Wednesday, November 07, 2012

When truths don't commute. Inconsistent histories.

A short introduction to Consistent Histories after some trivial appetizer

When the uncertainty principle is being presented, people usually – if not always – talk about the position and the momentum or analogous dimensionful quantities. That leads most people to either ignore the principle completely or think that it describes just some technicality about the accuracy of apparatuses.

However, most people don't change their idea what the information is and how it behaves. They believe that there exists some sharp objective information, after all. Nevertheless, these ideas are incompatible with the uncertainty principle. Let me explain why the uncertainty principle applies to the truth, too.




Every proposition we can make about objects in Nature and their properties may be determined by a measurement and mathematically summarized as a Hermitian projection operator \(P\),\[

P = P^\dagger, \quad P^2=P.

\] The first condition is the hermiticity condition; the second one is the "idempotence" condition (Latin word for "the same [as its] powers") that defines the projection operators. The second condition implies that eigenvalues have to obey the same identity, \(p^2=p\), which means that the eigenvalues have to be \(0\) or \(1\). We will identify \(0\) with "truth" and "true" while \(1\) will be identified with "lie" and "false".

In some sense, you could say that \(P^2=P\) is more fundamental and \(p\in\{0,1\}\) is derived. The very claim that there are two truth values, "true" and "false", may be viewed as a derived fact in quantum mechanics, a result of a calculation. This is a toy model of the fact that many seemingly trivial facts result from calculations in quantum mechanics and some of these facts are only approximately true under the everyday circumstances and they are untrue at the fundamental level.

The first condition, hermiticity, implies that eigenstates of \(P\) associated with eigenvalue "false" (\(0\)) are orthogonal to those with the eigenvalue "true" (\(1\)). This is what allows us to say that the probability that the state disobeys a condition if it obeys the condition is 0 percent, and vice versa. They are mutually excluding. The proof of orthogonality is\[

\eq{
\braket{\text{yes-state}}{\text{no-state}} &= \bra{\text{yes-state}} P^\dagger \ket{\text{no-state}} =\\
&= \bra{\text{yes-state}} P \ket{\text{no-state}} = 0,\\

}

\] In the first step, I used the freedom to insert \(P^\dagger\) in between the states because when it acts on the eigenstate bra yes-state, it yields \(1\) times this state because this state is an eigenvalue-one eigenstate (I used the Hermitian conjugate of the usual eigenvalue equation). In the second step, I erased the dagger which is OK because of the hermiticity. In the final step, I acted with \(P\) on the no-state ket vector to get zero – because the no-state is an eigenvalue-zero eigenstate. So I got zero. The opposite-order inner product is also zero because it's the complex conjugate number (or you may prove it by a proof that is mirror to the proof above).

Let me just give you examples of projection operators corresponding to different propositions. For example, the statement "\(x\) of a particle belongs to the interval \((a,b)\)" is represented by the projection operator\[

P_{a\lt x\lt b} = \int_a^b \dd x\,\ket x\bra x.

\] It's keeping the position-eigenstate components of any vector \(\ket\psi\) that belong to the interval and erases all others. Now, the proposition that the "electron's spin relatively to the \(z\)-axis is equal to \(\hbar/2\) i.e. up" is represented by the projection operator\[

P_{z,\,\rm up} = \frac 12+ \frac{J_z}{\hbar}.

\] It's a simple linear function that moves the values \(J_z=\mp \hbar/2\) to \(0\) and \(1\), respectively. I hope you are able to write down the projection operator for a similar "up" (or "right") statement relatively to the \(x\)-axis:\[

P_{x,\,\rm up} = \frac 12+ \frac{J_x}{\hbar}.

\] Now, is the electron's spin "up" relatively to the \(z\)-axis? Is it "up" relatively to the \(x\)-axis? Those are perfectly meaningful questions that may be answered by a measurement. Because the truth value is either "false" or "true", we may obtain classical bits of information by a measurement.

However, my point is that the truth values of "\(z\) up" and "\(x\) up" propositions can't be sharply well-defined at the same moment. Indeed, it's because the commutator of the two projection operators is nonzero:\[

[ P_{z,\,\rm up}, P_{x,\,\rm up}] = \frac{iJ_y}{\hbar} \in\{-\frac i2,+\frac i2\}.

\] The commutator of the two projection operators – that just represent the numbers \(0\) and \(1\) when the corresponding propositions about the spin is false or true, respectively – is equal to a multiple of \(J_y\) and the eigenvalues (i.e. possible values) of this multiple are \(\pm i/2\). Because zero isn't among the eigenvalues of the commutators in this case, there exists no vector \(\ket\psi\) for which \[

\nexists \ket\psi:\quad (P_z P_x-P_x P_z)\ket \psi = 0.

\] Yes, I started to omit "up" in the subscripts. This non-existence means that it can't possibly happen that \(P_x,P_z\) would simultaneously have some values from the set \(\{0,1\}\). While we may rigorously prove the logical statement that the only possible values of these (and other) projection operators are zero or one, we may also rigorously prove the statement that a physical system can't have a well-defined sharp answer to both questions, "\(z\) up" and "\(x\) up".

