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.