$Φ^4_3$ Theory from many-body quantum Gibbs states
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908493784547328 |
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| author | Nam, Phan Thành Zhu, Rongchan Zhu, Xiangchan |
| author_facet | Nam, Phan Thành Zhu, Rongchan Zhu, Xiangchan |
| contents | We derive the $Φ^4_3$ measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas. To be precise, starting from many-body quantum mechanics, where the problem is linear and regular but involving non commutative operators, we justify the emergence of the $Φ^4_3$ measure as a semiclassical limit which captures the formation of Bose--Einstein condensation just above the critical temperature. We employ and develop several tools from both stochastic quantization and many-body quantum mechanics. Since the quantum problem is typically formulated using a nonlocal interaction potential, our first key step involves approximating the $Φ^4_3$ measure through a Hartree measure with nonlocal interaction, achieved by developing new techniques in paracontrolled calculus. The connection between the quantum problem and the Hartree measure emerges through a variational interplay between classical and quantum models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $Φ^4_3$ Theory from many-body quantum Gibbs states Nam, Phan Thành Zhu, Rongchan Zhu, Xiangchan Mathematical Physics Analysis of PDEs Probability We derive the $Φ^4_3$ measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas. To be precise, starting from many-body quantum mechanics, where the problem is linear and regular but involving non commutative operators, we justify the emergence of the $Φ^4_3$ measure as a semiclassical limit which captures the formation of Bose--Einstein condensation just above the critical temperature. We employ and develop several tools from both stochastic quantization and many-body quantum mechanics. Since the quantum problem is typically formulated using a nonlocal interaction potential, our first key step involves approximating the $Φ^4_3$ measure through a Hartree measure with nonlocal interaction, achieved by developing new techniques in paracontrolled calculus. The connection between the quantum problem and the Hartree measure emerges through a variational interplay between classical and quantum models. |
| title | $Φ^4_3$ Theory from many-body quantum Gibbs states |
| topic | Mathematical Physics Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2502.04884 |