Editing Infrared divergence (section)

==Bremsstrahlung example==
When an [[electric charge]] is accelerated (or decelerated) it emits [[Bremsstrahlung radiation]]. Semiclassical [[electromagnetic theory]], or the full [[quantum electrodynamic]] analysis, shows that an infinite number of soft photons are created. But only a finite number are detectable, the remainder, due to their low energy, falling below any finite energy detection threshold, which must necessarily exist.<ref name=Kaku>{{cite book | last=Kaku | first=Michio | title=Quantum Field Theory: A Modern Introduction | year=1993 | location=New York | publisher=Oxford University Press | isbn=0-19-507652-4}}, pages 177-184 and appendix A6</ref> However even though most of the photons are not detectable they can't be ignored in the theory; quantum electrodynamic calculations show that the [[transition amplitude]] between ''any'' states with a finite number of photons vanishes. Finite transition amplitudes are obtained only by summing over states with an infinite number of soft photons.<ref name=Kaku/><ref name=IZ>{{cite book
 |author=Claude Itzykson, Jean-Bernard Zuber
 |year=1980
 |title=Quantum Field Theory
 |url=https://archive.org/details/quantumfieldtheo0000itzy/page/172
 |url-access=registration
 |publisher=[[McGraw-Hill]]
 |isbn=0-07-032071-3
 |pages=[https://archive.org/details/quantumfieldtheo0000itzy/page/172 172/3]
 }}</ref>

The zero-energy photons become important in analyzing the [[Bremsstrahlung radiation]] in the coaccelerated frame in which the charge experiences a thermal bath due to the [[Unruh effect]]. In this case, the static charge will only interact with these zero-energy (Rindler) photons in a sense similar to virtual photons in the coulomb interaction.<ref>{{Cite journal |last1=Higuchi |first1=A. |last2=Matsas |first2=G. E. A. |last3=Sudarsky |first3=D. |date=1992-05-15 |title=Bremssstrahlung and zero-energy Rindler photons |url=https://link.aps.org/doi/10.1103/PhysRevD.45.R3308 |journal=Physical Review D |volume=45 |issue=10 |pages=R3308–R3311 |doi=10.1103/PhysRevD.45.R3308|pmid=10014292 |bibcode=1992PhRvD..45.3308H |url-access=subscription }}</ref><ref>{{Cite journal |last1=Higuchi |first1=A. |last2=Matsas |first2=G. E. A. |last3=Sudarsky |first3=D. |date=1992-10-15 |title=Bremsstrahlung and Fulling-Davies-Unruh thermal bath |url=https://link.aps.org/doi/10.1103/PhysRevD.46.3450 |journal=Physical Review D |volume=46 |issue=8 |pages=3450–3457 |doi=10.1103/PhysRevD.46.3450|pmid=10015290 |bibcode=1992PhRvD..46.3450H |url-access=subscription }}</ref>