Maudlin on Wallace on the Problem of Probability

In this post I will give a quick summary of how Wallace deals with the problem of probability in the Everett interpretation followed by Maudlin’s response and finishing with some questions/discussion points.

In the Everett interpretation there is no collapse of the wavefunction during measurement, instead the state vector of the universe evolves unitarily always. Standard quantum mechanics primarily makes statistical predictions of the probabilities of various measurement outcomes following a measurement via the Born rule. The problem of probability is, hence, the problem of recovering the correct probabilistic predictions of quantum mechanics.

Wallace’s solution is that the branch weight ie the modulus-squared amplitude of the components of the universe’s state vector should be interpreted as objective probability in the Everett interpretation. He shows that, following from some rationality assumptions, a (rational) agent will use the branch weight as a subjective probability for making decisions. Then according to a doctrine of functionalism—that if some object is treated just like a particular type of thing then we should take it to actually be that type of thing—he reasons that we should interpret branch weight as objective probability as well.

Maudlin’s review starts by praising Wallace’s book’s thoroughness and consideration; goes on to complain about Wallace’s claim that the Everett interpretation is a `literal’ reading of quantum mechanics; raises objections to physical ontology being based on patterns in the wavefunction; and then moves onto what I will consider: the problem of probability.

According to Maudlin, the functionalism that Wallace employs is a double-edged sword since it also implies that if branch weight is not treated by rational agents as a subjective probability then nor should the interpreter of quantum mechanics consider it to be an objective probability. He then goes on to argue that a rational agent would not necessarily use branch weights as subjective probabilities (although I don’t believe he goes so far as to show that such an agent would not use them as such).

Maudlin takes issue with one of Wallace’s axioms: that the set of `rewards’ between which an agent may doesn’t allow for any considerations of several (classically mutually exclusive) things happening simultaneously in different branches. In particular an agent cannot consider the possibility of performing a quantum experiment in order to get two simultaneous outcomes (albeit in different branches). Maudlin suggests that if one takes the Everett interpretation seriously one should certainly consider in one’s decision-making that after a branching event several outcomes happen simultaneously. With this axiom false (according to Maudlin) the edifice upon which Wallace’s representation theorem rests falls, and Wallace cannot claim branch weights to be rational subjective probabilities nor, via functionalism, objective probabilities.

Discussion questions/points:

Wallace seems to imply (p148 first complete paragraph) that symmetry principles are more applicable to probabilities for an Everettian. Does his derivation of the Born rule in a special case in section 4.13 bear this out? It’s not obvious to me that his argument really requires Everettianism.

Maudlin’s claim that a rational agent might value two results occurring simultaneously (on different branches) over either one result occurring could be countered if Wallace could argue that such a preference would just not be rational. Consider someone who regularly paid for quantum experiments in order to get the `study both history and physics’ style outcomes that Maudlin describes. It seems to me that such a person might quickly come to decide that they are wasting their money after having conducted several such experiments and still only experienced one outcome each time. Would this behaviour be irrational or is it, as Maudlin seems to believe, an arrational and valid preference?

Maudlin draws a distinction between betting in the normal sense and betting on Everettian quantum experiments: `No wagering or betting is involved, so the thorny issue of decision-making under uncertainty never arises.’ (p801 final sentence of 2nd paragraph) This is reminiscent of the `probability requires genuine uncertainty/indeterminacy’ objection that Wallace deals with in section 4.2. Is it the same? or related? is it as easily dismissed?

Dynamical Collapse Theories

Hi everyone! Firstly, many apologies for this being late.

Since we never got around to lecture 5, I have focused this post on the end of lecture 4, namely the third proposed single world interpretation of quantum mechanics: the dynamical collapse theory.

I thought I’d summarise my understanding of this general interpretation, before focusing on a few specific examples, highlighting any problems that I had in interpreting them. Here goes…

Objective Collapse theories, also known as QMSL theories (Quantum Mechanical Spontaneous Localisation), are realistic, indeterministic and fundamentally reject hidden variables. The general idea is that the wave function associated with all sub-atomic particles has some ontological reality, along with a very small but finite probability of collapse. The time scale associated with the collapse of individual particles is thought to be of the order of 10⁸ years.  When an observer makes a measurement using some physical piece of apparatus (be that an eye, or lab equipment etc) then the observed particle becomes entangled with the observing equipment, which will necessarily be a very large collection of atoms >10⁸. The chance of any one of the large collection of particles collapsing is now very high, and since they are entangled, if one collapses so too must all the particles in the system.

