I’ve come to realize that, when it comes to the Many Worlds Interpretation (MWI) of quantum physics, there is at least one aspect of it that’s poorly understood. Since it’s an aspect that even proponents of MWI recognize as an issue, I thought I’d take a stab at explaining it. (If nothing else, I’ll have a long reply I can link to in the future.)
The issue in question involves what MWI does to probability. Essentially, our view of rare events — improbable events — is that they happen rarely, as we’d expect. Flip a fair coin 100 times; we expect to get heads roughly 50% of the time.
But under MWI, someone always gets 100 heads in a row.
Granted, the overwhelming bulk of experimenters do get roughly 50% heads after 100 flips of a fair coin.
[The “fair coin” here is an experiment with a photon passing through a half-silvered mirror. If the photon passes through the mirror to detector A, we call that “heads.” If it’s reflected into detector B, we call that “tails.”]
Essentially what we’re doing is creating a 100-bit number. When the coin is heads, we set the bit to 1. When the coin is tails, we set the bit to 0. We’re interested in how many 1s are in the resulting 100-bit number. We expect, on average, 50.
Which is what we observe if we perform the experiment.
Mathematically, determining what we should get involves determining how many outcomes are possible and categorizing those outcomes.
Our experiment with 100 coin flips (alternately: 100 random bits), involves a 50/50 individual outcome performed 100 times. Each of these is separate from all others. That means we multiply a two-outcome result 100 times:
2100 = 10∼30.1 = 1,267,650,600,228,229,401,496,703,205,376
Obviously a whomping big number. It’s the total number of distinct patterns 100 bits can have. (Alternately, it’s how high you can count with 100 bits.)
It therefore is also the (extremely long) odds of any specific sequence occurring. (Odds like that make one think they’re on the Heart of Gold.)
Let’s first consider two patterns: all heads, and all tails. Both are specific patterns, so both have the extremely low odds of occurring just mentioned. (The odds of getting either are only half that huge number above. Effectively, but not quite, zero.)
We’re interesting in how many heads occur in the pattern; in this case, one-hundred and zero, respectively. And, as mentioned, these patterns are distinct. There is only one pattern with 100 heads and only one with 0 heads.
How many patterns have two (and only two) heads? There’s an app for that, and by app I mean formula:
The formula is for a mathematical process often called n-choose-k. The idea is that you have n items, you choose k of them, and want to know how many patterns can be formed from those k items.
If we do the math, which involves some big numbers — n! has almost 160 digits — we get 4,950. There are that many 100-bit patterns with just two heads (the rest tails).
Recall there was one pattern with 100 heads and one pattern with 0 heads. That mirror symmetry persists: there are also 4,950 patterns with only two tails (the rest all heads).
(I skipped patterns with just one heads (or one tails). It should be obvious there are 100 each of those.)
The number of possibilities builds rapidly as we have more heads (or tails). With three, there are 161,700; with four, 3,921,225. By the time we get up to 10 heads, there are 17,310,309,456,440 combinations (and an equal number with 10 tails).
If we skip all the way to 50 heads (and 50 tails — the reflections meet), we find a very large number of patterns with an exact 50/50 mix:
You’d think a hefty 30-digit number like that would give pretty long odds on turning up in a random selection of bits. The odds are certainly higher than the near zero odds of a single bit pattern: 7.96%
Nearly an eight percent chance!
In contrast, a pattern with 30 heads (or 30 tails) also has what seems like an awfully large number of occurrences:
But it turns out the odds are still very nearly zero: 0.0023% (a tiny bit more than two one-thousandths of a percent).
So if there’s only an eight percent chance of getting a 50/50 pattern, why do “the overwhelming bulk of experimenters” get roughly 50% odds?
Because of the roughly.
Here’s a histogram showing the frequencies (number of occurrences) of a given number of heads (or tails) for patterns with 16 bits:
As you see, the 50/50 pattern (with 8 bits set) occurs most often, but what’s important here is that the 7-bit and 9-bit patterns — which are almost 50/50 — also occur many times.
