# Bound States as Density Matrices

Review: Previously, we showed that non Hermitian density matrices arise naturally as the result of using density matrices to define symmetry operations on density matrices, and that these sorts of things might be used in defining Feynman diagrams modeling bound states. We begin with a proton modeled as a set of three valence quarks held together by gluons:

We cut the gluon lines and look only at the valence quarks (as it is these that determine the quantum numbers of the bound state). This gives us a simplified model:

To model this from density matrices requires a simple sort of Feynman propagator, one that corresponds to a non Hermitian density matrix:

The corresponding non Hermitian density matrix is a product of two pure Hermitian density matrices. Let us assign to the colors red, green, and blue, the density matrices for spin in the +x, +y, and +z directions, respectively. Then, for example, the above propagator becomes:

In the remainder of this post we take Feynman diagrams like this, and assemble them into density matrices that represent the bound state. (They will be 3×3 matrices of Feynman diagrams or, equivalently, 3×3 matrices of non Hermitian density matrices.)

Our objective is to hook up non Hermitian density matrices so that they can model bound states such as the proton (or getting back to the E8 theory, elementary particles composed of simpler free particles). The first thing to note is that we must always have a vertex or node when a propagator changes color. For example:

For the example of the proton built from three colors of quarks, we need three colors and therefore three different (Hermitian) color propagators. For the vertices, we will have three choices of color for the incoming propagator and can choose either of the other two for the outgoing color. Thus there are 6 color changing propagators to add to the three normal, Hermitan, propagators. It is convenient to arrange these nine types of propagators / pure density matrices into an array as follows:

The array form is suggestive. We will be replacing these Feynman diagram propagators with density matrices, which of course are normally constructed from spinors. So let’s rewrite this array into a briefer form. We will keep the colors, but use shapes more reminiscent of spinor notation, i.e. |x><x| :

What happens if we have two such objects and multiply them together using the usual rules of matrix multiplication? We find that colors are handled just the way that we would like them, that is, red propagators hook up to red propagators, etc.

Suppose that we have two such matrices, A and B. The matrix multiplication will look like this:

In the above, we have blown up the (3,2) or (blue,green) entry to show that it is a sum over the three possible intermediate colors. This is reminiscent of how one stitches together Feynman diagrams.

Feynman Diagrams as Matrices

In fact, the analogy with Feynman diagrams is perfect, provided we realize that our Feynman diagrams are (a) in qubits so there is no dependency on spatial coordinates or momentum, and (b) are Fourier transformed so that we do not integrate over all possible intermediate energies (that is, the convolution over position is eliminated in the Fourier transform). What is left is simply multiplication and the matrices organize the propagators so that they fit correctly into each other.

In interpreting the matrix entries as Feynman diagrams, we need to assign each such diagram a complex number. We can think of the 3×3 matrices as containing 9 complex degrees of freedom, each of these degrees of freedom multiplying the appropriate propagator. This allows us to translate from the difficult language of Feynman diagrams to the simple language of matrix multiplication.

3×3 Complex matrices as Density Matrices

The requirements for a matrix to be a density matrix are that (a) it have trace 1, and (b) it be idempotent. For the case of the proton as composed of identical quarks, we can add a third requirement, (c) the three colors must be treated equally.

The requirement that the colors be treated equally greatly simplifies the problem of finding the density matrices among the 3×3 matrices. Cyclically permuting the three colors and demanding that the matrix be unchanged, we find that we have only 3 complex degrees of freedom — the matrices must be circulant. There are 8 circulant matrices that are idempotent. Of these only three have trace 1, they are:

where “w” is the complex cubed root of unity, $\exp(2 i \pi/3)$. For the calculations, see pages 39-40 and equation (3.39) of the author’s book on density matrix theory. Also see Marni Sheppeard’s note on circulant matrices and discrete Fourier transforms.

