Wednesday, May 18, 2016

Slowly for Peter Shor: Page 1 of Dirac

Most of the people who want to argue with me about quantum mechanics are marginal figures such as the psychopath named John Duffield who has picked Paul Dirac as the ultimate man who was "hampered because he didn't have an understanding of the electron".

Imagine, among tens of billions of humans who have ever lived or live on the Earth, Paul Dirac has clearly introduced much more clarity to our understanding of the electron than anyone else, but this Gentleman is willing to pick Paul Dirac as the "guy who was hampered by misunderstanding the electron". It often looks like these people are competing to write the most absurd or psychopathic sentence anyone can write down.

So it was refreshing to have some disagreement with MIT's Peter Shor again, a guy who is well-known because of a quantum algorithm. I think he's an expert on many related things and most (?) of his comments about foundations of QM are just fine. But like the postmodern feminist philosophers, he seems to believe that the theorems in mathematics and physics only apply when certain sociological criteria are obeyed.

The question was whether the first pages of Dirac's textbook of quantum mechanics ruled out classical theories that obey the technical definition of a "classical theory" (which includes Bohmian mechanics, as we will discuss in detail) or just some subclass of classical theories that are thrown under the bus by the social scientists according to the criteria of humanities. Shor clearly prefers the latter!

Shor has pleased us with the following statement:
Dirac does not intend classical to mean non-quantum mechanics; he intends classical to mean pre-quantum mechanics (a standard definition of classical is: traditional in style or form). So no, this says nothing about de Broglie–Bohm theory.

Dirac opens his paragraph with
  • The necessity for a departure from classical mechanics is clearly shown by experimental results. Dirac is not talking about any theory that one might call "classical" today. He is talking about the theory of classical mechanics that existed before quantum mechanics was formulated.
Wow, it would be so funny if this man weren't serious. If he only excludes theories in papers with a certain date written on them, do the papers written by women get an exemption as well? Didn't you forget your pills today, Peter?

Dirac obviously talks about classical theories in the technical sense of non-quantum theories, those that create an objective picture of the physical system associated with the spacetime that exists at every moment and is independent of any observers.

Sometimes in the context of discussions about relativity, the word "classical" has been used as "non-relativistic" (or "non-relativistic and non-quantum") but this usage has largely disappeared. The adjective "classical" used for a theory is exactly synonymous with "non-quantum" or "realist" or "pre-quantum" or "observer-independent" and with other synonyms.

The date of publication doesn't matter. The preface to the book (written by Dirac, just to be sure) starts as follows:

The methods of progress in theoretical physics have undergone a vast change during the present century. The classical tradition has been to consider the world to be an association of observable objects (particles, fluids, fields, etc.) moving about according to definite laws of force, so that one could form a mental picture in space and time of the whole scheme. This led to a physics whose aim was to make assumptions about the mechanism and forces connecting these observable objects, to account for their behaviour in the simplest possible way. It has become increasingly evident in recent times, however, that nature works on a different plan. Her fundamental laws do not govern the world as it appears in our mental picture in any very direct way, but instead they control a substratum of which we cannot form a mental picture without introducing irrelevancies. The formulation of these laws requires the use of the mathematics of transformations.
Even if you managed to read at least the first five sentences of Dirac's textbook, you would have no doubt that "classical physics" as used by Dirac was exactly the same as "classical physics" as used by any competent particle physicist or condensed matter physicist or any other person doing serious quantum physics. "Classical physics" means "non-quantum" physics, i.e. any physics where an objective picture of the "state of the physical system" exists at every moment, independently of any observations. That's the class of theories that Dirac identified as dead in his textbook.

It is spectacularly obvious that Bohmian mechanics or any attempt to replace proper quantum mechanics by a "realist" theory would be classified as "classical physics" by Dirac. But at the end, the terminology isn't what is important in physics. What is important in physics is the content.

In particular, whether or not you use the term "classical physics" for Bohmian mechanics, Bohmian mechanics satisfies the assumption of many theorems and methodologies in classical physics, so these theorems and methodologies may be applied to these theories (including Bohmian mechanics), and Dirac's conclusions – that classical physics (including Bohmian mechanics) is no good to describe the microscopic world – follow. I will discuss technical aspects of Dirac's disproofs of these theories later.