If we have an appropriate apparatus, we can immediately answer the question whether the spin was "up" relatively to a given axis. So the questions associated with the projection operators \(P_x\) and \(P_z\) are totally physical and operationally meaningful. Also, by rotational symmetry, both of them are clearly equally meaningful. Nevertheless, they can't simultaneously have sharp truth values!

These projection operators represent potential truths that don't commute with each other. If you talk about the truth values of \(P_{x}\), your logic is incompatible with the logic of another person who assigns a classical truth value to \(P_z\). It's just not possible for both propositions to be "certainly true". It's not possible for both of them to be "certainly false", either. However, it's also impossible for one of them to be "true" and the other to be "false". ;-) They just can't have classical truth values at the same moment!

Because some people often like to repeat Wheeler's notion that the information is more fundamental in physics – and yes, no, it doesn't really mean much although I sometimes philosophically agree with such a priority – they usually think that such vacuous clichés may protect the classical world for them because the information is surely behaving classically. But it isn't. Quantum mechanics says that the information is counted in quantum bits or qubits (the electron's spin above is mathematically isomorphic to any qubit in quantum mechanics) and the Yes/No answers to most pairs of questions don't commute with one another which means that they can't be simultaneously assigned truth (eigen)values for a given situation.

This was just a trivial introduction. We will use it by realizing that "consistent histories" that would mix "different logics", i.e. statements about \(J_z\) and \(J_x\) at the same moment, are clearly forbidden. We will see formulae that prohibit them, too.

Consistent Histories

Fine. We may finally start to talk about the Consistent Histories interpretation of quantum mechanics. Wikipedia and other sources start by screaming lots of nonsense that it's surely an attempt to debunk the Copenhagen interpretation. Such misconceptions have occurred because virtually all the people talking about "interpretations" are activists and imbeciles who have promoted the fight against quantum mechanics as defined by Bohr and Heisenberg to their life mission. Some of them say such silly things because they don't historically know what Bohr and Heisenberg were actually saying. Most of them are saying such silly things because they refuse to understand basic things about modern physics. Many people belong to both sets.



At any rate, once some of them start to understand what the Consistent Histories interpretation says, they realize that it's not the "weapon of mass destruction" used against the Danish capital that they were dreaming about. In some sense, the Consistent Histories interpretation is a homework exercise:
Apply the Copenhagen interpretation to a collection of arbitrary sequences of measurements at various times and discuss which collections are permissible as interpretations of alternative histories. With the help of decoherence, show that your formulae clarify all issues surrounding the so-called "measurement problem" i.e. that quantum mechanics in its Copenhagen interpretation is a complete theory that produces meaningful predictions for microscopic as well as macroscopic systems.
Of course, they feel utterly disappointed. The Consistent Histories approach is refusing to offer them the "classical mechanisms" and "classical information about the system" and "preferred, 'real' choices of bases and operators" and all other things they were expecting. Instead, the fathers of the Consistent Histories join Bohr and Heisenberg in announcing that quantum physics is different than any theory within the general classical framework. It doesn't assume any objective information about the reality. The probabilities are intrinsically incorporated to the foundations of the theory, just like they have always been. But there is no engine or mechanism that "produces" the probabilities in a way that could be fully described by a classical model. No surprise, the Consistent Histories interpretation was coined by mature physicists such as Murray Gell-Mann, James Hartle, Roland Omnès and Robert B. Griffiths.

I should get into some formulae. Readers are recommended to read e.g. this simple and pioneering 1992 text by Gell-Mann and Hartle. They use the Heisenberg picture and it makes the formulae simple. I agree with them it's more natural to use the Heisenberg picture, especially in such discussions (but also in other contexts), but because the people who tend to misunderstand the foundations of quantum mechanics almost universally prefer the Schrödinger picture, I will translate the Consistent Histories wisdom into the picture of the guy who didn't quite respect complementarity or uncertainty (either \(X\) or \(P\)) and who lived both with his wife and with his mistress. :-)

(Well, if you want to hear some defense, he's had children with three mistresses in total and he justified the relations by saying that he "sexually detested his wife Anny".)

It's not so hard to summarize the definition of "weakly consistent" and "medium consistent" histories. What is a history? In the picture we use, the operators are constant in time and the states evolve according to Schrödinger's equation. I will assume that the Hamiltonian is time-independent and the unitary evolution operators from time \(t_1\) through later time \(t_2\) will be denoted \[

U_{t_2,t_1} = \exp(H\frac{t_2-t_1}{i\hbar}).

\] Let's use the value \(t=0\) for the initial state and the time \(t=T\) with some \(T\gt 0\) for the "end of the history". From the beginning through the end, an initial pure state evolves as\[

\ket{\psi}_{\rm initial} \to \ket{\psi}_{\rm final} = U_{T,0} \ket{\psi}_{\rm initial}.

