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Intercontinental Wilson line as a proof of state dependence

Today, the hep-th arXiv offers us several noteworthy papers.

First, six authors including Heckman, Morrison, and Vafa study the little string theories – non-gravitational but non-local theories describing decoupled dynamics on NS5-branes (not to mention other equivalent definitions) – using compactifications of (Vafa's) F-theory in various geometric phases. They conclude that little string theories (at least those with two or more tensor multiplets) may be rather easily obtained from a six-dimensional superconformal field theory. Also, all six-dimensional superconformal field theories may be embedded in a little string theory. Little string theory's existence was pointed out almost 20 years ago and this beast seemed mysterious – and it's remarkable that people are befriending it and demystifying it in this way.

The remaining two preprints I will mention are dedicated to the black hole interior in quantum gravity.

First, Don Page and four Asian collaborators (sorry for this description, 4 folks) point out something rather obvious in their short paper. If Polchinski et al. really mean that there is a deadly firewall at the event horizon, it's pretty bad because the precise location of the event horizon may be affected by events that are occurring at distant places. In particular, they consider the Hawking radiation that shrinks the black hole. This radiation is random so its rate has statistical fluctuations which are enough to influence where the deadly firewall should sit and kill you. This would allow the infalling observer to acausally see something about the future – namely whether the amount of the Hawking radiation will be above or below the statistical average. The firewalls are in trouble for this reason (not to mention other reasons), especially because the fluctuations may separate the firewall from the event horizon by an arbitrary amount.

The most interesting paper – one referred to in the title above – is one by Monica Guica and Dan Jafferis,

On the construction of charged operators inside an eternal black hole
It's also an anti-firewall paper of a sort but importantly enough, it has a much more positive flavor (saying how things actually work, not how they don't). They point out that a new operator, a Wilson line connecting two boundaries in models of AdS/CFT, are an unavoidable part of the holographic dictionary, at least in the spacetime of an eternal black hole they focus on.

Because of this intercontinental Wilson line, it's clear that they have been partially scooped by Dan Harlow who published a paper on that topic three weeks ago. However, Harlow emphasized the implications of the Wilson lines for the Weak Gravity Conjecture and other things.

Monica and Dan have rather different priorities. They show that this intercontinental Wilson line construction basically proves the state dependence by Suvrat Raju and Kyriakos Papadodimas. In quantum gravity, local operators in the bulk only have well-defined matrix elements with respect to microstates that only differ from each other by the action of a limited number of local excitations. The parameterization of the Hilbert space in terms of the local fields just can't hold in the full generality. Whenever I write these things, I feel that they're so obvious, almost tautological. For example, it seems to me that the sentence "The param..." is equivalent to a rather uncontroversial statement that quantum gravity isn't a local field theory: the local fields are just not precise enough to parameterize every detail about the black hole microstates, they are just effective observables that work approximately in certain conditions.

For microstates that are "totally different", the form (matrix elements) of a local operator in the bulk has to be defined from scratch even though the operators seem to "do the same" in both cases because by the long-distance implications, the microstates may look indistinguishable. There are no operators that would represent the local fields well on the whole Hilbert space. This is no violation of the postulates of quantum mechanics. The full Hilbert space may still be described using the CFT or otherwise "microscopic" operators which have well-defined matrix elements on the full Hilbert space – but those just don't imply the exact bulk locality (or, almost equivalently, they don't allow to exactly and universally define the local bulk operators).

Now, sociologically, you could view this synergy as a conspiracy. Monica Guica, Dan Jafferis, Suvrat Raju, and Kyriakos Papadodimas were all Harvard PhD students when I was working at that place. So maybe the Harvard PhD students have conspired to promote certain views. But you know, there is an excuse. These four physicists had five different advisers ;-) – each of them had a different background.

OK, how do Monica and Dan prove the state dependence? They study quantum gravity on the background of an eternal black hole whose Penrose diagram is basically a square. Draw a maximum X cross into this square and you divide the spacetime to four regions: right, future (up), left, past (down). Monica and Dan then solve the following question:
What are the CFT operators dual to a charged bulk field \(\phi\) in these four quadrants?
Just to be sure, they assume that the bulk theory has not only gravity but a Maxwell field, too. In the right quadrant, the bulk field \(\phi_R\) may be written in terms of some fields in the "right CFT"\[

\phi_R = \int K_R {\mathcal O}_R

\] which is standard. The right boundary points that contribute are sitting in the "wedge" of points that are spacelike separated from the locus of \(\phi_R\) in the bulk. When only one boundary is relevant, the CFT has no problem to represent charged operators.

Similarly, when the charged field sits near the left boundary of the bulk, you get the same formula with \(L\) replacing \(R\). But what if you consider \(\phi_F\) at the points in the future quadrant? (One could similarly consider the past one but the future quadrant is more resilient because black holes that exist agree with the eternal black hole there; the past quadrant only exists in white holes that don't exist in the real world. BTW I renamed the quadrants from \(I,II,III,IV\) to \(R,F,L,P\), right future left past.)

