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You’ve signed up for a cloninmixag experiment. Gotta pay off them student loans, right? The
researchers put you to sleep and place you in their cloning machine. It clones you at a
subatomic level, creating two perfectly copies. As they had previously informed you, they even
take one of your clones (still asleep) to a room in LA and the other to a room in NY. As you wake
up, you remember everything about the experiment up until the point you were put to sleep. A
researcher in the room with you then asks you “What is the probability that we are in LA?”
You probably reply that there’s a 50% chance you’re in LA. After all,
you know there are two copies of you but you have no way to tell which one you are.
OK, so what if we split you into countless clones traveling the many quantum timelines predicted
by the Many Worlds interpretation of quantum mechanics. Can we figure out the probability that
you’re in one particular timeline? It turns out we can, and in the process derive the probabilities
for measurements of the quantum wavefunction – that is, to derive the famous Born Rule.
Flip a coin and cover it without looking. You know it’s definitely either heads or tails
before you look, and if it’s a fair coin the probability of it being either is 50%.
Now if this was a quantum coin, it’s still 50-50 that you’ll see heads or tails when you look,
but before you look the coin is not in either definite state.
See, quantum mechanics describes the world as wavefunctions—fuzzy wuzzy
distributions of possibility that evolve according to the Schrodinger equation.
Before the quantum coin is revealed, its wavefunction is neither heads nor tails,
but a quantum superposition of both. What does that even mean? I don’t know. No one knows. But
we do know that we have to describe the quantum world in terms of superpositions to
get the right answers when we calculate quantum processes. Upon observing the coin,
this superposition evaporates and the coin is once more definitely heads or definitely tails.
The mystery of what happens when we go from a superposition to a definite state is known as
the Measurement Problem, and it’s arguably the most mysterious outstanding problem in physics.
The different interpretations of quantum mechanics are really about solving the measurement problem.
The most mainstream answer—the Copenhagen interpretation—argues that upon measurement,
all but one of the superposition of states vanish, leaving only the
state we observe. We say the wavefunction collapses. Then we have Pilot Wave Theory,
which states that there’s an underlying sort of labeling system—specks of reality we’ll call
corpuscles—that defines which of the multiplicity of states in a superposition are actually real,
relegating the rest of the branches to a sort of empty, ghostly state. Both Copenhagen and Pilot
Wave need to add some additional mechanics to the Schrodinger equation in response to
the Measurement Problem. And then of course there’s the Many Worlds Interpretation, which
states that the Schrodinger equation is the entire story—there’s no collapse, no labeling of the one
real branch. Rather, the wavefunction evolves only according to the Schrodinger equation forever.
In our previous episode in this series we talked about how Pilot Wave Theory
may just be Many Worlds in disguise. But I promised we’d see if Many Worlds can
take us further. For example, does it solve the Measurement Problem?
In other words, if Many Worlds tells us that wavefunction doesn’t collapse,
why don’t we observe superpositions of macroscopic states—like the alive and
dead cat in Schrodinger’s thought experiment? Well, Many Worlds would say it’s because we
aren’t viewing the wavefunction from outside. Rather, our subjective experience is embedded
within a branch of the wavefunction where we can observe only a single outcome—say,
heads—which means there’s another you in another branch observing tails.
Whether or not this is true, it’s compelling enough for many physicists
to take it very seriously. The problem is that on the surface Many Worlds seems
unverifiable. There’s no way for us to ever interact with these other branches.
But there may be a way to at least improve our confidence in a theory without a direct test,
and that’s by seeing if the theory offers an explanation for something else confusing in
quantum mechanics. And there is one more thing that we’ve just sort of accepted since the
beginning of quantum mechanics but never fully explained, and that’s the Born rule. The Born
rule is how we find the probability of getting a particular result when we make a measurement. That
probability is just the square of the wavefunction for the property we’re measuring. The Born rule is
taken as a starting axiom in standard quantum mechanics. But if an interpretation could tell
us why the Born rule works the way it does, then that would be a point in its favour.
Well, today I’m going to present an argument that the Born rule falls naturally out of Many Worlds,
but not quite so naturally out of other interpretations.
Let’s start by looking at our quantum coin with some proper
quantum language. We can write its pre-measurement
wavefunction as a linear combination of a wavefunction for heads and for tails.
The alpha and beta are coefficients that represent the weight or strength of each of the basis
states. We can think of the wavefunction as an arrow—what we call a state vector—pointing in
some direction indicating how much “heads-ness” or “tails-ness” this wavefunction has. The
way the Schrodinger equation works, the length of this arrow is always the same,
so good ol’ Pythagorus tells us that its length is the sum of the squares of the coefficients.
