Spinor Info

Many popular science books and articles mention that the Standard Model of particle physics, the model that unifies three of the fundamental forces and describes all matter in the form of quarks and leptons, has about 18 free parameters that are not predicted by the theory.

Very few popular accounts actually tell you what these parameters are.

So here they are, in no particular order:

1. The so-called fine structure constant, \(\alpha\), which (depending on your point of view) defines either the coupling strength of electromagnetism or the magnitude of the electron charge;
2. The Weinberg angle or weak mixing angle \(\theta_W\) that determines the relationship between the coupling constant of electromagnetism and that of the weak interaction;
3. The coupling constant \(g_3\) of the strong interaction;
4. The electroweak symmetry breaking energy scale (or the Higgs potential vacuum expectation value, v.e.v.) \(v\);
5. The Higgs potential coupling constant \(\lambda\) or alternatively, the Higgs mass \(m_H\);
6. The three mixing angles \(\theta_{12}\), \(\theta_{23}\) and \(\theta_{13}\) and the CP-violating phase \(\delta_{13}\) of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which determines how quarks of various flavor can mix when they interact;
7. Nine Yukawa coupling constants that determine the masses of the nine charged fermions (six quarks, three charged leptons).

OK, so that’s the famous 18 parameters so far. It is interesting to note that 15 out of the 18 (the 9 Yukawa fermion mass terms, the Higgs mass, the Higgs potential v.e.v., and the four CKM values) are related to the Higgs boson. In other words, most of our ignorance in the Standard Model is related to the Higgs.

Beyond the 18 parameters, however, there are a few more. First, \(\Theta_3\), which would characterize the CP symmetry violation of the strong interaction. Experimentally, \(\Theta_3\) is determined to be very small, its value consistent with zero. But why is \(\Theta_3\) so small? One possible explanation involves a new hypothetical particle, the axion, which in turn would introduce a new parameter, the mass scale \(f_a\) into the theory.

Finally, the canonical form of the Standard Model includes massless neutrinos. We know that neutrinos must have mass, and also that they oscillate (turn into one another), which means that their mass eigenstates do not coincide with their eigenstates with respect to the weak interaction. Thus, another mixing matrix must be involved, which is called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. So we end up with three neutrino masses \(m_1\), \(m_2\) and \(m_3\), and the three angles \(\theta_{12}\), \(\theta_{23}\) and \(\theta_{13}\) (not to be confused with the CKM angles above) plus the CP-violating phase \(\delta_{\rm CP}\) of the PMNS matrix.

So this is potentially as many as 26 parameters in the Standard Model that need to be determined by experiment. This is quite a long way away from the “holy grail” of theoretical physics, a theory that combines all four interactions, all the particle content, and which preferably has no free parameters whatsoever. Nonetheless the theory, and the level of our understanding of Nature’s fundamental building blocks that it represents, is a remarkable intellectual achievement of our era.

Richard Feynman’s Lectures on Physics remains a classic to this day.

Its newest edition has recently (I don’t know exactly when, I only came across it a few days ago) been made available in its entirety online, for free. It is a beautifully crafted, very high quality online edition, using LaTeX (MathJax) for equations, redrawn scalable figures.

Perhaps some day, someone will do the same to Landau’s and Lifshitz’s 10-volume Theoretical Physics series, too?

A physics meme is circulating on the Interwebs, suggesting that any length shorter than the so-called Planck length makes “no physical sense”.

Which, of course, is pure nonsense.

The Planck length is formed using the three most fundamental constants in physics: the speed of light, \(c = 3\times 10^8~{\rm m}/{\rm s}\); the gravitational constant, \(G = 6.67\times 10^{-11}~{\rm m}^3/{\rm kg}\cdot{\rm s}^2\); and the reduced Planck constant, \(\hbar = h/2\pi = 1.05\times 10^{-34}~{\rm m}^2{\rm kg}/{\rm s}\).

Of these, the speed of light just relates two human-defined units: the unit of length and the unit of time. Nothing prevents us from using units in which \(c = 1\); for instance, we could use the second as our unit of time, and the light-second (\(= 300,000~{\rm km}\)) as our unit of length. In other words, the expression \(c = 300,000,000~{\rm m}/{\rm s}\) is just an instruction to replace every occurrence of the symbol \({\rm s}\) with the quantity \(300,000,000~{\rm m}\).

