Thursday, March 05, 2015

Taming the butterfly effect

The butterfly effect is a defining effect of chaos theory. As Edward Lorenz said, the flapping of a butterfly's wings in Beijing is needed to decide whether a hurricane is created and lands in New Orleans a few weeks later. (And it is not, as the porn gurus in the IPCC claim, because of the CO2 that the butterfly exhales and that leads to a climate cataclysm.)

Is that true? There are many other butterflies and effects that help to decide but in general, it is often true: major events (hurricanes) are often strongly affected by very small perturbations of the initial state (butterfly).

How strong the dependence is and how quickly the sensitivity may grow with time? Three famous physicists just argued that there is an upper limit. In their fresh new hep-th paper
A bound on chaos,
Juan Maldacena, Steve Shenker of Stanford, and Douglas Stanford of Shenker ;-) provide us with evidence in favor of a cute new inequality.

First, you should ask: How can we make these comments about beautiful butterflies quantitative? Won't butterflies always be known for their beauty and chaos theory for its impressive but purely verbal proclamations? Well, there are simple ways to quantify it.

Represent the perturbation – of the butterfly – by the Hermitian operator \(V(0)\) at time \(t=0\). The properties of the hurricane are represented by a later measurement, at \(t\gt 0\), of another operator \(W(t)\).

The fact that the butterfly \(V(0)\) matters for \(W(t)\) is mathematically encoded by the nonzero commutator\[

[W(t),V(0)] \neq 0.

\] The commutator would vanish if the two operators were spacelike-separated, for example. In that case, relativity would prohibit any influence of \(V\) on \(W\). OK, how strong the influence is? How large is the commutator? You surely want to ignore the signs and the phases which is why you want to square the commutator before you take the expectation value\[

C(t) = -\langle [W(t),V(0)]^2 \rangle

\] The minus sign is there to make \(C(t)\) positive: note that the commutator of the two Hermitian operators is anti-Hermitian, and its square is therefore negatively definite. The bracket \(\langle\dots \rangle\) is the thermal expectation value at temperature \(T\),\[

\langle \dots \rangle = \frac{ {\rm Tr}( \dots e^{-H/T} ) }{Z}

\] OK, schematically, this measure of the influence, \(C(t)\), has the units of \(VWVW\), Volkswagen Volkswagen (or Voskovec, Werich, Voskovec, Werich). There are different ways to place the brackets around \(VWVW\) and it may happen that \(C(t)\) is comparable to\[

C(t)\approx \langle VV \rangle \langle WW \rangle.

\] It means that the commutator is "maximized". Note that this is the case for operators \(x,p\) in the Gaussian wave packets that saturate the uncertainty principle (minimize the uncertainty). For those operators, \([x,p]=i\hbar\) so \(C(t)\) would be \(\hbar^2\) while \(\langle x^2\rangle = x_0^2\) and \(\langle p^2 \rangle = p_0^2\). The product of the latter two is also \( (\hbar/m\omega)(m\omega \hbar) = \hbar^2 \). I omitted the authors' factor of two in equation 1 because I don't see it there; and because we are talking about parametric estimates and \(2\approx 1\) in this sense.

OK, let's return to the time-dependent evolution. The influence of the flapping of the butterfly's wings decreases by itself, \[

\langle V(t)V(0) \rangle\sim \exp(-t / t_d)

\] and the half-time \(t_d\) (associated with the dissipation or the collision time) is expected to be comparable to the inverse temperature \(t_d \sim 1/T=\beta\). So the exponent above is \(-tT\). Now, to make the story short, Maldacena, Shenker, and Stanford conjecture that a similar exponential growth, not decrease, of the initial-stage size of the chaos-measuring commutator is at most \(2\pi\) times faster than that.

In other words, some difference \(F_d-F(t)\) which is a polished version of \(C(t)\) above will behave as\[

F_d - F(t) = \epsilon \exp(+\lambda_L t) + \dots

\] where the prefactor \(\lambda_L\) with the units of inverse time is called the Lyapunov exponent. Maldacena's and pals' key assertion is that\[

{\Large \lambda_L \leq 2\pi T = \frac{2\pi}{\beta}.}

\] So the "exponentially fast takeover by chaos" is at most \(2\pi\) times faster than the decrease due to the thermal dissipation etc.

If you worry that there's too much chaos in the world around, this is good news for you. The rise of chaos can't be too fast, at least if the temperature is low enough. At temperatures near absolute zero, chaos can only kick in very slowly – you would probably expect that.

