Monday, January 16, 2017

Book Review: "Making Sense of Quantum Mechanics"

After much delay I had found the time to finish reading Jean Bricmont's "Making Sense of Quantum Mechanics" book.

The book is the best presentation of Bohmian interpretation I have ever read. It masterly combines the philosophical ideas with a bit of math, famous quotes, and some historical perspective. 

After preliminary topics in chapter one, chapter two discusses the first quantum "mystery": superposition, while chapter four discuses the second "mystery": nonlocality.  It was chapter three, a philosophical "intermetzzo" which took me a very loooong time to read and prevented me to write this review much sooner: one one end I could not write this post without reading it, and on the other end I was loosing interest very quickly in it after a couple of pages of historical review. Then Bricmont proceeds into presenting Bohniam mechanics - the heart of the book. 

Let's dig a bit deeper into it. Chapter two is a very well written introduction into why quantum mechanics is counter-intuitive. This is presented in the style of modern quantum foundation undergrad classes. Chapter four main idea is this: to many physicists Bell's result proved the impossibility of non-contextual hidden variables (or local realism) while Bell should be understood in conjunction with EPR: EPR+Bell = nonlocality. But what does nonlocality mean? Is it just higher than expected correlations? Here Bricmont makes a very bold and provocative claim: 

"the conclusion of his [Bell's] argument, combined with the EPR argument is rather that there are nonlocal physical effects (and not just correlations between events) in Nature."

To support this chapter 4.2 discusses "Einstein's boxes" [I had a series of posts discussing why in my opinion they do not represent an argument for nonlocality. What EPR+Bell shows is that the composition of two physical systems into a larger physical system does not respect the rules of classical physics - parabolic composability but new rules - elliptic composability. Nature is not "nonlocal" but "non-parabolic composable"]. 

Onto the main topic, the presentation of Bohmian mechanics is standard and what it is surprising is the degree on which the underdetermination issue is addressed: there are an infinite number of alternative theories (like Nelson's stochastic theory) which are in the same realistic vein and which make the same predictions as Bohmian mechanics. Chapter three discussion is invoked here but I feel the argument is very weak (not even a handwaving).

Then the book talks about alternative approaches to Bohmian mechanics courageously taking (some well deserved some not) shots at alternative interpretations (like GRW, MWI, CH, Qbism), and wraps up with historical topics and sociological arguments.

Now onto what the book covers poorly: surreal trajectories, and quantum field theory in Bohmian mechanics. Surreal trajectories are mentioned in passing in a quote while they are the major objection to the interpretation. As I said before, the very name "surreal trajectory" was a masterful catchy clever title for a paper but it backfired in the long term because it was attaching a stigma to Bohmian mechanics which in turned allowed Bohmian supporters to summarily and unfairly dismiss the argument. I will revisit the argument in next post. The key point of surreal trajectories paper is that the particle is detected where Bohmian mechanics predicts it must not go, and since the only thing "real" in Bohmian mechanics is the position of the particle, it represents a fatal blow to the Bohmian ontology. Currently, to my knowledge, there is no consensus inside the Bohmian community on the proper answer the surreal trajectory paper: some deny it is a problem at all while others acknowledge the problem and propose (faulty) ideas on how to deal with it. This is similar with the situation inside the MWI camp where the big pink elephant in the room there is the notion of probabilities: some in MWI disagree it is an issue while others attempt to solve it (but fail). 

Quantum field theory in Bohmian mechanics is another sore point which is not properly discussed. My take on the topic is that a Bohmian quantum field theory is impossible to be constructed, and I want to be proven wrong by a consistent proposal: show me the money, show me the archive paper where the problem is comprehensibly solved.

Bad points aside, overall I liked the book, I find it stimulating, and I enjoyed very much reading it (except chapter 3 which invariably succeeded putting me to sleep). The book is a must read for any person seriously interested in the foundations of quantum mechanics.

Monday, January 9, 2017

(The nonsense of) Joy Christian Reloaded

I was preparing the first physics posts of the year when I got some comments and a question on Joy Christian on an old blog post. In the words of late Yogi Berra, this is "deja vu all over again". Probably the best description of Joy Christian is given by the Monty Python: The Dead Parrot sketch:

The question I got is the following:

"I would like to understand whether the equations (67) - (75) in Joy Christian’s paper “Local Causality in a Friedmann-Robertson-Walker Spacetime” make any sense at all. I don't understand how the mathematical limes operation are carried out."

