The Particle and the “Particle” (Part 2)

In my last post, I looked at how 1920’s quantum physics (“Quantum Mechanics”, or QM) conceives of a particle with definite momentum and completely uncertain position. I also began the process of exploring how Quantum Field Theory (QFT) views the same object. I’m going to assume you’ve read that post, though I’ll quickly review some of its main points.

In that post, I invented a simple type of particle called a Bohron that moves around in a physical space in the shape of a one-dimensional line, the x-axis.

• I discussed the wave function in QM corresponding to a Bohron of definite momentum P₁, and depicted that function Ψ(x₁) (where x₁ is the Bohron’s position) in last post’s Fig. 3.
• In QFT, on the other hand, the Bohron is a ripple in the Bohron field, which is a function B(x) that gives a real number for each point x in physical space. That function has the form shown in last post’s Fig. 4.

We then looked at the broad implications of these differences between QM and QFT. But one thing is glaringly missing: we haven’t yet discussed the wave function in QFT for a Bohron of definite momentum P₁. That’s what we’ll do today.

The QFT Wave Function

Wave functions tell us the probabilities for various possibilities — specifically, for all the possible ways in which a physical system can be arranged. (That set of all possibilities is called “the space of possibilities“.)

This is a tricky enough idea even when we just have a system of a few particles; for example, if we have N particles moving on a line, then the space of possibilities is an N-dimensional space. In QFT, wave functions can be extremely complicated, because the space of possibilities for a field is infinite dimensional, even when physical space is just a one-dimensional line. Specifically, for any particular shape s(x) that we choose, the wave function for the field is Ψ[s(x)] — a complex number for every function s(x). Its absolute-value-squared is proportional to the probability that the field B(x) takes on that particular shape s(x).

Since there are an infinite number of classes of possible shapes, Ψ in QFT is a function of an infinite number of variables. Said another way, the space of possibilities has an infinite number of dimensions. Ugh! That’s both impossible to draw and impossible to visualize. What are we to do?

Simplifying the Question

By restricting our attention dramatically, we can make some progress. Instead of trying to find the wave function for all possible shapes, let’s try to understand a simplified wave function that ignores most possible shapes but gives us the probabilities for shapes that look like those in Fig. 5 (a variant of Fig. 4 of the last post). This is the simple wavy shape that corresponds to the fixed momentum P₁:

where A, the amplitude for this simple wave, can be anything we like. Here’s what that shape looks like for A=1:

If we do this, the wave function for this set of possible shapes is just a function of A; it tells us the probability that A=1 vs. A=-2 vs. A=3.2 vs. A=-4.57, etc. In other words, we’re going to write a restricted wave function Ψ[A] that doesn’t give us all the information we could possibly want about the field, but does tell us the probability for the Bohron field B(x) to take on the shape A cos(P₁ x).

This restriction to Ψ[A] is surprisingly useful. That’s because, in comparing the state containing one Bohron with momentum P₁ to a state with no Bohrons anywhere — the “vacuum state”, as it is called — the only thing that changes in the wave function is the part of the wave function that is proportional to Ψ[A].

In other words, if we tried to keep all the other information in the wave function, involving all the other possible shapes, we’d be wasting time, because all of that stuff is going to be the same whether there’s a Bohron with momentum P₁ present or not.

To properly understand and appreciate Ψ[A] in the presence of a Bohron with momentum P₁, we should first explore Ψ[A] in the vacuum state. Once we know the probabilities for A in the absence of a Bohron, we’ll be able to recognize what has changed in the presence of a Bohron.

The Zero Bohron (“Vacuum”) State

In the last post, we examined what the QM wave function looks like that describes a single Bohron with definite momentum (see Fig. 3 of that post). But what is the QM wave function for the vacuum state, the state that has no Bohrons in it?

The answer: it’s a meaningless question. QM is a theory of objects that have positions in space (or other simple properties.) If there are no objects in the theory, then there’s… well… no QM, no wave function, and nothing to discuss.

[You might complain that the Bohron field itself should be thought of as an “object” — but aside from the fact that this is questionable (is air pressure an object?), the QM of a field is QFT, so taking this route would just prove my point.]

