The concept of on-shell mass : what is it?

[member 11137]

The concept of "on-shell mass": what is it?

Following the recommandation of MTd2, but after some hesitations, here are my questions.So, reading rapidly a first time the document “Curved Momentum Space and Relative Locality” [arXiv: 1205.1304v1 [hep-th] 7 May 2012 on another part of this forum], I find it interesting to see a discussion on the geometry of phase space. The concluding relation (19) was for me a kind of illumination in that sense that it had a formalism recalling a generic relation obtained in a (I am sorry for this remark) personal research. Reading the document a second time with more attention and paying attention to “details” I discovered the (for me) unknown the “on-shell mass” concept introduced in equation (3), page 3. I have in between “googled” on Internet with the hope to learn more via some free scientific articles. This is thus inducing my questions:
1°) why is the subscript “a” in the above equation going from 0 to 3 and not from 1 to 3?
2°) the “on-shell mass” concept seems to be strongly related to the actual analyzes of the LHC experiments (renormalization ...), all things far over my head but so fascinating when you look for unity in physics and when you try to understand how the nature works; is it exact?
3°) do we actually have a stabile theory concerning this concept or –as I guess it through the literature- are we yet condemned to test the accordance between some hypothesis via statistic methods (Monte Carlo...) and results of experiments? Example given: ...to test the energy-momentum relation which is in fact the relation involved in the above reference.

I beg my pardon if, as amateur, I should not ask such question (In that case feel free to delete this question from the forum) because it is a too “hot stuff”.

Best regards

 

[haushofer]

My 2 cents:

1) The subscript goes from 0 to 3, because it also involves time (the 0-component).
E.g. taking the speed of light c to be 1:

$$X^a = (X^0,X^1,X^2,X^3) = (t,x,y,z)$$

or, for the 4-momentum,

$$p^a = (E, p_x,p_y,p_z)$$

2) If special relativity holds to arbitrary energy scales, the relation is indeed exact

 

[member 11137]

My 2 cents: (Thank you for your investment)

1) The subscript goes from 0 to 3, because it also involves time (the 0-component).
E.g. taking the speed of light c to be 1:

$$X^a = (X^0,X^1,X^2,X^3) = (t,x,y,z)$$

or, for the 4-momentum,

$$p^a = (E, p_x,p_y,p_z)$$
After analyze this was finaly my conclusion too.
2) If special relativity holds to arbitrary energy scales, the relation is indeed exact Indeed

My remaining trouble is: the action contains thus a term the units of which are (energy or mass)². Is it correct? Is it normal?
Otherwise -and that was a part of my question- where does the name: "on-shell mass" come from? Is that kind of mass really on shell (of what; topologicaly?) or is it only a technical term?

 

[haushofer]

It is a technical term, only saying that you focus on a particular part of your phasespace. In classical physics this condition often (if you don't invoke auxiliary fields) follows from a primary constraint of your action.

I would say that the units of an action should be [energy]=[mass].

 

[Finbar]

On-shell simply means that the equations of motion apply. Off-shell means that the equations of motion don't apply.

If you think of the configuration space of a theory the "dynamical shell" is the sub space on which the equations of motion are satisfied. Equally you could view the equations of motion in momentum space where one will typically find a relation between momentum and mass when the equations of motion apply. That is what is meant by being on the mass shell. In a relativistic theory (e.g. the Klien-Gordon equation) one typical has a relation of the form p^2 = m^2.

 

[andert]

Right, the on mass shell concept becomes more visible when doing Feynman diagrams where you see the, e.g., scalar field propagator of the Klein-Gordon action:

$$D(x,y) = \frac{1}{(2\pi)^4}\int d^4k \frac{e^{-ik(x-y)}}{k^2 - m^2 \pm i\epsilon}$$ which gives the probability amplitude for a particle to travel from position x to position y.

The mass shell refers to when k²=m² where
$$k^2 = k_0^2 - k_1^2 - k_2^2 - k_3^2,$$ in Cartesian coordinates. That is when a particle is "real" instead of virtual---it has a real amplitude. Note that the propagator is largest when a particle is on mass shell and the farther off mass shell a particle is, the more of a penalty it pays in terms of probability.

 

[member 11137]

I get now a very clearer vision of the thematic and...

I realize the importance of what I am doing my self ...

If I am somewhere glad to understand that my old brain was fit enough to perform a new kind of mathematical tool (method) which can also be useful to test the Lorentz Einstein force and which is finaly able to give a lagrangian proportional to a variation of the squared mass ...

I suspect that I should leave this to professionnals... (I am certainly to old for a new start in my life with physics!)

Thank you very much for your help

 

[member 11137]

Perhaps does this thread appear to be a little bit obscure for some readers. In order to clarify the context it can be (e. g.) useful to refer to the work of A. Aharonovich and L. P. Horwitz in arxiv:1105.5498v1 [math-ph], 27 May 2011. Although the discussions take mainly place in a 5D space (citation: off-shel electrodynamics based on a manifestly covariant off-shell relativistic dynamics of Stueckelberg, Horwitz and Piron), the investigation concerns a test of the Lorentz Einstein force. My own approach is developped in a 4D context around the same thematic. As said in previous posts, it is a fascinating domain of research, I think.

I hope that it can help some of you.