Boolean logic and quantum mechanics
In the last physics post I made the following remarks:
which prompted this reply from Lubos Motl:
"Even more obviously, it is complete nonsense - as you state - that quantum mechanics violates the laws of Boolean logic."
I was in the process to systematically rebut point for point all prior objections from Lubos, starting with Bertlmann's socks. However, the importance all of other points pale in comparison with this incorrect statement of Mr. Motl. It makes no sense to split hairs on finer points of disagreement when basic well known facts about quantum mechanics are misunderstood.
To appreciate how blatantly incorrect is Lubos' objection, the implication that in quantum mechanics Boolean logic always applies amounts to denying the applicability of Hilbert spaces!!! Why? Because one can reconstruct the Hilbert space from orthomodularity, completeness, atomicity, and the covering property. But what does all this mean? Let's proceed.
The first thing we need to understand is that Boolean logic is a well defined mathematical structure with sharp axioms. In particular it needs to satisfy the so-called distributive property:
no ifs, ands, and buts. However this is not always true in quantum mechanics. A simple way to see this is by using the uncertainty principle. Let the three propositions A, B, and C be the following:
A = momentum of the particle is in between \(p_1\) and \(p_2\)
B = position of the particle is in between \(x_1\) and \(x_2\)
The first thing we need to understand is that Boolean logic is a well defined mathematical structure with sharp axioms. In particular it needs to satisfy the so-called distributive property:
\(A \wedge (B \vee C) = (A \wedge B)\vee (A \wedge C)\)
no ifs, ands, and buts. However this is not always true in quantum mechanics. A simple way to see this is by using the uncertainty principle. Let the three propositions A, B, and C be the following:
A = momentum of the particle is in between \(p_1\) and \(p_2\)
B = position of the particle is in between \(x_1\) and \(x_2\)
C = position of the particle is in between \(x_2\) and \(x_3\)
The left hand side reads:
"momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_1\) and \(x_3\)"
and the right hand side reads:
"momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_1\) and \(x_2\)
OR
momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_2\) and \(x_3\)"
So this seems to be identical, but if the \(x_1, x_2, x_3\) are close enough then they can be picked in such a way that left hand side obeys the uncertainty principle while the right hand side violates it. And therefore the distributive property can be violated by quantum mechanics.
After proving that Boolean logic is not applicable in quantum mechanics we need to figure out a replacement. After all, quantum mechanics is not in any way illogical. To do that we start with yes/no questions we can ask a physical system and attempt to organize it in a consistent way. Some questions are more general than others and when a general question is true, so are the particular ones. This means that we can define a partial order structure. Because we can ask nature no questions, or the trivial question of the existence of the physical system (which is always true) we have in fact a bounded lattice.
More can be said. Because the quantum logic area is rather dry, it is helpful to visualize the concepts using Hilbert spaces. The end goal is to reconstruct the Hilbert space from lattice properties, but we can start in reverse: from Hilbert space we will extract the properties of the lattices of propositions.
By the projection postulate, a measurement collapses the wavefunction to a 1 dimensional subspace of the Hilbert space corresponding to the eigenvector. Then one can decompose the Hilbert space into this subspace and its orthogonal complement. In general the complement is an involution and in terms of lattice of propositions we have what it is called an orthocomplement.
A lattice is called modular if it satisfies a weaker distributive law:
The left hand side reads:
"momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_1\) and \(x_3\)"
and the right hand side reads:
"momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_1\) and \(x_2\)
OR
momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_2\) and \(x_3\)"
So this seems to be identical, but if the \(x_1, x_2, x_3\) are close enough then they can be picked in such a way that left hand side obeys the uncertainty principle while the right hand side violates it. And therefore the distributive property can be violated by quantum mechanics.
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| No Distributivity Ghost of Classical Physics Allowed |
After proving that Boolean logic is not applicable in quantum mechanics we need to figure out a replacement. After all, quantum mechanics is not in any way illogical. To do that we start with yes/no questions we can ask a physical system and attempt to organize it in a consistent way. Some questions are more general than others and when a general question is true, so are the particular ones. This means that we can define a partial order structure. Because we can ask nature no questions, or the trivial question of the existence of the physical system (which is always true) we have in fact a bounded lattice.
More can be said. Because the quantum logic area is rather dry, it is helpful to visualize the concepts using Hilbert spaces. The end goal is to reconstruct the Hilbert space from lattice properties, but we can start in reverse: from Hilbert space we will extract the properties of the lattices of propositions.
By the projection postulate, a measurement collapses the wavefunction to a 1 dimensional subspace of the Hilbert space corresponding to the eigenvector. Then one can decompose the Hilbert space into this subspace and its orthogonal complement. In general the complement is an involution and in terms of lattice of propositions we have what it is called an orthocomplement.
A lattice is called modular if it satisfies a weaker distributive law:
if \(a \leq c\) then \(a \vee (b \wedge c) = (a \vee b) \wedge c\)
Also a lattice is called orthomodular if the modular condition holds only for b = orthocomplement of a. Therefore we have the following hierarchy:
distributivity => modularity => orthomodularity
[Exercise 1: come up with 3 propositions a, b, c about a quantum system for which modularity is violated due to the uncertainty principle. Hint: adapt the position and momenta example from above]
[Exercise 2: what happens if in your example for exercise 1 you replace modularity with orthomodularity? Why is the uncertainty principle not violated in this case?]
We still need more properties, that of completeness, atomicity, and the covering property to recover the Hilbert space.
[Exercise 2: what happens if in your example for exercise 1 you replace modularity with orthomodularity? Why is the uncertainty principle not violated in this case?]
When for all collection of elements in a lattice we have a infimum and a supremum the lattice is called complete.
An element is called an atom if \(0 \leq a \leq p\) implies either \(a=0\) of \(a = p\). A lattice is called atomic when every nonzero element majorizes at least one atom.
We say that a covers b if \(a > b\) and \(a \geq c \geq b\) implies either \(c=a\) or \(c=b\). An atomic lattice has the covering property if for every a and an atom p, such that \(a\wedge p = 0\), \(a \vee p\) covers a.
Now we have all the ingredients to reconstruct the Hilbert space for quantum mechanics. This was done by Constantin Piron in 1964 and the construction goes in three steps:
- Embed the orthomodular, complete, atomic, with the covering property lattice in a projective space
- Define an isomorphism from the projective space into a vector space
- Restrict the vector space to a subspace corresponding to the elements of the lattice.
The details are too technical, but in the end one obtains quantum mechanics over reals, complex, and quaternionic numbers.
So, what does all of this mean?
There are several conclusions:
- The questions we can ask a quantum system form an orthomodular, complete, atomic, and with the covering property lattice.
- There is a description duality: [QM in Hilbert space] - [orthomodular lattice + additional properties] as each can be derived from the other.
- The logic of QM is that of projection operators (or subspaces in a Hilbert space) and not the Boolean logic of ordinary sets and Venn diagrams. In fact it is easy to visualize quantum logic statements using ordinary 3D space and picturing intersections and unions of points, lines, and planes.
- Quantum OR and Quantum NOT are distinct from Classical OR and Classical NOT. Quantum OR corresponds to the superposition principle.
- Quantum AND is the same as Classical AND.
- The logic of QM does not always satisfy the distributivity property of Boolean logic.
Let's illustrate the point of 3D visualization trick with the modularity condition. First, let's prove that distributivity implies modularity:
From distributivity we get that \(a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c) = (a \vee b) \wedge c \) when \(a \leq c\). The point of modularity is that this weaker distributivity holds only for \(a \leq c\) and not in general.
Now the ordinary 3D Euclidean space is endowed with an inner product and it is in fact a Hilbert space. Sure, it is not a complex Hilbert space which ordinary quantum mechanics demands, but Piron's result is equally valid for the usual complex number formulation as well as for the unusual quaternionic and real number formulations of quantum mechanics. This visualization trick works only for 3 dimensional Hilbert spaces but for pedagogical purposes it is enough to quickly reject invalid quantum logic identities. We only use it as a visual tool to illustrate quantum logic statements and build a visual intuition instead of being confined in the abstract and dry land of logic.
Back to modularity, suppose a is a line and c is a plane which includes a. Hence \(a \leq c\). If b is a point outside the plane c, \(b \wedge c\) is the null set and the left hand side evaluates to \(a \vee (b\wedge c) = a \vee null = a\). For the right hand side, \(a \vee b\) defines another plane which intersects the plane c precisely at a and so \( (a \vee b) \wedge c = d \wedge c = a\) as well.
Please note what \(a \vee b\) corresponds to: it is the geometric figure spanned by the elements a and b which in this case above is the yellow plane. Alternatively it is the smallest subspace containing both a and b.
You can also try to draw yourself the case when b is a line which intersects the plane c in a point and figure out what the modularity relation would correspond to in this case. [Exercise 3: do it. Hint: the answer is that both sides of the modularity equality are c in this case.]
[Exercise 4: why is classical OR different than quantum OR? Hint: Quantum OR for a and b is the smallest subspace containing both a and b. How is this different for classical mechanics? The fact that in the quantum case there are points in between a and b for "a OR b" corresponds to the existence of continuous transformations between pure state. No such thing exists in classical mechanics.]
[Exercise 5: why is classical AND the same as quantum AND? Hint: compare Venn diagram intersections with k-dimensional objects intersections.]
[Exercise 6: why is classical NOT different than quantum NOT (the orthocomplement)? Hint: imagine a quantum proposition a which is a line and a system whose state is a point not in a and not in NOT a. Picture it using the 3D method from above. Can this happen in Boolean logic?]
Why are quantum properties like spin measurement outcomes not always representable as points in a Venn diagram (which was the lesson from d'Espagnat's pedagogical simplification of Bell's theorem)? Because the questions we ask a quantum system are sometimes represented as lines, planes, etc. which have a geometric structure richer than that of a point. And why is that? Because of superposition which is the novel physical property which exists in the quantum world and is not present in the classical case.
To understand quantum mechanics you have to free yourself from thinking classically in terms of points, sets, Boolean logic, and Venn diagrams (equivalently thinking that physical systems have sharply defined (dispersion free was an old term for this) properties before measurement). This is all classical baggage unfit to describe nature. The logic of nature is far richer and has a clear geometric representation. Don't let the unfamiliar nature of complex Hilbert spaces stand in the way of your visual intuition.
