Tuesday, July 30, 2013

ER-EPR correspondence and bipartite closed strings

This text is just an experiment. I want to know the number of viewers of a blog entry on a related topic and the composition and character of comments if there are any. The insight below is not necessary correct; and it is not necessarily fundamental. ;-)

The Maldacena-Susskind ER-EPR correspondence invites one to think that (non-traversable) wormholes are natural, sort of inevitable – because spacetimes with this topology are physically equivalent to ordinary spacetimes with entangled degrees of freedom in two regions.

One of the things I was thinking about was whether there are other dual descriptions of such spacetimes. Consider, for example, two faraway Strominger-Vafa black holes in type IIB stringy vacua with 5 large dimensions and connect them with the ER bridge. Now, reduce the coupling \(g_s\) to a very low value. What will you get?

The single Strominger-Vafa black hole's microstates are mapped to particular excited states of D1-branes, D5-branes, and momenta. All these excitations of collections of D-branes may be expressed as degrees of freedom carried by open fundamental strings.

Open strings have Neumann boundary conditions for some spacetime coordinates, e.g.\[

\partial_\sigma(X^0)|_{\sigma=0,\pi} = 0

\] and, as Joe Polchinski in particular has taught us, Dirichlet boundary conditions for other coordinates, e.g.\[

X^1_{\sigma=0,\pi} = X^1_\text{D-brane location}.

\] It seems that if you want to entangle the two Strominger-Vafa D-brane stacks, you have to create the same open string to the vicinity of both stacks. You literally want to "tensor square" the information in the microstates, so the state should contain each open string in pairs of copies. Just to be sure, this is different from having a single open string in a permutation-symmetric state.

These open strings end on the two stacks of D-branes and they correspond to objects connected by a bridge. So it seems natural to imagine that there is another object, a closed string that is split to two open strings, that has\[

X^1_{\sigma=0^\pm} = (X^1)_{\pm,\text{D-brane location}}

\] This was an example of what happens with a Dirichlet boundary condition; the two branes were labeled by the symbols \(\pm\). So the closed strings are allowed to have a discontinuity \(\Delta X^1\) at \(\sigma=0\) and \(-\Delta X^1\) at \(\sigma=\pi\) but only if these points of the string are sitting at the right position in space (locations of the D-brane stacks etc.).

Assuming that the dynamics away from \(\sigma=0\) and \(\sigma=\pi\) is ordinary, the only way to guarantee that these points of the closed string remain at the right place is to make the function \(X^1(\sigma)\) odd. If we ask what it means for the excited strings, it means that we only allow the excitations by\[

\alpha^1_{-n}\tilde \alpha^1_{-n}

\] i.e. by products (not sums – that would be a way to produce unorientable closed string states whose Hilbert space is "less reduced") of left-moving and right-moving oscillators. Needless to say, such a Hilbert space of "very special" closed string states is isomorphic to the Hilbert space of open string states (but the basis vectors are squared). One may see that the \(L_0=\tilde L_0\) level-matching condition isn't challenged because the left-moving and right-moving excitations are paired.

This picture generalizes the old Susskind's idea of an open string as a closed string whose half was stuck beneath the event horizon. Now, "deeply" beneath the event horizon, there's the other black hole, so an open string should really be one-half of a closed string whose other half is an open string on the opposite side of the wormhole or ER bridge.

Note that the individual D-branes represent a "topological defect" which allows the existence of open strings with the corresponding boundary conditions. The pairs of perfectly entangled D-brane stacks also changes the state of the spacetime. This topological defect allows the existence of the closed strings with the discontinuity that may only exist at points where the closed string hits a throat of the wormhole.

The closed string states considered above – in which the excitations are only allowed in the \(\alpha^1_{-n}\tilde \alpha^1_{-n}\) pairs – create extremely special states of the closed string and it is not allowed to "permanently" impose non-local identifications on the paths taken by closed strings. At any rate, if you allow this closed string looking like a "doubled open string" to interact with other, more ordinary strings, those that ignore the bridge, you will generically create closed strings that don't respect the \(\ZZ_2\) symmetry and that consequently deviate from the right value of \(X^1(\sigma)\) for \(\sigma=0,\pi\) at a later time.

I am confused what it could mean. The discontinuity allowing the closed string to jump from one throat to another is only allowed at the right location of the D-brane (stack) but if the closed string gets kicked, it will deviate from the place where the discontinuity is allowed. Does it mean that the bipartite strings aren't allowed at all? Is the picture inconsistent?

Quite generally, the Hilbert space of the connected pair should have the form\[

\HH_\text{one BH}\otimes \HH_\text{one BH}

\] but the closed-string visualization of this Hilbert space could give us a new natural basis optimized for the bridge, i.e. for the heavily entangled states.

