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Thursday, October 4, 2012

Evading quantum mechanics: again

Posted on 12:00 AM by Unknown
A reader named "the guy from Rhinoceros" has brought my attention to yet another article trying to evade quantum mechanics; it has clearly become a fashionable hobby of many people these days. Otherwise sensible semi-technical Ars Technica wrote an article with a deliberately offensive title,
Demolishing Heisenberg with clever math and experiments.
No kidding. It's a title of three lies because the authors aren't demolishing Heisenberg but working tightly within the framework he co-discovered; their maths isn't clever but rather completely trivial; and they have done no experiments.

The subtitle of the Ars Technica article is "Good, general measurement choices eliminate uncertainty." Thank God, we may return to the 19th century again.

The paper on which this hype is based upon is trying to do "pretty much the same general thing" as the guys abusing the misleading concept of a weak measurement. They just have a different name with different misinterpreted maths for the same thing, "quantum non-demolition experiments".

It's a concept that has, much like the "weak measurements", a meaningful definition, but all the actual applications of the concept that attract the media – because they "demolish Heisenberg" – are completely bogus once again. Let's look at the paper.




The paper is called
Evading Quantum Mechanics: Engineering a Classical Subsystem within a Quantum Environment Phys. Rev. X: X stands for excellence (no kidding)

Evading quantum mechanics (arXiv March 2012: a free copy) by Mankei Tsang and Carlton M. Caves.
Note that the paper has a dramatic title, too, although Ars Technica made it even more dramatic, more personal, and more dishonest.

The general dream of these people is to be able to measure systems in the real world so that the measurement doesn't disturb the system. This is, of course, impossible due to the Heisenberg uncertainty relationships. Whenever a physical system carries at least some information – whenever its Hilbert space is at least two-dimensional – there will always be observables that don't commute with a given one (unless it is a multiple of the unit matrix, \(\lambda\cdot {\mathbb 1}\): and these special matrices don't measure anything because their eigenvalues \(\lambda\) are constant and independent of the state of the physical system) simply because matrices \(2\times 2\) and larger don't commute with each other. It's that simple.

The other matrices that don't commute with a given one aren't artificial in any sense. They're pretty much as fundamental as the original matrix. Just think about the three Pauli matrices: they describe three components of the spin and all these three components are clearly equally natural; after all, they are related by the rotational symmetry. The most widespread example – position and momentum – really conveys the general story, too. The position operator and the momentum operator don't commute with each other,\[

[x,p] = i\hbar

\] and it has consequences even for position itself because the time-derivative of the position is the velocity \(v\) which is nothing else than a multiple of the momentum, \(v=p/m\). So the position doesn't commute with its own time derivative which implies\[

[x(t),x(t')]\neq 0.

\] The nonzero commutators in quantum mechanics that lead to the uncertainty principle don't relate "good observables" with some "artificial observables" you could dismiss, ignore, overlook, and ban. Even if you pick the most natural and essential observable that all the anti-quantum zealots like, the position of a particle (or anything), it just doesn't commute with itself at later times. Consequently, if you know the value of the position at time \(t\), the position at time \(t'\) which is later will only be known probabilistically. They can't have sharp values at the same moment, at least generically.

Now, the "quantum non-demolition measurements" are meant to be measurements of an observable \(O\) that obeys\[

[O(t),O(t')]=0.

\] This is the "dream" of the anti-quantum zealots because they may worship \(O(t)\) as one of the "nice observables" that has made the world classical again and that helped to demolish the evil Werner Heisenberg. But do such observables \(O(t)\) exist?

If you care about interesting, nontrivial, evolving ones in systems that can actually be realized and that are interacting, the answer is a resounding No. The reason is simple: the vanishing of the commutator above also means that \(O(t)\) commutes with its time derivative. If the spectrum of \(O(t)\) is discrete and non-degenerate, it means that an initial state which is an eigenstate of \(O(t)\) isn't allowed to evolve to anything else than a multiple itself.

Indeed, if it evolved into a state containing a admixture of states with different eigenvalues of \(O(t)\), it would mean that the Hamiltonian wouldn't conserve \(O(t)\). It would have to contain matrix elements that are off-diagonal in a basis of \(O(t)\) eigenstates. Their commutator with \(O(t)\) would therefore inevitably contain off-diagonal elements as well. This commutator would be proportional to \(dO(t)/dt\) which would therefore refuse to commute with \(O(t)\) due to these off-diagonal elements.

So you can't do it.

It means that under these assumptions, an operator only commutes with itself at all times if its value is essentially conserved. But in that case, you don't get any information if you make a later measurement. You will get the same value as you did at the beginning.

In the paragraphs above, I assumed that the spectrum of \(O(t)\) is discrete and non-degenerate. But morally speaking, I didn't have to. If the operator had a degenerate spectrum, there would be a loophole that allows states to evolve into superpositions of states that also contain other states with the same eigenvalue. But that wouldn't change the value of \(O(t)\), either, so it would still be true that the measurements are giving us no dynamical information.

What about my assumption of the discreteness? It isn't really reducing the strength of the argument in physically interesting situations, either. Momentum-like continuous variables become discrete in a box. Put a system in a large box, physics shouldn't really change, the spectrum will become discrete, and my argument will apply. For a position-like continuous observable, latticize the space to achieve the same thing.

Nevertheless, strictly speaking, the proof that \(O(t)\) must be time-independent if \([O(t),O(t')]=0\) won't be valid anymore and the authors think it's important to look for counterexamples. They find the following pathological two-dimensional harmonic oscillator:\[

H = \zav{\frac{kx_1^2}{2} + \frac{p_1^2}{2m}} - \zav{\frac{kx_2^2}{2} +\frac{p_2^2}{2m}}

\] The first parenthesis is a normal harmonic oscillator. The second one is normal as well except that the energy is counted with a minus sign. Note that the kinetic energy has the inverted sign as well so this is something else than a particle in the inverted potential (which has no discrete energy eigenstates): it is a negative-mass particle. The eigenvalues of the second parenthesis are the same discrete numbers as the eigenvalules of the first term except for the overall minus sign.

Now, one may easily find "quantum non-demolition observables" in this system – in fact, a broader set of "non-demolition observables" that they call "quantum mechanics free subsystem" and they even introduce an acronym QMFS for this manifestly useless concept. It is a set of observables that must obey\[

\forall j,k:\quad [Q_j(t),Q_k(t')]=0.