[Hard Exercise 7: after playing with drawings of points, lines, planes to visualize modularity and orthomodularity, attempt to describe probabilities in quantum mechanics in the geometric representation of quantum logic. Hint: probabilities correspond to a simple geometric concept. What is it?]
Coming back to the original statement: "it is complete nonsense [...] that quantum mechanics violates the laws of Boolean logic" we see now that it is incorrect and moreover this is known to be incorrect for more than 80 years. The complete understanding of what is going on is known for more than 50 years. The quantum logic field is still an active research area. Yet again, category theory plays a key role because it provides semantic-free mathematical objects for logic, meaning it provides uniqueness proofs for logic concepts.
From distributivity we get that \(a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c) = (a \vee b) \wedge c \) when \(a \leq c\). The point of modularity is that this weaker distributivity holds only for \(a \leq c\) and not in general.
Now the ordinary 3D Euclidean space is endowed with an inner product and it is in fact a Hilbert space. Sure, it is not a complex Hilbert space which ordinary quantum mechanics demands, but Piron's result is equally valid for the usual complex number formulation as well as for the unusual quaternionic and real number formulations of quantum mechanics. This visualization trick works only for 3 dimensional Hilbert spaces but for pedagogical purposes it is enough to quickly reject invalid quantum logic identities. We only use it as a visual tool to illustrate quantum logic statements and build a visual intuition instead of being confined in the abstract and dry land of logic.
Back to modularity, suppose a is a line and c is a plane which includes a. Hence \(a \leq c\). If b is a point outside the plane c, \(b \wedge c\) is the null set and the left hand side evaluates to \(a \vee (b\wedge c) = a \vee null = a\). For the right hand side, \(a \vee b\) defines another plane which intersects the plane c precisely at a and so \( (a \vee b) \wedge c = d \wedge c = a\) as well.
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| This is way cooler than flatlander's Venn diagram reasoning |
Please note what \(a \vee b\) corresponds to: it is the geometric figure spanned by the elements a and b which in this case above is the yellow plane. Alternatively it is the smallest subspace containing both a and b.
You can also try to draw yourself the case when b is a line which intersects the plane c in a point and figure out what the modularity relation would correspond to in this case. [Exercise 3: do it. Hint: the answer is that both sides of the modularity equality are c in this case.]
[Exercise 4: why is classical OR different than quantum OR? Hint: Quantum OR for a and b is the smallest subspace containing both a and b. How is this different for classical mechanics? The fact that in the quantum case there are points in between a and b for "a OR b" corresponds to the existence of continuous transformations between pure state. No such thing exists in classical mechanics.]
[Exercise 6: why is classical NOT different than quantum NOT (the orthocomplement)? Hint: imagine a quantum proposition a which is a line and a system whose state is a point not in a and not in NOT a. Picture it using the 3D method from above. Can this happen in Boolean logic?]
Why are quantum properties like spin measurement outcomes not always representable as points in a Venn diagram (which was the lesson from d'Espagnat's pedagogical simplification of Bell's theorem)? Because the questions we ask a quantum system are sometimes represented as lines, planes, etc. which have a geometric structure richer than that of a point. And why is that? Because of superposition which is the novel physical property which exists in the quantum world and is not present in the classical case.
To understand quantum mechanics you have to free yourself from thinking classically in terms of points, sets, Boolean logic, and Venn diagrams (equivalently thinking that physical systems have sharply defined (dispersion free was an old term for this) properties before measurement). This is all classical baggage unfit to describe nature. The logic of nature is far richer and has a clear geometric representation. Don't let the unfamiliar nature of complex Hilbert spaces stand in the way of your visual intuition.
[Hard Exercise 7: after playing with drawings of points, lines, planes to visualize modularity and orthomodularity, attempt to describe probabilities in quantum mechanics in the geometric representation of quantum logic. Hint: probabilities correspond to a simple geometric concept. What is it?]
Coming back to the original statement: "it is complete nonsense [...] that quantum mechanics violates the laws of Boolean logic" we see now that it is incorrect and moreover this is known to be incorrect for more than 80 years. The complete understanding of what is going on is known for more than 50 years. The quantum logic field is still an active research area. Yet again, category theory plays a key role because it provides semantic-free mathematical objects for logic, meaning it provides uniqueness proofs for logic concepts.
What I presented today is well documented in the literature. For the interested reader who wants to go in depth, an excellent entry point resource is the Beltrametti and Casinelli book: The logic of quantum mechanics.
All of this is just bullšit of yours. The debate whether classical logic is enough has been alive since the 1930s. Birkhoff and von Neumann proposed the "quantum logic" - a set of axioms claimed to generalize classical logic - in 1936. None of the leading folks in quantum mechanics thought it was useful and necessary.
ReplyDeleteWhenever one deals with propositions that are simultaneously allowed by the uncertainty principle and/or complementarity, and one is simply not allowed to do something else, all the rules always hold. For example, the distributive law holds because
A(B+C) = AB + AC
where the product is simply "and" and A,B,C are projection operators. The formula above is relevant for the case BC=0. If it's not, one may make a more correct "B or C" by subtracting BC.
The uncertainty principle *demands* that if we talk about "B or C", then B and C have to commute with each other, so your would-be counterexamples are nothing else than the canonical examples of wrong propositions violating the uncertainty principle - you are talking about the position and the momentum at the same moment! When this elementary mistake is avoided everywhere, everything in Boolean logic works perfectly.
"None of the leading folks in quantum mechanics thought it was useful and necessary."
DeleteThat is because the logic approach is hard and dry, not because it is not valid. But that was then. Today quantum logic offers valid insights in the information approach which is a hot topic in QM research.
Quantum logic is yet another equivalent formulation of QM: standard Hilbert space formulation, path integrals, Wigner's functions.
Lubos, you are confusing sometimes with always. Just because sometimes you are right about physics, this does not mean you are always right. Just because Boolean logic sometimes applies to QM, it does not mean that it always apply to QM. Boolean logic *ALWAYS* applies to classical mechanics and that is the key difference.
In the next comment you said:
"one can't simultaneously discuss the truth value of such propositions about the positions and momenta."
Here you replied to Andrei, but I think it also illuminates what is wrong with your take on quantum logic. As an argument against quantum logic this is invalid because in a logic system I have no knowledge of the meaning of the symbols; I have no rules preventing me to consider something, except the rules of the logic itself. Logic (and quantum logic) is pure abstract math.
Your mistake stems from sloppily mixing the physical and mathematical points of view. I am not able to measure with infinite precision position and momenta, but that does not prevent me to consider abstract statements which talk about position and momenta at the same time. I can consider any statements I want, like I am the king of physics, and the foundation people are all crackpots. But are those statements consistent, and do they describe reality? Quantum logic is consistent and describes nature darn well because it turns out that its propositions are precisely the projector operators in a Hilbert space.
Dear Florin,
ReplyDeleteThere is nothing wrong with classical logic. I think it is your interpretation of what QM is that leads you to such a conclusion.
Take for example the uncertainty principle. It doesn't state that a particle cannot have a well defined position and momentum at the same time. It only puts a limit on future predictions given possible measurements.
It is trivial to measure both momentum and position of a particle at the same time experimentally by simply placing a screen at an arbitrary large distance from the particle source. By operating the source for an arbitrarily small time interval you can compute the momentum with any accuracy and by increasing the distance from the source you can increase the accuracy in position. After the measurement, you can postdict the whole trajectory of the particle.
If you try to determine the position and momentum of some billiard balls by throwing other billiard balls in them you will also be unable to make an accurate picture of the system. Nevertheless, this is in no way an argument against logic.
Andrei
Sorry, Andrei, but the uncertainty principle surely *does* say that a particle cannot have a well-defined position and momentum at the same time - and that these values can't be simultaneously measured with the accuracies whose product is more accurate than hbar/2.
DeleteThat also implies that one can't simultaneously discuss the truth value of such propositions about the positions and momenta. But once one discusses propositions that are allowed to co-exist, classical logic applies to all such ensembles of propositions.
Dear Andrei,
DeleteYou state: “There is nothing wrong with classical logic.”.
I agree. There is nothing “defective” about it; the question is on its usefulness in describing nature. If you take a contextual point of view like Lubos does with his physical point of view arguments, Boolean logic applies beautifully.
You state:
“It is trivial to measure both momentum and position of a particle at the same time experimentally by simply placing a screen at an arbitrary large distance from the particle source. By operating the source for an arbitrarily small time interval you can compute the momentum with any accuracy and by increasing the distance from the source you can increase the accuracy in position. After the measurement, you can postdict the whole trajectory of the particle.”
Here I disagree. Simple counterargument: place a double slit between the source and detector. Then try to posdict which way the particle went.
Now let me make some general remarks about logic and contextuality. If you take a non-contextual (mathermatical) point of view this means that whatever statements you make are universally valid regardless of experiments. The wavefunction in the Hilbert space is an example. If I have the wavefunction I know how to obtain statistical predictions for *any* future experiment. Quantum logic is another example.
Opposed to this is the contextual (or physical) point of view of Bohr and Bell (and Lubos). Bohr said: “the impossibility of any sharp distinction between the behaviour of atomic objects and the interaction with the measurement instrument which serve to define the conditions under which the phenomena appear”. [So in his physical point of view Lubos is right but he is overreaching into the mathematical arena].
But now here is a rhetorical question: if I take a contextual point of view, why do I have to go to the Hilbert space description? I have good realistic explanations descriptions available like Bohmian QM.
Florin
I definitely side with Lubos on this, but I also have very little time to explain. The whole point is that non-commuting observables cannot be perfectly known at the same time, against what Andrei things...
DeleteAndrei (another Andrei however... how many Andreis are there?)
Dear atreus,
DeleteWhy do I have the distinct impression you are Lubos in disguise? I am not sure to whom you reply here so I'll say something. To my knowledge there is only one Andrei participating in this discussion. And I agree with Lubos on his physical points. In fact I must say his reply at his blog is one of his best posts on QM so far. My disagreement with Lubos is not on physics on this one, but on math. Just because there is no operational (physical) meaning to some quantum logic statements we should not discard the math. Cause if you do, you throw away the Hilbert space altogether and that is insane. Math lives by its own abstract rules.
