Regge Trajectories and Koide’s formula

As many of you know, I’m a proponent of the density operator formalism of quantum mechanics as opposed to the usual state vector / spinor formalism. The basic idea with density operator formalism is that quantum states should be represented by a density operator, or density matrix, rather than a state vector. This sort of idea is similar to the S-matrix theory that was part of the foundations of string theory, and I thought it would be worth exploring the similarities and differences.

The idea behind the state vector / spinor formalism is that quantum states are represented by state vectors or spinors. These sorts of objects are linear. In the state vector / spinor formalism, the spin state of an electron is typically represented by a Pauli spinor, a 2 element complex vector. Such a quantum state corresponds to spin-1/2 in some direction, \vec{v} = (v_x,v_y,v_z) . The operator for spin in this direction is typically written out as a 2×2 matrix using the Pauli spin matrices. One takes the dot product of the spin direction with the vector of Pauli spin matrices, giving:
Pauli spin operator in v direction

The spinor representing spin-1/2 in the v direction is a two element vector which is an eigenvector of the above matrix with eigenvalue +1. It is easy to see that for almost any 3 dimensional unit vector \vec{v} , such an eigenvector (in ket form) is given by:
Pauli spinor for spin 1/2 in direction v
The above formula gives a valid ket for any spin direction except (0,0,-1), but it isn’t normalized. To normalize it, divide by \sqrt{2+2v_z} . Higher spin states than spin-1/2, say spin-n/2, need spinors of size (n+1)x1 in size. So spin 3/2 will be 4×1 vectors, spin 2 will be 5×1, etc, but we won’t discuss these much.

The S-Matrix
A simple particle experiment is to take two particles, label them 1 and 2, and collide them. Suppose that two particles are produced in the collision, labeled 3 and 4. We have two quantum states involved; the initial state with particles 1 and 2, and the final state with particles 3 and 4. To represent a quantum state with more than one particle, one takes the direct product of the spinors. If the 1 and 2 particles are spin-1/2 each will be 2×1 vectors, so their direct product with one another will give a vector with 2×2 = 4 elements.

Quantum mechanics works by computing probabilities. The probabilities are always the squared magnitude of a complex number which is called the amplitude. For the case of transition probabilities, the amplitude is computed by putting an operator between the bra for the initial state, and a ket for the final state. Another way of saying this is that in quantum mecahnics, amplitudes are “bilinear” , that is, they are a linear function of the initial state, and also a linear function of the final state.

If you vary the spin orientations of the initial and or final states, you will vary the computed amplitude and there will be various probabilities for the various cases. But they are linear in both the initial and final states, and therefore you can always model the amplitude process as a matrix, the “S-matrix” where “S” stands for “scattering.”

If you arrange the energies and spins of a scattering experiment just right, the collision can produce a state that lasts a long time. The “long time” is interpreted as a state where the particles are temporarily bound together. Such a state is called a “resonance” or an unstable bound state. The states are typically unstable because they were entered with too much energy or angular momentum for them to last. Such a state might become stable by getting rid of its excess energy by emitting particles such as photons or whatever, or it might break apart one way or another.


As the reader probably knows, my opinion is that the known elementary particles are bound states; that is, they are composite particles made from simpler and more elementary particles. Bound states are difficult to analyze with the usual tools of quantum mechanics, but the analysis of resonances is a great place to start, and makes a good introduction to how I have modified the usual tools of quantum mechanics to make them amenable to the understanding of bound states.

Let’s look at the simplest resonance problem that we can: a temporary bound state composed of two particles of the same type (but perhaps with different spin or momentum) 1 and 2, which breaks up elastically into particles 3 and 4 which are also of that same type. Following the rules of quantum mechanics, this is represented by an S-matrix.

The way one usually does this sort of problem, the initial and final states are free states far from one another. This makes it possible to represent them as plane wave functions. This simplifying assumption is not in itself a requirement of quantum mechanics, but it makes particular sense for the situation typically seen in a particle experiment.

Regge Trajectories

Resonances were analyzed in the 1960s by Tullio Regge. The resulting theory became known as Regge theory. The central idea behind the theory was to use the basic properties of any S-matrix, unitarity (the fact that something has to happen, so the total probability is one), analyticity (the S-matrix can be expanded in a complex series), and crossing symmetry (what happens when you turn Feynman diagrams upside down etc.) to derive a description of how the S-matrix has to look when it is written over complex angular momenta. This is done using partial wave analysis to break up the interaction into different resonances. It is partial wave analysis that gives us the wonderful tables of resonances on the Particle Data Group website.

