Thursday, April 27, 2017

Like James Bond, Nature loves to walk a micron from the cliff

Inconsistencies, like the death of 007, are often avoided by microns or "almost" miracles. Nature abhors arbitrary bans on "potentially dangerous" things.

If you are a realist and you watch a movie about James Bond or a similar film, you may be annoyed by some scenes that seem very unlikely. In the real world, you may argue, the hero would have died about 20 times throughout the movie. It's implausible that the hero could have survived so many lethal challenges. And if he has survived them, it would be much more likely that he escaped them an hour, and not a second, before the last moment.



These movies may be said to be "unrealistic" and you could rephrase the adjective as "unnatural". James Bond's survival is "unnatural", some people could say. The word "unnatural" indicates that this is not how Nature normally behaves. And when you apply this reasoning to the laws of physics, you may be led to think that Nature actually avoids this last-minute and last-micron salvation, all the miracles that keep the likes of James Bond alive – and that allow him to beat the foes. Nature is insured and many layers of insurance policies are protecting Her from being fatally hurt – from being inconsistent.

But you would be completely wrong. Nature loves to survive – and Her laws are consistent – but She doesn't have numerous levels of insurance. She actually is and loves to be similar to James Bond. Her goal is to survive, not to "safely" survive, and She does it, often walking an infinitesimal distance away from a cataclysm.




In this blog post, I want to enumerate several examples of Nature's James-Bondness. But the main purpose of the blog post aren't these examples – the trees. The purpose is to convince you about the forest, about the general idea, because people's being wrong about the general idea is arguably a general invalid prejudice that makes them repeatedly wrong about many things, confused about others, and expecting lots of things that simply don't hold in physics.




Modern physics turned Nature to James Bond

There could exist examples of "borderline inconsistencies" of the laws of physics as they were imagined before the 20th century and if you think that you know nice examples, it could be fun if you explain them. But as far as I can see, the laws of physics started to enjoy "adrenaline sports" in the 20th century, with the rise of modern physics.

Well, it seems to me that the Newtonian physics was self-evidently consistent. Everything was assumed to follow from an objective state of affairs – objective positions and velocities of particles or magnitudes of fields. Some trajectories \(x^i(t)\) or histories \(\phi^i(x,y,z,t)\) were assumed to be right, others were wrong. The right ones obeyed some differential equations. Those could be easily seen to have a solution (and typically a unique solution) for nice enough initial conditions. How could anything go wrong?

The situation changed already with Einstein's special theory of relativity. The argument \(t\), time, of the functions \(x^i(t)\) and \(\phi^i(x,y,z,t)\), ceased to be absolute. Different inertial observers should use different coordinates \(t,t'\). The latter is a combination of the former and a spatial coordinate, and vice versa. It follows that the simultaneity or chronology of two events is observer-dependent. One inertial observer may conclude that the event \(A\) took place before \(B\). Another inertial observer may conclude that \(A\) took place after \(B\).

This "relativity of simultaneity" is a threat. The cause should precede its effects, the principle of causality says, and when the chronology is relative, there is some danger that people could change their past which would cause logical contradictions. Special relativity doesn't actually suffer from any problem. While the values of \(t,t'\) and the chronology depend on the inertial observer, the causal loops are avoided because the superluminal influences are forbidden for all observers. And their being forbidden for one observer is equivalent to their ban from the viewpoint of all others.

You should understand why you can't change your past, even in the subtle world of relativity, but I want to emphasize a more general lesson that follows from this example:
The correct physical theory, in this case special relativity, is ultimately logically consistent. But to understand why some potential contradiction doesn't become a real one, you need to think a little bit hard. The consistency isn't quite obvious to everybody.
In other words, Nature behaves like James Bond. Not everyone could survive all the challenges – and not everyone can immediately understand why and how the laws of physics avoid all the potential paradoxes. They do but they have to be as creative, skillful, and seemingly lucky as James Bond. The absolute time looked like an obvious and obviously safe way to avoid closed time-like curves. But from the relativistic viewpoint, this way is a wrong one. It was too cowardly one. Special relativity employs a seemingly "less safe" way which turns out to be consistent as long as you analyze it carefully.