\] Because the density matrix is a combination of ket-bra products\[

\rho = \sum_i p_i \ket{\psi_i}\bra{\psi_i},

\] we may also immediately write down the evolution for the (initial) density matrix:\[

\rho\to U_{T,0}\rho U^\dagger_{T,0}.

\] The daggered evolution operator at the end appeared because of the bra-vectors in the density matrix: they also evolve.

Now, the operator of a history will be the operator \(U_{T,0}\) with some extra, a priori arbitrary, projection operators inserted between the evolution over different intervals into which \((0,T)\) will be divided. We will search for "collections of coarse-grained histories". In the collection, individual elements i.e. histories will be labeled by the Greek letters such as \(\alpha\). Mathematically, the value of \(\alpha\) will store all the information about the moments at which we inserted projection operators as well as the information which projection operators.\[

\alpha\leftrightarrow \{ n_\alpha, \{t_{\alpha,1},t_{\alpha,2},\dots t_{\alpha,n_{\alpha}}\},
\{i_{\alpha,1},i_{\alpha,2},\dots i_{\alpha,n_{\alpha}}\}
\}

\] where \(i_\alpha\) are subscripts distinguishing all possible projection operators \(P_{i_\alpha}\) that we use in any history in the collection. Here, \(n_\alpha\) is the number of projection operators we are inserting in the \(\alpha\)-th alternative history, the labels \(t_j\) specify the value of time \(t\) where we are inserting the projection operators, and \(i_j\) say which projection operators we insert at the \(j\)-th insertion.

You may see that the history operator \(C_\alpha\) will be a generalization of \(U_{T,0}\) of this sort:\[

{\Large
\eq{
C_\alpha &= U_{T,t_{\alpha,n_\alpha}} P_{i_{\alpha,n_\alpha}}\cdot\\
&\cdot U_{t_{\alpha,n_\alpha},t_{\alpha,n_\alpha-1}} P_{i_{\alpha,n_\alpha-1}}\cdot
\\
&\quad \cdots\\
&\cdot U_{t_{\alpha,2},t_{\alpha,1}} P_{i_{\alpha,1}}\cdot\\
&\cdot U_{t_{\alpha,1},0}.
}
}

\] I increased the font size because of the nested subscripts and wrote it on many lines. But the operator is exactly what you expect. You cut \(U_{T,0}\) into \(n_\alpha+1\) evolution operators over intervals and insert the appropriate projection operators to the \(n_\alpha\) places. The insertions and evolution operators at the "earlier times" appear on the right side from their friends linked to "later times"; the usual time ordering holds because the operators on the right are the first ones that act on the initial ket state.

It was a messy formula and I won't write it again. (My formula in Schrödinger's picture differs by the usual transformations by evolution operators, times some possible additional evolution operator, from the Heisenberg-picture formulae in the paper by Gell-Mann and Hartle.)

How can you interpret the history operator? Well, it's like the evolution with \(n_\alpha\) "collapses" in between. However, instead of a discontinuous step in the evolution at which Schrödinger's equation ceases to hold (this totally wrong description occurs at many places, including the newest book by Brian Greene), you should interpret the inserted projection operators differently. They're insertions that are needed to calculate the probability that the history \(\alpha\) will be realized.

This interpretation is needed because the separation into the histories from the particular set is surely not unique, and therefore can't be objective. You may always make the splitting to the histories "less finely grained" and the formalism will calculate the probabilities of these "less finely grained" histories, too. It is clearly up to you – within some limitations – how fine and accurate questions you ask about the evolution which is why you surely can't consider the insertions of the projection operators to be "objective collapses".

Now, how do we calculate the probability that the particular history will take place? It's simple if we assume a pure initial state \(\ket\psi\). What happens with the state? Well, it evolves by the evolution operators \(U_{t_{j+1},t_j}\) over the intervals and at the critical points, the pure state is projected by the projection operators. It is kept non-normalized so we pick the multiplicative factor of the complex probability amplitude associated with the projection operator. We do so for every projection operator in the history so that gives us the product of the complex probability amplitudes associated with all measurements. Finally, we must square the absolute value of this product to get the probability out of the total amplitude.

If you think about the action of \(C_\alpha\) on the initial state as well as the usual Born rule to calculate the probabilities of various measurements (plus the product formula for probabilities of composite statements), you will realize that the probability of the history \(\alpha\) which I will write as \(D(\alpha,\alpha)\) is given by\[

D(\alpha,\alpha) = \bra{\psi} C_\alpha^\dagger\cdot C_\alpha \ket\psi.

\] The first, bra-daggered part of the product, is needed because we calculate the probabilities from the squared absolute values of the complex probability amplitudes that we picked from the projection operators. By the cyclic property of the trace, that can be rewritten as\[

D(\alpha,\alpha) = {\rm Tr}\zav{ C_\alpha \ket\psi
\bra{\psi} C_\alpha^\dagger}.

\] We may easily generalize this formula to a mixed state which is just some combination of \(\ket\psi\bra\psi\) objects. By linearity, we get:\[

D(\alpha,\alpha) = {\rm Tr}\zav{ C_\alpha \rho C_\alpha^\dagger}.

