## Saturday, October 16, 2004

### Causality and entanglement

I was just moderating a posting on sci.physics.strings that was proposing a mechanism involving branes to send signals faster than light. That provoked me to write another essay about this issue.

Because we're gonna talk about causality (and locality), it may be natural to start chronologically.

People have known for millenia that the cause precedes its effects. For example, before you were born, your parents had to do a certain thing. This fact, known as causality, became a part of Newtonian physics. Classical, non-relativistic physics had a universal time coordinate t that everyone could agree upon. The events and properties of the objects at time t was thought to only affect the events at times t' where t'>t.

Einstein's special relativity has revolutionized many properties of space and time, but the previous sentence remained true. In fact, it remained true for all inertial observers. Each reference frame has a different time coordinate (and time and space are getting mixed), but the statement always holds. What does the required causality in other frames implies for our frame? Well, it implies that the event B affected by the event A must not only come after A, but it must belong to the future light cone of A: physical influences (such as material objects that can influence something) are not only constrained to propagate from the past to the future, but they are not allowed to travel faster than light.

If you wait for a short time period t, you will only be able to affect the objects that are the distance s=ct from you, or closer. Therefore, roughly speaking, this relativistic version of causality is also called locality (because your influence remains local), and we won't distinguish between the concepts of locality and causality.

General relativity allowed the spacetime to be curved, and the notion of a future light cone had to be modified, too. But for sufficiently small patches of spacetime, the curvature can be neglected and general relativity must always reduce to special relativity for experiments in the "elevator" or other local environments. Therefore we will always mean "special relativity" if we mention that some theories or conclusions are "relativistic".

(In quantum gravity, the metric tensor is a quantum observable. Also, the metric tensor determines the structure of the light cones and the rules for causality, and therefore the causal structure becomes uncertain and confusing. At any rate, string theory is smarter than we are and it is able to avoid these conceptual problems. However, this is not the main topic of this essay.)

OK, what about quantum mechanics that emerged in the 1920s? It described the world in terms of wavefunctions associated with particles. Because the wavefunction is not a real wave, and the electron is always found at one point, Max Born successfully proposed that the wavefunctions are waves of probabilities. Einstein deeply believed that the world was deterministic, or at least the question about the "state of the Universe" was an objective question with a unique answer. Although he was one of the grandfathers of quantum theory, this feature of quantum mechanics was unacceptable for him, and therefore he tried to show that the leaders of quantum mechanics had to be wrong.

Meanwhile, these people like Heisenberg, Bohr, Dirac, Born, Pauli, and others knew perfectly how to predict experiments involving quantum mechanics. On the other hand, in order to "disprove" quantum mechanics, Einstein, Podolsky and Rosen (EPR) prepared a gedanken experiment whose result was supposed to show disagreement with quantum mechanics. Well, they were thinking about a system that splits into two subsystems A,B (a positronium decaying into two photons A,B, for example). In the future, A,B are highly separated. Nevertheless, the predictions what happens with B when we measure it depend on the type of measurement that we perform with A.

For example, the angular momentum conservation law implies that the total angular momentum of two photons in A,B must vanish. That means that either both of them are right-handed, or both of them are left-handed (the opposite momenta imply that two R's or two L's cancel). It would be fine with Einstein if the photons were RR or LL - one of these two choices was chosen already when the positronium decayed. However, quantum mechanics predicts that if we measure the linear polarization of the photon A and then B, we always get the opposite polarizations (either xy, or yx). The correlations exist if we measure the circular polarization but also if we measure the linear polarization.

EPR knew that this follows from quantum mechanics. Einstein found this behavior truly counter-intuitive: if the photons are already in the states RR - which is one of two choices that must occur, he thought that the probability to find the right-handed photon A to be x-polarized is 50 percent, much like the likelihood that it is y-polarized - but these two cases should be totally uncorrelated with the same predictions about the photon B that is very far.

Therefore, Einstein thought, if we measure the linear polarizations of both photons, all four choices xx,xy,yx,yy must have probabilities 25 percent. However, once again, quantum mechanics only predicted xy or yx with probabilities 50 percent. It was able to correlate the two photons in many different ways. If we measure the photon A to be R, then the photon B must suddenly be exactly R, so that the person who measures the circular polarization gets it right. But if we measure the photon A to be x, then the photon B must become linearly polarized in the y-direction, so that the second experimentalist gets the right result.