In QFT, by contrast, the “vacuum state” is perfectly meaningful and has a wave function. The full vacuum state wave function Ψ[s(x)] is too complicated for us to talk about today. But again, if we keep our focus on the special shapes that look like cos[P₁ x], we can easily write the vacuum state’s wave function for that shape’s amplitude, Ψ[A].

Understanding the Vacuum State’s Wave Function

You might have thought, naively, that if a field contains no “particles”, then the field would just be zero; that is, it would have 100% probability to take the form B(x)=0, and 0% probability to have any other shape. This would mean that Ψ[A] would be non-zero only for A=0, forming a spike as shown in Fig. 6. Here, employing a visualization method I use often, I’m showing the wave function’s real part in red and its imaginary part in blue; its absolute-value squared, in black, is mostly hidden behind the red curve.

We’ve seen a similar-looking wave function before in the context of QM. A particle with a definite position also has a wave function in the form of a spike. But as we saw, it doesn’t stay that way: thanks to Heisenberg’s uncertainty principle, the spike instantly spreads out with a speed that reflects the state’s very high energy.

The same issue would afflict the vacuum state of a QFT if its wave function looked like Fig. 6. Just as there’s an uncertainty principle in QM that relates position and motion (changes in position), there’s an uncertainty principle in QFT that relates A and changes in A (and more generally relates B(x) and changes in B(x).) A state with a definite value of position immediately spreads out with a huge amount of energy, and the same is true for a state with a definite value of A; the shape of Ψ[A] in Fig. 6 will immediately spread out dramatically.

In short, a state that momentarily has B(x) = 0, and in particular A=0, won’t remain in this form. Not only will it change rapidly, it will do so with enormous energy. That does not sound healthy for a supposed vacuum state — the state with no Bohrons in it — which ought to be stable and have low energy.

The field’s actual vacuum state therefore has a spread of values for A — and in fact it is a Gaussian wave packet centered around A=0. In QM we have encountered Gaussian wave packets that give a spread-out position; here, in QFT, we need a packet for a spread-out amplitude, shown in Fig. 7 using the representation in which we show the real part, imaginary part, and absolute-value squared of the wave function. In Fig. 7a I’ve made the A-axis horizontal; I’ve then replotted exactly the same thing in Fig. 7b with the A axis vertical, which turns out to be useful as we’ll see in just a moment.

Another way to represent this same wave function involves plotting points at a grid of values for A, with each point drawn in gray-scale that reflects the square of the wave function |Ψ(A)|², as in Fig. 8. Note that the most probable value for A is zero, but it’s also quite likely to be somewhat away from zero.

But now we’re going to go a step further, because what we’re really interested in is not the wave function for A but the wave function for the Bohron field. We want to know how that field B(x) is behaving in the vacuum state. To gain intuition for the vacuum state wave function in terms of the Bohron field (remembering that we’ve restricted ourselves to the shape cos[P₁ x] shown in Fig. 5), we’ll generalize Fig. 8: instead of one dot for each value of A, we’ll plot the whole shape A cos[P₁ x] for a grid of choices of A, using gray-scale that’s proportional to |Ψ(A)|². This is shown in Fig. 9; in a sense, it is a combination of Fig. 8 with Fig. 5.

Remember, this is not showing the probability for the position of a particle, or even that of a “particle”. It is showing the probability in the vacuum state for the field B(x) to take on a certain shape, albeit restricted to shapes proportional to cos[P₁ x]. We can see that the most likely value of A is zero, but there is a substantial spread around zero that causes the field’s value to be uncertain.

In the vacuum state, what’s true for a shape with momentum P₁ would be true also for any and all shapes of the form cos[P x] for any possible momentum P. In principle, we could combine all of those shapes, for all of the different momenta, together in a much more complicated version of Fig. 9. However, that would make the picture completely unreadable, so I won’t try to do that — although I’ll do something intermediate, with multiple values of P, in later posts.

