Abstract
We review some recent results on Sorkin’s spacetime formulation of the entanglement entropy (SSEE) for a free quantum scalar field both in the continuum and in manifold-like causal sets. The SSEE for a causal diamond in a 2d cylinder spacetime has been shown to have a Calabrese–Cardy form, while for de Sitter and Schwarzschild de Sitter horizons in dimensions \(d>2\), it matches the mode-wise von-Neumann entropy. In these continuum examples the SSEE is regulated by imposing a UV cut-off. Manifold-like causal sets come with a natural covariant spacetime cut-off and thus provide an arena to study regulated QFT. However, the SSEE for different manifold-like causal sets in \(d=2\) and \(d=4\) has been shown to exhibit a volume rather than an area law. The area law is recovered only when an additional UV cut-off is implemented in the scaling regime of the spectrum which mimics the continuum behaviour. We discuss the implications of these results and suggest that a volume-law may be a manifestation of the fundamental non-locality of causal sets and a sign of new UV physics.
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Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Notes
See [9] and references therein for a review of causal set theory.
This additional condition is not satisfied for example for a causal diamond in the \(d=2\) cylinder spacetime [18].
One may refer to [18] for details.
\(\lambda ^{cs}\) has the same physical dimensions as \(i\Delta \) while \(\lambda \) has the physical dimensions of \([length]^2\).
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Acknowledgements
We would like to thank Yasaman Yazdi and Maximillian Ruep for discussions. NX is supported by the AARMS fellowship at UNB.
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Mathur, A., Surya, S. & Nomaan, X. Spacetime entanglement entropy: covariance and discreteness. Gen Relativ Gravit 54, 74 (2022). https://doi.org/10.1007/s10714-022-02948-x
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DOI: https://doi.org/10.1007/s10714-022-02948-x