\] Can you find such observables in the two-dimensional harmonic oscillator above? Yes, you can. Take some positions and momenta in the light-like directions,\[

x_1+x_2, \quad p_1-p_2,

\] and you may check that these two operators commute with each other (at the same moment). The commutator \([x_1,p_1]=i\hbar\) simply cancels against \([x_2,-p_2]=-i\hbar\). They spend half a page with totally trivial manipulations and field redefinitions to convey this trivial point about the "light-like combination", showing that they're not really friends with basic linear algebra.

Also, you may check that the time derivative of \(x_1+x_2\), the first operator in the "quantum mechanics free subsystem", is proportional to \(p_1-p_2\), the second operator. The sign in front of \(p_2\) is negative because of the negative mass in the second "parenthesis" of the Hamiltonian. Also, the time derivative of \(p_1-p_2\) is proportional to \(x_1+x_2\).

So the time derivative of an operator commutes with the operator. It's true for both of them so the operators \(x_1+x_2\) and \(p_1-p_2\) will commute with copies of each other at all times. Great.

The only problem is that the Hamiltonian we introduced to realize this dream of a "quantum mechanics free subsystem" (the terminology does betray that those folks would love to be "liberated" from quantum mechanics) is unphysical. It is unbounded from below. No physical system you may construct in Nature may be well described by the Hamiltonian simply because it could make the energy arbitrarily low. A related problem is that all the energy levels of this two-dimensional harmonic oscillators are infinitely degenerate. You may get \(7\hbar\omega\) as the sum of two integers (or half-integers), a positive one and a negative one, multiplied by \(\hbar\omega\) in infinitely many ways, e.g. as \((1007.5-1000.5)\hbar\omega\). An experimental arrangement trying to emulate the Hamiltonian clearly has to have infinitely many degrees of freedom (atoms).

Equally disturbingly, the two-dimensional harmonic oscillator is too simple and non-interacting. Any extra terms that you introduce to your Hamiltonian – in particular, terms that are necessary for you to be able to actually measure the values of \(x_1+x_2\) or \(p_1-p_2\) using an apparatus – will violate the property that \(x_1+x_2\) commutes with its time derivative.

One must emphasize that there are still the other observables, \(x_1-x_2\) and \(p_1+p_2\), and \(x_1-x_2\) doesn't commute with \(p_1-p_2\) while \(x_1+x_2\) doesn't commute with \(p_1+p_2\). Of course that you're not escaping quantum mechanics in any way. Everything they're doing is done within the framework of quantum mechanics. Everything they do critically depends on the precious insights of Werner Heisenberg. So comments that this is "demolishing Heisenberg" are absolutely preposterous.

Let's stop discussions about the conceptual misinterpretations and invalid motivation of the work. Can the "quantum mechanics free subsystems" be useful for anything? I am less certain about the answer to this question but my guess is No. For example, you may want to use the operators above to measure the time including the phase which could be seen very accurately, something you can't do with a single ordinary photon. Is it possible to extract the phase accurately?

I don't think so. To be able to react to the "current phase" of \(x_1+x_2\) and \(p_1-p_2\) (which are rotating along some ellipses, just like in any harmonic oscillator) sufficiently quickly, the apparatus has to have strong enough interactions with the "bizarre two-dimensional harmonic oscillator", and with strong enough interactions, you will modify the commutators and the "quantum mechanics free subsystem" will no longer have sufficiently accurately vanishing commutators, so your attempts will fail, anyway.

Moreover, we know how to measure the phase precisely. If you have a coherent state of many photons, a macroscopic electromagnetic wave (coming from a LASER, for example), the intensity of the electric field may be measured rather accurately. We are redefining the "effective Planck's constant" (I mean a more general parameter measuring how important quantum mechanics is for certain questions) to \(\hbar/N^k\) where \(N\) is the number of photons in the same state and \(k\) is a positive exponent I don't want to calculate now.

These two very authors think that their construction will be useful to eliminate quantum noise. This is just a name for the effects resulting from the intrinsic and unavoidable probabilistic nature of quantum mechanics. I think that this whole reasoning is completely fallacious. First, the term "quantum noise" is misleading because it tries to give a negatively sounding emotional charge to a great and crucial property of quantum mechanics. Second, the "quantum noise" is both unavoidable and dependent on the situation you consider. So the only way how you may "avoid the quantum noise" is to change the situation. But then you're solving a different problem. This is particularly clear in the authors' own example. If you want to reduce the quantum noise in a particular optical device transmitting information to a NASA spaceship, the solution isn't to claim that the optical device should better be an exact two-dimensional harmonic oscillator with energy unbounded from below. It's not! ;-)

Combined with the fact that the required Hamiltonian seems physically impossible, it seems very hard for me to imagine that this construction could be useful for anything. However, I still think that the reason why such papers – despite their having no citations except for self-citations – get to various "excellent" Physical Reviews and are so frequently hyped by the non-expert media is that they seem like they are "demolishing Werner Heisenberg" which they are surely not. I am greatly annoyed by this dishonest activity – to which the researchers contribute mostly (but not only) dishonesty and the journalists contribute mostly (but not only) stupidity.

You won't demolish Heisenberg's insights because they're demonstrably true, idiots!

And that's the memo.



P.S.: If you have one hour of time and nerves for this kind of insanity, you may watch this January 2011 Google talk by Ron Garret, a guy formerly employed at Google and later turned into a professional armchair physicist of a sort (although, the speaker in this talk about quantum physics admits, it doesn't mean he is a physicist).



The video was sent to me by Lazăr Lung. Three sentences from the description summarize what category of a loon this Gentleman is:
The problem is that the vast majority of popular accounts of QM are simply flat-out wrong. They are based on the so-called Copenhagen interpretation of QM, which has been thoroughly discredited for decades. It turns out that if Copenhagen were true then it would be possible to communicate faster than light, and hence send signals backwards in time.
It's a Category 5 loon. Popular sources are full of anti-Copenhagen crackpot pseudoscience and of exactly this kind of bullshit – claims that the proper Copenhagen quantum mechanics implies superluminal or acausal effects (for the latter, see some largely confused fresh text by Nude Socialist about entangled photons in graves, London, and Beijing: this sort of stuff gets produced every minute, it seems) – but you may still find people who think that the presentation should be even more anti-Copenhagen.