Well, you could imagine a simple sum\[

\phi_F = \int K_R {\mathcal O}_R + \int K_L {\mathcal O}_L

\] of the two terms, the left term and the right term, corresponding to the two causal "beams". But such a sum is clearly wrong. It behaves badly when you slowly move from the right quadrant to the future one. It has wrong vanishing or non-vanishing commutators with the charges \(Q_L\) and/or \(Q_R\), or violates the charge conservation laws. Their discussion on page 3 is more coherent than my sketch.

Moreover, you may see that the sum above is sort of "discontinuous" and mixes apples with oranges. The two boundaries had initially degrees of freedom that had nothing to do with the other boundary so you can't expect a nice operator in a smooth bulk spacetime by abruptly switching from one to the other (or from one to the sum).

Their fix is clever enough. First, they modify the bulk operator by adding the factor of a bulk-to-boundary untraced Wilson line (the path-ordered exponential of the bulk gauge field)\[

\hat\phi(y) = \phi(y) P \,\exp\zav{ iq\int_y^{\hat x_R} A }

\] and then, not to add apples and oranges, they convert the "left" and "right" terms in the sum to their "common language" by multiplying the left term with the factor of \[

W_{LR} (\hat x_L|\hat x_R) = P\,\exp \zav{ iq\int_{\hat x_L}^{\hat x_R} A }

\] The "right" term in the expression for the operator \(\phi_F\) in the future quadrant was already expressed "relatively to the boundary point \(\hat x_R\)" when we added the Wilson line. The "left" term may be similarly translated relatively to a point \(\hat x_L\) on the left boundary and the \(L\)-\(R\) Wilson line translates it so that it's expressed relatively to the point \(\hat x_R\), too.

So the final expression for the charged bulk field in the future quadrant is\[

\hat\phi(y) &= \int d^d x_R K_R(y|x_R){\mathcal O}_R^{(j)}(x_R)+\\
&+ W_{LR}(\hat x_L|\hat x_R) \int d^d x_L K_L(y|x_L){\mathcal O}_L^{(j)}(x_L)

\] Now it makes sense. The commutators and charge conservation laws are right from the viewpoint of both boundaries, and so on. Monica and Dan discuss tons of issues in a 3-dimensional bulk as well as higher-dimensional bulks.

What does it have to do with state dependence?

Well, the appearance of the transcontinental Wilson line operator demands state dependence because this operator just can't be defined on the whole Hilbert space. It encodes some "relative phase" between two points at the two disconnected boundaries and the bulk charged operator \(\phi\) is sensitive to this relative phase. However, the phase is only defined modulo \(2\pi\). Moreover, it depends on how you exactly compare the phases in the two CFTs – it depends on the choice of the entanglement.

If you entangled the two boundaries differently, the form of the operator \(W_{LR}\) would unavoidably change as well, and so would the matrix elements of the charged field \(\phi\) relatively to the microstates of the two CFTs. And if the states of the two CFTs weren't (maximally) entangled at all, you couldn't define the charged field in the future quadrant at all.

This shouldn't be shocking. If the two boundary CFTs' states aren't entangled, it means – according to the ER-EPR logic that however goes back to Maldacena 15 years ago – that the boundaries aren't geometrically connected and if they are not geometrically connected, there isn't any "shared future quadrant" at all. Because \(W_{LR}\) is only defined for sufficiently entangled states and those are a "nonlinear subspace" of the full \(LR\) Hilbert space, it follows that \(W_{LR}\) can't have a global definition in terms of fixed matrix elements on the full Hilbert space.

They discussed the charged bulk field for the sake of simplicity. Because the bulk metric is "more or less analogous" to the bulk gauge fields and its curvature is sourced by energy-momentum rather than the charge, similar conclusions almost certainly hold for all bulk fields that carry energy-momentum (instead of the charge considered in this paper) and all fields do. It means that all bulk fields in the future quadrant have to be expressed by formulae that depend on the precise microstate, the precise form of the entanglement between the two boundaries.

At long distances, the bulk fields may "look" the same regardless of the black hole microstate. But this is really just an illusion. The precise description of the local fields is different for the different microstates situations and can't be reconciled into any universal "umbrella" operator. Bulk locality is emergent and approximate – valid to the extent to which it's clear what we mean by the causal structure of the background. The state dependence is a particular technical description of the "emergent character of the spacetime".

It is hard for me to imagine how Polchinski and others could continue to reject this state dependence. By now, it seems not only well-established but many rather explicit consequences expressed by formulae and mathematical properties have been written down, too. It's time to abandon flawed ideas such as the firewalls and embrace the rough intellectual skeleton of the explanation of the future quadrant and the black hole interior, a skeleton that requires the state dependence, among other things.

The Harvard graduate school's alumni have already made the switch and reached a "consensus". What about Santa Barbara schools and others? Will their children be left behind?

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