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Maldacena's bound on statistical significance

When Juan Maldacena began his Strings 2014 talk after so many speakers who have displayed their eloquence back in June,

JM: Geometry and Quantum Mechanics,
he was feeling like a soccer player who had to play against Argentina. ;-) The audience exploded in laughter; Juan is clearly an Argentine patriot. Despite his personal modesty on steroids, the 13-minute talk was filled with inspiring thoughts.

Many of them have been discussed on this blog repeatedly. But let me focus on a rather new thing that starts to be covered around 6:00.

Maldacena reminds us of the obvious and old observation that the spacetime inside the black hole interior (i.e. the lifetime and the Lebensraum of the poor infalling observers) is limited which inevitably seems to affect the accuracy and reliability of the experiments.

Such limitations are often described in terms of the usual uncertainty relations. Inside the hole, you can't measure the energy more accurately than with the \[

\Delta E = \frac\hbar{2 \Delta t}

\] error margin and similarly for the momentum, and so on. But Juan chose to phrase his speculative ideas about the universal bound in a more invariant and more novel way, using the notion of entropy.

A person who is falling into a black hole and wants to make a measurement must be sufficiently different from the vacuum. But after she is torn apart, hung by her balls, and destroyed (note that I am politically correct and "extra" nice to the women so I have used "she"), the space she has once occupied is turned into the vacuum. The vacuum inside a black hole of a fixed mass is more generic so the "emptying" means that the total entropy goes up.

Juan says that the relative entropy\[

S(\rho|\rho_{\rm vac}) = \Delta K - \Delta S \geq 0

\] Because we know that once she's destroyed at the singularity, the entropy jumps at least by her entropy, it is logical – and Juan is tempted – to interpret the life and measurements inside the black hole, and not just the fatal end, as a process in which she approaches the equilibrium.

So it's not possible to perform a sophisticated, accurate, and/or reliable experiment without sending something in. And if we send something in, the entropy will increase. An explicit inequality that Maldacena conjectured is the following inequality for the statistical significance:\[

p \gt \exp(-S)

\] That's a formula written in the convention where the \(p\)-value is close to zero. If you prefer to talk about "\(P=\)99% certainty", you would write the same thing as\[

P \lt 1-\exp(-S)

\] The certainty is less certain than 100% minus the exponential of the negative entropy and I suppose that by \(S\), Juan only means the entropy of the object. It's still huge which means that the statement above is very weak. The entropy of a human being exceeds \(10^{26}\) (in the dimensional units nats or, almost equivalently, in the less natural but more well-known bits) so the deviation from 100% is just \(\exp(-10^{26})\) which is a really small number morally closer to the inverse googolplex than the inverse googol.

There may be stronger inequalities like that. And I also suspect that many such inequalities could be applicable generally – outside the context of black hole interiors. Have you ever encountered such inequalities or proved them?

Note that the \(p\)-value encoding the statistical significance is the probability of a false positive. If we're constrained to live in a finite-dimensional Hilbert space where all basis vectors get ultimately mixed up with each other or something else, it's probably impossible to be any certain than your microstate isn't a particular one. But there are just \(\exp(S)\) basis vectors in the relevant Hilbert space and one of them may be right even if the "null hypothesis" holds, whatever it is. I am essentially trying to say that \(\exp(-S)\) is the minimum probability of a false positive.

If someone thinks that she can formulate such comments more clearly or construct some evidence if not a conclusive proof (or proofs to the contrary), I will be very curious.

If you allow me to return to the black hole interior issues: It seems to me that these "bounds on accuracy or significance" haven't played an important role in the recent firewall wars. But they're still likely to be a part of any complete picture of the black hole interior. For example, it's rather plausible that all the arguments (and instincts) directed against the state dependence violate these bounds.

