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Diversity of observables in quantum mechanics

A question at the Physics Stack Exchange has reminded me about a philosophical preconception that "helps" to prevent many people – including people such as Brian Greene – from appreciating and understanding the foundations of quantum mechanics and the wonderful ways in which they radically differ from the foundations of classical physics.

In classical physics, there are some objective physical quantities – such as positions and velocities of all particles; or the values of several fields at each point and their time derivatives – and all observers may agree about their values, at least in principle. Moreover, all other things that can be in principle measured are simple real functions of these real, commuting variables.




In quantum mechanics, observables must be represented by Hermitian linear operators that usually refuse to commute with each other. (I fell in love with the verb "refuse" as a replacement for "fail" in similar situations because if one says "fail", it suggests – and leads many impressionable people to an incorrect belief – that there is something wrong that should be fixed so that the "failure" doesn't occur again. But the word "refusal" correctly indicates that if you don't like the result, it's just your problem.)

The fact that the operators don't commute with each other means that they can't simultaneously possess sharp, objective values, not even in principle; things that have objective real values do commute with each other because the real values do. That's of course nothing else than the Heisenberg uncertainty principle.

People who remain uncomfortable about quantum mechanics just can't live with the uncertainty principle so they invent various ways to "fake" quantum mechanics by a classical system – that may ultimately be described by some objective quantities with real, unambiguous values. They believe that a "trick" such as classical random generators, hidden variables, gadget to collapse a wave, or many universes is good enough to emulate the new features of quantum mechanics.

But it ain't the case and it can't ever be the case. Many additional pilot waves, classical random generators, or multiple worlds won't change the fact that the functions of the "classical" quantities that describe them will commute with each other. But the observables in the actual quantum world don't commute with each other. What this actually shows is that the observable quantities don't have objective values! Random generators, shrinking waves, pilot waves, or multiple worlds have nothing to do with the quantumness of Nature.

The first new insight of quantum mechanics is that the actual observables – whatever they are for a given physical system – don't commute with each other which directly means that they can't be functions of any objective, classical quantities: those would commute with each other. That's why all existing – and quite certainly all future – attempts to return and squeeze the world into the classical straitjacket have been proved incorrect. However, the classical straitjacket is just a religious dogma for certain people so they will never give up, regardless of any amount of evidence that their efforts are fundamentally misguided.

But there exists a more general assumption that these people are making and that quantum mechanics has refuted. It's the assumption that there exist "the right basic quantities" the world is all about; and "the right ways to measure them".

For example, in mechanics, one may talk about the positions \(\vec x_j\) and momenta \(\vec p_j\) of all particles. Derived quantities such as the angular momentum may be written as simple functions\[

\vec L_\text{total} = \sum_j \vec x_j \times \vec p_j

\] The classical framework encourages you to think that you may equip the Universe with lots of devices that measure the particles' positions \(\vec x_j\); and many apparatuses that analogously measure \(\vec p_j\). All the measured results are real numbers and all other observables – including the angular momentum – are functions of the measured positions and momenta. So all the other quantities are "redundant". You don't need extra apparatuses to measure them – to measure the angular momentum, for example.

This assumption is self-consistent. After all, classical physics has tried to described the world in this way for centuries. However, this assumption is incompatible with Nature, the real world around us. It is simply not true that you may reduce the measurement of a general observable to the measurement of a predetermined set of "basic observables" followed by a calculation of the appropriate function. Quantum mechanics has irreversibly demonstrated that there are no "basic observables" such that their measurements would be enough to determine the values of all observables.

In quantum mechanics, the positions \(\vec x\) don't commute with the momenta \(\vec p\). But you may still think about devices that measure \(\vec x\) and devices that measure \(\vec p\). And you may (incorrectly) assume that things such as the total angular momentum may be measured by using the \(\vec x\) and \(\vec p\) devices many times. But it ain't the case.

