The previous few posts showed how a density matrix formalism gives a variety of quantum mechanics that naturally supports an interpretation of quantum states as symmetry operators on the quantum states. The method for doing this required ignoring the gauge bosons in bound states. For example, beginning with a complicated Feynman diagram for a bound state:
we simplified it by trimming off all the guage bosons and particle / anti-particle pairs created from gauge bosons. What’s left is just the valence fermions. We mark the points where these valence fermions change state with black dots and have:
This sort of thing will really annoy the old folks. It was the method we used to extend Koide’s charged lepton mass formula to the neutrinos. It may have something to do with the triality trick that Garrett Lisi used to fit the standard model particles to E8, and eventually we will return to the subject. But for now, I’d like to discuss the application of these trimmed diagrams as I was originally exposed to them; as a generic method of giving mass to massless particles. But first, a word about the philosophy behind the “new physics.”
The New Physics
Of course everyone has a different description (many not fit for family newspapers) of what the “new physics” is all about. My version is that it consists of rejecting the theory of the old physics, while keeping its calculations, and generalizing those calculations without regard to how badly this goes against the old theory. As in the past, eventually a new theory will arise, but for now, the important thing is to generalize the calculations.
Back when classical mechanics was king, the early electric companies gave Max Planck the very practical engineering problem of figuring out how to design light bulbs so that they would give the maximum amount of light for the power consumed. He began attempting to model the blackbody spectrum. The answer was not easy and “driven to an act of despair”, he came up with something that was valid mathematically, matched the experimental data, but was nonsense according to classical mechanics. And so quantum mechanics began.
At the present time physics is again troubled (Smolin). And again, in the dim footsteps of Planck, we are driven by despair to look for a simple way of modeling what, according to our old physics, should be very complicated, the masses of the elementary fermions.
We will treat the standard model as a set of calculational techniques that happen to give an accurate model of elementary particle interactions. We seek to generalize those calculational techniques in such a way that mass is treated like the forces that are so well modeled by the standard model.
Quantum Calculations and Mass
In 1982, Yoshio Koide noticed that the masses of the charged leptons fit the approximate equation
Since the masses of the electron and muon are known to high accuracy, the above formula gives a prediction for the mass of the tau which was not measured accurately in 1982. As the past 25 years have gone by, the error bars in the tau mass have steadily shrank, but the Koide prediction still holds well within the error bars.
In the standard model, the elementary particle masses are arbitrary and there is little reason to expect them to be related by a simple equation. The standard view is that elementary particle interactions should be simplest at high energies, not the low energies where mass is important. And Koide’s formula is not in a form that one sees in the rest of quantum mechanics, so it leaves little clue as to how it could fit into a deeper theory.
A few years ago, I took a look at Koide’s formula and found that it could be rewritten so that the electron, muon, and tau masses are proportional to the eigenvalues of an peculiarly simple 3×3 circulant matrix:
This form for Koide’s formula has several advantages over the previous. First, it puts the formula in an eigenvector / eigenvalue form that is used extensively in quantum mechanics. Second, the angle suggests that there is a perturbation series whose first term will be 2/9. Such a perturbation series will be related to how the particles are given their masses and since this is contrary to the obvious Higgs mechanisms it hints at new physics. Third, while Koide’s original formula is incompatible with the measurements of neutrino mass differences (see Nan Li and Bo-Qiang ma, Phys. Lett. B 609, 309 (2005) and Gerard, Goffinet and Herquet, Phys. Lett. B 633, 563 (2006)), the eigenvector form is compatible and allows predictions of the neutrino masses. Various authors worked with this extension of Koide’s formula to the neutrinos last year. More recently, Marni Sheppeard noticed that the circulant matrices are related to Fourier Transforms, which gives a clue that mass should be related to the Fourier transformation from position / time to momentum / energy.
The reason the original Koide equation is unexpected in the old physics is well described in a paper by Koide. There are three unusual aspects to the equation according to Koide: (a) mass is used in square root form which suggests a bilinear form instead of a Yukawa coupling, (b) the formula is invariant under exchange of the particles, and (c) in violation of what we expect from the renormalization group, the formula is exact at low energies instead of high.
The above three points predate the equation being rewritten in eigenvalue form. The square root mass issue is strengthened by the eigenvalue form as to fit the neutrinos requires that the lightest of the neutrinos take the opposite sign in the square root of its mass. This can only be done if it is the square roots of mass that are fundamental rather than the masses. Also, matrices are bilinear, so writing the equation in matrix form fulfills Koide’s guess.