The title of his talk was "The Quantum Conspiracy: What Popularizers of QM Don't Want You to Know". LOL. At the beginning of the talk, he says that the title was a joke – but the rest of the talk proceeds just like if he believed the title is serious and accurate. The talk is confused from the beginning to the end – not a surprise given the fact that the first claim is that a measurement must mean the detection of an underlying objective reality. A few minutes later, he says that there's no objective reality, but that's the main insight by the Copenhagen school that there isn't.

At the end, he says lots of correct things – at various points, he says that the measurements don't reflect an underlying objective reality; there are no hidden variables explaining the quantum randomness etc.; there is no collapse; Schrödinger's cat does evolve into linear superpositions, etc. – and a large part of his statements is a complete misunderstanding what the Copenhagen interpretation is – because the Copenhagen folks are the originators of many ideas he uses as a "replacement" of what he calls the Copenhagen interpretation. Also, he identifies the Copenhagen interpretation with the ideas about the real "collapse", something that no one in the proper Copenhagen school even championed. In particular, Heisenberg never used the term collapse, preferring to speak of the wavefunction representing our knowledge of a system, and collapse as the "jumping" of the wavefunction to a new state, representing a "jump" in our knowledge which occurs once a particular phenomenon is registered by the experimenter (i.e. when an observation takes place).

It's just a complete mess – this guy is completely confused about the history which is perhaps less irritating than the folks who are confused about the physics. Well, he's still confused about much of the physics, too. For example, he believes that the polarization relatively to a 45° slanted axis is the same thing as an "unpolarization". It's surely not. It's just another polarization, another pure state. The unpolarized light is a mixed state which is something else.

Around 26:00, he proposes Einstein-Podolsky-Rosen-Garret paradox (my instructor in Prague taught me not to declare myself a co-author with people who don't know about it, in the context of Roland Omnes, and I guess that EPR didn't approve Garret's message). In that paradox, he either proves superluminal action at a distance; or a violation of the complementarity/uncertainty. That's of course impossible to design such an experiment (the error turns out to be as trivial as not realizing that photons entangled with different states of other particles no longer interfere, and he seems to realize that). At that moment, I became totally unable to say which of his statements he made are meant seriously and which of them are ironic or statements he wants to disprove. I suppose he essentially understands those things, he just encapsulates them in a completely confused story.
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Posted in philosophy of science, science and society | No comments

Wednesday, October 3, 2012

SUSY with a colored adjoint chiral multiplet

Posted on 5:50 AM by Unknown
I discussed the same exciting possibility in November 2011 and May 2012 but because there's a new paper on the arXiv, I won't resist to make another comment.

The paper is called
Pushing the SUSY Higgs mass towards \(125\GeV\) with a color adjoint
and it was written by Gautam Bhattacharyya and Tirtha Sankar Ray. Even though the names may look like two names of the same ethnic origin, the paper is actually an outcome of an Indian-German-French-Australian collaboration. ;-)

Steven Weinberg has said many nontrivial propositions I fully subscribe to and one of them is
"Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough."
What could it mean in the case of supersymmetry? Well, it could mean that we're not trying to incorporate enough supersymmetry into our theories. The new Indian paper is another example showing why this criticism could be valid.




String/M-theory admits lots of vacua with 32 real supercharges; this "maximum" amount of supersymmetry is also referred to as \(\NNN=8\) in the four-dimensional notation. However, even 16 and 8 supercharges i.e. \(\NNN=4\) and \(\NNN=2\) is too many (too many new particles, too constrained interactions, no left-right asymmetry) so whenever physicists try to describe the world around us, they talk about models with 4 supercharges, i.e. \(\NNN=1\) supersymmetric theories.

That's fine but the reduction all the way down to 4 supercharges is only necessary because of some "problems" that are really problems for the matter fields – leptons and quarks (and perhaps the Higgs field) – only. For the gauge fields, one could perhaps survive a larger amount of supersymmetry.



Somewhat more elementary: Don Lincoln of Fermilab explains the Standard Model in 8 minutes. A new video.

The new Indian paper is another way to formulate a theory of this kind; they are adding a "chiral multiplet" transforming in the adjoint of the gauge group. Well, they mean the QCD \(SU(3)\) group only so they're cousins of the gluon and the gluino. What does it mean? It means that they are adding a complex scalar field and a Weyl fermion that transform in the same way as the gluons and gluinos and that mix with each other under the minimal \(\NNN=1\) supersymmetry.

A funny thing is that their charges and colors are the same as those of the gauge bosons' vector supermultiplets. And if you combine a vector multiplet and a chiral multiplet with the same charges and related quantum numbers, you may get an \(\NNN=2\) vector multiplet – a vector multiplet under a larger supersymmetry group, one that has 8 real supercharges.

Of course, the main reasons why I find this possibility attractive are
  • the more supersymmetry, the better;
  • the fact that string theory may make gauge fields live on branes that may preserve a higher amount of SUSY than the "brane intersections" where the matter fields live.
Those are theorists' reasons to be intrigued by the possibility. But the phenomenologists' reasons are cool, too. With the new chiral multiplet, one may predict the observed Higgs boson mass around \(125.7\GeV\) much more naturally than without it (aside from the new multiplet, they try to follow the rules of the cMSSM, so they are working within the so-called cMSSM+ framework where the plus sign represents the new multiplet).

Even more importantly, look at this graph of the predicted gluino (\(x\)-axis) and stop squark (\(y\)-axis) masses:



The blue region is the prediction from an ensemble of cMSSM models and they really want the gluino to start at \(2\TeV\) and the stop squark around \(1\TeV\). However, the red cMSSM+ regions nicely allow both superpartners to be as light as \(0.5\TeV\) or so. And that's quite something.

While I have never liked the wishful thinking of the model builders who always expected new physics right behind corner (they looked to me like if they tried to have a chance to get a Nobel prize as soon as possible which is not a legit motivation to direct physics research, I think), I do feel that a model that still allows the new particles to be this light – given the available data – is more natural, both in a vague and in the technical sense.

These models with some impressively light superpartners and "even more supersymmetry than we thought" remain viable and that's quite something. Maybe if the gauge fields come up with all the light superpartners of an \(\NNN=2\) multiplet, something even more dramatic awaits us in the gravitational sector. What about models with an \(\NNN=4\) graviton supermultiplet, to make it really ambitious? Maybe with such enhanced supermultiplets, we could get new tools and new cancellations to address some annoying problems including the cosmological constant problem.