The person asking the question at the Physics Stack Exchange presented a simple example: the operator \(xp\) in one-dimensional quantum mechanics. First, let us get rid of a technicality. We often want the observables to be defined as Hermitian operators. However, \(xp\) isn't Hermitian. Its Hermitian part is Hermitian; it is equal to\[

\eq{
\frac{xp+(xp)^\dagger}2 &= \frac{xp+p^\dagger x^\dagger}{2} =\\
&= \frac{xp+px}2 = \frac{xp+xp+i\hbar}{2} = xp+\frac{i\hbar}{2}
}

\] In other words, \(xp\) is equal to \(i\hbar/2\) plus a Hermitian operator. The eigenvalues of \(xp\) are equal to \(i\hbar/2\) plus a real number. Without a loss of generality, we may allow quantities such as \(xp\) to be considered "observables". Let me propose a more general criterion for an observable. It is a normal linear operator \(M\) i.e. one that commutes with its Hermitian conjugate,\[

MM^\dagger = M^\dagger M

\] This definition allows anti-Hermitian operators, operators that differ from Hermitian or anti-Hermitian ones by a \(c\)-number such as \(i\hbar/2\) above, unitary operators (because both products above are equal to \({\bf 1}\) for them), and others.

So the anti-Hermitian part of \(xp\) isn't a real problem. It is a universal \(c\)-number, \(i\hbar/2\), and we may simply subtract it. Quantum mechanics implies that \(xp-i\hbar/2\) is a Hermitian observable. In principle, it may be measured. The measured values must belong among the eigenvalues. The spectrum of this operator is \(\RR\), of course; all real numbers are possible. The positive values appear for states for which \(x,p\) are positively correlated; negative values appear for states with a negative correlation between \(x,p\).

But let me return to the main point. The main point is that in principle, there exists a device that measures the observable \(xp-i\hbar/2\) exactly. However, this device cannot be constructed as one that measures \(x\), \(p\), and makes some classical calculations with the results. If you wanted such a combined device to be precise, you would need the subdevice measuring \(x\) or \(p\) to be precise as well, but a single precise measurement by these subdevices would already make the value of the other quantity, \(p\) or \(x\), completely undetermined and fuzzy, so no full result for \(xp-i\hbar/2\) could ever be precise.

Even though \(xp\) seems to be made out of the "basic" operators \(x\) and \(p\), it isn't made out of them in the usual classical sense. What I mean is that it cannot be measured by measuring the "basic" operators and by doing a calculation with the results for these "basic" operators. You need a completely new device if you want to measure \(xp\).

Is it legitimate to say that \(xp\) is a function of operators \(x\) and \(p\)? Well, it is a function because we can write it as a simple function, a product in a particular order,\[

f(x,p) = x\cdot p.

\] But it's important to realize that this function \(f\) isn't equivalent to a function of two real variables. A function of two real variables \(f(a,b)\) couldn't distinguish \(ab\) and \(ba\) simply because \(ab-ba=0\) for commuting real variables. However, it's not true for operators such as \(x,p\). So the product \(xp\) is an "operator function" which is more general and remembers the ordering and other things.

There are many operators in quantum mechanics. If the Hilbert space is \(n\)-dimensional, there exists an \(n^2\)-real-dimensional space of Hermitian matrices acting on this \(n\)-dimensional Hilbert space. If you pick a random matrix or operator \(M\) in this set, there will be some operators that commute with \(M\) but most of the operators will refuse to commute with \(M\). It will be rather unlikely that you will be able to write \(M\) as a function of simpler yet mutually commuting operators. In other words, you will probably not be able to reduce \(M\) to any simpler operators.

(Only if you measure the multiples of \(M\), it may be achieved by the same apparatus.)

To summarize, for each observable – each matrix on the Hilbert space – you need a completely new apparatus to measure it. The measurement can't be reduced to the measurement of some "basic" operators followed by a calculation reflecting the function by which the "composite" operator is defined!

These elementary facts contradict the prejudice held by many people – including all the anti-quantum zealots – that there exist "basic" operators and without a loss of generality, we may and we should talk about them only. I've discussed similar misconceptions in the case of the Bohmian models and their attempts to segregate observables into primitive and contextual ones. No such segregation is possible in Nature, of course. All observables – all normal linear operators – are as contextual as all others.

In the case of a particle on a line, it may sound particularly counterintuitive that we cannot reduce all observables to \(x,p\) in the classical sense because \(x,p\) "look" much more elementary than all of their functions. However, there exist examples in which you would find the very same claim less controversial. Consider the angular momentum \(\vec J\).

It has three components, \(J_x,J_y,J_z\). They don't commute with each other. For example, one of the commutators is\[

[J_x,J_y] = i\hbar J_z.

