Abstract
We introduce the Causal Compatibility Conjecture for the Events, Trees, Histories (ETH) approach to Quantum Theory (QT) in the semi-classical setting. We then prove that under the assumptions of the conjecture, points on closed causal curves are physically indistinguishable in the context of the ETH approach to QT and thus the conjecture implies a compatibility of the causal structures even in presence of closed causal curves. As a consequence of this result there is no observation that could be made by an observer to tell any two points on a closed causal curve apart. We thus conclude that closed causal curves have no physical significance in the context of the ETH approach to QT. This is an indication that time travel will not be possible in a full quantum theory of gravity and thus forever remain a fantasy.
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Notes
In the standard formulation of Quantum Theory one considers closed isolated physical systems. In this case one has \(\mathcal {E}_{P}=\mathcal {E}\) for all P. \(\mathcal {E}\) is the algebra generated by all physical observables of the closed system S. Such a system does not feature any event and, in the Heisenberg picture, the state is fixed and time evolution is given by a unitary automorphism from \(\mathcal {E}\) into itself. A measurement is then an external intervention where an “observer” randomly selects a self-adjoint operator - an observable - in \(\mathcal {E}\) to be measured. This process is governed by Born’s rule and cannot be described from within the system itself. Furthermore the random selection of observables enables different “observers” to choose different observables which needn’t commute. This leads to a whole array of puzzles.
The relevant self-adjoint operator determining an event in the ETH approach to Quantum Theory, on the contrary, is defined uniquely from within the system itself as we will see below.
To simplify the discussion we here assume the algebras to be of type I and thus isomorphic to the bounded linear operators on a Hilbert space. In a generic setting with massless modes one expects the algebras to be of type III instead. For technical details see again [11].
Other sensible choices might of course exist. For an in-depth explanation of the motivation for the particular choice made here see again [11].
The algebra \(\mathcal {Z}_{\omega _P}(\mathcal {E}_P)\) is an abelian von Neumann algebra. On a separable Hilbert space, it is generated by a single self-adjoint operator G, whose spectral decomposition yields the projections, \(\lbrace \pi _{\xi }, \xi \in \mathfrak {X} \rbrace\), describing a potential event.
We use the same notation for the algebra \(\langle S\rangle\) generated by a set of operators S.
In the context of black hole spacetimes and cosmological spacetimes it is unclear, whether one should consider the entire future of x or only the part intersecting with the domain of outer communication. Though likely this will not change the answer to the conjecture.
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Acknowledgements
C.F.P. is funded by the SNSF grant P2SKP2 178198. I would like to thank Jürg Fröhlich for a patient explanation of his ETH approach and for many interesting discussions, further I would like to thank Isha Kotecha, Markus Strehlau, Felix Finster and Marco Oppio for valuable feedback.
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Paganini, C.F. No Events on Closed Causal Curves. Found Phys 52, 26 (2022). https://doi.org/10.1007/s10701-022-00542-4
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DOI: https://doi.org/10.1007/s10701-022-00542-4