Sean Carroll's description of the Many Worlds interpretation

[DaveBeal]

TL;DR

How is the Many Worlds interpretation more consistent with Schroedinger's equation than the Copenhagen interpretation?

In Sean Carroll's book The Biggest Ideas In The Universe: Quanta And Fields he gives a very brief description of Everette's Many Worlds interpretation (page 73). I think he's saying that in this interpretation, when you perform a measurement on a quantum system, the universe splits into multiple branches, one for each possible outcome of the experiment. For instance, if you were measuring the spin of an electron that was initially in a superposition, this would result in two branches, one for a spin-up result and one for a spin-down result. In each branch, the wave function of the electron now contains only the one term corresponding to the result of the measurement. So in the branch in which the electron measured spin-up, its wave function now contains only a spin-up term, and in the branch for the spin-down result, the wave function contains only the spin-down term. Do I have this correct?

Carroll seems to imply that this is somehow more consistent with the Schroedinger equation than the wave function collapse of the Copenhagen Interpretation, but I don't see why it's better. Either way, you have an instantaneous discontinuous change in the wave function which bears no resemblance to the continuous evolution described by the Schroedinger equation. Why is the Many Worlds interpretation better?

 

[PeterDonis]

Moderator's note: Thread moved to the QM interpretations subforum.

 

[PeterDonis]

Do I have this correct?

Sort of. There is only one wave function, which includes both branches; but in the wave function after the measurement, the electron is entangled with the measuring device so that in each branch, the measuring device registers the result corresponding to the electron's spin in that branch.

For example, if we measure the electron's spin in the up-down basis, the wave function before measurement would look like this:

$$
\left( \alpha \ket{\uparrow} + \beta \ket{\downarrow} \right) \ket{\text{ready}}
$$

where $\ket{\text{ready}}$ is the "ready to measure" state of the measuring device, where it hasn't yet registered a result, and $\alpha$ and $\beta$ are complex amplitudes whose squared moduli sum to $1$. And the wave function after measurement would look like this:

$$
\alpha \ket{\uparrow} \ket{\text{measured} \uparrow} + \beta \ket{\downarrow} \ket{\text{measured} \downarrow}
$$

The latter wave function is entangled between the electron and the measuring device, where the former is not.

Carroll seems to imply that this is somehow more consistent with the Schroedinger equation than the wave function collapse of the Copenhagen Interpretation

It is, because there is no collapse in the Schrodinger Equation; time evolution in that equation is always unitary. All you need is an appropriate interaction Hamiltonian to describe the measurement process and how it entangles the electron with the measuring device.

Either way, you have an instantaneous discontinuous change in the wave function

No, that's not the case with the MWI. The change between the two wave functions I described above happens over time according to unitary evolution as described by the Schrodinger Equation with an appropriate interaction Hamiltonian. There is no discontinuous change anywhere. The change just happens fast enough that our current technology can't spot it happening, we can only see the "before" and "after" states. That's what the MWI says, anyway, and whether or not you like it, it's perfectly consistent with the Schrodinger Equation.

 

[PeroK]

Carroll seems to imply that this is somehow more consistent with the Schroedinger equation than the wave function collapse of the Copenhagen Interpretation, but I don't see why it's better.

In the MWI, the universe is the universal wave-function. The universe doesn't split into two universes, as there is still the one wave-function. The measurement process becomes part of the evolution of the universal wave function.

In Copenhagen, the particle wave-function evolves before measurement and then "collapses" during measurement. And the measurement device is a separate, macroscopic system.

In MWI, the universal wave-function simply continues evolving, entangling the wave-function of the particle and the wave-function of the measuring device (considering as a quantum system also described by the universal wave-function).

In one sense, it is the simplest interpretation of QM.

Either way, you have an instantaneous discontinuous change in the wave function which bears no resemblance to the continuous evolution described by the Schroedinger equation. Why is the Many Worlds interpretation better?