The paper which got past the referees by trickery is on the archive: and there you see the full derivation of the main faulty claim. Minus some obfuscation techniques, Eqs. 67-75 are nothing but the one-pager Joy preprint:

The main hand-waving trick in the "derivation" is a conversion inside of a sum of \(\lambda^k\) from a variable into an index which amounts to adding apples to oranges and obtaining the incorrect result (see the bottom of page 8 on my preprint: 

The mistake happens on the transition from Eq 73 to Eq 74 because the L's belong to two distinct kinds of algebras: let's call them apples and oranges. Ignoring the axb, the troubled sum term is something like this:

\(L(\lambda^1) + L(\lambda^2)+L(\lambda^3)+L(\lambda^4)+L(\lambda^5)+...=\)
apple_1 + apple_2 + orange_3 +apple_4 + orange_5+...

with \(\lambda^1 = +1, \lambda^2\ = +1, \lambda^3 = -1, \lambda^4 = +1, \lambda^5 = -1...\)

and with the transformation rule: "apple = - oranges" when we convert to objects of the same kind (let's pick apples) we get:

\(apple(\lambda^1) + apple(\lambda^2)-apple(\lambda^3)+apple(\lambda^4)-apple(\lambda^5)+...=\)
\(apple(+1) + apple(+1)-apple(-1)+apple(+1)-apple(-1)+...=\)
\(apple+ apple+apple+apple+apple+...=\)

which no longer vanishes.

The preparation for this trick is on Eq. 49 which encodes the two distinct algebras (of apples and oranges) into a common formula.  In my preprint you can double check this by trying out the matrix representations of the two algebras (eqs 53-56).

Hopefully my explanation is clear enough. I know all Joy's mathematical tricks in all of his papers or in his blog debates, but I ran out of energy debunking his nonsense. Kudos to Richard Gill for pursuing this further. I was aware of the "Causality in a Friedmann-Robertson-Walker Spacetime" paper and it was on my to do list to write a rebuttal to it, but the journal withdrew it before I could get to it.

Sunday, December 18, 2016

Is the Decoherent Histories Approach Consistent?

One particular approach of interpreting quantum mechanics is Decoherent Histories. All major non-Copenhagen approaches have serious issues: 
- MWI has the issue of the very meaning of probability and without a non-circular derivation (impossible in my opinion) of Born rule does not qualify for anything but a "work in progress" status.
-GRW-type theories make different predictions than quantum mechanics which soon will be confirmed or rejected by ongoing experiments. (My bet is on rejection since later GRW versions tuned their free parameters to avoid collision with known experimental facts instead of making a falsifiable prediction) 
-Bohmian approach has issues with "surreal trajectories" which invalidates their only hard ontic claim: the position of the particle.

Now onto Decoherent Histories. I did not closely follow this approach and I cannot state for sure if there are genuine issues here, but I can present the debate. On one hand, Robert Griffiths states: 

"What is different is that by employing suitable families of histories one can show that measurement actually measure something that is there, rather than producing a mysterious collapse of the wave function"

On the other hand he states:

"Any description of the properties of an isolated physical system must consists of propositions belonging together to a common consistent logic" - in other words he introduces contextuality.

Critics of decoherent (or consistent) histories use examples which are locally consistent but globally inconsistent to criticize the interpretation.

Here is an example by Goldstein (other examples are known). The example can be found in Bricmont's recent book: Making Sense of Quantum Mechanics on page 231. Consider two particles and two basis for a two-dimensional spin base \((|e_1\rangle\, |e_2\rangle), (|f_1\rangle, |f_2\rangle))\) and consider the following state:

\(|\Psi\rangle = a |e_1\rangle|f_2\rangle + a|e_2\rangle|f_1\rangle - b |e_1\rangle|f_1\rangle\)

Then consider four measurements A, B, C, D corresponding to projectors on four vectors, respectively: \(|h\rangle, |g\rangle, |e_2\rangle, |f_2\rangle\) where:

\(|g\rangle = c|e_1\rangle + d|e_2\rangle\)
\(|h\rangle = c|f_1\rangle + d|f_2\rangle\)

Then we have the following properties:

(1) A and C can be measured simultaneously, and if A=1 then C=1 
(2) B and D can be measured simultaneously, and if B=1 then D=1
(3) C and D can be measured simultaneously, but we never get both C and D = 1
(4) A and B can be measured simultaneously, and sometimes we get both A and B = 1

However all 4 statements cannot be true at the same time: when A=B=1 as in (4) then by (1) and (2) C=D=1 and this contradicts (3).

So what is going on here? The mathematical formalism of decoherent histories is correct as they predict nothing different than standard quantum mechanics. The interpretation assigns probabilities to events weather we observe them or not, but does it only after taking into account contextuality. Is this a mortal sin of the approach? Nature is contextual and I don't get the point of the criticism. The interpretation would be incorrect if it does not take into account contextuality. Again, I am not an expert of this approach and I cannot offer a definite conclusion, but to state my bias I like the approach and my gut feeling is that the criticism is without merit.
PS: I'll be going on vacation soon and my next post will be delayed: I will skip a week.

