Two years ago, I wrote a blog post entitled PostBQP Postscripts, owning up to not one but *four* substantive mathematical errors that I’d made over the years in my published papers, and which my students and colleagues later brought to my sheepish attention. Fortunately, none of these errors affected the papers’ main messages; they just added interesting new twists to the story. Even so, I remember feeling at the time like undergoing this public repentance was soul-cleansing intellectual hygiene. I also felt like writing one big “post of shame” was *easier* than writing a bunch of separate errata and submitting them to journals, while also reaching a wider audience (and, therefore, doing an even better soul-cleansing job).

So I resolved that, anytime I’d saved up enough errata, I’d do another sackcloth-and-ashes post. Which brings us to today. Without further ado:

**I. Quantum Money Falling Down**

My and Paul Christiano’s explicit public-key quantum money scheme—the one based on low-degree polynomials—has now been fully broken. To clarify, our abstract hidden-subspace scheme—the one that uses a classical black-box to test membership in the subspaces—remains totally fine. Indeed, we unconditionally proved the security of the black-box scheme, and our security proof stands. In the paper, though, we also stuck our necks out further, and conjectured that you could instantiate the black box, by publishing random low-degree polynomials that vanish on the subspaces you want to hide. While I considered this superfluous, at Paul’s insistence, we also recommended adding completely-random “noise polynomials” for extra security.

Our scheme was broken in two stages. First, in 2014, Pena et al. broke the noiseless version of our scheme, using Gröbner-basis methods, over fields of characteristic greater than 2. Over F_{2}—the field we happened to use in our scheme—Pena et al. couldn’t quite prove that their attack worked, but they gave numerical evidence that at least it finds the subspaces in n^{O(log n)} time. Note that nothing in Pena et al.’s attack is specific to quantum money: indeed, their attack consists of a purely classical algorithm, which efficiently solves the general classical problem of recovering large subspaces from polynomials that hide them.

At that point, at least the *noisy* version of our scheme—the one Paul had insisted we include—was still standing! Indeed, the Gröbner-basis attack seemed to break down entirely when some of the polynomials were random garbage.

Later, though, Paul and Or Sattath realized that a quantum trick—basically, the single-copy tomography of Farhi et al.—can identify which polynomials are the noisy ones, provided we’re given a legitimate quantum money state to start with. As a consequence, the problem of breaking the noisy scheme can be reduced to the problem of breaking the noiseless scheme—i.e., the problem that Pena et al. already essentially solved.

As bad as this sounds, it has an interesting positive consequence. In our paper, Paul and I had actually given a security reduction for our money scheme based on low-degree polynomials. In particular, we showed that there’s no polynomial-time quantum algorithm to counterfeit our money states, *unless* there’s a polynomial-time quantum algorithm that finds a basis for a subspace S≤F_{2}^{n} of dimension n/2 with Ω(2^{-n/2}) success probability, given a collection of low-degree polynomials p_{1},…,p_{m} and q_{1},…,q_{m} (m=O(n)) most of which vanish on S and its dual subspace respectively, but that are otherwise random. So, running our reduction backwards, the only possible conclusion from the break is that there *is* such a quantum algorithm! Yet we would’ve had no idea how to find that quantum algorithm without going through quantum money—nor do we know a classical algorithm for the problem, or even a quantum algorithm with Ω(1) success probability.

In the meantime, the problem of designing a public-key quantum money scheme, with good cryptographic evidence for its security, remains open. It’s plausible that there’s some other, more secure way to instantiate my and Paul’s hidden subspace scheme, for example using lattices. And even before we’ve found such a way, we can use indistinguishability obfuscation as a stopgap. We could also seek cryptographic evidence for the security of other kinds of public-key quantum money, like Farhi et al.’s based on knot invariants.

A paper about all this is on our to-do stack. In the meantime, for further details, see Lecture 9 in my Barbados lecture notes.

**II. A De-Merlinization Mistake**

In my 2006 paper QMA/qpoly ⊆ PSPACE/poly: De-Merlinizing Quantum Protocols, the technical core of the complexity result was a new quantum information lemma that I called the “Quantum OR Bound” (Lemma 14 in the paper).

Basically, the Quantum OR Bound says that, if we have an unknown quantum state ρ, as well as a collection of measurements M_{1},…,M_{n} that we might want to make on ρ, then we can distinguish the case that (a) every M_{i} rejects ρ with overwhelming probability, from the case that (b) at least one M_{i} accepts ρ with high probability. And we can do this *despite* having only one copy of ρ, and despite the fact that earlier measurements might corrupt ρ, thereby compromising the later measurements. The intuition is simply that, if the earlier measurements corrupted ρ substantially, that could only be because some of them had a decent probability of accepting ρ, meaning that at any rate, we’re not in case (a).

I’ve since reused the Quantum OR Bound for other problems—most notably, a proof that private-key quantum money requires either a computational assumption or a huge database maintained by the bank (see Theorem 8.3.1 in my Barbados lecture notes).

Alas, Aram Harrow and Ashley Montanaro recently discovered that my proof of the Quantum OR Bound is wrong. It’s wrong because I neglected the possibility of “Zeno-like behavior,” in which repeated measurements on a quantum state would gradually shift the state far away from its starting point, without ever having a significant probability of rejecting the state. For some reason, I assumed without any adequate argument that choosing the measurements at random, rather than in a predetermined order, would solve that problem.

