By W. P. Healy
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Extra info for Non-Relativistic Quantum Electrodynamics
11, 1899. Healy, W. P. (1979). “Line-integral Lagrangians for the electromagnetic fields interacting with charged particles. “ Phys. R ev. A19, 2353. Jeffreys, H. and Jeffreys, B. S. (1956). “Methods of Mathematical Physics” . Cambridge University Press. Lee, T. D. and Yang, C. N. (1956). ” Phys. R ev. 104, 254. Lorentz, H. A. (1915). “The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat” . Teubner, Leipzig. Maxwell, J. C. (1891). A Treatise on Electricity and Magnetism” .
35) 4>(x91, ê) = i ê . 36) a(jc, t, ê) = J—oc J—oo Now the gauge of the potentials obtained from these by averaging over all directions of ê is the Coulomb gauge (Belinfante 1962). 37) where the last line results from an integration by parts and where the (positive) volume element d 3 x' is u2 d ild (—u). 37) is solenoidal, as may be verified by differentiating with respect to x under the integral sign. Thus the potentials in the Coulomb gauge are the isotropic averages of the potentials in those line integral gauges for which the integration paths consist of fixed parallel straight lines coming from infinity and ending at the field points.
76) preserve the local energy and momentum balance equations as well as the properties (i) to (iii). 74), the angular momentum balance is also preserved. 69), it follows that the angular momentum in V is increased by an amount §s x x s dS and that the total moment of the outward forces acting on S is decreased by just the time derivative of this quantity. This corresponds to the increase of §s s dS in the linear momentum and the decrease of (d/di)§s s dS in the net outward force. REFERENCES Belinfante, F.
Non-Relativistic Quantum Electrodynamics by W. P. Healy