The Particle and the “Particle” (Part 1)

Why do I find the word particle so problematic that I keep harping on it, to the point that some may reasonably view me as obsessed with the issue? It has to do with the profound difference between the way an electron is viewed in 1920s quantum physics (“Quantum Mechanics”, or QM for short) as opposed to 1950s relativistic Quantum Field Theory (abbreviated as QFT). [The word “relativistic” means “incorporating Einstein’s special theory of relativity of 1905”.] My goal this week is to explain carefully this difference.

The overarching point:

• in QM, an electron really is a particle, at least to the limited extent that QM can handle that notion;
• in QFT, an electron is at best a “particle”, with the word in quotation marks
  • (and I personally prefer the term wavicle.)

I’ve discussed this to some degree already in my article about how the view of an electron has changed over time, but here I’m going to give you a fuller picture. To complete the story will take two or three posts, but today’s post will already convey one of the most important points.

There are two short readings that you may want to dofirst.

• The first is an article about the distinction between physical space (within which all objects actually are located) and the abstract space of possibilities (which consists of all ways that the objects in a physical system could be arranged in physical space.) This issue, essential in quantum physics of any type, will be crucial below.
• The second is a short section of a blog post that explains how QM views an isolated object in a state of definite momentum — i.e. something whose motion (both speed and direction) is precisely known, and whose position is therefore completely unknown, thanks to Heisenberg’s uncertainty principle.

I’ll will review the main point of the second item, and then I’ll start explaining what an isolated object of definite momentum looks like in QFT.

Removing Everything Extraneous

First, though, let’s make things as simple as possible. Though electrons are familiar, they are more complicated than some of their cousins, thanks to their electric charge and “spin”, and the fact that they are fermions. By contrast, bosons with neither charge nor spin are much simpler. In nature, these include Higgs bosons and electrically-neutral pions, but each of these has some unnecessary baggage. For this reason I’ll frame my discussion in terms of imaginary objects even simpler than a Higgs boson. I’ll call these spinless, chargeless objects “Bohrons” in honor of Niels Bohr (and I’ll leave the many puns to my readers.)

For today we’ll just need one, lonely Bohron, not interacting with anything else, and moving along a line. Using 1920s QM in the style of Schrödinger, we’ll take the following viewpoints.

• A Bohron is a particle and exists in physical space, which we’ll take to be just a line — the set of points arranged along what we’ll call the x-axis.
• The Bohron has a property we call position in physical space. We’ll refer to its position as x₁.
• For just one Bohron, the space of possibilities is simply all of its possible positions — all possible values of x₁. [See Fig. 1]
• The system of one isolated Bohron has a wave function Ψ(x₁), a complex number at each point in the space of possibilities. [Note it is not a function of x, the points in physical space; it is a function of x₁, the possible positions of the Bohron.]
• The wave function predicts the probability of finding the Bohron at any selected position x₁: it is proportional to |Ψ(x₁)|², the square of the absolute value of the complex number Ψ(x₁).

A QM State of Definite Momentum

In a previous post, I described states of definite momentum. But I also described states whose momentum is slightly less definite — a broad Gaussian wave packet state, which is a bit more intutive. The wave function for a Bohron in this state is shown in Fig. 2, using three different representations. You can see intuitively that the Bohron’s motion is quite steady, reflecting near definite momentum, while the wave function’s peak is very broad, reflecting great uncertainty in the Bohron’s position.

• Fig. 2a shows the real and imaginary parts of Ψ(x₁) in red and blue, along with its absolute-value squared |Ψ(x₁)|² in black.
• Fig. 2b shows the absolute value |Ψ(x₁)| in a color that reflects the argument [i.e. the phase] of Ψ(x₁).
• Fig. 2c indicates |Ψ(x₁)|², using grayscale, at a grid of x₁ values; the Bohron is more likely to be found at or near dark points than at or near lighter ones.

For more details and examples using these representations, see this post.

To get a Bohron of definite momentum P₁, we simply take what is plotted in Fig. 2 and make the broad peak wider and wider, so that the uncertainty in the Bohron’s position becomes infinite. Then (as discussed in this post) the wave function for that state, referred to as |P₁>, can be drawn as in Fig. 3:

In math, the wave function for the state at some fixed moment in time takes a simple form, such as

• 

where i is the square root of -1. This is a special state, because the absolute-value-squared of this function is just 1 for every value of x₁, and so the probability of measuring the Bohron to be at any particular x₁ is the same everywhere and at all times. This is seen in Fig. 3c, and reflects the fact that in a state with exactly known momentum, the uncertainty on the Bohron’s position is infinite.

Let’s compare the Bohron (the particle itself) in the state |P₁> to the wave function that describes it.

