Field operators and their spectral properties in finite-dimensional quantum field theory

Miguel Navarro

ming Cheng

This work analyzes the fundamental distinction between finite-dimensional quantum systems and infinite-dimensional wavefunction systems. It introduces three mathematically independent notions of dimension—linear dimension, parameter dimension, and degree-of-freedom dimension—and shows that finite-dimensional systems are “triple-finite,” while wavefunction systems such as L^2(R) are “triple-infinite.” The paper distinguishes geometric continuity on finite-dimensional projective manifolds (e.g., the Bloch sphere) from structural continuity arising from the uncountable degrees of freedom of continuous configuration spaces. It further explains why infinite-dimensional systems cannot be reduced to finite-dimensional ones, drawing on functional analysis, spectral theory, and the properties of unbounded operators. Intermediate systems such as L^2([0,L]) and the extension to quantum field theory are also discussed, illustrating how spatial topology shapes spectral and dynamical behavior. The framework clarifies the categorical difference between finite- and infinite-dimensional quantum state spaces and highlights their complementary roles in quantum theory. 本文系统分析了有限维量子系统与无限维波函数系统之间的根本差异。文章提出三种彼此独立的“维度”概念——线性维度、参数维度与自由度维度,并指出有限维系统是“三重有限”,而波函数系统(如 L^2(R))是“三重无限”。有限维系统的连续性来自射影流形的几何结构(如 Bloch 球),而无限维系统的连续性来自连续构型空间所带来的不可数自由度。文章进一步利用泛函分析与谱理论论证了无限维系统对有限维系统的不可约化性,并讨论了中间系统 L^2([0,L]) 以及量子场论中的更高层次结构连续性。该框架揭示了有限维与无限维量子态空间在范畴上的本质差异,并说明它们在量子理论中所扮演的互补角色。

G. Rivlis

Vladimir A. Geyler, I. Yu. Popov

A solvable model of a double quantum electron layer in a magnetic field based on the theory of self-adjoint extensions of symmetric operators is suggested. The spectral problem reduces to a finite-dimensional one through the use of harmonic analysis on a magnetic translations group. Properties of a gap between symmetric and antisymmetric states are discussed. The model gives a linear dependence of spectral parameters on a magnetic field B.

John Edward Gough, Joachim Kupsch

Mikhail Nemirovsky

Abstract We formulate a minimal operator-theoretic criterion—the Spectral Confinement Class (SCC)—that any quantum theory exhibiting confinement-like behavior in four space-time dimensions should satisfy. This criterion consists of three necessary spectral properties: (i) a strict spectral gap above the vacuum, (ii) exponential clustering of connected correlation functions, and (iii) the absence of massless non-singlet excitations with respect to an internal compact symmetry (e.g., color). Within a general multichannel lattice framework, we prove that gate-induced spectral localization provides an explicit mechanism realizing SCC. Under minimal structural assumptions—local spectral rigidity with a gap m₀>0 uniform in the channel dimension N, bounded finite-range interactions, and a structural gate condition preventing spectral percolation—we derive a uniform lower bound on the spectral gap Δ ≥ m₀ - 2·z(d)·g·Vₘₐₓ where z(d)=2d is the lattice coordination number. The bound is uniform in system volume and channel dimension and exhibits a dimensional penalty: for fixed coupling g>0, the admissible gap window shrinks with increasing spatial dimension. We emphasize that SCC is necessary but not sufficient for confinement in the Yang–Mills sense. Its role is that of a spectral admissibility filter: candidate confining theories should satisfy SCC at the operator level; failure to do so presents obstacles to confinement regardless of gauge structure. In this sense, SCC functions as a spectral admissibility filter: candidate confining theories must satisfy it at the operator level, independently of gauge-theoretic or geometric mechanisms. Keywords Primary Keywords: spectral confinement class • spectral gap • gate condition • dimensional penalty • Yang–Mills theory • confinement • lattice Hamiltonians • multichannel systems • spectral localization Secondary Keywords: exponential clustering • local spectral rigidity • excitation projectors • coordination number • four-dimensional quantum field theory • gauge theories • Wilson loops • operator theory • variational methods • mass gap problem Mathematical Subject Classification (MSC 2020): Primary: 81T13 (Yang-Mills and other gauge theories), 81Q10 (Selfadjoint operator theory in quantum mechanics) Secondary: 82B20 (Lattice systems), 47A10 (Spectrum and spectral theory)