Once again, Einstein thought that this proved that quantum mechanics must send some "signals" that make it work, and it violated causality and special relativity, and therefore it had to be incorrect. However, today we know for sure that it was Einstein who was misled. Experiments done after Einstein died - for example those by Alain Aspect and his colleagues, and Anton Zeilinger and his colleagues - have clearly shown that all the correlations are there, exactly like quantum mechanics predicts.

There have been many other developments. The prince Louis de Broglie did not want to accept the probabilistic interpretation either, and therefore he proposed the pilot wave theory in the late 1920s. According to this theory, the particles have a well-defined position and momentum, but there also exists an objective wave associated with the particle. This wave creates a potential that affects the particle's motion in such a way that for a generic position of the particle in the wave at the beginning, the evolution will preserve the fact that the particle remains at a generic point given by the probability distribution associated with the wave, and therefore de Broglie's theory can give the same predictions as quantum mechanics in the simplest contexts, even though it is deterministic: the particle as well as the wave were the usual classical concepts, governed by some deterministic differential equations.

Such theories were called "hidden variable theories". De Broglie's theory was rediscovered in the 1950s by the communist David Bohm and it made him very famous, even though he was not the first one and even though the theory is misled. (The communists are always good in adopting things that do not belong to them.)

It was thought by the majority of physicists that the question whether the hidden variable theories are more true than the orthodox probabilistic quantum mechanics would remain a philosophical, religious question forever, and no physical experiment could ever resolve the dispute. Another advocate of these (crappy) hidden variable theories, namely John Bell, decided that it could not be the case, and he wanted to destroy orthodox quantum mechanics forever. He realized that there is a measurable difference. In the example with the two photons, we saw that quantum mechanics predicts "more correlation" related to "more types of measurements" than what our classical intuition finds natural. Bell quantified this observation, and he showed that every deterministic theory (or even a theory where the state of the photons is objectively and uniquely given already before the measurement is done) must lead to a combined correlation of various pairs of observables that always belongs to an interval. Quantum mechanics however often leads to higher (or smaller) correlations. Consequently, if you perform this experiment, the correlations should respect Bell's bounds - because the world is classical, as Bell believed - and quantum mechanics would be ruled out.

Unfortunately for Bell, the critical experiments were already done during his life. Instead of confirming his deterministic prejudices, they led to a spectacular confirmation of quantum mechanics and its very large correlations - and Bell's approach became one of the key insights showing that quantum mechanics can't really be messed with. These experiments probing entanglement, EPR phenomena, and quantum teleportation rule out all hidden variable theories whose dynamics is local and causal (that admit no faster than light signals). In other words, if you want to revive these hidden variable theories today, you must give up special relativity and the bound on the maximal speed of signals, and this will most likely lead you to contradictions with various experiments.

It may be useful to say a few more words why the hidden variable theories are crappy:
• They have serious problems to incorporate special relativity. Some "Bohmian fundamentalists" believe that they can construct Bohmian versions of quantum electrodynamics and perhaps the Standard Model, but it seems to only be their wishful thinking because the general argument above shows that the hidden variable theories that can agree with experiments don't respect causality, and therefore they will generically break special relativity
• They can't naturally explain physical notions such as the spin. In Bohmian theories, you need to assign an objective classical value to a complete system of observables. This must include a projection of the spin of every particle. But in that case, you must decide which component of the spin is allowed to have this classical value - and that will break rotational invariance. The only reason why the discreteness of the z-component of the spin in quantum mechanics does not break the rotational symmetry is that the amplitudes for the spin are probabilistic, not objective classical numbers.
• Even if we forget about these advanced subjects, the hidden variable theories go against the lessons that quantum mechanics taught us. For example, different observables on the Hilbert space (such as the position and the momentum; or spins with respect to different axes) are equally good observables - and the bases built from their eigenstates are equally good bases. It's not natural to pick the position (plus another random set of observables) and allow them classical values. Feynman's lectures in physics are good because they explain this "democracy" between different observables quite nicely.
• In reality, decoherence explains the "classical character of the position of macroscopic objects" dynamically. The fact that the Moon should be thought of as having a well-defined position follows from the Hamiltonian (and the decoherence calculations), not from a pre-determined special role of the position. Moreover, decoherence (combined with consistent histories) solves many other conceptual problems of the Copenhagen quantum mechanics (especially the emergence of the boundary between "classical" and "quantum"), and therefore the reasons to abandon quantum mechanics keep on converging to zero.

