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The definitions between on- and off-shell are given in Wikipedia.

Why is it so important in QFT to distinguish these two notions ?

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Some of the discussion in physics.stackexchange.com/q/17087 and physics.stackexchange.com/q/4349 may be useful for setting the scene here. –  dmckee Mar 28 '13 at 19:17
QFT needs this to be consistent with the conservation of energy over time.. that's all –  user12811 Mar 28 '13 at 20:49
@user12811 Sorry, QFT needs what ? I was asking about the difference between two things... –  FraSchelle Mar 29 '13 at 4:25
@dmckee Thanks for the good references. –  FraSchelle Mar 30 '13 at 10:11
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It's important to distinguish them because on-shell and off-shell are opposite to each other, in a sense. On-shell describes fields that obey the equations of motion and real particles; off-shell describes fields that don't have to obey the equations of motion and virtual particles.

On-shell are momenta with $p^2=m^2$ with the right $m^2$ for the given field; off-shell are momenta that don't obey the condition.

Amplitudes with external particles that are on-shell, i.e. on-shell amplitudes, express the scattering amplitudes and may be directly used to calculate cross sections etc. Off-shell amplitudes i.e. amplitudes with external off-shell momenta encode much more general correlators.

In some theories, i.e. quantum gravity i.e. string theory in the flat space, only on-shell amplitudes for the external particles such as gravitons may be calculated. On the contrary, the analogous quantities to these on-shell amplitudes in the AdS space may be expressed by off-shell correlators in the corresponding CFT.

It's always important to know whether the 4-momenta etc. we are attaching are obliged to be on-shell or not, i.e. whether they're on-shell or off-shell. If we substitute off-shell momenta to on-shell-only formulae, we get meaningless or ill-defined results. If we only substitute on-shell momenta to off-shell formulae, we miss a significant portion of the physical information.

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Ha, the on/off shell issue in the context of AdS/CFT is very interesting for me, I did not hear about that before ... –  Dilaton Mar 28 '13 at 22:21
@LubosMotl Thanks a lot. Would you please precise what "amplitudes" refers to on your third paragraph (the sentence "off-shell amplitudes i.e. amplitudes with external off-shell amplitudes" makes no sense to me) ? Fourth paragraph: is there a fundamental reason why one can only calculate on-shell amplitudes in quantum gravity and string theory ? Last paragraph: what about the substitution you are talking about (I thought the momenta comes from the process involved, it looks you can replace as you want the momenta in your calculation, why is it the case) ? Thanks again. –  FraSchelle Mar 29 '13 at 4:44
Dear @Oaoa, it should have been "amplitudes with external off-shell momenta", fixed. Amplitudes are the complex numbers - like values of the wave functions - whose squared absolute values are probabilities or probability densities. Yes, there is a fundamental reason why covariant quantum gravity theories admit no off-shell amplitudes. Off-shell amplitudes may be Fourier transformed to get correlators $D(x^\mu,y^\mu...)$ of operators but the locations $x,y$ in GR-like theories are gauge-dependent because of the diffeomorphism symmetry so the whole function becomes ill-defined. –  Luboš Motl Mar 29 '13 at 4:57
"The Noether current is conserved on-shell" means that $\partial_\mu j^\mu=0$ follows from equations of motion - field equations in the QFT. So if you don't assume these laws of physics, the Noether charge isn't conserved, if you do, then it is. It's important because it's important to know whether something is conserved and what assumptions are needed for it to be conserved. On-shell and "equations of motion hold" are equivalent because equations of motion away from interactions are e.g. $(-\Box-m^2)\Phi=0$ i.e. (in momentum space) $(p^2-m^2)\Phi$=0, giving the on-shell condition. –  Luboš Motl Mar 29 '13 at 4:59
Otherwise this kind of "what is the importance" questions is also a bit loaded. I have never said anything about the degree of importance of that statement. It's surely important for something - if one wants to know the very thing that the statement is saying - but I didn't say whether it was among the top 3 most important statements in QFT; the answer would almost certainly be No. The "what is the importance" is like an invitation for a scientist to apply for grants and only say the positive things - i.e. to say not the whole truth. –  Luboš Motl Mar 29 '13 at 5:18
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I) The meaning of the word on-shell depends on context. We are aware of at least three meanings.

  1. The original meaning of the word on-shell refers to (as Lubos Motl writes in his correct answer) that the mass-shell condition $$ p_0^2 - p_1^2-p_2^2-p_3^2 ~=~m_0^2 $$ is satisfied for a 4-momentum $p_{\mu}\in \mathbb{R}^4$. The mass-shell is therefore a 3-dimensional hyperboloid in 4-dimensional energy-momentum space $\mathbb{R}^4$.

  2. Nowadays more generally, the word on-shell is also used in the sense that equations of motion of the system are satisfied.

  3. In constrained systems, the word on-shell refers to the constrained surface of configurations of the dynamical variables that obey the constraints.

II) The word off-shell originally means not on-shell. (Note however that the word off-shell sometimes effectively means not on-shell and on-shell. E.g. an off-shell formulation also describes physics on-shell.)

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