# What do we mean when we say on shell particles?

## Main Question or Discussion Point

What do we mean when we say "on shell" particles?

Hello!

What do we mean when we say a particle in on shell? What is off shell? I gather it is related to energy and momentum conservation to say if a process is possible, but I've no idea what it actually is. Is it something to do with writing E^2 = p^2 + m^2?

Cheers,

Samantha

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By on-shell one means "satisfying the equations of motion". For relativistic particles this indeed includes the property that the particle satisfies the energy-mass relation: E^2 = p^2 + m^2.

But it's a broader concept than that. For instance, Noether's theorem links conserved quantities to symmetries of the system. But this theorem only holds on-shell.

By on-shell one means "satisfying the equations of motion". For relativistic particles this indeed includes the property that the particle satisfies the energy-mass relation: E^2 = p^2 + m^2.

But it's a broader concept than that. For instance, Noether's theorem links conserved quantities to symmetries of the system. But this theorem only holds on-shell.
When one makes calculations of scattering initially and finally free particles, there involved solutions of the "free" equations of motion. In these solutions the energy is determined with the particle momentum. They say such particles are "on shell", meaning E(p) equation is a constraint to four-vector p. The Noether's theorem gives conserved values if they are expressed via solutions rather than via unknown variables.

If one makes calculations with help of Fourier representation, they are expressed via Fourier variables denoted historically also as p (also four variables used in the Fourier integrals). As soon as these variables are independent of each other, they are not bound by any relationship, so the combination E(p) takes non-physical values (E(p) is just meaningless). They say "off shell" or virtual particles. So any actual calculation includes on- and off-shell "particles".

Bob.

Last edited:
tiny-tim
Homework Helper
Hello!

What do we mean when we say a particle in on shell? What is off shell? I gather it is related to energy and momentum conservation to say if a process is possible, but I've no idea what it actually is. Is it something to do with writing E^2 = p^2 + m^2?

Cheers,

Samantha
Hello Samantha! (try using the X2 tag just above the Reply box )

On shell means real (physical), and is E2 = p2 + m2.

Off shell means not real (not physical), and is only used in the maths.

Off shell occurs only in the middle of calculations, never at the beginning or end.

Energy and momentum conservation applies to both on shell and off shell equations (but E2 = p2 + m2, which applies to each particle individually, does not apply off shell).

Unless you're doing Feynman diagrams and the Dyson expansion (a math technique in Quantum Field heory), you won't come across off shell. Thank you all very much for your help. :-) However, how could I tell if a process is possible for on-shell particles? So... picking a simple process a photon annihilating to give an electron and positron. How can I determine if this is possible for on-shell particles?

Cheers, Samantha