Bohmian mechanics threatend by Occam's razor

Last semester, I have been running a seminar on "Foundation of Quantum Mechanics" (wiki page) for TMP students that had been disappointed that "Mathematical Quantum Mechanics" was not on foundations.

Overall, I am quite satisfied with the outcome. We had covered several approaches to foundational issues, in particular the relation of quantum to classical physics and here specifically the "measurement problem" (which I am convinced is not a problem but is explained withing quantum theory by decoherence). We will produce a reader with all the contributions and I myself will write some introduction (which I will post here as well once it is finished).

But today, want to discuss Bohmian mechanics which was one of the topics and which has strong support by some local experts. I never really cared about this approach (being one of the Gallic villages where a small group of people know they are doing it better than the rest of the ignorant world, much like algebraic QFT or loop quantum gravity) being satisfied with quantum physics without any extras.

But now was the time to find out what Bohmian mechanics is really about and in this post I would like to share my findings. The big question everybody asks really is "do they make any predictions that differ from usual quantum mechanics i.e. can it be distinguished by some sort of experiment or is it just an alternative interpretation?" but unfortunately I do not have a final answer. But more below.

Before I start, let me put it a bit in perspective: Inequalities of Bell type (an I would include the Kochen-Specker theorem and GHZ type experiments) show in effect that the world cannot be both "realistic" and "local". Realistic means here that all properties have values at any instant of time irrespective of whether they are measured or not while local means that any decision I take here and now (for example whether I measure the or component of the spin of my half of an EPR singlet state) cannot influence measurements that are so far away that they cannot be reached even at the speed of light.

Thus one has to give up either realism or locality. The common interpretation of quantum mechanics gives up realism, the component of the spin does not have a value when I measure the component but is local. In some of the popular literature you will find statements to the contrary but they are mistaken: It is true, there can be non-local correlations. But this is no different from classical physics: Most of the time the color of the sock on my right foot is correlated with the color of the sock on the left foot, even at the same instant of time (when they are space-like to each other). But the question of locality is not about states (which are always global) it is about operators or measurements. And measuring the color (as compared for example to the size) of one of the socks does not influence the other sock, the local operators do commute.

Bohmian mechanics insists on realism and the price it has to pay is to give up is locality. It does not violate causality in an obviously measurable way but doing the - or -measurement here influences what happens far far away. But enough of these philosophical remarks, let's look at some formulas.

In its pure form, Bohmian mechanics is about non-relativistic systems of particles with Hamiltonian of the form . Everybody knows that the norm-squared wave function in position representation gives the probability distribution of finding particle 1 at , particle 2 at etc. and there is a conserved current for this density. That is if you start with some distribution at an initial time then wait a bit while you flow according to the current you end up with the new at a later time.

The new thing for the Bohmians is to interpret this current as an actual current of particles with velocities . According to the Bohmians, these particles with joint coordinates are dots that for example show up on the screen of a double slit experiment. Obviously, if you start with a probability distribution of particle positions given by at an initial time and follow the deterministic flow equation for above, then at any later time the particles will be distributed according to . The Bohmians claim, that there are really particles and at any instant of time their position is and the velocity is no matter whether they are measured or not. That's it.

A few trivial remarks: This theory is non-local as the velocity of the -th particle does depend via the wave function on the positions of all the other particles. Bohmians say that this is not to worry about since their theory is non-relativistic and this is like for example the Coulomb interaction in non-relativistic quantum mechanics where the force on one electron depends on the instantaneous positions of the other charged particles.

The next remark is that in Bohm's theory there is also the wave function that follows the same Schroedinger equation as in usual quantum mechanics. Thus any question involving only the wave function trivially gives the same answer as in quantum mechanics. The equation of motion for the particle positions which are the new ingredient in the Bohm theory depend on the wave function but not the other way around. There is no feed-back and the wave function does not know about the . Any quantum mechanical measurement that in the end measures position (like for example Stern-Gerlach) gives the same result as the follow the wave function that determines the outcome in the usual interpretation.

All observables that are functions of the coordinates at one instant of time do commute with each other and one can thus give them all sharp values at that instant of time. Thus there is no problem with claiming those positions are even in the usual interpretation.