They offer some motivation for that conjecture, especially one based on gauge theories, AdS/CFT, stringy and higher-derivative corrections to various actions, and a unitarity bound on the scattering amplitudes. And the middle portion of the paper boasts a near proof of their inequality whose essence seems to be highly mathematically robust although the connection with complex enough physical systems may hide some problems.

A technical detail I liked was that they considered operators located at various points of the thermal circle. Note that the circumference of this circle is \(\beta=1/T\) and they considered operators at \(t\pm i\beta/4\) and \(t\pm i\beta/2\) – at four points uniformly distributed along the circle.

They present these things as convenient choices but I believe that those points will ultimately play a much more fundamental role – operators extrapolated to places such as \(t\pm i\beta/2\) are everywhere in Papadodimas-Raju papers, among others – in the future conceptual understanding of the information, black hole interior, the emergence of spacetime, chaos, and other things.

My prophesy is that there will be some very natural, universal, and robust statements involving the extrapolation of operators to these points.

You may also view the result by Maldacena, Shenker, and Stanford as an example of a family of insights that link the periodic functions with the exponential ones. Dissipation and the rise of chaos are connected with the exponential rise or decrease in time. On the other hand, the thermal circle is periodic. Periodic and exponential functions differ by the extra \(i\) in the exponent. You may think that the functions \(\exp(at)\) and \(\exp(ibt)\) have very little to do with each other, they're just mathematically similar but they're "physically" very different things. But \(a,b\) may be extrapolated to imaginary or complex values which is why there exist analytical results that relate these functions.

Also, this new result is somewhat analogous to the AdS/CFT bounds on viscosity. I would even say that the "bound on chaos" is deeper and more interesting because its applicability is much more universal. It may be fair to say that this paper has been decoupled from any dependence on AdS/CFT, string theory, or quantum gravity. You may apply it to any (quantum) physical system. But it's likely that without the experience with the string or quantum gravitational physical systems, this inequality wouldn't have been found (or have some non-string theorists scooped them?).

No Heisenberg-Coulomb effect

By the way, I am happy to be able to endorse most of Sabine Hossenfelder's comments about a (primarily German) media Blitz hyping the promised looming detection of the gravitational Casimir effect.

The Casimir effect of the electromagnetic field depends on having boundary conditions \(\phi={\rm const}\) at the surface of the conductor. This boundary condition follows from Ohm's law, \(\vec j =\gamma \vec E\). Whenever there is an electric field inside the conductor, it makes charges carriers move, and they neutralize the electric field. At the end, there is \(\vec E = 0\) inside the metal which allows you to say that the electrostatic potential \(\phi={\rm const}\) in the metallic bulk (including the surface).

The Casimir effect is a force between two metallic plates that may be derived from the zero-point energy in the vacuum between the plates. Only modes going like \(\phi(z)\sim \sin \pi N z / L\) for \(N\in \ZZ\) and \(0\leq z \leq L\) are allowed in between the metals whose distance is \(L\), thanks to the boundary conditions, and their zero-point energy is proportional to \(\sum_{N=1}^\infty N^3=\zeta(-3)=+1/120\) (a fact analogous to \(1+2+3+\dots = -1/12\)) times \(1/L^3\) and other factors. The divergent part of the sum cancels against the zero-point energies that you would get in the vacuum without any constraints such as \(\phi={\rm const}\). For that reason, the attractive Casimir force per unit area \(F/A\) will be something like \(1/120L^4\), among some extra factors such as \(\pi\) and \(2\) – and \(\hbar\) and \(c\).

Now, nothing like the "metals" exists for the gravitational field. You can't make the counterpart of charge carriers, the mass carriers, move in the opposite direction because the mass of anything is never negative. So even though the gravitational waves would experience the Casimir effect as well if there were analogous boundary conditions, it is a big If and you won't be able to "fully" enforce such boundary conditions.

Real-world materials – whose density is nearly zero, relatively to the Planck density – may only "partly distort" the gravitational field's boundary conditions, and that's way too little for observable consequences. The gravitational force is extremely weak because \(G_N\) is so small. It means that any gravitational objects between very small objects are hard to measure. There just can't be a measurable force due to a gravitational Casimir effect between two special "plates".

Now, the media Blitz is based on a new paper by James Quach which is pretty good. But it takes something as input, namely a 2009 paper by Minter and 2 pals that coined what they called the Heisenberg-Coulomb effect (the two famous physicists have nothing to do with this particular effect!).