\] Here, \(\rho\) is the initial state at \(t=0\). So the Consistent Histories interpretation allows us to pick a collection of histories and calculate the probability of each history in the collection by the formula above. Finally, I must say what it means for the histories to be "consistent".

Well, if we "merge" two nearby (or any two) histories \(\alpha\) and \(\beta\), we get a less fine history called "\(\alpha\) or \(\beta\)". I have assumed that all the histories in the set are mutually exclusive and the total probability is guaranteed to be one. The probability of "\(\alpha\) or \(\beta\)" must be equal to the sum of probabilities, \(D(\alpha,\alpha)+D(\beta,\beta)\), but even this "\(\alpha\) or \(\beta\)" thing is a history so its probability must be given by the same formula for \(D\), one involving the history operator\[

C_{\alpha\text{ or }\beta} = C_\alpha + C_\beta.

\] Because \(D(\gamma,\gamma)\) is bilinear in \(C_\gamma\) and/or its Hermitian conjugate, the addition formula needs the mixed \(\alpha\)-\(\beta\) terms to cancel. The additivity of the probabilities therefore requires\[

{\rm Re}\,D(\alpha,\beta) = 0.

\] The imaginary part doesn't have to be zero because it cancels against its complex conjugate term. The condition above, required for all pairs \(\forall \alpha,\beta\) in the collection of histories, is known as "weak consistency" (originally "weak decoherence") condition.

Now, it's very unnatural to require that just the real part of the off-diagonal entries \(D(\alpha,\beta)\) for the histories' probability vanishes. The reason is that the phase of \(C_\alpha\) is really a matter of conventions and in realistic situations, the phases of \(C_\alpha\) and \(C_\beta\) may even change independently, almost immediately. So instead of the "weak consistency" condition, it is more sensible to demand the "medium consistency" condition\[

\forall \alpha\neq \beta:\quad D(\alpha,\beta) = 0.

\] The matrix of probabilities for the histories, \(D(\alpha,\beta)\), must simply be diagonal and the diagonal entries calculate the probability of each history for us. It's that simple.

Any collection of alternative histories satisfying the medium consistency condition may be "asked" and quantum mechanics gives us the "answers" while all the identities for the probabilities of composite propositions such as "\(\alpha\) or \(\beta\)" will hold as expected. So one will be able to use "classical reasoning" or "common sense" for the answers to all these questions.

It's important to realize that the job for quantum mechanics isn't to "calculate the right questions" or the "right collection of alternative histories" for us. There is no canonical choice. To say the least, there's clearly no preferred "degree of fine or coarse graining" we should adopt. Too coarse graining will be telling us too little; too fine graining will lead us to a conflict with the consistency condition – this conflict really has the origin in the uncertainty principle. You simply can't expect too many things to be specified too sharply. If you tried to fine-grain the histories "absolutely finely", the histories would resemble the classical histories summed in Feynman's approach to quantum mechanics. But they're clearly not consistent. In particular, we know that they can't be mutually excluding because even in the classical limit, many histories in the vicinity of the classical solution contribute to the evolution, as Feynman taught us. This fact also manifests itself by nonzero diagonal entries between histories that are too close to each other (e.g. because the projection operators on states or "cells in the phase space" are clearly not mutually exclusive if the two cells overlap).

The right attitude is somewhere in between – collections of coarse-grained histories for which the consistency condition holds accurately enough, i.e. histories that obey the uncertainty principle etc. sufficiently satisfactorily, but also histories that are fine enough for us to be satisfied with the precision we need. The precise location of the "compromise" clearly cannot be objectively codified. To choose how accurately we want to distinguish histories is clearly a subjective choice. It's up to the observer.

It should be obvious to the reader that there can't exist any "only right degree of coarse-graining". So there can't exist any "only right set of consistent histories". The choice of the right questions, alternative answers, and the degree of accuracy is up to the observer who chooses the logic. It is inevitably subjective and non-unique. The projection operators don't represent any "objective collapse". Instead, the way how they're inserted encodes the question that an observer asked – and I have written down the explicit formula for the answer, namely the probability of a given history, too.

All physically meaningful questions may be summarized as the questions about the probabilities of different alternative histories in a consistent collection, given a known initial state encoded in a density matrix. If you find several collections of consistent histories, good for you. You may perhaps succeed even if there won't be any "unifying finely grained collection" that would allow you to fully answer all the questions from the two collections. The collections may perhaps look at the physical system from a totally different angle. But if they're consistent, they're allowed.

This is clearly a complete and consistent interpretation of quantum mechanics. It tells you exactly what you may ask and what you're not allowed to ask, and for the things you may ask, it tells you how to calculate the answers. They agree with the experiments. All the criticism of this interpretation is clearly pure idiocy and bigotry.

Let me just mention two representative examples of histories that are not consistent.

Start with Schrödinger's cat described by the density matrix \(\rho\). Let the killing device evolve. At the end, try to define two histories that project the cat to some random macroscopic superpositions of the "common sense" dead and alive stated such as\[

0.6\ket{\rm dead}+0.8i\ket{\rm alive},\quad 0.8i\ket{\rm dead}+0.6\ket{\rm alive}.

