Waves in an Impossible Sea

# Chapter 17 — The Mass of a Wavicle

## Supplemental Material

Standing Waves: A key concept in this book is an unfamiliar type of “standing wave”, which appears in Chapter 17 and Chapter 20. It’s relation to guitar strings’ standing waves is sketchily depicted in Figure 41. The difference between this type of standing wave and more familiar standing waves is described in this post, along with animations (and some further links to the math background) on this page.

## Endnotes

Note 1: The W boson’s lifetime and mass
• Quote: Thus, for each elementary field of nature, there’s a corresponding type of wavicle, with a definite rest mass. For the electron field, there are electrons, every one of which has a rest mass identical to that of all the others. For the electromagnetic field, there are photons, all with zero rest mass. The W field >has W bosons, each of which has the same mass.

• Endnote: Strictly speaking, this is true to within the requirements of the cosmic certainty limit.

• Discussion:

I have stated that all W bosons have the same rest mass. But thanks to quantum physics, this isn’t quite true.

A W boson has a finite lifetime — on average, it exists for just 3×10−25 seconds. Quantum physics, via Heisenberg’s uncertainty principle, then makes it impossible to define its energy precisely. As a result, if you measure many W bosons (by studying the objects they decay to), you’ll find their rest masses are very similar but not exactly the same.

The same is true for the Z boson. Because it has some very simple decay modes, a measurement of its mass is much more precise, and we can see and understand the effect more immediately. A measurement of the masses of a large number of Z bosons shows the result in the figure below. (The measurement is more precise that the width of the spike.) The cause of the width of the spike is just what I described — the finite lifetime reduces the time that the the particle may live, making the time of its demise more certain. This in turn introduces uncertainty in its energy, including its internal energy, which translates into uncertainty in its rest mass.

Quantitatively, the uncertainty in the Z boson rest mass $\Delta m_Z$ is related to its lifetime $\tau$ via Planck’s constant $\ h$ as follows:

$\Delta m_Z = \frac{h/2\pi}{\tau} \approx \frac{6.6 \times 10^{-25} \rm{\ GeV \ sec}}{2.5 \times 10^{-25} {\rm \ sec}} \approx 2.5 \ {\rm GeV}$

The uncertainty in the Higgs boson’s rest mass is 1/500 that of the Z boson’s rest mass, and its lifetime correspondingly more than 500 times longer. This is thanks to details of the Higgs field’s interactions, as compared to the simpler interactions of the Z field (see Chapters 21-22.)

Note 2: The proton’s size
• Quote: This difference between electrons and protons confirms that the latter has a measurable size, while the former (up to now) apparently does not.

• Endnote: More technically, what one looks for is proof that the internals of the proton can themselves start vibrating, like the sloshing of milk in a shaken container; this is called an excited state of the proton. There are simpler but more subtle methods that can measure the proton’s size even more easily.

• Discussion

Note 6: Absorption and emission of light
• Quote: When the atoms in your body absorb light, each does so one photon at a time.

• Endnote: When a photon is absorbed, the photon strikes the atom and disappears, while its energy is taken up by the atom. Emission of light is the exact reverse; the atom gives up some excess energy and transfers it to a spontaneously created photon.

• Discussion (coming soon)

Note 7: Breadth of a stationary electron
• Quote: “That gives us the best image I know of: a high-frequency low-amplitude extended standing wave stretching off in all directions.”

• Endnote: How broad does this standing wave’s crest really need to be to count as a “stationary” electron? For our purposes, a lot wider than an atom but narrower than a human. If it’s a millimeter across or more, any associated uncertainty in its motion has slowed below walking speed, which is far less than needed for a precise rest mass measurement.

• Discussion: I addressed this issue recently in this post (see this page for animations) and in this post (which also has animations).

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