Minter et al. claimed that in the presence of gravitational waves, the positively charged nuclei move along the geodesics but the negatively charged di-electron Cooper pairs are not allowed to do so, and huge discrepancies in the distribution of the positive and negative charges is therefore created which leads to a huge plasma-like effect and an insanely amplified impact of the gravitational wave.

This whole separation of the positive and negative charges is nonsense, as you would probably guess.

The separated positive and negative charges carry a huge energy and the energy conservation law guarantees that this won't happen just because of some nearby low-energy-carrying gravitational wave. Just keep on describing the superconductor using the exact Schrödinger's equation for many electrons and many nuclei if you get confused by the effective descriptions that sometimes simplify things for superconductors but that may confuse you when it comes to unusual questions. And indeed, things get confusing because the Cooper pairs aren't "localized". But they're made of electrons whose operators \(\hat x\) are still well-defined so any miraculous prediction based on the "non-localizability of the Cooper pairs" is bound to be wrong.

Of course that Schrödinger's equation for the many electrons and many nuclei will guarantee a pretty much exact local cancellation of the charges (think about the expectation value of the charge density – but the vanishing is really a full approximate operator equation). If your superconductor-optimized effective equations for the Cooper pairs violate the local compensation of charges, then these effective equations are just wrong!

The right solution for a superconductor in the background of a gravitational wave must start with the environment – the Cooper pairs are moving in the background of the electrostatic and gravitational forces. As always, the former ones dominate while gravity is negligible. Whatever the spacetime geometry is, the charge density will be effectively neutralized in every region of the superconductor. The equations for the field creating/annihilating the Cooper pairs will be modified by the gravitational wave but the corrections will never violate the local compensation of the electric charge density.

So Hossenfelder is obviously right that no one will observe this "quantum gravitational" effect. No one will observe any other quantum gravitational effect that she and others have written lots of papers about, either, but you can't expect her to be so much right. ;-)


  1. Yes I skipped a line, sorry about that. How about this: the government agreed to bail out the banks, based on false figures , and when it ran into trouble as the debts were higher than quoted and the world economy faltered, the ECB forced it to continue with an even greater bailout
    It still sucks ;)

  2. Dear Lubos, other than the article itself I also found your comments about relationship with string theory interesting. Weinberg says that quantum field theory is the framework for theories that satisfy cluster decomposition principle at low energies. Minwalla has said that he believes that when string theory is fully understood it will be the framework for all computations :) (Minwalla said this in his video lectures on string theory.) . Do you have some insights about this ?

  3. Hi,
    I am wondering if the literal butterfly effect can really be true.
    If there is no initial macroscopic variation (one temperature, no flows) the butterfly is prevented from "producing its own macroscopic motion" by the second law.
    Now if there is macroscopic movement and variation already how much can it be perturbed and thereby effected by the butterfly?
    In a simulation of the weather using classical physics one can indeed get hypersensitivity to the initial conditions (chaos), but with quantum uncertainty the initial conditions can be determined this accurately anyway. What I mean is, wouldn't the flapping of the wings influence on the air some meters away from there have gone through such a "cascade" of quantum uncertainty that one really can't say how it flapped its wings (chaos in reverse ;-) ) and therefore its "influence" on how a hurricane manifest is rendered moot, or in other words, it really did not impact how it manifested because the uncertainty or differences it produced has been washed away by quantum uncertainty.

  4. Got derailed from reading all this because you started by citing Goldman Sachs, the bank that helped the Greek government falsify data so it could join the Eurozone. Then you continued with babble about money as a store of wealth, as if money were the wealth of nations. I had to quit reading, for fear I would read something that implied the dollar is the world reserve currency because of the superior economic productivity of the US or...well, after a beginning like that, who knows?

    You talk about money as a store of wealth as if, for example, retirees could actually save up what they need, for food and clothing and medical care in money form. Apparently you believe that if the money is hard, it's like all the food and clothing and medical system is packed away into the cellar. Except of course to a first approximation all these things come from current production. Money has no supernatural powers over time.

    A country with hard currency and low economic production is not a wealthy country.

  5. What is the right way to think about the infinities involved in the Casimir effect? Is there a positive infinity canceled by a negative infinity or simply two infinities canceling out so that one is left with 1/120{3} or -1/12{1}? I don't quite understand where the cancelation is, how it occurs.

  6. I'm usually sceptical on 'good news' from the government, but this bit of good news sounds promising:

    I hope it turns out.