OK, let me now emphasize that quantum mechanics - for example quantum mechanics extended to the relativistic world, such as in quantum field theory - respects not only all the rules of quantum theory (the probabilistic character of the amplitudes; the possibility to entangle distant objects; the superposition principle for the wavefunction; the possibility to have higher correlations than Bell's bound), but it also respects the rules of special relativity.

It is useful to think in the Heisenberg picture. The field operators evolve according to the same equations as their classical counterparts (classical fields). The classical field is only affected by the values of this and other fields in the past light cone, and correspondingly, all correlators, expectation values (which includes the probabilities of various things, because the probability is an expectation value of a projector, and a projector is a function of the other observables) of an operator will only be affected by the observables constructed from the operators in its past light cone. The commutation relations respect the Lorentz invariance, too. This implies that no superluminal signals are possible in quantum field theory (also called "relativistic quantum field theory" or "local quantum field theory").

Even if you "feel" that something "seems" to propagate faster than light between A,B, you will never be able to use the EPR effects to inform someone who lives on the Sun about your new Nobel prize faster than in 8 minutes.

It should be repeated that the only reason why superluminal signals are not possible in QFT is that the outcomes of the experiments are probabilistic. Consider the example with two photons from the beginning. Why can't we send a signal from A to B superluminally? It's because the results of the B measurement are always 50:50 and the person A just can't affect it. If we forget about A altogether, the probabilities to get L or R for the photon B are always 50:50 percent, much like for x:y, even if the person A jumps like mad. If we think about both A and B, they can be correlated, but it's not correct to say that the outcome in B was a consequence of anything done at A. The measurements at A,B can be space-like separated, and it would be foolish to talk about a causal relation between them.

Another safety rule of quantum mechanics is that the probabilistic nature of quantum theory prevents the person A from commanding her own photon. Even A herself will get random results. If A were able to force her photon to be measured as R (or L, depending on A's thinking), and if the correlations were preserved, the person B would have to get the same result as A, and A could therefore send bits of information faster than light. But once again, it's not possible in reality because the specific results in A as well as B are unpredictable. Both A and B will know that the results of their measurements are correlated - if they measure the same type of property of their photons - but they will never be able to affect in advance what these results are, and therefore they won't be able to use these strange features of quantum theory for superfast communication.

You can see that the orthodox quantum mechanics is an ally of special relativity. They work together, but a modification of the quantum theory (such as the hidden variable theories) would spoil relativity, too.

Finally, let me say that a more complete theory underlying quantum field theory - in other words, string/M-theory - may predict some subtle violations of locality and causality. But their reach will probably be very short (the string scale or the Planck scale); it may become macroscopic in the presence of horizons. Nevertheless, there are also arguments that show that string/M-theory preserves locality and causality (and their major consequences) exactly, at least in some contexts and formalisms. See a recent paper by David Gross and Ted Erler, for example. The lessons from quantum field theory can therefore be extrapolated quite seriously even to a deeper theory underlying QFT.

#### 13 comments:

1. The difference between Bell and crappy string theorists is that Bell was able to make his prejudices experimentally decidable. String theory remains undecidable. This undecidability is being mistaken for truth. If no conceivable experiment can rule out string theory, then it must be true, right?!

2. To the anonymous poster (let me identify him as a "bastard" because there could be many anonymous posters which may become confusing): yes, in some sense you are right.

The difference between the crappy hidden variable theories and string theory is that the former just can be proved wrong (and obsolete, and later as misguided), while string theory has all the required features to continue as a very serious candidate for a complete theory, even though it is not proved correct yet.

We don't know for sure whether string theory is the right theory of our Universe, so unlike you, we are not using the word "truth" for this major question, but the reason why we continue with it is that it is an appealing candidate that has not been ruled out yet.

Do you think that we should only study theories that have already been ruled out? We study those that still remain candidates, and string theory is the only game in town (among the candidates for a theory that goes beyond quantum field theories).