\] The functional \(D(1,2)\) will be nonzero because the matrix of probability – the final density matrix after decoherence – is off-diagonal in this "uncommon sense" basis.

In principle, you could think that if the probability of "dead" and "alive" will be exactly equal, the matrix \(D\) will be a multiple of the identity matrix – and the identity matrix has the same form in all bases, including bases of unnatural superpositions. In principle, it's right and you have the freedom to rotate the bases arbitrarily.

In practice, you can't rotate them because the evolution of the cat will be producing and affecting lots of environmental degrees of freedom. If you choose a slightly more fine-grained history for the "dead portion" of the evolution than for the "alive portion", or vice versa, the relevant part of \(D(\alpha,\beta)\) will cease to be a multiple of the identity matrix: the entries on the diagonal of \(D\) will be divided to smaller pieces in the "dead cat branch" of the matrix. Because you want your calculation to be independent of the precise level of coarse-graining or the number of degrees of freedom that you treat as the environment, even in the special case when some of the diagonal entries of \(D\) are exactly equal, you won't really be allowed to rotate the basis while preserving the consistency condition "robustly".

Conventional low-energy situations won't really allow you "qualitatively different choices of the collection of consistent histories" that wouldn't be just some "coarse-graining of quantum possibilities around some classical histories". However, the black hole complementarity actually represents a great example of non-uniqueness of the solution to the condition of consistency of histories. The infalling and outgoing observer are using qualitatively different consistent collections of history operators acting on the same (or overlapping) Hilbert space.

Finally, let me also mention that the consistency condition may seemingly allow you to choose "\(z\) up" and "\(x\) up" histories from the beginning of the article in the same collection. The Consistent Histories formalism simplifies dramatically if we only want to show this simple point. The evolution may be completely dropped, the history operators reduce to simple projection operators, and we essentially consider\[

D(x,z) = {\rm Tr} \zav{P_x \rho P_z}.

\] If you write \(\rho\) as a combination of the identity matrix and three Pauli matrices (or, equivalently, multiples of \(P_x,P_y,P_z\)), you will find out that the trace above vanishes as long as \(\rho\) contains no contribution from \(P_y\). So if the expectation value of \(J_y\) in the initial state vanishes, the off-diagonal elements will be zero. (The latter claim may also be easily seen by calculating \(D(x,z)-D(z,x)\) from the commutator \([P_z,P_x]\).)

However, such a collection of histories will fail to obey the logical condition I haven't mentioned yet:\[

\sum_\alpha C_\alpha = {\bf 1}.

\] This should be valid as an operator equation so it's stronger than \(\sum_\alpha D(\alpha,\alpha)=1\). So it's not allowed to consider "alternatives" that aren't really orthogonal to each other.

In practice, the equation \(D(\alpha,\beta)=0\) is never "quite accurate" so we always ask questions about alternative histories that are only approximately consistent although the accuracy quickly becomes sufficient for all practical and most of impractical purposes. That's a manifestation of the fact that classical physics – and classical reasoning in general – never kicks in quite exactly.

Let me mention that aside from the "weak decoherence" and "medium decoherence" conditions above (the medium one clearly implies the weak one), Gell-Mann and Hartle also discussed a "strong decoherence" condition which would imply both of the previous two but which is too strong and would kill almost all choices of "history collections" whenever the initial state is highly mixed. The condition said that one could express all products \(C_\alpha \rho\) as\[

C_\alpha \rho = R_\alpha \rho

\] where \(R_\alpha\) is a projection operator, a "record projection". So one wants to work with the "medium decoherent" histories.

38 comments:

  1. An impressive summary, I think I got the gist, although I didn't understand it all. But let's not fool ourselves, consistent histories may appeal as one of the better "interpretations", but like all the others it's not really getting at the (non-classical!) "nuts and bolts" of nature, is it?

    I understand consistent histories to say that the universe has had a unique, but probabilistic, evolution path. Which seems more rational than MWI for example.

    ReplyDelete
  2. Great article, Lubos. I really like to learn about the consistent histories in more detail.


    I don't get one thing though:
    The spin x and spin z eigenstates aren't mutual exclusive and also eg. Tr(Px*rho*Pz) isn't zero eg. for rho being the identity matrix.

    ReplyDelete
  3. Ouch! ;-\ - Re: "it's not really getting at the (non-classical) "nuts and bolts" of nature, is it?". ;-)

    ReplyDelete
  4. @jamesg

    (consistent histories)--" it's not really getting at the (non-classical!) "nuts and bolts" of nature, is it?"


    ???? does your statement mean anything or is it as vacuous as it sounds? What are the "nuts and bolts" of nature? Sounds like hidden variables to me....

    ReplyDelete
  5. Dear Mikael, be sure that a state can't be both a J_x eigenstate and J_z eigenstate because these two operators don't commute and the commutator, i.hbar.J_y, doesn't have 0 among its eigenvalues.