Koide’s second observation, that the formula is unchanged under the exchange of generations, is slightly modified when the equation is rewritten in eigenvalue form. Circulant 3×3 matrices share the same three eigenvectors (eigenkets):
In the Koide formula, the real eigenvector corresponds to the tau, and the two complex eigenvectors correspond to the electron and muon. If we write the three eigenvectors in bra form as , where n=1,2,3 for generations 1, 2, and 3, respectively, then we see that as per Koide’s second observation, the three generations are treated in a sort of equal manner. However, it is also the case that the electron and muon eigenvectors are complex conjugates of each other, while the tau eigenvector is its own complex conjugate (we’ve chosen its normalization to make it real in the above). Thus the three eigenvectors are in the form of a singlet and a doublet.
Thus, under the action of complex conjugacy, the electron and muon generations are swapped, while the tau generation is left alone. None of us thought anything of this until Garrett Lisi wrote his recent paper fitting the elementary particles into E8. In that paper, he put one generation in and got the other two generations as “triality partners”. These involve rotations by just as the eigenvector version of the Koide formula (see page 12). And the resulting generation structure matches the Koide eigenvector form in that there is a singlet associated with the tau generation, and a doublet, associated with the electron and muon generations.
Koide’s third observation, that the mass formula is accurate at low energies instead of high energies is sharpened by the conversion to eigenvalue form. The angle is close to a rational fraction and this suggests that it should be expanded in a perturbation series; a series that is converging very quickly. To get a fast converging perturbation series, we expect a very small number of applicable Feynman diagrams. Under the usual methods of working these problems out, this would occur at very high energies, but the Koide formula applies instead at the lowest possible energies.
One way that we could obtain a perturbation series that would quickly converge at low energies is if the elementary fermions are composites and it so happens that we can approximate their bound states in such a way that we can find a perturbation expansion around those approximations. Bound states are the last place one would think that standard Feynman diagarams could be applied. To obtain perturbation expansions around bound states, one begins by assuming that the bound state is exactly stable over time (i.e. does not decay). One then solves the bound state Feynman diagram problem. The perturbation involves adding new diagrams to the bound state to allow its decay.
Bosons as Composite Particles
An even number of fermions, when bound together, make a boson. Consequently, it’s natural to suppose that the bosons we think of as fundamental in the Standard Model are composites made from a more fundamental fermion. This sort of thing happens in particles that we know to be composite, for example, a pion (which is a boson) can mediate an interaction between two baryons (which are fermions, the usual example is a neutron or proton). Drawn as a Feynman diagram among the quarks, the interaction looks like:
If the baryons and pion are treated as point particles, the interaction looks like two fermions exchanging a gluon:
Of course we now know that pions and protons are not point particles. We learned this by running experiments at a high enough energy that we could detect the degrees of freedom internal to these particles. Ignoring those degrees of freedom is a suitable approximation at low energies, and similarly if we are looking to unify the elementary particles by assuming that they are composites from some deeper, fermionic, particle.
HOWEVER, in making the assumption that the observed gauge bosons are composite, we run into a difficulty of the sort of “which comes first, the chicken or the egg?” That is, to make bound states (both fermions and bosons) from fermions alone, we already need gauge bosons. In the drawing above, the quarks that make up the baryons and pion are not shown as held together any force. In fact, we need gluons to hold them together. But that puts the gluons on the same level as the quarks as far as being fundamental particles. And that means that we can’t take the simple assumption that everything is built from some fundamental fermionic preon. Unless…
A simple way out of the “chicken and egg” gauge boson problem is to assume that at the deepest level, the fermions interact with each other with gauge bosons that we can ignore. This is the same assumption that our calculations on this blog have used. One way of describing it physically is that we will be assuming that the binding force between the preons is so incredibly strong that it cannot be spread over more than a single point in spacetime. And therefore the annihilator and creator for the force take the same point in space time and therefore can be ignored. It’s as if the fermions interacted with each other directly, with no gauge bosons needed to carry the force because the force doesn’t have to be carried any distance at all.
The foundations of quantum mechanics should be simple, but is this too simplistic? Let’s next take a look at what Feynman had to say about this idea.
Feynman Gives Mass to the Massless
Feynman died a few years ago and so cannot be placed on my list of brilliant physicists who think I’m a complete idiot. And in addition he isn’t here to comment on the concept of modeling fermion bound states without the benefit of gauge bosons to glue them together. However, he did put a passage in a book that showed that one can give mass to massless propagators by assuming an illegal sort of perturbation diagram, a diagram that has no force giving gauge boson. The book is QED: The Strange Theory of Light and Matter.
The book is unfortunate in that it is directed to a non mathematical audience and therefore easy for the professionals to ignore. Since I know my audience well, and am sure that they are too lazy / arrogant to dropy by a bookstore and too broke to buy a copy, so you can verify my quotations by comparison with the photographs I’ve pasted here.