Sorry. I posted this question in the previous blog by mistake. It belongs here. Question for you. Are ideas of points, sets, Boolean logic, and Venn diagrams equivalent or can there be difference between them?
ReplyDeleteKashyap,
DeleteFor discussion sake let's restrict ourselves to finite sets. Then Venn diagrams are an universal visual way to represent set membership. An element is either inside of a set or outside one. It is this "inside-ness" which is clearly marked by the Venn diagram boundary.
So in the finite dimensional case by an abuse of language we can say that the things you mention are equivalent. It is an abuse of language because the elements of the sets are like words, and Boolean logic is like grammar. Venn diagrams are like books with words written in a particular grammar. Books, words, and grammar are distinct concepts, but they play well together.
Florin,
DeleteThanks. Yes. I am abusing language, but I do not know how to put it more precisely! So, are you saying that any set which can be represented by Venn diagrams obeys Boolean logic? This seems to be so because QM objects fail to obey inequalities following from Venn diagrams and hence you conclude that QM does not follow Boolean logic. Also in your terms is Boolean logic the same as classical logic which is applied to everyday classical set of objects? Then again many situations in QM require infinite dimensional Hilbert space.
Kashyap,
DeleteBoolean logic is the everyday classical logic. More precisely one talks about a Boolean algebra.
"any set which can be represented by Venn diagrams obeys Boolean logic?" No, it is the other way around, because sets are the foundation of mathematics and they can be used by all "logics".
Boolean logic operations: union, intersection, complement are representable by Venn diagrams. But in those diagrams the element's additional mathematical properties do not interact with the (boolean) algebra operations. Boolean algebra is blind to the internal structure of the elements. Not so in quantum logic: the geometrical structure of the elements determine the meet and join (AND and OR) outcome.
Oops, I forgot to answer about infinite dimensional Hilbert space. Here you need to know functional analysis to understand what is going on. In particular you need to know about rigged Hilbert spaces, theory of distributions, and bounded and unbounded operators.
DeleteBut let me bypass all of this and use the approach of Mackey. An example of unbounded operator is the position operator. This requires an infinite dimensional Hilbert space because now we need to consider functions of position f(x) and you cannot specify a continuous function using a finite number of elements. So how do you generate yes/no questions about the position of a particle? You ask "was the particle detected in between x1 and x2? Yes or no. And so yet again quantum logic applies for infinite dimensional Hilbert spaces as well.
Lubos:
ReplyDelete"Sorry, Andrei, but the uncertainty principle surely *does* say that a particle cannot have a well-defined position and momentum at the same time - and that these values can't be simultaneously measured with the accuracies whose product is more accurate than hbar/2."
So, what is wrong with my example? What stops you from detecting the position of a particle with a accurately known momentum?
Andrei
Florin:
ReplyDelete"Here I disagree. Simple counterargument: place a double slit between the source and detector. Then try to posdict which way the particle went."
I do not see how this experiment is a counterargument. I have given you an example when both position and momentum can be accurately known simultaneously. I have not claimed that you can do this for every imaginable experiment.
In my example the momentum of the particle remains the same until detection, so it is known at the time of detection. In the double slit experiment the particle trajectory is determined by its interaction with the slits. To postdict this trajectory one needs to know exactly the force acting on the particle as it passes through the slit. For an electron this force would be generated by the EM field and gravitational field. Unfortunately, classical electromagnetism is not correct at this scale (it cannot describe atoms for example).
Unfortunately I cannot understand your point about contextuality. The physics describing our universe is certainly contextual but this is nothing new. Classical theories, like classical electromagnetism or general relativity are also contextual when they are not studied outside of their domain. My example with billiard balls used to measure other billiard balls is also a case of contextuality. Nevertheless I cannot see any reason to blame classical logic. You only need to be careful to give the correct interpretation to measurement results. They are not independent properties of the studied system but the result of an interaction between the system and the instrument.
Andrei,
DeleteYou have very good questions and I was expecting your first one:
"I do not see how this experiment is a counterargument."
To this question I can do no better than Zee in his QFT in a nutshell book http://www.amazon.com/Quantum-Theory-Nutshell-Edition-nutshell/dp/0691140340 which is my personal inspiration of how to do explain physics. So I'll adapt his first or second chapter, I don't recall which one and make it into my answer. Here we go.
First I assume the answer to my question about which way is no, you cannot tell which way, otherwise you would have provided an answer already. (And if you say you can tell which way then you have the problem of explaining the interference pattern)
Now let's add an additional slit. Can you tell which way? Still no. Add another slit. Can you tell which way Still no. etc, etc. Add as many slits as needed UNTIL THERE IS NO SCREEN LEFT. What you get? You get Feynman's path integral formulation of QM. But still you cannot tell which way, and therefore there is no real path qed.
On contextuality, perhaps it is not a perfect choice of a word, but it is the best I could find. Take a look at the third Escher picture at: http://harrowakker.webnode.nl/escher/ which is the up and down staircase. On each side the picture makes sense, but not together: the staricase is "contextual". Locally consistent, globally inconsistent. Boolean logic is just like that for QM: it makes sense only within the confines of an experiment when observables commute. But it does not make sense when you consider all possible experiments.
Lubos' point is this: hey, I don't care about all possible experiments because there is no single experimental setup to which I can put it to the test. You should *only* talk about what you can actually test. Because position and momenta obey the uncertainty principle, you cannot measure both of them better than hbar/2 so it makes no sense to talk about it at the same time. If you do you are insane. Something like: "what are the color of the eyes of the Emperor of USA?" - a perfect nonsensical statement because although each word has a precise meaning, together is hogwash. And in taking this position he follows in the footsteps of positivism which is perfectly suited for QM. Lubos point of view is perfectly consistent, so how can he be wrong then?
He is wrong in that he does not understand the quantum logic approach as he is stuck with ancient knowledge in pre Constantin Piron era arguments. In quantum logic, as in the Hilbert space formalism of QM (to which it is dual) we talk about any experiments regardless of how we actually perform them. This is a non-contextual, global point of view. Non-contextual in the sense that I ignore each and all experimental setups. And that was my point about the wavefunction. The wavefunction for a particular quantum system is the best information I have which allows me to make predictions about ANY experiment. In quantum logic the propositions are projection operators. I can consider any projection operator I want: for position, for momenta, for spin x, for spin y, for spin z. I just don't care I cannot measure spin on the x and y directions simultaneously, that does not prevent me from writing sigma_x sigma_y. Quantum logic is abstract math. Lubos is confusing math with physics and imposes sensible physics demands where they don't belong. In the process he is screwing up the math and arrives at this mathematically incorrect conclusion: "it is complete nonsense [...] that quantum mechanics violates the laws of Boolean logic".
Sure, in quantum logic there are nonsensical statements on par with "what are the color of the eyes of the Emperor of USA?". But they don't matter. In the end when you test its agreement with nature you have only sensible statements.
Logically why should one be able to distribute among quantities that do not commute. Luboš impolite but quite right.
ReplyDelete"Logically why should one be able to distribute among quantities that do not commute"
DeleteI agree. That is why there is no distributivity in quantum logic, only orthomodularity.
Lubos' objection is different though. He does not want to allow statements about say position and momenta at the same time because they are nonsense and he contends quantum logic is garbage because of this.
By the same kind of argument he should say QFT and string theory are garbage because you can talk about ghosts and tachyons in them. String theory and QFT are not garbage, and neither is quantum logic.
Florin:
ReplyDelete"First I assume the answer to my question about which way is no, you cannot tell which way, otherwise you would have provided an answer already. (And if you say you can tell which way then you have the problem of explaining the interference pattern)
Now let's add an additional slit. Can you tell which way? Still no. Add another slit. Can you tell which way Still no. etc, etc. Add as many slits as needed UNTIL THERE IS NO SCREEN LEFT. What you get? You get Feynman's path integral formulation of QM. But still you cannot tell which way, and therefore there is no real path qed."
I am not sure, but the answer is probably no, here is why:
The experiment I proposed requires one particle (say an electron) being emitted and subsequently detected. The detection gives you both position and momentum, and the trajectory can be inferred because the momentum is conserved between emission and detection (this might not be exactly true, as both the source and screen do produce EM fields as they are composed by charged particles, but for simplicity let's assume we can ignore those fields).
Now let's increase the complexity a little by performing the same experiment using two electrons. Again we know their initial and final positions, but we cannot calculate the momentum anymore because the two electrons interact, changing their initial momentum. We can calculate some sort of mean momentum but I am quite sure the problem will be underdefined. we simply cannot extract enough data by performing measurements so that we may solve the trajectories. It might be the case that with some numerical computation one could aproximate the trajectories but I realy don't know that.
There is also the problem that classical electrodynamics is not right, and there is no known claasical replacement for it ( a hidden variable theory). But even if such a theory is found we are still limited by the uncertainty in the amount of data we can extract.
So, the problem is exactly solvable for 1 particle, the two body problem is probably not solvable. Now, the double, triple,....whatever -slit experiment is equivalent to a n-body problem, where n is something like 10^23. Is it realy amazing that such a problem cannot be solved? What can this prove? That we have too many unknows/equations ratio, not because there are not trajectories.
Florin:
"On contextuality, perhaps it is not a perfect choice of a word, but it is the best I could find. Take a look at the third Escher picture at: http://harrowakker.webnode.nl/escher/ which is the up and down staircase. On each side the picture makes sense, but not together: the staricase is "contextual". Locally consistent, globally inconsistent. Boolean logic is just like that for QM: it makes sense only within the confines of an experiment when observables commute. But it does not make sense when you consider all possible experiments."
That picture cannot be correct, the perspectives must be wrong. I bet that an architect can tell you where the error is and that error should be locally detectable as well.
Nevertheless, I like the analogy, I think it applies well to QM. It might give you the illusion that you cannot have an objective, consistent picture of reality, but if you think carefully you will see that no contradiction between the statistical predictions of QM and classical framework can be found
Dear Andrei,
DeleteYour strategy with many electrons to avoid the problem is not valid. All you have to do is to dim the electron source until you emit say one electron per second. Then you are only in the 1 particle case. Adding or subtracting slits does not change the 1-particle nature of the argument.