3. Two points:

a) I am not sure QFT and QM are the same as far as entanglement is concerned.

There have been claims that EPR does not hold in field theory, since the number of particles is not conserved
( quant-ph/0202175 ).

Perhaps as a related issue, information in quantum fields
has very different characteristcs from information in
"first quantized" systems: For example, the Von Neumann
entropy theorem (Entropy, defined as rho ln (rho), where
rho is the density matrix) does not hold for quantum fields.

The whole topic, in short, is controversial, and I was in
fact wondering if string theory has something to say about it.

b) The reference to Bohm as a "communist" is, simply,
offensive.
Bohm was accused of being a communist during the McCarthy
era, and censured by the House of Un-American
activities
committee due to his principled refusal to testify (an
extremely
honorable thing to do under the circumstances)
For this, he was fired from Princeton and effectively
stripped of his US citizenship.

The whole thing was a completely shameful affair.
It does you no credit to blame the victim.

4. Entanglement from QM does not hold in QFT? Are you joking?

http://www.arxiv.org/abs/quant-ph/0202175

It's just a crackpot paper, the author definitely has not quite mastered quantum field theory, and if you at least clicked at "cited by" to see a single self-citation, you might get the rough idea without the need to write your confused comment.

Quantum field theory - namely the Standard Model - is *the* theory that describes everything that we've ever observed (except for gravity, which you must account for by adding general relativity). If I say "everything", it of course includes all EPR, entanglement and teleportation experiments that have ever been done.

Saying that QFT disagrees with the usual entanglement in the usual EPR experiments is just stupid - entanglement has nothing to do with conservation of the number of particles or anything like that.

5. There is absolutely nothing controversial about the existence of EPR entanglement and the result of the experiments, and what it means for any theory that wants to agree with reality, and about the fact that the correct EPR entanglement is predicted by QM and QFT.

Instead of "controversial", we may say that it is "difficult" for many people, and therefore there exist hundreds of crackpot papers that try to give "alternative" proposals of various types, but their authors simply don't understand quantum physics. Their ignorance does not make quantum physics scientifically controversial.

Concerning Tommasini, you might have looked more carefully

http://www.arxiv.org/find/quant-ph/1/au:+Tommasini_D/0/1/0/all/0/1

to see that 8 of his 10 citations are self-citations. You know, if you don't understand certain topic in science, you almost always get a better idea about whom you should believe if you check the citations.

Young David Bohm was *objectively* involved with radical left-wing politics.

Like many young idealists in the late 1930s (including Oppenheimer himself), Bohm and his colleagues were attracted to alternative models of society, and were active in organizations like the Young Communist League, the Campus Committee to Fight Conscription, and the Committee for Peace Mobilization (all would be branded Communist fronts by the FBI under J. Edgar Hoover).

6. Concerning McCarthyism in general.

The House... HUAC was established to fight against Nazi espionage, and later Soviet espionage. I find it totally obvious that *any* responsible US politician would have to create an institution like that during these difficult times, in one form or another. Even Al Qaeda is next to nothing compared to the threats of Hitler and Stalin.

I also fully understand the tough approach during McCarthy's era. This really started in 1948. You may want to know that it was exactly 1948 when the communists established the totalitarian system in Czechoslovakia, among other places. The communists then executed thousands of people and forced tens of thousands to leave our small country. They stole all assets of the "capitalists". The communists have devastated our economy, our values, our - more or less everything.

It would be nice if you kept your irresponsible judgements of the anticommunist defense of the USA after the war for your friends. For me it is a traumatic point from our national history.

If we had a person like McCarthy in Czechoslovakia who would be able to go after the communists, all these things could have been avoided.

I just find it highly unfair to drag McCarthyism from the context. It should be compared to what was being done by Stalin and other communists at that time, and if a fair person sees all these things from this fair perspective, the American politicians including McCarthy were - relatively speaking - holy angels of innocence.

I also don't find anything nice about David Bohm's refusal to testify for HUAC.

It's not trivial at all that democracy is preserved - there is a lot of experimental support for this claim - and sometimes it needs tough measures.

7. Regarding QFT/hidden variables:

If your only argument against that paper is that It/the author
have not been cited enough, that's not very scientific.

