## Symmetries and solutions

Here is an underemphasized way to frame the relationship between trajectories and symmetries (in the sense of Noether’s theorem)You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).a  . Consider the space of all possible trajectories for a system, a real-valued Lagrangian functional on that space, the “directions” at each point, and the corresponding functional gradient in each direction. Classical solutions are exactly those trajectories such that the Lagrangian is stationary for perturbations in any direction , and continuous symmetries are exactly those directions such that the Lagrangian is stationary for any trajectory . That is,

(1)

There are many subtleties obscured in this cartoon presentation, like the fact that a symmetry , being a tangent direction on the manifold of trajectories, can vary with the tangent point it is attached to (as for rotational symmetries). If you’ve never spent a long afternoon with a good book on the calculus of variations, I recommend it.

### Footnotes

(↵ returns to text)

1. You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).

• Popular-level introduction to the five methods used to identify exoplanets.
• Another good profile of the SEP.
• ArXiv gets some money to improve stuff.
• Flying fish are hard to believe. It’s something of a tragedy that fish capable of long-distance flight never evolved (that we know of?). They are so bird like it’s startling, and this ability has evolved independently multiple times.
• In addition to Russia and China, the US also at one time had ICBMs deployed by rail.
• For nuclear power plants governed by the United States Nuclear Regulatory Commission, SAFSTOR (SAFe STORage) is one of the options for nuclear decommissioning of a shut down plant. During SAFSTOR the de-fuelled plant is monitored for up to sixty years before complete decontamination and dismantling of the site, to a condition where nuclear licensing is no longer required. During the storage interval, some of the radioactive contaminants of the reactor and power plant will decay, which will reduce the quantity of radioactive material to be removed during the final decontamination phase.

The other options set by the NRC are nuclear decommissioning which is immediate dismantling of the plant and remediation of the site, and nuclear entombment which is the enclosure of contaminated parts of the plant in a permanent layer of concrete.Mixtures of options may be used, for example, immediate removal of steam turbine components and condensors, and SAFSTOR for the more heavily radioactive containment vessel. Since NRC requires decommissioning to be completed within 60 years, ENTOMB is not usually chosen since not all activity will have decayed to an unregulated background level in that time.

• The fraction of the federal budget devoted to NASA peaked in 1966, three years before the Moon landing.

## How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution

[Other parts in this series: 1,2,3,4,5,6,7,8.]

You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.

After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.

Now let’s hear another joke from renowned physicist Chris Jarzynski:

The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]

Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well.[continue reading]

• Elephants are secretly wearing high heels.

• Cost per unit hard-drive space is flattening (for consumer models).
• Crux was known to the Ancient Greeks due to the fact that it can be seen from southern Egypt; Ptolemy regarded it as part of the constellation Centaurus. It was entirely visible as far north as Britain in the fourth millennium BC. However, the precession of the equinoxes gradually lowered its stars below the European horizon, and they were eventually forgotten by the inhabitants of northern latitudes. By AD 400, most of the constellation never rose above the horizon for Athenians.

• Zotero 5.0 has significant changes and is out now.
• Mainland China has 36 nuclear power reactors in operation, 21 under construction, and more about to start construction.” See also Wikipedia and this long piece on Chinese investment in Namibia. In comparison, the US gets essentially all nuclear power from reactors built at least 30 years ago, and has just 4 new reactors under construction.
• Sentience Institute: “In discussions of effective animal advocacy (EAA) — the field of study for how we can most effectively help animals, also known as effective altruism for animals — there are several important, challenging, and sometimes controversial foundational questions that come up over and over. This post attempts to summarize and catalog the key evidence cited by EAA supporters on each side of these debates for easy reference.”
• Third black hole merger detected by LIGO. No neutron stars yet. Binary BH distribution might be more massive and have more misaligned spins than popular models. Nothing revelatory.
• Vulcan aerospace unveils airplane with world’s largest wingspan (by far) as part of air-launch orbital rocket service.

## Selsam on formal verification of machine learning

Here is the first result out of the project Verifying Deep Mathematical Properties of AI SystemsTechnical abstract available here. Note that David Dill has taken over as PI from Alex Aiken.a   funded through the Future of Life Institute.

