coupling constatn dimensions

[eljose79]

givne a theory with a given lagrangian..how do you obtain the dimension of coupling constatn?..in fact how do you know that for electromagnetism is g=1/137 or that for gravitation have units of m**-2?

Can the coupling constant be rescaled to get always a dimensionless one?

[vanesch]

Using these dimensions, one has to know that this makes sense when we put c = 1 and h = 1. This makes L = T and M L^2 T^(-1) = 1. So M = L^(-1).
Now, the action needs to be dimensionless (same unit as h = 1). The action is the 4-dim integral of the lagrangian density, so L has to have dimensions L^(-4) = M^4. Mass still has, well, M as dimension. A typical mass term for a scalar field is m^2 phi^2, so it follows that a scalar field has to have dimensions M. As such, you can puzzle together the dimensions of different fields from their "free field" terms. The dimensionality of the interaction term coefficients (the coupling constants) is very important because it determines the renormalizability of the theory.

cheers,
Patrick.