Partial Differential Equations and Quantum States in Curved Spacetimes
In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admiss...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/16/1936 |
| _version_ | 1797523032281972736 |
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| author | Zhirayr Avetisyan Matteo Capoferri |
| author_facet | Zhirayr Avetisyan Matteo Capoferri |
| author_sort | Zhirayr Avetisyan |
| collection | DOAJ |
| description | In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called <i>Hadamard states</i>—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution. |
| first_indexed | 2024-03-10T08:37:40Z |
| format | Article |
| id | doaj.art-feff3a6ffbad49a5802634a9e502348b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:37:40Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-feff3a6ffbad49a5802634a9e502348b2023-11-22T08:34:07ZengMDPI AGMathematics2227-73902021-08-01916193610.3390/math9161936Partial Differential Equations and Quantum States in Curved SpacetimesZhirayr Avetisyan0Matteo Capoferri1Department of Mathematics, University of California at Santa Barbara, South Hall, Santa Barbara, CA 93106, USASchool of Mathematics, Cardiff University, Senghennydd Rd, Cardiff CF24 4AG, UKIn this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called <i>Hadamard states</i>—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.https://www.mdpi.com/2227-7390/9/16/1936quantum field theorypartial differential equationshyperbolic propagatorsHadamard states |
| spellingShingle | Zhirayr Avetisyan Matteo Capoferri Partial Differential Equations and Quantum States in Curved Spacetimes Mathematics quantum field theory partial differential equations hyperbolic propagators Hadamard states |
| title | Partial Differential Equations and Quantum States in Curved Spacetimes |
| title_full | Partial Differential Equations and Quantum States in Curved Spacetimes |
| title_fullStr | Partial Differential Equations and Quantum States in Curved Spacetimes |
| title_full_unstemmed | Partial Differential Equations and Quantum States in Curved Spacetimes |
| title_short | Partial Differential Equations and Quantum States in Curved Spacetimes |
| title_sort | partial differential equations and quantum states in curved spacetimes |
| topic | quantum field theory partial differential equations hyperbolic propagators Hadamard states |
| url | https://www.mdpi.com/2227-7390/9/16/1936 |
| work_keys_str_mv | AT zhirayravetisyan partialdifferentialequationsandquantumstatesincurvedspacetimes AT matteocapoferri partialdifferentialequationsandquantumstatesincurvedspacetimes |