Juan tends to say that the rules of quantum mechanics may become approximate or inaccurate or emergent inside the black hole, and so on. He even says that "because the time is emergent inside, so is probably the whole quantum mechanics". Well, the answer may depend on which rule of quantum mechanics we exactly talk about. But quite generally, I don't believe that there can be any modification of quantum mechanics, even in the mysterious black hole interiors. In particular, the inequalities sketched by Maldacena himself might be derivable from orthodox quantum mechanics itself. And I would be repeating myself if I were arguing that ideas like ER-EPR and state dependence agree with all the postulates of quantum mechanics.

Also, if we sacrifice the exact definition of time as a variable that state vectors or operators depend on – and we do so e.g. in the S-matrix description of string theory – it doesn't really mean that we deform quantum mechanics, does it? If we lose time, we no longer describe the evolution from one moment to another and we get rid of the explicit form of the Heisenberg or Schrödinger equations. But the "true core" of quantum mechanics – linearity and Hermiticity of operators, unitarity of transformation operators, and Born's rule – remain valid.

What breaks down inside the black hole is the idea that exactly local degrees of freedom capture the nature of all the phenomena. But unlike locality, quantum mechanics doesn't break down. I should perhaps emphasize that even locality is only broken "spontaneously" – because the black hole geometry doesn't allow us to use the Minkowski spacetime as an approximation for the questions we want to be answered.

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reader R T Deco said...

They're government workers. Of course 80% are going to require an operating system that was designed for mental defectives! Frankly, I'm surprised that the number is not even higher. I guess that's an indication of the partial success that the LiMux developers had in dumbing down the system to government-worker level -- a difficult task.

I guess that Munich, in anticipation of the change, is transferring the budget for hiring a competent IT staff to purchasing third-party virus-protection software.

"Penguins belong to the South Pole, not to European or American buildings."

Except, apparently, Google datacenters. You do know that Google Web Server (which feeds this blog) runs on Linux, don't you?

reader Uncle Al said...

Linux is fast and tight, Windows is pretty. I did a 3-month calculation of a growing crystal lattice. Knoppix (boot from CD) ran 30% faster than Windows, AMD ran 30% faster than Intel. Knoppix in AMD still ran three months - but the log-log plot of the output was longer,

Past 32 A radius ran in blades. Theoretical slope is -2. The fun is in the intercept (smaller is better) and the bandwidth.

Unix is not unfriendly, but it is selective about who its friends are. "the Linux solution is very expensive because it requires lots of custom programming." Bespoke vs. off the rack.

reader Holger said...

Nope, I am using Linux since over 20 years, and I am in trouble only whenever I have to use a computer with Windows installed :-)

reader Luboš Motl said...

This is silly. Germany is (unlike Greece and others) a very well functioning country with a healthy equilibrium between the commercial and government sector. So the people who work for the government are in principle the very same kind of people who work in the private sector, too. The government sector has a different way how it's funded - it's stealing money from the productive citizens via the so-called "taxes" - but that doesn't really affect the work that the employees are doing there.

I think that the Google web server running this server should be moved to the South Pole, too. ;-)

reader Gene Day said...

I just cannot envision any modification of quantum mechanics whatsoever. I’ll bet that lubos is correct here.

reader Crazy Eddie said...

"Time" is a whore concept. No reason to believe QM depends on its survival.

reader Uncle Al said...

Interesting point that one cannot perform a measurement absent a source and a sink. If everything is at equilibrium, one can build a thermometer and read it, but not calibrate it to assign the output meaning.

reader Alice Cheshire said...

Sadly, Windows taught people that (1) Computers should be pretty and should be so easy a 3 year old could use them and (2) Computers should crash all the time. People expect lousy performance and don't care, as long as Facebook and Twitter come up most of the time. I don't use Windows at all now. I use open source software. I fully admit that most people have not the training nor the ambition to do this. I pay nothing for my software and my computer works the way I want it to. I find Windows too confining. On the other hand, for those who want pretty, sparkly screens, and no thought required, Windows is the way to go.

reader Luboš Motl said...

OK but having used Linux for 20 years should be classified as a medical disorder. ;-)

reader Luboš Motl said...