Florin
Florin,
DeleteI wasn't speaking about the electrons coming from the electron source. You can send them one at the time, sure. But that single electron moves in the field produced by the electrons and atomic nuclei from the material used to make the slits. Being a macroscopic object, it will contain 10^23 particles or so. The incoming electron interacts with these particles and therefore changes its momentum. So, in order to calculate its trajectory you need to describe in detail that interaction involving 10^23 particles which ain't easy.
Andrei
I see. Now I understand your position (but I still disagree with it).
DeleteThe uncertainty principle is something intrinsic about 1 particle, it does not depend on how many other particles are around to interact. Adding additional particles does not change the nature of the argument.
Florin:
Delete"The uncertainty principle is something intrinsic about 1 particle, it does not depend on how many other particles are around to interact."
I know that this is what you believe, but what arguments do you have in favor of this interpretation?
You still did not provide a justification for why you reject my 1-particle example as a valid proof against your interpretation of uncertainty. Do you think that my proposed experiment cannot be done? What exactly is it wrong with it?
"Adding additional particles does not change the nature of the argument. "
Of course it does. As I have previously demonstrated the trajectory of the electron cannot be calculated because of some well understood reasons that have nothing to do with the presumed non-classical behavior of the particle. Therefore not being able to determine the electron's path is irrelevant to the meaning of uncertainty.
Andrei
Dear Andrei,
DeleteI could present several arguments. I could say that in my argument I am changing the boundary condition, but I don't think that will convince you. I could say the uncertainty principle follows from simple Fourier analysis but if you don't believe in the wavefunction it does not carry much weight, I could revert back to early Heisenberg microscope arguments, but they are imprecise, or I could show you my mathematical derivation of QM which implies the uncertanty principle, but that is too abstract.
So let's reset the discussion and restate your position/argument. Let's simplify as much as possible it to get to the bottom of it. How do you propose to show the uncertainty principle does not hold for one particle? Yes, you stated that before, but please indulge me and let's start again.
Florin
Dear Florin,
DeleteI will restate my arguments, sure.
I do not deny the HUP (Heisenberg uncertainty principle). I only claim that it has to be understood as a limitation of the amount of data that can be extracted from a system, and not as a fundamental property of that system. I accept the Heisenberg microscope argument. It is exactly what I try to argue here. Measurements induce a perturbation in the system therefore limiting our ability to fully describe it.
There is no way Heisenberg microscope experiment can be used as an argument against the electron having a well defined trajectory.
If you try to measure the momentum the electron had before entering the microscope you are limited indeed by the uncertainty. However, you can make a change to this experiment. Use the most energetic photons possible so that you get a perfect knowledge of electron's position. Say, you make this measurement at time t1 and the measured position is x1. Repeat the measurement so that you get a second pair t2/x2. According to HUP, the first measurement renders the momentum the electron had before t1, say p0, completely unknown, the particle acquiring a new momentum, p1, as a result. The second position measurement will change again p1 with an unknown momentum p2. So, we don't know p0 and neither p2, as HUP predicts. But what about p1? Its value is (x2-x1)/(t2-t1)×m. Therefore, for the time between t1 and t2 we know perfectly the electron's trajectory (both position and momentum). This information cannot be used to predict what the particle will do in the future, because p2 is completely unknown.
So, HUP has to be understood as a limitation to acquire information about a system in certain situations, like before t1 and after t2 and not as a general disproof of the classical reality of the quantum world.
Andrei
Dear Andrei,
DeleteThank you for your reply. I was busy today preparing the next post and I could not answer but I promise I'll reply tomorrow.
Florin
Dear Andrei,
DeleteSorry for the delay. If you do not disagree with the uncertainty principle then I don't think we have a disagreement. The Bohmian interpretation demands the particle to have a well defined position at any time and although I do not agree (for several key reasons) that Bohmian interpretation is the correct QM interpretation, there is nothing inconsistent about this point of view.
On your expanded explanation I can agree with it in the Ehrenfest theorem spirit (https://en.wikipedia.org/wiki/Ehrenfest_theorem) because you are talking about averages and not the instantaneous values.
Florin
Dear Florin,
DeleteI do not see how Ehrenfest theorem applies here. In my example there is no uncertainty at all; the particle does not only approach a classical trajectory but it has a very well defined classical trajectory.
You say that the momentum of the particle, calculated from the two position measurements is only an average momentum. Yes, but in this case, as no force is acting on the particle between measurements, the average momentum is identical with the instantaneous momentum. The momentum just stays the same between x1 and x2. The vector is on x1-x2 line and the magnitude is (x2-x1)/(t2-t1)×m.
Here you have another example which shows even better why your interpretation of HUP is wrong. Perform a perfect momentum measurement first and then a perfect position measurement. At the time of the position measurement you know both momentum and position perfectly. It is obvious that HUP cannot be applied to the past (because the position measurement cannot erase the momentum the particle had) but only to the future. After the position measurement you don't know the momentum. But, for the time between these measurements you know perfectly the particle's trajectory.
Andrei
Dear Andrei,
DeleteI mentioned Ehrenfest theorem only for the purposes of averages, nothing else. But I agree with your point "as no force is acting on the particle between measurements, the average momentum is identical with the instantaneous momentum."- point taken.
Here is where I start to disagree: "Perform a perfect momentum measurement first and then a perfect position measurement." How do you perform a "perfect momentum measurement"?
Florin
Dear Florin,
DeleteIn practice one does not actualy measure the momentum, but prepare the particle (or an ansamble of particles) in a momentum eigenstate. I have found this article:
http://www.ing.unibs.it/~beretta/www.quantumthermodynamics.org/ParkBand-FoundPhys-22-657-1992.pdf
Preparation procedures for momentum eigenstates for neutrons and electrons are presented.
So, performing a position measurement on these particles gives you both momentum and position for the time between preparation and measurement.
Andrei
Dear Andrei,
DeleteYou can assign a "fuzzy" trajectory to a particle the same way you see one in a cloud chamber. As long as the de Broglie wavelength is small, this works the same was as in geometrical optics.
However I disagree with the paper assessment:
"Practical limits obviously exist to the purity of the momentum thus
secured, apart from the fact that the finite time between cut-offs inevitably prohibits monochromatic purity of the de Broglie wave representing the neutron. But at least in principle one can obtain a pure monochromatic wave, hence generate a pure momentum state ensemble of neutrons"
The part that I disagree is this: "in principle one can obtain a pure monochromatic wave". In practical terms what happens when you have a perfect momenta or position preparation, you get no particle for subsequent use. To get to a perfect momentum state you have to wait an infinite amount of time. To get to a perfect position measurement you have to use an infinitely energetic photon, and this is like measuring the position of a fly with a cannon ball: there is no fly left after the collision, and there is no particle to be found after the encounter with a very energetic photon.
No matter what you do for measurement purposes, you cannot get a trajectory which beats the uncertainty principle.
You say: "performing a position measurement on these particles gives you both momentum and position for the time between preparation and measurement." to which I qualify with: "gives you both momentum and position with position and momenta uncertainty whose product is larger than hbar/2" If you accept this qualification, then I have no objection.
Florin
Dear Florin,
DeleteI have four points to make:
1. You seem to be concerned with the impossibility of getting perfect measurements/preparations. Nevertheless, do you agree that they are theoretically possible from the point of view of the QM formalism? If we agree on this then I would say that I have proven my point. We have a case for which QM does not forbid knowledge of both momentum and position with any accuracy.
2. You didn't have any more objections with my first example (with two position measurements). So, would you agree that I have proven the possibility of knowing both momentum and position at the same time?
3. Let's see a real-world example of momentum/position experiment:
http://www.adichemistry.com/jee/qb/atomic-structure/1/q8.html
The first example shows that for an electron for which the speed is 300 m/s , accurate up to 0.001% the HUP predicts an uncertainty in position at about 2cm. Now, if you accept the posibility of getting the electron's momentum with the above accuracy, what stops me to subsequently measure its position with 1 mm accuracy? This is easily achievable with an old TV screen.
4. The way I imagined the experiment, the momentum preparation and position measurement are independent procedures. There will be experimental limitations in regards to the their accuracy but I see no reason for assuming that these errors are releted by the uncertainty relation. Why would you think that by improving the accuracy of momentum preparation would decrease the resolution of the screen?
Andrei
Dear Andrei,
DeleteI enjoy this debate, and let me start with a joke to have some fun:
Mr. Heisenberg was driving a car and was pulled over by a cop for speeding.
-Do you know how fast you were travelling?
-No, but I know precisely where I am.
-You were going 120 km/h in a 100 km/h zone.
-Now I am lost!
Now back to the debate.
"You seem to be concerned with the impossibility of getting perfect measurements/preparations. Nevertheless, do you agree that they are theoretically possible from the point of view of the QM formalism?"
It is possible to measure with better and better accuracy, but you cannot reach perfection.
Now onto the 300m/s electron example. First I want to clarify that the position uncertainty refers to the uncertainty along the direction of travel, not in a plane orthogonal to it. But still, an old TV screen will be able to localize the electron within 1 cubic millimeter.
So now I think you will say something along the lines: I have an electron with a speed accurate to 0.001%, and when it was detected the position uncertainty was 5% of the minimum position uncertainty demanded by HUP. At the detection time the electron had a position and momenta with uncertainties under hbar/2. Counterfactually I can extend this argument back along the entire trajectory and hence the electron had a well-defined trajectory.
I don’t want to speculate and give my argument for this and potentially go in a wrong direction, so I want to ask you to confirm that this is your position, and if not, please correct my statement. Then I’ll take it from there.
Florin
Dear Florin,
DeleteNice joke! Thanks!
You have understood perfectly my position. I only have an objection regarding the word "counterfactually". I think that the trajectory determination is based on perfectly valid measurements and solid physical assumptions (momentum conservation). There is no need for unperformed measurements to be included in the calculation.
Andrei
Dear Andrei,
DeleteExcellent. I actually agree that "At the detection time the electron had a position and momenta with uncertainties under hbar/2." and moreover Heisenberg himself agreed with those kind of violations in the past. However he argued that they are meaningless. I will attempt to argue the same thing using my own take on it.
First a disclaimer: it is possible to assign in a non-contradictory way a position and a trajectory to the electron, and this happens in Bohmian interpretation. Heisenberg uncertainty principle does not disprove Bohmian quantum mechanics (but I have other objections against it).
