About This Website
This website is one of five websites founded and maintained by Carl Brannen. Three of
these websites are devoted to educating the public in three subjects that I think are
important to the foundations of physics, quantum mechanics and especially elementary
particle physics for spin-1/2 particles. There are many possible ways of defining the
foundations of quantum mechanics. While the broad ideas presented here are not new,
the synergies that emerge when they are used in combination is.
The purpose of www.DensityMatrix.com
is to promote the understanding of the density matrix formulation of quantum mechanics. As
quantum mechanics is usually taught, density matrices are derived functions that are mainly
useful for modeling mixed states. In this website, I show that the derivation can go both
ways, at least for the spin-1/2 fermions. Here we assume that the pure density matrices (pure
density matrices which are those which are unchanged by squaring) are
fundamental. From them, we derive the spinors. In a certain sense, the density matrix
formulation is a return to
Von Neumann's program for an algebraic foundation
for quantum mechanics. The modification is that we are now connecting
the algebra to the geometry of spacetime by way of David Hestenes'
geometric calculus or geometric algebra. In that
sense the roots of the theory date to the earliest days of quantum mechanics, and the objective of the theory is to achieve Einstein's goal of
geometrizing quantum mechanics. This is in distinction to Hestenes'
method, which is to
geometrize the spinor rather than the density matrix. The primary
advantage of choosing the density matrix formalism for geometrization
is that it avoids unphysical gauge freedom, a subject of
and or confusion.
The usual use of density matrices is well known so my website concentrates on how one
gets the spinors from pure density matrices. In doing this, we see that the U(1)
gauge symmetry appears as a geometric choice. It is the objective of the rest of these
websites to expand this geometrization to the remaining gauge symmetries.
The purpose of www.MeasurementAlgebra.com
is to promote the understanding of Julian Schwinger's "measurement algebra". The
measurement algebra is a particularly elegant method of defining the foundations of
quantum mechanics that dates to the 1950s. Schwinger was looking for a way of
defining quantum mechanics that was based on the known facts of microscopic
measurements of elementary particles. The act of making such a measurement
will typically destroy previous knowledge of the particle.
Schwinger then looks at what happens when various experiments are consecutively
conducted on an elementary particle (or on a beam of them). For example, if
the measurements are of the spin of spin-1/2 particles, then the measuring
apparatus is the Stern-Gerlach experiment. If we were to restrict our particles
to be only electrons and we only measured spin, then the measurement algebra
would represent each particle with the projection operator corresponding to
the orientation of the Stern-Gerlach apparatus that gives that spin. But we
can be more general than this and imagine an experiment that separates, for example,
muons from electrons.
If a measurement is repeated on the same particle, the result of the measurement
is unchanged. Schwinger represents this by saying that such a measurement, when
squared, gives itself. This is identical to the definition of the pure density
matrices -- therefore, we can use Schwinger's measurement algebra to extend the
density matrix formulation to more general particles. For example, instead of
working with the (single particle) density matrix for an electron, we can consider
a single particle density matrix for a lepton, a particle that is either an electron, a
muon, a tau, or one of the neutrinos. This is useful in flavor physics.
The purpose of www.CliffordAlgebra.com
is to promote the understanding of Clifford algebra, particularly in its
application to the density matrix formulation extended by Schwinger's measurement
algebra. Physicists already use two Clifford algebras in the standard model,
the Pauli algebra and the Dirac algebra. The subject is usually taught by
defining particular representations of these algebras; these are known
as the Pauli spin matrices and the Dirac gamma matrices.
The density matrix formulation of quantum mechanics is written in the Pauli or
Dirac algebra. In extending this formulation with Schwinger's measurement algebra
we need a larger algebra than even the Dirac algebra and this is where Clifford
algebra comes in.
The usual way quantum mechanics is presented has the particles represented by
vectors while the operators are represented by matrices. The equations one
solves are of the form M|m> = m|m>, where M is an operator, and |m> is an
eigenvector with eigenvalue m. These are linear equations.
In the density matrix formulation, there is no splitting of the elements into
operators and eigenvectors. Instead, all elements are operators, or matrices.
The equations one solves are of the form (|m>
But the mathematics of Clifford algebras is sufficiently advanced that it
is possible to solve these sorts of equations. While more difficult, it is
here that I hope to find the foundations of the elementary particles. To do
this, a knowledge of Clifford algebras is necessary.
There are plenty of sources of information on Clifford algebras on the web.
A good place to begin is
Wikipedia on Clifford Algebra.
These sources generally treat the "canonical basis vectors" as fundamental,
and define the rest of the algebra in terms of these. This follows the
mathematical definition of the algebras. But for the purpose of applying
Clifford algebra to Schwinger's measurement algebra, we need an analysis of
Clifford algebras from the point of view of particles. That is, we need an
analysis that concentrates on the elements of the algebra that satisfy
M M = M. These turn out to be related to the elements that square to one.
www.Snuark.com is for my preon theory
of elementary particles. It is based on the above observations. To
understand it, you will first need to learn the appropriate mathematics and
get used to my notation.
www.Brannenworks.com was originally
started so that I could sell porcelain on the web. This engrossing hobby was
driven out by a desire to better understand the foundations of physics. Since
early 2003, my website has been mostly about physics. I will continue to keep
my papers there, along with various human interest items, Java applets, etc.
How Am I doing???
It is very difficult to get much attention in the elementary
physics community. In particular, trying to get a busy physicist to read a
paper that will require learning new mathematical techniques is essentially
impossible for an amateur. The only paper I've written that has received
much attention was one that had all the "adult" mathematics removed from it.
is on the masses of the leptons, particularly the neutrinos. It provided
an extension of Yoshio Koide's famous empirical formula
for the masses of the charged leptons.
The way I got the paper read was by carefully eliminating all the things that
would prevent a professional physicist from reading it. First I got rid of all
the references to alternative foundations for physics. Then I got rid of any
new mathematics that anyone would find inconvenient. Then I boiled the result
down to just a few very simple equations. This got through, and as a result
my formula for the neutrino masses appears in
If you are interested in finding other places my formula is mentioned on
the web, a useful google search is
At the same time that I am typing up the above three informational websites, I am
also working on my next paper. This one will bring back the adult mathematics
into the lepton mass formulas, and is intended to derive the standard model
from a very small number of very simple principles. In the present environment,
with very few people who understand the mathematics involved, the paper is
pointless. As far as the vast majority of possible readers, it will consist
of some empirical formulas, and a bunch of confusing symbols. Thus this paper
is a joint effort with these websites.
Upon Julia's Clothes
Whenas in silks my Julia goes,
Then, then, methinks, how sweetly flows
That liquefaction of her clothes.
Next, when I cast mine eyes, and see
That brave vibration each way free,
Oh, how that glittering taketh me!
Robert Herrick, 1591-1674