Now, I might actually be *right* that randomizing the measurements is enough to solve the Zeno problem! That remains a plausible conjecture, which Harrow and Montanaro could neither confirm nor refute. In the meantime, though, Harrow and Montanaro were able to recover my QMA/qpoly⊆PSPACE/poly theorem, and all the other conclusions known to follow from the Quantum OR Bound (including some new ones that they discover), by designing a *new* measurement procedure whose soundness they can prove.

Their new procedure is based on an elegant, obvious-in-retrospect idea that somehow never occurred to me. Namely, instead of just applying M_{i}‘s to ρ, one can first put a control qubit into an equal superposition of the |0〉 and |1〉 states, and then apply M_{i}‘s *conditioned* on the control qubit being in the |1〉 state. While doing this, one can periodically measure the control qubit in the {|+〉,|-〉} basis, in order to check directly whether applying the M_{i}‘s has substantially corrupted ρ. (If it hasn’t, one will always get the outcome |+〉; if it has, one might get |-〉.) Substantial corruption, if detected, then tells us that some M_{i}‘s must have had non-negligible probabilities of accepting ρ.

**III. Almost As Good As True**

One lemma that I’ve used even *more* than the Quantum OR Bound is what I’ve called the “Almost As Good As New Lemma,” and what others in the field have called the “Gentle Measurement Lemma.”

I claimed a proof of the AAGANL in my 2004 paper Limitations of Quantum Advice and One-Way Communication (Lemma 2.2 there), and have used the lemma in like half a dozen later papers. Alas, when I lectured at Barbados, Sasha Razborov and others discovered that my proof of the AAGANL was missing a crucial step! More concretely, the proof I gave there works for pure states but not for mixed states. For mixed states, the trouble is that I take a purification of the mixed state—something that always exists mathematically—but then illegally assume that the measurement I’m analyzing acts on the particular purification I’ve conjured up.

Fortunately, one can easily fix this problem by decomposing the state ρ into a mixture of pure states, then applying my earlier argument to each pure state separately, and finally using Cauchy-Schwarz (or just the convexity of the square-root function) to recombine the results. Moreover, this is exactly what other people’s proofs of the Gentle Measurement Lemma *did *do, though I’d never noticed it before Barbados—I just idly wondered why those other proofs took twice as long as mine to do the same work! For a correct proof, see Lemma 1.3.1 in the Barbados lecture notes.

**IV. Oracle Woes**

In my 2010 paper BQP and the Polynomial Hierarchy, I claimed to construct oracles A relative to which BQP⊄BPP_{path} and BQP⊄SZK, even while making only partial progress toward the big prize, which would’ve been an oracle relative to which BQP⊄PH. Not only that: I claimed to show that *any* problem with a property called “almost k-wise independence”—one example being the Forrelation (or Fourier Checking) problem that I introduced in that paper—was neither in BPP_{path} nor in SZK. But I showed that Forrelation *is* in BQP, thus yielding the separations.

Alas, this past spring Lijie Chen, who was my superb visiting student from Tsinghua University, realized that my proofs of these particular separations were wrong. Not only that, they were wrong *because I implicitly substituted a ratio of expectations for an expectation of ratios* (!). Again, it might still be *true* that almost k-wise independent problems can be neither in BPP_{path} nor in SZK: that remains an interesting conjecture, which Lijie was unable to resolve one way or the other. (On the other hand, I showed here that almost k-wise independent problems *can* be in PH.)

But never fear! In a recent arXiv preprint, Lijie has supplied correct proofs for the BQP⊄BPP_{path} and BQP⊄SZK oracle separations—using the same Forrelation problem that I studied, but additional properties of Forrelation besides its almost k-wise independence. Lijie notes that my proofs, had they worked, would also have yielded an oracle relative to which BQP⊄AM, which would’ve been a spectacular result, nontrivial progress toward BQP⊄PH. His proofs, by contrast, apply only to worst-case decision problems rather than problems of distinguishing two probability distributions, and therefore don’t imply anything about BQP vs. AM. Anyway, there’s other cool stuff in his paper too.

**V. We Needed More Coffee**

This is one I’ve already written about on this blog, but just in case anyone missed it … in my, Sean Carroll, and Lauren Ouellette’s original draft paper on the coffee automaton, the specific rule we discuss *doesn’t* generate any significant amount of complexity (in the sense of coarse-grained entropy). We wrongly thought it did, because of a misinterpretation of our simulation data. But as Brent Werness brought to our attention, not only does a corrected simulation not show any complexity bump, one can rigorously *prove* there’s no complexity bump. And we could’ve realized all this from the beginning, by reflecting that pure random diffusion (e.g., what cream does in coffee when you don’t stir it with a spoon) *doesn’t* actually produce interesting tendril patterns.

On the other hand, Brent proposed a different rule—one that involves “shearing” whole regions of cream and coffee across each other—that *does* generate significant complexity, basically because of all the long-range correlations it induces. And not only do we clearly see this in simulations, but the growth of complexity can be rigorously proven! Anyway, we have a long-delayed revision of the paper that will explain all this in more detail, with Brent as well as MIT student Varun Mohan now added as coauthors.

