- #1
Frank Castle
- 580
- 23
What is the intuitive reasoning for requiring that a Lagrangian describing a free-field contains terms that are at most quadratic in the field?
Is it simply because this ensures that the EOM for the field are linear and hence the solutions satisfy the superposition principle implying (at least in the classical) sense, that wavepackets do not interfere with one another as they propagate past one another, i.e. they are free-fields?!
Furthermore, what is the motivation for including the term ##\frac{1}{2}m^{2}\phi^{2}## in the free-field case? I get that the parameter ##m## is attributed to the mass of the field a posteriori, but is the reason for the inclusion of such a term in the first place? Is it simply because a priori there is no reason not to - one should include all possible terms up to quadratic order?! Or is there also some physical motivation as well, in that from quantum mechanics, the wave function of a relativistic particle (of mass ##m##) should satisfy the Klein-Gordon equation?!
Is it simply because this ensures that the EOM for the field are linear and hence the solutions satisfy the superposition principle implying (at least in the classical) sense, that wavepackets do not interfere with one another as they propagate past one another, i.e. they are free-fields?!
Furthermore, what is the motivation for including the term ##\frac{1}{2}m^{2}\phi^{2}## in the free-field case? I get that the parameter ##m## is attributed to the mass of the field a posteriori, but is the reason for the inclusion of such a term in the first place? Is it simply because a priori there is no reason not to - one should include all possible terms up to quadratic order?! Or is there also some physical motivation as well, in that from quantum mechanics, the wave function of a relativistic particle (of mass ##m##) should satisfy the Klein-Gordon equation?!