标量理论中的非单射场重定义与量子不等价
Non-injective field redefinitions and quantum inequivalence in scalar theories
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中文总结 AI 辅助
研究通过特定多项式场重定义得到的标量理论，以三变量非单射映射和四标量映射为例，分析其场空间度规、原像数量及谱重数等，区分局部微扰等价与全局量子等价。
中文摘要 AI 辅助
我们研究通过具有恒定单位雅可比行列式的多项式场重定义拉回自由大质量多重态得到的标量理论。主要例子使用阿尔波格最近公布的三变量非单射映射。经线性归一化后，它定义了一个具有平坦、单位体积场空间度规和三个孤立真空的三标量西格玛模型。每个真空由质量为m的三个自由模式局部描述，精确运动方程在每张片上局部简化为自由克莱因 - 戈登方程。但全局理论并非单个自由理论：场空间度规不完整，目标空间上实原像数量变化，对易位置算符具有非常数的联合谱重数。这排除了场重定义的全局幺正实现和形式正则动量的正则外尔指数化。接着分析具有一般五次纤维的四标量映射，它展示了相同机制且有一个额外场控制纤维多项式。这两个例子区分了所选局部片上的微扰等价与全场空间的全局量子等价。
英文摘要
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.
