8.6 The Many-Worlds Interpretation

The Schrödinger equation has been enormously successful, but it describes the wave function as always smoothly evolving in time, in apparent contra­diction to its postulated collapse in the orthodox inter­pretation. So, it would seem to be extremely inter­esting to examine the solution of the Schrödinger equation for measurement processes more closely, to see whether and how a collapse might occur.

Of course, if a true solution for a single arsenic atom already presents an unsur­mountable problem, it may seem insane to try to analyze an entire macroscopic system such as a measurement apparatus. But in a brilliant Ph.D. thesis with Wheeler at Princeton, Hugh Everett, III did exactly that. He showed that the wave function does not collapse. However it seems to us humans that it does, so we are correct in applying the rules of the orthodox inter­pretation anyway. This subsection explains briefly how this works.

Let’s return to the experi­ment of section 8.2, where a positron is sent to Venus and an entangled electron to Mars, as in figure 8.6.

Figure 8.6: Bohm’s version of the Einstein, Podolski, Rosen Paradox.

The spin states are uncertain when the two are send from Earth, but when Venus measures the spin of the positron, it miraculously causes the spin state of the electron on Mars to collapse too. For example, if the Venus positron collapses to the spin-up state in the measurement, the Mars electron must collapse to the spin-down state. The problem, however, is that there is nothing in the Schrödinger equation to describe such a collapse, nor the super­luminal communi­cation between Venus and Mars it implies.

The reason that the collapse and super­luminal communi­cation are needed is that the two particles are entangled in the singlet spin state of chapter 5.5.6. This is a 50% / 50% proba­bility state of (electron up and positron down) / (electron down and positron up).

It would be easy if the positron would just be spin up and the electron spin down, as in figure 8.7.

Figure 8.7: Nonentangled positron and electron spins; up and down.

You would still not want to write down the super­colossal wave function of everything, the particles along with the observers and their equipment for this case. But there is no doubt what it describes. It will simply describe that the observer on Venus measures spin up, and the one on Mars, spin down. There is no ambiguity.

The same way, there is no question about the opposite case, figure 8.8.

Figure 8.8: Nonentangled positron and electron spins; down and up.

It will produce a wave function of everything describing that the observer on Venus measures spin down, and the one on Mars, spin up.

Everett, III recognized that the solution for the entangled case is blindingly simple. Since the Schrödinger equation is linear, the wave function for the entangled case must simply be the sum of the two non­entangled ones above, as shown in figure 8.9.

Figure 8.9: The wave functions of two universes combined

If the wave function in each non­entangled case describes a universe in which a particular state is solidly established for the spins, then the conclusion is undeniable: the wave function in the entangled case describes two universes, each of which solidly establishes states for the spins, but which end up with opposite results.

This explains the result of the orthodox inter­pretation that only eigen­values are measurable. The linearity of the Schrödinger equation leaves no other option:

Assume that any measurement device at all is constructed that for a spin-up positron results in a universe that has absolutely no doubt that the spin is up, and for a spin-down positron results in a universe that has absolutely no doubt that the spin is down. In that case a combin­ation of spin up and spin down states must unavoidably result in a combin­ation of two universes, one in which there is absolutely no doubt that the spin is up, and one in which there is absolutely no doubt that it is down.

Note that this observ­ation does not depend on the details of the Schrödinger equation, just on its linearity. For that reason it stays true even including relativity.

The two universes are completely unaware of each other. It is the very nature of linearity that if two solutions are combined, they do not affect each other at all: neither universe would change in the least whether the other universe is there or not. For each universe, the other universe “exists” only in the sense that the Schrödinger equation must have created it given the initial entangled state.

Non­linearity would be needed to allow the solutions of the two universes to couple together to produce a single universe with a combin­ation of the two eigen­values, and there is none. A universe measuring a combin­ation of eigen­values is made impossible by linearity.

While the wave function has not collapsed, what has changed is the most meaningful way to describe it. The wave function still by its very nature assigns a value to every possible configur­ation of the universe, in other words, to every possible universe. That has never been a matter of much controversy. And after the measurement it is still perfectly correct to say that the Venus observer has marked down in her notebook that the positron was up and down, and has transmitted a message to earth that the positron was up and down, and earth has marked on in its computer disks and in the brains of the assistants that the positron was found to be up and down, etcetera.

But it is much more precise to say that after the measurement there are two universes, one in which the Venus observer has observed the positron to be up, has transmitted to earth that the positron was up, and in which earth has marked down on its computer disks and in the brains of the assistants that the positron was up, etcetera; and a second universe in which the same happened, but with the positron everywhere down instead of up. This description is much more precise since it notes that up always goes with up, and down with down. As noted before, this more precise way of describing what happens is called the “relative state formul­ation.”

Note that in each universe, it appears that the wave function has collapsed. Both universes agree on the fact that the decay of the $\pi$-meson creates an electron/positron pair in a singlet state, but after the measurement, the notebook, radio waves, computer disks, brains in one universe all say that the positron is up, and in the other, all down. Only the unobservable full wave function “knows” that the positron is still both up and down.

And there is no longer a spooky super­luminal action: in the first universe, the electron was already down when send from earth. In the other universe, it was send out as up. Similarly, for the case of the last subsection, where half the wave function of an electron was send to Venus, the Schrödinger equation does not fail. There is still half a chance of the electron to be on Venus; it just gets decomposed into one universe with one electron, and a second one with zero electron. In the first universe, earth send the electron to Venus, in the second to Mars. The contra­dictions of quantum mechanics disappear when the complete solution of the Schrödinger equation is examined.

