# Advances in Chemical Physics, Vol.119, Part 3. Modern by Myron W. Evans, Ilya Prigogine, Stuart A. Rice

By Myron W. Evans, Ilya Prigogine, Stuart A. Rice

Major advances have happened within the box because the past version, together with advances in mild squeezing, unmarried photon optics, section conjugation, and laser know-how. The laser is basically liable for nonlinear results and is largely utilized in all branches of technology, undefined, and medication.

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**Sample text**

W. evans and s. jeffers III. CLASSICAL LEHNERT AND PROCA VACUUM CHARGE CURRENT DENSITY In this section, gauge theory is used to show that there exist classical charge current densities in the vacuum for all gauge group symmetries, provided that the scalar field of gauge theory is identified with the electromagnetic field [O(3) level] or a component of the electromagnetic field [U(1) level]. The Lehnert vacuum charge current density exists for all gauge group symmetries without the Higgs mechanism.

There are several experimental reasons [42,47–61] for preferring O(3) over U(1) for electrodynamics. The vacuum charge density is also structured in general, but in the plane wave, first approximation is given by J0 ¼ k2 A0 m0 c ð191Þ because by definition, the time component of the vector A is zero. This is how it differs from the 4-vector Am, and why it is an independent variable in the method of functional variation used to derive Eq. (183) from an O(3) invariant Lagrangian. The vacuum transverse current densities are also structured, and in general they are J1 ¼ g g2 q1 A Â A þ A Â ðA Â A1 Þ m0 c m0 c ð192Þ J2 ¼ g g2 q2 A Â A þ A Â ðA Â A2 Þ m0 c m0 c ð193Þ In the plane-wave first approximation, they reduce to J1 ¼ Àg2 A1 A22 i J2 ¼ g2 A21 A2 j ð194Þ ð195Þ using the vector triple products: A Â ðA Â A1 Þ ¼ ÀA1 A22 i ð196Þ A Â ðA Â A2 Þ ¼ ÀA21 A2 j ð197Þ In SI units, the transverse vacuum current densities are given in the plane-wave first approximation by J1 ¼ Àg2 J 2 ¼ g2 A1 A22 i m0 c A21 A2 j m0 c ð198Þ ð199Þ 36 m.

100) produces the following result (reduced units) by functional variation: qn F mn ¼ ÀigðBÃ Dm B À BDm BÃ Þ À pﬃﬃﬃ pﬃﬃﬃ g2 m2 m A þ 2 2g2 aB1 Am þ 2agqm B2 l ð156Þ The term Àg2 m2 Am =l implies that the electromagnetic 4-potential Am has acquired mass. Simultaneously there appear two other terms. All four vacuum charge current densities produce vacuum energy through the equation ð EnðvacÞ ¼ J m ðvacÞAm dV ð157Þ Alternatively, Eq. (156) can be written from Eq. (146) in terms of the scalar A: qn F mn ¼ ÀigðAÃ Dm A À ADm AÃ Þ À g2 m2 pﬃﬃﬃ pﬃﬃﬃ Am þ 2 2g2 aA1 Am þ 2agqm A2 l ð158Þ Therefore, spontaneous symmetry breaking of the vacuum on the U(1) level produces new vacuum charge current densities that act as sources for the electromagnetic field and produce energy inherent in the topology of the vacuum.