I am thinking about brane worlds that make such things pretty natural. In fact, they're extremely close to the "toy theories of everything" that Barton Zwiebach uses in his textbook based on his lectures of string theory for the undergraduates. Wouldn't it be fun if the stuff taught to the youngest kids was actually the most physically relevant one? ;-)
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Posted in experiments, LHC, string vacua and phenomenology | No comments

EU bureaucrats' new strategy to close Czech nuclear power plant

Posted on 12:28 AM by Unknown
Somewhat off-topic: Fred Singer has pointed out that Martin Fleischmann, a co-father of problematic cold fusion claims, died a month ago. RIP (although I don't believe that fusion can be cold). He was born in Czechia.
The Czech Republic whose population is, according to surveys, the most pro-nuclear-energy nation in the world is producing about 1/3 of electricity in nuclear power plants. Plans to move this figure closer to 1/2 are underway.

One is located in Dukovany (Southeast of the country) and produces about 13.4 TWh per year. A newer one, one in Temelín (South/Southwest of the country) produces 11.4 TWh per year.



Temelín – with its combined Russian-American design – was opened after the fall of communism, in 2002 (although the construction began in 1981), and it was a frequent target of attacks by the Austrian Luddite activists. However, Dukovany (constructed started 1974, opened in 1985-1987) which has apparently invited almost no opposition just came under a vicious assault by the EU bureaucrats.




The nuclear power plants are classified as safe in every meritocratic evaluation; the latest one appeared today. Many experts agree that Dukovany with its VVER-440 is one of the safest nuclear power plants in Europe.

Just an hour ago, the Czech media informed about a stunning development in the EU headquarters:
Brussels is going after the neck of [trying to close] Dukovany: because of Russian uranium (autom. translation, week.cz)

ČEZ [the utility running the power plants] faces pressure to close Dukovany due to the uranium from Russia (autom. translation, idnes.cz)

Reuters press digest
We are learning that the Europeans are not allowed to buy uranium enriched outside of the EU due to some strange paragraph agreed upon at the 1994 EU Corfu Summit (island in Greece). Holy cow. How many shocking ghosts of this magnitude does the EU have? We weren't members of the EU at that time and the citizens who were deciding about our EU membership in a referendum were not told that "Yes" could mean that some stunning assholes could get a weapon to close our nuclear power plants because of some silly sentence okayed by some drunk and corrupt jerks at an island belonging to a country that shouldn't have been in the EU at all. If this information were the case, I would consider the referendum to be fraudulent.

Moreover, ČEZ argues that it had signed the contracts about the purchase of the uranium with the Russians before we entered the EU – and this contract was approved by the ESA Euroatom's Supply Agency. Let's assume that at least the legality of this contract will be respected by the EU assholes. However, this contract only lasts through 2018 and although both sides are ready to sign a new contract that de facto extends the current one beyond 2018, the EU assholes could see a new opportunity to stop the purchase. They are already trying to create problems by saying that Finland which has a power plant (Loviisa) in a similar situation did manage to get an exemption – and we may just "fail" to get it because the anointed ones are no longer in a good nuclear mood.

I find the restriction on the nuclear suppliers to be an unbelievable violation of the basic rules of the free markets and economic freedom of the countries in general. Note that the Corfu Agreement I have never heard of bans not only Russian nuclear fuel but also, for example, the American one. By this declaration, the EU is trying to present itself as the world's self-sufficient and ultimate dominant leader in nuclear technologies which it's surely not. Europe just can't afford similar bans of business relationships with the world, especially when it comes to hi-tech industries in which it is arguably not the world's #1.

How many time bombs of this sort are there in the impenetrable laws and treaties that the EU has managed to sign at various random islands over the decades? I find it scary. Only if an interested citizen such as myself had the capacity to read a summary of all these contracts, I would view them as legitimate. If the European Commission is really serious about pushing this insanity, I want my country to leave the EU. It's not just about the billions of dollars per year that the EU wants to steal from a major Czech company – one that is clearly too big and too important to fail. It's about our right to provide ourselves with the basic needs for modern life such as electricity.
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Posted in Czechoslovakia, Europe, markets, politics | No comments

Tuesday, October 2, 2012

Miracles prove the divine power of string theory

Posted on 9:20 AM by Unknown
No, I didn't mean Jesus Christ.

Pete Wilton, an Oxford science writer, wrote a piece introducing the new website WhyStringTheory.com:
Pulling the strings (Oxford science blog)
Three creative folks behind the website say various things. Edward Hughes, a Cambridge UK student, says that string theory is fundamental, beautiful, and he wanted to communicate the excitement.

Charlotte Mason, an Oxford student, says that she has mixed feelings about string theory. The picturesque ideas are beautiful but she describes the maths exactly in the way you would expect from a somewhat randomly chosen girl. No, Charlotte, the true beauty of string theory may only be revealed and understood when all the relevant maths is added. It should be done peacefully and beautifully – and not necessarily in the Greg-Moore-like heavy formalism way – but the maths is still critical for the beauty.

Finally, Joseph Conlon is the only "senior" person behind the project – he's at Oxford faculty. And he says some interesting – although not quite new – things that I want to spend some time with. It has something to do with the miracles.




Conlon says that strings are too small to be seen directly, so lots of good luck or theoretical or experimental breakthroughs may be needed to see them. He says the usual things about string theory's ability to cure the short-distance problems of quantum field theory by replacing points with strings; unify gravity with other forces and matter; about the correct counting of black hole entropy string theory provides us with, and a few other achievements in which string theory remains (and, most likely, will always remain) unmatched.

However, two of the paragraphs have a kind of a cool religious spin:
Joseph Conlon of Oxford University, another member of the team, explains that part of the theory's appeal lies in 'string miracles', these are 'calculations that look like they are going to fail and show that the theory is inconsistent, but then something comes in and suddenly saves the day. Once you see this happening several times you realise that the theory has a very deep structure and your understanding of it only scratches the surface.'

...

Yet string theory has a habit of turning up surprises, as Joseph says: 'Working on it is also good for humility, you are perennially aware that the theory is smarter than you.'
You might say that these sentence describe string theory by similar words that other people use for God. And in some sense, you would be right. The only difference is that the power of God is supported by personally and verbally communicated superstitions – sorry, believers – while the power of string theory boils down to objectively functional calculations that everyone may verify and that reveal a striking degree of internal coherence and compatibility with all known qualitative concepts and phenomena observed in Nature.

String theory is also able to turn water into wine – well, its nuclear physics approximation is enough for that because you just need to produce some carbon nuclei from the hydrogen and oxygen nuclei, aside from a few trace elements.

