Many Worlds? The Measurement Problem

Interpretation

In the last article on Many Worlds, we saw that quantum mechanics is a weird and counterintuitive theory of how the world works on the microscopic level. Despite its strange framework however, experimental evidence weighs convincingly on its side, so it is a theory that we should take seriously. If we accept quantum mechanics as an accurate description of the world at the microscopic level, then we must translate its mathematical framework into an interpretation.

What do I mean by this? Take Newton’s Laws of Motion. The laws themselves are just mathematical equations, there’s no necessary link between them and the physical world. To make the link, we need to interpret the mathematical symbols physically. Let’s take a look at the high-school statement of Newton’s Second Law:

Leyes de Newton no hay más que una | Cuentos Cuánticos

If we take it at face value, it means very little. All it tells us is that two quantities multiplied together gives us a different quantity. This equation only becomes meaningful when we link each of its quantities to something physical – we need to provide these symbols with an interpretation.

Any high-school physics student will know how we interpret this equation. The “F” represents the resultant force on a body, the “m” represent the mass or “quantity of matter” of the body, and the “a” represents how quickly the body’s velocity is changing – its acceleration. Once we have an interpretation of the equations, we can form an interpretation of what the theory says about what there is in the world. In the Newtonian worldview, we have particles bouncing around with mass, forces that act on them, and time flowing absolutely from past to future.

Since we’ve agreed that quantum mechanics provides a description of our microscopic world, we now want to interpret the theory and what it says constitutes fundamental reality. What does it say everything is made out of, and what does it say about how it behaves? As it turns out, how to interpret the mathematical framework of quantum theory has stumped physicists and philosophers for over a century. All of the difficulty boils down to a particular conundrum – the Measurement Problem.

The Measurement Problem

The central problem in the interpretation of quantum mechanics is the Measurement Problem, which stems from the perplexing fact that quantum particles seem to behave following two, contradictory sets of rules:

  1. Schrödinger Equation
  2. Wavefunction Collapse

The first set of rules is given by the Schrödinger equation, which applies whenever we’re not observing or measuring a quantum system. When we leave particles alone, they are governed by the Schrödinger equation. This is a deterministic, continuous equation that makes the quantum world run like clockwork in the same way that Newton’s Laws of Motion would theoretically determine the future of the entire universe (if we knew the position and velocity of every particle in the universe). In the context of quantum mechanics, this kind of deterministic behaviour is called “unitary evolution”. Whenever the Schrodinger equation applies, the system evolves nice and unitarily.

File:Schrodinger Equation.png - Wikipedia
When we don’t observe quantum systems, they evolve obeying this differential equation. This evolution is nice and continuous and is called “unitary” evolution.

It’s important to point out that this equation doesn’t determine the positions and velocities of particles per se. What it actually applies to it development of the quantum states. Quantum states are expressed as sums of states that represent possible measurement results. This sum is called a “superposition”. Each of these states in the sum has a coefficient called a “probability amplitude” which, if we square it, tells us the probability of obtaining that particular measurement result if we were to measure the system.

Here’s an example of a quantum state:

\vert Spin\rangle= \frac{1}{\sqrt{2}}\vert +\rangle+ \frac{1}{\sqrt{2}}\vert- \rangle

This state could represent the spin of an electron. It tells us that the spin is a superposition of two possible measurement results: the electron having spin +, or the electron having spin -. The coefficient in front of the states representing the possible measurement results tells us the square root of the probability of each measurement. This tells us that the probability of observing + or – in this case is 50%. The Schrödinger equation determines the evolution of these probability amplitudes. What this means is that when we’re not observing a quantum system, we can use the Schrödinger equation to determine the probabilities of different measurement results were we to measure the system.

What about the other set of rules – the wavefunction collapse? What happens when we try to measure things?

When we measure a quantum system, the rule governing its behaviour changes completely! Instead of a smooth evolution of a superposition of states, the system abruptly jumps into a particular state. Which state it jumps into is a probabilistic affair, the probability that we obtain a particular measurement result is determined by the quantum state as discussed above. u This jump is known as the “collapse of the wavefunction”.

The Copenhagen and Many Worlds Interpretations of Quantum Mechanics
When we try to measure a system, it changes abruptly! Instead of remaining as a superposition, the quantum state “collapses” onto one particular value – the value obtained by the experiment.

The measurement problem asks how we are supposed to reconcile these two completely contradictory types of behaviour. It is a perplexing problem that has puzzled physicists and philosophers since the early twentieth century! To put it in terms of quantum states:

    • When we don’t look at a system, the Schrödinger equation means that its quantum state always looks like a superposition of different measurement outcomes:

\vert Spin \rangle=a\vert+ \rangle+b\vert - \rangle

    • However, when we look at the system, the quantum state “collapses” into a definite state:

\vert Spin \rangle=\vert + \rangle

On the one hand we have a deterministic, smooth evolution governed by a first order differential equation, while on the other we have an indeterministic, discontinuous jump not governed by any equation at all. How are we supposed to reconcile these two contradictory behaviours?

