When Worlds Branch¶
So far, we have made occasional allusions to picturing different entangled components as existing in “different worlds.” Let’s discuss this point a little more clearly. After all, we are hoping to understand the “Many Worlds” interpretation.
When it comes to the Many Worlds interpretation, some people love it and some people hate it. Sometimes both the love and hate stem from the same sorts of misconceptions. What do you imagine when you hear about “Many Worlds”? Somehow, new universes being created from nothing every time a subatomic particle can go one way or another, perhaps. What sort of sound does that make? Is it cataclysmic, like new Big Bangs happening millions of times per second? Or do the universes just pop into existence, like an index finger popping out of your cheek (“Loll-i-POP!”) Where does the matter and energy come from to make all these universes? Can we visit them? Is there a universe where I am a rock star and/or Supreme Court justice?
If you have read the preceding sections of this tutorial, you will have noticed that there is nothing like the above absurdities. There is just the proposition that particles are described by wave functions that evolve according to the Schrödinger equation (sorry if it has been boring). This proposition includes the idea that the wave function for multiple particles describes an amplitude for all the different configurations of the particles together, not separately.
In fact, we could even picture a wave function for all the particles in the universe. Granted, it is difficult to actually picture such a wave function, as it has an astronomical number of dimensions. But somehow, we can think of blobs floating around, interacting with each other, splitting apart, interfering, etc. It is worth noting that the wave function doesn’t have to look like discrete blobs, it can have any continuous spatial dependence in general. In any case, we can posit that there is such a wave function, and it evolves in time according to the Schrödinger equation.
The picture of many worlds emerges from this singular wave function and its evolution. In the preceding examples, we have often seen a wave function described by two or more blobs. The blobs exist in a space with a number of dimensions equal to the sum of the spatial dimensions for each particle. (In all the examples, the particles only move in one dimension, but real particles move in three dimensions so such wave functions would need three dimensions for each particle.)
It has often been convenient to consider the different blobs of a wave function separately. This has raised the question, can we consider the different blobs to exist in different worlds, completely distinct from each other? We have seen that the answer is “not always”. If the blobs are aligned along some dimension, then if the components of the wave function in that dimension come back together, then interference will occur and our assumption of separate worlds will be spoiled. If the blobs are not aligned along any dimension, we might still worry that they might become aligned, either by chance, or by something like a Quantum Eraser experiment.
The real question then is, “When can we really safely assume that two blobs are never going to overlap?” Then we can say with confidence that they are in separate worlds. This would likely be the case once the blobs are misaligned along a large number of dimensions. This happens quite commonly in the case of decoherence, discussed in the previous section (Amplification and decoherence).
If we consider a blob getting randomly buffeted by collisions with particles in the environment, we might picture its trajectory as a random walk. The standard idea of a random walk is a path where steps of fixed size are taken, each in a random grid direction. In one dimension, each step would be randomly forward or back, in two dimensions, randomly north, south, east, or west, etc. This staggering path is also called the drunkard’s walk. It has been shown that, the higher the dimension, the less likely the path will be to return to the starting point, and the more likely to head off somewhere never to return. In one and two dimensions, the path will always eventually return to the starting point. In three dimensions and higher, the path will always eventually head off towards infinity. In three dimensions, before heading off to infinity, there is a 34% chance that the path will return to the starting point at least once. By eight dimensions, that probability is down to 7%. For decoherence of a macroscopic system, we might expect something like a mole of particles to become entangled with our system, on the order of \(10^{23}\). That is, a pair of blobs would become misaligned in \(\sim 10^{23}\) dimensions. The odds that it would ever randomly come back into alignment are unimaginably miniscule. Especially so if you consider that more particles keep on getting entangled, increasing the number of misaligned dimensions further. Likewise, it is unimaginable how one could intentionally engineer a system to bring all those dimensions back into alignment. The point is: in a very large number of dimensions, there is a tremendous amount of space to get lost in. In such a case, we can be very confident in saying those blobs are now separate, and we might as well consider them to be in separate worlds.
We have argued that the process of macroscopic decoherence can lead to sufficient separation of two components of a wave function that we can safely consider them to be in separate “worlds”. This argument does imply that there is no hard line between one world and two worlds. If we consider the proposed “marble run” measurement device in Fig. 8.4, we could ask at what step in the process does one world branch irreversibly into two? We might start the original particle to be measured in a superposition of packets in the two positions. This results in two blobs, but these blobs are only separated along one dimension initially. As the two packets separate and start knocking out the increasingly heavy particles on either side, the blobs start getting misaligned in increasing numbers of dimensions. If particles from the environment start becoming entangled with the particles of the apparatus, then the blobs become misaligned in even more dimensions. According to this picture, there is no specific definition of when the worlds irreversibly branch. (Of course, we can always consider two blobs to have branched into separate worlds if we are OK with a chance that the two worlds may influence each other via interference.)
Just to emphasize what happens when two blobs are sufficiently misaligned to be considered to be in separate worlds. It means that you can focus on just one blob, ignoring all others, and predict how that blob will evolve. Recall that a single blob in many dimensions represents the motion of many particles. Blobs can and will pass by each other in one dimension or another, even if there is an extremely small chance of ever actually overlapping. When this happens, a given particle is existing in the same place at the same time, but with no effect of that crossing whatsoever. It is worth further emphasizing several points:
The separate “worlds” are not composed of different particles. They are composed of the same particles. It’s just that according to the Schrödinger equation, particles can and will do multiple distinct things simultaneously.
The separate “worlds” are not in distinct locations. In real, 3D space the different “worlds” can and do overlap. A particle that overlaps another particle (or itself) in another world is none the wiser.