Oh, and I mustn’t forget to flash a warning: everything I’ve just told you and will tell you for the rest of this post is limited to a child’s version of QFT. I’m only describing what the vacuum state looks like for a “free” (i.e. non-interacting) Bohron field. This field doesn’t do anything except send individual “particles” around that never change or interact with each other. If you want to know more about truly interesting QFTs, such as the ones in the real world — well, expect some things to be recognizable from today’s post, but much of this will, yet again, have to be revisited.

The One-Bohron State

Now that we know the nature of the wave function for the vacuum state, at least when restricted to shapes proportional to cos[P₁ x], how does this change in the presence of a single Bohron of momentum P₁?

The answer is quite simple: the wave function Ψ(A) changes from to (up to an overall constant of no interest to us here.) Depicting this state in analogy to what we did for the vacuum state in Figs. 7b, 8 and 9, we find Figs. 10, 11 and 12.

Notice that the one-Bohron state is clearly distinguishable from the vacuum state; most notably the probability for A=0 is now zero, and its spread is larger, with the most likely values for A now non-zero.

There’s one more difference between these states, which I won’t attempt to prove to you at the moment. The vacuum state doesn’t show any motion; that’s not surprising, because there are no Bohrons there to do any moving. But the one-Bohron state, with its Bohron of definite momentum, will display signs of a definite speed and direction. You should imagine all the wiggles in Fig. 12 moving steadily to the right as time goes by, whereas Fig. 9 is static.

Well, that’s it. That’s what the QFT wave function for a one-Bohron state of definite momentum P₁ looks like — when we ignore the additional complexity that comes from the shapes for other possible momenta P, on the grounds that their behavior is the same in this state as it is in the vacuum state.

A Summary of Today’s Steps

That’s more than enough for today, so let me emphasize some key points here. Compare and contrast:

• In QM:
  • The Bohron with definite momentum is a particle with a position, though that position is unknown.
  • The wave function for the Bohron, spread out across the space of the Bohron’s possible positions x₁, has a wavelength with respect to x₁.
• In QFT:
  • The Bohron “particle” (i.e. wavicle) is intrinsically spread out across physical space [the horizontal x-axis in Figs. 9 and 12] and the Bohron itself has a wavelength with respect to x.
  • Meanwhile the wave function, spread out across the space of possible amplitudes A (the vertical axis in Figs. 7a, 8, 10 and 11) does not contain simply packaged information about how the activity in the Bohron field is spread out across physical space x; both the vacuum state and one-Bohron states are spread out, but you can’t just read off that fact from Figs. 8 and 11.
  • And note that the wave function has nothing simple to say about the position of the Bohron; after all the spread-out “particle” doesn’t even have a clearly defined position!

Just to make sure this is clear, let me say this again slightly differently. While in QM, the Bohron particle with definite momentum has an unknown position, in QFT, the Bohron “particle” with definite momentum does not even have a position, because it is intrinsically spread out. The QFT wave function says nothing about our uncertainty about the Bohron’s location; that uncertainty is already captured in the fact that the real (not complex!) function B(x) is proportional to a cosine function. Indeed physical space, and its coordinate x, don’t even appear directly in Ψ(A). Instead the QFT wave function, in the restricted form we’ve considered, only tells us the probability that B(x) = A cos[P₁ x] for a particular value of A — and that those probabilities are different when there is a single Bohron present (Fig. 12) compared to when there is none (Fig. 9).

I hope you can now start to see why I don’t find the word particle helpful in describing a QFT Bohron. The Bohron does have some limited particle-like qualities, most notably its indivisibility, and we’ll explore those soon. But you might already understand why I prefer wavicle.

We are far from done with QFT; this is just the beginning of our explorations. There are many follow-up questions to address, such as

• Can we put our QFT Bohron into a wave packet state similar to last post’s Fig. 2? What would that look like?
• Do these differences between QM and QFT have implications for how we think about experiments, such as the double-slit experiment or Bell’s particle-pair experiment?
• What do QFT wave functions look like if there are two “particles” rather than just one? There are several cases, all of them interesting.
• How do measurements work, and how are they different, in QM versus QFT?
• What about fields more complicated than the Bohron field, such as the electron field or the electromagnetic field?

We’ll deal with these one by one over the coming days and weeks; stay tuned.