• In the state |P₁>, the Bohron’s location is completely unknown. Still, its position is a meaningful concept, in the sense that we could measure it. We can’t predict the outcome of that measurement, but the measurement will give us a definite answer, not a vague indefinite one. That’s because the Bohron is a particle; it is not spread out across physical space, even though we don’t know where it is.
• By contrast, the wave function Ψ(x₁) is spread out, as is clear in Fig. 3. But caution: it is not spread out across physical space, the points of the x axis. It is spread out across the space of possibilities — across the range of possible positions x₁. See Fig. 1 [and read my article on the space of possibilities if this makes no sense to you.]
• Thus neither the Bohron nor its wave function is spread out in physical space!

We do have waves here, and they have a wavelength; that’s the distance between one crest and the next in Fig. 3a, and the distance beween one red band and the next in Fig. 3b. That wavelength is a property of the wave function, not a property of the Bohron. To have a wavelength, an object has to be wave-like, which our QM Bohron is not.

Conversely, the Bohron has a momentum (which is definite in this state, and is something we can measure). This has real effects; if the Bohron hits another particle, some or all of its momentum will be transferred, and the second particle will recoil from the blow. By contrast, the wave function does not have momentum. It cannot hit anything and make it recoil, because, like any wave function, it sits outside the physical system. It merely describes an object with momentum, and tells us the probable outcomes of measurements of that object.

Keep these details of wavelength (the wave function’s purview) and the momentum (the Bohron’s purview) in mind. This is how 1920’s QM organizes things. But in QFT, things are different.

First Step Toward a QFT State of Definite Momentum

Now let’s move to quantum field theory, and start the process of making a Bohron of definite momentum. We’ll take some initial steps today, and finish up in the next post.

Our Bohron is now a “particle”, in quotation marks. Why? Because our Bohron is no longer a dot, with a measurable (even if unknown) position. It is now a ripple in a field, which we’ll call the Bohron field. That said, there’s still something particle-like about the Bohron, because you can only have an integer number (1, 2, 3, 4, 5, …) of Bohrons, and you can never have a fractional number (1/2, 7/10, 2.46, etc.) of Bohrons. This feature is something we’ll discuss in later posts, but we’ll just accept it for now.

As fields go, the Bohron field is a very simple example. At any given moment, the field takes on a value — a real number — at each point in space. Said another way, it is a function of physical space, of the form B(x).

Very, very important: Do not confuse the Bohron field B(x) with a wave function!!

• This field is a function in physical space (not the space of possibilities). B(x) is a function of physical space points x that make up the x-axis, and is not a function of a particle’s position x₁, nor is it a function of any other coordinate that might arise in the space of possibilities.
• I’ve chosen the simplest type of QFT field: B(x) is a real number at each location in physical space. This is in contrast to a QM wave function, which is a complex number for each possibility in the space of possibilities.
• The field itself can carry energy and momentum and transport it from place to place. This is unlike a wave function, which can only describe the energy and momentum that may be carried by physical objects.

Now here’s the key distinction. Whereas the Bohron of QM has a position, the Bohron of QFT does not generally have a position. Instead, it has a shape.

If our Bohron is to have a definite momentum P₁, the field must ripple in a simple way, taking on a shape proportional to a sine or cosine function from pre-university math. An example would be:

where A is a real number, called the “amplitude” of the wave, and x is a location in physical space.

At some point soon we’ll consider all possible values of A — a part of the space of possibilities for the field B(x) — so remember that A can vary. To remind you, I’ve plotted this shape for A=1 in Fig. 4a and again for A=-³/₂ in Fig 4b.

Initial Comparison of QM and QFT

At first, the plots in Fig. 4 of the QFT Bohron’s shape look very similar to the QM wave function of the Bohron particles, especially as drawn in Fig. 3a. The math formulas for the two look similar, too; compare the formula after Fig. 3 to the one above Fig. 4.

However, appearances are deceiving. In fact, when we look carefully, EVERYTHING IS COMPLETELY DIFFERENT.

• Our QM Bohron with definite momentum has a wave function Ψ(x₁), while in QFT it has a shape B(x); they are functions of variables which, though related, are different.
  
  
• On top of that, there’s a wave function in QFT too, which we haven’t drawn yet. When we do, we’ll see that the QFT Bohron’s wave function looks nothing like the QM Bohron’s wave function. That’s because
  • the space of possibilities for the QM wave function is the space of possible positions that the Bohron particle can have, but
  • the space of possibilities for the QFT wave function is the space of all possible shapes that the Bohron field can have.
• The plot in Fig. 4 shows a curve that is both positive and negative but is drawn colorless, in contrast to Fig. 3b, where the curve is positive but colored. That’s because
  • the Bohron field B(x) is a real number with no argument [phase], whereas
  • the QM wave function Ψ(x₁) for the state of definite momentum has an always-positive absolute value and a rapidly varying argument [phase].
• The axes in Fig. 4 are labeled differently from the axis in Fig. 3. That’s because (see Fig. 1)
  • the QFT Bohron field B(x) is found in physical space, while
  • the QM wave function Ψ(x₁) for the Bohron particle is found in the particle’s space of possibilities.
• The absolute-value-squared of a wave function |Ψ(x₁)|² is interpreted as a probability (specifically, the probability for the particular possibility that the particle is at position x₁. There is no such interpretation for the square of the Bohron field |B(x)|². We will later find a probability interpretation for the QFT wave function, but we are not there yet.
  