How many readers who aren't shy are there who can give sensible answers or observations about similar questions? String theorists who are in e-mail contact with me may send me an e-mail message, too, of course. ;-)


  1. Hi Lubos,
    as you say yourself, the closed string you describe is in an extremely special microstate encoding quantum entanglement. It should therefore not be surprising that this entanglement is rapidly lost when you let this state interact with some generic nearby degrees of freedom of comparable energy -- that's just decoherence.
    You may want to consider entanglement in macroscopic properties which are more robust to decoherence. You would have to specify exactly what kind of observables you want to entangle and see if you can describe the resulting system in terms of elementary string excitations. If you can find a robust, working example of this, I think it would be a very interesting system to study with the potential of illuminating some of the inner workings of ER-EPR.
    Thanks for this stimulating post.
    Best, Dan

  2. Thanks, Dan, but the smooth geometric bridge/wormhole is only created if all microscopic degrees of freedom of the two systems are appropriately entangled! In other words, the entanglement entropy has to be large for the inner area of the correcting bridge to be larger and thus smoother than Planckian etc.

  3. Hi Lubos,

    Probably I have misunderstood something basic about your above construction but in which way are these closed strings of yours different from ordinary closed strings wrapped over a nontrivial 1-cycle? The boundary condition seems the same as that of closed strings with a winding number w=2 along a S^1 parametrized so as to have X=0 identified with X=Delta X^1, and two extra conditions fixing X=0 to sigma=0 and X=Delta X^1 to X=pi. However I do not see how these conditions can be compatible with the dynamics; the closed strings may "slide" along the EPR bridge, so that X=0,Delta X^1 will not correspond to sigma=0,pi anymore but to different values of sigma.



  4. Thanks, Miguel. I think that the total winding number is zero (because the closed string goes once from throat A to throat B and once back), so the closed string is internally (when it comes to its tower of excited states) even more identical to the ordinary untwisted closed string than you suggest.

  5. I +1ed this, because from my very (too) naive understanding I was asking myself too of kicking the closed string could lead to a loss of entanglement or decorrelation.

    Happily you stuck your head out first, such that I just can join your comment ... :-P

    (Off topic, are you Dan at Physics SE too http://physics.stackexchange.com/users/3936/dan ...?)

  6. Can it mean that decoherence effectively prohibits large WHs?

  7. Hi Lubos,

    Thanks for your kind reply and apologies, I obviously didn't get your setting. Even so, I still do not understand what's going on at sigma=0,pi.

    If these are fixed at X=X-,X+, then the split closed string is equivalent to two independent open strings, one for each D-brane. The requirement that X be odd means that at sigma=pi there is a boundary and one can define new oscillators for the (0,pi) and the (pi, 2pi) intervals, and these are regular, uncoupled open string oscillators. Any state constructed with these oscillators would, when expressed in terms of the original oscillators, satisfy the property of being a product of tilded and untilded oscillators.

    If they are not, it seems an ordinary closed string and I do not see how you manage to prevent it from sliding through the ER bridge which connects the branes and just fly away.

    Thanks again,


  8. Wow. That should allow us to travel from galaxy to galaxy via the black holes. Just slide down the cosmic strings stuck in the N/S poles of the black holes and come out the other end of the other black hole.

  9. I have a naive question :Suppose I represent, at fixed time, a string as a point-particle evolving with a pseudo-time \sigma. At fixed time, I may have a X1-\sigma graph. The branes are localized, say at x1 =+- L. The horizon is at x1=0. Now, at \sigma = 0 and \sigma = pi, I put 2 frontiers, that is, between 0 and pi, this is a Minkowski space time, and, admitting "extending \sigma", for \sigma < 0 and \sigma > pi, this is an euclidean spacetime. So the "movement" of the pseudo-particle (below sigma =0 and above sigma =pi) could be seen as a kind of half-circle joining the discontinuity. So, the total graph would be a closed string. Is this a non-sense representation or/and a non-useful representation, or has it some interest ?

  10. Dear Trimok, I feel that such reasoning has to be useful and insightful but I am not sure what's the right way to unleash its powers.

    The usual approach that would declare your reasoning inappropriate is to remove the coordinate singularities. So if you have horizons, you must choose better singularities around that locus, and this will mean that the signature isn't ever really flipped. So in the usual approach, one may always identify analytic continuation when it occurs.

  11. Lubos,

    Another naive question, but does this er = epr business have anything to say about that "Exotic Branes" paper last year by deBoer and Shigemori? When talking about closed strings being half inside the horizon and then entangled to another black hole, does a supertube or exotic brane come into play? I am not a physicist, only a blog reader, and this comment is strictly for the lols. Thank you for all of your work.

  12. Dear Dan, exotic branes are more a technical development for Mathur's fuzzballs. Ultimately, there may be a link between fuzzballs and ER=EPR, too, but it's not known today AFAIK. The former is about states of a single black hole, the latter is about two.

  13. Hi Lubos,
    Maybe the question is how to define interactions between one of these bipartite closed strings and an ordinary closed string.
    For a free bipartite closed string, you can think of it as a tubular worldsheet with 2 lines painted on it - these lines are the places where the X fields are discontinuous.
    For tree level scattering of one bipartite closed string with an ordinary closed string, you'd have the usual diagram that looks like a sphere with 4 punctures, but now you'd also have to keep track of where the 2 lines of discontinuities begin and end.