If any of my colleagues feel inspired to write up their own “litanies of mathematical error,” they’re welcome to do so in the comments! Just remember: you don’t earn any epistemic virtue points unless the errors you reveal *actually* embarrass you. No humblebragging about how you once left out a minus sign in your paper that won the Fields Medal.

Today was the day of finite temperature and density, on which the general review talk was delivered by Heng-Tong Ding. While in the meantime agreement has been reached on the transition temperature, the nature of the transition (crossover) and the equation of state at the physical quark masses, on which different formulations differed a lot in the past, the Columbia plot of the nature of the transition as a function of the light and strange quark masses still remains to be explored, and there are discrepancies between results obtained in different formulations. On the topic of U(1)_{A} restoration (on which I do have a layman's question: to my understanding U(1)_{A} is broken by the axial anomaly, which to my understanding arises from the path integral measure - so why should one expect the symmetry to be restored at high temperature? The situation is quite different from dynamical spontaneous symmetry breaking, as far as I understand), there is no evidence for restoration so far. A number of groups have taken to using the gradient flow as a tool to perform relatively cheap investigations of the equation of state. There are also new results from the different approaches to finite-density QCD, including cumulants from the Taylor-expansion approach, which can be related to heavy-ion observables, and new ways of stabilizing complex Langevin dynamics.

This was followed by two topical talks. The first, by Seyong Kim, was on the subject of heavy flavours at finite temperature. Heavy flavours are one of the most important probes of the quark-gluon plasma, and J/ψ suppression has served as a diagnostic tool of QGP formation for a long time. To understand the influence of high temperatures on the survival of quarkonium states and on the transport properties of heavy flavours in the QGP, knowledge of the spectral functions is needed. Unfortunately, extracting these from a finite number of points in Euclidean point is an ill-posed problem, especially so when the time extent is small at high temperature. The methods used to get at them nevertheless, such as the maximum entropy method or Bayesian fits, need to use some kind of prior information, introducing the risk of a methodological bias leading to systematic errors that may be not only quantitative, but even qualitative; as an example, MEM shows P-wave bottomonium to melt around the transition temperature, whereas a newer Bayesian method shows it to survive, so clearly more work is needed.

The second topical talk was Kurt Langfeld speaking about the density-of-states method. This method is based on determining a function ρ(E), which is essentially the path integral of δ(S[φ]-E), such that the partition function can be written as the Laplace transform of ρ, which can be generalized to the case of actions with a sign problem, where the partition function can then be written as the Fourier transform of a function P(s). An algorithm to compute such functions exists in the form of what looks like a sort of microcanonical simulation in a window [E-δE;E+δE] and determines the slope of ρ at E, whence ρ can be reconstructed. Ergodicity is ensured by having the different windows overlap and running in parallel, with a possibility of "replica exchange" between the processes running for neighbouring windows when configurations within the overlap between them are generated. The examples shown, e.g. for the Potts model, looked quite impressive in that the method appears able to resolve double-peak structures even when the trough between the peaks is suppressed by many orders of magnitude, such that a Markov process would have no chance of crossing between the two probability peaks.

After the coffee break, Aleksi Kurkela reviewed the phenomenology of heavy ions. The flow properties that were originally taken as a sign of hydrodynamics having set in are now also observed in pp collisions, which seem unlikely to be hydrodynamical. In understanding and interpreting these results, the pre-equilibration evolution is an important source of uncertainty; the current understanding seems to be that the system goes from an overoccupied to an underoccupied state before thermalizing, making different descriptions necessary at different times. At early times, simulations of classical Yang-Mills theory on a lattice in proper-time/rapidity coordinates are used, whereas later a quasiparticle description and kinetic theory can be applied; all this seems to be qualitative so far.

The energy momentum tensor, which plays an important role in thermodynamics and hydrodynamics, was the topic of the last plenary of the day, which was given by Hiroshi Suzuki. Translation invariance is broken on the lattice, so the Ward-Takahashi identity for the energy-momentum tensor picks up an O(a) violation term, which can become O(1) by radiative corrections. As a consequence, three different renormalization factors are needed to renormalize the energy-momentum tensor. One way of getting at these are the shifted boundary conditions of Giusti and Meyer, another is the use of the gradient flow at short flow times, and there are first results from both methods.

The parallel sessions of the afternoon concluded the parallel programme.

This was followed by two topical talks. The first, by Seyong Kim, was on the subject of heavy flavours at finite temperature. Heavy flavours are one of the most important probes of the quark-gluon plasma, and J/ψ suppression has served as a diagnostic tool of QGP formation for a long time. To understand the influence of high temperatures on the survival of quarkonium states and on the transport properties of heavy flavours in the QGP, knowledge of the spectral functions is needed. Unfortunately, extracting these from a finite number of points in Euclidean point is an ill-posed problem, especially so when the time extent is small at high temperature. The methods used to get at them nevertheless, such as the maximum entropy method or Bayesian fits, need to use some kind of prior information, introducing the risk of a methodological bias leading to systematic errors that may be not only quantitative, but even qualitative; as an example, MEM shows P-wave bottomonium to melt around the transition temperature, whereas a newer Bayesian method shows it to survive, so clearly more work is needed.