Position measurements at different times in general do not commute and thus they have no common meaning. Thus the only hope to find disagreement is in experiments that in the Bohmian interpretation require sharp positions at different instants of time.

When it comes to spin I have the impression that the Bohmians cheat a bit: They declare that "spin is no a property of a point-like particle" meaning that realism does not apply to the different components and like in the usual interpretation, the components do not have a meaning unless measured. One can read this as a manifestation of the preferred role the Bohmians give to observables that are a function of the position operators over all other operators. In effect they claim only those position observables deserve realism.

Of course, one can reformulate the Bell type experiments mentioned above in terms of positions (e.g. by translating spins into positions via Stern-Gerlach set-ups) but then the non-local flow equation seems to prevent any obvious contradictions with quantum mechanics.

There are more formal problems: For time-reversal invariant Hamiltonians, one can always choose the eigenfunctions of the Hamiltonian to be real. Thus for the wave-function to be such an eigenfunction , the particles don't move, even in i.e. the Coulomb field of a hydrogen atom. You may say that this is not the classical world but the quantum world and there are other equations of motion but I must say I find particles standing still even in the presence of forces a bit strange.

That that brings us to my main criticism: It is not clear to me how to observe the particle at . Do experiments measure the wave function (via ) or do they measure ? And if so, can I prepare (and later measure) without significantly disturbing the wave function? If that is the case I can of course check whether I put an electron in a hydrogen atom in an energy eigenstate at some and later check whether I find it at some other place (which quantum mechanics would predict).

There are if course ways to wiggle out: You could argue that this experiment is impossible since I would always disturb the wave function significantly by placing a particle at and thus everything get screwed up.

But this excuse is pretty much equivalent to "you cannot observe (directly)". But then we are adding something (the particles at positions ) to our theory which is not observable. And that sounds to me to be directly threatened by Occam's razor.

Anyway. Unless somebody explains to me how to measure , I maintain that adding to the theory is as good as adding invisible angels.

Update: The promised write-up is here.

Picking the bigger number of two even if one is unknown

Here is a nice problem from the xkcd blog: Two real numbers, A and B, are drawn using some unknown, possibly probabilistic process and written on papers that go into two envelopes. You randomly pick one and open it to find some number on it. You now have to decide whether you want to receive that number as an amount in dollars or rather the number that is in the other envelope (which is still sealed). Can you come up with a process that with probability >50% picks the larger amount?

Think about it.

SPOILER ALERT

You can. You will need some function f that maps the real numbers to the open interval (0,1) in a strictly monotonic way. You could for example take f(x) = (1+tanh(x))/2. Assume the number that you found in the first envelope was X. Then throw an unfair coin such that with probability f(X) you keep X and otherwise take the other envelope. Obviously (?), if you started with the envelope with the smaller number you are more likely to switch than if you had started with the larger number.

This sounds a bit counter intuitive. How can you increase your expected payoff if you know nothing about the number in the second envelope?

You might have the idea that something fishy is going on. What comes to mind is for example that you can produce paradoxes when assuming that there is a uniform probability distribution on the reals (or integers). But I believe, that this is not what is going on here since I did not say how the numbers were picked. They could have been picked with any perfectly fine probability measure on the reals, nobody said all numbers were equally likely. Below I will compute the expected outcome for any probability distribution that might have been used and it always works, not just in average.

More precisely, I think this is unrelated to a similar puzzle: In that second puzzle, there are also two envelopes that contain numbers representing payout but none is opened but instead it is known that the number in one envelope is twice the number in the other. You just don't know if you have the half oder double. There, assuming your envelope contains X then you could be tempted to argue that with probability 50% the other contains 2X and with 50% it contains X/2 and thus the expectation value is 50% x 2 X + 50% x X/2= 5/4 X and thus you could increase your expectation by 25% by switching. But then you could increase it by another 25% by using the same argument again and switching back.