The most immediate approach to physicist was is to say that measurement processes involve billions and billions of atoms, and are quantum mechanically very complex systems. It’s no wonder then that we seem to have one rule for an unobserved quantum system containing just a few particles, and a different rule for observations which involve billions of particles. Theoretically then, if we modeled the entire system using the Schrödinger equation, then it might turn out that the “wavefunction collapse” rule is just a special case of the Schrödinger equation applied to a very complex, macroscopic system.

Cue the infamous Schrödinger’s Cat experiment.

Schrödinger’s Cat

Physicists hoped that if we applied the Schrödinger equation to measuring apparatus, then the measurement problem could be solved. The infamous thought experiment Schrodinger’s Cat however, tells us a very different story. Interestingly, this thought experiment was proposed by Schrödinger himself, who wanted to warn people against following his own equation too far! Schrödinger was very wary of the implications of his equation, and the deal with the cat was meant to demonstrate the absurdities of taking his idea too far.

I’ve already written about the thought experiment here, and there are many great descriptions of it you can find online. In this article, I will just outline the details of it that are important for this discussion.

Image result for schrodinger's cat
Schrodinger’s Cat. In the experiment, a radioactive quantum particle has a 50/50 chance of decaying and triggering a mechanism that kills the cat. According to quantum theory, this means that the cat is in a superposition of being both dead and alive until an observer intervenes.

What’s important to us is that quantum theory tells us that the state of the cat before anyone observes it is a superposition:

|State of system>=|Dead Cat> + |Alive Cat>

But what does this actually mean? In terms of measurement outcomes, this tells us that there is an equal chance of an observer seeing the cat dead or alive when they open the box. But what about before the box is opened? Is the cat really in a superposition of death and life before someone else observes it?

Suppose we include the observer in our quantum state. According to quantum theory, this would mean that the state of the system is instead:

|State of system>=|Person sees dead cat> + |Person sees alive cat>

Which means that even the observer is supposed to be in a superposition of seeing a dead cat and an alive cat until someone else observes the observer! At this point, it should be clear that there is something wrong with our interpretation. In real life, we never observe ethereal superpositions, we always see things in definite states. Is it really the case that systems exist in superposition until we observe and “collapse” them, or is there something missing from our picture?

Introducing Many-Worlds

There are many different interpretations of quantum mechanics that try to solve this problem in all sorts of ways. In fact, virtually all interpretations of quantum mechanics can be classified purely on how they try to solve the measurement problem. I’ve discussed a few different approaches in my article here. Some try to modify the Schrodinger equation so that it can generate both smooth, unitary evolution of the wavefunction, and wavefunction collapse. Some deny that the quantum state is objectively real, and is actually just a codification of what we can know about a system. Some much more radical theories claim that measurements and consciousness must take a primary role in quantum mechanics in order to explain the problem. The interpretation that I want to focus on however is the Many-Worlds Interpretation.

The mantra of the Many-Worlds Theorists is that they are the only ones who “take quantum mechanics seriously”. While other interpretations seek to change the equations of quantum mechanics, or reinterpret what the quantum state actually is, Many-Worlders claim to take the theory at face-value. After all, the predictions of quantum theory have been verified countless times by experiment. Surely we should accept that quantum physics fundamentally reflects how nature operates, and we should try and interpret it in an unmodified form.

The aim of the MWI is to take the quantum state as a real description of reality and that the Schrodinger equation is the one true way it evolves. The reason we only ever see one outcome in our experiments then, is not because there’s some process of “wave-function collapse” running contrary to the Schrodinger evolution, but because we only inhabit one branch of the superposition, one world out of many.

Many-worlds interpretation - Wikipedia
We never see objects in superposition because the superposition describes all the possible worlds, whereas we just live in one of them.

In the MWI, each state in the superposition of states that makes up the overall quantum state of a system is a separate world. The quantum state is real, but it doesn’t just describe one world. It describes the multitude of worlds that are out there. The reason we only ever see one experimental outcome is because we are just one of these worlds. The reason perhaps that MWI has such notable support in academia is because of how neatly it seems to resolve the measurement problem without altering any of the mathematical structure of quantum theory itself. At face-value, the MWI is an extremely compelling interpretation compared to many others on the table. But we need to make a closer examination of the theory to decide whether it is really the most convincing interpretation, or whether it doesn’t stand up to scrutiny.

Over the next few articles we’ll explore some of the problems that the MWI has faced, and overcome, and also some of the problems and issues that still hamper the theory today.

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