The second topical talk was Kurt Langfeld speaking about the density-of-states method. This method is based on determining a function ρ(E), which is essentially the path integral of δ(S[φ]-E), such that the partition function can be written as the Laplace transform of ρ, which can be generalized to the case of actions with a sign problem, where the partition function can then be written as the Fourier transform of a function P(s). An algorithm to compute such functions exists in the form of what looks like a sort of microcanonical simulation in a window [E-δE;E+δE] and determines the slope of ρ at E, whence ρ can be reconstructed. Ergodicity is ensured by having the different windows overlap and running in parallel, with a possibility of "replica exchange" between the processes running for neighbouring windows when configurations within the overlap between them are generated. The examples shown, e.g. for the Potts model, looked quite impressive in that the method appears able to resolve double-peak structures even when the trough between the peaks is suppressed by many orders of magnitude, such that a Markov process would have no chance of crossing between the two probability peaks.

After the coffee break, Aleksi Kurkela reviewed the phenomenology of heavy ions. The flow properties that were originally taken as a sign of hydrodynamics having set in are now also observed in pp collisions, which seem unlikely to be hydrodynamical. In understanding and interpreting these results, the pre-equilibration evolution is an important source of uncertainty; the current understanding seems to be that the system goes from an overoccupied to an underoccupied state before thermalizing, making different descriptions necessary at different times. At early times, simulations of classical Yang-Mills theory on a lattice in proper-time/rapidity coordinates are used, whereas later a quasiparticle description and kinetic theory can be applied; all this seems to be qualitative so far.

The energy momentum tensor, which plays an important role in thermodynamics and hydrodynamics, was the topic of the last plenary of the day, which was given by Hiroshi Suzuki. Translation invariance is broken on the lattice, so the Ward-Takahashi identity for the energy-momentum tensor picks up an O(a) violation term, which can become O(1) by radiative corrections. As a consequence, three different renormalization factors are needed to renormalize the energy-momentum tensor. One way of getting at these are the shifted boundary conditions of Giusti and Meyer, another is the use of the gradient flow at short flow times, and there are first results from both methods.

The parallel sessions of the afternoon concluded the parallel programme.

Following the canonical script for lattice conferences, yesterday was the day without plenaries. Instead, the morning was dedicated to parallel sessions (including my own talk), and the afternoon was free time with the option of taking one of several arranged excursions.

I went on the excursion to Salisbury cathedral (which is notable both for its fairly homogeneous and massive architectural ensemble, and for being home to one of four original copies of the Magna Carta) and Stonehenge (which in terms of diameter seems to be much smaller than I had expected from photos).

Today began with the traditional non-lattice theory talk, which was given by Monika Blanke, who spoke about the impact of lattice QCD results on CKM phenomenology. Since quarks cannot be observed in isolation, the extraction of CKM matrix elements from experimental results always require knowledge of the appropriate hadronic matrix elements of the currents involved in the measured reaction. This means that lattice results for the form factors of heavy-to-light semileptonic decays and for the hadronic parameters governing neutral kaon and B meson mixing are of crucial importance to CKM phenomenology, to the extent that there is even a sort of "wish list" to the lattice. There has long been a discrepancy between the values of both |V_{cb}| and |V_{ub}| extracted from inclusive and exclusive decays, respectively, and the ratio |V_{ub}/V_{cb}| that can be extracted from decays of Λ_{b} baryons only adds to the tension. However, this is likely to be a result of underestimated theoretical uncertainties or experimental issues, since the pattern of the discrepancies is not in agreement with that which would results from new physics effects induced by right-handed currents. General models of flavour violating new physics seems to favour the inclusive value for |V_{ub}|. In b->s transitions, there is evidence for new physics effects at the 4σ level, but significant theoretical uncertainties remain. The B_{(s)}->μ^{+}μ^{-} branching fractions are currently in agreement with the SM at the 2σ level, but new, more precise measurements are forthcoming.

Ran Zhou complemented this with a review talk about heavy flavour results from the lattice, where there are new results from a variety of different approaches (NRQCD, HQET, Fermilab and Columbia RHQ formalisms), which can serve as useful and important cross-checks on each other's methodological uncertainties.

Next came a talk by Amy Nicholson on neutrinoless double β decay results from the lattice. Neutrinoless double β decays are possible if neutrinos are Majorana particles, which would help to explain the small masses of the observed left-handed neutrinos through the see-saw mechanism pushing the right-handed neutrinos off to near the GUT scale. Treating the double β decay in the framework of a chiral effective theory, the leading-order matrix element required is a process π^{-}->π^{+}e^{-}e^{-}, for which there are first results in lattice QCD. The NLO process would have disconnected diagrams, but cannot contribute to the 0^{+}->0^{+} transitions which are experimentally studied, whereas the NNLO process involves two-nucleon operators and still remains to be studied in greater detail on the lattice.

