Bird (Vanderbilt) and I pair-coded a likelihood function for the age–velocity relation in the Milky Way Disk. After that, I coded up a likelihood function for the star-formation history in the Milky Way disk, given very noisy age estimates (from *The Cannon*). By the end of the day, Bird was successfully MCMC sampling his parameters, and my inference was going massively unstable. I am sure I have a simple bug.

Price-Whelan, Rix, and I discussed a brute-force search for stellar streams in stellar catalogs called *KingKong*. It samples randomly in orbit space for a trial or toy potential and asks, for each orbit, how many stars have observations that are consistent with being on that orbit. The method is simple. But it is all about implementation details.

The laser pistol (or similar personal directed energy weapon) is a staple of science fiction, and you can imagine why: No pesky ammunition to carry, possible some really cool light/sound effects, near-speed-of-light to target (though phasers and blasters and the like are often shown in TV and movies with laughably slow velocities, so the audience can see the beam or pulse propagate*). Still, does a laser pistol make sense as a practical weapon? This isn't "nano", but it does lead to some interesting science and engineering.

I'm a complete amateur at this topic, but it seems to me that for a weapon you'd care about two things: Total energy transferred to the target in one shot, and the power density (energy per area per time). The former is somewhat obvious - you need to dump energy into the target to break bonds, rip through armor or clothing, etc. The latter is a bit more subtle, but it makes sense. Spreading the energy transfer out over a long time or a big area is surely going to lessen the damage done. Think about slapping a board vs. a short, sharp karate chop.

From the internet we can learn that a typical sidearm 9mm bullet has, in round numbers, a velocity of about 400 m/s and a kinetic energy of about 550 J. If at bullet stops about 10 cm into a target, you can use freshman physics (assuming uniform deceleration) to find that the stopping time is half a millisecond, meaning that the average power is 1.1 megawatts (!).

So, for a laser pistol to be comparable, you'd want it to transfer about 550 J of energy, with an average power of 1.1 MW spread over a beam the size of a 9 mm bullet. Energy-storage-wise, that's not crazy for a portable system - a 1 kg Li-ion battery when fully charged would contain enough energy for several hundred shots. Batteries are not really equipped for MW power rates, though, so somehow the energy would probably have to be delivered to the beam-producing component by some exotic supercapacitor. (Remember the whine as a camera flash charges up? That's a capacitor charging to deliver a high-wattage pulse to the flash bulb.) The numbers for portability there don't look so good there - megawatt power transfers would likely require many liters of volume (or interchangeable, many kg of mass). Of course, you could start with a slower optical pulse and compress it - that's how facilities like the Texas Petawatt Laser work. Fascinating science, and it does get you to ~ 100 J pulses that last ~ 100 femtoseconds (!!). Still, that requires a room full of complicated equipment. Not exactly portable. Ahh well. Interesting to learn about, anyway.

(*The beam weapons in sci-fi movies and TV are generally classic plot devices: They move at whatever speed and have whatever properties are required to advance the story. Phasers on Star Trek can disintegrate targets completely, yet somehow their effects stop at the floor, and don't liberate H-bomb quantities of energy. The stories are still fun, though.)

I'm a complete amateur at this topic, but it seems to me that for a weapon you'd care about two things: Total energy transferred to the target in one shot, and the power density (energy per area per time). The former is somewhat obvious - you need to dump energy into the target to break bonds, rip through armor or clothing, etc. The latter is a bit more subtle, but it makes sense. Spreading the energy transfer out over a long time or a big area is surely going to lessen the damage done. Think about slapping a board vs. a short, sharp karate chop.

From the internet we can learn that a typical sidearm 9mm bullet has, in round numbers, a velocity of about 400 m/s and a kinetic energy of about 550 J. If at bullet stops about 10 cm into a target, you can use freshman physics (assuming uniform deceleration) to find that the stopping time is half a millisecond, meaning that the average power is 1.1 megawatts (!).

So, for a laser pistol to be comparable, you'd want it to transfer about 550 J of energy, with an average power of 1.1 MW spread over a beam the size of a 9 mm bullet. Energy-storage-wise, that's not crazy for a portable system - a 1 kg Li-ion battery when fully charged would contain enough energy for several hundred shots. Batteries are not really equipped for MW power rates, though, so somehow the energy would probably have to be delivered to the beam-producing component by some exotic supercapacitor. (Remember the whine as a camera flash charges up? That's a capacitor charging to deliver a high-wattage pulse to the flash bulb.) The numbers for portability there don't look so good there - megawatt power transfers would likely require many liters of volume (or interchangeable, many kg of mass). Of course, you could start with a slower optical pulse and compress it - that's how facilities like the Texas Petawatt Laser work. Fascinating science, and it does get you to ~ 100 J pulses that last ~ 100 femtoseconds (!!). Still, that requires a room full of complicated equipment. Not exactly portable. Ahh well. Interesting to learn about, anyway.