It's only strange because the "technical people" have been penetrated by anti-market zealots who suppress everyone else.

It's much stranger to be a fan of such a thing. Unix is a system from the 1960s that should be as obsolete today as the cars or music from the 1960s. But it's not obsolete especially because its modern clones have been promoted by a political movement.

Unix, like Fortran and other things, should share the fate of Algol, Cobol, Commodore 64 OS, and many other things, and go to the dumping ground of the history where it has belonged for quite some time.

reader Luboš Motl said...

There is nothing wrong for a system to be usable by a 3-year-old. Coffee machines, toasters, and vacuum cleaners have the same property. Kids are ultimately the best honest benchmarks to judge whether software is constructed naturally. When kids may learn it, it really means that an adult is spending less energy with things that could also be made unnecessarily complicated, and it's a good thing.

My Windows 7 laptop hasn't crashed for a year since I stopped downloading new and new graphics drivers etc. I had freezes due to Mathematica's insane swapping to the disk - when it should say "I give up" instead - but that's a different thing.

reader R T Deco said...

"So the people who work for the government are in principle the very same kind of people who work in the private sector, too."

Ah ... so can you show me the private sector equivalent, in principle, of the Potsdam Institute for Climate Impact Research? ;-)

The United States also is a very well-functioning country with a healthy equilibrium between the commercial and government sector. (In fact, I would argue that the US is less socialist than Germany.) Surely, during your time in the US you must have been forced to deal with the New Jersey or Massachusetts DMV? (Here I use the generic term -- in New Jersey it's called the MVC, while in Massachusetts it's the RMV.) If not, consider yourself very fortunate.

There's a little bit of Greece in every government bureaucracy. (In the US, we have to tell them not to defecate in the hall -- http://www.newser.com/story/189036/epa-to-workers-stop-pooping-in-the-hall.html -- yeah.) These are the folks who prefer a platform that is better suited for gaming, entertainment, and viruses than getting quality work done. Hence, I agree with you, I think that Munich is leaning toward making the right decision.

reader Luboš Motl said...

Sure, I can. The commercial sector is literally drowning in similar šit, too. Try e.g.


reader AJ said...

Your taking of COBOL out to the dumping ground of history may be a bit premature. It's still actively being used in bluechip industries such as banking, insurance, and telecommunications. As far as new development goes it's rarely (if ever) used in GUI type applications but remains popular for high volume backend transaction processing in the bluechip industries. My guess is that your recent Bank of America transactions were touched by COBOL at some point, most likely in the mission critical application of updating your account.

Not that I don't agree with your sentiment, it's just that it's incredibly difficult to get rid of. The business case for replacing existing backend systems with a more modern platform are usually weak.

reader Swine flu said...

Keyboards and mice should theoretically be obsolete too, but after playing with tablets for a couple of years, many people are moving back to laptops and even desktops for "real work". Linux having its origins in the 1960's is not an argument at all against it.

reader Luboš Motl said...

LOL, right, it surely feels like the two debit cards were attempted to be sent to me by a COBOL robot. ;-)

I understand it's hard to get rid of things when lots of stuff has been written in an old framework.

reader Eelco Hoogendoorn said...

'What I am really stunned by is the unbelievably complicated culture of installing things on Linux.'
Indeed. The only thing such accomplishes is making people feel clever because they haxxored their computer with 1337 compilars. In the real world of people trying to get stuff done, such nonsense is known as a lack of encapsulation, which is simply objectively bad software design.

reader FlanObrien said...

Wow, what a highly emotional and non-factual piece. I come here for science news, but the credibility of the blog just plummeted. So three year old user friendliness is the main criterion for municipal desktop operating systems? Where did this criterion come from? If valid, there are several Linux distributions dedicated to three year olds. Dou Dou, for example. Come on Lubos you can de better. Where is the meat (facts)?

reader orbifold said...