In the second puzzle, it is really the implied uniform distribution of X's that is the origin of the paradox: You can see this by giving the additional information that both numbers are definitely smaller than 100 trillion dollars. That sounds like a trivial information but note that the calculation of the expectation value changes: If X is greater than 50 trillion dollars, you know with certainty that the other number cannot be 2X and thus the expectation of taking the other envelope is nor 125%X but X/2. If you now carefully go through the expectation value calculation you will find that averaged over all values of X the expectation for switching is the same as for keeping the first envelope.

Some of my readers will notice that the second puzzle is related to recent arguments that were made in the Landscape scenario about the imminent end of the world.

Back to the first game. Let's do some calculation to compute the expectation of the outcome. We will assume that the numbers were picked according to some probability measure rho(x)dx and that has a finite expectation value, i.e. the integral E=E(X)=int x rho(x)dx converges.

Then the expected outcome of the strategy above is X with probability f(X) and E with probability (1-f(X)) (as in that case we take the number in the second envelope).

We can now compute the expectation E(f(X) X + (1-f(X))E)= E(f(X)X) + E - E E(f(X)). For simplicity assume that E=0. Otherwise we could pay out E immediately and then subtract E from all number in the envelopes. Thus the expected payout of our strategy is E(f(X)X) but it is easy to see that this is positive (and thus we make more than the average E=0): In computing

E(f(X)X) = int f(x) x rho(x) dx

we can for x<0 overestimate f(x) by f(0) and for x>0 underestimate f(x) by f(0) and then conclude (unless rho(x) = delta(x) and we always spit out 0s)

E(f(X)X) > f(0)E(X) = 0

Thus we do better on the average than by deciding for one envelope or the other not taking into account the contents of the first one.

Not that the difference to the first puzzle is that this works for any rho(x), we did not have to assume some (non existent) uniform rho(x) and the effect does not go away as soon as a cut-off is introduced contrary to the other puzzle.

Is there a prisoner's dilemma in vaccination?

Recently, I have been thinking about vaccination strategies as I was confronted with opinions which I consider at least very risk inviting. Not to spill oil in the fire I will thus anonymize illnesses and use made up probabilities. But let me assure you that for the illness I have in mind the probabilities are such that the story is similar.

Let's consider illness X. For simplicity assume that if you meet somebody with that illness you'll have it yourself a bit later with 100% probability. If you have X then in 1 in 2000 cases you will develop complication C which is lethal. But C itself is not contagious.

Luckily, there exists a vaccination against X that is 100% effective, i.e. if vaccinated you are immune to X. But unfortunately, the veccination itself causes the deadly C in 1 in a million cases.

So, the question is: Should you get vaccinated?

Unfortunately, the answer is not clear: It depends on the probability that if not vaccinated you will run into somebody spreading X. If X is essentially eradicated there is no point in taking the vaccination risk but if X is common it is much safer to vaccinate.

The break even is obviously when that probability is 1 in 500. If it's less likely to meet somebody then it would be to your advantage not to vaccinate.

Unfortunately, the probability of meeting an X infected person depends on how well people are vaxinated: As X is so contagious, if the vaccination rate drops the probability of meeting somebody with X dramatically increases. That is, not vaccinating yourself might be profitable for you but if everybody follows the same strategy the vaccination rate might drop and the society as a whole will see many more cases of C.

If you assume in addition that your information is not perfect and you might be wrong in estimating the probabilities involved it is not clear to me to which fixed point this system evolves.

But it seems likely to me that there are situations where for the society as a whole it is much better if you get vaccinated even if this increases you personal risk of encountering C.

Opinions?

Tensor factor subalgebras question

After some serious arm-twisting performed by some TMP students I accepted to run a seminar on Foundations of Quantum Mechanics under the condition that it would be a no-nonsense class. This is in addition to the String Theory Lectures I have to teach as well.

After coming back from our vacation and workshop I found myself in the situation that I had much more fun doing some reading for the seminar (reviews on decoherence etc) than preparing for the string class (given that I have already twice taught Intro to Strings and that there are David Tong's wonderful lecture notes which give you the impression that you could take a few pages of those and be well prepared for class).

In the preparation, I came across a wonderful video of a lecture by a well known physicist (I will link this later as I might take part of this as a quiz for the first session. Let me just mention that it contains the clearest version of Bell's inequality that I a am aware of).