\] In fact, you may write the same commutator in a more suggestive form:\[

J_z = \frac{1}{i\hbar}\zav{J_x J_y - J_y J_x}.

\] In this form, you see that \(J_z\) is an "operator function" of the first two components! In classical physics, you would never think that \(J_z\) depends on \(J_x,J_y\): they are three independent components of an unconstrained vector, after all. But in quantum mechanics, if you allow the "operator functions", each of the three components may be written as a function of the remaining two (a multiple of their commutator).

But even though \(J_z\) is an "operator function" of the first two components, you would never think that the right way (or a possible way) to measure \(J_z\) is to measure \(J_x\) and \(J_y\) in various combinations and do a calculation involving the outcomes of these measurements. That would be stupid. \(J_z\) is as elementary as \(J_x\) and \(J_y\). After all, they are related to each other by rotations. You need a special gadget, e.g. a magnetic field in the \(z\)-direction, to measure \(J_z\).

Those people who are incapable of understanding why quantum mechanics is conceptually new and who are dreaming about squeezing physics into the 19th century classical straitjacket again are forced to admit that there exist some objective "preferred variables" in Nature and everything else that may be talked about are just "classical functions" of these "preferred variables". But the example of the angular momentum – much like pretty much any other example of operators in quantum mechanics – is enough to show that this viewpoint is indefensible.

At the end, your theory has to make predictions about the values of \(J_z\) that may be measured. To deny that the angular momentum may be measured and should be predictable isn't a path to make progress in physics. Because "preferred classical variables" is everything that your theory offers, all components of \(\vec J\) have to be some classical functions of these "preferred classical variables" (which may include corrections from "classical random generators"). But if they were classical functions of them, they would have to commute with each other. But they don't commute with each other. So they can't be "classical functions". Whether you are imagining that your would-be fundamental realist (i.e. classical) theory contains pilot waves, multiple universes, gadgets to shrink a wave function, or anything else, you won't be able to make coherent predictions for as simple observables as the components of the angular momentum.

And the angular momentum isn't an exception; it is a rule. A pair of random observables almost never commutes!

Quantum mechanics tells you that you can't reduce all knowledge to some elementary apparatuses that measure "basic" observables only. For the same reason, you should never think about "preferred bases" of the Hilbert space. Instead, quantum mechanics tells you that a physical system may have many features – described by arbitrary linear operators – and for each Hermitian operator, you may decompose the Hilbert space into a basis of its eigenstates. Every basis is as good as every other basis - but they are mutually incompatible.

One might speculate that this feature – the diversity of qualities that a physical system may have – is important for Nature's ability to produce complicated structures, living forms, intelligent life. A world where everything would boil down to classical functions of \(x,p\) could look "dull" and indeed it could be dull and sterile, too. Chemistry and biology depends on atoms' and molecules' discrete spectra – on the finite number of possible excitation energies that an atom or a molecule may have. It's important that the operator representing the energy, the Hamiltonian, is a "new object" that dynamically depends on the context. Even though the energy in simple quantum mechanical models is an "operator function" of \(x,p\), the value of the energy is independent of the values of \(x,p\) one could measure. That's obviously a necessary condition for the spectrum of the Hamiltonian to be discrete or mixed: the set of possible values of \(p^2/2m+V(x)\) with \(x,p\) eigenvalues substituted for \(x,p\) would obviously be continuous which would be bad.

Let me mention that I discussed two operators, \(xp\) and \(J_z\), that may be written as "operator functions" of some other operators, either \(x,p\) or \(J_x,J_y\), respectively. In the first case, it looked more natural to "reduce" \(xp\) to a "more basic" observable than in the second case. However, it's actually equally unnatural in both physical situations and one could construct various isomorphisms between the sets of operators in various limits.

So quantum mechanics has taught us that there aren't "preferred questions about Nature" and "preferred gadgets" that would make all other conceivable apparatuses redundant. For the same reason, there aren't preferred bases of the Hilbert space and preferred representations of the state vector. One may construct infinitely many observables. For each observable, we need a different device to measure it. Quantum mechanics is able to predict the probabilities that we get any result for any observable but these calculations cannot be reduced to any algorithm respecting a classical framework simply because Nature isn't classical, stupid.

