By Donald Melrose

The box of quantum plasmas has an extended and numerous culture and is turning into of accelerating present curiosity, stimulated by way of purposes to micro-electronics and to targeted high-power lasers. during this publication, plasma kinetic concept is constructed a covariant (4-tensor) notation, which enables generalizations to incorporate all relativistic and electromagnetic results. Relativistic quantum results are integrated by means of synthesizing the idea with quantum electrodynamics (QED), together with the plasma responses in QED, and utilizing QED to calculate the plasma reaction 4-tensors.

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**Example text**

2. 3. 4. 6. 7. B. Melrose: Covariant theory of wave dispersion, Lect. Notes Phys. 2), for the 4-potential, plus a subsidiary equation that relates the Maxwell tensor to the 4-potential. This single equation is identiﬁed as the wave equation for the medium. In this section the wave equation is written down in covariant form and the corresponding Green’s function or photon propagator is derived from it. 2), viz. 5). The change from a theory for the electromagnetic ﬁeld in vacuo to a theory for the electromagnetic ﬁeld in a medium involves including this induced term on the left hand side of the equation.

2) The current, J µ (k), is separated into an induced (ind) part that describes the response of the medium and an extraneous (ext) part that acts as a source term: µ µ J µ (k) = Jind (k) + Jext (k). 3) is not uniquely deﬁned, and how the separation is to be made needs to be speciﬁed in any speciﬁc theory for the response of a medium. 2 Weak-turbulence expansion The weak-turbulence approximation involves assuming that the induced current is suﬃciently weak that an expansion in terms of the amplitude Aµ (k) of the electromagnetic ﬁeld converges rapidly.

As this name implies, when inside an integral over ω, ℘{1/(ω − ω0 )} implies that the Cauchy principal value of the integral is to be taken. 4 Plemelj formula The unit, step and sign functions are not independent, and the relation between them implies a relation between their Fourier transforms. 18) when Fourier transformed, implies 1 1 = ℘ − iπ δ(ω). 20) this Plemelj formula is interpreted as a separation into a nonresonant part, identiﬁed with the principal value part, and a resonant part, identiﬁed with the part involving the δ-function.