When you examine a particular resonance that comes in a series of small to large angular momentum (which is deterimined by the partial wave analysis), you find that they form a straight line if you graph the square of the angular momentum against the mass. This straight line is a “Regge Trajectory” , and Regge Theory gave a theoretical explanation for this.

Bound States as Resonances

Now the strange thing about Regge theory is that you’ve apparently got two distinct types of quantum states running around. The initial and final states are represented by spinors or direct products of spinors. But the resonance state, which is a temporary bound state, is represented by a matrix instead of a spinor. And these temporary bound states can be things as prosaic as an excited state of a proton. A proton is a stable bound state, something we’d be more inclined to represent with a state vector instead of a matrix. (Regular readers of this blog already have had drilled into them, in quantum mechanics, virtual states are represented by density matrices. This is just another example.)

Now suppose one of the initial or final states is a muon. Muons aren’t stable. Following the method of Regge theory, they should be represented not by a spinor, but instead by an S-matrix. And of course we can always turn the spinor for a stable state into a pure density matrix and so represent such a stable initial or final state as a matrix too. This is half the idea of density operator formalism: quantum states should be represented in matrix form, S-matrix for resonances, and density matrix for stable particles. (The other half of the idea is to use the operator algebra rather than soiling one’s hands by choosing an arbitrary representation in matrices, but that idea will be less familiar to the audience and I will leave it for some other post.)

The usual idea in Regge theory is to have the initial and final states be quantum states that are orthogonal in that they describe particles that are widely separated in space. However, quantum theory is quite generally bilinear, and the same reasons that show that the amplitude for this sort of temporary bound state can be described by a complex matrix, also applies to the particles that make up a permanently bound state. Since “S” stands for scattering, let us call such a matrix a “B-matrix” where “B” stands for “bound.” Since it’s bound, the initial and final states are not going to be far apart, but instead are going to have to be the same states. The unitarity, analyticity, and crossing symmetry of the usual S-matrix translate into similar statements about the matrix that gives the amplitude for activity inside the bound state. In short, the B-matrix tells us how the bound state is glued together.

In a bound state matrix, unitarity means that probability is preserved. For a density matrix, this means that the matrix must square to give itself: \rho^2 = \rho . Since we are interested in bound states that combine to give point particles, we need only consider finite numbers of degrees of freedom. Therefore the requirement of analyticity means that our matrix is finite. Another way of describing the difference is that Regge theory is concerned with various energies and angular momenta. We’re only concerned with one energy and angular momenta; the ground state itself. This converts the anylticity requirement into a finite matrix requirement where analyticity is trivial. And the crossing symmetry needs to be applied on a case by case basis. For instance, if we are dealing with three identical particles (that are in different but equivalent states such as red, green and blue), then we require that the S-matrix be circulant.

So it should not be too much of a surprise that Koide’s mass formulas for the leptons can be put into a particularly simple circulant form. This arises directly from an assumption that they are composite particles made up of three parts with circulant symmetry.

Koide Equations for the Baryon Resonances

The above discussion was a little scattered in that we began talking about S-matrix theory, which would apply, for example, to resonances such as the baryons, but then we segued on to the subject of the Koide mass relations, which applies to the generations. The implication is that we should be able to apply the B-matrix analysis that worked for the charged leptons and neutrinos to the baryon resonances themselves. How does the Koide formula interact with Regge trajectories?

In the B-matrix theory of the leptons, the three generations come from the assumption that each lepton is composed from three preons. Writing the B-matrix as a circulant 3×3 matrix, one finds that there are eight solutions for the idempotency equation \rho^2 = \rho . Of these 8 solutions, only 3 are “primitive” the rest are uninteresting solutions such as the zero matrix, the one matrix, and the three matrices you get by summing pairs of the primitive circulant idempotents. These three circulant primitive idempotent matrices are:
The three 3x3 circulant primitive idempotent matrices
for n=1, 2, 3 . For a more complete discussion and derivation of these matrices, see Chapter 3, Section 3 of my incomplete book on density operator theory.