For convenience we always set the length of this vector to equal one. So in this case we
alpha and beta are 1/2.
For our quantum coin, the Born rule tells us that the probability that we’ll measure heads or tails
is the square of the coefficient associated with that state. Alpha-squared it’s heads,
beta-squared it’s tails. If this is a fair coin then the probabilities
are the same—both alpha-squared and beta-squared need to equal a half,
so alpha and beta are the square root of a half. Our fair-quantum-coin wavefunction looks like this
If we’re committed to the coefficients defining the probabilities—which honestly we don’t have
a good reason for just yet—then the Born rule is the only way for those probabilities to be equal
for equal coefficients. But we can do better than this—we can show that the probabilities
HAVE to be governed by the coefficients and that the Born rule applies even if they aren’t equal.
Let’s look at what happens when we measure our quantum coin to see if we can find the place
where coefficients turn into probabilities. To do this we need to place ourselves into
the wavefunction and track the series of interactions between the flipped quantum
coin and our awareness of the result of that flip. Because the universe is
fundamentally quantum, the classicalness emerges
from the combination of an extremely large number of quantum interactions.
Let’s write the wavefunction of the universe right after the coin flips but before the
result is known. We include the state of the measurement apparatus by just multiplying its
own wavefunction into the coin’s superposition. As well as the observer’s wavefunction, and the
wavefunction of the rest of the universe—E for the environment or for everything else.
As information about the flipped coin travels out, each of the miriad individual quantum
constituents of these external entities eventually becomes entangled with the
flipped coin. Each quantum element of the surrounding world enters its superposition
of the two possible states—having interacted with a result of tails and having interacted
with a result of heads. Assuming we don’t collapse any wavefunctions, then the measurement device,
the observer, and everything else enters the horrifically complex superposition.
The Measurement Problem is really the question of why we never observe such
macroscopic superpositions, but rather only one the definite results within. Many Worlds
claims to solve this by stating that we are not outside the superposition. We’re here,
or here, inside one of the branches. We see the superposition component that we are inside,
not the whole thing. The many quantum states that make up your brain and
hence your conscious experience split with the rest of the world.
Many Worlds can’t say why we observe one specific state over another state—rather
it tells us why we observe only a single state. And it can also tell
us why we measure the probabilities that we do—why we get the Born rule.
Just like in our first thought experiment where a
clone could state a probability of finding themselves in LA versus NY,
perhaps we can figure out the probability of finding ourselves in one “world” versus another.
In their reasoning, our clone applied something called the Principle of Indifference,
which states that “in the absence of any relevant evidence, agents should distribute their credence,
or 'degrees of belief', equally among all the possible outcomes under consideration.” So,
two outcomes, no evidence to distinguish them,
their probability, evenly divided should add to one—ergo 50-50 LA versus NY. If there was a
third clone sent to Buenos Aires then the probabilities would by ⅓ for each city.
Let’s do our quantum coin experiment again, but now asking our observer to close their eyes
until the coin, the measurement device, and the environment know the result of the flip.
There’s going to be some little neural circuit in the brain of the observer corresponding to
knowing the result of the coin flip. We define the wavefunction of that circuit as the observer,
and it is identical after the flip but before becoming aware of the result of the flip.
So, the observer asks “what’s the probability that when I open my eyes I see heads versus
tails?” We have two possible worlds, so is it 50-50? Well,
that would be true if alpha and beta are the same. It’s possible to show very rigorously that when
branches have equal coefficients the Principle of Indifference applies and the credence an observer
should have of being in the tails branch or the heads branch is the same—and I’ll
put a link to that proof in the description. The probability of you observing one of any
number of equally-weighted outcomes is one over the number of outcomes—one over the number of
“worlds”--which is the coefficient of your world squared. And that is, of course, the Born rule.
But what if the coefficients for the quantum coin are different? Say, and squqre
root(1/3) and square root(2/3)? If there are still only two worlds, heads and tails
then it seems intuitive that the probability should still be 50-50. Why should the coefficients
even matter? But the Born rule says that the probabilities are ⅓ and ⅔ for heads versus
tails in this case. So does that mean Many Worlds is inconsistent with the Born rule?"
Actually, no. There is a way to go from quantum states to the correct probabilities even in
the case of unequal coefficients without ever assuming the Born rule. We can still count worlds.
But before we do that, let’s count cards. I deal you 6 cards, and only one of them
is a joker. I shuffle and deal out two stacks of three—one for you and
one for me. What’s the probability that the joker is in your stack? You know it’s 50%,
and you know that because you intuitively apply the principle of indifference.
