Taming non-analyticities of QFT observables
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arXiv
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| Format: | Preprint |
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2024
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| author | Grimm, Thomas W. Ravazzini, Giovanni van Vliet, Mick |
| author_facet | Grimm, Thomas W. Ravazzini, Giovanni van Vliet, Mick |
| contents | Many observables in quantum field theories are involved non-analytic functions of the parameters of the theory. However, it is expected that they are not arbitrarily wild, but rather have only a finite amount of geometric complexity. This expectation has been recently formalized by a tameness principle: physical observables should be definable in o-minimal structures and their sharp refinements. In this work, we show that a broad class of non-analytic partition and correlation functions are tame functions in the o-minimal structure known as $\mathbb{R}_{\mathscr{G}}$ - the structure defining Gevrey functions. Using a perturbative approach, we expand the observables in asymptotic series in powers of a small coupling constant. Although these series are often divergent, they can be Borel-resummed in the absence of Stokes phenomena to yield the full partition and correlation functions. We show that this makes them definable in $\mathbb{R}_{\mathscr{G}}$ and provide a number of motivating examples. These include certain 0-dimensional quantum field theories and a set of higher-dimensional quantum field theories that can be analyzed using constructive field theory. Finally, we discuss how the eigenvalues of certain Hamiltonians in quantum mechanics are also definable in $\mathbb{R}_{\mathscr{G}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Taming non-analyticities of QFT observables Grimm, Thomas W. Ravazzini, Giovanni van Vliet, Mick High Energy Physics - Theory Mathematical Physics Many observables in quantum field theories are involved non-analytic functions of the parameters of the theory. However, it is expected that they are not arbitrarily wild, but rather have only a finite amount of geometric complexity. This expectation has been recently formalized by a tameness principle: physical observables should be definable in o-minimal structures and their sharp refinements. In this work, we show that a broad class of non-analytic partition and correlation functions are tame functions in the o-minimal structure known as $\mathbb{R}_{\mathscr{G}}$ - the structure defining Gevrey functions. Using a perturbative approach, we expand the observables in asymptotic series in powers of a small coupling constant. Although these series are often divergent, they can be Borel-resummed in the absence of Stokes phenomena to yield the full partition and correlation functions. We show that this makes them definable in $\mathbb{R}_{\mathscr{G}}$ and provide a number of motivating examples. These include certain 0-dimensional quantum field theories and a set of higher-dimensional quantum field theories that can be analyzed using constructive field theory. Finally, we discuss how the eigenvalues of certain Hamiltonians in quantum mechanics are also definable in $\mathbb{R}_{\mathscr{G}}$. |
| title | Taming non-analyticities of QFT observables |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2407.08815 |