  
• Both Fig. 4 and Figs. 3a, 3b show curves with a wavelength, albeit along different axes. But they are very different in every sense
  • In QM, the Bohron has no wavelength; only its wave function has a wavelength — and that involves lengths not in physical space but in the space of possibilities.
  • In QFT,
    • the field ripple corresponding to the QFT Bohron with definite momentum has a physical wavelength;
    • meanwhile the QFT Bohron’s wave function does not have anything resembling a wavelength! The field’s space of possibilities, where the wave function lives, doesn’t even have a recognizable notion of lengths in general, much less wavelengths in particular.

I’ll explain that last statement next time, when we look at the nature of the QFT wave function that corresponds to having a single QFT Bohron.

A Profound Change of Perspective

But before we conclude for the day, let’s take a moment to contemplate the remarkable change of perspective that is coming into our view, as we migrate our thinking from QM of the 1920s to modern QFT. In both cases, our Bohron of definite momentum is certainly associated with a definite wavelength; we can see that both in Fig. 3 and in Fig. 4. The formula for the relation is well-known to scientists; the wavelength λ for a Bohron of momentum P₁ is simply

• 

where h is Planck’s famous constant, the mascot of quantum physics. Larger momentum means smaller wavelength, and vice versa. On this, QM and QFT agree.

But compare:

• in QM, this wavelength sits in the wave function, and has nothing to do with waves in physical space;
• in QFT, the wavelength is not found in the field’s wave function; instead it is found in the field itself, and specifically in its ripples, which are waves in physical space.

I’ve summarized this in Table 1.

Let me say that another way. In QM, our Bohron is a particle; it has a position, cannot spread out in physical space, and has no wavelength. In QFT, our Bohron is a “particle”, a wavy object that can spread out in physical space, and can indeed have a wavelength. (This is why I’d rather call it a wavicle.)

[Aside for experts: if anyone thinks I’m spouting nonsense, I encourage the skeptic to simply work out the wave function for phonons (or their counterparts with rest mass) in a QM system of coupled balls and springs, and watch as free QFT and its wave function emerge. Every statement made here is backed up with a long but standard calculation, which I’m happy to show you and discuss.]

I think this little table is deeply revealing both about quantum physics and about its history. It goes a long way toward explaining one of the many reasons why the brilliant founding parents of quantum physics were so utterly confused for a couple of decades. [I’m going to go out on a limb here, because I’m certainly not a historian of physics; if I have parts of the history wrong, please set me straight.]

Based on experiments on photons and electrons and on the theoretical insight of Louis de Broglie, it was intuitively clear to the great physicists of the 1920s that electrons and photons, which they were calling particles, do have a wavelength related to their momentum. And yet, in the late 1920s, when they were just inventing the math of QM and didn’t understand QFT yet, the wavelength was always sitting in the wave function. So that made it seem as though maybe the wave function was the particle, or somehow was an aspect of the particle, or that in any case the wave function must carry momentum and be a real physical thing, or… well, clearly it was very confusing. It still confuses many students and science writers today, and perhaps even some professional scientists and philosophers.

In this context, is it surprising that Bohr was led in the late 1920s to suggest that electrons are both particles and waves, depending on experimental context? And is it any wonder that many physicists today, with the benefit of both hindsight and a deep understanding of QFT, don’t share this perspective?

In addition, physicists already knew, from 19th century research, that electromagnetic waves — ripples in the electromagnetic field, which include radio waves and visible light — have both wavelength and momentum. Learning that wave functions for QM have wavelength and describe particles with momentum, as in Fig. 3, some physicists naturally assumed that fields and wave functions are closely related. This led to the suggestion that to build the math of QFT, you must go through the following steps:

• first you take particles and describe them with a wave function, and then
• second, you make this wave function into a field, and describe it using an even bigger wave function.

(This is where the archaic terms “first quantization” and “second quantization” come from.) But this idea was misguided, arising from early conceptual confusions about wave functions. The error becomes more understandable when you imagine what it must have been like to try to make sense of Table 1 for the very first time.

In the next post, we’ll move on to something novel: images depicting the QFT wave function for a single Bohron. I haven’t seen these images anywhere else, so I suspect they’ll be new to most readers.