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reader Frederick Bloggs said...

And that's the memo.


Excellent distillation of QM


reader anna v said...

That last word must include the author of this new paper, linked in a question at SE: http://arxiv.org/abs/1207.3612


reader Shannon said...

I do like these type of articles. Thank godness Nature is surprising... :)


reader quantumish said...

I think you are being a bit close-minded about quantum mechanics.
Gerard 't Hooft is certainly showing that this is still a very open question with his most recent paper: http://arxiv.org/abs/1207.3612


reader Luboš Motl said...

Well, I think that what he is showing by this paper is something completely different and not exactly flattering towards the author. ;-)


Quantum mechanics plays an essential role in string theory, both on the world sheet and spacetime. Without quantum mechanics, the spectrum of particles wouldn't be discrete, the conformal symmetry and modular invariance wouldn't work. Dualities wouldn't exist, unitarity would break, all hell would break loose.


The same applies to continuity (i.e. non-discreteness) of the worldsheet variables that are essential for conformal symmetry which is essential for consistency as well, and so on. The paper is complete garbage.


reader Eugene S said...

(Apologies for off-topic) To TRF commenter SteveBrooklineMA: I can't find the article again in which you asked me a question that I was unable to answer at the time, so I am using this comments section instead.


This webpage giving a high-level overview of the EPR paradox and Bell's Inequality gives what I believe to be the answer, in the sixth paragraph:


(Example: the neutral pion is a scalar particle—it has zero angular momentum. So the two photons must speed off in opposite directions with opposite spin. If photon 1 is found to have spin up along the x-axis, then photon 2 must have spin down along the x-axis, since the total angular momentum of the final-state, two-photon, system must be the same as the angular momentum of the initial state, a single neutral pion. You know the spin of photon 2 even without measuring it.


reader Dilaton said...

This is the question
http://physics.stackexchange.com/q/32203
and it has not yet an (appropriate) andwer ... ;-)


reader Eugene S said...

Shannon, did you know that your avatar is a Kippbild... alternating between the head of a lioness and a close-up of a camel's face?


reader Shannon said...

Eugene, are you sure it has nothing to do with your eyesight ? ;-)


reader Mephisto said...

While reading this article, I remembered the chapter 8-3 in the Feynman lectures III. In this chapter Feynman talks about the base states of our world.
Maybe I am wrong, but my own ideas are following. For every quantum system, there are two kinds of observables - the commuting ones and the not commuting ones. The commuting ones form CSCO - complete set of commuting observables. The commuting matrices preserve each others eigenspaces. So the commuting observables share a common Hilbert space, they are only expressed in a different basis. But how is it with non-commuting observables? Is there something like a complete set of non-commuting observables? I feel this has probably something to do with group theory. The non-commuting observables remind me of the generators of Lie groups.


reader Luboš Motl said...

Sorry, Mephisto, you have clearly misunderstood everything. It's remarkable - haven't you already tried to study quantum mechanics for several years?


"Commuting" is not a property of a single observable, it's a mutual relationship between two observables. It means that they satisfy AB=BA.


You can't divide observables into two groups. And there are always infinitely many ways to choose a CSCO. Feynman focuses on this point - freedom to pick the observables and bases - as much as I do. Most of his discussion of quantum mechanics of two-level systems is about this point.


reader Mephisto said...

Yes, I know. I expressed myself incorrectly. Once you choose a particular basis in a Hilbert space, you get some observables that commute with it (CSCO) and others that do not commute with it. And in this sense there are two groups. It is not an absolute division, but relative to a given basis. Once you choose a basis in a Hilbert space, the two groups are given.


reader Luboš Motl said...

Except that it isn't possible for anyone to "commute with a basis", OK.


reader Chris Walsh said...

I know you're not supposed to base your science on what you "like" but personally I don't know why people are so unhappy about quantum mechanics. I think the way nature really works is much more interesting! I think even the distinction between "operationalist" and "realist" is anthropocentric. What can be more real than reality? Who are we to say that the way nature works according to quantum mechanics (without any "extra" filler metaphysics) isn't real?


reader Mephisto said...