In fact, the reason the paper I put AS AN EXAMPLE was not
cited is simply that it's main point is pretty obvious, and it
does not invite further research.
It was only an example.
An example from a better cited author is
http://www.slac.stanford.edu/spires/find/conf/www?rawcmd=fin+cnum+C92/11/10

You are right that QED, in the "non-relativistic" limit,
gives back Bell Inequality (but not really the paradox, if you
look at it from the point of view of "sum-over-paths". In this picture, everything is local, and "entanglement" arises
from some paths being canceled by the symmetries of the Lagrangian, in this case Angular momentum).

But you recover Bell's inequality only because the electron
mass and absence
of asymptotic freedom regulate infrared divergences.
If the electron were massless, OR if there was asymptotic
freedom, there would be large Bell-violating effects coming
from higher-order Feynman diagrams.

In general we still do not know how to treat the information
within a quantum field not in equilibrium (see my paragraph
about Von Neumann's entropy, which you skipped in your reply),
especially if there are non-perturbative effects present.
Even if the Quantum field Lagrangian is well known.

Out-of-equilibrium quantum fields are an active research field, even if their papers have fewer citations than
string theory.

8. No, the number of citations is not MY argument against the nonsensical theories.

The number of citations is my recommended criterion for the people who don't know physics - a pretty reliable way for THEM to figure out whether a paper is probably serious or not. Certainly more reliable than anything else that these people have shown so far.

This paper is not serious because of scientific criteria, and if you read and understand my essay, you will also be able to find a more solid explanation why the paper is not serious.

QED is a quantum theory, so it does NOT respect Bell's inequality. Why do you always choose the incorrect version of these statements?

The violation of Bell's inequalities has nothing to do with out-of-equilibrium physics or loop diagram or any other nonsense that you listed. Violation of Bell's inequality follows from the very interpretation of quantum mechanics, combined with the fact that its Hilbert space has entangled states. Every quantum theory is able to violate these inequalities.

Note that I did not have to count your citations to explain why your text was garbage.

9. Regarding Bohm/HUAC/etc.

What struck me most about your reply is that you manage to decry communist tyranny and, at the same time, show complete contempt for the freedoms that tyranny denied.

Yes, the HUAC was supposedly there "to fight Soviet espionage", and it "was Harsh".

The problem with HUAC was not that it was harsh on Soviet
spies (as far as we know, it failed to catch any!)

The problem is that it's activities had nothing
to do with finding out whether the people it interrogated
were Soviet spies or not, and everything to do with persecuting people for their peacefully held and constitutionally protected beliefs. Like you say 2 posts
above, there is a difference between spin and propaganda.

Bohm's is a case in point:
He was treated pretty much like Soviet dissidents were treated in the '70s and the 80s: Subjected to an official
show-trial, stripped of his livelihood, and eventually of his
US passport.

Was he "lucky" because "it would have been worse under
Stalin?
Sure, in the same way Vaclav Havel was "lucky" that
he conducted his pro-democracy activities in '70s and '80s Chekoslovakia and not in a "free country" like '70s and '80s El Salvador.
(this is not a hyperbole: Salvadoran Jesuit priests explicitly remarked that he'd have been brutally murdered a week after publishing his first
pro-democracy pamphlet, see, e.g., here)

You claim he "deserved it", motivating it with the fact that
"Young David Bohm" was a member of organizations "classified
as communist fronts" by J. Edgar Hoover (most organizations
supporting the civil rights movement fall into this
category), and because he refused to participate in what
was a blatantl abuse of power.

I can not argue against this, or any other,
moral judgement, since it is based on assumptions.
I hope for you, however, that your ideas will never be subjected to some future "House of Un-American Activities committee", formed of course to protect "freedom" against
"totalitarianism".

10. I repeat the obvious fact that the postwar period was a tough struggle against Stalin, and the idea of preserving 100 percent of all rights of everyone, without any exceptions, including all suspicious communists etc. - and all this fundamentalist crap that you are writing here - is absolutely unrealistic. It's just plain stupid.

The very nature of the democratic systems was threatened after the war. Not just threatened - in Central and Eastern Europe, for example, the systems were really destroyed. It is impossible to live with the perfect fairy-tale democracy 100 percent of the time. A U.S. politician who would say that the development in Europe did not matter, and that they could have continued their lives without any change, would be as irresponsible as a politician who would say that 9/11 did not matter.