There is no discontinuity if we consider the measuring device continuously becoming entangled with the particle.

Note that the downside of MWI is how to explain the probabilities (or apparent probabilities) of the various measurement outcomes (especially if they are not 50-50). And, you have to accept that you are not just described by a universal wave-function, but that you are literally a part of that wave-function and exist only as a piece of mathematics in a giant Hilbert space!

 

[Peter Morgan]

In that book, Sean Carroll specifically focuses on Quantum Field Theory, with MWI as a side hustle. There are multiple ways to approach MWI, but, in a very idealized way, we can think of QFT as aiming or claiming to describe everything, so there is no room in its world for measurements from outside that would collapse the wave function. Hence there is only unitary evolution of the quantum state and there is no discontinuity of the usual measurement kind.
Because there are no state collapses, this requires a measurement theory that either does not include incompatible/noncommuting measurements (in which case it is arguably a classical measurement theory) or that accommodates incompatible/noncommuting measurements in some way that does not include state collapse, which is usually through decoherence.
Decoherence takes that QFT model of everything and chooses a way to think of it as a tensor product of

• a model of a measured system,
• a model of a measurement apparatus, and
• a model of everything else.

Then we use a partial trace to construct a model that knows nothing of what the full model knows about 'everything else'. In that reduced model, which no longer knows everything, there is an effective 'collapse' to a mixed state that we can interpret as a classical probability measure, even though in the ideal model of everything there is not. There is an analogy here with classical thermodynamic transitions, which are analytically smooth if we work with only a finite system whereas for an infinite system there can be sharp thermodynamic transitions.
Note that because the evolution is only unitary, we don't have to think of there being any 'splitting' in this way of thinking of MWI. A unitary evolution is reversible, so that if we are 'on a particular branch' now, we always were on that same branch and always will be. We can think of the infinity of worlds in which there is something like us (or that are completely different) as being exactly as real as 'our' world or we can think of them as only possible and unreal (which are both ways of thinking that we can also adopt for classical probability.)

All that is my individual take on MWI, just as Sean Carroll's is his. I suppose everybody either comes to terms with MWI in their own way or they decide mostly to ignore it: it's taken me decades to arrive at this current rapprochement with it, so you have my best wishes for your own journey. For more on the philosophical aspects of MWI, you could try the Stanford Encyclopedia of Philosophy page about MWI, or David Wallace's 2012 book, The Emergent Multiverse, Oxford University Press.

 

[DaveBeal]

Sort of. There is only one wave function, which includes both branches; but in the wave function after the measurement, the electron is entangled with the measuring device so that in each branch, the measuring device registers the result corresponding to the electron's spin in that branch.

For example, if we measure the electron's spin in the up-down basis, the wave function before measurement would look like this:

$$
\left( \alpha \ket{\uparrow} + \beta \ket{\downarrow} \right) \ket{\text{ready}}
$$

where $\ket{\text{ready}}$ is the "ready to measure" state of the measuring device, where it hasn't yet registered a result, and $\alpha$ and $\beta$ are complex amplitudes whose squared moduli sum to $1$. And the wave function after measurement would look like this:

$$
\alpha \ket{\uparrow} \ket{\text{measured} \uparrow} + \beta \ket{\downarrow} \ket{\text{measured} \downarrow}
$$

The latter wave function is entangled between the electron and the measuring device, where the former is not.

Thanks for your response, Peter. You say that, after the measurement, your second (entangled) wave function applies in both branches. So say you're in the branch where the electron just measured spin-up. If the entangled wave function holds and you immediately measure again, there will be a probability of beta squared that you get a spin-down result. Is that consistent with experiment? I thought that if you make two consecutive measurements, you were guaranteed to get the same result.

 

[PeterDonis]

after the measurement, your second (entangled) wave function applies in both branches

No, it is one single wave function with two terms in it--each term is referred to as a "branch". There's no question of whether it "applies"--it just is the physical reality.