Have people who struggled with Linux run Windows computers for a long time before switching to a different operative system? Are there people who have always run Linux machines and never used Windows, but still feel unhappy about the Linux user experience. Just wondering because my mother started using computers when she was 60 yo, and she always found it pretty straightforward to use. Only time she tried to use Windows she found it pretty disgusting and user-unfriendly.

reader lcs1956 said...

Lubos is a theorist. All theorists use Windows, while most all experimentalists use Linux (Scientific Linux is the official OS of Fermilab and CERN). I'll let someone else explain the reasons.

reader FlanObrien said...

I think I get it already. Theorists tax the Operating System as lightly as a three-year-old, whereas experimentalists need the system for real work.

reader Luboš Motl said...

Dear Eelco, thanks for making these observations clear with some adult terminology! ;-)

reader Luboš Motl said...

I think it is true to some extent and there is nothing to be ashamed of.

Of course that theorists often use computers in similar ways as writers (of literature), not really to compute, and they don't want to waste their time by forcing computers to do elementary things because computers are supposed to make things simpler, not harder. Experimenters do lots of complicated things with computers so they may sacrifice some friendliness without increasing the amount of wasted time by too high a percentage.

For the Kaggle contest,


I had to recreate an Ubuntu virtual machine because it seemed like the most plausible if not only way to install software that helps one produce competitive scores.

By now, someone has ported it to Windows. I would probably prefer it but my experience with things like Visual Studio etc. is really non-existent, due to my Linux training, so the Linux path could have been easier for me due to the historical coincidences, too.

reader FlanObrien said...

"it's been my point for years that the movement to spread Linux on desktop is an ideological movement"

The reverse is true.

Computing in the free world is subject to market forces. Linux has won hands down everywhere except for the Desktop where MS Office addicted persons obstruct innovation. Political and objective reasoning has placed Linux everywhere except the desktop.

Grandmothers, children and some theorists have been well served on Desktop Linux for a decade or more. I invite you to drill down to the objective reasons why that is.

We will probably never know the truth about Munich IT management decisions, but the wider market tells a clear and dramatic story in favour of open (but profit making) systems.

If you find being called out for lack of meat obnoxious then I am sorry. This article happens to be the the first protein lacking I have seen by you,
Thank you for the Reference Frame.

reader Luboš Motl said...

Desktop - and increasingly more often, mobile platforms - are the places where the actual work is being done and where the actual relevant features of operating systems are being tested.

It's unambiguously clear that for the operating systems to do their work well, they should be profit-driven, company-protected systems. Whether the source is open or closed isn't too important. What's important is that a company has a financial interest to make it work. So Apple is doing the same thing for iOS and Google for Android that Microsoft is doing for Windows. The underlying mechanisms that make all these things usable are completely analogous and they require capitalism.

reader FlanObrien said...

You call the sharing of IT ideas, architecture and open core modules "socialism". By the same token you are a rabid socialist for openly discussing your physics theories.
By all means let Apple and Microsoft tinker with buttons and pixels to accommodate the increasingly dumbed down populations, but let the core architecture be defined by the Open Source world. This massively benefits the corporate world as well as the rest of humanity, which is why the corporate world all use Open solutions in one way or another.

reader Luboš Motl said...

Yes, I am an insane socialist donating intellectual assets of multi-million values to others for free.

But that's less unethical than to be forcing others to use unusable products.

reader Casper said...

It may be several hundred thousand generations behind the most obsolete flying saucer dimensional transfer management system in the galaxy, but .NET is the greatest thing in the known universe for sure. Do the Linux bug dwellers have anything remotely like this? I don't know since I haven't looked but I seriously doubt it. Congratulations to the officials of Munich city who have belatedly achieved common sense.

reader Gordon said...

Hmm, think you have been brainwashed by microsoft, Lubos---there are plenty of uses for Linux...even Google uses a lightly morphed version, as does Android, etc...here is a partial list of surprising adopters from Wikipedia:
--lots of free compilers as well for developers and programmers.

reader Luboš Motl said...