Let me mention that these days, thousands of bigots and crackpots love to say that there is something wrong about the dependence of quantum mechanics on observers – which is pretty much a defining property of this new framework of physics. Well, things were very different in Dirac's famous times. While he discussed that the new framework of physics may be harder for a student because it differs from all the things he had previously learned, Dirac also wrote:
Further progress lies in the direction of making our equations invariant under wider and still wider transformations. This state of affairs is very satisfactory from a philosophical point of view, as implying an increasing recognition of the part played by the observer in himself introducing the regularities that appear in his observations, and a lack of arbitrariness in the ways of nature, but it makes things less easy for the learner of physics.
The first sentence could be used today to celebrate things like string dualities. But I added the bold face to the sentence showing that Dirac considered the increased role of the observer to be "very philosophically satisfactory". Either you have a better taste in physics than Dirac had (you don't) or maybe the very basics of your thinking about the Universe in general and observers in particular are just garbage, crackpots and bigots!

It's philosophically pleasing that 90 years ago, physicists began to discover the laws of physics that can't work without the pre-identification of "observations" because that proves that people were reaching the fundamental layer of the laws of Nature. Why? Because all the knowledge about Nature ultimately comes from our observation and the "objective reality" was clearly an auxiliary object to organize these observations. Most of "auxiliary objects" are likely to be fundamentally misleading (just like the concepts in effective and otherwise approximate theories) so if the new theory forces us to find the relationships between the observations directly, without the assumption about the auxiliary object, it seemed clear that people were getting rid of another approximation in their thinking about Nature – the most important, classical approximation.

OK, let's look at the first pages of the first chapter and apply them to Bohmian mechanics.

The first chapter of the Principles of Quantum Mechanics is titled The Principle of Superposition and its first section is:
1. The need for a quantum theory

CLASSICAL mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems, including the electromagnetic field in interaction with matter. The underlying ideas and the laws governing their application form a simple and elegant scheme, which one would be inclined to think could not be seriously modified without having all its attractive features spoilt. Nevertheless it has been found possible to set up a new scheme, called quantum mechanics, which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than the classical scheme. This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive, as a result of which all these features can be incorporated in the new scheme.
That was the first paragraph. Imagine we're high school students in a literature class. Could you please rephrase the paragraph in your words?

Yup. Dirac says that classical physics had been a big success and like a macroscopic mutation of DNA, a serious modification of classical physics is likely to make it unattractive or lethally sick. It almost looked like no major conceptual advance was possible except that people found exactly one loophole. There exists a framework that is totally inequivalent to classical physics, namely quantum mechanics, where things are done carefully so that the attractiveness or viability doesn't decrease. Instead, the resulting theory is actually more elegant and satisfying than the old, classical theories in physics. Moreover, it may be seen why the qualitatively new, quantum theory contains the old classical physics as a limit.

In this paragraph, Dirac also made it clear that there are only two "schemes" and no finer fundamental way to divide theories than to classical (=non-quantum) and quantum (=non-classical) exists, directly disagreeing with the convoluted claims about additional "pre-quantum" theories and "theories with a date" by Peter Shor.

Once you read and master the second paragraph of the first chapter, you will have swallowed the whole first page of the first chapter of Dirac's book and you may congratulate yourself:
The necessity for a departure from classical mechanics is clearly shown by experimental results. In the first place the forces known in classical electrodynamics are inadequate for the explanation of the remarkable stability of atoms and molecules, which is necessary in order that materials may have any definite physical and chemical properties at all. The introduction of new hypothetical forces will not save the situation, since there exist general principles of classical mechanics, holding for all kinds of forces, leading to results in direct disagreement with observation. For example, if an atomic system has its equilibrium disturbed in any way and is then left alone, it will be set in oscillation and the oscillations will get impressed on the surrounding electromagnetic field, so that their frequencies may be observed with a spectroscope. Now whatever the laws of force governing the equilibrium, one would expect to be able to include the various frequencies in a scheme comprising certain fundamental frequencies and their harmonics. This is not observed to be the case. Instead, there is observed a new and unexpected connexion between the frequencies, called Ritz's Combination Law of Spectroscopy, according to which all the frequencies can be expressed as differences between certain terms, the number of terms being much less than the number of frequencies. This law is quite unintelligible from the classical standpoint.
This is the paragraph that Peter Shor was asked to respond to before he ludicrously said that Dirac's arguments only apply if one is discussing papers written before 1925.

