Non-injective field redefinitions and quantum inequivalence in scalar theories

We study scalar theories obtained by pulling a free massive multiplet back through a polynomial field redefinition with constant unit Jacobian. Our main example uses the three-variable noninjective map recently announced by Alpöge. After a linear normalization, it defines a three-scalar sigma model with a flat, unit-volume field-space metric and three isolated vacua. Each vacuum is locally described by three free modes of mass m, and the exact equations of motion reduce locally on each sheet to free Klein--Gordon equations. The global theory is nevertheless not a single free theory: the field-space metric is incomplete, the number of real preimages changes across target space, and the commuting position operators have nonconstant joint spectral multiplicity. This rules out a global unitary implementation of the field redefinition and a regular Weyl exponentiation of the formal canonical momenta. We then analyze a four-scalar map with a generic quintic fiber. It exhibits the same mechanism with an additional field that controls the fiber polynomial. The two examples separate perturbative equivalence on a chosen local sheet from global quantum equivalence of the full field space.

Source: arXiv:2607.18166v1 - http://arxiv.org/abs/2607.18166v1 PDF: https://arxiv.org/pdf/2607.18166v1 Original Link: http://arxiv.org/abs/2607.18166v1