In the above, the Koide formulas are supposed to come from the leptons being composed of three identical preons. The baryons are known to be composed of three quarks, so one expects that baryons will also arrive in three “generations.”

Regge trajectories apply to series of baryon resonances that have different angular momentum quantum numbers. A good set of notes with an application to the baryon resonances are the class notes of Stephen D. Elliswhere the example is given of the series \Delta(1232), \Delta(1950), \Delta(2420) . In the Particle data book entry on the baryons these are listed, respectively, as having quantum numbers of P_{33}, F_{37} and a H_{3,11} . The second digit of these three states, “3”, “7”, and “11”, gives twice the total angular momentum, i.e. 3/2, 7/2, and 11/2.

On the other hand, the Koide formula applies to particles in different generations, but which are otherwise identical. Such particles have identical quantum numbers. And the baryon resonances do include such sequences. For example, there are three Delta P_{33} states in the particle data book: \Delta(1232), \Delta(1600), \Delta(1920) . In those few instances where the Particle Data book includes 4 resonances with identical quantum numbers, one of them is marked with a single asterisk and is not well supported in the literature. As is traditional in physics theory, I will simply assume that these are experimental error and ignore them.

The natural next step is to take a look at all the sequences of baryon resonances and see if they can be fit to Koide type formulas. Preliminary work on this is at Physics Forums and later posts. One of my tasks on my long list of things to do is to write the baryon resonances up in Koide form in a better format. And it’s natural to ask if there is a way of combining Regge trajectories and Koide trajectories to give a combined formula that works for more than one quantum number.

Combining Regge and Koide Trajectories

In Regge trajectories, the square of the mass is proportional to the angular momentum plus a constant:
Regge trajectory mass equation
One typically writes masses in Koide’s formula as the square of a cosine function:
Koide trajectory, traditional (exact) form
However, recently I’ve had some second thoughts about the Koide formula, and I suppose that I should mention them here as it could effect how one would fit the two trajectories into a single formula.

The basic idea is to suppose that there are two different types of fields that contribute to the lepton masses. One type of field, call it the electric field, is associated with a very small mass, such as the electron mass. The other type of field, call it the magnetic field, is associated with a very large mass, such as the mass of a (fictional) magnetic monopole. The two fields contribute unequally to the mass of the particle, but the underlying dynamics come from particles converting between the two types. Thus the contribution from the electric field gets squished in favor of the contribution from the magnetic field.

Now the Koide formula for the charged leptons uses an angle, 0.22222204717(48) that is very close to 2/9. What I wondered is if you could rewrite the Koide formula so that it would use exactly 2/9, but that there would be an orthogonal contribution to the energy of a very small amount. That is, I wondered if you could replace the formula:
Koide trajectory formula in usual (perfect circulant matrix) form
with a formula that included contributions from two orthogonal fields with characteristic energies \mu_b >> \mu_e :
Koide's formula in field equation form
so that the upper term would contribute most of the mass, and the lower term would contribute the remainder. Another way of describing the above, is that I wondered if I could replace the arbitrary angle in the Koide formula with an arbitrary ratio of two field strengths. It turned out that this works. The Koide formula CAN be rewritten as a sum of squares of field strengths. And when you do this, the strength you get for the weaker field turns out to be fairly close the neutrino masses.

Ellis gives a “derivation” of the Regge relationship between angular momentum and mass (or energy) roughly as follows: Assume a gluon flux tube of length 2R connecting a quark and antiquark. Assume both particles travel at speed c in a circle and that the flux tube has energy density \kappa along its length. Computing the angular momentum one finds J = 0.5\kappa \pi R^2 and total energy is E = M = \kappa \pi R . This gives J = M^2 / (2\pi\kappa) or putting mass M = E as a function of angular momentum, M = \sqrt{2\pi\kappa J} . Of course this relationship does not depend on the details of the flux tube geometry, just that it is 1-dimensional with total length R. The total enegy is then proportional to R, but angular momentum (assuming rotation perpendicular to the symmetry axis of the flux tube) is proportional to R squared.

The Koide formula applies to quantum states with the same angular momentum J, but with different masses. Looked at from the point of view of the above “stringy” derivation of the Regge trajectory formula, the Koide formula implies that the generation difference in mass M comes from differences in \kappa , and one has
Koide equation for flux density in gluon tube
for the flux density in the gluon tube.