More seriously, the mathematical miracles behind string theory are numerous and initially unbelievably surprising. Later, they may be understood as consequences of a smaller number of technical properties of string theory's maths. While those explanations reduce the seemingly "supernatural" character of the miracles, the ultimate "conceptual explanation" always boils down to the existence and consistency of string theory.

So the ultimate miracle – the very existence of this rich mathematical structure that contains all good ideas and physics but remains fully consistent despite its incredibly richness – remains a miracle even today. We may spend years by philosophical musings on the reasons why the Universe exists at all; however, it's much more likely that we may discover a shocking and valuable insight if we ask why string theory exists at all.

The partial, individual miracles were parts of string theory's CV from its very birth. In fact, the first characteristically stringy formula – the Veneziano amplitude (for the tree-level scattering of four open-string tachyon, using the modern terminology) was found by Veneziano by demanding the first miracle, the "world sheet duality" (that's the modern terminology; they would call it just "duality" in the late 1960s, a word that is used somewhat differently today, and that's why string theory was initially known as "dual models"). What was this zeroth miracle of string theory?

Imagine that you collide 2 particles elastically. So there are 4 external legs in the Feynman diagram. You may write down diagrams in which a particle is exchanged in the \(s\)-channel. That will make the amplitude depend on \(s\). However, this intermediate particle may also have derivative interactions with the 4 external particles and in this way, the amplitude may acquire some \(t\)-dependence, too. You may get a rather general function of \(s\) and \(t\).

Then there are also diagrams with the \(t\)-channel exchange in which the role of \(s\) and \(t\) is interchanged. Veneziano boldly demanded that the sum of all \(t\)-channel diagrams is actually the same as the sum of all the \(s\)-channel diagrams. It turned out it was possible even though the "channel" variable enters through denominators and the "derivative interaction" variable enters through numerators. The Euler Beta function that Veneziano finally found in a library had the required property. In fact, you should only count the \(s\)-channel diagrams or only \(t\)-channel diagrams, otherwise you're double-counting the amplitude.

It was a miracle, something that couldn't appear in a quantum field theory with finitely many particle species (or fields). The Euler Beta function was soon derived from the assumption that the particles were actually open strings – and a closed string counterpart of the Veneziano amplitude, the 4-closed-string-tachyon Virasoro-Shapiro amplitude, was soon found as well. Once you know that the particles are open strings, the Veneziano miracle has a simple geometric interpretation: Both the \(s\)-channel and \(t\)-channel diagrams may be viewed as the particle limits of a string diagram, a disk with four strips coming out of it. The topology is the same for \(s\)-channel and \(t\)-channel diagrams so it's not too surprising that a naturally calculated "full tree amplitude" already contains both \(s\)-channel and \(t\)-channel terms.

You probably don't understand the previous paragraph – unless you have understood it for years. In that case, I want to say that there is a geometric explanation why certain sums of quantum-field-theory-like diagrams end up being \(s,t\)-symmetric even though such a symmetry looks insane from the quantum-field-theoretical viewpoint.

But much more typical examples of miracles in string theory have something to do with fully canceled anomalies, divergences, or harmful discontinuities. The first superstring revolution started with one of these miracles: Green and Schwarz found the precise cancellation of all the gravitational, mixed, and gauge anomalies in type I string theory in \(d=10\) assuming that the gauge group is \(SO(32)\). You're only allowed to choose one number, \(n=32\) "half-colors" of the quarks carried by the open string endpoints, and the theory manages to cancel five coefficients in front of these anomaly terms even though there are lots of contributions and the numerical constants that contribute are as complicated as \((n-496)/725760\) – and indeed, the dimension of the \(SO(32)\) group is \(496\). Thank God, or thank string theory, more precisely.

A sixth-order invariant in \(SO(32)\) from a gaugino loop seemed impossible to cancel in general. However, Green and Schwarz found a sub-miracle, the Green-Schwarz mechanism. They noticed that the sixth-order invariant factorizes to the product of a fourth-order and second-order invariants exactly when the group was \(SO(32)\) – and to cancel the remaining product of these fourth- and second-order terms, they discovered a previously neglected Feynman diagram resulting from unusual gauge transformation laws for fields – laws you wouldn't expect in low-brow perturbative quantum field theories.

So the Green-Schwarz miracle was five-fold and it had sub-miracles, too. You could think that no sensible scientist would assume this many miracles but you would actually be wrong. As Schwarz has repeatedly recalled, Green and Schwarz have done the tedious calculation because they already lived in the string theory "belief system". String theory was beautiful, they reasoned, so it couldn't possibly have any anomalies. And indeed, a long calculation with miraculous cancellations has shown that the belief was true even though they weren't able to write a crisp mathematical proof of the belief at that time.

Of course, within two years, this miracle was largely demystified, too. For the \(SO(32)\) gauge group, one may show that the "disk" cancels the "projective sphere" – some simple world sheet diagrams contributing world sheet anomalies of a kind. So the theory is free of world sheet anomalies and it's enough to prove that the resulting spacetime amplitudes will have all the desired physical properties including the spacetime anomaly cancellation, too. The spacetime calculation looks complicated because, in some sense, it's not the easiest or most fundamental one here: the world sheet objects and anomalies are simpler and more profound in those calculations.

In the following year, in 1985, the heterotic string was discovered. Exactly when you combine left-movers from bosonic string theory and right-movers from the only other known credible string theory, superstring theory, you obtain a hybrid string whose extra left-moving bosons have exactly enough freedom to produce the weight lattice of \(Spin(32)/\ZZ_2\), the right way to write \(SO(32)\) in this context, and exactly this weight lattice miraculously turns out to be even and self-dual. That's a new way to construct a string theory with an \(SO(32)\) symmetry, one that was shown to be equivalent (S-dual) to type I string theory ten years later – and this S-duality boils down to lots of miracles because all the objects and couplings that must match do match. Also, the heterotic string theory of 1985 had another version, based on the other self-dual even lattice, and it gives an \(E_8\times E_8\) gauge group in the spacetime which (now already less miraculously) cancels all the spacetime anomalies as well and that miraculously produces realistic spectra when compactified on Calabi-Yau manifolds.