    The trace is zero whenever rho contains no sigma_y Pauli matrix. It's because among the four-dimensional basis of 2x2 matrices, namely {1,sigma_x,sigma_y,sigma_y}, only the first one, identity, has a nonzero trace, and it translates to rho proportional to sigma_y (the multiplication by Px and Pz from left and right only permutes the basis of the 4 matrices and multiplies them by powers of i).

    ReplyDelete
  6. This is a very nice introduction to consistent histories :-)


    I did not exactly know what it is before ...
    And it somwhow inspires me to ask something (here or at Physics SE),

    but I'll first have to think about it a little more ...

    ReplyDelete
  7. I mean it's a nice philosophical aid to thinking about QM (Nature) - but it doesn't add any new scientific ideas.

    Classical (deterministic) hidden variables don't exist which is why I added "(non-classical!)" in front of the expression "nuts and bolts" which is an english idiom meaning something like "the basic working components"

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  8. Dear Lubos,
    please calculate Px*Pz explicitly and see that it isn't traceless.


    You argument works for sigma_x*rho*sigma_z.

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  9. You're right, apologies for the glitch, that's what I meant. At any rate, in the 4-dimensional space of density matrices, there's 1 condition that will guarantee that the trace is zero.

    ReplyDelete
  10. I am familiar with English idioms, being a native speaker---my comment was ironic. I too would like non-classical "nuts and bolts",or even classical ones-- ie some sort of quantum underlying reality prior to measurement, but it seems that this is ruled out as well by the Copenhagen interpretation combined with Bell's Inequalities. I also would like a less operational and more explanatory theory, but quantum is the best we have and may be an inviolable limit to what we have access to.

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  11. Dear Gordon, concerning the adjective "explanatory", believe me (you won't): one can't properly scientifically explain things that don't exist or that are untrue. And a "[classical] mechanical explanation" of the facts that quantum mechanics guarantees is, together with the tooth fairies, among the things that don't exist. So if one wants science to be "explanatory" in this sense, he wants the science to be dead wrong.

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  12. Yes, Lubos, I said I want those things, but that reality doesn't work that way. We are not disagreeing I don't think.

    ...just like I would like to fly etc.
    I think that most scientists (except extreme operationalists and positivists) agree that explanations are good whenever they can be found. That is the end result of curiosity.

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  13. Dear Gordon, I agree with your wise but common words about explanations and curiosity.


    What I disagree is your statement that the conclusion "operational, subjective, intrinsically probabilistic rules of quantum mechanics are fundamental" isn't an explanation of the fundamental layer of Nature's inner workings - as well as an explanation of the utterly embarrassing failure of all those who want to find something more classical, more "mechanistic".


    It *is* an explanation, a right one, and arguably the most important one in the 20th century science. By saying that it is not an explanation, one is only exposing his dogmas and bias.

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  14. Well, yes, the operational approach provides important components of explanation....prediction and consequences, and to some degree, context. But it doesn't try to look into causes, or the why something occurs.
    It is nice to be able to provide a cause for an effect. Quantum mechanics is like

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  15. No, Gordon, your text is bunk. Quantum mechanics explains the causes of all the facts we know about the Universe. You just don't fucking like the actual causes because of your preconceptions.

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  16. I guess, in Gordon's mind, the statement : "in QM the result of a measurement doesn't reflect an underlying reality that existed before the measurement" means that there is no cause explaining the result.

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  17. I think so, too, Shannon, but it's a clearly invalid identification. A cause is something else - something more general - than an objective state of affairs prior to the measurement. In the same sense as a cause doesn't have to be "God". Quantum mechanics and the evidence backing it up irrevocably prove that the cause behind all the experimental results is something else than a pre-existing reality of any objective sort.

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  18. Dear James, Nature isn't man-made - it is natural - so it doesn't contain any "nuts and bolts" and every correct fundamental theory agrees with the non-existence of such "nuts and bolts".


    What you and millions of others dislike about quantum mechanics (and/or its proper interpretation, which is really an inseparable part of QM) isn't any genuine incompleteness but incompatibility with your irrational, debunked, obsolete preconceptions. But Nature is in charge so if you don't like it doesn't have any nuts and bolts, you should better apply for asylum in a different universe, a man-made one that has nuts and bolts that you would like. Our Nature isn't like that.

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  19. Once again you are being black and white and twisting what I said. I said that I would "like" to see explanations of causes. By this I mean say, similar to General Relativity. By this I mean an intuitive understanding of the theory. This does not mean that I think that QM is incomplete or needs improvement or changing. This is similar to what Bohr has said about understanding QM, to what Feynman has said about understanding QM, and what Ed Witten has said about string theory ( that we need some intuitive understanding to more deeply understand it).

    I do not understand why or how you come to the conclusions about what I write. I don't believe in tooth fairies as well. Maybe I want to, but as I said, Reality doesn't work that way.

    I love mathematical structures, but I also am curious about phenomena. We are using the word "explanation" in different ways.