  7. Dear John, there are many ways to think about this cancellation.

    For example, you may start with the ordinary empty vacuum. It's full of quantum fields and they have zero-point energies, like harmonic oscillators in quantum mechanics.

    When you sum these contributions to the energy density of the vacuum, you get some divergent term, plus some finite corrections.

    Because we observe the vacuum to be basically zero, up to the tiny cosmological constant, it means that the total energy density of the vacuum must be written as

    rho = Lambda_0+ sum of the zero-point energies

    You could naively think that the vacuum energy density is "just" the sum of the zero-point energies, but that would yield a huge - or divergent - energy density of the vacuum. That's clearly not what we observe. We observe nearly zero. So the correct formula can't be just the sum. There must be an extra additive term that cancels whatever you get from the sum, especially the divergent pieces, and I wrote it as Lambda_0.

    The fun of the Casimir effect is that you may do similar calculations in the presence of the metallic plates, and in the absence of the metallic plates. Whatever scheme you use, the calculation of the total energy in the case with the metallic plates will have the same divergent part as without the plates.

    For simplicity, imagine that you fill the space with infinitely many parallel thin conductive plates separated by the distance L. Then you may still talk about the average energy density. This density will be

    rho(with_plates) = Lambda_0 + sum of zero-point energies (with plates)

    The fun thing is that the divergent, dull part of the sum of the zero-point energies will be the same as the sum in the absence of the plates above, and it will therefore cancel Lambda_0. However, there will still be the finite part left, and this part is exactly equal to -1/12 (for the power 1) or +1/120 (for the power 3).

    The cancellation between Lambda_0 and the sum may be viewed as the "simplest" case of renormalization, the renormalization of the constant term in the Lagrangian, the vacuum energy density. The bare value in the classical Lagrangian is "correctly" divergent so that it cancels the loop diagrams which also have a divergence, leaving (almost) zero.

    Physicists normally subtract these pieces "automatically". Effectively, they consider energy differences etc.

    The discussion would depend on the precise regularization and renormalization schemes - there were various ways to calculate the sum of integers mentioned e.g. at

    It's a good exercise to discuss all these things in detail. But in all cases, the healthy heuristic attitude simply works. The sums may be assigned finite values and you may claim that the divergent piece never existed for a while, it was just a defect of the "human visualization" of the sums, while 1+2+3+... is "really" equal to -1/12, and 1+8+27+64+125+... is "really" equal to +1/120. Everything that is experimentally verifiable is consistent with these assumptions.

  8. kashyap vasavadaMar 5, 2015, 5:57:00 PM

    Nice provocative article! My understanding of the chaos phenomena , whether classical or quantum, is that it arises when the basic equations are highly non linear resulting in extreme dependence on the initial conditions. Thus every phenomenon is not chaotic. Thank God!! Does the non vanishing of the commutator has the same implications?

  9. Yup, I do believe we are completely in accord, Lubos.

  10. Please, give me a break with this hyperstinky communist garbage.

    Goldman Sachs may have done some professional work it was hired to do, and if it was hired to make some data look compatible with some conditions, it did the work. That's what companies normally do. They are seeking profit.

    None of these things changes the fact that Greece has lived beyond its means, produced amazing debt by real consumption and overspending, and the entry of Greece to the EU as well as the Eurozone occurred because it was favored by the political elites on both sides as well as the bulk of the Greek nation. So some work done by Goldman Sachs, whatever it was, was a pure formality that has nothing to do with the bad key things that have occurred.

    Goldman Sachs is an eminent financial company and its top minds are clever than you and your fellow nutjobs and anti-banking conspiracy theorists combined.

    I won't read your comment beyond the first sentence because it is a clear waste of time.

  11. Dear Michael,

    in quantum mechanics, you can't have the exact value of x and p at the same moment, due to the uncertainty principle. But the quantum mechanical counterpart of the exact point (x,p) in the phase space is the exact pure initial state, and the final state still depends on the choice of the initial pure state finely, analogously to the dependence in classical physics.

    In a system with many degrees of freedom, the thermalization (the whole thing that looks like increasing entropy etc.) is a part of the butterfly effect!

    If you pick sufficiently general final operators to measure - those that are sensitive to the relative motion of all the atoms etc., and not just some "overall average operators" (like the number of atoms in a cubic millimeter) that we often talk about, then these final operators will depend on the initial microstate.

