# Non Hermitian Density Matrices

Recently Garrett Lisi’s used a broken E8 symmetry to classify the known elementary particles. I believe that E8 arises naturally from composite particles. The outline of the argument (as covered in the first post) amounts to noticing that E8 is unique among the simple Lie algebras in that “in the context of linear algebra, E8 is most simply described as the group of symmetries of its own Lie algebra!”

In the second post in this E8 series, I showed how particle states could be used as symmetry generators for certain symmetries of the particle states. This was done using pure density matrices. We began with a certain symmetry of the Clifford algebra of the Pauli algebra, and turned that symmetry into one of the pure density matrices of the Pauli algebra. In effect, we got the quantum states to operate on themselves to make other quantum states, exactly the sort of thing that is necessary to build E8.

We want to add composite or “bound” states to the “free” states that are represented by the usual pure density matrices. We will then add these bound states to the usual pure density matrices to create a larger algebra. (In that algebra, multiplication of states with differing numbers of particles will give zero.) The bound states will also be represented as density matrices. In this post we will lay the foundations for describing bound states by very simple Feynman diagrams, it will be collections of these Feynman diagrams, formed into matrices, that will create the density matrices for the bound states.

Point Particles and Qubits
The known elementary particles are point particles. Therefore, in explaining why these particles can arise in a composite model, we need only worry about bound states that can be represented as point particles. For that reason, our quantum field theory will be very simple; it will be a quantum field theory using qubit quantum states. For an arXiv paper on qubit quantum field theory using Feynman diagrams see Bialynicki-Birula and Sowinski’s “Quantum Electrodynamics of Qubits”, or my blog post on why qubits apply to this sort of thing, Feynman Diagrams for the Masses.

In order to build an algebra that contains both elementary and composite particles on an equal footing, we need to have a way of describing bound states that is similar to how we describe the free states. Our method will be with density matrices. As you discovered by reading the above linked papers on qubit Feynman diagrams, the Feynman diagrams for propagators in a qubit theory are density matrices, or as the mathematicians say, primitive projection operators.

To put these ideas into practical mathematics, let us examine the proton as a bound state of three quarks. The quarks come in three colors, red, green and blue, but their colors change as they exchange gluons, which bind the three quarks together. A diagram for the process might look like this:

Simplified Bound State Calculations

In analyzing bound states, the simplification we will make is to ignore the gluons. This is a gross simplification, but it is why our theory will only give a broken E8. The symmetry breaking is not in nature, but instead is in our model. We will ignore the gauge bosons (gluons in the above diagram) that bind the particles. In addition, we will also ignore the particle / antiparticle pairs that are created by the gauge bosons (the loop on the bottom right of the above diagram). Instead, we will only model the “valence” quarks.

In making this simplification, we will be unable to determine whether the bound states we model really are bound. What information can we get? It turns out that the quantum numbers of a bound state of quarks only depends on the quantum numbers of the valence quarks. Thus we hope that our gross simplification will be good at calculating quantum numbers. And since Garrett Lisi’s fitting of the known elementary particles to E8 was done through their quantum numbers only, the simplification will be a natural one in terms of supporting a broken E8. Nature is perfect and unbroken, it is our assumption of exact symmetry (in this model the ignoring of the gauge bosons) that is broken.

Having removed the gluon propagators from the above, we are left with the following grossly simplified Feynman diagram:

To build it from smaller Feynman diagrams, we need two kinds of Feynman diagrams, the usual quark propagators, and we need the gluon interaction vertices (but without the gluon propagators). We need:

that is, we need (a) free propagators, which amount to density matrices, and (b) gluon vertices surrounded by two free propagators, which amount to products of two density matrices. While the above proton diagram could be built from objects of just the sort (b), we need the (a) case for the proton diagram when there is no interaction.

Both the above diagrams are “propagators” in that they have one quark going in and one quark going out. The propagators of type (a) you already know are represented by density matrices. The propagator of type (b) is different from the usual in that its beginning state is different from its ending state.

