Paper preview: A Simple Unified Charge Space for Fermions

14 Sep

I’m an open-source kind of person, so I deeply believe in free access to and sharing of ideas. In keeping with that philosophy, here’s a peek at a paper I’m working on to help folks navigate the Standard Model more visually. For those of you who want to see the story first in colorful graphics first, here are the three main figures:

Xi Fermion Charge Space
Color Hexagon, Q View
Color Hexagon, Q-bar View

Related, but enormously more comprehensive (and lots of fun to use) is:

Garrett Lisi’s Elementary Particle Simulator

which anyone interested in particle physics should try.

Fully understanding the Standard Model of particle physics requires years of study in some fairly intense mathematical topics, including group theory, or the study of symmetries in complex systems.

However, the fact that the Standard Model is all about symmetries means there are points within it in which complexity suddenly snaps into perspective, providing a simpler, more intuitive way of understanding certain aspects of the model. This potential for uncovering elegant, more insightful views of the Standard Model and beyond (including gravity) is delightfully exemplified by the Elementary Particle Explorer, a free application from Garrett Lisi, Troy Gardner, and Greg Little. The Explorer allows anyone to explore different perspectives of Garrett Lisi’s E8 grand unified model of particle physics, and through that exploration understand ways in which the model snaps into simpler viewpoints. (See again Lisi’s simulator at http://deferentialgeometry.org/epe/.)

Computer science encourages viewing patterns not in terms of particles, but rather as indications of abstract structures that need to be reduced to their simplest possible forms. A good example of this principle is the decades-old particle concept of “rishons,” which attempts to construct fundamental fermions from still smaller particles. For multiple reasons, the symmetries that led to the rishon idea lead to inconsistent and poorly defined particles. However, if those same symmetries are instead analyzed and reduced to their simplest possible forms, they lead to a rather unexpected destination: A three-dimensional space whose axes represent the Red, Green, and Blue (RGB) color charges of the strong force.

That is, the “rishon” concept is nothing more than a reflection of structure of the color force, distorted by attempting to force-fit that structure into a full particle model. But conversely, if the rishon concept is used to guide the representation of color charges as quantized vectors within a special three-dimensional space, the color force suddenly snaps into a cleaner perspective that makes it much easier to visualize and comprehend.

Surprisingly, a large part of this simplification comes from treating the electric and color charges as non-orthogonal components of a single unified charge vector. Certain orientations of these vectors in the oriented (anisotropic) three-dimensional charge space become pure electric charge, while other orientations mix electric and strong charges. Weak force transitions become vertical, one-unit moves parallel to pure electric vectors.

My name for the integrated three-dimensional charge space is Xi, which in the math world is pronounced  “zah-ee”, like “sigh” with a z in front instead of an s.

What is remarkable about Xi space is that by predefining just four vectors {R, G, B, Q} and their inverses, all legal charges for fermions and anti-fermions within a generation can be defined by simply rearranging and simplifying the additive terms of the charge equation RGBQ=0, which is just a short form for R+G+B+Q=0.

Xi even captures baryon construction if three rules are added: (1) “Chaining” of the four predefined vectors R, G, B, and Q and their inverses is permitted, which results in jagged paths along the edges and surfaces of the cubes; (2) The chained paths must begin and end on the color-free Q “axis” (body diagonal), and; (3) Adding a +Q vector to the chain also always adds +1/2 spin, while adding a -Q (Q-bar) vector adds -1/2 spin.

With these three rules the standard baryons and their spins pop out easily. Examples included the spin 1/2 protons and neutrons and the spin 3/2 delta+ baryons. Even the delta++ and delta– baryons show up by adding another layer of cubes above and below the ones shown. Spin 1/2 combinations of three down quarks are forbidden not by Pauli exclusion, but by the impossibility of reversion the spin of one of the three down quarks without at the same time reversing its charge.

Two important additional references:

(1) In 1979, S.L. Glashow published without comment the upper cube of Xi space in “The Future of Elementary Particle Physics.” He mentions it simply as a mnemonic for remembering the particles in a generation! I did not find out about his use of this model until a couple years ago.

(2) Starting in 2008, Piotr Zenczykowski began publishing a series of articles that redefine the rishon concept in terms of Clifford algebras, rather than as particles. Unlike the Glashow cube, I found and looked at one of his papers while attempting to refactor the rishon idea down to its basics. I am not aware of them making a direct link to the strong force, but his papers very much capture the need for a deeper, less “particle first” examination of these remarkable symmetries in the fermion family. I heartily recommend his papers for anyone interested in looking for deeper fermion family structure.

Here’s what the Xi Fermion Charge Space looks like graphically.

—– Addendum 2016-09-14.14:51 ET —–

One of the points of Xi space is that adding the Q vector as a non-orthogonal diagonal within a fully orthogonal 3D color space enables a much simpler and more elegant navigation of two main charge characteristics of fermions, electric and color. Even the weak force shows up in simplified form as Q or Q-bar transitions that remain within a single color, e.g. R to RQ.

However, if the goal is only to track color charge and ensure color balance, it turns out that the best perspective is not the full Xi space, but its projection onto a plane perpendicular to the Q diagonal. In this plane, Red, Green, and Blue become three non-orthogonal points on an equilateral triangle, with their anti-colors occupying another similar triangle. The result is a hexagonal view in which the quarks and colors that make up matter stay in one triangle, the RGB triangle, while antimatter and anticolors occupy the other triangle. The Q subscript keeps track of the exact vector relationships, but for the purpose of the color hexagon it can simply be ignored, since Q transitions only change the non-color electric force.

The first view of the Color Hexagon is looking down from the positive Q direction, and the second view is looking up from the negative Q (Q-bar) direction. These two views show exactly the same vectors and vector relationships as the Xi space figure, but using octahedrons rather than cubes to frame the vectors. The octahedrons in these perspectives better capture the three-dimensional origins of the axis projections:

Color Hexagon, Q View

Color Hexagon, Q-bar View

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