The Algebra of Orthogonal Spin-1/2 States

Everybody knows that the quantum states of spin-1/2 particles come in two and only two independent spin orientations, for example spin +z and spin -z (up and down). My snuark theory uses spin-1/2 calculations to compute the Koide mass formulas, so it is a little inconvenient that snuarks need to come in six independent quantum states, +x, -x, +y, -y, +z and -z.

Three snuarks (no two antiparallel) combine to make one chiral handed lepton, the other three combine to make another chiral handed lepton, but traveling in the opposite direction. One chiral handed lepton could be a left handed electron moving in the +u direction, the other would be a left handed electron moving in the -u direction. Thus the six snuark degrees of freedom get turned into two spin-1/2 degrees of freedom.

In terms of classical probabilities, the six snuark states are grouped in three pairs, (+x, -x), (+y, -y), and (+z, -z), with the elements of each pair being mutually exclusive in terms of transition probabilities, and elements between different pairs having transition probabilities all equal to 1/2, and hence independent.

As a physical definition, this requirement, three axes that give transition probabilities of 1/2, will always give three pairs of states on perpendicular axes. The snuarks are then defined by picking one element from each axes. This is the largest number of spin-1/2 quantum states that can be chosen given the requirement that the transition probabilities all be equal to 1/2.


In the mass interaction, a left handed chiral particle becomes a right handed one. In the usual field theory notation this is written as something like m e^\dag_Le_R . That is, a right handed electron is annihilated and a left handed electron is created. We will follow this notation, but we need to distinguish between orientation (which we treat as precolor).

When Feynman first saw field theory calculations with creation and annihilation operators he rejected it as unphysical because he knew that particles cannot be created one at a time. For example, if an electron is created from the vacuum, a positron must also be created.

Eventually Feynman took up the new notation, but we will follow Feynman’s hesitation and use a notation that does not allow us to write isolated creation or annihilation operators. Instead, we will always follow an annihilation opeartor with a creation operator for the same particle.

This very restricted notation allows the modification of a quantum state so long as the particle type does not change. An example of this would be the conversion of an electron with spin in the +y direction into an electron with spin in the +x direction. This is done by writing the annihilation and creation of the +x electron on the left side of the creation and annihilation of the +y electron: e^\dag_{+x} e_{+x}\;\;\;e^\dag_{+y} e_{+y}

The above product is is an operator that annihilates a +y electron and creates a +x electron. It is tempting to follow the notation of Schwinger’s measurement algebra and replace this with something like e^\dag_{+x} e_{+y} . We will not do so because writing the creation and annihilation opeartors separately in this way implies a choice of complex phase (which appears in complicated products), and it is not possible to choose complex phases for all the creation and annihilation operators as a continuous function.

The Basis for the Snuark Algebra

The Snuark Algebra is a discrete subset of a measurement algebra of the sort invented by Julian Schwinger in the 1950s. To change a quantum state vector into a measurement, you turn it into a density matrix. For example, when you turn the state vector |+x> into a measurement you get |+x><+x|, which in our reduced notation is the same as (1 + x) /2. A measurement algebra is generated by all possible complex multiples, products, and sums of a set of basis measurements. In the case of the Snuark Algebra, the basis measurements are made from the 3 independent spin-1/2 qubit states, +x, +y, and +z, and therefore are (1+x)/2, (1+y)/2, and (1+z)/2.

Complex multiples” means that you can take an arbitrary complex multiple of a measurement and what you get is still a measurement. For example, (1+x)/2 is a measurement and therefore so is (2+i)(1+x)/2. “Products” means that if A and B are measurements, then so is AB.

Given the three independent measurements (1+x)/2, (1+y)/2, and (1+z)/2, there are an infinite number of products that can be produced from them. But it turns out that these can be written as complex multiples of a complete set of only 9 basis measurements.

The reader may be more familiar with basis sets in linear algebra where the basis elements are orthonormal. In such a case, normal means that the inner product of the elements with themselves give one. In our case, “normal”, means that the basis measurements all square to themselves, that is, they are all idempotent, or that they be projection operators. The idempotency requirement allows us to choose the scale of the basis measurements. With this choice, the 9 Snuark basis measurements turn out to be:
NIne Basis Measurements os Snuark Algebra

It should be clear that the above nine measurements are necessary. That is, each one is a product of the (1+x)/2, (1+y)/2, and (1+z)/2 measurements we started out with. What I’ve not shown is that any further products give stuff in the above list (which is in the form of a matrix for a good reason, this is where you will see the Koide mass matrix come from). To verify that the nine are sufficient, I need only compute the 81 products among them, and verify that all 81 of these are complex multiples of the basis set. Ouch.

Hmmm. A simpler way to do this is to show that any product of three of the original measurement algebra elements, (1+x)/2, (1+y)/2, (1+z)/2, can be reduced to one of the nine above. Then, by recursion, the above nine are complete. There are 27 such products.

Since (1+x)/2, (1+y)/2, and (1+z)/2 are idempotent, there is no need to consider products that have the same measurement in a row. This reduces the 27 products to 12. A few of these are calculated here:
Snuark products reducing 3 to 2 measurements.

For example, the third line of the above calculation, when written out explicitly in Pauli sigma matrices, computes as:
zyx calculation done explicitly with Pauli matrices
where (1-i)/2 = \exp(-i\pi/4)/\sqrt{2}

Looking forward to the Koide mass formula, the square root of 2 in the formulas will arise from the difference between the 1/2s and 1/sqrt(2)s in these three examples, while the extra phase angle of exp(i\pi/12) in the neutrino sector comes from the cubed root of the exp(i/\pi/4) phase factors in the last two. And the cubed root arises when we convert the Feynman diagrams into a complex matrix algebra, next post.

I have to review a business plan for an ethanol plant so I’m posting now and continuing this on later.


Filed under physics

2 responses to “The Algebra of Orthogonal Spin-1/2 States

  1. Kea

    Wonderful post! I assume you know about the too-wide latex problem. Good to see interesting numbers like 9, 27 and 12 all appearing in such a natural way. Good luck with the business.

  2. carlbrannen

    Actually, when I reviewed this, the too-wide Latex image was getting inserted below the bottom right edge of “recent posts”, so I thought I could get away with it. In general, with this css, my pictures have to be 500 pixels wide.

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