In part 1, we briefly discussed how Feynman diagrams show up in QFT with an example taken from QED (quantum electrodynamics). The calculation for the first order correction to the photon propagator (i.e. vacuum polarization) turned out to be a rather messy integral:
QFT’s little Infinity Problem
This is a mess because it (a) involves the trace of a product of four 4×4 matrices, (b) requires eight definite integrals, each from , and (c) gives infinity. Of these problems, the worst was the fact that the calculation gives infinity. This would make one conclude that the theory was nonsense, but there were some other calculations that worked, and what’s more important, the results of certain calculations could be evaluated as and k was a number which gave an excellent match to experiment.
The infinities in the usual QFT arise from the terms in the denominator of the above sample QFT calculation, for the photon propgators, terms such as , for the electron propagator, terms such as . The problem arises when one integrates over all possible position or momentums around the loops. The loops show up in the higher order corrections for propagators as well as pretty much any other useful calculation, but the place where they are the worst is in computing bound states. For example:
The above diagram shows three quarks bound together into a baryon. Gluons are exchanged between the quarks and change the colors of the quarks. Three loops are labeled A, B, and C. Loop A is from the creation of a quark / anti-quark pair. This is a correction to the gluon propagator similar to the corrections to the photon propagator that gave the calculation discussed in the previous post. Loops B and C are loops caused when two quarks exchange gluons more than once.
The basic idea of elementary particle theory is that as one increases the energy or temperature, matter becomes simpler, and this gives clues to the structure of matter. As far as the loops go, at higher temperatures fewer of them are needed. At the most extreme energies, tree diagrams, that is, diagrams with no loops at all are sufficient. So in a certain sense, the loop infinity problem of QFT appears only when considering matter in interactions that are not elementary.
Eventually Feynman and Schwinger “solved” the infinity problem. The method was a technique that has grown into a variety of methods generally called renormalization. Most of the older generation of physicists, the generation that invented QFT, was never happy with the renormalization solution. Dirac said:
This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!
Feynman felt similarly:
The shell game that we play … is technically called ‘renormalization’. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.
(Quotes from wikipedia’s article on renormalization.)
Okay, I’m not quite an “older generation physicist”, but I agree with their arguments. One can accept that our present QFTs are only effective field theories, and therefore the infinities are acceptable, but this implies to me that the unified field theory is still out there. I realize that the quest for the unified field theory has turned more than its share of young physicists into burned out old men, but hey, I’m already a burned out old man. What have I got to lose?
Preons and Quantum Field Theory.
Ideally, a unified field theory should also explain the elementary particles. Earlier, I posted a short description of Bilson-Thompson’s helon (braid) model of the elementary particles. This had the fermions as composed of three or six preons. What I’d like to do now is to describe how the usual infinity-prone QFT can be modified to model preons of point particles, and how this eliminates the infinities.
Experimentally, the standard model elementary particles are point particles, that is particles with no substructure. At one time, the proton was thought to be a point particle, but experiments with high enough energy showed it to have components, the quarks. Perhaps we are unable to detect helons (braid preons) for this same reason; our energies are too low. Assuming this is the case, we have a choice. We can try to model the standard model elementary particles as composite particles the usual way, (and end up with a lot of nasty infinities that are iffy to fix, plus a non perturbative bound state problem that we are clueless as to how to solve), or we can make the assumption that they are in an “s-state,” that is, a state that is spherically symmetric, and that the interaction that binds them together keeps them in that state.
When the baryons were found to be composed of three quarks each, there was a problem in fitting them into regular quantum mechanics because it was expected that the wave function for three identical fermions would have to be antisymmetric. This was incompatible with experimental observations until an extra quantum number, color, was added to the quarks so that they would not be identical, and therefore would not have to have an antisymmetric wave function. In assuming that the three preons that make up a fermion are in an s-state, which must be spatially symmetric, we are faced with a similar issue. We will therefore assume that the preons are not identical, and have another quantum number similar to color, which we will call “precolor”.
The Quantum Field Theory of Qubits.
Rather than defining a complete QFT for precolor, since we are assuming that precolor is stuck in s-state bound states, we need only define a precolor QFT that works with particles that are in an s-state. The only degree of freedom available to a spin-1/2 fermion that is in an s-state is its spin. The quantum model of such a state is a “qubit.” Fortunately for us, the condensed matter folks have beat us to publication and have written a paper giving the quantum field theory of qubits:
Quantum Electrodynamics of qubits
Iwo Bialynicki-Birula, Tomasz Sowinski
The above paper is largely devoted to the subject of QFT for an electron in an s-state. Since this would otherwise be a trivial paper, the authors break the symmetry of the two available states by assuming an external magnetic field. They then derive the electron and photon propagators for the QFT theory on the qubit space. Spontaneous symmetry breaking also was used first by condensed matter theorists, and only later in elementary particle papers.
The result of restricting QFT to qubits is that the infinities go away. As Bialynicki-Birula and Sowinski state in section V of their paper:
Owing to the absence of the space components of momentum vectors, the calculation of radiative corrections is much simpler here than in the full-fledged QED. There is no need to combine denominators á la Feynman and Schwinger. All integrations with respect to the loop variables etc. can be evaluated analytically by the residue method in any order of perturbation theory.