I think at least some people are not happy with quantum mechanics because they cannot imagine any kind of sensible reality behind the mathematical structure. In general relativity, you have the Einstein equation and you can imagine some spacetime continuum that is bent by matter (or stress energy tensor, to be exact) - so it is a concrete picture behind the equation, the interpretation is straghtforward. No so in quantum mechanics. You have abstract Hilbert spaces, operators on the spaces, and a complex wave function (the complex coefficients of the basis vectors) whose square is the probability. I like the mathematics, it is ingenious and elegant. Only it is difficult to imagine what kind of reality it describes. It doesn't necessarily need to be a classical kind of reality. If I study for example quantum scattering theory (the Born approximation, partial wave scattering etc), it really looks like some kind of waves in spacetime (just like waves on a pond of water). And if you square these waves, you get the probability of finding the scattered particle. So you ask the question what is waving? Because the mathematics really looks like real waves. Maybe the Copenhagen school is right and there is no reality prior to measurement (although this statement in iself sounds absurd, how could there be no reality, at least in the form of some quantum potential? - no, I do not like Bohm's pilot wave). So the problem is that you cannot link the mathematics to any underlying reality. And there are people who find it hard to accept, even very good physicists. Even Feynman was very careful
"I suspect there's no real problem, but I'm not sure there's no real problem"


reader G. 't Hooft said...

I suggest that you read my "complete garbage" paper.
Your comments above show that, at best, you leafed through it without attempts to understand what it says. Yes, it contradicts some of those things you are so sure about in this blog. But check the math and be prepared for a shock.
Or just continue to sleep peacefully.


reader Luboš Motl said...

Dear Prof 't Hooft,

thanks for your options, especially the last one (to keep on sleeping peacefully) that I will kindly prefer over the other options.


What your deterministic quasi-QM papers contradict is not only things "I am so sure about" but primarily things that can be easily and rigorously demonstrated - and most of them have been demonstrated 85 years ago.


I have already read several of these papers of yours, won a bet in which I claimed that the papers would remain in the zone of marginal stuff for five more years (converted the bounty to a hotel-like one-day treatment in New Jersey that I needed), and I simply won't read every new paper of this sort that you write down unless there exists a new reason to think that they contain something "really different" than the old ones whose reasons of invalidity are understandable to me very well because it is a waste of time.


Because many people are interested in what you have to say, I will offer you to post a guest blog on this blog.


Best regards
Luboš


reader G. 't Hooft said...

This reply is directed not to Lubos, who went back to sleep, but to others who might be reading this blog.
I would be happy to bet that my papers on QM are basically correct, and important, but not on how much time people will need to realise this; my estimate was twenty years, not five.


- The usual objections against my CA theories are based on Bell's inequalities; these objections are erroneous, because of what they say on their page one, line one: their assumptions.
It is assumed that if state A is a CA state, and state B is a CA state, a superposition of these two is also a CA state. Not true. The allowed ontological states of the Universe are only the well-defined modes of a CA; there are no superpositions there. The particles and states we talk about in QM are templates, and so are their superpositions, but none of these represent the entire universe. This means that, if you make a superposition of two allowed CA states, you only get an allowed CA state if you make further changes somewhere else. This could be deduced by (intelligent) readers of my paper on the collapse of the wave function and Born's rule. I explain Schroedinger's cat there.


- Motl understood nothing of what I wrote about the superstring. My point is that the superstring is not modified at all; there is a lattice on the world sheet, but that can be taken to the continuum limit. So, I am talking about the QUANTIZED superstring, with all its discrete string levels and its quantum constraints. My first message is about the string with infinite length. The independent degrees of freedom there are the D-2 transverse oscillations in target space. I make a simple mathematical observation: that thing is mathematically equivalent to a CA. The transverse degrees of freedom then all get restricted to a D-2 dimensional lattice. The lattice mesh size is 2 Pi Sqrt(alpha'), so it never goes to zero.
Even though the CA is classical, one can retrace all the genuine quantum states of the superstring and check that these evolve correctly.
The transformations that rotate or Lorentz transform these states in target space, are quantum transformations, but this does not change the fact that we have a CA here. To prove Lorentz invariance, one needs exactly the string constraints usually imposed on the quantum string, so that D has to be 26 or 10 and the intercept a is fixed, as usual.
Next, I observe that the string exchange interactions, which add closed strings with finite length to the system, are also deterministic if the string coupling constant g_s has a precisely defined value.
Perhaps I should be thankful to Motl for rubbing it in my face that I should do more to get my message across.
G. 't H.


reader Luboš Motl said...