After the coffee break, Agostino Patella reviewed the hot topic of QED corrections to hadronic observables. There are currently two main methods for dealing with QED in the context of lattice simulations: either to simulate QCD+QED directly (usually at unphysically large electromagnetic couplings followed by an extrapolation to the physical value of α=1/137), or to expand it in powers of α and to measure only the resulting correlation functions (which will be four-point functions or higher) in lattice QCD. Both approaches have been used to obtain some already very impressive results on isospin-breaking QED effects in the hadronic spectrum, as shown already in the spectroscopy review talk. There are, however, still a number of theoretical issues connected to the regularization of IR modes that relate to the Gauss law constraint that would forbid the existence of a single charged particle (such as a proton) in a periodic box. The prescriptions to evade this problem all lead to a non-commutativity of limits requiring the infinite-volume limit to be taken before other limits (such as the continuum or chiral limits): QED_{TL}, which omits the global zero modes of the photon field, is non-local and does not have a transfer matrix; QED_{L}, which omits the spatial zero modes on each timeslice, has a transfer matrix, but is still non-local and renormalizes in a non-standard fashion, such that it does not have a non-relativistic limit; the use of a massive photon leads to a local theory with softly broken gauge symmetry, but still requires the infinite-volume limit to be taken before removing the photon mass. Going beyond hadron masses to decays introduces new IR problems, which need to be treated in the Bloch-Nordsieck way, leading to potentially large logarithms.

The 2016 Ken Wilson Lattice Award was awarded to Antonin Portelli for his outstanding contributions to our understanding of electromagnetic effects on hadron properties. Antonin was one of the driving forces behind the BMW collaboration's effort to determine the proton-neutron mass difference, which resulted in a*Science* paper exhibiting one of the most frequently-shown and impressive spectrum plots at this conference.

In the afternoon, parallel sessions took place, and in the evening there was a (very nice) conference dinner at the Southampton F.C. football stadium.

I went on the excursion to Salisbury cathedral (which is notable both for its fairly homogeneous and massive architectural ensemble, and for being home to one of four original copies of the Magna Carta) and Stonehenge (which in terms of diameter seems to be much smaller than I had expected from photos).

Today began with the traditional non-lattice theory talk, which was given by Monika Blanke, who spoke about the impact of lattice QCD results on CKM phenomenology. Since quarks cannot be observed in isolation, the extraction of CKM matrix elements from experimental results always require knowledge of the appropriate hadronic matrix elements of the currents involved in the measured reaction. This means that lattice results for the form factors of heavy-to-light semileptonic decays and for the hadronic parameters governing neutral kaon and B meson mixing are of crucial importance to CKM phenomenology, to the extent that there is even a sort of "wish list" to the lattice. There has long been a discrepancy between the values of both |V

Ran Zhou complemented this with a review talk about heavy flavour results from the lattice, where there are new results from a variety of different approaches (NRQCD, HQET, Fermilab and Columbia RHQ formalisms), which can serve as useful and important cross-checks on each other's methodological uncertainties.

Next came a talk by Amy Nicholson on neutrinoless double β decay results from the lattice. Neutrinoless double β decays are possible if neutrinos are Majorana particles, which would help to explain the small masses of the observed left-handed neutrinos through the see-saw mechanism pushing the right-handed neutrinos off to near the GUT scale. Treating the double β decay in the framework of a chiral effective theory, the leading-order matrix element required is a process π

After the coffee break, Agostino Patella reviewed the hot topic of QED corrections to hadronic observables. There are currently two main methods for dealing with QED in the context of lattice simulations: either to simulate QCD+QED directly (usually at unphysically large electromagnetic couplings followed by an extrapolation to the physical value of α=1/137), or to expand it in powers of α and to measure only the resulting correlation functions (which will be four-point functions or higher) in lattice QCD. Both approaches have been used to obtain some already very impressive results on isospin-breaking QED effects in the hadronic spectrum, as shown already in the spectroscopy review talk. There are, however, still a number of theoretical issues connected to the regularization of IR modes that relate to the Gauss law constraint that would forbid the existence of a single charged particle (such as a proton) in a periodic box. The prescriptions to evade this problem all lead to a non-commutativity of limits requiring the infinite-volume limit to be taken before other limits (such as the continuum or chiral limits): QED

The 2016 Ken Wilson Lattice Award was awarded to Antonin Portelli for his outstanding contributions to our understanding of electromagnetic effects on hadron properties. Antonin was one of the driving forces behind the BMW collaboration's effort to determine the proton-neutron mass difference, which resulted in a

In the afternoon, parallel sessions took place, and in the evening there was a (very nice) conference dinner at the Southampton F.C. football stadium.

Back from family vacation in Greece. Tiny notes/memories: I have a heuristic that Americans fly the national flag much more than Europeans do, but in Greece, the Greek flag is all over the place. Greeks really like, or Greeks think people in hotels and restaurants really like, soft-rock covers of hits from the 80s. Maybe […]

Back from family vacation in Greece. Tiny notes/memories:

- I have a heuristic that Americans fly the national flag much more than Europeans do, but in Greece, the Greek flag is all over the place.
- Greeks really like, or Greeks think people in hotels and restaurants really like, soft-rock covers of hits from the 80s. Maybe both! We heard this mix CD
*everywhere*.If you don’t feel like an hour and a half of this, at least treat yourself to James Farelli’s inexplicably fascinating acoustic take on “Owner of a Lonely Heart.”