The technical details for why this works to all orders of perturbation theory are given in the paper:
The numerator of the integrand corresponding to each Feynman diagram is a polynomial in the integration variables. The denominator is a product of first-order polynomials in the integration variables, each factor leading to a simple pole. All integrations can easily be done by the standard residue method. Note that after each successive integration the integrand retains its rational form. Therefore, it will continue to be amenable to the same treatment as during the first integration.
Propagators as Density Matrices
Bialynicki-Birula and Sowinski give a set of rules for Feynman diagrams for qubits. Their rules assume that the two quantum states are split by, for example, a magnetic field. In applying the QFT of qubit technology to the preons we will not have to assume that the quantum states are split by an external field. For us, the Feynman rules will be even simpler than their, already much simplified, version of QFT.
The propagator for the qubit electron is given in Bialynicki-Birula and Sowinski’s equation (38a):
In the above, refer to the “excited” and “ground” states. In our context, the states are equivalent and we will use + and – instead. Next, the denominators include the electron masses for the excited and ground states. The Helon (braid) model we discussed a couple posts ago is a preon-like model for the left and right handed fermions. In this sort of model, mass appears as an interaction; the left and right handed fermions are massless. So for us, . The last new portion of the denominator, the variable , is the energy of the particle. For the handed particles, energy really doesn’t make a lot of experimental sense.
To obtain a purely left handed electron we would have to accelerate a regular electron to an infinite energy. But the reason we would have to do this is because otherwise the mass interaction would mix the left and right handed portions. In a similar manner, the quarks of a baryon are permanently locked inside and to discuss their energy is iffy (but elementary particle theory does just this). So for our purposes, the energy of the handed fermions need not be equivalent to the usual measures of energy, but we will not worry too much about this.
The numerators, are projection operators into the excited and ground states. For the Pauli algebra, these are just the density matrices. Using our notation, we replace with and write these operators as:
The fact that virtual propagators are equivalent to density matrices is briefly discussed in Landau and Lifshitz’s Course on Theoretical Physics, Volume 4, Quantum Electrodynamics by Lifshitz, Pitaevskii, and Berestetskii. It is possible that I will find my copy, look up the passage, and replace this sentence some day.
Clifford algebra and Qubits
The Standard Model uses the symmetry group SU(3)xSU(2)xU(1). Of these, Qubits would seem to have applications only to the SU(2) group. But the theory of qubits comes from the traditional (i.e. non relativistic) theory of spin; that is, the traditional use of the Pauli spin matrices. This traditional use of the Pauli spin matrices is tied to the nature of spacetime. When one rotates a quantum spin-1/2 state one uses the Pauli spin matrices to adjust the representation of the quantum state. It’s not so clear what is being rotated in the weak isospin SU(2).
From a physical point of view, the SU(2) of the SU(3)xSU(2)xU(1), is an internal or intrinsic symmetry as contrasted with the Lorentz symmetry that provides the Dirac spinors that are natural to associate with qubits. And of course the SU(3) and U(1) portions are not immediately equivalent to a qubit model. Preon models such as the Helon (braid) model explain these symmetries as due to the structure of the preons.
The relativistic generalization of the Pauli spinors are the Dirac spinors. These unify the treatment of particles and antiparticles, (Dirac used them to predict the positron), and it is the Dirac spinors that make up the elementary particles of the Standard Model. It is natural that we replace our qubits with the Dirac equivalent. This suggests that we should briefly discuss the mathematics behind density matrices.
Only certain 2×2 matrices can be interpreted as density matrices in the Pauli theory. In my unfinished article presently titled “Density Operators, Spinors, and the Particle Generations”, I showed that a useful characterization is that the density matrices are the Hermitian matrices that are also primitive idempotents. “Idempotent” means that the matrix satisfies the equation , and “primitive” means, roughly, that the matrix is as small as possible but not zero. For matrices, “primitive” turns out to mean that the trace is 1. The generalization of Pauli spinors and Dirac spinors is are the spinors of an arbitrary Clifford algebra. So to understand how to generalize the QFT theory of Qubits to preons, we are led to the subject of the primitive idempotent structure of Clifford algebras.
Quantum Field Theory as a purely Algebraic Theory
In the usual wave function form, the characteristic mathematical operation is the integral. It is these integrals that produce the infinities that plague the theory. When one converts the theory to a theory on qubits, as Bialynicki-Birula and Sowinski showed, the integrals become trivial and the theory becomes one that lives mostly within the Pauli algebra (for qubits), the Dirac algebra, (for the relativitstic generalization), or a more general Clifford algebra. The theory ceases being a calculus theory, and becomes an algebraic theory. This makes the theory much simpler and easier to use.
Instead of requiring renormalization to get sensible results, QFT on qubits automatically gives finite results. Renormalization gives only approximate predictions, QFT on qubits gives very precise equations. One can solve problems this way that would be arduous or impossible in wave theory.
In addition to the absence of infinities, one also has the advantage of working with density matrices instead of spinors. Density matrices have many advantages over spinors. A primary advantage is that density matrices can be written in density operator form as objects with immediate and physical geometric interpretation. I’m writing a book on density operator theory (when I’m not working on alternative energy projects, and not writing on this blog).
When one is in possession of a different way of looking at quantum mechanics, oen naturally looks around for ways to apply it that take advantage of the easier calculations. These sorts of ideas provide one with what amounts to an unfair advantage over the competition. In a later post, I will describe how an old amateur, with no PhD and no academic connections, used the ideas described above to find an equation for the masses of the neutrinos that in its first year received four citations in peer reviewed physics journals.