OK, let's discuss these "reasons making quantum mechanics unavoidable", with a special focus on the question whether Bohmian mechanics could be enough.

Dirac wrote that classical electromagnetism coupled to classical charges didn't explain the material constants and the stability of the atoms and asked whether the addition of new forces could fix the problems in classical physics. His answer is obviously "No". If you have read this blog post so far, you have no doubts that Dirac's "No" was surely meant to apply to Bohmian mechanics – because that theory also has an objective mental picture of the state of objects. The pilot wave is just a new type of a field (a multi-local one) and the particles are just driven by new forces from the pilot wave (it's the velocity, not acceleration, that is computed but that changes nothing about the essence because a phase space still exists, even without a division to positions and momenta).

It's obvious that Dirac meant that realist theories such as Bohmian mechanics were no good. The only question is whether Dirac's claim that all theories such as Bohmian mechanics suffer from the problems is correct. Be sure, it is correct.

To be specific, let us look at the simplest Bohmian description of one hydrogen atom. In the approximation where the nucleus is static, the electron gives us the degrees of freedom \(\ket\psi\), the ket vector, which is however rebranded as a classical field, plus a classical trajectory \(\vec x(t)\) whose velocity is determined from the Schrödinger's probability current as \(\vec v = \vec j / \rho\).

The first thing that Dirac wrote was that one may apply very general methods of classical physics to this system and derive that the emitted spectrum won't have the right properties. Are these flaws of the spectrum enumerated by Dirac there even if your classical field "emulates" the wave function? You bet.

Dirac says that near the ground state, any classical system will oscillate with some frequencies. How does it work in Bohmian mechanics? First, there is no "ground state" for the Bohmian trajectory. They're completely artificial lines where the electron is moving in arbitrary ways while it avoids the nucleus. When the wave function (sorry, "pilot wave") is real, the electron doesn't move at all but it sits at a random place. If it is complex, the Bohmian electron moves but no position is closer to a "ground state" than any other.

Following Dirac, we want to know what electromagnetic radiation is emitted by the system. First, experiments prove that the coupling of the atom to the electromagnetic field is nonzero: atoms sometimes emit or absorb light. Can it be done by the Bohmian trajectories? No. These Bohmian electrons are chaotically revolving around the nucleus and accelerating all the time. If the nonzero acceleration of the Bohmian particle were enough to emit the electromagnetic radiation, we would be back to the instability of classical physics. The electron would be losing energy, and it would have to fall towards the nucleus. Either the atoms would be unstable with a tiny lifetime, or some force would lift the electron back, and the energy conservation law would be violated because the electron would keep on accelerating and emitting.

So it's clear that the only viable option is that all the radiation emitted by the atom is calculated from the pilot wave \(\ket\psi\) (now a set of classical degrees of freedom), similarly to quantum mechanics, and not from the Bohmian trajectory. It's just one example of the fact that if you want Bohmian mechanics to avoid the most obvious contradictions with the observations (such as the prediction of a constantly emitting or collapsing atom), you must make the Bohmian trajectory pretty much inconsequential.

But if you make it inconsequential for the radiation, you will hit another lethal problem: the photons' positions are not going to be correlated with the actual Bohmian trajectory, so charged particles will emit radiation not only "where they actually are" but where they "had a potential to live". It means that the synchrotron radiation may be separated away from the Bohmian particle that you detect – it's like a soul flying away from the body etc. None of these things work and the Bohmian advocates must really suffer from a brain defect if they're incapable of seeing these "stop signs" for years or decades but I want to return to Dirac's specific statements.

Dirac says that the classical physical system will reduce to classical harmonic oscillators with some frequencies and those will be imprinted to the electromagnetic field, along with their higher harmonics. Is that true for Bohmian mechanics? Yes, it is.

The pilot wave is mathematically a ket vector that obeys Schrödinger's equation and may be decomposed to\[

\ket \psi = \sum_n c_n(t) \ket{n}, \quad c_n(t)=c_n(0) \exp(E_n t / i\hbar)

\] Near the ground state which we label with \(n=1\) (I will use \(n\) for the whole \(n,l,m\) triplet, I am sure you could write things more rigorously), we have \(c_n(t)\sim 1\). Let's additively shift the energy by \(13.6\eV\) so that the ground state looks like \(E_1=0\) – just a phase change to simplify our matters a little bit. But I will still use \(E_n\) for the regular energies \(-13.6\eV/n^2\).