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  20. "Quantum mechanics and the evidence backing it up irrevocably prove that
    the cause behind all the experimental results is something else than a
    pre-existing reality of any objective sort. The cause of any outcome is
    some known information about the state and/or observables (from some
    previous measurements) combined with the quantum mechanical
    probabilistic laws that translate the initial state (i.e. results of
    previous measurements) into probabilistic predictions (i.e. about future
    measurements). Nothing else can be a cause. In the quantum world, the
    dynamical laws directly translate previous measurements to the
    probabilities of future measurements. No extra redundant intermediary is
    allowed."


    Now your above statement is the sort of explanation and answer that I wanted and would start to accept.
    It helps me to understand things better from a QM viewpoint.

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  21. Shannon--I look for causes, that doesn't mean I will find them. An example is radioactive decay. One could say its "cause" is the weak force, but that is like what Feynman raged against in a California high school physics text he was reviewing when he came across "What makes things move?" And the answer was "Energy makes things move."

    You don't have to agree with everything Lubos says. I have found his bark is worse than his bite. Of course, it is foolish to seriously challenge his physics since his "intuition" is so good, like Einstein had when he was young (that is a compliment, Lubos. I know you are a Bohr adherent :) )

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  22. Somewhat OT: I am writing too much, but I am starting to understand your position and find myself absorbing it and agreeing with it to a large degree. The theory has discovered a truth about the mathematical structure of the universe and the equations themselves reflect laws of nature and are the best explanation of "cause". Ultimately, laws of nature are fundamental and looking for causes of them are like looking for causes of axioms in a proof.

    Am I correct in this? If so, I accept this.

    I am an arm chair physicist, not a professional one, and my training was in pre-quantum General Relativity, which was classical, geometric, and quite Platonic.

    I also like mathematics quite apart from physics, and think that probability theory and statistics is fascinating and important. Particularly Bayesian thinking...

    http://www.xkcd.com/1132/

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  23. Look, by "nuts and bolts" I'm suggesting that what Heisenberg (and Schrödinger) actually discovered in the mid 1920s is that the universe is a huge linearly evolving system of probability amplitudes (complex numbers).


    But I have to introduce the (controversial) postulate of discrete time evolution to make it work.


    At worst my model would be able to approximate matrix string theory to any desired level (if ST was worth approximating), except I've made the sensible decision to discard continuous geometrical models and just accept that the universe is not so obliging to our silly human minds, it's just a hugely complex discrete system - BUT with continuous probability seeding the evolution.


    So my model is subtle with regard to discreteness/continuity.


    Well, I need to work out at least a weak field approx to GR so I can understand everyone's doubts.

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  24. The most important thing we agree on is that Nature is probabilistic, it's why I have so much respect for you, you're one of the few capable scientists on the internet making this fact again and again and again.


    I don't know how the details will work out after that, and perhaps I am a bit cheeky in dismissing certain ideas - but the fact that Nature is probabilistic is the most impotant that must be rammed up the ass of everyone.

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  25. I deleted the above comment but it's still appearing under 'Guest' - I deleted it becuse it's too twee

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  26. I think that cause is an ill defined concept that deserves the Clinton "meaning of is" award.
    This is a blog - so at risk of "causing" annoyance to Lumo or those who have comments here, here is what I think:

    1. A measurement by an observer given any prior state accessible (per special relativity) to that observer can only result in a "real" eigenvalue with a positive probability;
    2. Special relativity precludes any observation from being inconsistent with previously observed states accessible to that observer;
    3. QM can say nothing about "why" a particular possible outcome was observed except that it must be within the possible outcomes assuring a total probability of 100%. That is it
    tells us nothing about why or how nature chooses the particular new state and affirmatively tells us it cannot tell us why;
    4. "reality" describes what observers (who may communicate within constraints of relativity) can agree upon as observed.
    5. QM gives us probabilities that observers (in position to communicate in future per relativity) will agree was observed.
    Also, perhaps more tongue in cheek:
    6. Philosophy believes it can answer cause questions we know can never be answered;
    7. Faith believes it has answered cause questions we know can never be answered;

    9. Science believes it should answer questions that can be answered ;
    10. Religion believes that ignoring what science can answer is most important.




    So correct or not?

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  27. Sure, reality is fundamentally random, but it is still legitimate
    and very important to ask why.

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  28. OK, it's not. Nature just has certain basic properties and they're the most fundamental facts about Nature. By your loaded question, you are implicitly suggesting that there is something "derived" about the randomness and/or other postulates of quantum mechanics, and you want to replace these would-be "derived" facts by some would-be "fundamental" ones, by some classical mechanism etc.


    But there can't be anything "more fundamental" than them.

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  29. I assure you, I harbor no such thought. I fully believe in QM random's nature. I know there is no deterministic reason of any form or shape as base to QM. But I do believe That there is a reason that clarifies why nature is as such. It is based inherently on a design that randomness is forced upon it, or you can say it had to be that way, imperative if you will. Yes Naturally, i.e. for a very good mathematical reason.

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  30. Apologies if it is inconvenient news but science isn't about "beliefs" in one way or another. Before the 20th century, we didn't have evidence against determinism, but since the early 20th century, we've had such evidence backing up all the new and novel universal features of QM, so that's the end of the beliefs in anything else as long as one is doing science.


