Strange solutions to field theories in one spatial dimension

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Abstract
Many models of interacting particles rely heavily for their solution on restriction to one-dimensional motion and a linearized kinetic energy. We examine this in detail, and find that the linearization can lead to patently strange and possible spurious solutions in first quantization. The usual, correct solutions are obtained only in second quantization. The strange solutions do not reduce to the usual plane wave determinantal solutions, even when the interactions are extinguished, and have the character of a condensed phase — a sort of Wigner lattice—for arbitrary interactions.