By Mark Burgess

ISBN-10: 0521813638

ISBN-13: 9780521813631

This publication discusses the classical foundations of box conception, utilizing the language of variational tools and covariance. It explores the bounds of what could be accomplished with merely classical notions, and indicates how those classical notions have a deep and critical reference to the second one quantized box concept, which follows on from the Schwinger motion precept. Its pragmatic view of box thought makes a speciality of concerns that are often passed over from quantum box concept texts and catalogs effects that are usually tough to discover within the literature.

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**Example text**

However, in gaseous matter or plasmas, this is not the case: the thermal motion of atoms relative to one another can be important, because of the Doppler effect. This fact can be utilized to good effect; for example, in laser cooling the motion of atoms relative to a laser radiation field can be used to bring all of the atoms into a common rest frame by the use of a resonant, frequency-dependent interaction. A Galilean-covariant expression can be written by treating the field as one co-moving mass, or as a linear super-position of co-moving masses.

53) The delta-function constraint in eqn. 52) ensures that the combination of waves satisfies the dispersion relation in eqn. 51). If we use the property of the delta function expressed in Appendix A, eqn. 15), then eqn. 52) may be written Aµ (x) = ˆµ 1 dn+1 k i(ki x i −ωt) ∂ω e A(k) (2π )n+1 c ki ∂k i √ √ × δ(k0 − k2 ) + δ(k0 + k2 ) . 55) where ω and ki satisfy the dispersion relation. 56) known as the group velocity. By choosing the coefficient C(k) for each frequency and wavelength, the super-position principle may be used to build up any complementary (steady state) solution to the Maxwell field.

13) is chosen to be zero, but, of course, any constant would do. Making this arbitrary (but not random) choice, is called choosing a gauge. It partially fixes the freedom to choose the scalar field s in eqns. 8). Specifically, eqn. 13) is called the Lorentz gauge. This common choice is primarily used to tidy up the equations of motion, but, as noted above, at some point one has to make a choice anyway so that a single pair of vector potentials (scalar, vector) corresponds to only one pair of physical fields (E, B).

### Classical covariant fields by Mark Burgess

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