Uncovering the Mystery Behind SM Lagrangian Sums

[jouvelot]

Hello all,

I'm a bit baffled by the fact that the various quite different components of the SM Lagrangian (or other systems, btw) are simply summed up, without even one ponderation coefficient, in the total Lagrangian. I know one reason it is like that is that... it works in practice, but I keep thinking there must be a more profound and/or mathematical reason for this (a rapid Google search on this didn't provide anything really conclusive).

Thanks for your help.

Bye,

Pierre

 

[ChrisVer]

I don't quiet understand the question... can you give an example of a Lagrangian you are talking about?
For example I've seen $L = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu}$ ...

 

[dextercioby]

But each component of the sum has different coefficients. Therefore it has an individuality, even if one can show that the so-called interaction terms are derived from the so-called free fields.

 

[jouvelot]

Well, for instance, the EM and Higgs Lagrangian subexpressions have their own parameters, of course. But, when one combines them, one just adds them, plus the possible interaction term, without introducing any additional coefficients, something like $$\alpha L_{EM}+\beta L_{Higgs}.$$ I find this sort of miraculous.

 

[Vanadium 50]

I find this sort of miraculous.

Why? Specifically, why do you find this miraculous in the quantum case but not the classical case?

 

[jouvelot]

Sure :) As I alluded to in my first post, I have the same issue with classical systems. Just adding stuff, without ponderation or additional parameters, and then minimizing it in the action "just" works in all cases. I find it amazing... Maybe I'm too impressionable :)

 

[Vanadium 50]

Maybe this should be moved to General or Classical then, because the exact same thing happens classically. If I have two free particles, the Lagrangian is T1 + T2. If I add an interaction, it's T1 + T2 - U_A. If I add another interaction, it's T1 + T2 - U_A - U_B. Another and it's T1 + T2 - U_A - U_B - U_C.

 

[jouvelot]

Thanks for the suggestion. I'll put my question there then.

Bye,

Pierre

 

[fresh_42]

Thanks for the suggestion. I'll put my question there then.

Bye,

Pierre

I moved it for you, because I didn't want to lose the already existing part of the discussion.
Please do not create a second one. Thanks.

 

[jouvelot]

Thanks.

Pierre

 

[jouvelot]

Just thinking a little bit more about this, and answering my own question :)

it seems to me that the coefficients I was looking for are, in some sense, embedded in the coupling constants that occur in the various terms of the lagrangian. And then, using non-weighted addition between these terms comes from the additive nature of the notion of energy