If we did this in the definition of \(G\), we get a new value: \(G’ = G/c^2 = 7.41\times 10^{-28}~{\rm m}/{\rm kg}\).

Splendid, because this reveals that the gravitational constant is also just a relationship between human-defined units: the unit of length vs. the unit of mass. It allows us to replace every occurrence of the symbol \({\rm kg}\) with the quantity \(7.41\times 10^{-28}~{\rm m}\).

So let’s do this to the reduced Planck constant: \(\hbar’ = \hbar G/c^3 = 2.61\times 10^{-70}~{\rm m}^2\). This is not a relationship between two human-defined units. This is a unit of area. Arguably, a natural unit of area. Taking its square root, we get what is called the Planck length: \(l_P = 1.61\times 10^{-35}~{\rm m}\).

The meme suggests that a distance less than \(l_P\) has no physical meaning.

But then, take two gamma rays, with almost identical energies, differing in wavelength by one Planck length, or about \(10^{-35}~{\rm m}\).

Suppose these gamma rays originate from a spacecraft one astronomical unit (AU), or about \(1.5\times 10^{11}~{\rm m}\) from the Earth.

The wavelength of a modest, \(1~{\rm MeV}\) gamma ray is about \(1.2\times 10^{-12}~{\rm m}\).

The number of full waves that fit in a distance of \(1.5\times 10^{11}~{\rm m}\) is, therefore, is about \(1.25\times 10^{23}\) waves.

A difference of \(10^{-35}~{\rm m}\), or one Planck length, in wavelength adds up to a difference of \(1.25\times 10^{-12}~{\rm m}\) over the \(1~{\rm AU}\) distance, or more than one full wavelength of our gamma ray.

In other words, a difference of less than one Planck length in wavelength between two gamma rays is quite easily measurable in principle.

In practice, of course we’d need stable gamma ray lasers placed on interplanetary spacecraft and a sufficiently sensitive gamma ray interferometer, but nothing in principle prevents us from carrying out such a measurement, and all the energy, distance, and time scales involved are well within accessible limits at present day technology.

And if we used much stronger gamma rays, say at the energy level of the LHC (which is several million times more powerful), a distance of only a few thousand kilometers would be sufficient to detect the interference.

So please don’t tell me that a distance less than one Planck length has no physical meaning.

I don’t normally comment on crank science that finds its way into my Inbox, but this morning I got a really good laugh.

The announcement was dramatic enough: the e-mail bore the title, “Apparent detection of antimatter galaxies”. It came from the “Santilli foundation”, who sent me some eyebrow-raising e-mails in the past, but this was sufficiently intriguing to make me click on the link they provided. So click I did, only to be confronted with the following image:

What’s that, you ask? Why, a telescope with a concave lens. Had I paid a little bit more attention to the e-mail, I might have been a little less surprised; they did include a longer title, you see, helpfully typeset in all caps, which read, “APPARENT DETECTION OF ANTIMATTER GALAXIES VIA SANTILLI’S TELESCOPE WITH CONCAVE LENSES”.

Say what? Concave lenses? Why, it’s only logical. If light from an ordinary galaxy is focused by a convex lens, then surely, light from an antimatter galaxy will be focused by a concave lens. This puts this Santilli fellow in the same league as Galileo; like his counterpart five centuries ago, Santilli also invented his own telescope. But wait, Santilli is also a modern-day Newton: like Newton, he invented a whole new branch of mathematics, which he calls “isodual mathematics”. Certainly sounds impressive.

So what does Einstein’s relativity have to say about all this? Why, it’s all a “century of scientific scams by organized interests on Einstein […] to discredit opposing views”. It’s all “sheer dishonesty and scientific gangsterism”. But it is possible “for the United Stated of America to regain a minimum of international scientific credibility”. All that is needed is to “investigate the legality of the current use of public funds by the Department of Energy and the National Science Foundation on research based on the current mandate of compatibility with Einstein’s theory” and the US of A will cease to be bankrupt.

Oh, and you also need some telescopes with concave lenses.