If you mean (as you appear to) that both terms in the wave function "apply" in both branches, that is wrong. See below.

say you're in the branch where the electron just measured spin-up

"You" are "in" all the branches. If you want to include yourself in the model, you need to add degrees of freedom describing you, and a ket for those degrees of freedom will then appear in the wave function. If you observe something about the measurement, you get entangled with the other things.

If the entangled wave function holds and you immediately measure again, there will be a probability of beta squared that you get a spin-down result.

No, there won't. In the spin-up branch, the electron is spin-up and no future measurement can change that. (In a more technical treatment we would explicitly include decoherence in our model to show why branches can never interfere with each other, which is what it would take for something like what you're describing to happen.)

What happens if you measure again is that you need to include the second measuring device in your model, so you get this before the second measurement:

$$
\left( \alpha \ket{\uparrow} \ket{\text{measured} \uparrow} + \beta \ket{\downarrow} \ket{\text{measured} \downarrow} \right) \ket{\text{second measurement ready}}
$$

which then evolves to this after the second measurement:

$$
\alpha \ket{\uparrow} \ket{\text{measured} \uparrow} \ket{\text{second measurement} \uparrow} + \beta \ket{\downarrow} \ket{\text{measured} \downarrow} \ket{\text{second measurement} \downarrow}
$$

There is no chance of a spin down measurement in the spin up branch. Note that all this works the same if you yourself are the "second measuring device", observing for yourself what the first measurement result was.

Note, by the way, that interpreting the squared moduli of $\alpha$ and $\beta$ as "probabilities" is extremely problematic in the MWI. The standard interpretations of probability don't work, and while various MWI proponents have proposed ways to interpret them as probabilities, none of them have general acceptance among physicists. So this is a key open issue with the MWI.

 

[PeterDonis]

@DaveBeal, as my responses should show you, you can't just wave your hands about what the MWI says. You have to do the actual math, using unitary evolution everywhere, and taking proper account of how measurement interactions entangle things. If all you know of the MWI is treatments that don't show you the explicit math, much of what you think you know about the MWI might well be wrong. There is no good way in ordinary language to describe what is happening in the math that I wrote down; it doesn't match any of our intuitive concepts of how "reality" works.

 

[PeroK]

Thanks for your response, Peter. You say that, after the measurement, your second (entangled) wave function applies in both branches. So say you're in the branch where the electron just measured spin-up. If the entangled wave function holds and you immediately measure again, there will be a probability of beta squared that you get a spin-down result. Is that consistent with experiment? I thought that if you make two consecutive measurements, you were guaranteed to get the same result.

The $\alpha$ and $\beta$ apply initially to the state of an electron, but after the interaction with a measuring device, they apply to a much larger quantum system. These branches decohere and effectively cannot interfere with each other. Under MWI it then becomes difficult to say what you mean by the measurement apparatus or the physicist overseeing the experiment. Without realising it, the original physicist has become a superposition across two branches and there are now, presumably, two separate, non-interfering consciousness entities - both representing the original physicist. Although, in fact, this has happened to all of us throughout our lives and continues to happen. And, indeed, this has happened though all human history. That's why I said that according to MWI you are more like a complex mathematical entity than something which you might describe as "physical".

In any case, a further measurement in one branch is separate and distinct from a further measurement in the other branch. Subsequent probabilities are relative to the initial branching. In any case, as already pointed out, there would be one branch with two consecutive up measurements and one branch with two consecutive down measurements. As the two branches do not interfere with each other.

The real problem here, IMO, is why $|\alpha|^2$ and $|\beta|^2$ represent probabilities. Almost everything on MWI quietly assumes that $|\alpha|^2 = |\beta|^2 = 1/2$ and thus pulls the wool over you eyes. But, what if $|\alpha|^2 = 1/3$? Why does the physicist appear to find herself in the $\alpha$ branch less often than in the $\beta$ branch?

If you are looking for a flaw in MWI, that is where to look!