For Galilean relativity, a sphere expanding at a given speed will remain so after a boost, but will have a net velocity. Densities will remain the same. A moving source shouldn't matter -- any material will itself will set the rest-frame, and must to have a non-infinite propagation velocity. Whereas, emitting particles, these will have the velocity of the source. Ah! if it's a complex wave equation, you can put a varying phase on everything encoding the velocity. But moving to a complex field is in some sense putting more degrees-of-freedom in.
For Einsteinian relativity, a sphere expanding at light speed remains a sphere in any reference frame, but the density along it changes. A moving source should have that same pattern, so it would seem that local differences (i.e. derivatives) could "point out" the velocity of the source. This behaves the same for particles being emitted. Is there a first-order Lorentz invariant scalar wave-equation, or only second-order, using the d'Alembertian? And that seems to introduce another degree-of-freedom as well.
I'm still really confused. How do pions behave? Is it reasonable to consider them as a fundamental scalar field when they're modeled as pairs of quarks? Particularly when they have excited states that are spin-1?
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Hmm. Now I need to think about it a bit more.
Suppose we have a scalar field.
For Galilean relativity, a sphere expanding at a given speed will remain so after a boost, but will have a net velocity. Densities will remain the same. A moving source shouldn't matter -- any material will itself will set the rest-frame, and must to have a non-infinite propagation velocity. Whereas, emitting particles, these will have the velocity of the source. Ah! if it's a complex wave equation, you can put a varying phase on everything encoding the velocity. But moving to a complex field is in some sense putting more degrees-of-freedom in.
For Einsteinian relativity, a sphere expanding at light speed remains a sphere in any reference frame, but the density along it changes. A moving source should have that same pattern, so it would seem that local differences (i.e. derivatives) could "point out" the velocity of the source. This behaves the same for particles being emitted. Is there a first-order Lorentz invariant scalar wave-equation, or only second-order, using the d'Alembertian? And that seems to introduce another degree-of-freedom as well.
I'm still really confused. How do pions behave? Is it reasonable to consider them as a fundamental scalar field when they're modeled as pairs of quarks? Particularly when they have excited states that are spin-1?