With this choice, we see that the coefficients\[

c_n(t)=c_n(0)\exp[(E_n-E_1)t /i\hbar], \quad n\geq 2

\] The amplitudes in quantum mechanics just change their phase as a function of time, with the frequency \((E_n-E_1)/\hbar\). But this circular motion of \(c_n\) in the complex plane is isomorphic to the circular (or, in general units, elliptic) motion of a harmonic oscillator in the \((p,x)\) phase space. So the real and imaginary part of the coefficient \(c_n(t)\) – which is now a classical degree of freedom – may be said to define a classical harmonic oscillator, just like Dirac wrote in general.

It had to work like that – those conclusions are really general all over classical physics and Bohmian mechanics obviously is classical physics when it comes to these basic mathematical and philosophical properties.

Now, we only had one electron. How do you couple it to the electromagnetic field? Bohmians won't tell you. But whatever the emission of radiation is, it should better depend on these classical degrees of freedom \(c_n(t)\), and if the "knowledge of the Schrödinger's equation" is supposed to be used to produce the photons of the actually correct frequencies, the oscillating \(c_2(t)\) should better imply the production of photons of frequency \((E_2-E_1)/\hbar\) because that's exactly the frequency at which this degree of freedom oscillates.

So unsurprisingly, we have a chance to get really close. The atom is a classical system which contains some harmonic oscillators that oscillate at these frequencies \((E_n-E_1)/\hbar\), so if these oscillations are imprinted to the electromagnetic field, you may get some frequencies in the resulting spectrum correctly. Needless to say, to have a chance to produce a viable theory of the light emission, you will need to upgrade the pilot wave \(\ket\psi\) to the full second-quantized \(\ket\Psi\) that also includes the wave functional for the quantized electromagnetic field, and so on. There won't be any room for new beables – similar to the Bohmian positions. But I have already proved many times that these beables are unusable, anyway. If anything depends on them, you will deviate from the predictions of quantum mechanics or observable facts dramatically.

OK, we got some frequencies \((E_n-E_1)/\hbar\) which look good as frequencies of the photons because the system was really carefully chosen. But what about the other frequencies such as \((E_3-E_2)/\hbar\)? Can Bohmian mechanics explain the Ritz Combination Law that for two frequencies \(\omega_1,\omega_2\), the frequency \(\omega_1+\omega_2\) is often a part of the spectrum as well, and consequently, the photons' frequency may be written as the differences \((E_m-E_n)/\hbar\) of the energy levels?

The answer is that Dirac is right again. There exists no way in which the emitted photons of right frequencies are calculated from the classical degrees of freedom \(c_n(t)\). Proper quantum mechanics shows its magical muscles and produces the general frequencies \((E_m-E_n)/\hbar\) out of "two indices \(m,n\)" because the probability amplitude for the \(m\to n\) transition is proportional to the matrix element\[

\bra m q\vec r \ket n

\] where \(q\vec r\) is the operator of the electric dipole. In quantum mechanics, it's wonderful because we have transitions between two states that are expressed by matrix elements and matrix elements have two indices. But if \(\ket\psi\) is a set of classical degrees of freedom, such matrix elements in between two states become physically meaningless.

In quantum mechanics, the evolution operator \(U\) has matrix elements \(\bra f U \ket i\) that determine the probability amplitudes. But in Bohmian mechanics, probabilities of the fundamental transitions – between an initial combination of the "pilot wave, Bohmian positions" and the final combination are either 0% or 100%. It's still a classical, in fact deterministic theory that maps initial states to final states in a one-to-one fashion. So it's simply physically meaningless to consider matrix elements of anything in Bohmian mechanics at all. The actual evolution always brings a particular precise initial state to a particular final state after time \(t\). Like always in classical physics, the question isn't what are the probabilities but what exact final state arises from a given initial state.

