Møller Maps for Dirac Fields in External Backgrounds

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2024-07-24 DOI:10.1007/s11040-024-09487-8

Valentino Abram, Romeo Brunetti

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Abstract

In this paper we study the foundations of the algebraic treatment of classical and quantum field theories for Dirac fermions under external backgrounds following the initial contributions already present in various places in the literature. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions \(d\ge 4\) and to external fields modelled with trivial principal bundles. In particular, we construct the classical Møller maps intertwining the configuration spaces of charged and uncharged fermions, and we show some of its properties in the case of a U(1) gauge charge. In the last part, as a first step towards a quantization of the theory, we explore the combination of the classical Møller maps with Hadamard bidistributions and prove that they are involutive isomorphisms (algebraically and topologically) between suitable (formal) algebras of functionals (observables) over the configuration spaces of charged and uncharged Dirac fields.

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外部背景中狄拉克场的默勒图

在本文中，我们研究了外部背景下狄拉克费米子经典和量子场论代数处理的基础，这是继文献中不同地方已有的初步贡献之后的又一研究。这种处理仅限于维数（d）的全局双曲性质的可收缩时空，以及用琐碎主束建模的外部场。特别是，我们构建了交织带电和不带电费米子构型空间的经典莫勒映射，并展示了它在U(1)规电荷情况下的一些性质。在最后一部分，作为理论量子化的第一步，我们探讨了经典莫勒映射与哈达玛德双分布的结合，并证明它们是带电和不带电狄拉克场构型空间上合适（形式）的函数（观测值）代数学和拓扑学同构。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.