Now the “gluon tube” only makes immediate sense in the meson case, but the general idea should follow over to the baryons. This seems to say that the energy density of the gluon tube has to follow the 4th power of a field strength. Any ideas why this might be so?

The obvious physical interpretation is that the radius of the flux tube is not constant, but instead is proportional to the field strength. This makes the area proportional to the square of the field strength, and since the energy is proportional to the area times the square of the field strength, you get that the energy is proportional to the fourth power of the field strength.


Filed under physics

13 responses to “Regge Trajectories and Koide’s formula

  1. nc

    This is a very informative post, and is written at a level I should be able to grasp with some study of the details, so thank you.

    For now, can I say that the result at the end that the gluon energy is proportional to the fourth power of the gluon field strength, is a very interesting way of looking at the difference between strong (gluon) force fields and electromagnetism.

    The flux tubes and relationship of energy density to field strength are things I’ve come across in electromagnetic theory, since Faraday invented the idea of tubes of flux connecting charges, and Heaviside/Poynting discovered that the energy density of an electromagnetic field is proportional to the square of the field strength.

    For me the question is the physical connection underlying this. The “flux tubes” in electromagnetic theory seem physically to correspond to paths taken by gauge bosons being exchanged between charges. Gluons, being massive (unlike electromagnetic field quanta) and having three types of colour charge, are interesting because they each comprise of a colour and an anti-colour but the 3 colours only give 8 gluons not 3*3 = 9, as Baez comments at

    This is a very interesting topic, one I hope to study carefully.

  2. carlbrannen

    I’m glad you liked it. I’m going to write up more posts that give introductions to various things I’m working on at a level that people can understand more easily, maybe starting tonight. And having reread this post, I’m going to fix a few things…

  3. Pingback: Koide formulas and Qubit / Qutrit MUBs « Mass

  4. AngryPhysicist

    I just found this paper on composite and elementary particles which you may find interesting; it is relatively old (from 1964!)…

    On the equivalence of elementary and composite particles
    B. W. Lee, K. T. Mahanthappa, I. S. Gerstein and M. L. Whippman

  5. carlbrannen

    AP, the abstract is interesting enough I will put it on my list of papers to download the next time I am over at the university.

    It has the smell of a boot-strap paper, but those ideas are very appropriate for the kind of things I’m doing. It’s hard to find free papers that talk about the N/D method. here is one.

  6. AngryPhysicist

    Oh, one last thing! I found a great introduction to Regge theory on pages 48-50 of “Fundamental theory of matter: a survey of results and methods.” by Abdus Salam (freely available here:

    …and this article seems to be good about it too (also free): “On the Zero energy behavior of Regge poles and residues” by Abdus Salam and J.A. Strathdee, available here:

    Sorry if I’m leaving too much stuff!

  7. angryphysicist

    This thesis appears to have a fairly good explanation of the N/D method on pages 7-8 “A Method of Calculating Regge Trajectories” (available

  8. carlbrannen

    Wow, that thesis makes me feel young. 1966, back when 8 minutes of CPU time on an IBM 7094 was a fairly expensive use of computing resources. And the thesis dates to back before LaTeX.

    The new paper I’m working on has a reference to Regge trajectories and Koide mass formulas, but it does not go into the details of the above calculations. I’ll link in a temporary copy here. I’m still working on the section dealing with goodness of fit. Then I have to go through and add references. And after rewriting a couple times, go through and verify that it meets the journal requirements.

  9. Kerry McKenzie

    Hi there – I was psyched to read that there’s still someone around who doesn’t believe in elementary particles! I’m looking at the S-matrix theory for a philosophy project and am hacking my way through a load of Regge theory to try to understand what it is about these trajectories that implies that – if something lies on one – it must be composite. I’m finding it hard to hit the nail on the head. Does anyone know of an approachable introduction to the connection? (By ‘approachable’, I mean I have a [bad] undergraduate physics degree!)
    Thanks for your post!

  10. carlbrannen


    Well by hanging around here you are definitely on the fringes of physics. However, as to why a Regge trajectory implies non elementary, that’s because Regge trajectories are for orbital angular momentum (i.e. J), and to have an orbit you have to have one thing orbiting another.

    But in the standard model, the hadrons are not elementary so this is not anything different from the usual.

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