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It's commonly said K-G equation describes spinless particles, but we know any solution of dirac eqn is a solution of K-G eqn, then can't we say K-G could possibly describe a Dirac spinor field? But if so, it's not spinless any more.
The field equation fully characterizes the field. It encompasses both the spin and the mass, which are the 2 Casimir invariants of the Poincare algebra.I see, so the field equation can hardly characterize the field, instead we need the structure of the field (scalar, spinor, vector....) and how they transform under change of reference frame, to characterize the field, am I right? Then I'm a bit curious, what's the role of field equation in QFT, any comment?
The correct statement is: The ONE-COMPONENT KG equation describes spinless particles. Higher-spin wave functions have more than one component, even if each of them satisfies the KG equation.It's commonly said K-G equation describes spinless particles, but we know any solution of dirac eqn is a solution of K-G eqn, then can't we say K-G could possibly describe a Dirac spinor field? But if so, it's not spinless any more.
"The field equation" is what you get from the Lagrangian. For the Dirac field, that's the Dirac equation. The field satisfies the field equation. Its components satisfy the Klein-Gordon equation, but since the field components aren't scalar fields, the method I described before won't give you a spin-0 representation.Yeah I agree, but now it's a bit fuzzy to me "what's the role of field equation in QFT"
Well you solve the field equation to find the Green's function (i.e., free-field propagator) by setting the RHS equal to a delta function.Hmm, if that's all, it seems field equations do not really play a big role in QFT
The field equation is[/i] the (free) dynamics, expressed as a space-time constraint, which also expresses in particular how the system transforms under time translation. It is thus used in QFT to construct the propagator for the quantized fields.I see, so the field equation can hardly characterize the field, instead we need the structure of the field (scalar, spinor, vector....) and how they transform under change of reference frame, to characterize the field, am I right? Then I'm a bit curious, what's the role of field equation in QFT, any comment?
They do, in the Heisenberg picture.Hmm, if that's all, it seems field equations do not really play a big role in QFT