Demise of realism in quantum mechanics

The far more difficult change that physics has undergone is the transition to quantum mechanics. Our observations are no longer reflections of some objective state of affairs. Instead, they must be considered subjective information arising from subjective observations – and all these observations help to quantify quantities that are no longer commuting \(c\)-numbers. Instead, the observables become \(q\)-numbers, i.e. non-commutative yet associative generalizations of numbers, or operators on the Hilbert space.

Someone who is used to think classically could be shocked, and most laymen are still shocked: How can something like that work at all? Quantum physics should still be able to produce similar predictions as classical physics did. How can you achieve it from the subjective observation-dependent, non-commuting operators? Well, it does work very well, thanks for asking. For example, the operators don't have to be commutative because an order of projection operators is always preferred – because observers always make their observations chronologically ordered in a certain way – but the sequence of operations (e.g. observations) \(A,B,C\) still doesn't depend on the brackets. And quantum mechanics agrees with that because the product of operators is still associative.

The observations are dependent on an observer's perspective and a beginner may be afraid of new inconsistencies that it could cause. Won't this observer dependence contradict the agreement between several humans who have observed a given experiment? There's no contradiction, however. From the viewpoint of Alice, Bob announces the same result of his observation of Object because according to Alice, Bob's observation of Object made Bob and Object entangled. And it's this Bob-Object entanglement that guarantees the correlation (or identity) of Alice's own observation of Object and Bob's announcement of his observation of Object.

In classical physics, the agreement between Alice and Bob was explained trivially – both Alice and Bob were just irrelevant observers added on top of the shared, objective reality which was the only thing that "invariantly mattered". Both of their observations just reflected the shared, objective reality which explains all the agreement. Quantum mechanics abandons the shared, objective reality but it still agrees with the agreement between the two observers' testimonies. However, the quantum explanation of this agreement isn't the same, trivial one that we had in classical physics. Objective reality was a sufficient condition for the agreement between two people's observations – but it's not a necessary condition. And in quantum mechanics, the assumption of the objective reality is rejected, indeed.

You could formulate and people have formulated hundreds of potential paradoxes in quantum mechanics using various types of the language. For example, Feynman's path integral seems to use \(c\)-numbers for the observables \(x(t)\). Don't they violate the fact that the positions and velocities should be non-commuting? Again, there's no contradiction. It's the non-differentiability of the generic trajectories that reconciles the uncertainty principle with the path integrals.

Dozens of TRF blog posts have shown that there is no paradox in quantum mechanics. Everything works perfectly. But again, the general lesson is that:
Quantum mechanics is ultimately internally consistent and consistent with all the observations. But the reasons behind this consistency may be subtle, hard to understand, and seemingly risky. They're different than the "obvious reasons" in the older theories.
Classical physics was "more obviously" logically consistent – it had lots of "safe spaces" in between itself and any inconsistency – while quantum mechanics, like James Bond, isn't afraid of walking on the tightrope and save its life in the last millisecond. But there's nothing wrong about the walk on a tightrope. If you can't do it well, you're recommended not to do it, but James Bond and Nature can do these things well, and you shouldn't criticize them for the stunts just because they are unsafe for you!

At the end, it's very natural for Nature to be courageous in this sense – to exploit all the possibilities that are still compatible with the survival. When something is possible and/or compatible with a logically consistent theory of Nature, it will almost certainly be exploited by Nature. Cowardliness is anthropomorphic and it's just silly to assume that Nature is afraid of the same things as beginners who start to learn modern physics.

Quantum field theory: divergences

When you calculate the loop corrections in quantum field theory, you obtain divergent expressions. The total amplitudes seem to be infinite. It's another contradiction if true. However, if you apply the renormalization procedure correctly, you substitute the numbers of the form \(1/137+\infty\) for the bare constants as well, and when the infinities from the loops are added, all the infinities cancel and you obtain finite results for quantities that are actually measurable!