Dynamical collapse theories originated out of attempts to get around the measurement problem in quantum mechanics, which inexplicably gives the observer in any system a special position, in that they are solely responsible for the collapse of said system. Such a view, as held in the popular ‘Copenhagen Interpretation’, leaves much room for interpretation in what constitutes an observer or what qualifies as an observation. DCTs neatly remove this issue altogether by attributing the collapse of the wave function to random processes with no contribution by the observer other than providing the measuring equipment.

**As an aside, I was wondering how you might justify the view that particles just so happen to follow a theory of random collapse, but after further thought I realised that there are many examples in physics of this, spontaneous atomic decay in nuclear fission being the most obvious example. I suppose this view still suffers from the issue that Einstein took with QM (god does not play dice) in that it offers no explanation for the basis of the random collapse in the first place, to which the only retort I am aware of, is that this is just fundamentally how the universe operates, and we as humans are not in a position to suppose it should be any other way. Incidentally if anyone has anything to say on this then I would be keen to discuss, as I don’t find it altogether that satisfying.**

Many theories fall under the dynamical collapse umbrella, but as far as I am aware there are two main variations, based on the mechanics of the wave-function collapse:

1.     In the first group the collapse is found ‘within’ the wave-function. I.e. the formula which guides the evolution of the system (the Schrodinger eq), which under normal formulation describes the state of the system as some linear superposition of the basis states, is adjusted in some non-linear manner. This adjustment means the governing equation actually describes the collapse itself. The original GRW paper is a good example of this view.

2.     Evolution of the wave function remains unchanged, with the additional collapse process added in. The most famous example would be the ‘Penrose Interpretation’.

The first variation is the most accepted version of the theory I believe, although it has been modified a little since it’s first inception by GRW in the 80’s. The original paper didn’t respect the symmetries of many particle systems (didn’t conserve parity?), so a modified version of the theory was created, known as CSL (Continuous Spontaneous Localisation), which amends the shortcomings mentioned above.

Despite the many attempts to reconcile dynamical collapse with our present understanding of QM, there remains a fundamental issue with variation 1). Since the collapse occurs from within the wave function, the collapse cannot be complete or else the principle of conservation of energy would be broken. Why this must be is not completely clear to me, so I would like to discuss this tomorrow if possible. A non-complete collapse implies that the wave function has some non-zero amplitudes for states not described by the collapse, and so in theory there is a non zero chance of the system (or particle) simultaneously jumping to another state. This is fairly counter-intuitive and has not been verified by experiment thus far.

I will now discuss the interpretation of the second variation of DCT, namely the ‘Penrose Interpretation’. Penrose theorises that the wave-function collapses to it’s basis states when the space-time curvature of the quantum states attains a significant level, which he speculates to be around 1 graviton, (the hypothesised but yet to be discovered force carrying particle in quantum gravity).

To explain this, let’s consider a solitary particle, which has multiple basis states that it could occupy (eg the position of an electron around an atom). The theory states that each of the electron’s states exists simultaneously, and so the electron creates a gravitational field in all the positions that it can exist according to its basis states. A field intrinsically contains energy, and requires energy to be maintained, thus the larger the mass (energy) of the particle, the higher the required energy that is needed to maintain all the fields that exist curtsey of all the simultaneously existing states. The next step is to say that for a small particle, the energy required is small enough that it can exist as a superposition indefinitely. (I couldn’t find a precise reason for this, but I think you can use the uncertainty principle ∆E∆t>h/2π and say that for a very small mass, the time can be extremely large).

Likewise for larger particles, their super positions require too much energy to simultaneously exist for any significant period of time (again, presumably found from the U-P). Since it would require less energy to maintain one gravitational field than all the countless super positions, Penrose believes that this energy requirement causes larger mass systems to collapse to a single state, with the probability of collapsing to a specific basis state given by the square of the amplitude as usual. Penrose speculates that the transition from macroscopic to the sub-atomic domain would happen at around the mass of a dust particle.

To summarise, both variations of DCT attractively side step the measurement problem, giving no special position to an observer. They also rule out many world theories by holding the collapse to curtail the branching of the wave function, removing unobserved behaviour.

Discussion Points:

Personally I find the Penrose interpretation to be the most attractive of all the interpretations we have seen so far. The Copenhagen interpretation throws up countless issues with the interpretation of what the wave function represents and why observers should be gifted such a special position in the universe.

Variation 1. of the DCT seems a little like GRW have moulded the solution to fit the answer. I don’t like resorting to spontaneous and randomly collapsing wave functions, but as I mentioned previously there are other example in physics of this kind of thing.

I am also rather unclear on how the wave function can have any physical reality, and what that would mean or look like.

If you’ve got to the bottom of this then thanks for staying with me, I’m sure some of this is quite unclear so looking forward to discussing tomorrow!

Best,

Jamie