We begin with a paragraph spanning pages 90-91:
The second action fundamental to quantum electrodynamics is: An electron goes from point A to point B in space-time. (For the moment we will imagine this electron as a simplified, fake electron, with no polarization — what physicists call a “spin-zero” electron. In reality, electrons have a type of polarization, which doesn’t add anything to the main ideas; it only complicates the formulas a little bit.) The formula for the amplitude of this action, which I will call E(A to B) also depends on (X_2 – X_1) and (T_2 – T_1) (in the same combination as described in note 2) as well as on a number I will call “n,” a number that, once determined, enables all our calculations to agree with experiment. (We will see later how we determine n’s value.) It is a rather complicated formula, and I’m sorry I don’t know how to explain it in simple terms. However, you might be interested to know that the formula for P(A to B) — a photon going from place to place in space-time — is the same as that for E(A to B) — an electron going from place to place — if n is set to zero.
The above quote references footnote #3, which is the crux of Feynman’s observation:
The formula for E(A to B) is complicated, but there is an interesting way to explain what it amounts to. E(A to B) can be represented as a giant sum of a lot of different ways an electron could go from point A to point B in space-time (see Figure 57): the electron could take a “one-hop flight,” going directly from A to B; it could take a “two-hop flight,” stopping at an intermediate point C; it could take a “three-hop flight,” stopping at points D and E, and so on. In such an analysis, the amplitude for each “hop” — from one point F to another point G — is P(F to G), the same as the amplitude for a photon to go from a point F to a point G. The amplitude for each stop is represented by , n being the same number I mentioned before which we used to make our calculations come out right.
The formula for E(A to B) is thus a series of terms: P(A to B) [the “one-hop” flight] + P(A to C)*n*n*P(C to B) [the “two-hop” flights, stopping at C] + P(A to D)*n*n*P(D to E)*n*n*P(E to B) [ “three-hop” flights, stopping at D and E] + … for all possible intermediate points C, D, E, and so on.
Note that when n increases, the nondirect paths make a greater contribution to the final arrow. When n is zero (as for the photon), all terms with an n drop out (because they are also equal to zero), leaving only the first term, which is P(A to B). Thus E(A to B) and P(A to B) are closely related.
Mass for Scalar Particles
If you’ve had a QFT class, the above should be pretty clear by itself. For those learning the subject, let me translate the above into more standard terminology. As usual, we will work in the momentum representation. The propagator for a massless scalar particle is given by:
This is what we will use for the “scalar photon” propagator, what Feynman calls “P(A to B)” . To make it into a massive propagator, we add a Feynman diagram for an intermediate point with an amplitude of . The basic interaction diagram, along with its translation into mathematics, is:
The “sum over all intermediate points” becomes an infinite sum of Feynman diagrams as follows:
Translating these into mathematics using the simple scalar interaction, we have:
The above is an example of a geometric series and is easily summed to give . This gives the massive scalar progator, when we set .
Handed Particles and the Dirac Equation
While Feynman states that the above is “interesting”, it does not directly read on the electron propagator in that the electron is a spin-1/2 particle rather than a scalar. In addition, the standard model is built from handed particles. We can build the standard massive Dirac propagator from massless propagators for the right and left handed states in a manner similar to the above. The basic idea is to take massless propagators for the left and right handed particles in spinor form, and combine them with a pair of interactions that take a left handed particle to right handed form and back.
The elemental Feynman diagrams, and their mathematical values, are:
A typical complex Feynman diagram assembled from these portions, and it’s mathematical equivalence, is:
The above shows a propagator that converts a left handed propagator to a right handed form. There are four such sorts of things, left to left, left to right, right to left, and right to right. And when we assemble the fundamental propagators, any particular assembly will fall in one of these four cases.
We will begin with the complex propagators that start and begin with the same handedness. These propagators need to have an even number of nodes, so the terms will have factors of from these, and will have a one larger number of propagators, contributing a factor of . With the usual rules for products of Feynman “slashed” momenta, these will multiply to give . The sum over terms gives:
This is the propagator for left to left and for right to right. The complex diagrams that convert between left and right have odd numbers of nodes and an even number of propagators. Thus these terms look like . These sum as follows:
In the context of Dirac particles built from massless spinors, the usual Dirac spinor is not the propagator of a single particle, but instead is a propagator built from a number of parts. The various complex numbers give the relative amplitudes of the various parts. To unite the four propagators we’ve computed above, that is, the LL, LR, RL and RR propagators into a single propagator, we put them into a matrix. The incoming and outgoing states are now 2 element vectors instead of 1 element complex numbers. We assemble them as follows:
The usual massive Dirac propgator is given as:
To math this with the form we’ve derived, we need to do two things. First, we put . Second, we note that in the usual Dirac propagator, the term converts left to right handed chiral states back and forth to each other, while the term preserves handedness. Thus we’ve derived the Dirac propagator from a massless propagator, as desired.