嵌套积分生成定理：从算符恒等式到精确积分恒等式族
Nested Integral Generator Theorem: From Operator Tautologies to Families of Exact Integral Identities
AI总结：
本文利用嵌套积分生成定理，从算符恒等式出发生成精确积分恒等式族，建立算符等式到积分恒等式的映射，导出内积与博赫纳积分交换条件，给出算符函数积分表示，并通过多种算符及非平凡例子进行说明。
AI中文摘要：
在整个量子理论、量子场论及相关数学物理领域，插入单位分解是表示态和算符的标准技术。本文通过嵌套积分生成定理将此过程提升到一个严格的、与表示无关的框架。该定理从算符恒等式出发，通过连续插入单位分解并投影到任意正交基向量上，系统地生成精确的多重积分恒等式。所得构造适用于作用于任意目标态的闭算符的任意有限组合，建立了从算符等式到精确积分恒等式族的一般映射。利用向量值积分理论，导出了内积可与博赫纳积分严格交换的显式且可验证的充分条件。作为直接结果，该定理给出了单个算符函数的精确积分表示。通过单模算符、多项式和解析算符以及单模和双模高斯酉算符进行了说明，还应用于超出标准高斯计算的两个非平凡例子：克尔压缩相干态重叠的精确积分表示以及用双变量埃尔米特多项式表示的复合双模高斯网络的精确福克基矩阵元。
英文摘要：
Inserting resolutions of the identity is a standard technique for representing states and operators throughout quantum theory, quantum field theory, and related areas of mathematical physics. This paper elevates this procedure to a rigorous, representation-independent framework through the Nested Integral Generator Theorem. Starting from operator tautologies, the theorem systematically generates exact multi-fold integral identities by successive insertions of continuous resolutions of the identity followed by projection onto arbitrary orthonormal basis vectors. The resulting construction applies to arbitrary finite compositions of closed operators acting on arbitrary target states and establishes a general mapping from operator equalities to families of exact integral identities. Using the theory of vector-valued integration, explicit and verifiable sufficient conditions are derived under which inner products may be interchanged rigorously with Bochner integrals, thereby placing a step that is often left implicit in the physics literature on a firm mathematical foundation. As an immediate consequence, the theorem yields exact integral representations for individual operator functions. Its scope is illustrated through elementary, polynomial, and analytic single-mode operators, as well as single- and two-mode Gaussian unitaries, including squeezing and beam splitting. The framework is further applied to two nontrivial examples beyond standard Gaussian calculations: an exact integral representation of a Kerr-squeezed coherent-state overlap and exact Fock-basis matrix elements for a composite two-mode Gaussian network expressed in terms of bivariate Hermite polynomials.
