When reading quantum mechanics, it often feels like the more you know, the less you understand. My reading of the Yukawa theory of force, as an exchange of virtual particles (see my previous post), must have left you with many questions. Questions I can’t answer because… Well… I feel as much as a fool as you do when thinking about it all. Yukawa first talks about some potential – which we usually think of as being some *scalar *function – and then suddenly this potential becomes a wavefunction. Does that make sense? And think of the mass of that ‘virtual’ particle: the rest mass of a neutral pion is about 135 MeV. That’s an awful lot – at the (sub-)atomic scale that is: it’s equivalent to the rest mass of some 265 electrons!

But… Well… Think of it: the use of a static potential when solving Schrödinger’s equation for the electron orbitals around a hydrogen nucleus (a proton, basically) also raises lots of questions: if we think of our electron as a point-like particle being first here and then there, then that’s also not very consistent with a static (scalar) potential either!

One of the weirdest aspects of the Yukawa theory is that these emissions and absorptions of virtual particles violate the energy conservation principle. Look at the animation once again (below): it sort of assumes a rather heavy particle – consisting of a d- or u-quark and its antiparticle – is emitted – out of nothing, it seems – to then vanish as the antiparticle is destroyed when absorbed. What about the energy balance here: are we talking six quarks (the proton and the neutron), or six plus two?Now that we’re talking mass, note a neutral pion (π^{0}) may either be a uū or a dđ* *combination, and that the mass of a u-quark and a d-quark is only 2.4 and 4.8 MeV – so the *binding *energy of the constituent parts of this π^{0 }particle is enormous: it accounts for most of its mass.

The thing is… While we’ve presented the π^{0 }particle as a *virtual *particle here, you should also note we find π^{0 }particles in cosmic rays. Cosmic rays are particle rays, really: beams of highly energetic particles. Quite a bunch of them are just protons that are being ejected by our Sun. [The Sun also ejects electrons – as you might imagine – but let’s think about the protons here first.] When these protons hit an atom or a molecule in our atmosphere, they usually break up in various particles, including our π^{0 }particle, as shown below.

**So… Well… How can we relate these things? What is going on, really, inside of that nucleus?**

Well… I am not sure. Aitchison and Hey do their utmost to try to explain the pion – as a *virtual *particle, that is – in terms of *energy fluctuations *that obey the Uncertainty Principle for energy and time: ΔE·Δt ≥ *ħ*/2. Now, I find such explanations difficult to follow. Such explanations usually assume any measurement instrument – measuring energy, time, momentum of distance – measures those variables on some discrete scale, which implies some uncertainty indeed. But that uncertainty is more like an imprecision, in my view. Not something fundamental. Let me quote Aitchison and Hey:

“Suppose a device is set up capable of checking to see whether energy is, in fact, conserved while the pion crosses over.. The crossing time Δt must be at least *r*/*c*, where r is the distance apart of the nucleons. Hence, the device must be capable of operating on a time scale smaller than Δt to be able to detect the pion, but it need not be very much less than this. Thus the energy uncertainty in the reading by the device will be of the order ΔE ∼ *ħ*/Δt) = *ħ·*(*c*/*r*).”

As said, I find such explanations really difficult, although I can sort of sense some of the implicit assumptions. As I mentioned a couple of times already, the E = m·*c*^{2} equation tells us energy is mass in motion, somehow: some weird two-dimensional oscillation in spacetime. So, yes, we can appreciate we need some *time unit *to *count *the oscillations – or, equally important, to measure their *amplitude*.

[…] But… Well… This falls short of a more *fundamental *explanation of what’s going on. I like to think of Uncertainty in terms of Planck’s constant itself: *ħ or h* or – as you’ll usually see it – as *half *of that value: *ħ*/2. [The Stern-Gerlach experiment implies it’s *ħ*/2, rather than *h*/2 or *ħ or h *itself.] The physical dimension of Planck’s constant is *action*: newton times distance times time. I also like to think action can *express* itself in two ways: as (1) some amount of energy (ΔE: some force of some distance) over some time (Δt) or, else, as (2) some momentum (Δp: some force during some time) over some distance (Δs). Now, if we equate ΔE with the energy of the pion (135 MeV), then we may calculate the *order of magnitude *of Δt from ΔE·Δt ≥ *ħ*/2 as follows:

Δt = (*ħ*/2)/(135 MeV) ≈ (3.291×10^{−16 }eV·s)/(134.977×10^{6 }eV) ≈ 0.02438×10^{−22 }s

Now, that’s an *unimaginably *small time unit – but *much and much *larger than the Planck time (the Planck time unit is about 5.39 × 10^{−44} s). The corresponding distance *r *is equal to *r *= Δt·*c* = (0.02438×10^{−22 }s)·(2.998×10^{8 }m/s) ≈ 0.0731×10^{−14} m = 0.731 fm. So… Well… Yes. We got the answer we wanted… So… Well… We should be happy about that but…

Well… I am not. I don’t like this indeterminacy. This randomness in the approach. For starters, I am very puzzled by the fact that the lifetime of the *actual* π^{0 }particle we see in the *debris *of proton collisions with other particles as cosmic rays enter the atmosphere is like 8.4×10^{−17} seconds, so that’s like 35 *million *times longer than the Δt = 0.02438×10^{−22 }s we calculated above.

Something doesn’t feel right. I just can’t see the logic here. Sorry. I’ll be back.