Sunday, December 11, 2016

Measurement = collapse + irreversibility

I got a lot of feedback from last 2 posts and I need to continue the discussion. Even Lubos with his closed mind unable to comprehend anything different than the textbooks from 50 years ago and his combative style said something worth discussing.

But first let me thank Cristi for the picture below which will help clarify what I am trying to state. Let me quickly explain it: the interferometer arms are like the two sides of Einstein's box and once the particle was launched -for the duration of the flight- you can close the input and output of the interferometer, open the exit just in time and still have the interference. So this seems to contradict my prediction. But does it?

This time I do need to dig a bit deeper into the mathematical formalism. First, the role of the observer is paramount: no observer=no measurement. Second, the observer is described by quantum mechanics as well: there is the wavefunction of the quantum system, and there is the wavefunction of the observer. Now here is the new part: while we can combine the quantum system and the observer by tensor product and do the usual discussion of how unitary evolution does not predict an unique outcome, we need to combine the quantum system and the observer using the Cartesian product. This is something new, not present in standard quantum mechanics textbooks. However this follows naturally from the category theory derivation of quantum mechanics from first principles. There are equivalent Cartesian products corresponding to potential measurement outcomes:

\((|collapsed ~1 \rangle, | observer~ see~1\rangle ) \equiv ( |collapsed~2\rangle, | observer~see~2\rangle)\) 

This equivalence exists in a precise mathematical sense and respects the three properties of the equivalence relationship: reflexivity, symmetry, and transitivity. Break the Cartesian pair equivalence by any mechanism possible and you get the collapse of the wavefunction. 

Closing the interferometer, or cutting Einstein's box in half kills the equivalence and the wavefunction collapses. However while the particle is still in flight the process is reversible!!! Open the interferometer's exits in time and you restore the equivalence, undo the collapse still get the interference (Han Solo kills the stormtrooper 100% of the time). 

However there is a way to make the collapse permanent: just wait enough time with the ends closed such that the energy-time uncertainty relation allows you to reduce the energy uncertainty to the point that you can detect the particle inside by weighing the half-boxes or the arms of the interferometer. Suppose the ends of the interferometer is made out of perfect mirrors. If you wait long enough (even though you are not physically weighing anything) and then reopen the exits will result in loss of interference: this is my prediction.

But what happens if you only wait a little bit of time and you are in between full interference and no interference? You get a weak measurement.

Now let me discuss Lubos objection and then come back to the nonlocality point Bricmont was making.

First Lubos was stating: "If you place some objects (a wall) at places where a particle is certain not to be located, the effect on the particle's future behavior is obviously non-existent". The objection is vacuous. Obviously I don't disagree with the statement, but his entire line of argument is incorrect because collapse is not a dynamic process. If collapse would have had a dynamic origin then we would have had a unitary explanation for it we would have had to talk about the "propagation" of collapse. What the Cartesian pair mathematical framework does is first getting the rid of the consciousness factor, and second clarifying the precise mathematical framework of how the observer should be treated within the formalism. Contextuality is paramount in quantum mechanics and cutting the box changes the context of the experiment.

Now onto Bricmont argument. Andrei stated in his comments: " I still do not see the relevance of all this in regards to the locality dilemma.". It has deep relevance as I will explain. And by the way, the rest of Andrei's comments were simply not worth answering-nothing personal: I don't have the luxury of enough free time to answer each and every comment.

Bricmont's point on Einstein's boxes was this: "either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete "

Let's discuss the two options:
  1. opening B1 changes the situation at B2
  2. or the particle was in B2 all along and quantum mechanics is incomplete
Option one is clearly not the case. Wait long enough and the interference will no longer happen. At that point the particle IS in either B1 or B2 and shipping one box far away changes nothing. But how about option 2? Is quantum mechanics incomplete? Bohmian supporters think so because they augment the wavefunction with a hidden variable: the particle's initial position. Do we actually need this initial condition to make predictions? Not at all. Last thing to consider: was the particle in say B2 all along? If yes, there is no interference because the which way information. What about weak measurements? This is a case of even more examples of "surrealistic trajectories": combine two interferometers and you can obtain disjoint paths!!! The only thing which makes sense is that the particle does not have a well defined trajectory.

My question to Bohmian interpretation supporters is as follows: In the above picture close the arms long enough. What happens to the quantum potential? Does it dissipate? If yes how? If no, do you always get interference after opening the stormtrooper end regardless of the wait time? 

