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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.03279 |
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| _version_ | 1866909999303753728 |
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| author | Dereziński, Jan Gaß, Christian |
| author_facet | Dereziński, Jan Gaß, Christian |
| contents | We discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold $M$. Typically, we assume that $M$ is globally hyperbolic. The term propagator here refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Quantum Field Theory.
The off-shell setting is based on the Hilbert space $L^2(M)$. It leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often coincide with the so-called in-out Feynman and out-in anti-Feynman propagator. On some special spacetimes, the sum of the operator-theoretic Feynman and anti-Feynman propagator equals the sum of the forward and backward propagator. This is always true on static stable spacetimes and, curiously, in some other cases as well.
The on-shell setting is based on the Krein space $\mathcal{W}_{\rm KG}$ of solutions of the Klein-Gordon equation. It allows us to define 2-point functions associated to two, possibly distinct, Fock states as the Klein-Gordon kernels of projectors onto maximal uniformly positive subspaces of $\mathcal{W}_{\rm KG}$.
After a general discussion, we review a number of examples. We start with static and asymptotically static spacetimes, which are especially well-suited for Quantum Field Theory. Then we discuss FLRW spacetimes, reducible by a mode decomposition to 1-dimensional Schrödinger operators. We compare various approaches to de Sitter space where, curiously, the off-shell approach gives non-physical propagators. Finally, we discuss the universal cover of anti-de Sitter spaces, where the on-shell approach may require boundary conditions, unlike the off-shell approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Propagators in curved spacetimes from operator theory Dereziński, Jan Gaß, Christian Mathematical Physics We discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold $M$. Typically, we assume that $M$ is globally hyperbolic. The term propagator here refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Quantum Field Theory. The off-shell setting is based on the Hilbert space $L^2(M)$. It leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often coincide with the so-called in-out Feynman and out-in anti-Feynman propagator. On some special spacetimes, the sum of the operator-theoretic Feynman and anti-Feynman propagator equals the sum of the forward and backward propagator. This is always true on static stable spacetimes and, curiously, in some other cases as well. The on-shell setting is based on the Krein space $\mathcal{W}_{\rm KG}$ of solutions of the Klein-Gordon equation. It allows us to define 2-point functions associated to two, possibly distinct, Fock states as the Klein-Gordon kernels of projectors onto maximal uniformly positive subspaces of $\mathcal{W}_{\rm KG}$. After a general discussion, we review a number of examples. We start with static and asymptotically static spacetimes, which are especially well-suited for Quantum Field Theory. Then we discuss FLRW spacetimes, reducible by a mode decomposition to 1-dimensional Schrödinger operators. We compare various approaches to de Sitter space where, curiously, the off-shell approach gives non-physical propagators. Finally, we discuss the universal cover of anti-de Sitter spaces, where the on-shell approach may require boundary conditions, unlike the off-shell approach. |
| title | Propagators in curved spacetimes from operator theory |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2409.03279 |