Dear Prof 't Hooft,

concerning your first sentence, maybe you should have appreciated that when someone sleeps, it is not necessarily an eternal process. In the morning, it often happens that I wake up again and I did so today, too. ;-)


I am eager to make a bet against you that your papers that you began to produce in the late 1990s won't be considered "correct as well as important" 20 years after they're written, 30 years after they're written, or any other time period that makes any sense to talk about.


Otherwise what I can do against your passionate assault against the superposition principle? The superposition principle in quantum mechanics says that if a physical system may be in states psi1 as well as psi2, Nature must inevitably also allow the system to be found in any of the form state a*psi1+b*psi2 (times the appropriate normalization constant if we want states to be normalized). It's not just a hypothesis, it's an established law, something that's been demonstrated by nearly a century of experiments. In any sufficiently well-defined a system, we may actually describe the operational procedure to prepare the system in the superposition state. And the superposition principle plays an absolutely vital role in the mathematical structure of any quantum mechanical theory - everywhere .If you have forgotten about the reasons, I recommend you e.g. the chapter 1 of Dirac's textbook on quantum mechanics.

Your denial of the superposition principle is utterly irrational. You don't have a slightest glimpse of evidence that there is anything wrong with the superposition principle. You just become happy about writing things that are manifestly wrong. You say that you are talking about the quantized superstring but you want to deny the superposition principle at the same moment. This is clearly an indefensible position. The quantized string is a quantum mechanical system and *every* quantum mechanical system has to obey the superposition principle.


Best regards
Lubos


reader G. 't Hooft said...

Dear Lubos,

Whether my papers will be considered correct and important any given time from now I do not know; all know is that that's what they are.
You still haven't understood AT ALL my position towards the superposition principle. I am not "against" superposition; quite to the contrary, it is a powerful and essential element in the construction of models that describe the world as we experience it. In particular, this is true if : not every hairy detail of a system, such as its initial state, is known to us. If such is the case, we can describe it as a superposition of CA states. You see, if a CA evolves deterministically, with an evolution operator containing only ones and zeros, then any superposition of any number of its states, can be interpreted as a probabilistic distribution (we use probabilities because we do not know any better). But our superposition, any superposition, then evolves with the same evolution operator, and this evolution obeys the laws of probability exactly, as precisely as it obeys a Schroedinger equation.
This is why, after any basis transformation described by any unitary matrix, linear superposition will still hold: the linear superposition of two CA states evolves exactly as prescribed both by QM and by probability theory.
So the superposition principle holds exactly whenever we look at states where the initial conditions are not exactly known, like in all conceivable experiments.
However, Bell's arguments are not (only) about what we can measure, because we can never do measurements of two mutually non-commuting operators at the same time; Bell's argument is about "what is really going on?". "We measure , what would have "really" been the outcome if we measured instead?" In my theories, this question is as ill-posed as in standard qm, but the description of what happens at the ultralocal scale is different. What is really going on is that the universe is in only one CA state, not in a superposition. This difference is not because I want to write "wrong papers"; it is because it can be done, it explains a lot about qm, and it is not wrong. It is highly illuminating.
The point is that this simple logic goes so far back into the 19th century that all these wise guys, indoctrinated as they are in the mysticism of QM, can't follow it anymore.
What I am saying is NOT against qm as we experience it in thousands of experiments, it is against the arguments behind Bell's inequalities. My theory reproduces exact qm, including our experimental findings concerning superpositions, without any changes; just like my description of the superstring does not require any changes in the theory, except perhaps that I come with a restriction concerning the coupling strength.
There are several reasons why this is important. One is to put all these silly discussions concerning Bell's inequalities to an end; I admit I did not succeed here. An other is that, this way, new leads may be found for model building, such as string theory. String theory is in need of better foundations; my insights may help there. Thirdly, the discussions on the topic of "quantum cosmology". What does it mean to have a probabilistic distribution of "measurements" in an evolving universe, where every measurement can only be done once? So, clearing up such issues will be important for model building in cosmology.
Good night.

G. 't Hooft.


reader Luboš Motl said...