Just to be pedantic, let’s formally justify the principle of indifference in this case. If I
swap the stacks everything looks the same, so your credence that you have the joker
shouldn’t change. But also your credence that the joker is in your new stack after the swap
should be the same that it’s in my stack before the swap because they’re the same stack. So the
probability of the joker being in your stack pre-swap is the same as it being in my stack
pre-swap. Those two probabilities have to add to one, so they’re each a half.
But what happens if, after dealing, I take move card from your stack to mine. Now you have 2 cards
and I have 4. Now what’s the probability that you have the joker? The Principle of
Indifference can’t be applied directly because if we swap the stacks things look different.
But there’s a way around this. I just split my deck in 2. Now we
have 3 stacks of 2 cards. Any swapping of these stacks looks exactly the same,
so the principle of indifference tells us that there’s equal probability that the joker is in
any one of them. So the probability that the joker is in your one stack is ⅓. It’s also ⅓
that it’s in either particular one of my stacks, and ⅔ overall that it’s in one of my two stacks.
Now we could have got the same answer just with frequentist arguments about
there being one joker in 6 cards of which you have 2 cards. But
that’s equivalent to the swapping argument with stack sizes of one card.
So how do we apply this to Many Worlds? Unlike with the cards we don’t know,
and perhaps can’t know, how many discrete worlds
there are. But that doesn’t mean we can’t divide the worlds into stacks.
Let’s go back to the case of our unfair quantum
coin. Leaving out the measurement device and the environment for now,
the wavefunction after flipping the coin and prior to opening our eyes looks like this.
We know from the principle of indifference that these two branches should NOT have
equal probability. So let’s break up the second term just like we split
our stacks of cards. Don’t worry just yet if we’re even allowed to do this. We
say that the tails state is really two equal sub-states |T_1> and |T_2>
The coefficient of sqrt(½) comes from thinking of these sub-states as orthogonal
vectors that make up the tails state, with lengths given by Pythagorus' theorem.
We could do any number of substates,
but two gives us the right coefficient so that when we expand this out…
All coefficients are now equal. Now we can apply the principle of indifference to say that the
probability of all these branches is ⅓. That’s ⅓ for heads and for either of the tails substates,
or ⅔ for the sum of tails substates. And there’s your Born rule—the coefficients squared are the
probabilities. And that’s just from the Principle of indifference forcing you to
equally weight your credence about which branch you’re in when the coefficients are equal.
OK, some very serious questions immediately come to mind. Like, are we really allowed to break up any
state in any way we want? Are those divided components also valid quantum states? Well,
remember that we never observe a quantum object directly. We are observing the
macroscopic effect of that quantum object on a measurement device and embedded in an
environment. The full wavefunction of the quantum coin in the world looks like this.
The measurement device and the environment are made of countless quantum objects,
each of which can also exist in multiple states. So after the coin flip,
there are many, many states in the device and the environment that all correspond to heads,
and many also to tails. No matter the coefficients of the quantum event, we can split the states of
the world so they have equal coefficients between those states. Then we can apply the principle of
indifference to say there’s equal probability of being in one of these “stacks of worlds”. Then
just sum the stacks that are associated with a particular outcome of a quantum event. You’ll
always find the resulting probability is equal to the square of the coefficient for that outcome.
We should also ask: Is this argument is really exclusively valid for Many Worlds? Can we use
this to derive the Born Rule for Copenhagen or Pilot Wave Theory for example? Well,
some parts are valid for other interpretations. For example, as long as we imagine that the state
we’re measuring is in some way course-grained—so, composed of other, finer states, then we can split
up the state so that the overall superposition is a chain of states with equal coefficients.
But it’s not then clear what it means to apply the principle of indifference in the other interpretation.
In Copenhagen, we have to say that the likelihood of branches collapsing is
governed by the coefficient of the branch, such that equally weighted branches have
equal likelihood of surviving the collapse. Same with Pilot Wave Theory, but with respect to the
locations of the corpuscles rather than branch collapse. If we’re happy with that then sure,
we can apply the principle of indifference to get the Born rule. But we still have this
axiomatic requirement linking the coefficients to some mysterious quantum event that’s not part
of the Schrodinger equation, even if it’s a more subtle axiom than the full Born rule.
In Many Worlds, the application of the principle of indifference is the correct way to decide
which branch—or family of branches—we belong to, and so we can say that the Born rule does
follow naturally from this interpretation. Many-Worlds is rather unique amongst the
popular interpretations because it has a direct path to finding probabilities,
rather than having to bake it into the theory by axiom. We just need to ask what is the chance
of finding ourselves in this or that part of the wavefunction of an endlessly branching Space Time.
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