I could tell you hundreds of examples of Conlon's general theme that many calculations may proceed for a long time and until the very end, they may keep an infidel physicist who does the calculation worrying that the final answer will be meaningless, ambiguous, inconsistent, full of divergent, ambiguous, regulator-dependent, and ill-defined factors, bullocks. But right before you're finished, all the sources of problems get canceled against others, usually many others (and new objects that you may have stupidly overlooked because they are purely stringy although they always admit a rather comprehensible explanation) and string theory's perfect coherence is saved.

For example, string theory implies that the topology of spacetime – of the extra dimensions, to be concrete – may continuously change. However, such a change would lead to a discontinuous jump of masses of some string modes, something that isn't allowed, because the masses depend on intersection numbers of various "cycles" and those change if the topology gets transformed (through a singular Calabi-Yau space whose topology is ambiguous). However, there's a contribution from world sheet instantons that string theory clearly tells you to include as well (it's a part of the Feynman path integral) and with these world sheet instantons, the continuity of the masses is restored. Also, on a similar manifold, one may be afraid of a singular formula for the Hamiltonian before she realizes that it's exactly what one gets from integrating out massless D3-branes wrapped on a "shrunk 3-cycle", thus proving that the full theory before the D3-branes are integrated out is completely smooth near that point in the configuration space. The new "stringy building block" – either the world sheet instanton or the D3-brane – has exactly the right properties that are needed to cure a disease that would be certainly lethal in quantum field theory or any other generic "related" theory which isn't quite the full string theory. But these new stringy building blocks aren't ever added in an ad hoc way. They're always the same blocks, satisfying the same unambiguous laws in all the environments. There's nothing to adjust about these building blocks and the laws they obey; nevertheless, their properties are always exactly right to cure all potential diseases.

The possibility to describe string/M-theoretical vacua via Matrix theory, matrix string theory, AdS/CFT correspondence, and the correct black hole behavior, including the thermodynamic properties, much like the very fact that the path integral summing over stringy histories fully reproduces general relativity including the non-linear corrections aren't just slogans. They are totally well-defined calculations and each of them presents numerous examples of Conlon's miracle theme. Tons of things could go wrong and pretty much "any" generic theory containing fields, particles, or other pieces of string theory – but mixing them in a slightly different way so that it is not "quite" string theory – would almost inevitably end up with an anomalous, ambiguous, inconsistent, non-unitary, or otherwise pathological result. But string theory always passes without any flaw. You may always take it to the limits. Any limits. It gets connected with some new degrees of freedom, new possible processes, possibly new branches of maths, and the answer always makes sense at the very end.

String theory isn't the first example of a physical theory that has this property. Quantum field theory itself was showing a similar internal strength – and it wasn't the first framework, either. However, in string theory, these miracles are much more diverse, multi-dimensional (in the figurative or imaginative sense), and they make the theory resilient under a much wider spectrum of inequivalent tests than any previous framework in physics. It's hard to imagine that this perfectly functional mathematical structure doesn't actually fulfill any functions in the inner workings of our Universe.

As I have said many times, many of the "miracles" have been demystified. They have been reduced to some maths whose statements we can prove and understand even without any "religious" assumptions on string theory. But it's a somewhat analogous situation to mathematicians' ability to prove Fermat's Last Theorem but only for certain exponents. What is really shocking – and unproved so far, assuming you can't use "religious arguments" in the proof – is that whatever you do with string theory, whatever new vacua or objects or processes you find by taking the previous ones to the limits or finding new solutions to the exact equations that constrained the old well-known objects, you will always cancel all the candidate inconsistencies. So far it's been the case.

I said that the "universal miracle" – the omnipresent cancellation of all anomalies and pathologies in all sectors of string theory – is somewhat analogous to Fermat's Last Theorem (which had to be proved for all exponents simultaneously). But it's probably much deeper and more important than that; Fermat's Last Theorem is a piece of recreational mathematics in comparison (despite the abstract concepts that Wiles' proof had to introduce). And the proof may be much harder or non-existent – or it may also be much easier, although no one can imagine what such a simple proof of string theory's "universal miracle" could look like. If it exists, it's bound to be conceptually profound.

The "partial proofs of stringy miracles" that we can actually write down have been compared to exponents in Fermat's Last Theorem; you could also compare them to theorems proved for individual patches of a manifold. However, we don't have any proof that the "miracles" apply to the whole manifold. After all, this manifold isn't just a simple manifold – it's a much richer structure (I am talking about string/M-theory) whose "global definition" remains elusive although we have always been able to extend the manifold from the known "patches" to the adjacent ones.

There are lots of things to write here but this text has already gotten pretty long. I would love to understand "why" string theory really exists and remains well-behaved regardless of the directions (not only on its configuration space) in which we take it. Because string theory's spectrum and parameter spaces and lists of objects etc. (and even lists of relevant branches of mathematics) change as you move on its moduli space (or as you switch in between dual descriptions), the proof – if there's any – must say something about the continuity and closedness of a highly flexible, chameleon-like mathematical structure. I don't know what the proof is and whether it exists at all. But despite the absence of a complete proof – i.e. despite the fact that we have just hundreds of "anecdotal pieces of evidence" – I am a clear believer. The coherence of string theory is perfect, is here with us to stay, and implies that curious, sufficiently mathematically talented people will always be intrigued by it and will continue to do research of it.

Of course, I am less certain about the continuing existence of curious and sufficiently mathematically talented (and motivated) people. ;-)

None of those miracles rigorously proves that string theory is the right physical theory describing this Universe. But I find it implausible that a theory which shows this degree of internal coherence, precision, and co-operation between its pieces as well as this degree of qualitative agreement with the features of Nature as we know them – including the general relativistic and quantum field-theoretical approximations, and beyond – exists by an accident. It would just sound utterly bizarre. It would be like if you found an alien rocket that seems to contain a perfectly streamlined and sophisticated engine that is capable of going through most of the difficult stages of an interstellar flight (while for some of them, we can't verify whether it can do what it needs to do) – but learning that this alien rocket has never been outside the Earth. Well, the analogy isn't perfect. Someone may produce a fake alien spaceship and impress people who know less than the constructor. But string theory clearly has no anthropomorphic constructor. All the new pieces of its engine are "objectively out there". So a better analogy would be that the alien spaceship seems capable of doing all the tasks even to the best constructors on Earth. If we found such a spaceship, it would have to mean it's an alien spaceship, wouldn't it?
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Posted in mathematics, philosophy of science, string vacua and phenomenology, stringy quantum gravity | No comments

Higgs: living near the cliff of instability

Posted on 7:22 AM by Unknown
Jester posted an interesting text on the instability of the Higgs field:
What's the deal with vacuum stability?
Of course, this topic has been covered many times on this blog, including on July 3rd and July 17th this year, but it's interesting to read Jester's twists on the story, anyway.