What I want to show is that the fact that the electron violated HUP in the past does not *necessarily* mean that the electrons have a well defined path. Here we go:
Consider a system of reference moving in the direction of the electron beam with 300 m/s. Since the velocity uncertainty is 3mm/s, the position uncertainty is 2cm. Upon encountering a detector screen (coming at us with a speed of 300m/s) the position uncertainty is reduced to 1mm. So what do we have? We have the standard collapse of the wavefunction. To argue that the electron MUST HAVE HAD a definite path from the fact that the electron is simply detected someplace (when the wavefunction collapses) is an incorrect conclusion. Just because the electron violated HUP in the past this does not imply WITH CERTAINTY that the electron had a definite path. In this way Heisenberg has called this kind of violation of HUP in the past meaningless. You cannot draw any definite conclusion from it, you cannot use it as initial condition to make future predictions, etc.
But just as HUP in the past does not disprove the positivist position, the HUP does not prove the positivist position either. One one hand you have the Copenhagen (and its variants) interpretation, and on the other hand you have Bohmian QM. If you assume a definite electron path then you must use Bohmian interpretation, there is no other possibility. But is Bohmian interpretation desirable? I could find many faults with it, but this is a much larger discussion.
Florin
Dear Florin,
Delete1. You agree now that HUP can be violated in certain experiments, in the past. This proves that, contrary to your original point, a particle CAN have a well-defined position and momentum at the same time.
Now, you seem to argue that we cannot be sure that the particle always has these properties. It seems to me that this goes against Ocham's razor. Why should we assume a different behavior of the particle at other times?
2. I think that, even in the absence of a direct experimental proof for the existence of particle trajectories, the evidence we have points strongly in their favour.
So, let's see what the denial of particle trajectories implies:
We know for sure that the particles move from one place to another, that's beyond doubt. How do they move? We have three possibilities (if you have more, I will be happy to discuss them):
-particles move along well-defined trajectories
-particles are extended objects (matter-waves)
-particles dissapear from the universe and appear again when detected.
Only the first posibility is in agreement with the known lows of physics. The second option implies non-locality. The third implies both non-locality and violations of conservation laws (charge, mass).
Particle trajectories folow directly from the momentum conservation law. If no force acts on it it must travel in a straight line. Once the position has been established, you have a well-defined line. This seems to me the logical default position. If you argue for a different type of mass transfer, some strong evidence should be provided.
Andrei
Dear Andrei,
DeleteI think blogger.com has some software bug, I am not sure why your reply was not posted right away.
Now on point 1 on Occam's razor, this is a misleading argument. Beauty is in the eye of the beholder. If the electron has a definite position, explain why the Hydrogen atom does not radiate energy according to electromagnetism. In Bohmian interpretation the electron is actually stationary. How is this natural? I find it bizarre.
On point 2, there is no definite evidence in favor of the trajectory. If there were, Bohmian interpretation would have already won the day.
On the three options:
-particles move along well-defined trajectories
-particles are extended objects (matter-waves)
-particles dissapear from the universe and appear again when detected.
"Only the first posibility is in agreement with the known lows of physics".
I disagree. This and the other 2 options do not exhaust the possibilities because they imply realistic points of view. There are positivist epistemic points of view which you did not consider.
"The second option implies non-locality."
So does the first option. See Bohmian's quantum potential there and the trajectories which go faster than the speed of light but without violation of no-signaling.
"The third implies both non-locality and violations of conservation laws (charge, mass)"
This is a crazy proposal which nobody entertains it.
"Particle trajectories folow directly from the momentum conservation law."
I disagree. if you *assume* particle trajectories and momentum conservation you get straight lines. But momentum conservation alone is not enough to establish distinct paths.
Dear Florin,
Delete"Now on point 1 on Occam's razor, this is a misleading argument. Beauty is in the eye of the beholder."
True, but this is not a case of beauty vs. ugly. It is an experimental fact that quantum particles are only detected as point particles. There is no experiment in which a particle is detected in more than one place (like a classical wave). The size of elementary particles, as determined by state of the art accelerators is practically 0 (no non-zero size has been found).
It is therefore a natural option to assume that they remain point particles all the time. If you claim the contrary you need some evidence to back that up, describe what form those particles take when not observed, what size they have, how they move, etc.
If I were to tell you that your car becomes a spaceship when safely closed in a garage, you wouldn't take my word for it, right?
"If the electron has a definite position, explain why the Hydrogen atom does not radiate energy according to electromagnetism."
Classical electromagnetism is a de-facto statistical theory. It is based on experimental evidence involving a great number of charges. The field equations of this theory describe the average fields produced by macroscopically charged objects, not the fields around isolated particles. Therefore, the theory cannot be extrapolated to atomic scale. This fact doesn't contradict in any way the fact that quantum objects are point particles as all experimental evidence suggests.
"In Bohmian interpretation the electron is actually stationary. How is this natural? I find it bizarre."
I am not a fan of this interpretation, but I find nothing particularly wrong with this result. For a hydrogen atom the force generated by the quantum potential compensates the electrostatic force. It is no more amazing than the fact that you can make a magnet levitate above other magnets. But again, even if Bohm's theory is wrong this doesn't mean that particles cannot be material points.
"This and the other 2 options do not exhaust the possibilities because they imply realistic points of view. There are positivist epistemic points of view which you did not consider."
I'd very much like to hear such a point from you.
"So does the first option. See Bohmian's quantum potential there and the trajectories which go faster than the speed of light but without violation of no-signaling."
Again, Bohm's theory cannot be assumed to be the only possible embodiment of point-particles theories.
"This is a crazy proposal which nobody entertains it."
I agree, but this seems to be implied by the "epistemic" theories. We will go to that later.
"I disagree. if you *assume* particle trajectories and momentum conservation you get straight lines. But momentum conservation alone is not enough to establish distinct paths. "
I only need to assume point particles + momentum conservation. A point particle with a constant momentum gives you a trajectory.
Andrei
Dear Andrei,
DeleteYou say: "Bohm's theory cannot be assumed to be the only possible embodiment of point-particles theories." I know no alternative if you assume a well defined particle position. Please explain.
Let's clarify this first to prevent the discussion to go in too many tangents and become unmanageable.
[Side note: I think I state this before (but maybe not in the context of this discussion), my interpretation of QM is closest to QBism but not identical. I will expand on this in upcoming posts at this blog.]
Florin
Dear Florin,
DeleteYou can observe that the planets go in elliptical orbits without having a theory explaining that. There were many proposed explanations: angels pushing the planets, a non-local gravitational force or a space-time curvature. So, you have a fact (planetary orbits) and various theories to explain them.
The same holds true for quantum particles. They appear as points. This is a fact. Bohm's theory may be the only known one describing their motion but it is not necessary the only one. I see no reason why I should embrace this theory. Denying Bohm's interpretation does not necessarily imply a denial of the idea of point-particles moving in space.
The reason I do not think the de Broglie-Bohm theory is on the right track is as folows:
QM consists of two parts: a physical part (which is represented in the Hamiltonian) and a probability theory part (I think the Q-Bists would agree on that). The physical part is based on classical physics which, as I have said before, is a de-facto statistical theory, a theory that fails to be extrapolated to quantum realm. So, QM takes a statistical theory, applies a probability calculation on it and produces true statistical predictions.It is unlikely that you could reverse-engineer this theory and get to the bottom of things. What is needed is not to replace a statistical theory by another statistical theory but find out what was wrong in the first place. It might be the case that the electric field around an electron is not spherical, does not decrease with the square of the distance, it might be directional, periodic in space or/and time, etc.
However, even in the absence of a theory explaining the particles' motion I think there is good evidence for well-defined particle trajectories. Basically all you need is:
1. Partcles are points (there is no evidence opposing this)
2. Particles exists in this universe at all times (you should agree on this, because you considered the opposite absurd)
From these two premises the particles' trajectories necessarily follow.
Andrei
Dear Andrei,
DeleteBohmian QM is nothing but the complex QM decomposed in its real and imaginary parts. Hence its predictions are the same as the predictions of standard QM. If you want another theory in which particles have well defined trajectories, but this theory is not Bohmian QM, then you violate in one form or another Bohmian's quantum potential. And then your predictions are different than what we observe in nature.
Quantum potential reintroduces the weirdness of QM into the theory. A classical theory of well defined trajectories for particles (without the quantum potential) is unable to explain the discreetness of the energy spectra. In such a world the atoms would not be stable.
Now why do I insists on Bohmian QM? Because I can show you the warts of this interpretation which may change your mind about the reality of the trajectory. In short my argument is this:
-trajectory is real, but no Bohmian QM? Then show me how you get discrete energy spectra.
-trajectory is real but in Bohmian QM? Then I have some "surreal" trajectories for you.
Florin
Dear Florin,
Delete" If you want another theory in which particles have well defined trajectories, but this theory is not Bohmian QM, then you violate in one form or another Bohmian's quantum potential."
I do not think that the quantum potential is a fact of nature. It is an implication of one theory. If Bohm's theory is wrong there is no quantum potential.
"A classical theory of well defined trajectories for particles (without the quantum potential) is unable to explain the discreetness of the energy spectra. In such a world the atoms would not be stable. "
Here is our main disagreement. I'd like you to back up this claim. Can you prove that no classical, yet to be discovered, theory could ever explain atomic stability and spectra? Here you have some articles that bring evidence to the contrary:
http://arxiv.org/pdf/1402.1423
http://arxiv.org/pdf/1307.6051
http://arxiv.org/pdf/1307.5986
Please do not assume that I am committed to the idea that Nature is a vibrating oil bath. This is just one counterexample to your above claim.
It seems to me that you are trying to shift the burden of proof here. It is not necessary for me to provide a classical theory explaining the atoms. It is you that needs to prove the impossibility of constructing such a theory.
I will restate my argument here:
1. Partcles are points
2. Particles exists in this universe at all times.
From these two premises the particles' trajectories necessarily follow.
In order to contradict my argument you need to show that atomic spectra/stability is incompatible with at least one of the above premises. If you cannot do that, my argument stands.