I am convinced that a lot of possible confusion about quantum physics together with locality (let me only mention the three letters E, P and R) comes from the fact that people confuse the roles of observables and states: Observables can be local and causality is built in by asking operators localised at space like distances to commute while states are always global objects. There is nothing like "the wave function of electron 1" or only in the approximation where you ignore all the other particles. You cannot use it when talking about correlations etc. But this is not bad, even in classical (statistical) physics, there are non-local correlations, like the colors of the socks on my two feet. The fact that in addition to correlations, there can be entanglement in the quantum theory does not change that.

Furthermore, I find it helpful to think (of course I did not come up with this approach) of the Hilbert space (and its wave functions) as a secondary object and take the observables as a starting point (and not derived as the operators acting on the wave functions). Those then are the elements of a (C*)-algebra and the Hilbert space only arises as a representation of that algebra. Stone and von Neumann for example then tell you that there is essentially a unique representation if the algebra is that of canonical commutation relations.

States are then functionals w that map each observable A to a complex number w(A) (interpreted as the expectation value). This linear function has to be normalised, w(1)=1 and positive meaning that for all A one has w(A^* A)>=0 (did I tell you that formuals are broken?). Then the GNS construction is similar to a highest weight representation: Using w and the algebra, one can construct a Hilbert space: As a vector space you can take the algebra. It is a representation after defining the action to be simply left multiplication. The scalar product of the elements A and B can be given by w(A^* B). Positivity of w tells you this is at least positive semi-definite. One can quotient out the zero-space to obtain something potitive definite and then employ some C*-magic to show that the action by left multiplication can be lifted to the quotient. I have suppressed some topological fine-print here like taking completions etc.

The states correspond in general to density matrices (or reducible representations) and as always can be convex combined as x w1 + (1-x) w2, the extremal states corresponding to irreducible representations.

In quantum information applications (as well as EPR and decoherence), one often starts with a Hilbert space that is a tensor product H = H1 x H2. Restricting attention to the first factor only corresponds to taking the partial trace over H2 and in general turns pure states on H into mixed states on H2. This has the taste of "averaging over all possible states of H2" but in the algebraic formulation if becomes clear that one is only restricting a state w to the subalgebra of operators of the form A1 x id where id is the identitiy on H2.

What I do not understand yet and where I am asking your help is the following: How does the splitting into tensor factors really work on the algbraic side? In particular, assume I have a C*-algebra C and a pure state w. Now I take some subalgebra C1 of C and obtain a new state w1 on C1 by restricting w to this subalgebra. What is the relation of the two Hilbert spaces H and H1 I obtain from the GNS construction on w and and w1 respectively? What is a sufficient condition on C1 that I can regard H1 as a tensor factor of H as above?

A necessary condition are obviously dimensions in the finite dimensional case: Here, the C*-algebras are just the complex matrix algebras of size n x n and the irreducible representation is on C^n. This is only a non-trivial tensor product if n is not prime. But nothing stops me for example to start with the big algebra being the 17x17 matrices and the subalgebra being those matrices that have the last row and column filled with zeros. But C^16 is definitely not a tensor-factor of C^17.

Back from silence

It has been very silent here recently (or not so recently) but there is no particular reason for this except that I have been busy with other things (including an update on my facebook relationship status) and small things have been posted to Twitter rather than the blog.

And if one is not constantly taking care of things they tend to degrade. So is this blog. What happened is that mathphys.jacobs-university.de, the computer that I have been using to host (background and other) images and the mimetex installation that was serving formulas for this blog as well as a number of other CGI scripts has died or at least is being turned off. Anyway, I have to relocate these things and I am still looking for a good solution. It should be a computer with a static, routed IP address on which I can install programs and in particular cgi-scripts of my liking. Here at LMU, this is probably not going to happen for reasons of security paranoia on the sysadmin side. In addition, mathphys was handling my email traffic, meaning that currently spam reaches my inbox and messages are threatened to be deleted by well meaning service providers. But this just means that the suffering is strong enough that I will be looking for a solution in the very near future. The solution will most likely be renting some virtual linux server. Suggestions in this direction would be more than welcome.

