Mass terms in quantum field theory

[voltan]

Does anyone know why in quantum field theory mass terms are quadratic in fields (not contaning derivatives)?

Thanks!

 

[Avodyne]

That's just what comes out of quantizing a field. You find that the excited states can be interpreted as particles with a mass corresponding to the coefficient of a quadratic term with no derivatives. I don't know of any good heuristic explanation for this; sometimes you just have to do the calculation and see what comes out.

 

[PRB147]

I think that it is determined by Einstein Mass-Energy Relation.

 

[BruceG]

Yes - start with Einstein, since QFT is just the quantization of special relativity.

So with c=1, start with E^2 - p^2 = m^2

In terms of 4-momentum this is p^2 = m^2

In quantum theory with h= 1, p -> id/dx where x = (x,t) p = (p,E)

So we get the wave equation, [(d/dx)^2 + m^2] A(x) = 0

The Lagrangian L which produces this when we do DL/DA = 0 is then

L = A(x)[(d/dx)^2 + m^2]A(x)

So in summary we can say: quadratic Lagrangians give linear equations of motion.

At this point you should consider the advice we were given in our undergraduate math course: "the world is Linear because that is the only type of equations mathematicians know how to solve".