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Remark on charge conjugation in the non relativistic limit. (English) Zbl 1106.81029
Summary: We study the non relativistic limit of the charge conjugation operation \(C\) in the context of the Dirac equation coupled to an electromagnetic field. The limit is well defined and, as in the relativistic case, \(C\), \(P\) (parity) and \(T\) (time reversal) are the generators of a matrix group isomorphic to a semidirect sum of the dihedral group of eight elements and \(\mathbb Z_2\). The existence of the limit is supported by an argument based in quantum field theory. Also, and most important, the limit exists in the context of galilean relativity. Finally, if one complexifies the Lorentz group and therefore the galilean spacetime \(x_\mu\), then the explicit form of the matrix for \(C\) allows to interpret it, in this context, as the complex conjugation of the spatial coordinates: \(\vec{x} \to \vec{x}^*\). This result is natural in a fiber bundle description.

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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