Understanding the Mathematical Formalisms of Everetts MWI

Question

Link to article: http://jamesowenweatherall.com/SCPPRG/EverettHugh1957PhDThesis_BarrettComments.pdf

I'm writing an essay on Everettian MWI and its incompatibility with Born Rule probabilities. I understand the concept I'm explaining quite well (I think) but I fear without direct reference to the math behind the idea I will not a) convey my point as effectively as possible and b) understand the problem at hand fully.

My understanding: Everettian MWI can not be compatible with the Born Rule because each vector has a a P<1 probability of occurring YET at each interaction multiple universes (one for each vector in the superposition) are created with 100% probability. This is especially evident in the case of Schrodingers Cat if there you are in a world where there was no decay after 1 hour the probability that it decays in the next hour is exponentially higher yet once again the worlds will split evenly one in which the state decayed (and cat dead) occurs and the other in which non-decay (cat alive occurs) all possibilites occur with 100% probability.

What I'm missing:

I believe that this is the part of the article that lays out the math involved

(pages 11 & 12 in the linked article)

But I can't for the life of me really figure out what it's saying as I have no background in physics or math. If someone would be so kind as to breakdown the equations it would be appreciated.

Answer 1

I think you may be starting with the wrong premise. MWI has troubles deriving the Born rule because it does not use it from the outset. So you cannot even talk about probabilities within this interpretation before providing a rationale for what they are.

At any given time a Schrodinger cat would be in a superposed state alive and dead with various (time-dependent) weights assigned to either possibilities.

However, at the same time, decoherence ensures that such a superposition cannot be observed in practice (we cannot make these two incompatible states interfere with each other); that's the universe splitting phenomenon in MWI.

Now, from a naive frequentist approach, the probability of the cat to have survived after a time interval $t$ after the experiment has started is something like the "number" of histories compatible with it to be still alive over the total "number" of histories associated to the state of the cat. These two "numbers" are likely to be a) infinite and b) very hard to estimate, provided the frequentist reasoning makes any sense at all! Notice, however, that the ratio I have used to define the probability is a priori equal to or less than one, regardless of the time elapsed after the experiment has started.

Now, this is a very rough explanation on why the splitting of the MWI does not entail incompatibility with probability theory or the Born rule and people have been thinking quite a lot about it recently (and often adopted a Bayesian view point for probability to obtain a tractable strategy to obtain the Born rule).

I recommend you take a look at David Wallace's work on it, which I often find quite enlightening on the subject (although not always an easy journey).