用于描述量子场论（QFT）中重整化福克空间的扩展态空间
Extended State Space for describing renormalized Fock spaces in QFT
AI总结：
针对一类简单极化子模型的非微扰重整化中奇异性导致的无限自能等问题，研究通过域扩张构建扩展态向量空间，严格定义相关算子，得到福克空间上的稠定自伴重整化哈密顿量。
摘要：
我们重新研究了一类含静止费米子的简单极化子模型的非微扰重整化。所考虑的色散关系和形状因子允许具有高度奇异性，从而可能出现无限自能和波函数重整化，且对角化的缀饰变换可能无法在福克空间上实现。我们未采用截断方法，而是严格地将自能和波函数重整化解释为合适向量空间的元素，以及复数的一个域扩张。此外，我们在该域扩张上定义了两个扩展态向量空间（ESSs），它们包含福克空间的一个稠密子空间，同时还纳入了非平方可积波函数。这些扩展态向量空间的元素可被视为虚粒子的态，以数学上严格的方式进行描述。无截断的哈密顿量、形式上为无穷大的抵消项以及缀饰变换随后可被定义为两个福克空间扩张的特定子空间之间的线性算子。通过这种方式，我们得到了一个重整化的哈密顿量，它可实现为福克空间上的稠定自伴算子。
英文摘要：
We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space.
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