In quantum physics, we must always distinguish physical space — the space we live in, and in which exist the objects that make up a physical system — from the space of possibilities, a more abstract space that let’s us understand what this physical system can potentally do. This issue also arises in pre-quantum physics, but in the quantum setting it is elevated from merely useful to absolutely critical. I’ve given concrete examples of this distinction (here and here and here).
In this article I want to focus on this issue carefully. Since this issue doesn’t specifically have to do with quantum physics, I’m going to mostly address it using pre-quantum language, that of Isaac Newton and the later physicists of the 17th-19th centuries. The quantum applications of these ideas will mostly be covered on other pages. To do this, I’ll look at simple physical systems consisting of one, two or three particles that are moving along a line in physical space.
You might think the case of one particle would be easiest, but in fact it is the most easily misunderstood. So we’ll start with two particles, and return to the one-particle system only at the end.
Two Particles on a Line (Pre-Quantum)
Imagine that we have two objects — small and simple objects, which I’ll refer to as particles — which are situated on a line, and can move around to the left or right, as at the top of Fig. 1. I’ll refer to this line as the x-axis. In a pre-quantum world, each particle has a position; we’ll call the first particle’s position we call X1. Said differently, at the top of Fig. 1, the particle exists in physical space at the location x=X1. Similarly, the second particle has position x=X2.
In sum, in physical space the system consists of two particles with definite positions along the x-axis, specifically x=X1 and x=X2.
At the bottom of Fig. 1, we see the same system as at the top of Fig. 1, with the same two particles in the same locations, depicted in the space of possibilities as the red star. But what is the space of possibilities? And in what sense does the star represent the two particles on the line?
Answer to the first question: in any arrangement of the two particles, not necessarily the one shown at the top of Fig. 1, particle 1 must be located at some location x1, and particle 2 must be located at some location x2. So the full set of possibilities consists of choosing a pair of numbers x1 and x2. That is the same as choosing a single point anywhere in a two-dimensional plane, whose axes are the x1 axis and the x2 axis. In other words, any possibility for the system of two particles corresponds to a point whose coordinates are (x1, x2).
Answer to the second question: the speicific arrangement of the two particles shown at the top of Fig. 1 corresponds to choosing the specific point marked by the star, the point whose (x1, x2) coordinates are (X1, X2).
It’s crucial to notice that each axis in the space of possibilities indicates the possible locations of one of the two particles. Neither one is the actual x–axis, the axis whose points are those of physical space.
Three Particles
What if there are three particles? Physical space is still just the line consisting of the x-axis, and the three particles are located at positions x=X1, x=X2, and x=X3. But the possible arrangements of the particles are now given by three choices for the particles’ locations, and so consist of all possible points in three dimensions, with coordinates (x1, x2, x3) along the x1-axis, the x2-axis, and the x3-axis. The specific arrangement shown at the top of Fig. 2 corresponds to the location of the star at the bottom of Fig. 2, which has the specific coordinates (X1, X2, X3).
Here physical space is always one-dimensional, consisting of the x-axis, no matter how many particles we have in the system. But the more particles in the system, the larger the space of possibilities. When we have n particles moving on a line, each particle’s position is a single number, so the space of possibilities is then n-dimensional and is spanned by the x1-axis, the x2-axis, and so on up to the xn-axis. [More generally, if the particles move around in a d=dimensional physical space, the dimension of the space of possibilities is n times d.]
The Tricky Case of a Single Particle
The most confusing case — and also, unfortunately, the one most often emphasized, leading to widespread misconceptions — is when we have just one particle. In that case both the physical space and the space of possibilities are one-dimensional. This makes them look the same. And that makes it easy to forget that they are actually very different! This is shown in Fig. 3. The crucial difference is that the physical space is the x-axis, the physical points where all objects could potentially be located, while that of the space of possibilities is the x1-axis, which are the possible locations of particle 1.
This might seem like a minor detail, but it turns out not to be.
How This Applies in Quantum Physics
So far, everything I have said applies to pre-quantum physics, in which the particles always have definite location. But in quantum physics of the 1920s (“quantum mechanics”), the system of particles (no matter how many) is described by a single wave function on the space of possibilities. For instance, if there are two particles, the wave function takes the form Ψ(x1,x2).
Take note:
- Because there are two particles, it is a common misconception that there should be two wave functions, one per particle, which are functions of physical space: Ψ1(x) and Ψ2(x). But this is not the case; a system of objects that is temporarily isolated from the world requires one and only one wave function.
- The wave function is not a function of physical space locations x [that is, it is not of the form Ψ(x)] so we cannot ask “what is the wave function at this point x in space?” That question would be meaningless. Instead, we can only ask, “what is the wave function if particle 1 is located at the position x1 and particle 2 is located at the position x2?” Rephrased slightly, “what is the wave function at this particular location in the space of possibilities?” The probability that the system is in that arrangement, with the particles located at x1 and x2 in physical space is proportional to the absolute-value-squared of the complex number Ψ(x1,x2).
The case of one particle easily leads to confusion, an issue which is extremely common in books, articles and even classes on quantum physics; few textbooks emphasize the distinctions.
- The wave function is now a function of the single particle’s position, Ψ(x1).
- But because it is a function of one variable, it is easy to mistake the wave function for a function that exists in physical space, of the form Ψ(x).
To avoid this confusion is not so easy, so it’s best to always keep the two-particle case in mind. That’s the simplest case for which such confusion is impossible, because (as in most systems) physical space and the space of possibilities have different numbers of dimensions, as in Fig. 1. The wave function exists on the latter, not the former.
To summarize the main points:
- In physical space, the x axis describes where any and all particles might be located along the line.
- In the space of possibilities, the x1 axis tells us only where particle 1 is located.
- Quantum wave functions are functions on the space of possibilities, not on physical space.
and organized into a slogan:
- Objects live in physical space; wave functions live in possibility space.
When we get to quantum field theory, the distinction between these two spaces will become even more dramatic — because even though there is still physical space, there won’t be particles or locations, at least not in the way I’ve described here. So it’s important to be comfortable with the space of possibilities in 1920’s quantum physics before we jump to the 1950’s and beyond.