First, given the current best estimates of the top quark mass and the Higgs mass (other parameters don't matter too much), the quartic (=fourth power) coupling describing the Higgs self-interaction \(|H|^4\) almost certainly goes negative beneath the Planck scale.



Taken from May 2012 paper by Degrassi et al.

The best estimate is that it goes negative at an intermediate scale near \(10^{10}\GeV\) but because of the uncertainties in the masses, the "crossover" may be very different. So our Universe is probably (at 98% confidence level) unstable.

But that's just the first part of the story.




When it's unstable, it doesn't mean that it has to decay immediately. In fact, above the scale when the quartic coupling \(\lambda\) goes negative, the sixth-order \(|H|^6\) and other couplings kick in. They restore the positivity of the potential but the true minimum of the potential only occurs for \(|H|\) comparable to a high-energy scale (either the Planck scale or at least something comparable to the crossover energy scale).

Now, it's debatable whether such a setup is compatible with the observations. I would say that it is not. From the viewpoint of the high-energy fundamental theory, "our" vacuum with a small \(|H|\) isn't a stable vacuum, so you can't expand around it. And you can't really expand around a maximum of the potential, even though it is also a stationary point. It's because one may sit at a maximum at most for a logarithmically long time.

So the possibility to sit at a small value of \(|H|\) seems to be a mere illusion, a theoretical artifact of the approximate low-energy effective theory. But if there were a minimum with a small \(|H|\) and if another minimum were far enough, we would talk about metastability. If the lifetime were longer than the current age of the Universe (or so), then the metastable Universe could survive for quite some time before it would quantum tunnel and it would be compatible with the observations. That's the situation people describe as "metastable".

I think this doesn't apply to our situation because "our" minimum with a small \(|H|\) doesn't even exist and the quantum tunneling story inconsistently mixes the true minima of the underlying high-energy theory with the "apparent ones" that only exist in its low-energy approximation. But let's ignore this subtlety; I may be wrong and overlooking another nearby minimum. If you divide the regions to stable, metastable, and unstable, the colors look like this:



Our measured masses indicate that our Universe lives in the rectangle in the yellow strip! Note that if I am right, the yellow strip should really be included to the red region. In either case, you see that there is something very special about our Universe. It lives on (or near) the thin boundary between the stable and unstable worlds. In this graph, it looks like another parameter of our Universe seems to be fine-tuned. But maybe this fine-tuning is actually a cure for the "ordinary" fine-tunings we have known – we just don't know how the cure works yet.

In a Cold War metaphor, the unstable (plus nonperturbative and probably inconsistent, on the right side) red region is the Soviet bloc while the stable West is depicted in green. The measurements indicate that we live in Czechoslovakia, the crossroad connecting the West and the East. Well, during the Cold War, Czechoslovakia would belong squarely to the red Soviet bloc, I would say but some people give it a special color, yellow. Doesn't matter.

What are the chances that we end up in the thin yellow region? The percentage of the diagram that is filled with the yellow color is surely small. Note that for these – for the actually measured – "marginally unstable" values, the quartic coupling is close to zero near the Planck scale. But its derivative with respect to the (logarithm of the) scale, i.e. the \(\beta\)-function, is very small, too.

If you ask me what are the chances that I would happen to be a Czechoslovakia-born guy, a member of a medium-size nation, I would answer that the chances are rather high (after all, it's true), especially if you include the look-elsewhere effect and add the chances to be a citizen of any other nation of a comparable size. But the believers in some dogmatic anthropic principle who are Chinese and who believe that they belong to all majorities in the world could be surprised by the observation that the Universe is actually Czechoslovak. It's almost like if you managed to find Peter Higgs – who can be anywhere on Earth – just one meter from the LHC tube. How likely is that?



I agree with Jester that this fact that "we seem to be near the cliff" seems to tell us something. It doesn't have to be a coincidence. Let me add that various "asymptotic safety" may play with the idea that such couplings are zero near the Planck scale but none of these theories really succeeds (so far) in giving us a UV-consistent model incorporating gravity as well as non-gravitational matter and force fields. But there could be a related explanation. It could be linked to quantum gravity, inflation, ideas about the Conformal Standard Model, or just some technical vanishing of the quartic coupling and other things in a leading approximation of string theory (near the string scale) which may have a geometric explanation.

Maybe we should spend more time with this "coincidence".
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Posted in string vacua and phenomenology | No comments

Harvard's divestments: Israel and fossil fuels

Posted on 4:04 AM by Unknown
Harvard-related fun news: Next Tuesday, on October 9th at 10 am, Czech President Václav Klaus will host Larry Summers at the Prague Castle. Two important politically incorrect economists who know me in person. ;-)
When I was at faculty of Harvard, I also got familiar with many undergraduate students, went to the (usually Harvard) Pub with some of them, and deduced a realistic picture what they look like and what they care about.



There is a very high percentage of highly talented young people among them. (Philip Streich from Howard Georgi's house who tragically died a week ago in a family farm accident was probably one of them: an Intel Foundation bronze medal winner, a graphene company CEO since his teenage.) On the other hand, when it comes to aptitudes, I believe that the average Harvard undergraduate doesn't differ "strikingly" from the average student at other colleges that are just OK. Those folks are later unusually successful as well – but I tend to think that the Harvard diploma (and the contacts they develop over the years in the college) may be more important for that than their actual skills and hard work.

While the Harvard faculty is insanely super duper left-wing, Harvard students are much more moderate. This is manifested in many ways. For example, they would largely endorse Larry Summers when the far left (for readers who are U.S. conservatives: Larry doesn't belong to this set, according to Harvard's conventions!) organized the witch hunts against him. Of course, Harvard students are far less ideological and more practically oriented than the Harvard faculty. After all, we could say that they're normal kids with pretty normal interests.

While they drink stuff and have lots of sex, they are publishing an "adult" daily newspaper, The Harvard Crimson, too. I feel that the students – because everyone knows that they pay tuition etc. – have a significant impact on the Harvard policies. And Harvard University is a sort of a role model for many other U.S. universities, institutions, and even corporations. So you may want to follow what those kids think. They are kids who are programmed to "control the world" and to be pro-actual-establishment in every single dimension you may think of. So they're still highly politically correct – especially the self-proclaimed spokespeople of the student body. When it comes to students whom the Harvard environment naturally converts into spokesmen, think of slick, superficial, and self-centered folks like Sean Carroll; he used to be a Harvard graduate, not undergraduate student, but you may still get the idea.