    This includes the dependence on the exact microstate of the butterfly wings. This exact pure state of the wings does affect the generic final-state operators that "feel" the microscopic correlations. And if there's a way for these operators to get translated to "big event" operators such as the existence of a hurricane, and there is such a translation in the real world, then this hurricane depends on the pure initial state of the wings, too.


  12. I could not resist. Here is a practical example. ;-)

  13. Eh didn't the government pay Merrill Lynch 7M euros for a report where they underestimated the cost of the irish bank bailout by 45bn? The sad story was there was at least one guy there who knew about the impending banking crisis but they told him to shut up. I worked there for a while in Sandyford. I remember one project manager, who directed projects with such evocative names as the "Cayman Island" project, who boasted to his colleagues that he was leaning about decoherence by reading "How To Teach Quantum Physics To Your Dog". Hehe

  14. "Goldman Sachs may have done some professional work it was hired to do, and if it was hired to make some data look compatible with some conditions, it did the work."

    Exactly the problem, since you don't know when they're just doing the work.

  15. Thanks for the answer.

    What I meant by the first part of my comment was something like this:
    Say you have a huge big room where there is no macroscopic motion of the air, and a single uniform temperature. Now a butterfly you forgot in the room starts flapping around. It will never create a hurricane in there.

    If on the other hand you already have variations and movement in there sustained by some energy source, how they further manifest can be effected by small perturbations as simulations of chaos show.

    "and the final state still depends on the choice of the initial pure state finely, analogously to the dependence in classical physics."

    Right, I can only agree with that, but I still feel the quantum case makes a difference. Say we use the huge "room" and you have the ability to calculate from the initial quantum state the probabilities for measuring this hurricane or that hurricane or no hurricane at all a month later. Now you do that with and without the butterfly. In the classical case you could have an initial state where there never was hurricane (and a small perturbation of the butterfly would have changed things), but in the quantum case - with the butterfly or without - you have probabilities for hurricanes here and there, manifested like this or that or none at all in both cases, and the butterfly never really makes any difference. The classical case changed completely because of the butterfly but in the quantum case you had probabilities for hurricanes with it and without it. Therefore it had no impact. Its possibility to be the "decider" of the hurricane was destroyed.

  16. Ok, I see the argument in your text now, Lubos. Might be interesting to look at the GDP separately because it may already have been inflated in itself to unhealthy dimensions with unproductive state expenses. Also I think that as a member of the Euro zone they could borrow money cheaper for a while, which can be hidden in the graph.

  17. Very interesting , Lubos.
    It is another reminder that “human visualization” is even more defective than that small part of it that underlies the anti-QM zealot’s ignorance.

  18. What else are they doing? WTF?

  19. That's not the main problem, Mikael. Even if Greece borrowed totally cheaply i.e. at zero interest rate, the debt would already be pathological.

    Even without any interest rate, it may be hard to repay a big loan. When the interest is zero and stays zero, it is obvious that one may repay it in principle -it's enough to reduce the debt at least a little bit every year.

    But in practice, the same is true if one pays the interest as long as the interest payments are smaller (or significantly smaller) than the whole GDP of the economy. Whenever it is so, one may make the payments and some extra payments to reduce the debt, and use the rest to live.

    In all these discussions about the government, the problem is that they don't *want* and they think that they may avoid paying the debt. Individual borrowers know that they won't be treated this generously. If an individual borrower just can't repay, for whatever reason and regardless of any excuses, the bank simply confiscates all the collaterals and kicks the guy to the street to become homeless. It's clearly a #1 priority to repay the debt. The problem with government-borrowers is that they have their own muscles by which they may defend their otherwise indefensible behavior, and by which they may place their own comfort above their duty to repay the debt.

    I agree that the (Greek) GDP is heavily inflated because it contains lots of production that may only be consumed by a wasteful government - because it's otherwise uncompetitive in the market.

  20. You're right that it doesn't matter which units you use to weight the pizza: the underlying economic reality remains unchanged. Greece will still have s**t-tons of trouble. Actually, the s**t-ton looks to me like the appropriate unit of measure in this case.

    What is funny is that Miller (of Modigliani-Miller propositions fame) resorts frequently to the very same joke to explain why it also doesn't matter *who owns* which slice. See

    He even got a Nobel price for his pizza joke.

  21. Greeks are eternal debt slaves. They'll NEVER recover inside the Euro
    because they can't devalue their own currency. That , plus :
    1 - low labor productivity.
    2 - Far-left , socialist (crypto-communist) government.