And I personally do believe that David Bohm was working for the international communists. I do believe that he and similar people were dangerous, and I would even say that it is partly a matter of coincidences that the totalitarian system was organized by Soviet communists like Lenin and Stalin and not the US communists like Bohm.

I would have even found it reasonable if he were preventively arrested, and I insist that you will respect my beliefs. I am not violating anyone's rights - I just believe that these communists were dangerous bastards.

Your comparisons with the Soviet dissidents is absolutely outrageous, because this is implicitly a comparison of democracy with communism.

If you think that it is the same thing to fight against the free American (capitalist) society - as to fight against stalinism - be my guest. You are free to believe whatever you want, but I am free to be convinced that the people with this sort of belief are human garbage. Sorry.

11. You have really no idea what the communists were doing at the same time. Your comments can serve as an example of these left-wing academic people living in a cage, isolated from the real world, who are being paid and fed by the rest of the world and who complain about the "civil liberties" violations every time their chocolate from the government is not sweet enough.

But your comrades communists in Eastern Europe were just executing thousands of people (including intellectuals) after thousands of other people. Your criticism is just taken out of the context. Dr. Milada Horakova was a great brave Czech woman, accused from treason, and executed. She was really a right-of-center mainstream woman. The question whether Princeton wanted to have Bohm, a well-known communist activist, as a professor during Stalin's era or whether it was better for them to recommend him a different place is a complete detail compared to the things that were happening.

The obvious goal of the responsible US politicians had to be to preserve as much "expectation value of" safety of the USA, and as much freedom in other countries as possible. Obviously, some sacrificies had to be made. If they accused someone who was really innocent, or if we now think that McCarthy was tougher on some people than was necessary, that's sad - but it's very easy to criticize it today once we know all these things. At that time, they were facing serious threats - probably more serious threats than the terrorists today, and they had to do real decisions.

Stalin eventually copied the nuclear bomb, using his Russian spies. There have been so many spies around (on both sides, of course) that it just sounds weird to say that no US communists should have been treated as suspects. If they could prevent the Soviet Union from getting the nuclear know-how, it would have been great and the cold war could have finished 30 years earlier. But they were just not systematic enough in their struggle against the Soviet spies.

12. OK, let me spell it out for you.

QM:
If the detectors are oriented in the same direction, the 2 photons will ALWAYS have correlated polarizations due
to angular momentum conservation.

Local Hidden Variable theory:
Since no hidden variable can travel between the two detectors faster than the photons reach the detectors, and since
the detectors decide what direction of polarization to measure
after photons are emitted, the 2 polarizations can only be uncorrelated.

QED:
Correlation is not fully there: violations arise due to
the possibility of producing EXTRA photons, which are
not detected. We can have a sphere of detectors and try
to measure ALL photons under consideration. HOWEVER, our
detectors will always have a lower momentum resolution, and
photons can be produced at arbitrarily low frequency (where
they can still produce a spin violation, since all photons carry a spin 1).

HOWEVER, these correlation-breaking processes are suppressed
by the fact that charged particles have a mass (so virtual loops are regulated in the infrared), as well as by the asymptotic behaviour of \lambda_{QED}.

A general quantum field theory:
Now we allow massless charged particles, AND possible IR divergences. So we DON'T know what happens to entanglement
in the infrared limit.

13. Once a positronium decays to two photons (in otherwise empty space), there is just kinematically no way how the number of photons can change. A photon cannot decay (neither in reality nor in QED - these are equivalent), not even to other soft photons.

Loop diagrams do not mean that the number of physical photons is changing with time. If you define the number of photons physically, a two-photon state (with two separated photons) is preserving the number of photons in the time evolution.

If you try to address some infrared divergences, they must be accounted for before you define your final state. Yes, there is a nonzero amplitude to produce three-photon (or four-photon) final state, and so forth. There are the events that must be removed, much like the events in which the final photons go in the wrong direction. The EPR effect is only considering the decays into two antipodal photons.

There are all the usual uncertainty rules - you must define in advance which energies of the soft photons you are able to measure, and this will influence the counting which events you include as pairs of 2 antipodal photons (plus minus some sufficiently soft photons).