So the logical inconsistencies that could have arisen from some infinite terms ultimately evaporate because all the infinities cancel at the very end. But isn't it unsafe to have infinities in the intermediate calculations? It may be unsafe for you but it's safe for James Bond and for Nature. They are not afraid of these things because they can reliably escape the threats. Once again:
The infinities in quantum field theory could cause logical paradoxes. These paradoxes of Nature could be avoided by banning infinities in all intermediate calculations as well. Such a solution would look "safe". But it is not a necessary condition. It's enough for the final result to be finite and consistent and adventurous Nature happily uses this observation and affords lots of infinities in the intermediate results.
I've conveyed the same point many times. The point is that some people might have been indoctrinated by the precautionary principle and they would recommend Nature to act "really safely", so that it's "really obvious" that Nature can't become a victim of any lethal inconsistency. But Nature doesn't like the precautionary principle. If it were šitting into Her pants like you are, dear fan of the precautionary principle, it couldn't ever achieve anything impressive. Nature needs to do extreme steps, the consistency of the final results is the only true condition, and Nature just doesn't try to be safer than what is needed.

Anomalies, ghosts, holography, topology change, ...

In quantum field theories with chiral (left-right-asymmetric) fermions coupled to gauge fields such as the Standard Model, gauge anomalies may arise. Anomalies are quantum effects (basically one-loop diagrams) that would contribute nonzero terms to continuity equations for currents – and that would break symmetries that we assumed at the classical level. Gauge symmetries are needed for the consistency of the gauge theory (in particular, for the decoupling of the negative-norm polarizations of the gauge boson) so they must cancel.

In the Standard Model, the anomalies ultimately cancel. Many of them only cancel when you include the contributions of both leptons and quarks. A theory with leptons only or a theory with quarks only wouldn't be consistent. Again, a person who likes the precautionary principle could say that there's just too much risk in these anomalies – they can render a theory inconsistent. She could propose:
It's just too risky for these chiral fermions to contribute anomalies. Something goes wrong and the theory is inconsistent. To be safe, Nature should better ban the chiral fermions – and anomalies they contribute – altogether.
I just had a Quora exchange with a schoolgirl who was brainwashed by the European Union and who was whining that the very fact that the schools brainwash kids is something that no one should be allowed to say because it may sound insulting to the politically correct schoolkids. Holy cow.

OK, the European Union and the kids brainwashed by it would be eager to ban the chiral fermions in Nature, as potential sources of anomalies – just like they banned Edison's light bulbs, strong vacuum cleaners, or plastic bags, and critiques of Mohammedanism, among many other things. But Nature had to achieve something, instead of being a bunch of useless and distasteful losers and cowards such as the European Union, so Nature simply couldn't ban the chiral fermions.
According to Nature's spokeswoman, chiral fermions are great. They allow left-right-asymmetric phenomena and much lighter fermions than what would be possible in non-chiral theories. Nature would kindly insist that the consistency of the final predictions is the only constitutional requirement and to demand something on top of that is an unconstitutional assault on Nature's basic freedoms.
In other words, Nature despises the European Union, the signatories of Paris, Kyoto, Munich, and other cowardly agreements and similar champions of bans and precautionary principles who are šitting into their pants 24 hours a day.

The number of examples where Nature – quantum mechanics, quantum field theory, string theory etc. – seems "bold" or "bolder than some people would expect" seems immense. When you read a textbook of these modern subjects, you may see numerous examples in every chapter. String theory in a particular formulation seems to contain lots of negative-norm and/or zero-norm states but fortunately, all of them get decoupled. Holography seems strange but everything ultimately works. String theory also remains consistent when the spacetime develops an orbifold, flop, or conifold singularity, and so on, and so on.

Again, the message of this blog post is more general:
Don't repeatedly assume that Nature tries to be consistent for simple reasons – and to be "safe" even in the intermediate results where the absence of "seemingly dangerous things" (such as the infinities) isn't strictly speaking required. "Seemingly dangerous" aspects of the intermediate calculations are allowed by the laws of Nature and Nature uses them all the time.

The laws of physics are ultimately consistent but they're often consistent for different reasons and you need to work hard, deal with subtleties, cancel them carefully, and be "afraid for a while" before you conclude that the laws work just fine. There's nothing wrong about the elevated adrenaline level during the calculation, verification, or argumentation.

If you are imagining Nature as a European Union regulator who wants to ban almost everything, just to be safe, and who is šitting into Her pants, you will have a hard problem to understand modern physics because Nature despises these individuals at least as much as I do. Nature ultimately stays safe and consistent but the elevated adrenaline level is often needed before you see this happy end.
And that's the memo.

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