The 6-bit and 10-bit patterns show the drop off, but still occur many more times than the outlying areas. The 0-bit and 16-bit patterns, which we know can occur just once each, don’t even register. (The 1-bit and 15-bit patterns we also know will have just 16 occurrences each.)
Going back to the 100-bit pattern, when we include a range of roughly 50/50 patterns, the odds approach “overwhelming bulk” percentages. If we include all bit patterns from 40 through 60, we end up a 31-digit number:
Being more than an order of magnitude larger than the 50/50 number, the odds also jump by more than an order of magnitude: 96.48%
(Note it’s close to the total possible: 1.223 versus 1.268.)
So there are very good odds on getting roughly a 50/50 mix of heads and tails if we flip a fair coin 100 times.
So what does all this have to do with MWI?
If we performed this 100-flip experiment, we’d be very surprised to get all-heads (or all-tails) on the first trial.
To be clear, if we named any specific sequence, we’d be very surprised to get it the first time we did the experiment. Remember that the odds of any specific sequence are nearly zero.
I’ll explain below, but the punchline is that, under MWI, there is always a clone who gets all heads, and always a clone who gets all tails, every time the experiment is performed.
If we perform this this experiment enough times, it eventually produces all bit patterns. If we do it enough times we will get an all-heads pattern.
Given the near-zero odds, we wouldn’t expect it often or at first, though. It’s not impossible, but it’s not likely.
Now consider what happens under MWI when Alice performs the 100-flip experiment.
MWI involves an exponential explosion of realities, but we’ll focus only on the ones caused by the experiment. Therefore we start with one Alice (ignoring all her other clones from other quantum splits).
She pushes the button to start the experiment. The device will send 100 photons, one after the other, and record whether they hit the A or B detector. This will create a 100-bit number that is the result of the experiment.
The first photon causes Alice to split. We can label them Alice(0) and Alice(1) to reflect the building bit string a given Alice sees occurring.
The second photon splits both Alices, giving us four clones: Alice(00), Alice(01), Alice(10), and Alice(11).
The third photon creates eight Alice clones, the fourth creates sixteen, and so on, doubling the number of Alice clones each time. Each clone sees a different bit string — different pattern of heads and tails — build up bit by bit.
As this continues, most of the clones see roughly an equal mix of heads and tails. Some of them are a little surprised at the imbalance of heads or tails. A small number are very surprised they’re seeing all heads or all tails.
Two of the clones, Alice(1111111…) and her reflection, Alice(0000000…), are increasingly surprised. Their surprise continues all the way to the end of the experiment when the former ends up with all heads while the later ends up with all tails.
This happens every time the experiment is performed.
It creates 1030.1+ Alice clones, two of whom are very surprised. (And many trillions more are quite surprised at the striking imbalance of heads and tails.)
If we perform an experiment enough times, we would expect to see long odds things happen. Rare things happen rarely; this seems mathematically sane.
Under MWI rare things always happen. Every time. It’s only given branches of splitting that are somehow rare, which is the real problem. Since these branches always occur, in what sense are they rare?
To some extent the answer is in the relatively large numbers of expected outcomes compared to the rare ones. (Everett’s original paper leans heavily into statistical analysis, which might be his attempt to answer this. I can’t quite tell how he’s applying it, yet.)
The fact remains that, under MWI, rare events always happen to someone, whereas we normally expect that rare events rarely happen to anyone.
FWIW, Jim Baggott, in his book Farewell to Reality (see this post) has a version involving four flips of a weighted coin. He expects to see heads only 25% of the time with that coin, so the expected outcome of four flips three tails and one heads.
His experiment creates 16 Jim clones, only 25% of whom get the expected outcome (one of: 0001, 0010, 0100, 1000). The other twelve all see an unexpected outcome. (The issues MWI introduces with probability become even more urgent when a given quantum event is not 50/50.)
As Baggott says, that seems hard to reconcile with our experience.
Stay improbable, my friends!