(*The beam weapons in sci-fi movies and TV are generally classic plot devices: They move at whatever speed and have whatever properties are required to advance the story. Phasers on Star Trek can disintegrate targets completely, yet somehow their effects stop at the floor, and don't liberate H-bomb quantities of energy. The stories are still fun, though.)

At MPIA Galaxy coffee, there were great talks about the Milky Way disk by Gail Zasowski (JHU) and Jonathan Bird (Vanderbilt). Zasowski showed results on the kinematics of the MW disk ISM based on diffuse interstellar bands. She used the three-d dust map from Schlafly *et al* to figure out the mean distance to each absorber, and sees a consistent story. Bird showed that the age–metallicity relationship in the disk (briefly, that older stars have higher velocity dispersion) is a product not just of disk heating but also of disk formation, if the disks form in the cosmological context as expected. Late in the day, Bird and I (with help from Rix) formulated a way to measure the age–metallicity relationship and its dependence on Galactocentric radius via a likelihood function (and therefore Bayes). We vowed to try to do this with the *APOGEE* data plus stellar ages from Ness's recent work with *The Cannon*.

Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts. As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms: Theorem 1 (Inverse theorem for Gowers norms) Let and […]

Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.

As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms)Let and be integers, and let . Suppose that is a function supported on such thatThen there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that

This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:

Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms)Let be an integer, and let be an ergodic, countably generated measure-preserving system. Suppose that one hasfor all non-zero (all spaces are real-valued in this post). Then is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree nilsystems, that is to say systems of the form for some degree filtered nilmanifold and a group element that acts ergodically on .

It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of -actions, the connection is less clear.

One can split Theorem 2 into two components:

Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms)Let be an integer, and let be an ergodic, countably generated measure-preserving system. Suppose that one hasfor all non-zero , where . Then is a

factorof an inverse limit of ergodic degree nilsystems.

Theorem 4 (Pro-nilsystems closed under factors)Let be an integer. Then any factor of an inverse limit of ergodic degree nilsystems, is again an inverse limit of ergodic degree nilsystems.

Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)

The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:

As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.

To show that is a factor of another system up to almost everywhere equivalence, it suffices to obtain a unital algebra homomorphism from to that intertwines with , and which is measure-preserving (or more precisely, integral-preserving). On the other hand, by hypothesis, is generated (as a von Neumann algebra) by the dual functions

for , where

indeed we may restrict to a countable sequence that is dense in in the (say) topology, together with their shifts. To obtain such a factor representation, it thus suffices to find a “model” associated to each dual function in such a fashion that

for all and , and all polynomials . Of course it suffices to do so for those polynomials with rational coefficients (so now there are only a countable number of constraints to consider).

We may normalise all the to take values in . For any , we can find a scale such that

If we then define the exceptional set

then has measure at most (say), and so the function is absolutely integrable. By the maximal ergodic theorem, we thus see that for almost every , there exists a finite such that

for all and all . Informally, we thus have the approximation

for “most” .

Next, we observe from the Cauchy-Schwarz-Gowers inequality that for almost every , the dual function is anti-uniform in the sense that

for any function . By the usual structure theorems (e.g. Theorem 1.2 of this paper of Ben Green and myself) this shows that for almost every and every , there exists a degree nilsequence of complexity such that

(say). (Sketch of proof: standard structure theorems give a decomposition of the form

where is a nilsequence as above, is small in norm, and is very small in norm; has small inner product with , , and , and thus with itself, and so and are both small in , giving the claim.)

For each , let denote the set of all such that there exists a degree nilsequence (depending on ) of complexity such that

From the Hardy-Littlewood maximal inequality (and the measure-preserving nature of ) we see that has measure . This implies that the functions

are uniformly bounded in as , which by Fatou’s lemma implies that

is also absolutely integrable. In particular, for almost every , we have

for some finite , which implies that

for an infinite sequence of (the exact choice of sequence depends on ); in particular, there is a such that for all in this sequence, one has

for all and all . Thus

for all in this sequence, all , and all ; combining with (2) we see (for almost every ) that

and thus for all , all , and all we have

where the limit is along the sequence.

For given , there are only finitely many possibilities for the nilmanifold , so by the usual diagonalisation argument we may pass to a subsequence of and assume that does not depend on for any . By Arzela-Ascoli we may similarly assume that the Lipschitz function converges uniformly to , so we now have

along the remaining subsequence for all , all , and all .

By repeatedly breaking the coefficients of the polynomial sequence into fractional parts and integer parts, and absorbing the latter in , we may assume that these coefficients are bounded. Thus, by Bolzano-Weierstrass and refining the sequence of further, we may assume that converges locally uniformly in as goes to infinity to a polynomial sequence , for every . We thus have (for almost every ) that

for all , all , and all . Henceforth we shall cease to keep control of the complexity of or .

