whose energy we take to zero. The reason to consider soft photons is that whenever you evaluate the ${\cal S}$-matrix with one soft photon, either in the in or out state, there happens a factorization in which the scattering amplitude becomes a product of one universal soft factor and the scattering amplitude without that soft photon. In equations $$\langle \text{out}|a_+^{\rm out}(q){\cal S}|\text{in}\rangle=e\left[\sum_{k=1}^m \dfrac{Q_{k}^{\rm{out}}p_k^{\rm out}\cdot\varepsilon^+}{p_k^{\rm out}\cdot q}-\sum_{k=1}^n \dfrac{Q_{k}^{\rm{in}}p_k^{\rm in}\cdot\varepsilon^+}{p_k^{\rm in}\cdot q}\right]\langle{\rm out}|{\cal S}|{\rm in}\rangle+O(q^0),$$
where $|\text{in}\rangle$ is the incoming state with $n$ particles with momenta $p_k^{\rm in}$ and charges $Q_k^{\rm in}$ while $\langle\text{out}|$ is the outgoing state with $m$ particles with momenta $p_k^{\rm out}$ and charges $Q_k^{\rm out}$. Here $q$ and $\varepsilon^+$ are the momentum and polarization of the photon which is inserted into the out state and taken to be soft. Notice that $\langle{\text{out}}|a_+^{\rm out}(q)$ is the out state with an additional photon. The fact that we are taking this photon to be soft is clear in the fact that the above equation is an expansion in the energy $q^0$ in which only the leading order term has been included.
This result was known for many years, see in particular this paper by Weinberg in which he extended this result to gravitons https://journals.aps.org/pr/abstract/10.1103/PhysRev.140.B516. Nevertheless, in the last decade several interesting things were discovered related to this.
In particular, the soft photon theorem can be seem as the Ward Identity of a so-called asymptotic symmetry (https://arxiv.org/abs/1506.02906). Indeed, the whole point is that a gauge transformation which does not vanish at infinity and acts non-trivially on the asymptotic data of the theory, is not a mere redundancy in description, but a true physical symmetry with a conserved charge and measurable effects. This leads to some interesting developments. For example, infrared divergences can be understood in this context as a consequence of non-conservation of the asymptotic charge (https://arxiv.org/abs/1705.04311). Indeed, in the standard QFT story, QED ${\cal S}$-matrix elements suffer from IR divergences which end up setting them to zero. Realizing the connection between the soft theorem and asymptotic symmetries gives a new point of view: the usual ${\cal S}$-matrix elements are set to zero because asymptotic charge is not being conserved. Scattering with asymptotic states conserving the asymptotic charge is IR finite. Moreover, the measurable effects of the asymptotic symmetry are encoded in a so-called memory effect (https://arxiv.org/abs/1505.00716).
This is one example of what is known as one Infrared Triangle: a triangular equivalence relation between a soft theorem, an asymptotic symmetry and one memory effect. There are several such triangles: for instance in gravity or non-abelian gauge theorey there are such triangles as well. Moreover, one can go to further subleading order in the expansion in $q^0$ and find relations between subleading soft theorems, other asymptotic symmetries and other memory effects. This was particularly important in gravity.
In particular, let me briefly tell you that this IR triangle story is at the heart of a proposal for flat space holography known as celestial holography which is very interesting and which is a very active field of research today.
If you are interested in learning more about all of that I strongly suggest you study Andy Strominger's "Lectures on the Infrared Structure of Gravity and Gauge Theory". In particular, Chapter 2 gives a very pedagogic treatment of soft photons and the associated IR triangle to the soft photon theorem. In particular a very well-explained derivation of the soft photon theorem is included.
A: A soft photon is a photon, carrying very low energy compared to the remaining energy scales (fermion masses, fermion energies, other photon energies, etc) of the interaction, during which the it is produced.
The Weinberg's soft theorem is a statement describing the low energy limit of a photon emission during a generic QED scattering process. It is more general, to be honest, but you asked for the simplest version of it. The theorem relates the radiative scattering amplitude of the interaction with the non-radiative (or elastic) one. The radiative amplitude of a scattering procedure is the amplitude of a scattering procedure, in which additional radiation is emitted (in our case one additional soft photon). Furthermore, the soft photon theorem states that this behavior of the radiative amplitude (in the soft limit) is universal (i.e. does not depend on the internal degrees of freedom of the emitting particles).
The motivation behind all this is that, as showed by Bloch and Nordsieck, the infrared radiation (and more specifically an infinite number of soft photons) must accompany every scattering procedure, such that all the experimentally observed quantities are finite. It is derived by considering all the ways in which a scattering process can emitt additional radiation and then by expanding all the collected terms of the amplitude corresponding in that radiative process, in powers of the radiated momentum.
I hope this helps.