    If the 2 lines begin at one puncture and then both end up at another puncture, then that is like the process open + closed -> open + closed.
    If the 2 lines begin at one puncture and then end up on 2 different punctures, then that is like open + closed -> open + open.

    In either case, you sum over all worldsheets with some given discontinuity specified on the 2 lines.

  14. Florin MoldoveanuAug 16, 2013, 10:59:00 PM

    I am no string theory expert but I am an expert in QM [and I am in the process of slowly introducing a new interpretation - the correct one ;) - see http://fmoldove.blogspot.com/ ] and I can confidently state that EPR~ER is utterly ludicrous.

    QM correlations are space independent, and even the position operator does not naturally belong to a Hilbert space because it is unbounded (unbounded operators do not satisfy polarization identities allowing them to recover the inner product). Another way to see this is from spectral theory or from C* algebras. Hilbert spaces are the natural framework for bounded operators.

    And do I have to mention that time is not even an operator in QM?

    Any explanation or "duality" of QM (or QM correlations) with commutative geometry is crackpot. QM is best explained in the non-commutative geometry framework as the part that describe the internal degrees of freedom, but then the notion of distance changes from an infimum to a supremum.

  15. Dear Florin, I have registered your name in the discussions about Joy Christian's idiotic papers


    but now it seems to me that if I had actually read your papers in length, I would have probably found out that you were a Joy Christian's soulmate in those exchanges. ;-)

    Entanglement works generally, regardless of objects' geometric interpretation or positions in spacetime. But if you have a theory with a spacetime, then every degree of freedom is associated with some location - the location and time don't have to be and aren't operators in a QFT; they are variables that operators (fields) depend upon (so all your "paradoxes" of this kind are completely nonsensical) - and Maldacena, Susskind say that if you have an entanglement in a theory where degrees of freedom result from local fields, the entanglement is equivalent to a nontrivial Einstein-Rosen-bridge-like topology. In other words, entangled states in theories that live in a spacetime have a separate wormhole-like spacetime visualization, not just the ordinary visualization as superpositions of unentangled states in a spacetime of a trivial topology.

    There's no paradox here and there can't be any paradox here.

  16. Florin MoldoveanuAug 17, 2013, 3:59:00 PM

    Dear Lubos,

    You say: “if you have an entanglement in a theory where degrees of freedom result from local fields, The entanglement is equivalent to a nontrivial Einstein-Rosen-bridge-like topology”

    In a poetic sense, sure, but in a mathematical setting where there is an underlying mathematical structure related to both cases, it is crackpot. No such mathematical structure exists and it can be proven that it cannot exist. I know this because I know how to derive QM from natural postulates (http://arxiv.org/abs/1303.3935) and how to recover the full QM formalism
    step by step. I also know how to generalize QM to naturally arrive at field theory in one particular case, and I am working to obtain the complete generalization classification.

    But forget about fancy arguments, entanglement exists because superposition which exists because Hilbert spaces. And the wavefunction lives in a configuration space which is not the same as the ordinary space (but many people are tricked by the 1-particle problems like the Hydrogen atom and mistake the configuration space for real space). I will not name names, but I was surprised to discover that even well established QM experts sometimes forget about this and promote
    invalid arguments based on it.

    Still not convinced? “Einstein-Rosen-bridge-like topology” Topology demands the idea of neighborhood in the most general sense. Local realism is long dead and cannot be an explanation for QM correlations.

    “I would have probably found out that you were a Joy Christian's soulmate in those exchanges. ;-)”

    Funny you should say this. I can probably say the same thing about you and Lisi (http://motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html) regarding Coleman-Mandula ;) What
    does C-M say? “impossibility of
    combining space-time and internal symmetries in any but a trivial way. “ Gee, have they looked at the Einstein-Rosen-bridge-like
    topology and how it can “explain entanglement”? Of course some may argue that C-M is for flat space, but in GR the manifold has well defined tangent spaces and non-traversability does not invalidate C-M.

    I am grateful for your support of fighting Joy, but why don't you read this in depth (http://arxiv.org/abs/1109.0535). You may most likely conclude that Joy was probably not a crackpot but a con artist who planned the whole thing like a hoax to rescue his failed academic career. I am not saying he is a con artist, this is not provable, but I cannot reconcile the fancy math setting and his quick identification of mistakes in his opponents' arguments with the elementary mathematical mistakes he made. My favorite Joy math error was: epsilon/epsilon = 0 when epsilon goes to zero because the numerator goes to zero.

  17. Dear Florin, good to hear about some agreement concerning Joy Christian. That's where the agreement ends.

    Your arguments against ER-EPR correspondence are illogical and if there's something in your March 2013 preprint besides infantile colorful flow diagrams and circular and wrong comments about simple postulates of QM, you will have to hire someone else who will sell the discovery to people like me. ;-)