- The city of Akrotiri in the Aegean islands, a thousand years older than classical Greece, was buried under 200 feet of ash by the massive eruption of Santorini. They’ve only just started to dig it out. There are wall frescoes whose paint is still colorful and fresh. But these wall frescoes aren’t on the walls anymore; they fell during the earthquake preceding the eruption and lie in fragments on the floors. Our guide told us that they don’t try to reconstruct these using computers; archeologists put the pieces together by hand. I was perplexed by this:
*why*don’t they digitize the images and try to find matches? It seemed to me like exactly the sort of thing we now know how to do. But no: it turns out this is a problem CS people are already thinking about, and it’s hard. Putting together pottery turns out to be a computationally much easier problem. Why? Because pots are surfaces of revolution and so their geometry is much more constrained! - The 2-star Michelin molecular gastronomy restaurant Funky Gourmet, run by a member of the El Bulli disapora, is just as great as advertised. But how can you run a molecular gastronomy restaurant in Athens and not call it Grecian Formula…?

The loss of the 750 GeV diphoton resonance is a big blow to the particle physics community. We are currently going through the 5 stages of grief, everyone at their own pace, as can be seen e.g. in this comments section. Nevertheless, it may already be a good moment to revisit the story one last time, so as to understand what went wrong.

In the recent years, physics beyond the Standard Model has seen 2 other flops of comparable impact: the faster-than-light neutrinos in OPERA, and the CMB tensor fluctuations in BICEP. Much as the diphoton signal, both of the above triggered a binge of theoretical explanations, followed by a massive hangover. There was one big difference, however: the OPERA and BICEP signals were due to embarrassing errors on the experiments' side. This doesn't seem to be the case for the diphoton bump at the LHC. Some may wonder whether the Standard Model background may have been slightly underestimated, or whether one experiment may have been biased by the result of the other... But, most likely, the 750 GeV bump was just due to a random fluctuation of the background at this particular energy. Regrettably, the resulting mess cannot be blamed on experimentalists, who were in fact downplaying the anomaly in their official communications. This time it's the theorists who have some explaining to do.

Why did theorists write 500 papers about a statistical fluctuation? One reason is that it didn't look like one at first sight. Back in December 2015, the local significance of the diphoton bump in ATLAS run-2 data was 3.9 sigma, which means the probability of such a fluctuation was 1 in 10000. Combining available run-1 and run-2 diphoton data in ATLAS and CMS, the*local *significance was increased to 4.4 sigma. All in all, it was a very unusual excess, a 1-in-100000 occurrence! Of course, this number should be interpreted with care. The point is that the LHC experiments perform gazillion different measurements, thus they are bound to observe seemingly unlikely outcomes in a small fraction of them. This can be partly taken into account by calculating the *global* significance, which is the probability of finding a background fluctuation of the observed size anywhere in the diphoton spectrum. The global significance of the 750 GeV bump quoted by ATLAS was only about two sigma, the fact strongly emphasized by the collaboration. However, that number can be misleading too. One problem with the global significance is that, unlike for the local one, it cannot be easily combined in the presence of separate measurements of the same observable. For the diphoton final state we have ATLAS and CMS measurements in run-1 and run-2, thus 4 independent datasets, and their robust concordance was crucial in creating the excitement. Note also that what is really relevant here is the probability of a fluctuation of a given size in *any *of the LHC measurement, and that is not captured by the global significance. For these reasons, I find it more transparent work with the local significance, remembering that it should *not *be interpreted as the probability that the Standard Model is incorrect. By these standards, a 4.4 sigma fluctuation in a combined ATLAS and CMS dataset is still a very significant effect which deserves a special attention. What we learned the hard way is that such large fluctuations do happen at the LHC... This lesson will certainly be taken into account next time we encounter a significant anomaly.

Another reason why the 750 GeV bump was exciting is that the measurement is rather straightforward. Indeed, at the LHC we often see anomalies in complicated final states or poorly controlled differential distributions, and we treat those with much skepticism. But a resonance in the diphoton spectrum is almost the simplest and cleanest observable that one can imagine (only a dilepton or 4-lepton resonance would be cleaner). We already successfully discovered one particle this way - that's how the Higgs boson first showed up in 2011. Thus, we have good reasons to believe that the collaborations control this measurement very well.

Finally, the diphoton bump was so attractive because theoretical explanations were plausible. It was trivial to write down a model fitting the data, there was no need to stretch or fine-tune the parameters, and it was quite natural that the particle first showed in as a diphoton resonance and not in other final states. This is in stark contrast to other recent anomalies which typically require a great deal of gymnastics to fit into a consistent picture. The only thing to give you a pause was the tension with the LHC run-1 diphoton data, but even that became mild after the Moriond update this year.

So we got a huge signal of a new particle in a clean channel with plausible theoretic models to explain it... that was a really bad luck. My conclusion may not be shared by everyone but I don't think that the theory community committed major missteps in this case. Given that for 30 years we have been looking for a clue about the fundamental theory beyond the Standard Model, our reaction was not disproportionate once a seemingly reliable one had arrived. Excitement is an inherent part of physics research. And so is disappointment, apparently.