Not long ago, I have been attending the 40th incarnation of the Ahrenshoop Symposium once more organised by the Humboldt Uni crowd. This get together had a particularly interesting selection of talks many of which I really enjoyed. In particular I learned a lot and updated my options on F-Theory GUTs and AdS-Condensed Matter. Many thanks to the organisers! As you would expect, PDFs are online except for Sean's who gave a flip-chart talk (on four flip charts).

At that meeting I was asked what had happened to this blog and this post is supposed to be the answer to this question. I hope of course that more content will be here, soon. I was also asked to mention that it was Martin Rocek who got all the soap.

How to obtain a polymer Hilbert space

On Monday, I will be at HU Berlin to give a seminar on my loop cosmology paper (at 2pm in case you are interested and around). Preparing for that I came up with an even more elementary derivation of the polymer Hilbert space (without need to mention C*-algebras, the GNS-construction etc). Here it goes:

Let us do quantum mechanics on the line. That is, the operators we care about are and . But as you probably know, those (more precisely, operators with the commutation relation ) cannot be both bounded. Thus there problems of domains of definition and limits. One of the (well accepted) ways to get around this is to instead work with Weyl operators and . As those will be unitary, they have norm 1 and the canonical commutation relations read (with the help of B, C and H) . If you later want, you can go back to and similar for .

Our goal is to come up with a Hilbert space where these operators act. In addition, we want to define a scalar product on that space such that and act as unitary operators preserving this scalar product. We will deal with the position representation, that is wave functions . and then act in the usual way, by translation and by multiplication . Obviously, these fulfil the commutation relation. You can think of and as the group elements of the Heisenberg group while and are in the Lie algebra.

Here now comes the only deviation from the usual path (all the rest then follows): We argue (motivated by similar arguments in the loopy context) that since motion on the real line is invariant under translation (at least until we specify a Hamiltonian) is invariant under translations, we should have a state in the Hilbert space which has this symmetry. Thus we declare the constant wave function to be an element of the Hilbert space and we can assume that it is normalised, i.e. .

Acting now with , we find that linear combinations of plane waves are then as well in the Hilbert space. By unitarity of , it follows that , too. It remains to determine the scalar product of two different plane waves . This is found using the unitarity of and sesquilinearity of the scalar product: . This has to hold for all and thus if it follows that the scalar product vanishes.

Thus we have found our (polymer) Hilbert space: It is the space of (square summable) linear combinatios of plane waves with a scalar product such that the are an orthonormal basis.

Now, what about and ? It is easy to see that when defined by a derivative as above acts in the usual way, that is on a basis element which is unbounded as can be arbitrarily large. The price for having plane waves as normalisable wave functions is, however, that is not defined: It would be . But for the two exponentials in the denominator are always orthogonal and thus not "close" as measured by the norm. The denominator always has norm 2 and thus the limit is divergent. Another way to see this is to notice that would of course act as multiplication by the coordinate , but times a plane wave is no longer a linear combination of plane waves.

To make contact with loop cosmology one just has to rename the variables: What I called for a simplicity of presentaion is the volume element in loop cosmology while the role of is played be the conjugate momentum .

If you want you can find my notes for the blackboard talk at HU here (pdf or djvu

Entropic Everything

The latest paper by Eric Verlinde on gravity as an entropic force makes me wonder whether I am getting old: Let me admit it: I just don't get it. Is this because I am conservative or lack imagination or too narrow minded? If it were not for the author, I would have rated it as pure crackpottery. But maybe I am missing something. Today, there were three follow-up papers dealing with cosmological consequences (the idea being roughly that Verlinde uses the equipartition of energy between degrees of freedom each getting a share of 1/2 kT which is not true quantum mechanically at low temperatures as there the system is in the ground state with the ground state energy. As in this business temperature equals acceleration a la Unruh this means the argument is modified for small accelerations which is a modification of MOND type).

Maybe later I try once more to get into the details and might have some more sensible comments then but right now the way different equations from all kinds of different settings (Unruh temperature was already mentioned, E=mc^2, one bit per Planck area, etc) are assembled reminds me of this: