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.

Sunday, November 20, 2016

Gleason's Theorem

It feels good to be back to physics, and as a side note going forward I will do the weekly posts on Sunday. Today I want to talk about Gleason's theorem. But what is Gleason's theorem?

If you want to assign a non-negative real valued function \(p(v)\) to every vector v of a Hilbert space H of dimension greater than two, then subject to some natural conditions the only possible choice is \(p(v) = {|\langle v|w \rangle |}^{2}\) for all vectors v and an arbitrary but fixed vector w.

Therefore there is no alternative in quantum mechanics to compute the average value of an observable A the standard way by using:

\(\langle A \rangle = Tr (\rho A)\)

where \(\rho\) is the density matrix which depends only on the preparation process. 

Gleason's theorem is rather abstract and we need to unpack its physical intuition and the mathematical gist of the argument. Physically, Gleason's theorem comes from three axioms:

  • Projectors are interpreted as quantum propositions
  • Compatible experiments correspond to commuting projectors
  • KEY REQUIREMENT: For any two orthogonal projectors P, Q, the sum of their expectation values is the expectation value of P+Q: \(\langle P \rangle + \langle Q\rangle = \langle P+Q\rangle\)
In an earlier post I showed how violating the last axiom (which is the nontrivial one), in the case of spin one particles, can be used to send signals faster than the speed of light and violate causality. But how does Gleason arrives at his result?

Let's return at the original problem: to obtain a real non-negative function p. Now add the key requirement and demand that for any complete orthonormal basis \(e_m\) we have:

\(\sum_m p(e_m) = 1\)

For example in two dimensions on a unit circle we must have:

\(p (\theta) + p(\theta + \pi/2) = 1\)

which constrain the Fourier expansion of \(p (\theta)\) such that only components 2, 6, 10, etc can be non zero. In three dimensions the constraints are much more severe and this involves rotations under SO(3) and spherical harmonics. I'll skip the tedious math, but it is not terribly difficult to show that the only allowed spherical harmonics must be of order 0 and 2 which yields: \(p(v) = {|\langle v|w \rangle |}^{2}\).

The real math heavy lifting is on dimensions larger than three and to prove it Gleason first generalizes  \(\sum_m p(e_m) = 1\) to \(\sum_m f(e_m) = k\) where k is any positive value. He names this "f" a "frame function". Then he proceeds to show that dimensions larger than three do not add anything new.

If you are satisfied with the Hilbert spaces of dimension 3, the proof of the theorem is not above undergrad level, and I hope it is clear what the argument is. But what about Many Worlds Interpretation? Can we use Gleason's theorem there to prove Born rule? Nope. The very notion of probabilities is undefined in MWI, and I am yet to see a non-circular derivation of Born rule in MWI. I contend it can't be done because it is a mathematical impossibility and I blogged about it in the past. 

Sunday, November 13, 2016

The Open Society and Its Enemies

Today I had planned to return to physics and talk about Gleason's theorem, but as US politics still brutally interjects into our life I want to explain my take of the events and to cleanse and de-Trumpify my life before resuming the physics topics. Gleason's theorem s not going anywhere and I will postpone the topic for a week.

Let me start with a "disclaimer": I am manipulated/incited by media, I am a professional protester paid by Soros, I am a hard core liberal/socialist in love with Hilary, I hate the silent majority of blue color hard working people who makes my life comfortable, and I think they are all bigots and racists. (for the record this was tongue-in-cheek).

Until my second year in college I lived under a totalitarian system and I experienced a lengthy decade of a transition to democracy after 1989 in Romania. What I observe today in the US is the process in reverse. So how did we get here?

To shorten the history, let me start with the end of second Bush presidency and the inauguration of Obama. At that time all inter-bank landing came to a screeching halt and it was as if someone had hit the turn-off switch on the economy.  To jolt the system back to life, Obama resorted to the ides of Keynes and introduced the stimulus package. But Obama had one fault: he was black and this startled the rednecks who organized into what later became the Tea Party. This was America's reactionary ideology which cannot expressed their opposition to Obama's skin color due to political correctness, and instead went after him on fiscal ideas. The mainstream republicans had a love/hate relationship with the Tea Party because on one hand they fear it as something which cannot be controlled, but on the other hand they draw their support from the same electoral pool.