If you know your foundations of quantum mechanics extremely well, you might know that this sort of propagator is not Hermitian. If not, you can think of the propagator as modeling a sort of decay, in the above example from blue to red. A decaying particle violates the symmetry one otherwise assumes for translations in time and this causes the Hamiltonian to be complex (i.e. non Hermitian).

Non Hermitian Density Matrices

In our efforts to add composite particles to the density matrices, we wish to model these composite particles as (more complicated than the usual) density matrices. To do this consistently (that is, to write everything in terms of density matrices), we need to build our composite objects out of density matrices. Fortunately, it turns out that the gluon type propagators illustrated above are also forms of density matrices, but simply non Hermitian ones.

Let’s begin by writing down the product of the propagators for spin in the +x and +z direction:

Note that while the product is non Hermitian it does have real eigenvalues on both the left and right sides. The non Hermiticity appears in the fact that the eigenkets for the right side are not the Hermitian conjugates of the eigenbras for the left side.

The product misses being a valid density matrix only in that when it is squared, instead of giving itself, it gives half of itself. Getting rid of the leading 1/2, so that the trace of the 2×2 matrix is 1, gives a non Hermitian pure density matrix:

With this new theory under our belts, let us now return to the symmetry group generated by the density matrices as discussed in the previous post. We found that the Pauli spin matrix $\sigma_x$ generated the following transformation on the Clifford algebra:
To convert this transformation into a transformation by a density matrix, we can replace $\alpha\sigma_x$ with $\alpha(1+\sigma_x)/2$. The $\alpha/2$ factor is real and commutes with the Clifford algebra. And since the transformation uses two exponentionals, one as exp(+) and one as exp(-), this real factor does not change the transformation — we get it for free. To convert the $\alpha\sigma_x$ to $\alpha\sigma_x/2$ all we need do is divide $\alpha$ by two in the formula for the transformation. We find:

where I have written “X” for the pure density matrix for spin in the +x direction.

To convert this into a transformation of density matrices, I need to replace the $\sigma_x, \sigma_y, \sigma_z$ with $X=(1+\sigma_x)/2, Y=(1+\sigma_y)/2, Z = (1+\sigma_z)/2$. Since the transformation maps 1 to 1, the 1/2s are free, and since the transformation is linear, to get the halves we just divide by two giving:

Now a Miracle Occurs

Let us take the example of the middle line of the above transformation and write it out as a 2×2 matrix. The product $\sigma_y\sigma_x = -i\sigma_z$. Substituting in the Pauli matrices we have:

Sure enough, the above has trace 1. A little algebra verifies that it squares to itself. And it is clearly non Hermitian. It is a non Hermitian pure density matrix. [Of course this also follows from the fact that the transformation preserves unity, zero, addition and multiplication of the Clifford algebra.] So the transformations of the usual pure Hermitian density matrices gave us just the building blocks (propagators with ignored gauge boson vertices in between) that we need to write a very simplified form of the bound states. And so we end this post with some theorems for the reader to prove.
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A Mathematical Note on Non Hermitian Pure Density Matrices.

Theorem: Let V be any pure Hermitian density matrix. Let K be any Hermitian operator. Then VKV is a real multiple of V.

Proof: Use spinor notation. Let V = |v><v| = |v><v| = V = tr(VK) V.

Theorem: Let U and V be pure Hermitian density matrices. Let kV = VUV, k a non zero real number. Then UV/k is a pure density matrix (not necessarily Hermitian).

Proof: We need to show that UV/k is a pure density matrix, that is, we need to show that it is a primitive (i.e. trace = 1) idempotent. Idempotency: UV/k UV/k = U (VUV) / (kk) = U V k / (kk) = UV/k. Trace: tr(UV/k) = tr (U VV) /k = tr (VUV)/k = k/k = 1.

Theorem: Let W be an arbitrary pure density matrix. Then W is the product of two unique Hermitian pure density matrices and a real constant.