The non-relativistic limits of the Maxwell and Dirac equations: the role of Galilean and gauge invariance

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Peter Holland, Harvey R. Brown
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The aim of this paper is to illustrate four properties of the non-relativistic limits of relativistic theories: (a) that a massless relativistic field may have a meaningful non-relativistic limit, (b) that a relativistic field may have more than one non-relativistic limit, (c) that coupled relativistic systems may be ?more relativistic? than their uncoupled counterparts, and (d) that the properties of the non-relativistic limit of a dynamical equation may differ from those obtained when the limiting equation is based directly on exact Galilean kinematics. These properties are demonstrated through an examination of the non-relativistic limit of the familiar equations of first-quantized QED, i.e., the Dirac and Maxwell equations. The conditions under which each set of equations admit non-relativistic limits are given, particular attention being given to a gauge-invariant formulation of the limiting process especially as it applies to the electromagnetic potentials. The difference between the properties of a limiting theory and an exactly Galilean covariant theory based on the same dynamical equation is demonstrated by examination of the Pauli equation.

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Peter R. Holland

Peter R. Holland is an English theoretical physicist, known for his work on foundational problems in quantum physics and in particular his book on the pilot wave theory and the de Broglie-Bohm causal interpretation of quantum mechanics. Holland was educated at Hazelwick Compre...

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