Noisy data, non-convex objectives, model misspecification, and numerical instability can all cause undesired behaviors in machine learning systems. As a result, detecting actual implementation errors can be extremely difficult. We demonstrate a methodology in which developers use an interactive proof assistant to both implement their system and to state a formal theorem defining what it means for their system to be correct. The process of proving this theorem interactively in the proof assistant exposes all implementation errors since any error in the program would cause the proof to fail. As a case study, we implement a new system, Certigrad, for optimizing over stochastic computation graphs, and we generate a formal (i.e. machine-checkable) proof that the gradients sampled by the system are unbiased estimates of the true mathematical gradients. We train a variational autoencoder using Certigrad and find the performance comparable to training the same model in TensorFlow.

Q: Is the correctness specification usually a fairly singular statement? Or will it often be of the form “The program satisfied properties A, B, C, D, and E”? (And then maybe you add “F” later.)

Daniel Selsam: There are a few related issues: how singular is a specification, how much of the functionality of the system is certified (coverage), and how close the specification comes to proving that the system actually does what you want (validation).… [continue reading]

## Reeh–Schlieder property in a separable Hilbert space

As has been discussed here before, the Reeh–Schlieder theorem is an initially confusing property of the vacuum in quantum field theory. It is difficult to find an illuminating discussion of it in the literature, whether in the context of algebraic QFT (from which it originated) or the more modern QFT grounded in RG and effective theories. I expect this to change once more field theorists get trained in quantum information.

The Reeh–Schlieder theorem states that the vacuum is cyclic with respect to the algebra of observables localized in some subset of Minkowski space. (For a single field , the algebra is defined to be generated by all finite smearings for with support in .) Here, “cyclic” means that the subspace is dense in , i.e., any state can be arbitrarily well approximated by a state of the form with . This is initially surprising because could be a state with particle excitations localized (essentially) to a region far from and that looks (essentially) like the vacuum everywhere else. The resolution derives from the fact the vacuum is highly entangled, such that the every region is entangled with every other region by an exponentially small amount.

One mistake that’s easy to make is to be fooled into thinking that this property can only be found in systems, like a field theory, with an infinite number of degrees of freedom. So let me exhibitMost likely a state with this property already exists in the quantum info literature, but I’ve got a habit of re-inventing the wheel. For my last paper, I spent the better part of a month rediscovering the Shor code…a   a quantum state with the Reeh–Schlieder property that lives in the tensor product of a finite number of separable Hilbert spaces:

As emphasized above, a separable Hilbert space is one that has a countable orthonormal basis, and is therefore isomorphic to , the space of square-normalizable functions.… [continue reading]

## Abstracts for July 2017

• Modewise entanglement of Gaussian states
Alonso Botero and Benni Reznik
We address the decomposition of a multimode pure Gaussian state with respect to a bipartite division of the modes. For any such division the state can always be expressed as a product state involving entangled two-mode squeezed states and single-mode local states at each side. The character of entanglement of the state can therefore be understood modewise; that is, a given mode on one side is entangled with only one corresponding mode of the other, and therefore the total bipartite entanglement is the sum of the modewise entanglement. This decomposition is generally not applicable to all mixed Gaussian states. However, the result can be extended to a special family of “isotropic” states, characterized by a phase space covariance matrix with a completely degenerate symplectic spectrum.

It is well known that, despite the misleading imagery conjured by the name, entanglement in a multipartite system cannot be understood in terms of pair-wise entanglement of the parts. Indeed, there are only pairs of systems, but the number of qualitatively distinct types of entanglement scales exponentially in . A good way to think about this is to recognize that a quantum state of a multipartite system is, in terms of parameters, much more akin to a classical probability distribution than a classical state. When we ask about the information stored in a probability distributions, there are lots and lots of “types” of information, and correlations can be much more complex than just knowing all the pairwise correlations. (“It’s not just that A knows something about B, it’s that A knows something about B conditional on a state of C, and that information can only be unlocked by knowing information from either D or E, depending on the state of F…”).