There remains a question whether we really needed 500 papers... Well, of course not: many of them fill an important gap. Yet many are an interesting read, and I personally learned a lot of exciting physics from them. Actually, I suspect that the fraction of useless papers among the 500 is lower than for regular daily topics. On a more sociological side, these papers exacerbate the problem with our citation culture (mass-grave references), which undermines the citation count as a means to evaluate the research impact. But that is a wider issue which I don't know how to address at the moment.

Time to move on. The ICHEP conference is coming next week, with loads of brand new results based on up to 16 inverse femtobarns of 13 TeV LHC data. Although the rumor is that there is no new exciting anomaly at this point, it will be interesting to see how much room is left for new physics. The hope lingers on, at least until the end of this year.

In the recent years, physics beyond the Standard Model has seen 2 other flops of comparable impact: the faster-than-light neutrinos in OPERA, and the CMB tensor fluctuations in BICEP. Much as the diphoton signal, both of the above triggered a binge of theoretical explanations, followed by a massive hangover. There was one big difference, however: the OPERA and BICEP signals were due to embarrassing errors on the experiments' side. This doesn't seem to be the case for the diphoton bump at the LHC. Some may wonder whether the Standard Model background may have been slightly underestimated, or whether one experiment may have been biased by the result of the other... But, most likely, the 750 GeV bump was just due to a random fluctuation of the background at this particular energy. Regrettably, the resulting mess cannot be blamed on experimentalists, who were in fact downplaying the anomaly in their official communications. This time it's the theorists who have some explaining to do.

Why did theorists write 500 papers about a statistical fluctuation? One reason is that it didn't look like one at first sight. Back in December 2015, the local significance of the diphoton bump in ATLAS run-2 data was 3.9 sigma, which means the probability of such a fluctuation was 1 in 10000. Combining available run-1 and run-2 diphoton data in ATLAS and CMS, the

Another reason why the 750 GeV bump was exciting is that the measurement is rather straightforward. Indeed, at the LHC we often see anomalies in complicated final states or poorly controlled differential distributions, and we treat those with much skepticism. But a resonance in the diphoton spectrum is almost the simplest and cleanest observable that one can imagine (only a dilepton or 4-lepton resonance would be cleaner). We already successfully discovered one particle this way - that's how the Higgs boson first showed up in 2011. Thus, we have good reasons to believe that the collaborations control this measurement very well.

Finally, the diphoton bump was so attractive because theoretical explanations were plausible. It was trivial to write down a model fitting the data, there was no need to stretch or fine-tune the parameters, and it was quite natural that the particle first showed in as a diphoton resonance and not in other final states. This is in stark contrast to other recent anomalies which typically require a great deal of gymnastics to fit into a consistent picture. The only thing to give you a pause was the tension with the LHC run-1 diphoton data, but even that became mild after the Moriond update this year.

So we got a huge signal of a new particle in a clean channel with plausible theoretic models to explain it... that was a really bad luck. My conclusion may not be shared by everyone but I don't think that the theory community committed major missteps in this case. Given that for 30 years we have been looking for a clue about the fundamental theory beyond the Standard Model, our reaction was not disproportionate once a seemingly reliable one had arrived. Excitement is an inherent part of physics research. And so is disappointment, apparently.

There remains a question whether we really needed 500 papers... Well, of course not: many of them fill an important gap. Yet many are an interesting read, and I personally learned a lot of exciting physics from them. Actually, I suspect that the fraction of useless papers among the 500 is lower than for regular daily topics. On a more sociological side, these papers exacerbate the problem with our citation culture (mass-grave references), which undermines the citation count as a means to evaluate the research impact. But that is a wider issue which I don't know how to address at the moment.

Time to move on. The ICHEP conference is coming next week, with loads of brand new results based on up to 16 inverse femtobarns of 13 TeV LHC data. Although the rumor is that there is no new exciting anomaly at this point, it will be interesting to see how much room is left for new physics. The hope lingers on, at least until the end of this year.

Today Gail Zasowski (JHU) showed up at MPIA for a week or so. Hans-Walter Rix brought up the possibility that we should consider—for the next generation of *SDSS* projects—using the *BOSS* spectrograph in concert with the *APOGEE* spectrograph to study the detailed chemical abundances of millions of stars. He pointed out that both observationally (Anna Ho's work) and theoretically (Yuan-Sen Ting's work) we believe that we can get many precise chemical abundances out of high signal-to-noise spectra at low (2000-ish) resolution.

Late in the day I worked on text for a large NSF Physics Frontier Center proposal we are putting in at NYU to build new data-science methods for the physical sciences of cosmology, planetary and stellar dynamics, and particle physics. This is being led by Kyle Cranmer (NYU) but includes many other luminaries, including our new hires at (that is, coming to) NYU: Anthony Pullen, Yacine Ali-Haïmoud, and Kat Deck!

On September 14 2015, something really huge happened in physics: the first direct detection of gravitational waves happened. But measuring a single gravitational wave was never the goal—.though freaking cool in and of itself of course! So what is the purpose of gravitational wave astronomy?

The idea is that gravitational waves can be used as another tool to learn more about our Universe and its components. Until the discovery of gravitational waves, observations in astrophysics and astronomy were limited to observations with telescopes and thus to electromagnetic radiation. Now a new era has started: the era of gravitational wave astronomy. And when the space-based eLISA observatory comes online, it will begin an era of gravitational wave *cosmology*. So what is it that we can learn from our universe from gravitational waves?