We can lift the polynomial sequence up to a linear sequence (enlarging as necessary), thus

for all , all , and some . By replacing various nilsystems with Cartesian powers, we may assume that the nilsystems are increasing in and in the sense that the nilsystem for is a factor of that for or , with the origin mapping to the origin. Then, by restricting to the orbit of the origin, we may assume that all the nilsystems are ergodic (and thus also uniquely ergodic, by the special properties of nilsystems). The nilsystems then have an ergodic inverse limit with an origin , and each function lifts up to a continuous function on , with . Thus

From the triangle inequality we see in particular that

for all and all , which by unique ergodicity of the nilsystems implies that

Thus the sequence is Cauchy in and tends to a some limit .

If is generic for (which is true for almost every ), we conclude from (4) and unique ergodicity of nilsystems that

for , which on taking limits as gives

A similar argument gives (1) for almost every , for each choice of . Since one only needs to verify a countable number of these conditions, we can find an for which all the (1) hold simultaneously, and the claim follows.

Remark 6In order to use the combinatorial inverse theorem to prove the full ergodic inverse theorem (and not just the weak version), it appears that one needs an “algorithmic” or “measurable” version of the combinatorial inverse theorem, in which the nilsequence produced by the inverse theorem can be generated in a suitable “algorithmic” sense from the original function . In the setting of the inverse theorem over finite fields, a result in this direction was established by Tulsiani and Wolf (building upon a well-known paper of Goldreich and Levin handling the case). It is thus reasonable to expect that a similarly algorithmic version of the combinatorial inverse conjecture is true for higher Gowers uniformity norms, though this has not yet been achieved in the literature to my knowledge.

Filed under: expository, math.CO, math.DS Tagged: characteristic factor, Gowers uniformity norms, inverse conjecture, nilmanifolds, nilsequences

It remains embarrassing that physicists haven’t settled on the best way of formulating quantum mechanics (or some improved successor to it). I’m partial to Many-Worlds, but there are other smart people out there who go in for alternative formulations: hidden … Continue reading

It remains embarrassing that physicists haven’t settled on the best way of formulating quantum mechanics (or some improved successor to it). I’m partial to Many-Worlds, but there are other smart people out there who go in for alternative formulations: hidden variables, dynamical collapse, epistemic interpretations, or something else. And let no one say that I won’t let alternative voices be heard! (Unless you want to talk about propellantless space drives, which are just crap.)

So let me point you to this guest post by Anton Garrett that Peter Coles just posted at his blog:

It’s quite a nice explanation of how the state of play looks to someone who is sympathetic to a hidden-variables view. (Fans of Bell’s Theorem should remember that what Bell did was to show that such variables must be nonlocal, not that they are totally ruled out.)

As a dialogue, it shares a feature that has been common to that format since the days of Plato: there are two characters, and the character that sympathizes with the author is the one who gets all the good lines. In this case the interlocutors are a modern physicist Neo, and a smart recently-resurrected nineteenth-century physicist Nino. Trained in the miraculous successes of the Newtonian paradigm, Nino is very disappointed that physicists of the present era are so willing to simply accept a theory that can’t do better than predicting probabilistic outcomes for experiments. More in sorrow than in anger, he urges us to do better!

My own takeaway from this is that it’s not a good idea to take advice from nineteenth-century physicists. Of course we should try to do better, since we should alway try that. But we should also feel free to abandon features of our best previous theories when new data and ideas come along.

A nice feature of the dialogue between Nino and Neo is the way in which it illuminates the fact that much of one’s attitude toward formulations of quantum mechanics is driven by which basic assumptions about the world we are most happy to abandon, and which we prefer to cling to at any cost. That’s true for any of us — such is the case when there is legitimate ambiguity about the best way to move forward in science. It’s a feature, not a bug. The hope is that eventually we will be driven, by better data and theories, toward a common conclusion.

What I like about Many-Worlds is that it is perfectly realistic, deterministic, and ontologically minimal, and of course it fits the data perfectly. Equally importantly, it is a robust and flexible framework: you give me your favorite Hamiltonian, and we instantly know what the many-worlds formulation of the theory looks like. You don’t have to think anew and invent new variables for each physical situation, whether it’s a harmonic oscillator or quantum gravity.

Of course, one gives something up: in Many-Worlds, while the underlying theory is deterministic, the experiences of individual observers are not predictable. (In that sense, I would say, it’s a nice compromise between our preferences and our experience.) It’s neither manifestly local nor Lorentz-invariant; those properties should emerge in appropriate situations, as often happens in physics. Of course there are all those worlds, but that doesn’t bother me in the slightest. For Many-Worlds, it’s the *technical* problems that bother me, not the philosophical ones — deriving classicality, recovering the Born Rule, and so on. One tends to think that technical problems can be solved by hard work, while metaphysical ones might prove intractable, which is why I come down the way I do on this particular question.

But the hidden-variables possibility is still definitely alive and well. And the general program of “trying to invent a better theory than quantum mechanics which would make all these distasteful philosophical implications go away” is certainly a worthwhile one. If anyone wants to suggest their favorite defenses of epistemic or dynamical-collapse approaches, feel free to leave them in comments.