The mainstream republicans are masters of duplicity: they draw their support from the poor rural, uneducated part of America by praising their self-reliance and tickling their self-esteem, while they push policies which actually hurt their mass constituency while enriching the big business donors. To pull this trick they rely on a disgusting propaganda machine: Fox News. What Trump did was to break the republican lies and come out in the open with a full display of racism and intolerance. Trump was running not only against Hillary but against mainstream republicans as well.

Now fast forward to today. I don't think Trump is stupid: he is a master manipulator, an immoral con man who plays on others core beliefs. Trump lacks any core beliefs/moral compass and this makes him extremely dangerous: a narcissistic psychopath bully now with nukes. You only fool yourself into giving him a chance/benefit of the doubt. The only thing Trump respects is raw power and a forceful push back.

But what about the republican electorate? Almost half of them are brainwashed imbeciles: in June of 2016 41% of registered republicans thought Obama was not born in US. Moreover 31% did not know what to think and only 27% of them were on the sane side!!!

So how can we deal with Trump and his constituency? Let's go back to basics, the US constitution. Trump stated that the second amendment is under  siege, but he is now attacking the first amendment:

"Prohibits Congress from making any law respecting an establishment of religion, impeding the free exercise of religion, abridging the freedom of speech, infringing on the freedom of the press, interfering with the right to peaceably assemble or prohibiting the petitioning for a governmental redress of grievances."

The main characteristic of a totalitarian state was brilliantly captured by Popper in his book: 
"The Open Society and Its Enemies"

The other day Trump tweeted: "Now professional protesters, incited by the media, are protesting. Very unfair!"

He tried to intimidate the media and interfered with the rights of the people to protest. Trumps wants to build physical and economic walls, deport millions of people, and wants to turn America into a closed totalitarian society. I find this unacceptable:

Trump is not my president.

Trump is not alone. He has a cohort of bad supporters. First the shame list of totalitarians willing to trample your rights:

- Trump: see above
- Rudy Giuliani: advocates locking up political opponents
- Chris Christie: did political revenge
- Stephan Bannon: antisemitic racist; the Goebbles of Trump

Next this is followed by opportunistic, lying, immoral, disgusting individuals:
-Mike Pence: does not blink while lying to your face
-Reince Priebus: no slimy job is too slimy for him
-Bill O'Reilly

Then plain toxic people:
-Sarah Palin
-Ted Cruz

Then are the deplorables, and here I name a troll of this site: Lubos Motl. In Romania there is one guy Radu Moraru with his TV station Nasul TV who got involved into US politics and the Romanian vote here (in a covert effort to unseat the head of the Romanian anti-corruption agency). On that TV station I respect only one guy: Grigore Cartianu. Then there are the brainwashed.

I have no respect for the people above and I draw the line here. 

The rest of people who voted for Trump are not racists or brainwashed and I have a meaningful polite conversation with them. I have several friends who voted for Trump and I have no ill feeling towards them and while we disagree we do it in the boundary of decency.

I only insist on one point: if you voted for Trump you are personally responsible for the consequences of Trump's presidency.

Friday, November 11, 2016

The beginning of the end of US democracy

It's already happening...

Trump just visited Obama in the White House today and as a result he felt embolden to complain about protests against him. 

If you voted for Trump you are personally responsible for the consequences of Trump's presidency. 

Wednesday, November 9, 2016

Donald Trump proves Time Travel does not exist

I have been proven wrong, Donald Trump won the election and apparently it is not wise to underestimate people's stupidity. So how did we got here?

One one hand, the establishment IS corrupt and there was a deep need for fresh air. Another Bush or another Clinton was a insupportable option. But on the other hand, racism still runs deep and while it was driven underground by politically correctness, it came back with a vengeance. While Trump's disgusting brand of politics is nothing new in Europe, US did not have the antibodies to combat it and people will lean the hard way how to do it. 

If you voted for him you are personally responsible for the consequences of Trump's presidency. 

Tuesday, November 8, 2016

Waiting for US election results

I will comment tomorrow on the election outcome...