If the atom isn't coupled to anything, the complex coefficients \(c_n\) will oscillate forever. Once you couple it to some other classical objects, the theory will be at most capable of producing the radiation at frequencies \((E_n-E_1)/\hbar\) where \(E_1\) is the ground state energy. At most, you may design awkward modified laws where \(E_1\) will be replaced by another fixed level. But you will not be able to emit all the frequencies \(\omega_{mn}\) because the classical theory simply doesn't have this many harmonic oscillators. It only has the classical harmonic oscillators that are extracted from the coefficients \(c_n(t)\) and those only have one index, not two.

So exactly as Dirac says, Bohmian mechanics is simply incapable of explaining the Ritz Combination Law. The atom has a particular set of frequencies and their sums can't appear in the spectrum.

I must emphasize that if you extend the atom's \(\ket\psi\) to the quantum field theory's \(\ket\Psi\) that you rebrand as a "second-quantized pilot wave", you may obviously extend the theory to "mathematically coincide" with the whole quantum field theory. But the meaning of the symbol \(\ket\Psi\) is completely different than in quantum field theory and if you want to preserve the "realist" i.e. classical character of your theory, you must ultimately say that some observable quantities are functions of the classical degrees of freedom \(\ket\Psi\).

In this way, you're just postponing the moment when you admit that it doesn't work at all. At the end, you know that the only correct interpretation of the complex numbers in \(\ket\psi\) or \(\ket\Psi\) is that they are probability amplitudes that determine the probabilities of otherwise random outcomes, via the Born rule. But Bohmian mechanics ultimately wants to say that \(\ket\psi\) or \(\ket\Psi\) are classical degrees of freedom that should be observable "directly", in a single experiment. It's easy to experimentally demonstrate for every single physical system we know that no continuous amplitudes included in \(\ket\psi\) can be directly measured by any experiment (single repetition) – because only the bilinear things in \(\ket\psi\) may be measured and only as probabilities (by a big repetition). But to make the final step and connect e.g. the "grand pilot wave" \(\ket\Psi\) to the observations, you will need to do exactly something that I said to be impossible in the previous sentence.

So the Ritz Combination Law will unavoidably be broken once you connect the "grand pilot wave" to observations in any classical way.

Dirac also said that you should get "higher harmonics" in the electromagnetic field. Why did he say it? Our harmonic oscillators \(c_n(t)\) didn't produce any higher harmonics, did they? It's because he considered more general, anharmonic equations for these perturbations. When you Fourier transform a more general periodic function, you get the basic frequency as well as its higher harmonics.

For the linear equation governing the pilot wave \(\ket\psi\) of the electron and assuming a linear dependence of the probability amplitudes on \(\ket\psi\), the higher harmonics don't arise. But this is really a mathematical property coming from linearity – a fundamental postulate of quantum mechanics (which gave the name to Chapter 1 of Dirac's book because it's really the main topic).

In a classical theory describing the pilot wave \(\ket\psi\), a set of classical degrees of freedom, there is really no reason whatever to assume that the fundamental equations controlling these degrees of freedom are linear. Everything that is not forbidden is allowed. And because this nonlinearity isn't reducing the consistency of this non-quantum theory at all, one must assume that it exists – both in the normal evolution of the pilot wave as well as in its impacts on other parts of the system that we use to measure the atom (e.g. the electromagnetic field). This is an example of Dirac's claim that quantum mechanics is far less arbitrary than classical physics: the linearity of the evolution and other operators (observable) is needed for consistency (because the linearity basically coincides with the linearity rules in the probability calculus) while no similar linearity constraint may ever be justified in a classical theory.

So if you consider a general enough, not contrived and infinitely special, theory of this classical type, you do predict the appearance of the higher harmonics postulated by Dirac as well, in a direct conflict with observations. The absence of these higher harmonics only results from Bohmian mechanics when it's infinitely fine-tuned – it's infinitely fine-tuned to emulate predictions from another theory.