First of all, the first detection aka GW150914 was already super interesting:

- It was the first observation of a binary black hole system (with unexpected masses!).
- It put some strong constraints on the allowed deviations from Einstein’s theory of general relativity.

What is next? We hope to detect a neutron star orbiting a black hole or another neutron star. This will allow us to learn more about the equation of state of neutron stars and thus their composition. But the authors in this paper suggest another exciting prospect: observing so-called black hole kicks using gravitational wave astronomy.

So, what is a black hole kick? When two black holes rotate around each other, they emit gravitational waves. In this process, they lose energy and therefore they get closer and closer together before finally merging to form a single black hole. However, generically the radiation is not the same in all directions and thus there is also a net emission of linear momentum. By conservation of momentum, when the black holes merge, the final remnant experiences a recoil in the opposite direction. Previous numerical studies have shown that non-spinning black holes ‘only’ have kicks of ∼ 170 km per second, but you can also have “superkicks” as high as ∼5000 km per second! These speeds can exceed the escape velocity of even the most massive galaxies and may thus eject black holes from their hosts. These dramatic events have some electromagnetic signatures, but also leave an imprint in the gravitational waveform that we detect.

The idea is rather simple: as the system experiences a kick, its gravitational wave is Doppler shifted. This Doppler shift effects the frequency

with *v* the kick velocity and *n* the unit vector in the direction from the observer to the black hole system (and *c* the speed of light). The black hole dynamics is entirely captured by the dimensionless number *G f M*/*c*^{3} with *M* the mass of the binary (and *G* Newton’s constant). So you can also model this shift in frequency by using the unkicked frequency *f*_{no kick} and observing the Doppler shift into the mass. This is very convenient because this means that you can use all the current knowledge and results for the gravitational waveforms and just change the mass. Now the tricky part is that the velocity changes over time and this needs to be modelled more carefully.

A crude model would be to say that during the inspiral of the black holes (which is the long phase during which the two black holes rotate around each other – see figure 1), the emitted linear momentum is too small and the mass is unaffected by emission of linear momentum. During the final stages the black holes merge and the final remnant emits a gravitational wave with decreasing amplitude, which is called the ringdown phase. During this latter phase the velocity kick is important and one can relate the mass during inspiral *M*_{i} with the mass during the ringdown phase *M*_{r} simply by

The results of doing this for a black hole kick moving away (or towards) us are shown in fig. 2: the wave gets redshifted (or blueshifted).

This model is refined in various ways and the results show that it is unlikely that kicks will be measured by LIGO, as LIGO is optimized for detecting black hole with relatively low masses and black hole systems with low masses have velocity kicks that are too low to be detected. However, the prospects for eLISA are better for two reasons: (1) eLISA is designed to measure supermassive black hole binaries with masses in the range of 10

**Further Reading**

- The websites (LIGO / eLISA) of the ground-based gravitational wave interferometer LIGO and the large scale space mission eLISA have great descriptions about their mission and the science they do: worth checking out!
- Tushna Commissariat’s article in Physics World (Feb 11, 2016): A nice (non-technical) article on the first detection of gravitational wave
- Gravitation by Misner, Thorne and Wheeler. The ‘bible’ of general relativity and gravitational waves is aka as MTW (after its authors) and is a great start for a more solid background on the basics, but does not cover black hole kicks
- arXiv:1010.5260: A review on numerical methods (including applications to kicks)

*[I was on vacation for a few days.]*

Just before I left, Melissa Ness discovered that instrumental fiber number is a good predictor of whether or not two stars will get similar abundances in *APOGEE*, either with *The Cannon* or with the standard pipeline! This is perhaps not a surprise: The different fibers have different line-spread functions, and sit on different parts of the detector. We discussed how to mitigate this, and looked at the dependence of the issues on fiber number and line-spread function FWHM separately.

For the nth time, I re-wrote my abstract (that is, the scope of a possible paper) on what you could learn about a star's intrinsic properties from a *Gaia*-like parallax measurement. I think the focus perhaps should be the subjectivity of it: What you can learn depends on what you know and believe.

Hans-Walter Rix decided that my talk at the end of this week should be on the graphical model as a tool for data analysis. I hope he is right!

DFM and I worked through issues remaining in our MCMC *Data Analysis Recipes* paper, which I would like to post on the arXiv this month (or next!). We also worked through some remaining issues in his long-period transiting exoplanet paper, in which he discovers and estimates the population of very long-period planets in the *Kepler* data.

David Weinberg (OSU) gave a nice talk about how stellar populations come to chemical equilibria, making use of nucleosynthetic models. He looked at how star-formation events might appear in the metallicity distribution. He also showed the beautiful data on the alpha-abundance bimodality in the *APOGEE* data, but in the end did not give a confident explanation of that bimodality, which really *is* intriguing.

I also had a substantial chat with Matthias Samland about his project to constrain the directly emitted infrared spectrum of an exoplanet using multiple data sources. He has the usual issues of inconsistent data calibration, correlated noise in extracted spectra, and the simultaneous fitting of photometry and spectroscopy. It looks like he will have lots of good conclusions, though: The specta are highly informative.