    3 - Tax evasion (a national sport over there).

    4 - Unpayable debts.

    Euro is a monumental failure. Because Europe will never be America , no
    matter how much the EU tries to destroy national , ethnic , cultural
    identities... This currency has no serious institutional structure ,
    It's a political currency , not an economic project.

    The EU is
    the west's version of the soviet union. They're making the exact same
    mistakes. They expended too much. Included countries that could never
    realistically compete with Germany's productivity. Unfortunately , it'll
    all crumble....Because Eurocrats have a "too big to fail" mentality.
    So , just like the Ex-soviets , they'll continue what they're currently

    That's why the continuous pauperization of Europe is irreversible.

    Interesting fact...

    American government documents show that the US intelligence community
    ran a campaign in the Fifties and Sixties to build momentum for a united

    It funded and directed the European federalist movement.
    documents confirm suspicions voiced at the time that America was
    working aggressively behind the scenes to push Britain into a European
    state. One memorandum, dated July 26, 1950, gives instructions for a
    campaign to promote a fully fledged European parliament. It is signed by
    Gen William J Donovan, head of the American wartime Office of Strategic
    Services, precursor of the CIA.
    The documents were found by Joshua
    Paul, a researcher at Georgetown University in Washington. They include
    files released by the US National Archives. Washington's main tool for
    shaping the European agenda was the American Committee for a United
    Europe, created in 1948. The chairman was Donovan, ostensibly a private
    lawyer by then.

    The vice-chairman was Allen Dulles, the CIA
    director in the Fifties. The board included Walter Bedell Smith, the
    CIA's first director, and a roster of ex-OSS figures and officials who
    moved in and out of the CIA. The documents show that ACUE financed the
    European Movement, the most important federalist organisation in the
    post-war years. In 1958, for example, it provided 53.5 per cent of the
    movement's funds.
    The European Youth Campaign, an arm of the European
    Movement, was wholly funded and controlled by Washington. The Belgian
    director, Baron Boel, received monthly payments into a special account.
    When the head of the European Movement, Polish-born Joseph Retinger,
    bridled at this degree of American control and tried to raise money in
    Europe, he was quickly reprimanded.

    The leaders of the European
    Movement - Retinger, the visionary Robert Schuman and the former Belgian
    prime minister Paul-Henri Spaak - were all treated as hired hands by
    their American sponsors. The US role was handled as a covert operation.
    ACUE's funding came from the Ford and Rockefeller foundations as well as
    business groups with close ties to the US government.
    The head of
    the Ford Foundation, ex-OSS officer Paul Hoffman, doubled as head of
    ACUE in the late Fifties. The State Department also played a role. A
    memo from the European section, dated June 11, 1965, advises the
    vice-president of the European Economic Community, Robert Marjolin, to
    pursue monetary union by stealth.

    It recommends suppressing debate until the point at which "adoption of such proposals would become virtually inescapable""

  22. A complete equivalent for devaluing one's currency - when it comes to all the future events - is simply to lower all the salaries and pensions etc. by the same factor. It does the same job.

    The omnipresent comment about the biggest loss of freedom that "one can't devalue its currency" is highly oversold.

  23. And indeed that is what Ireland did. Of course this is painful and takes a crisis before any government will do it (as opposed to the more 'painless' version of devaluation, which seems to almost happen by itself, so harder to 'blame' someone). As you know, Greece already did it but the riots were not pretty. So while it can be done 'successfully', it is not as easy to sell it to the population. Still, if it needs to be done, it needs to be done... Portugal also tried tax increases as another 'equivalent' to devaluation.

  24. A tax increase is in no way equivalent to the reduction of salaries and public incomes - in some sense, they're the opposite things because lower salaries regulated by the government mean less state redistribution while higher taxes means more redistribution.

  25. I'm not sure what you mean by 'redistribution'. Do you mean the government sucking money from both public and private employees to pay the government debt, whereas only public employees should be hit? The tax intake would go to pay the debt, so wouldn't be redistributed (in that sense) within the economy as is usual for tax money. Portugal wanted to make the tax relatively easier for company employees, but I lost track of how far they got due to all the protests and u-turns.

  26. It doesn't matter that the taxes are used to pay the government's debt - they're still redistribution - in the sense of partial communism.

    If one raises taxes to pay the government debt, it means that the current productive people, especially the high earners (and because the taxes are mostly progressive, the high earners contribute more than proportionally to their income), are paying for the well-being of (mostly) the public employees in the past (who created the debt).