Abstract
As presaged in Chap. 1, we will primarily be interested in understanding entanglement in holographic field theories. But before we get to this particular set of quantum systems, it is useful to build some intuition in a more familiar setting. In this and the next section, we will therefore focus our attention on getting some insight into the concept of entanglement and learn some of the techniques which are used to characterize it. The discussion here will also serve to build some technical machinery which will be useful in the holographic context.
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Notes
- 1.
As we will be primarily interested in relativistic systems, we will use d to indicate the total spacetime dimension of the field theory. When necessary to explicitly distinguish the spatial dimension, we will resort to the notation d = d s + 1, with d s denoting the number of spatial dimensions.
- 2.
The domains of dependence are causal sets which are determined simply where a given set of points can communicate to or be communicated from, etc. For instance, \(D[\mathcal{A}]\) is defined as the set of points in \(\mathcal{B}\) through which every inextensible causal curve intersects \(\mathcal{A}\). A technical complication to keep in mind is that we take \(\mathcal{A}\) to be an open subset of \(\Sigma \); consequently, \(D[\mathcal{A}]\) is an open subset of \(\mathcal{B}\). We refer the reader to [32] for a discussion of these concepts.
- 3.
To belabor the obvious, \(\rho _{\mathcal{A}}\log \rho _{\mathcal{A}}\) is not a linear operator on the Hilbert space.
- 4.
This is also known as the closed time-path formalism or the in-in formalism. A closely related discussion for open quantum systems appears in [36].
- 5.
When it is necessary to distinguish this construction, we will refer to the Lorentzian geometry with an explicit subscript, viz., \(\mathcal{B}_{\text{Lor}}\).
- 6.
There are some other pitfalls, which we will get to later. For instance, the replica symmetry may itself be broken dynamically.
- 7.
Once again modulo the fact that we need to supply suitable caveats to discuss theories with gauge invariance in which spatial regions do not necessarily allow for such a factorization.
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Rangamani, M., Takayanagi, T. (2017). Entanglement in QFT. In: Holographic Entanglement Entropy. Lecture Notes in Physics, vol 931. Springer, Cham. https://doi.org/10.1007/978-3-319-52573-0_2
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