The three solutions to the circulant 3×3 density matrix problem have been associated with the three generations of particles, and the Koide mass formulas for the charged leptons and neutrinos, but we will leave that story for another day. Instead, since we derived this simple result (that three generations arise from a binding force among three particles that can be modeled by qubits), we will apply it to the problem we used to motivate the construction, that is, the proton, or more generally, the baryons and their resonances.

The Resonance Structure of the Baryons

The density matrix representation of bound states was written down under the assumption that (a) we could ignore the gluon interactions of a baryon and (b) the interactions could be dealt with qubits, which ignores momentum or position information. But the calculation is non perturbative and we might expect it to work okay on the baryons. The quantum numbers are determined by the valence quarks, and the three primitive idempotent circulant matrices given above have their diagonals exactly alike. So we expect the baryons, like the elementary particles, to come in 3 “generations.”

Baryons with identical quantum numbers but differing mass do appear. They are called “resonances” instead of “generations.” They are well described on the Particle Data Group (PDG) website. The resonances are labeled according to their quark content and their isospin and angular momentum quantum numbers, for example $L_{2I,2J}$ for the Nucleons and Deltas (which includes the proton and neutron and are made from up and down quarks), and somewhat confusingly, $L_{I,2J}$ for the Lambda and Sigmas (which are made from up and down quarks, plus one strange quark). I will make the translate the isospin index from $L_{I,2J} \to L_{2I,2J}$ to make the quantum numbers of the Lambdas and Sigmas match those of the nucleons and Deltas.

The first reason for looking at the baryon resonances is that the density matrix representation of bound states presented here would be somewhat undermined if these resonances came with more than 3 for the same set of quantum numbers.

Baryons with higher energy quarks, such as charm, bottom exist, but none of them are seen with more than 1 or 2 resonances. The failure of the experimentalists detecting all three predicted resonances for a baryon could be due to a number of things other than this theory being wrong. If two of the resonances happen to have very similar masses, the experimentalists will be unable to distinguish between them. And if one or two of the resonances is very weak, it will be hard to detect.

And in fact, there are only two instances where the PDG lists four resonances with the same quantum numbers. For the nucleons, there are four $P_{11}$ resonances (see nucleon and Delta Acrobat file linked above): N(939), N(1440), N(1710), and N(2100). However, the N(2100) comes with only 1 star, which indicates it is not well documented. When we look in the PDG description of the N(2100), we find that it reads only: “OMITTED FROM SUMMARY TABLE — The latest GWU analysis (ARNDT 06) finds no evidence for this resonance.”

Similarly, among the Sigma resonances, there are four $P_{21}$ states (labeled as $P_{11}$ in their unnecessarily confusing nomenclature), the Sigma(1193), Sigma(1660), Sigma(1770), and Sigma(1880). Of these, the Sigma(1770) has 1 star and the PDG text reads “OMITTED FROM SUMMARY TABLE — Evidence for this state now rests solely on solution 1 of BAILLON 75, (see the footnotes) but the Lambda Pi partial-wave amplitudes of this solution are in disagreement with amplitudes from most other Lambda Pi analyses.”

So if we eliminate two iffy experimental observations / calculations, we are left with a set of baryon resonances that agree with the density matrix model of the bound state of three particles, namely that there will be three resonances.

While this density matrix analysis is iffy on the baryons, it does not apply at all to more weakly bound states such as nuclei or atoms. These much more weakly bound states require momenta / position information and cannot be put into qubit form.

Next, we show how these bound states generate a symmetry of themselves and the free states. After that we are one post away from completing the derivation of E8 as a construction of bound state qubit theories. That last post will cover how one converts a symmetry into a model of a bound state. And after that? We will return to the problem of categorizing the baryon masses.

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Mathematical sidenote: In manipulating Feynman diagrams in this way there is an echo of Kapranov’s paper “Noncommutative geometry and path integrals”. Of course pointing out this sort of subtlety is due to the attention of the mathematical physicist, Marni Sheppeard (aka Kea) who has extended the theory to Fourier transforms of the particle masses.