Toward quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis
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arXiv
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| Format: | Preprint |
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2024
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| author | Steingasser, Thomas Kaiser, David I. |
| author_facet | Steingasser, Thomas Kaiser, David I. |
| contents | We revisit the path integral description of quantum tunneling and lay the groundwork for its generalization to excites states through real-time path integral techniques. For clarity, we focus on the simple toy model of a point particle in a double-well potential, for which we perform all steps explicitly. Instead of performing the familiar Wick rotation from physical to imaginary time -- which is inconsistent with the requisite boundary conditions when treating tunneling from states other than the false vacuum -- we regularize the path integral by adding an infinitesimal complex contribution to the Hamiltonian, while keeping time strictly real. We find that this gives rise to a complex stationary-phase solution, in agreement with recent insights from Picard-Lefshitz theory. We then show that there exists a class of analytic solutions for the corresponding equations of motion, which can be made to match the appropriate boundary conditions in the physically relevant limits of a vanishing regulator and an infinite physical time. We provide a detailed discussion of this non-trivial limit. We find that, for systems without an explicit time-dependence, our approach reproduces the picture of an instanton-like solution defined on a finite Euclidean-time interval. Lastly, we discuss the generalization of our approach to broader classes of systems, for which it serves as a reliable framework for high-precision calculations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00099 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toward quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis Steingasser, Thomas Kaiser, David I. High Energy Physics - Theory High Energy Physics - Phenomenology Quantum Physics We revisit the path integral description of quantum tunneling and lay the groundwork for its generalization to excites states through real-time path integral techniques. For clarity, we focus on the simple toy model of a point particle in a double-well potential, for which we perform all steps explicitly. Instead of performing the familiar Wick rotation from physical to imaginary time -- which is inconsistent with the requisite boundary conditions when treating tunneling from states other than the false vacuum -- we regularize the path integral by adding an infinitesimal complex contribution to the Hamiltonian, while keeping time strictly real. We find that this gives rise to a complex stationary-phase solution, in agreement with recent insights from Picard-Lefshitz theory. We then show that there exists a class of analytic solutions for the corresponding equations of motion, which can be made to match the appropriate boundary conditions in the physically relevant limits of a vanishing regulator and an infinite physical time. We provide a detailed discussion of this non-trivial limit. We find that, for systems without an explicit time-dependence, our approach reproduces the picture of an instanton-like solution defined on a finite Euclidean-time interval. Lastly, we discuss the generalization of our approach to broader classes of systems, for which it serves as a reliable framework for high-precision calculations. |
| title | Toward quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis |
| topic | High Energy Physics - Theory High Energy Physics - Phenomenology Quantum Physics |
| url | https://arxiv.org/abs/2402.00099 |