Finally back to measurement. There is no (strong) measurement without collapse. Collapse happens when a particular equivalence relationship no longer holds. Mathematically it can be proven that the wavefunction is projected on a subspace of the original Hilbert space. Moreover uniqueness can be proven as well: that this is the only mathematically valid mechanism in which projection can occur. Interaction between the quantum system and the measurement device can break the equivalence, but changing the experimental context can achieve the same thing as well. A measurement does not happen however until irreversibility occurs: there could be amplification effects, or as above enough time passes such that the energy uncertainty is low enough and the "which way" information becomes available regardless if we seek this information or not.

Sunday, December 4, 2016

A measurement can be more than an observer learning the value of a physical observable

Last post created quite a stir and I want to expand on the ideas from it. This will also help me get out of an somewhat embarrassing situation. For months now Lubos Motl tried to get revenge on his bruised ego after a well deserved April Fool's joke and became a pest at this blog. The problem is that although I have yet to see a physics post at his blog that is 100% correct, we share roughly the same intuition about quantum mechanics: I agree more much more with his position than say with the Bohmian, GRW, or MWI approaches. The differences are on the finer points and I found his in depth knowledge rusty and outdated. For his purpose: to discredit the opposite points of view at all costs this is enough, but it does not work if you are a genuine seeker of truth.

So last time he commented here: "A measurement is a process when an observer actually learns the value of a physical observable" which from 10,000 feet is enough. However this is not precise enough, and now I do have a fundamental disagreement with Lubos which hopefully will put enough distance between him and me. 

More important than my little feud with Lubos, I can now propose an experiment which will either validate or reject my proposed solution to the measurement problem. I do have a novel proposal on how to solve the measurement problem and this is distinct from all other approaches. I was searching for months for a case of a novel experimental prediction, but when I applied it to many problems I was getting the same predictions as standard quantum mechanics. Here is however a case where my predictions are distinct. I will not work out the math and instead let me simply present the experiment and make my experimental claim.

Have a box with a single particle inside. The box has a middle separator and also two slits A and B which can be placed next to a two-slit screen. We can then carry two kinds of experiments:

  1. open the two slits A and B without dropping the separator allowing the particle to escape the box and hit a detector screen after the two-slit screen.
  2. drop the separator and then open the two slits A and B allowing the particle to escape the box and hit a detector screen after the two-slit screen.
Next we repeat the experiments 1 or 2 enough times to see the pattern emerge on the final screen. Which pattern would we observe?

For experiment 1 we already know the answer: if we repeat it many times we obtain the interference pattern, but what will we get in the case of experiment number 2?

If dropping the separator constitutes a measurement, the wavefunction would collapse and we get two spots on the detector screen corresponding to two single slit experiments. If however dropping the separator does not constitute a measurement, then we would get the same interference pattern as in experiment 1.

My prediction (distinct from textbook quantum mechanics) is that there will be no interference pattern.

Monday, November 28, 2016

Are Einstein's Boxes an argument for nonlocality?

(an experimental proposal)

Today I want to discuss a topic from an excellent book by Jean Bricmont: Making Sense of Quantum Mechanics which presents the best arguments for the Bohmian interpretation. Although I do not agree with this approach I appreciate the clarity of the arguments and I want to present my counter argument.

On page 112 there is the following statement: "... the conclusion of his [Bell] argument, combined with the EPR argument is rather that there are nonlocal physical effects (and not just correlations between distant events) in Nature". 

To simplify the argument to its bare essentials, a thought experiment is presented in section 4.2: Einstein's boxes. Here is how the argument goes: start with a box B and a particle in the box, then cut the box into two half-boxes B1 and B2. If the original state is \(|B\rangle\), after cutting the state it becomes:


Then the two halves are spatially separated and one box is opened. Of course the expected thing happens: the particle is always found in one of the half-boxes. Now suppose we find the particle in B2. Here is the dilemma: either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete because \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\) does not describe what is going on. My take on this is that the dilemma is incorrect. Splitting the box amounts to a measurement regardless if you look inside the boxes or not and the particle will be in either B1 or B2.  

Here is an experimental proposal to prove that after cutting the box the state is not \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\):

split the box and connect the two halves to two arms of a Mach-Zehnder interferometer (bypassing the first beam splitter). Do you get interference or not? I say you will not get any interference because by weighing the boxes before releasing the particle inside the interferometer gives you the which way information.

If we do not physically split the box, then indeed \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\), but if we do physically split it \(|B\rangle \neq \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a hidden assumption in Einstein's boxes argument: realism which demands non-contextuality. Nature and quantum mechanics is contextual: when we do introduce the divider the experimental context changes. 

Bohmian's supporters will argue that always \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a simple way to convince me I am wrong: do the experiment above and show you can tune the M-Z interferometer in such a way that there is destructive interference preventing the particle to exit at one detector.