Dear Prof 't Hooft,


please accept my apologies if you don't want to hear the truth but the actual reason why I haven't understood your remarks about the superposition principle before your latest (red) comment as well as after that comment, and why nobody else has understood what you're saying about the superposition principle, is that it makes no sense whatsoever.


The superposition principle holds and must hold without any conditions and there exists no way in which it could "half-hold". In particular, your sentence "So the superposition principle holds exactly whenever we look at states where the initial conditions are not exactly known, like in all conceivable experiments." is completely nonsensical because when we talk about the states psi1, psi2, and a*psi1+b*psi2 relevant for the superposition principle, all these three states describe initial (or other) conditions that are *exactly* known. They are pure states and a pure state, by definition, describes a state of our maximum knowledge about a physical system. So it is completely nonsensical to talk about superposition of two (or more) states in the Hilbert space that are "not known". Such states don't exist. Your comments about "states that are not known" is completely vacuous and nonsensical.


Your later sentence in your newest comment What is really going on is that the universe is in only one CA state, not in a superposition. is in a complete contradiction with the superposition principle whose very *point* is that all linear superpositions are equally real or equally allowed as states of any physical system in quantum mechanics. This is not just some formality. It is not true that in practice, one may always talk about some preferred basis. After all, all the "simply constructible" would-be preferred bases are bases of some elementary fields. However, the actual states that are most directly usable as initial conditions are close to eigenstates of the Hamiltonian; so the most typical states we encounter anywhere in quantum mechanics refuse to be close to eigenstates of any basic operators. Eigenstates of the Hamiltonian are rather complicated state vectors. Just look at the energy eigenstates for a Hydrogen atom; complicated enough states. It's complete nonsense to claim that the Universe could objectively be in an eigenstate of any basic "lattice" operators. After all, we may measure sharp energy rather easily and when we do so, we know that the object is in an energy eigenstate which can't possibly be related to any classical-like cellular automaton state. If you deny any of these things, that superpositions are as real as psi1, psi2 and that the energy eigenstates are more "realistic" initial conditions than any "CA-like" states, then your framework obviously disagrees with totally basic features of quantum physics, with totally rudimentary properties of all of modern physics.


It's demonstrably nonsensical for you to say that you reproduce *anything* about the established quantum mechanical theories. Any framework obeying the properties you have mentioned *disagrees* with everything we know about physics, whether it's from direct experiments or from theoretical frameworks that have been deduced from previous experiments. If you want to build a model of the world in which the world "is" objectively in an eigenstate of some simple elementary operators, you're denying everything we know about modern physics, you must start to build a theoretical description of all things we know about physics from scratch because your assumptions have clearly nothing to do with modern physics, and there are tons of theorems that guarantee that such an attempt of yours to build a physical description of the world from scratch is guaranteed to disagree with observations.


Best regards
Lubos


reader G. 't Hooft said...

It makes no sense to you because you categorically refuse to even imagine that our world might be just a cellular automaton. As soon as you drop that prejudice, you will understand everything I say, including my statements about superposition. Perhaps even you will understand why Bell's inequalities don't apply, and also why there is absolutely no disagreement with any observations. Quantum mechanics, the thing we all love, including the Born rule and the `collapsing wave function', does apply even to cellular automata, but I give up, you will never understand that. Good night again,
G. 't H


reader Luboš Motl said...

Dear Prof 't Hooft,

the reason why I categorically refuse to imagine that our world is a cellular automaton is that such a hypothesis fragrantly disagrees with observations. In the so-called science, it's a basic rule that if a hypothesis disagrees with observations, it doesn't matter who the author is, what the name is, how philosophically pleasing it is to him: it's just wrong.


Also, your constant reference to the name "Bell" in your attempts to defend the indefensible suggests that you haven't read any text of mine. For example, in this blog entry, this comment is the first time I used the name "Bell".


There is no "collapse of the wave function".


Best regards
Lubos


reader G. 't Hooft said...

On your paragraph 1: I hope the other readers of this blog will understand that this argument is a circular one. On paragraph 2: Glad you learned a new word. I just use Bell to characterize this category of arguments against CA. On the rest: I terminate my discussion here; I continue on another blog where people have more constructive remarks.


reader Luboš Motl said...

Dear Prof 't Hooft,


be sure I haven't learned the term "Bell's inequalities" from you. I've taught this stuff at Harvard. I just pointed out that I hadn't used the phrase in my blog entry as a proof that you are incapable of reading and listening texts that don't confirm your misconceptions.