I will discuss divestments – decisions to sell all holdings related to XY whenever XY becomes politically inconvenient or politically incorrect.




In 2002, a community of Harvard's anti-Semites and outcast Semites decided to ventilate their anger with the most functional country in the Middle East by proposing that Harvard isolates itself from the "dirty Jewish state" and that its management company should sell all its holdings related to Israel.

Many people at Harvard supported the move. Alan Dershowitz was one of the brave exceptions who were fighting the bigots. The discussion continued for years and the divestment wasn't coming. However, it was suddenly revealed in August 2010 that Harvard had sold all its Israeli holdings even though two days later, Harvard stated it wasn't divestment (because Harvard stated it wasn't one), it was just financially indistinguishable from one. ;-)

Clearly, the management company chose to please the anti-Semites while verbally pretending to be a "neutral party". Still, Harvard has helped to legitimize the efforts to treat Israel in a bad way – in a way that no rogue state in the region could ever be treated. The direct impact of the decision on Israel was negligible, of course; however, the indirect impact caused by the "legal" anti-Israel attitudes seen at Harvard may be significant.

Now, the climate

In 2009, some Harvard students founded a college chapter of Students for a Just and Stable Future which – despite the vague progressive name – is just another climate alarmist organization. They want Harvard to sell all its holdings that are related to fossil fuels and perhaps any other carbon-dependent industry, too.

Five days ago, the Crimson published an editorial which said that Harvard students should help the alarmist cause by all other means but the divestment was a bad idea because in the past, all proposals for a divestment were restricted to companies linked to human rights violation (I guess that the implicit assertion was that the most respectable country in the Middle East when it comes to human rights, Israel, may also be criticized for human rights violation).

However, two students who are fighting for the "just and stable future" were allowed to publish a response today. So they scream that the divestment is necessary, repeat some usual alarmist talking points, and – now, this is the shocking point – they argue that global warming is a human rights issue, indeed. Holy cow.

I know that there are many sensible people among the students at Harvard but the environment and its habits just immediately downgrades them to "also students" who are obliged to be quiet. Be sure that I have met a couple of Harvard undergraduates who complained that they were heavily harassed for their world view – especially those who were practicing Christians. This is apparently the "good", semi-officially sponsored harassment.

Kids who propose to launch a war against fossil fuel companies – which belong among the pillars of the modern civilization – should be intensely spanked by their parents at least for half an hour. At Harvard, some of these deeply confused, spoiled frats not only fail to be spanked but they indirectly influence the thinking in the whole American society. Many of us are underestimating how much harm stupid kids at disproportionately influential places may do.

And that's the memo.
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Posted in climate, education, markets, Middle East, politics, science and society | No comments

Monday, October 1, 2012

Nima Arkani-Hamed attracts India to physics

Posted on 6:42 AM by Unknown
Nima Arkani-Hamed, a winner of the inaugural Milner prize who just sent me a kind and interesting e-mail, is visiting India.



He gave an interview to the Times of India:
Idea of space and time needs to be replaced: Arkani-Hamed
In the interview, he talks about the Higgs discovery, the future of particle physics, evolution of quantum mechanics, unity of string theory and particle physics, validity of the Big Bang theory, and benefits of the Milner prize for physics. You may tell everyone what you think about the questions and Nima's answers (yes, I agree with every word of his here, so my comments wouldn't be too interesting).




So the questions are pretty interesting for you to look at the interview. Try to guess which of the questions contains the topics referred to by the title. Yes, it's the QM question. ;-)

It's October 1st so aside from bureaucracy with monthly and quarterly payments (including the value-added tax), one more thing happened to this blog. The Haloscan/JS-Kit/Echo comments have finally come out of business, kind of gradually: look what a typical high-traffic Echo comment page looks like now.

All of the 70,000+ comments should have been transferred to the new DISQUS 2012 platform, with some relatively minor defects (especially the by-DISQUS ignored "parent comments" nested structure of the replies) that most readers won't even notice and the more careful ones will surely forgive them as signs of the fact that we don't live in an ideal world and we don't read a blog with infinite resources. ;-)

Due to the inconsistent and time-dependent conventions used in the XML file with the Echo comments, the conversion process has been rather difficult – thanks to Wolfram Mathematica for the help – and with all my characteristic modesty, I must tell you that I don't know another blogger on this planet who has successfully migrated from Haloscan+JS-Kit+Echo full evolution tree to DISQUS.

P.S. on health: It's probably the first day today when one could meaningfully argue that I actually feel healthier and more full of energy than when I downgraded my body as "out of proper balance" sometime in mid August. Yesterday, I played floorball again and biking and other sports don't reveal any disorders. For a week, my brown by-products have passed the strictest visual tests. Starting from yesterday or so, symptoms including the bad sweetish and sugar-transforming taste have been reduced to nearly invisible, homeopathic traces of what they were two or three weeks ago. The diet without sugar, gluten, yeast, vinegar, coffee, alcohol, caffeine etc. is brutal – over 90% of things in the supermarket are suddenly "not eligible" – but my discipline is superior and I don't mind looking at these things that I can't have. Of course, all the things that are forbidden may be pretty much compensated and "redone" from other things; they're convertible. Some things I wouldn't be buying in the past – like vegetables (at most, I would be buying fruits) or, if I mention a random product, a cheese with horseradish – are actually fun. And of course, white yoghurts became a really frequent consumable, due to their content of the friendly bacteria cultures, my new small individualist friends (who arguably share the foes, the collectivist yeasts from the kingdom of Fungi, with me: but I downgraded the latter from would-be plants to obnoxious large molecules). Among the hopefully "intensely anti-yeast things", I bought Candix at Candix.cz (mainly Capryllic acid, together with the probiotic cultures and vitamin C), 300 ml of coconut oil, and some goat milk (and yoghurts based on it) which is about 5 times more expensive than the cow milk counterpart(s) but is rich in the capryllic and similar acids that suppress the kingdom of Fungi within our animal body. Your hypothetical contributions (to the piglet at the bottom left of this blog) below $20 will be almost certainly used to buy similar "luxurious" counterparts of usual groceries.
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Posted in science and society, string vacua and phenomenology, stringy quantum gravity | No comments
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