Next, let’s examine why the results would seem to be covered by rules of chance, even though the Schrödinger equation is fully deterministic. To do so, assume earth keeps on sending entangled positron and electron pairs. When the third pair is on its way, the situation looks as shown in the third column of figure 8.10.

Figure 8.10: The Bohm experiment repeated.

The wave function now describes 8 universes. Note that in most universes the observer starts seeing an apparently random sequence of up and down spins. When repeated enough times, the sequences appear random in practi­cally speaking every universe. Unable to see the other universes, the observer in each universe has no choice but to call her results random. Only the full wave function knows better.

Everett, III also derived that the statistics of the apparently random sequences are propor­tional to the absolute squares of the eigen­function expansion coefficients, as the orthodox inter­pretation says.

How about the uncertainty relationship? For spins, the relevant uncertainty relationship states that it is impossible for the spin in the up/down directions and in the front/back directions to be certain at the same time. Measuring the spin in the front/back direction will make the up/down spin uncertain. But if the spin was always up, how can it change?

This is a bit more tricky. Let’s have the Mars observer do a couple of additional experi­ments on one of her electrons, first one front/back, and then another again up/down, to see what happens. To be more precise, let’s also ask her to write the result of each measurement on a blackboard, so that there is a good record of what was found. Figure 8.11 shows what happens.

Figure 8.11: Repeated experiments on the same electron.

When the electron is send from Earth, two universes can be distin­guished, one in which the electron is up, and another in which it is down. In the first one, the Mars observer measures the spin to be up and marks so on the blackboard. In the second, she measures and marks the spin to be down.

Next the observer in each of the two universes measures the spin front/back. Now it can be shown that the spin-up state in the first universe is a linear combin­ation of equal amounts of spin-front and spin-back. So the second measurement splits the wave function describing the first universe into two, one with spin-front and one with spin-back.

Similarly, the spin-down state in the second universe is equivalent to equal amounts of spin-front and spin-back, but in this case with opposite sign. Either way, the wave function of the second universe still splits into a universe with spin front and one with spin back.

Now the observer in each universe does her third measurement. The front electron consists of equal amounts of spin up and spin down electrons, and so does the back electron, just with different sign. So, as the last column in figure 8.11 shows, in the third measurement, as much as half the eight universes measure the vertical spin to be the opposite of the one they got in the first measurement!

The full wave function knows that if the first four of the final eight universes are summed together, the net spin is still down (the two down spins have equal and opposite amplitude). But the observers have only their blackboard (and what is recorded in their brains, etcetera) to guide them. And that infor­mation seems to tell them unambiguously that the front-back measurement “destroyed” the vertical spin of the electron. (The four observers that measured the spin to be unchanged can repeat the experi­ment a few more times and are sure to eventually find that the vertical spin does change.)

The unavoidable conclusion is that the Schrödinger equation does not fail. It describes the observ­ations exactly, in full agreement with the orthodox inter­pretation, without any collapse. The appearance of a collapse is actually just a limitation of our human observa­tional capabilities.

Of course, in other cases than the spin example above, there are more than just two symmetric states, and it becomes much less self-evident what the proper partial solutions are. However, it does not seem hard to make some conjectures. For Schrödinger’s cat, you might model the radioactive decay that gives rise to the Geiger counter going off as due to a nucleus with a neutron wave packet rattling around in it, trying to escape. As chapter 7.12.1 showed, in quantum mechanics each rattle will fall apart into a transmitted and a reflected wave. The transmitted wave would describe the formation of a universe where the neutron escapes at that time to set off the Geiger counter which kills the cat, and the reflected wave a universe where the neutron is still contained.

For the standard quantum mechanics example of an excited atom emitting a photon, a model would be that the initial excited atom is perturbed by the ambient electro­magnetic field. The perturb­ations will turn the atom into a linear combin­ation of the excited state with a bit of a lower energy state thrown in, surrounded by a perturbed electro­magnetic field. Presumably this situation can be taken apart in a universe with the atom still in the excited state, and the energy in the electro­magnetic field still the same, and another universe with the atom in the lower energy state with a photon escaping in addition to the energy in the original electro­magnetic field. Of course, the process would repeat for the first universe, producing an eventual series of universes in almost all of which the atom has emitted a photon and thus transitioned to a lower energy state.

So this is where we end up. The equations of quantum mechanics describe the physics that we observe perfectly well. Yet they have forced us to the uncomfortable conclusion that, mathemati­cally speaking, we are not at all unique. Beyond our universe, the mathematics of quantum mechanics requires an infinity of unobservable other universes that are non­trivially different from us.

Note that the existence of an infinity of universes is not the issue. They are already required by the very formul­ation of quantum mechanics. The wave function of say an arsenic atom already assigns a non­zero proba­bility to every possible configur­ation of the positions of the electrons. Similarly, a wave function of the universe will assign a non­zero proba­bility to every possible configur­ation of the universe, in other words, to every possible universe. The existence of an infinity of universes is therefore not something that should be ascribed to Everett, III {N.15}.

However, when quantum mechanics was first formulated, people quite obviously believed that, practi­cally speaking, there would be just one universe, the one we observe. No serious physicist would deny that the monitor on which you may be reading this has uncertainty in its position, yet the uncertainty you are dealing with here is so astronomi­cally small that it can be ignored. Similarly it might appear that all the other substantially different universes should have such small proba­bilities that they can be ignored. The actual contribution of Everett, III was to show that this idea is not tenable. Non­trivial universes must develop that are substantially different.

Formulated in 1957 and then largely ignored, Everett's work represents without doubt one of the human race's greatest accom­plishments; a stunning discovery of what we are and what is our place in the universe.