Andrei
Dear Andrei,
DeleteVery interesting arguments. Indeed I think that the burden of proof is on you, but this is now a moot point as it just happens that I have a proof of the uniqueness of classical and quantum mechanics:
http://arxiv.org/abs/1505.05577
If one wants for the laws of nature to be:
1) stable under time evolution
2) stable under physical system composition
then you have only 3 solutions possible:
- elliptic composition (quantum mechanics)
- parabolic composition (classical mechanics)
- hyperbolic composition (hyperbolic QM)
Condition 1 means that the laws of nature are the same today as they were yesterday and they will be tomorrow. In the infinitesimal case this leads to the Leibniz identity
Condition 2 means that if physical system A is described by an universal theory of nature T, system B is described by the same universal theory of nature T, then the composed system A+B is described by the same universal theory of nature T. In other words, there are no island universes with different laws of nature.
There is some uncertainty about the 3-rd option, hyperbolic QM. I am pondering if it is useful to describe the interior of black holes, or it is completely nonphysical, but this is work in progress.
So considering only QM and CM, now we are only talking about various mathematical representations. In CM there is no discreteness of spectra, only in QM (which comes from eigenvalue problems). That is why if you like particle trajectories then I insist on Bohmian's interpretation.
Now onto your 2 statements:
1. Particles are points
2. Particles exists in this universe at all times.
I would change 1 to:
1Prime) When detected particles are points
and so our disagreement is of countefactual nature: what are particles when we do not measure them? Welcome to the measurement problem. This has many solutions:
Bohmian: particles have well defined positions at all times
QBism, Copenhagen, positivism: questions about what cannot be measured are irrelevant
MWI: the appearance as points comes from the universe splitting into other universes
etc.
Florin
Dear Florin,
DeleteIn my previous post I have said:
"Can you prove that no classical, yet to be discovered, theory could ever explain atomic stability and spectra? Here you have some articles that bring evidence to the contrary:
http://arxiv.org/pdf/1402.1423
http://arxiv.org/pdf/1307.6051
http://arxiv.org/pdf/1307.5986
"
So, I am not proposing a theory that is not-classical and not-quantum, but a classical theory that is different from the classical electrodynamics.
You are saying that particles are points when detected. Right, but this is true for any scientific observation. A car is a car when detected, a planet is a planet when detected, and so on. It is logically possible that all those objects are different when not detected. But, if you make such a claim, you need to back it up somehow. You argued that classical point-particles are incompatible with atomic stability and discrete spectrum. However, this is only true for classical electrodynamics. You have no general proof that it is impossible to construct a different classical theory free of those problems. In the articles I have linked, you have an example of orbit quantification in a pure classical setup. So, let me clarify my argument:
P1. When detected particles are points
P2. There is no reason to assume that when not detected, the particles are not points
C1. Particles are points
P3. Particles exists in this universe at all times.
C2. Trajectories exist.
You cannot deny P2 just because classical electrodynamics doesn't work. Some other classical theory might work.
Now, I would like you to clearly state your position here. Do you embrace the QBist point that the nature of the particles is irrelevant?
Andrei
Dear Andrei,
DeleteLet me give you a simple example: when I look at a my computer screen I see continuous letters, but if I look at it with a powerful magnifying glass I see the individual pixels. It is a similar coarse graining which makes the particles appear to have continuous trajectories.
When I increase the magnification into the realm of quantum mechanics scales, a new physical phenomena starts to become apparent: superposition. The point of positivism is that of freeing the physical description from the dictates of classical physics in between measurements and allowing the new description of QM in Hilbert space to take hold. Your P2 is the problem. P2 is a rigid dictate of using classical physics at the expense of QM in Hilbert space. Then to restore agreement with experiment you have 2 options:
1) quantum potential in Bohmian QM
2) Wigner's functions for QM in phase space.
Form here you have to pick your poison: superluminal motion, or negative probabilities.
But both of them are equivalent with QM in Hilbert space and you did not really escape QM in Hilbert space.
Now on on other theory than classical electrodynamics, this is voodoo talk. Quantum field theory is one of the best theory of nature so far with impressive unmatched agreement between theory and experiment. I am not aware of any aspect of classical electrodynamics or QFT which is not well understood and verified experimentally.
Of course you may say that the burden of proof is on me. I don't agree, but I did gave you last time a very strong uniqueness argument. Your counter-example fails to prove your point, and here is why: I too have a similar example from fiber optics.Light propagation is described by Maxwell's equations. In the appropriate limits, those equations give rise to nonlinear Schrodinger equation, a nonlinear parabolic partial differential equation which has a very different character than the original Maxwell's equations. So how can I say then that I have a uniqueness proof then? The essential ingredient is in the assumptions. From my physical assumptions I derive QM and CM and the proof is mathematically iron-clad. You have to attack the assumptions to make the argument invalid. You have to make a case against the 2 physical principles: invariance under time evolution and invariance under composition, and that cannot be done. Sure, in certain limits you can get effective theories that may reproduce some QM behavior, but at core, nature can only be either quantum or classical mechanical and this is a mathematical statement coming from the 2 physical principles.
On QBism, I am in agreement with Asher Peres: Quantum phenomena happen in a lab and not in a Hilbert space. QM in Hilbert space is simply a description of real physical phenomena. I simply don't know where the electron is in between measurements. If you want to consider something other than position, like spin, then positivism is a much more natural position. And this is due to the K-S theorem. One of the faults (in my eyes) of Bohmian mechanics is the way they treat the reality of spin comparing with the reality of the position. To me this does not make any sense, but they are forced to do it like that to maintain agreement with QM.
Florin
Dear Florin,
Delete1. By classical theory I understand a theory where all the variables commute. Do you agree on this?
2. Do you claim that classical electrodynamics is the only possible classical theory of point particles?
I need to be sure your position on these points, and I will fully answer to all your arguments.
Andrei
Dear Andrei,
DeleteSorry for the delay, I was without internet connection this weekend.
On 1, this is not 100% correct. QM in phase space has position and momenta commuting and this is a counterexample. A classical theory is a theory for which composite system always factorizes.
2: Up to a certain energy scale quantum electrodynamics (https://en.wikipedia.org/wiki/Quantum_electrodynamics) is the correct theory of electromagnetism. In the appropriate limit this gives rise to classical electrodynamics. Correct means agreement with nature and mathematical correctness. For higher energies one has electroweak theory which is also correct. Higher still one would expect a grand unified theory and later on string theory.
Florin
Dear Florin,
DeleteCurrently I am on holiday, I am traveling a lot and my access to internet is sparse. So, please excuse my delayed answer.
Regarding your definition of classical theories, is not factorisability of a composite system a condition for locality more than for classicality?
Wouldn't it be better to say that a classical theory is a theory of local beables? Classical non-local theories like Newtonian gravity or electrostatics also fit in this definition.
I still do not understand your position on the second point (uniqueness of classical electrodynamics as a point-particle theory). Let me try to clarify the reason I claim the burden of proof is on you.
1. Given the fact that we detect particles as points, the most parsimonious assumption is that they remain points all along. To say that they become something else when not observed looks like a conspiracy theory to me. You cannot present any direct evidence for this mysterious transformation (because such an experiment would presumably force the particles take their point-state), you do not seem to have any physical mechanism in mind for how this change may occur, not the slightest idea about what these point-particles might change into. So you need to present some good reasons for believing this. As far as I can understand, your argument is that a point-particle theory cannot explain stable orbits or energy quantification. However, this is only true in classical electrodynamics. This is not a generic mathematical proof. Different classical theories, like Newtonian gravity, allow for stable orbits. You have also not presented evidence that the concept of energy quantification is mathematically incompatible with any theory of point-particles. So, at this point your argument is a straw man. You have found a theory that supports your view (classical electrodynamics) and then have tried to generalize this conclusion for any theory of point particles. Needless to say, this is a fallacious argument.
I do not see the relevance of your argument proving that a theory can only be classical or quantum. Assuming we reach an agreement on what we mean by classical, I suppose the point-particles can be accommodated by such a theory. No need for a different type of theory.
Andrei
Dear Andrei,
DeleteSorry for the delay, I was very busy. I will try to reply during the weekend.
Best,
Florin
Dear Andrei,
DeleteI am not sure what a theory of beables is. Bell started this project but his unfortunate death left the idea in limbo. The unfinished theory makes no mathematical sense. If you mean by beables a generic ontic theory, I am OK with that.
>Regarding your definition of classical theories, is not factorisability of a composite system a condition for locality more than for classicality?
Looks like it, but it is not, (see "Is Nature is Local or Nonlocal?"). I have to go into too much technical details to explain it rigorously.
>uniqueness of classical electrodynamics
When quantum effects are irrelevant for classical electromagnetism, Maxwell's equations work perfectly. If you change them you get predictions which are not in agreement with experiments.
>Given the fact that we detect particles as points, the most parsimonious assumption is that they remain points all along. To say that they become something else when not observed looks like a conspiracy theory to me.
I agree, it is a very natural assumption, and it looks like a conspiracy, but is it true? Nature does not exist to make us happy.
>You cannot present any direct evidence for this mysterious transformation (because such an experiment would presumably force the particles take their point-state), you do not seem to have any physical mechanism in mind for how this change may occur, not the slightest idea about what these point-particles might change into.
The problem is stated backwards. If QM is correct one needs to explain individual outcomes. This is the famous measurement problem. And I do have a concrete physical mechanism which I will attempt to publish. Also I will have experimental predictions as well. I am building up a series of blog posts to present my proposal.
>As far as I can understand, your argument is that a point-particle theory cannot explain stable orbits or energy quantification.
This is only one of many arguments I can present. Here is another one: quantum tunneling: according to classical physics if the particle have a well defined path, tunneling is simply forbidden.
>Different classical theories, like Newtonian gravity, allow for stable orbits.
This is because there is instataneous action at a distance in Newtonian gravity and hence there are no gravity waves in that theory. Any theory which obeys special relativity demands the change in fields to propagate with finite speed and in turn implies that energy is lost through waves.
>You have also not presented evidence that the concept of energy quantification is mathematically incompatible with any theory of point-particles.
Indeed, because Bohmian QM uses point particles.But Bohmian QM is still QM. There ca only be QM or classical mechanics. And in classical mechanics there is no energy quantification.
>I do not see the relevance of your argument proving that a theory can only be classical or quantum.
