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Additional resources for Advansed plasma theory
24) This can be written in an explicitly covariant form by means of the ﬁeld tensor Fµν = ∂µ Aν − ∂ν Aµ . 25) We ﬁnd 1 Lem = − F µν Fµν . 26) 4 The tensor Fµν is invariant under gauge transformations, as one can verify directly using eq. 19). 27) from which it is immediately clear that Fµν , and therefore Lem , are gaugeinvariant. We should stress that Lem is the only possible term that depends on Aµ and its ﬁrst derivatives, compatible with gauge and Lorentz invariance, C and P invariance, and renormalizability.
19) is ill deﬁned. There is an obvious physical reason behind this situation, that arises from the fact that spin rotation matrices are two-valued functions of the rotation angle. There is also a mathematical reason, that provides a deeper understanding of this phenomenon. We will limit our discussion to rotations, since they are responsible for the double-valuedness of L(Λ). The crucial point is that the Lorentz group and the rotation group are continuous groups, which means that the action of a rotation on a physical system or reference frame is a continuous operation, resulting from the combination of a sequence of inﬁnitesimal transformations.
Unitarity is recursively restored taking radiative corrections into account. In particular, the one-loop contribution corresponds to the diagram with two internal lines and two vertices, and hence one loop: 1 Because of time-reversal invariance. 46) and second, the amplitude carries a combinatorial factor, given by the inverse of the number of symmetry transformation of the diagram. In the present case this factor is 1/2, because the diagram is symmetric under permutation of the internal lines.