And it goes on. The third paragraph of Dirac's Chapter 1 says:
One might try to get over the difficulty without departing from classical mechanics by assuming each of the spectroscopically observed frequencies to be a fundamental frequency with its own degree of freedom, the laws of force being such that the harmonic vibrations do not occur. Such a theory will not do, however, even apart from the fact that it would give no explanation of the Combination Law, since it would immediately bring one into conflict with the experimental evidence on specific heats. Classical statistical mechanics enables one to establish a general connexion between the total number of degrees of freedom of an assembly of vibrating systems and its specific heat. If one assumes all the spectroscopic frequencies of an atom to correspond to different degrees of freedom, one would get a specific heat for any kind of matter very much greater than the observed value. In fact the observed specific heats at ordinary temperatures are given fairly well by a theory that takes into account merely the motion of each atom as a whole and assigns no internal motion to it at all.
I have discussed the specific heat extensively in two blog posts, in 2015 and yesterday. In our Bohmian mechanics case, we had the classical harmonic oscillators \(c_n(t)\). Each of them will contribute \(kT\) to the thermal energy. Just like the classical electromagnetic field suffers from the "ultraviolet catastrophe", Bohmian mechanics contributes \(kT\) to the energy – and therefore \(k\) to the heat capacity – for every index \(n\) labeling an energy eigenstate of the atom because each energy eigenstate is associated with a coefficient \(c_n(t)\) that helps to determine the "pilot wave" (rebranded wave function).

As I have said many times, the heat capacity of the hydrogen atom (or any atom) obviously ends up being infinite because there are infinitely many coefficients \(c_n(t)\) and each of them behaves as a classical harmonic oscillator. The classical theory of electromagnetism suffered from the ultraviolet catastrophe – an infinite heat capacity hiding in the very high frequencies of the modes of the field. The Bohmian mechanics suffers from the large \(n\) heat catastrophe which is even more pathological.

Why? As I have mentioned, this infinite heat (the correct heat should be at most of order \(k\) for the whole atom) isn't even extensive in electrons and atoms. The "pilot wave" for many electrons will be given by a state in the tensor product Hilbert space and the relevant coefficients will be, in the case of two electrons, \(c_{mn}\). So if \(\infty k\) represents the heat capacity of one electron, the heat capacity of two electrons won't be \(2\infty k\) but \(\infty^2 k\) because the two indices \(m,n\) of the coefficient \(c_{mn}\) have \(\infty^2\) possible values. The additivity of the heat capacity is totally broken.

I must emphasize that you can't "remove" the pilot wave from the computation of entropy in any way. The pilot wave really contains the "bulk" of the physics. It knows about all the energy levels. As we have seen many times, the Bohmian trajectories are a garbage that you shouldn't allow to influence physics much if you want to avoid absolutely immediate catastrophes and contradictions.

At any rate, Bohmian mechanics is a theory where the pilot wave and the particle position are equally "real". So if a physical system reaches the equilibrium, it spends an "equal amount of time" at every possible region of the total phase space. The logarithm of the total volume of the phase space may be called the entropy and the heat capacity of objects in equilibrium will be unavoidably dictated by the derivatives of this entropy.

I don't need to call Bohmian mechanics "classical physics" but even if I or you use different words, mathematics still works, all the well-established methods of classical physics may be applied, and one may derive all the wrong conclusions discussed by Dirac.

Sociologically, it's really insane to think that folks like Dirac could have missed a loophole that would be conceptually classical. They have tried to find the right theory of the atom – fixing Bohr's semi-quantum toy model – for some 15 years and they knew classical physics very well – that was the bulk of their physics education and the previous 200+ years in physics. Despite their background in classical physics, they were smart enough to find a completely new, logically consistent framework of physics – quantum mechanics.

Be sure that if there could be a viable, fundamentally classical theory – something like Bohmian mechanics – they would have found it before quantum mechanics. But instead, these people did their job right and found the proof that classical mechanics was no good. Bohmian mechanics obeys all the properties that allow us to use the classical tools and one may see that it's no good, either.

Everyone who still thinks in 2016 that something as 17th-century-like as Bohmian mechanics could be good enough to describe the fundamental laws of the Universe is a hopeless moron. It's sad if Peter Shor is one of them but his membership in that society can't change that it is a society of morons and I am ashamed of the living generations because the mankind has clearly deteriorated in these matters since the times of Dirac et al.

But let's not exaggerate the pessimism: I don't hide that I believe that most of the Internet-silent physicists at MIT and elsewhere actually know all these basics – and I basically wrote this blog post to bring them the data showing that Peter Shor is losing it. However, he's fine at the Q&A server. Seven crackpots have upvoted his answer so his score is actually above zero and 6 crackpots have downvoted mine, so mine is actually below zero.

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