It's very relevant: both of them support point particles. If classical mechanics is your point particle theory, then you run afoul of experiments. If on the other hand you prefer point particles in QM setting, then you have a choice: pick Bohmian QM, or pick QM in phase space. Both of them have ugly features and they will challenge your intuition.
Florin
Dear Florin,
DeleteYes, a theory of beables is a generic ontic theory.
"When quantum effects are irrelevant for classical electromagnetism, Maxwell's equations work perfectly. If you change them you get predictions which are not in agreement with experiments."
If this argument were correct, improving physical theories would always be impossible. Newtonian gravity is pretty accurate for low mass density systems, but this did not stop Einstein to find a better classical theory, general relativity. You have provided no reason why this should be impossible with electromagnetism.
"This is because there is instataneous action at a distance in Newtonian gravity and hence there are no gravity waves in that theory. Any theory which obeys special relativity demands the change in fields to propagate with finite speed and in turn implies that energy is lost through waves."
Is there a generic proof that this is true for any field geometry? For general relativity there are mass distributions for which no energy loss exists. Even for our Solar system the loss is virtually nonexistent.
"in classical mechanics there is no energy quantification."
Again, what proof do you have for this assertion? Show me that no classical theory can have energy quantification. Agreed, no such theory is known, but why believe that this is impossible?
"The problem is stated backwards. If QM is correct one needs to explain individual outcomes."
QM is a statistical theory. Explaining individual outcomes requires an exact theory of particles (or whatever entities might exist). I'm looking forward to see your proposal.
"This is only one of many arguments I can present. Here is another one: quantum tunneling: according to classical physics if the particle have a well defined path, tunneling is simply forbidden."
What is known in a tunneling experiment is just the average value of the potential barrier. There is some probability that in certain places at certain times the potential can be much lower, so that the particle can pass with a lower energy than the one required. Just like in the case of the double slit experiment one must analyze the tunneling in terms of an interaction between the incoming particle and the particles that are responsible for the barrier.
"If classical mechanics is your point particle theory, then you run afoul of experiments."
Again, this is yet an unsupported assertion.
Andrei
Dear Andrei,
DeleteSorry for the delay, first I was busy preparing my next post than I was traveling.
> If this argument were correct, improving physical theories would always be impossible. Newtonian gravity is pretty accurate for low mass density systems, but this did not stop Einstein to find a better classical theory, general relativity. You have provided no reason why this should be impossible with electromagnetism.
Indeed, any physical theory is defined within a range of validity and electromagnetism is no exception. QED supersedes electromagnetism, electroweak theory supersedes QED, etc. However I think this misses my point. You asked for a theory different than electromagnetism for effects *in the range of validity of electromagnetism* which would predict different things than electromagnetism, and any such theory would contradict the experiments.
>Is there a generic proof that this is true for any field geometry?
I am not sure what you mean by “any field geometry”. Is it any field theory?
>For general relativity there are mass distributions for which no energy loss exists.
I am not a general relativity expert, but I am not aware of any stationary solutions. If you know one, please point it to me. Einstein once thought incorrectly that by adding the cosmological constant he would be able to produce a stationary solution.
>Even for our Solar system the loss is virtually nonexistent.
This is irrelevant: electromagnetism is stronger than gravity by many many orders of magnitude (by 37 orders of magnitude!!!) and the energy loss is hugely magnified.
In general any classical field theory with (1) no instantaneous action at a distance, and (2) infinite range experiences waves which carry away energy.
>"in classical mechanics there is no energy quantification."
Again, what proof do you have for this assertion? Show me that no classical theory can have energy quantification. Agreed, no such theory is known, but why believe that this is impossible?
Again we have a disagreement on the burden of proof. But I think it might be possible to produce such a proof in classical mechanics (although I would find that totally uninteresting). Classical mechanics has not only the usual phase space formulation, but also a much less known Hilbert space formulation. And we know why there are discrete spectra in Hilbert spaces in QM: they come from spectral theory. All we would need to do is investigate the spectral theory of the Hilbert space formulation of classical mechanics.
>"The problem is stated backwards. If QM is correct one needs to explain individual outcomes."
QM is a statistical theory. Explaining individual outcomes requires an exact theory of particles (or whatever entities might exist). I'm looking forward to see your proposal.
Please keep reading my blog, I will slowly present my approach there.
>What is known in a tunneling experiment is just the average value of the potential barrier. There is some probability that in certain places at certain times the potential can be much lower, so that the particle can pass with a lower energy than the one required. Just like in the case of the double slit experiment one must analyze the tunneling in terms of an interaction between the incoming particle and the particles that are responsible for the barrier.
Nice try but I am not buying it. Take the example of nucleus radioactive decay by quantum tunneling. From scattering experiments we know the form of the particle forces, and we also know the dimension of the nucleus. Even under the worst case scenario in terms of particle distribution, in the nucleus the height of the barrier is extremely high But QM can tunnel even through infinitely high potential barriers provided the barrier is now infinitely wide. Moreover the decay rate matches the QM computation precisely.
Florin
Dear Florin,
DeleteLet me clarify the burden of proof issue.
You've said:
"If classical mechanics is your point particle theory, then you run afoul of experiments"
and previously:
"A classical theory is a theory for which composite system always factorizes."
If you still agree with the above statements, it necessarily follows that you need to prove the following statement:
Any particle theory for which composite system always factorizes contradicts experimental evidence.
I am still waiting for that proof.
If by "any particle theory" you mean known, already falsified classical theories, like Maxwell's electromagnetic theory then we are in agreement, but this is not the usual understanding of the word "any".
As far as I know, the only generic argument used to exclude classical theories comes from Bell's theorem and its variants (Free-will theorem, etc). All of them assume indeterminism or at least non-correlation between distant systems, assumptions that are false for classical deterministic theories. Once this argument is put to rest, the possibility of reproducing QM by means of a classical deterministic hidden-variable theory is very real. Until either such a theory is found or a sound proof of non-existence is put forward, it is difficult to know the truth. Such a theory would explain all quantum phenomena including atomic stability, tunneling, energy quantification, etc.
Now, let me answer in more detail to your points.
" You asked for a theory different than electromagnetism for effects *in the range of validity of electromagnetism* which would predict different things than electromagnetism, and any such theory would contradict the experiments. "
Obviously, atoms are not in the range of validity of classical electromagnetism, because this theory fails to explain them, right? But anyway, you are wrong. General relativity can be successfully applied in those situations that are in the range of validity of Newtonian gravity. The only condition required for replacing an old theory is that the new theory also reproduces the correct predictions of the old one. One can for example modify the formula for the electric field at the atomic scale but keep it in the macroscopic limit.
"I am not a general relativity expert, but I am not aware of any stationary solutions. If you know one, please point it to me. "
In GR energy is lost only when a change in the quadrupole moment occurs, unlike electromagnetism where radiation is produced by a change in the dipole moment. This implies that uniformly accelerated particles do not radiate in GR. This is the reason we do not have unstable planetary orbits (a small planet around a much bigger star has an almost constant acceleration).
I think that the argument can go further. In a deterministic theory, it is in principle possible to know the position of the particles in advance. So, you can make the particles accelerate towards the instantaneous position of the other particles without resorting to non-locality. So, at least in principle, classical deterministic theories with perfectly stable orbits should be possible.
"Classical mechanics has not only the usual phase space formulation, but also a much less known Hilbert space formulation. All we would need to do is investigate the spectral theory of the Hilbert space formulation of classical mechanics."
How would you write down the Hamiltonian for any possible classical theory? How would you calculate the potential energy for a generic classical theory?
Tunneling always occurs in a system of many particles. Due to their mutual interaction, enough energy can be transferred to one of them to make the "jump". An asteroid doesn't have enough energy to leave the solar system, but a passing planet can give it a kick.
In the case of the alpha decay we don't have a classical theory describing it so the actual forces involved are not known. The measured forces are probably only statistically averages.
Andrei
Dear Andrei,
DeleteSorry for the delay, here is my reply in 2 parts due to size limitation
“Any particle theory for which composite system always factorizes contradicts experimental evidence. I am still waiting for that proof.”
The proof is in two parts:
1) Factorization implies classical mechanics. The sketch of the proof can be found in http://arxiv.org/pdf/1505.05577v1.pdf What you get are Poisson manifolds but the paper is not expanded in the CM direction but in the QM area.
2) Classical mechanics cannot explain superposition in a non-contextual manner. CM obeys Bell’s inequalities and recently the first loophole free experiment was performed certifying the violation of Bell inequalities. Even if particle trajectories are retained as in Bohmian QM, factorizability is stilled violated.
>As far as I know, the only generic argument used to exclude classical theories comes from Bell's theorem and its variants (Free-will theorem, etc).
K-S theorem does the trick too in my opinion.
>All of them assume indeterminism or at least non-correlation between distant systems, assumptions that are false for classical deterministic theories. Once this argument is put to rest, the possibility of reproducing QM by means of a classical deterministic hidden-variable theory is very real. Until either such a theory is found or a sound proof of non-existence is put forward, it is difficult to know the truth. Such a theory would explain all quantum phenomena including atomic stability, tunneling, energy quantification, etc.
Easier said than done. ‘t Hooft is pushing some attempts particular using cellular automata (with some motivation from quantum gravity) but this has yet to produce believable models. Superdeterminism bypasses Bell’s theorem.
>Obviously, atoms are not in the range of validity of classical electromagnetism, because this theory fails to explain them, right? But anyway, you are wrong. General relativity can be successfully applied in those situations that are in the range of validity of Newtonian gravity. The only condition required for replacing an old theory is that the new theory also reproduces the correct predictions of the old one. One can for example modify the formula for the electric field at the atomic scale but keep it in the macroscopic limit.
Here I disagree as the situation is subtler. Yes, classical electromagnetism does not apply to the atom, but the standard repulsive force between 2 electrons is valid at distances much shorter than the atomic distances. Also we know the correct theory to a wider range of (smaller) distances, it is QED and there is no mystery at the atomic level.
Part 2 of 2:
Delete>In GR energy is lost only when a change in the quadrupole moment occurs, unlike electromagnetism where radiation is produced by a change in the dipole moment. This implies that uniformly accelerated particles do not radiate in GR. This is the reason we do not have unstable planetary orbits (a small planet around a much bigger star has an almost constant acceleration).