I have never communicated with Microsoft or read any of its opinions - unfortunately, I would say - so I couldn't have been "brainwashed by Microsoft".

I am not saying that people aren't using all kinds of other products, and so am I. Concerning mobile OSes, I have devices with iOS, Android, as well as Windows Phone, and Android is the most expensive one. I am just warning against the political movement that is trying to force different systems upon desktop users whose majority clearly and voluntarily prefers Microsoft Windows as the market conditions unambiguously show.

reader NikFromNYC said...

Unlike benchtop chemistry and biology, physics can be mostly taught online, with engineers later being hired to do experiments. I sure would like Lubos to join an online university to create video lectures, at both advanced and entry level physics.

-=NikFromNYC=-, Ph.D. in chemistry (Columbia/Harvard)

reader Michael said...

Honest question: What's so great about it? Can you explain or give an example? Thanks.

reader Swine flu said...

I have to say that I fail to see the Linux world as some sort of sinister kabal that is forcing innocents to use unusable systems. Look at the Linux desktop market share, and you can at least say that they have failed.

Windows is great for Microsoft-style word processing and spreadsheets. Perhaps it's even OK for TeX/LaTeX, if there's a decent and easy to install distribution for it (I know there is one for OSX, not sure about Windows).

Linux seems popular for scientific computing, and where such users want a more polished and easy to use system for their work laptop/desktop, they choose OSX, which gives you Unix underneath and a polished user interface on top.

That's why a progressive household would have all three operating systems on their computers. I know mine does. :)

reader John Archer said...


Which reminds me ...

I'm feeling nostalgic. It's many decades since every other word in those horrible computer trade magazines seemed to be about the 'goto' statement and 'spaghetti code'. Now all is silent — as far as I know anyway. Oh, how I miss the tedium of it all!

Anyone care to rekindle the exquisite ennui?

Hey, how about a discussion on punched cards versus paper tape? :)

Incidentally, as far as operating systems go, I mostly use Windows simply because, reluctantly, that was all that was made available to me at one point (more accurately it started with that awful DOS), but I got used to it and I can do all I need to do with it.

But most of all I use it these days because I'm buggered if I'm going to spend any time looking up the kind of stuff that I lost interest in and forgot about years ago just to make a change for the sake of Greater F#cking Spartan. Also VBA behind Excel can be very handy for a quickie, a little like a fast shag behind the bicycle shed. Just the ticket sometimes. :)


P.S. Many years ago, but again long past my interest date, I surprised myself by reading Bjarne Stroustrup's book on the genesis of C++ (I forget the title) and found it fascinating. I'm pretty sure I'm fully cured now though. :)

reader Holger said...

I just noticed that Microsoft is currently in the process of shifting its German operational center to - München, Schwabing. Now that they are becoming a big tax payer over there, it seems inconvenient for the municipal government to run on Linux. After all, Linux won't finance any pleasure ('amigo') trips for the local politicians, Microsoft perhaps does ...

reader Jim Z said...

Absent a source and a sink of time... everything happens? Or nothing happens?

The event horizon is when happening stops? Can entropy be static?

reader FlanObrien said...

"Suggestions the council has decided to back away from Linux are wrong, according to council spokesman Stefan Hauf."

Some meat:


reader Luboš Motl said...

Dear FlanObrien, the committee to review the computing in the city was probably built by the executive power in the city which is why one should also respect the interpretation of the executive power, and not the council, why it was done.

reader Alice Cheshire said...

Believing the world should run on the level of three-year-olds is really very disturbing. It may also explain why social has become more and juvenile over time. I figure if you need pretty pictures and shiny baubles, you're not really looking for a computer. More like an electronic playmate. It's interesting that your Vista computer worked so well. Mine crashed, despised the peripherals (all of which I replaced) and drove me to buy an Apple to escape the Microsoft curse. Maybe I just really use my computer more than most and expect it to function like I want it, not like a three-year-old wants it. I'm a grown-up now. I want a grown-up computer.