The separate “worlds” do not suddenly pop into existence. There is a blurry dividing line between components of an entangled state that may still interact or interfere, and when those components are sufficiently well separated that we can safely take them to be separate.
In a way, it is lucky that we can separate states out into different worlds. If we couldn’t, all particles would be hopelessly entangled, making things quite difficult to handle. As it is, if we can think of a blob as having branched into a separate world, we can ignore the other entangled components. We can consider the particles’ components in that blob as the only ones that exist, now in a new, non-entangled state.
At last, we will touch upon a subject that we have so far studiously avoided. That is how, if all these worlds exist simultaneously branching off from each other, we have never seen it. The short answer is that you cannot see the blobs, because you are in a blob!
But before we get all freaked out, let’s step back and consider a less fraught topic, like a computer memory. Something that is clearly mechanistic. We could take our “marble run” measurement device from Fig. 8.5 and hook up some circuitry so that a computer can detect which way the pointer goes. We’ll write some computer code that tells the computer to take the signal it receives from the pointer, and store it in its memory. If that memory is on a standard hard disk drive, that means the computer will flip the orientation of some magnetic elements to store this information.
If we use the marble run device to measure a particle initially in a superposition of two positions, we have seen that the final macroscopic wave function is two very misaligned blobs, one corresponding to a world with the pointer pointing left, and the other corresponding to a world with the pointer pointing right. Clearly, bringing the computer into the scenario just adds more particles to each blob: one blob (or world) with the computer’s magnetic memory elements pointing one way, the other with them pointing another way.
But is it possible for the computer to somehow detect that there were really two different outcomes? What could it try? We could have some computer code to retrieve the result from its memory. But this is clearly just going to return the same result as was originally measured. That is, the result of the memory retrieval will be correlated with the original result recorded in each world. Maybe we could use some additional sensors to double check the outcome of the measurement. A separate measurement device could remeasure whether the particles on the left or right side had been knocked from their divots. For example, a camera could collect photons bouncing off the apparatus, and the image could be analyzed to see whether the particles were knocked out on the left or right. The problem with all of these ideas is that the results will just be correlated with whichever result was first obtained in that blob. In the world where the computer recorded “left”, the image will show all the particles on the left displaced, and vice versa.
From within one world, every possible check, whether looking back to check the memory or looking externally to verify the subsequent state, looks just as if that were the only outcome that occurred. It’s as if there are two versions of the computer, one that recorded one outcome, and one that recorded the other.
So what is different between the computer described above, and our brains? We don’t know everything about how the brain works, and how it gives rise to consciousness. But it is plausible that the same type of arguments should apply to a brain as to a computer. You can check your memory, or you can check external consequences, but they will all be correlated with the original measurement outcome you took in through your senses. You really exist in multiple worlds simultaneously, experiencing distinct things.
So is there a world where I am a rock star, and another where I am a Supreme Court justice, and another where I am both? The examples shown here have been rather contrived examples, designed to illustrate the point. In the “marble run” measurement of a split wave packet, we argue that there are two distinct worlds: one where you see the pointer go to the left and one where you see it go to the right. Can we extrapolate from this simple question to answer the questions we really want to know about? I don’t have all the answers, but here are a few thoughts:
We have already seen that worlds do not branch off with every possible outcome. In the marble run example, we do not see a branch that has the pointer in a superposition of left and right. Only branches that the environment has selected for occur with any substantial amplitude.
Branches certainly don’t occur with impossible outcomes. The pointer can point left or right – it does not suddenly turn into a cat, or teleport to the moon.
We had to intentionally design the marble run device (or, say, a photomultiplier) to amplify a microscopic state that may be in a superposition of the measured observable to the macroscopic scale. Likewise, the Schrödinger’s cat scenario requires some clever device to translate a microscopic quantum superposition (e.g. decay of a nucleus) into macroscopic worlds where the cat is alive and dead. It is not clear if we can expect this to happen regularly by chance. For example, say we have a machine that flips a coin. A coin sits on a motorized lever that, when activated, flips the coin into the air. Is there a world where the coin lands heads and a world where it lands tails? I suspect that generally only one world exists with any significant amplitude, heads or tails. As the coin flies through the air, lots of particles will be colliding with it, such as air molecules and photons. It is certainly possible that the air molecules or photons will be in a superposition state when they collide, but this will not result in appreciable entanglement because the coin is too heavy (as in the heavy mirror example of Fig. 3.3). It is conceivable that many such collisions could each nudge the coin’s position slightly such that the sum total is enough the create a macroscopic flip of the coin. However, we would expect these random collisions to average out on all sides of the coin, and therefore in the vast, vast majority of cases not result in a change in outcome from heads to tails or vice versa. It seems more plausible that unintentional branching might occur in chaotic systems. The definition of a chaotic system is one in which a small change in an initial state gets amplified to a difference in a macroscopic state. The question of how chaotic systems cross over from quantum to classical has a long history, and is still an active area of research.
There remains a significant plot hole in this story. In several arguments above, we have said things like “only one world exists, with any significant amplitude.” But what does amplitude have to do with it? Either a blob exists or it doesn’t. In pretty much all of the examples shown so far, we have shown cases where the blobs have the same amplitude. This was just to keep it simple. Blobs can have different amplitudes. This could show up in a plot where one blob is bright yellow and another is dim yellow. Of course if a blob is completely absent and the wave function is zero in some region, we would be happy to say that no world exists there. But what if the wave function is just really, really small in that region? Does that mean a world exists? It seems like maybe a world exists, but… less so. This is a tricky question, and one that we have relegated to the final section of this tutorial.