Note: This article was originally posted on July 25, 2006, but is being reposted now that embedded LaTeX is working.
This post is on quantum mechanics, but is intended to be understandable by anyone. The small amount of mathematical notation that I introduce should be fairly digestible to anyone without a physics background.
A couple of nights ago I had a dinner table conversation with a fellow physicist, who raised the highly emotive, almost religious issue of the interpretation of quantum mechanics. For the non-physicists, quantum mechanics has some very unusual properties that we are not used to encountering in the world around us (see my previous post on quantum mechanics). The most well known of these unusual properties is the notion of a superposition, whereby something can assume multiple states simulatenously. A state is simply a property of a system. For example, the position of a particle is represented by a state. A superposition in this context means that the particle is in multiple positions simultaneously. Obviously this is very counterintuitive indeed. This, as well as numerous other counterintuitive properties of quantum mechanics give rise to the argument over the interpretation of quantum mechanics. That is, how do we reconcile such strange behavior with our inherent human desire for cartoon descriptions of everything.
Mathematical background (for non-mathematicians and non-physicists)
Let me now expand on this. I’ll do this by introducing the mathematical tools that physicists actually use, but in a way that can hopefully be understood to anyone without such a background.
Suppose we have a system, which we will label A. This could be anything – a particle, a cow, a person, a banana. Using quantum mechanical notation we will represent the state of a system as , where s represents the actual state, and the subscript A denotes the system we are referring to. For example, suppose we have a two-level cow, which can be in one of two states – it can be eating, or not eating. The mathematical descriptions of these two states are and . In quantum mechanics we denote a superposition using a ‘+’ symbol. So a superposition of two states a and b is just denoted .
This completes the quantum mechanical notations we use for representing the states of systems. Next we’ll turn our attention to the evolution (i.e. the dynamics) of a quantum system. In quantum mechanics there are precisely two (or possibly just one, as we will discuss later) types of dynamics. The first is so-called unitary evolution. The technical definition of this is irrelevant here. Just think of it as an operation that maps one state to another. For example the unitary evolution describing a two-level cow that is initially standing around and then decides to have dinner, is described by , where denotes the unitary evolution operation.
The second type of evolution that is possible is the so-called projective measurement. As the name suggests, this type of evolution takes place when we measure a system. As an example, suppose we start with a system in a superposition of two states, , and we measure it to establish whether the system is in state a or state b. When we perform this measurement we always get either one answer or the other. Our measurement device never says ‘you have a superposition of a and b‘. This is why we don’t observe quantum mechanical effects in the world around us. So the question now is, if we have a superposition, which measurement result do we get? The answer is that our result is random. Truly random, in the sense that it is impossible, even in principle, to predict it. So 50% of the time our measurement device will say ‘you have a‘, and the other 50% of the time it will say ‘you have b‘. After the measurement, the state of the system collapses (according to the standard interpretations of quantum mechanics) into either or . Notice that after a projective measurement, the system is no longer in a superposition. In other words, the strange quantum mechanical properties dissapear. To describe this mathematically we’ll introduce corresponding measurement operations. If we perform a measurement and get outcome x, we denote the measurement operation as . So, referring to the discussed example we have
It turns out that this kind of evolution cannot be described by the unitary evolutions I mentioned before.
The measurement problem
The previous discussion gives rise to an obvious question. What constitutes a measurement? We know from numerous experiments that if we have an isolated system of say, atoms interacting with one another, then the system evolves unitarily. It is only when we look at it that this projective evolution takes place. In general, isolated small systems evolve unitarily, but as soon as they interact with a macroscopic system they undergo projective dynamics. The distinction seems very arbitrary. What is it that actually causes this transition from one type of dynamics to another? This is the so-called ‘measurement problem’, and is one of central points of contention in the interpretation of quantum mechanics.
One interpretation of quantum mechanics, which resolves the measurement problem, is the so-called ‘many worlds’ interpretation, which I will now describe. Suppose we begin with two systems, an observer and an atom. We assume the atom is in a superposition of two states, 0 and 1. The observer could be you or I, and can assume one of three possible states: ‘doing nothing’, ‘I think I measured 0’ and ‘I think I measued 1’. We’ll assume the observer is initially in the ‘doing nothing’ state. So, the state of the entire system can be expressed
Expanding the brackets this is equal to
Now we are going to apply a unitary evolution that performs the following mapping
In other words, this evolution maps an idle observer to the state where he believes he has measured whatever the respective state of the atom is. So what happens when we apply this evolution to the initial state?
We now have a state which is a superposition of the atom being in state 0 and the observer thinking that the atom is in state 0, and the atom being in state 1 and the observer think the atom is in state 1. This is a so-called entangled state, a superposition where the states of distinct systems are correlated within the superposition. Entanglement is another one of the quirky features of quantum mechanics, which gives rise to all sorts of strange phenomena. But I won’t go into that now.
With a little bit of algebraic rearrangement, it can easily be seen that this state is equal to
In this expression we have a superposition of an observer who thinks he measured a 0 and 1 respectively, and the state of the atom is described by the respective projection. Note that we have not inserted projection operators explicitly at any stage, it arises naturally as a result of the entanglement between the observer and the system he is measuring. This is how the ‘many worlds’ interpretation of quantum mechanics arises. There is no such thing as a projective measurement. Instead, whenever we (an observer) become entangled with a system we enter a superposition with it. Note that from the view-point of a particular term in the superposition, projective measurement has taken place. Thus, according to this interpretation, the reason we observe projective dynamics in the real world is because we are simply one of the terms in the superposition.
This interpretation resolves several issues. First, there is no arbitrary scale at which interactions become projective. Instead it arises very naturally as a result of our entanglement with that which we are observing. Second, the laws of physics are once again completely deterministic, since these non-deterministic projections no longer exist. Unfortunately, this interpretation also raises a somewhat depressing and overwhelming possibility. That is, we are simply one term in an almost infinitely large superposition of parallel worlds. Every time you look at an atom and observe a 0, there’s another you in a parallel world that observed a 1. For some reason I find this quite distressing.