Editing On shell and off shell (section)

==Mass shell==
[[Image:Hyperboloid Of Two Sheets Quadric.png|thumb|upright|right|Points on the hyperboloid surface (the "shell") are solutions to the equation.]]
Mass shell is a synonym for '''mass hyperboloid''', meaning the [[hyperboloid]] in [[energy]]&ndash;[[momentum]] space describing the solutions to the equation:

:<math>E^2 - |\vec{p} \,|^2 c^2 = m_0^2 c^4</math>,

the [[Energy–momentum relation|mass–energy equivalence formula]] which gives the energy <math>E</math> in terms of the momentum <math>\vec{p}</math> and the [[rest mass]] <math>m_0</math> of a particle. The equation for the mass shell is also often written in terms of the [[four-momentum]]; in [[Einstein notation]] with [[metric signature]] (+,−,−,−) and units where the [[speed of light]] <math>c = 1</math>, as <math>p^\mu p_\mu \equiv p^2 = m_0^2</math>. In the literature, one may also encounter <math>p^\mu p_\mu = - m_0^2</math> if the metric signature used is (−,+,+,+).

The four-momentum of an exchanged virtual particle <math>X</math> is <math>q_\mu</math>, with mass <math>q^2 = m_X^2</math>.  The four-momentum  <math>q_\mu</math> of the virtual particle is the difference between the four-momenta of the incoming and outgoing particles.

Virtual particles corresponding to internal [[propagator]]s in a [[Feynman diagram]] are in general allowed to be off shell, but the amplitude for the process will diminish depending on how far off shell they are.<ref>{{cite journal|last1=Jaeger|first1=Gregg|title=Are virtual particles less real?|journal=Entropy |volume=21 |issue=2|page=141|date=2019|doi=10.3390/e21020141|pmid=33266857|pmc=7514619|bibcode=2019Entrp..21..141J|url=http://philsci-archive.pitt.edu/15858/1/Jaeger%20Are%20Virtual%20Particles%20Less%20Real_%20entropy-21-00141-v3.pdf|doi-access=free}}</ref> This is because the <math>q^2</math>-dependence of the propagator is determined by the four-momenta of the incoming and outgoing particles. The propagator typically has [[mathematical singularity|singularities]] on the mass shell.<ref>Thomson, M. (2013). ''Modern particle physics''. Cambridge University Press, {{ISBN|978-1107034266}}, p.119.</ref>

When speaking of the propagator, negative values for <math>E</math> that satisfy the equation are thought of as being on shell, though the classical theory does not allow negative values for the energy of a particle.  This is because the propagator incorporates into one expression the cases in which the particle carries energy in one direction, and in which its [[antiparticle]] carries energy in the other direction; negative and positive on-shell <math>E</math> then simply represent opposing flows of positive energy.