The falsification of a theory by its failure to agree with the superposition principle isn't circular in any sense.

Best regardsLubos


reader anna v said...

whew :( . I just realized "t'Hoof is contributing to physics at SE too.


reader Luboš Motl said...

Dear Anna, is his username SchrodingersGhost


http://physics.stackexchange.com/questions/18586/deterministic-quantum-mechanics


or do you have a different detailed theory? ;-)


reader Luboš Motl said...

Dear Anna, is his username SchrodingersGhost


http://physics.stackexchange.com/questions/18586/deterministic-quantum-mechanics


or do you have a different detailed theory? ;-) Oh, he has an account under his real name:


http://physics.stackexchange.com/users/11205/g-t-hooft
http://physics.stackexchange.com/users/9822/questionanswers


I added the second account, too, because I am unable to hide that I tend to think that it is a sock puppet account. Incidentally, Prof 't Hooft gave about 5 answers to a particular question. Dear Prof 't Hooft, the right number of answers that a given user should give to a particular question is one! ;-)


reader anna v said...

http://physics.stackexchange.com/questions/30065/why-do-people-rule-out-local-hidden-variables

He states his thesis in his answer.


reader Luboš Motl said...

Dear Anna, right, you found the hypothesized sock puppet. But look at my links above: G. 't Hooft also has an SE account under his own name, and 6 (completely nonsensical) answers (5 of which are under the same question, to make things worse).


reader anna v said...

he does give the official web page for the professor, though. If it is a sock puppet as you say it is a matter for the moderators. I do not know whether his list of his thesis is accurate or not and cannot judge from that. The number of answers in the other question is due to his not knowing the rules of the site, which he admits. Again a matter for the moderators, though I suspect they are too awed to intervene.


reader Luboš Motl said...

Dear Anna, there is some information noise here. I am saying that the user "QuestionAnswers" seems to be a sock puppet of Gerard 't Hooft. The account "Gerard 't Hooft" (and it is the only one that links to 't Hooft's web page) clearly isn't a sock puppet; it is his legitimate account.


I am convinced that being "awed" isn't an acceptable excuse for a moderator to fail to do his or her duties and if some moderators fail to enforce the rules of the site, they should be removed.


reader Dilaton said...

Huh, Prof. 't Hooft is sock puppeting at physics SE ?!
I almost cant believe that ...

Ts ts ts ts ... (if this is true)


reader Luboš Motl said...

Dear Dilaton, I really don't claim it's inevitably so. It may be true or not. There's no robust proof in either way. However, users with one purpose (writing positive things about papers by one particular author) - and QuestionAnswers is arguably a textbook example of that - are always suspicious.


reader Dilaton said...

Ah ok, everything else would have astonished me very much ... ;-)
(Somebody thought my comment above was over the top, I did not mean to insult Prof. t'Hooft of course ...)


reader George Christodoulides said...

Dilaton how in the world did you insult anyone?


reader George Christodoulides said...

i am sure t Hooft knows how to ssh


reader Dilaton said...

I usually only insult trolls and sourballs who higly deserve it ... :-)

From the downvote I received it seems somebody was a bit annoyed about my comment responding to the question if Prof. t'Hooft could entertain an additional account on physics SE ...
I've changed this comment a little bit now because the original version was probabbly a bit too bold (?).


reader Mikael said...

Dear Prof 't Hooft,
if you are still following this discussion I would ask you to accept Prof. Motl's offer for a guest blog here. I would be very, very interested. I can assure you from my own observation that Prof. Motl (and the community reading this) will treat guest bloggers fairly and respectfully even if he has a maximally different opinion.


reader Mikael said...

Lubos, I remember that Tony Banks mentioned in his guest blog that even in classical mechanics you can construct observables with a probabilistic nature. Prof. Banks never tried to assign any "meaning" to them so I guess it is impossible. In any case if this is true maybe the argument can be turned around and quantum mechanics is a deterministic theory in the "wrong" variables.


reader MarkusM said...

I just discovered this:

http://www.youtube.com/watch?v=F65n-C7MheU

Here is a topic that got me interested in QM & determinism:
http://arxiv.org/abs/hep-th/0207081

Best