    Your attitude is biased. Before QM was discovered and people used deterministic theories, there had been no obsession with constant complaints How does Nature dare to be deterministic, where does the determinism come from. So there isn't a good reason for this obsession today when we know that classical determinism doesn't hold at the deepest level.


    In fact, in the past, people *should* have questioned determinism and asked why it was there. They didn't. Well, if they had, QM would probably be found just at most a few decades earlier.

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  31. Hi Lubos,
    can you please delete the above comment by 'Guest', and my reply explaining why I triedto delete it and this comment too (obviously).


    Otherwise it makes me appear like a rambling loon :-)

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  32. Sorry, I don't see the other copy so I don't understand why I should selectively filter the history of this exchange.

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  33. Very sneaky, I'll remember this :-)

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  34. Well, it is very understandable why people were content with
    determinism, their world looked like that and all of their science was
    basically based on experiments close to that nature. But of course they knew
    something was wrong and that is the reason why they debated endlessly about the nature of what science was saying. As time went by and things got even more complicated they became content with that “the business of science is to describe and not explain”, but that does not mean they were not burning for an explanation. Like GR working wonderfully but no clear mechanism.

    And who could blame them with vast amount of work at blistering speed coming their way, the detour was too risky, but some did try
    to be bold. A problem was started when the scientific establishment became an institution with rituals that were highly regarded, since it did lead to good
    progress. But the trend has slowed somewhat lately as relative stagnation
    became evident. And so more scientists started to come out of the closet with genuine bold ideas, but also an avalanche of genuine crackpots, but all of that is good and a natural reaction.



    Now, let me ask you something before I go into why I “believe”.
    Don’t you want to know why particles have the masses they do, the coupling
    values and CC origins, what is exactly space/time/matter … etc, and knowing these things can lead the final resolution of why probabilities. Well, the reason I say I believe is that I have some good evidence. My
    theory/idea/fantasy or whatever is that reality is a mathematical structure.
    And there is only one structure based on fundamental entity that is dynamic
    that leads to our reality. This structure is inherently based on random numbers, you do not get to choose if you want a dynamic structure(reality).


    While it not an open shut case, but it has significant value at least to the fundamental issue at hand . I have a lot of other results that I have not shown and a lot more development is certainly possible. I guess it goes without saying to save your “nice” comments(if any) until you get at least the basic concept. The website and the info are not in best shape, but the concept is simple and should be a cinch for you.here it is.

    http://www.qsa.netne.net

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  35. Just like people observed deterministic motion of planets etc., since Marie Sklodowska, they have been observing random decays of nuclei, random places of electrons, random everything, so our experimental knowledge of indeterminism is at least as direct and tangible as our ancestors' knowledge of determinism.

    Your comments about "missing explanations" seem to repeat something that you, Gordon, and many others have written millions of times and that doesn't make any sense, as I have explained millions of times as well. There is nothing such as a "missing explanation" behind quantum mechanics.

    Concerning general relativity, one may say that one can find a deeper "explanation" - because it's an explanation of its Hamiltonian as a limit of a theory that is justified by something more robust. String theory is an explanation of GR. But it's an explanation at a deeper level that would really be incomprehensible to someone who can't work with the concepts, principles, and maths of GR.

    The same thing holds for the postulates of QM, after all. If something conceptually new will be found, it will be even more detached from the everyday life, classical physics, and laymen's experience. You are clearly hoping in something closer to the laymen's intuition and no such valid "explanation" of QM exists. Get used to it, all the millions of bigots. It becomes a huge source of junk on my blog and it has wasted lots of hours of my life. Is there some way to ask all the laymen in the world to stop spitting this shit at least on my blog? It is very annoying, repetitive, stupid, prejudices, dishonest, and just plain idiotic, thanks.

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  36. I will try to make it short as not to pollute your site. I could care less about the laymen’s requirement for “explanation”. I am only interested in things that make sense to me whether it is by complicated math or not. I have Masters in EE (Sussex) and my thesis’s math was along the lines of QED math and higher. It just happen that what I discovered coincided with DR. Tegmark’s conjecture and the simplicity of nature conjecture By Wheeler, using similar techniques of Wolfram (and others), without knowing anything about these good people at that time.

    You have the majority view; the above people are minority (at least I read them as the explanation crowd). The laymen’s view has nothing to do with it, they simply don’t count. Of course I do accept all the standard theories like GR and QM, but I respect all the people working in the scientific community no matter what their views are (the divergent ones). I think both STRING and “others” are important, with some bias based on my evolving knowledge of them.

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  37. One final post. I am a layman and a crackpot. what do you think of those people I mentioned. Are they after "explanation" or not in your opinion. How do you characterise their conjectures.

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  38. Steven Weinberg's Lectures On Quantum Mechanics (published this week) discusses interpretations in section 3.7. He discusses Copenhagen and MWI as well as Consistent Histories, and thinks they are all unsatisfactory, here are his concluding remarks (from amazon book preview)

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