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From "Le Bellac, Quantum and statistical field theory, 10.5.2-Massive vector field":

"The longitudinal part of the propagator [itex]k_{\mu}D^{\mu\nu}[/itex] has no pole at

[itex]k^2=m^2[/itex], so the longitudinal part doesn't constitute a dynamical degree of freedom."

I have two questions:

1) Why the propagator doesn't represent a dynamical degree of freedom if it hasn't any pole?

How do you demonstrate that physical particles correspond to the pole of the propagator?

2) The propagator [itex]D^{\mu\nu}[/itex] is a rank-2 tensor. The longitudinal part is [itex]k_{\mu}D^{\mu\nu}[/itex] and it is a vector, so, how can it be a propagator?

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# Poles of the propagator

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