Locally, in GR one has the equivalence principle, but this cannot be extended to large areas and there you have Ricci and Weyl curvature if I recall correctly. I don’t think your argument applies to the stability of the planetary motion: wasn’t a Nobel prize awarded for studying decaying orbits of binary stars due to gravitational waves? I think it is the smallness of the gravitational interaction which explains the stability of planetary motion. A similar infinitesimal effect is the loss of all planetary atmospheres since technically an atmosphere is not thermodynamically in equilibrium around a planet and energetically enough molecules do escape in outer space. Still the Earth is in no imminent danger of loosing its air, and its orbit does not significantly decay around the Sun either.
>How would you write down the Hamiltonian for any possible classical theory?
All you need is the actual Hamiltonian.
>How would you calculate the potential energy for a generic classical theory?
Tunneling always occurs in a system of many particles. Due to their mutual interaction, enough energy can be transferred to one of them to make the "jump".
This is handwaving. In QM tunneling works for any finite width barrier even if the potential energy of the barrier is infinite. In this case no amount of a kick from nearby particles can push it over the potential barrier. Sure, we are talking of physical systems where we deal only with finite quantities, but QM predicts correctly the tunneling probability, so why bother with convoluted alternative explanations?
>In the case of the alpha decay we don't have a classical theory describing it so the actual forces involved are not known. The measured forces are probably only statistically averages.
Not true: the classical analog of the Yang Mills SU(3) gauge theory corresponding to the strong force is relatively straightforward to obtain, and it is a simple generalization of electromagnetism. Here I worked it out for SU(2): http://fmoldove.blogspot.com/2014/08/what-if-electromagnetism-were-su2-yang.html The generalization to SU(3) is trivial: just replace the F_1, F_2, F_3 SU(2) generators with the generators of SU(3).
Florin
Dear Florin,
DeleteI think I should start with this:
„QM predicts correctly the tunneling probability, so why bother with convoluted alternative explanations?”
As you have said, QM only predicts a probability. It does not predict the actual observed experimental results, nor does it provide an explanation of them. This is enough reason to search for an exact theory of particle motion.
The only reason to not pursue this search would be to prove that such a theory cannot be found which brings us again to the no-go theorems (Bell, K-S, Free-will).
„Superdeterminism bypasses Bell’s theorem.”
Yes, but the main point I want to stress here is that all classical field theories are superdeterministic. So, if we ignore non-contextual theories like the mechanics of the rigid body, which nobody would propose as fundamental theories anyway, we can say that all no-go theorems are pretty much irrelevant. All promising classical theories (classical field theories) are immune, by virtue of being contextual/superdeterministic.
This means that your 2-steps proof:
„1) Factorization implies classical mechanics. The sketch of the proof can be found in http://arxiv.org/pdf/1505.05577v1.pdf What you get are Poisson manifolds but the paper is not expanded in the CM direction but in the QM area.
2) Classical mechanics cannot explain superposition in a non-contextual manner. CM obeys Bell’s inequalities and recently the first loophole free experiment was performed certifying the violation of Bell inequalities. Even if particle trajectories are retained as in Bohmian QM, factorizability is stilled violated.„
fails for all classical field theories.
Now, let me provide some counterexamples to the existence af a generic proof against classical stable atoms or classical energy quantification.
Assume the electric field changes direction at certain distances around the nucleus. The electrons cannot fall into the nucleus because the attractive force becomes repulsive. They also cannot occupy any position because of the same reason. Such an atom would resemble an ionic crystal, and would certainly be stable, with discrete energy levels.
End of part 1
Part 2
DeleteIf you want moving electrons, you need to remember that a deterministic local theory can be made to behave just like a non-local one. You can calculate the future position of all particles, include this calculation in the laws of motion and make the particles accelerate to the instantaneous position of the other particles. Just like in Newtonian gravity, you would have no energy loss.
Such a (approximate) mechanism is involved in GR:
http://arxiv.org/ftp/physics/papers/9910/9910050.pdf
http://arxiv.org/pdf/gr-qc/9909087.pdf
The conclusion of the first paper reads:
„Van Flandern is correct in his observation that gravitational attraction is directed towards the instantaneous (unretarded) position of a moving body. We have shown that this fact can be explained without a revision of physics to include superluminal propagation. „
In the abstract of the second you can find:
„The observed absence of gravitational aberration requires that “Newtonian” gravity propagate at a speed cg > 2*10^10c. By evaluating the gravitational effect of an accelerating mass, I show that aberration in general relativity is almost exactly canceled by velocity-dependent interactions, permitting cg=c. This cancellation is dictated by conservation laws and the quadrupole nature of gravitational radiation.
Now, about tunneling. I accept QM’s prediction of tunneling for infinite potentials. Nevertheless I would ask you to provide an example of an such an experiment (very high potential, not infinite, of course). I need to see the details in order to provide an answer.
In regards to the proof against classical physics, based on alpha decay you say:
„the classical analog of the Yang Mills SU(3) gauge theory corresponding to the strong force is relatively straightforward to obtain, and it is a simple generalization of electromagnetism. Here I worked it out for SU(2): http://fmoldove.blogspot.com/2014/08/what-if-electromagnetism-were-su2-yang.html The generalization to SU(3) is trivial: just replace the F_1, F_2, F_3 SU(2) generators with the generators of SU(3).”
So the argument is:
1. You propose a classical model for the weak force.
2. The model is found to be wrong (does not explain tunneling).
3. You conclude that the classical physics cannot explain alpha-decay.
My answer is that you need to search for a better classical model.
Andrei
Dear Andrei,
DeleteAlthough I do not agree, I respect your points of view because they are carefully argued (I get the distinct sensation you are a professional physicist). So I have a proposal: would you be interested in a guest post at this blog arguing your position? I have counter-arguments to your positions above, but I think this debate deserves better visibility instead of being buried in the commenting section of an old post.
If you want, you can contact me privately at fmoldove@gmail.com if you have any questions/concerns about a guest post.
If you are not interested in a guest post, I'll reply to your points above after I get your answer.
Best,
Florin
Dear Florin,
DeleteSorry to disappoint you, but I am not a professional physicist. I did study some physics, including QM as part of my chemistry degree.
I agree to write a guest post, but tell me please how to post it.
It is going to take me 2-3 days as I have to refine my text.
Thank you,
Andrei
Dear Andrei,
DeleteYou simply email your text to me and I'll upload it. You can write equations in Latex, and I'll convert them for the post. Also specify the links and pictures you want to include.
I am thinking to do this as a debate: you write your best points in your post and I'll reply to them in a subsequent post. I think this format would generate more interest and then readers could then go back and compare the two posts.
Best,
Florin
Quantum logic was not demonstrated to be wrong, but Bohr and others dismissed it as unnecessary. Bohr had an insistence that physics must be discussed in ordinary language. This in part went with his Copenhagen Interpretation, which means that quantum mechanics is understood from a classical perspective. Quantum logic though has found a small niche in quantum physics.
ReplyDeleteLubos' argument might be compared to the argument that quantum physics is not really complex valued because one can perform numerical computations on a computer that works with real 0s and 1s. Complex numbers or variables can be easily written as a pair of reals with special multiplication rules and conjugation rules. As such we can consider quantum physics as not at all complex valued. In some sense we can interpret complex numbers as just pairs of reals, quaternions as pairs of complex numbers and octonions … .
One can well enough work with quantum mechanics without ever concerns that Boolean logic does not apply to quantum propositions. On the other hand if one wants to look into some subtle matters of locality and nonlocality, which Lubos seems to also reject, quantum logic can be a tool in your toolbox. The structure of C* algebras with meets and joins is also quantum logical.
LC
I agree.
DeleteThe challenge for me in writing the post was to make it understandable to as large an audience as possible. This is why I came up with the exercises to keep the reader engaged through the dull abstractions.
Florin,
ReplyDeleteI am reading with lot of interest about this controversy on quantum logic. Once before, I had heard in a lecture by an “important man (!!!) “ that contrary to what many people believe, QM follows Boolean logic. Whatever that may be, my knowledge of formal logic is close to zero. So I cannot really participate in the debate. I will let scholars debate it out. I suppose it may be like and closely related to the 90 year old debate on interpretation on QM. (BTW, what is your favorite interpretation? Have you written about it?). I would say however that in my opinion controversies are good for science (hopefully civil!!). That is how science advances. It was good to hear about debate on locality and now I understand it is mostly semantics, an unfortunate wording by Bell. I sometime participate in blog comments on science and religion. I have used the word *illogical* for QM, explaining that I am using the word “logic” in the sense used in our everyday life to take various decisions. But perhaps I should be more careful and stop at saying that QM is *counterintuitive* as compared with our everyday life which is strictly classical.
Kashyap,
DeleteMy disagreement with Lubos is not about the physics, and in fact I agree with his reply blog post at TRF. My disagreement with him is about him overreaching into the math formalism. Just because you can construct sentences about tachyons in string theory this does not make string theory invalid. Just because you can construct nonphysical quantum logic sentences this does not mean quantum logic is junk. If it is junk and you need to throw it away, you just killed the Hilbert space formalism. This was the main lesson from Constantin Pirion. And that is why I bolded this sentence:
"the implication that in quantum mechanics Boolean logic always applies amounts to denying the applicability of Hilbert spaces!!!"
Lubos was not aware of entangelment swapping and I can bet good money he was not aware of Piron's result. I say this because he seemed to have latched onto Bohr's and other founding fathers early positions when there was genuine confusion about quantum logic.
In terms of QM interpretations, I resonate most with the intuitions of Bohr, Peres, Zurek, and Belavkin, and I am building up my own interpretation. In fact my approach can be traced directly to Bohr: I am expanding on an approach suggested by Bohr himself. This interpretation is closest to qbism, but the main difference is in how I attempt to solve the measurement problem. This is all work in progress and I am in the process of publishing the key results needed to support the interpretation. If you follow my blog you will learn all about my interpretation.
QM is genuinely counterintuitive because it allows you to do things which seem to contradict common sense (and people are now making money in commercial applications using counterintuitive quantum effects). But this is not the fault of QM, it the the fault of biological evolution for favoring classical intuition to secure survival. Quantum teleportation was not needed by the lion to eat his prey.
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