This website, www.MeasurementAlgebra.com is about Schwinger's Measurement Algebra (SMA). The SMA is a formulation of quantum mechanics that was defined by Julian Schwinger in the 1950s.

Two papers by Schwinger define the heart of the measurement algebra: The Algebra of Microscopic Measurement (1959) and The Geometry of Quantum States. Reading these may give you new insight into the foundations of quantum mechanics.

In brief, the measurement algebra consists of the measurements that you can perform on a beam of particles. Each measurement is of the pass or fail type, one can imagine a sort of filter. For example, we might arrange to measure whether or not the particle is an electron. Or we could measure its spin in the +z direction and only pass those particles that have spin+1/2.

Measurements can be more or less specific. A measurement of 0 is a beamstop. No particles pass such a measurement. A measurement of 1 is a free beam, all particles pass. The sum of two measurements means two separate filters, with the outputs of the two filters combined together. Measurement is associative but is not commutative.

If the beam we start out with consisted just of electrons, a set of measurements we might make upon them is their spin in various directions. Algebraically, a spin measurement is a projection operator. Such a projection operator is a pure density matrix.

For this reason, the Measurement Algebra has a close relationship with the density matrix formulation of quantum mechanics. But since the Measurement Algebra is more general, it provides a method of generalizing density matrices. This has applications to elementary particle theory because when one forms the usual density matrix, one eliminates the U(1) gauge freedom of the spinors. One can therefore hope that more general density matrices will allow one to eliminate more general gauge freedoms.

If you want to find books and papers about the SMA, click the "wiki" button on the left. You are invited to add links to your own papers related to the Measurement Algebra, along with a brief explanation. If someone is willing to type in a quick description of the SMA you are welcome to do it, but be warned that I will eventually get around to typing one up. Also, to put LaTex formulas into the Wiki, simply leave your comment in (correct) LaTex and I will convert it and pop it in later.

Schwinger's efforts with the Measurement Algebra were directed at showing how quantum mechanics can be derived from this very elegant foundation. This website will be devoted to the problem of geometrizing quantum mechanics. For this, Schwinger's Measurement Algebra provides a very natural basis.

As an example, consider the state defined by spin +1/2 in the +z direction. When one defines this as a spinor, one must first choose a coordinate system, then a representation of the Pauli algebra, one then solves the eigenvector equation, and finally one chooses an arbitrary complex phase. This is a long and relatively complicated sequence. From the density matrix point of view, one need only choose a coordinate system. One need not choose a representation of the Pauli algebra, but instead one can represent the state as:

m(+z) = (1 + S_z) / 2,

where the S_z is the spin operator in the +z direction, and 2 provides normalization so that m(+z) m(+z) = m(+z). Note that the representation of the state as a density matrix is fully geometric and has no gauge freedom.

It is thought by many that the spinor formalism must be more fundamental than the density matrix formalism because density matrices are defined from spinors and not vice versa, or because one can form linear superpositions with spinors but not with density matrices. Neither of these is true. See Physics Forums for a discussion.

With regard to linear superpositions, suppose I have two spin-1/2 states, |a> and |b>. How do I define their sum? In the spinor formalism, I must first choose a coordinate system, then a representation of the Pauli algebra, next solve the eigenvector problems, finally choose two arbitrary complex phases. The result is that the sum can be fairly arbitrary.

This example illustrates that the linear superposition of the spinor formalism is more a part of the mathematics than a part of the physics. One can perform a similar operation to add two density matrices X and Y, but it is far simpler. One need only choose an arbitrary pure density matrix O. Then define X+Y as

X+Y == (X + Y) O (X + Y)

It is easy to show that the above sum is a complex multiple of a pure density matrix, if X, Y and O are such. (See Physics Forums link for the proof.) By the way, if you happen to know a reference for the above fact, please inform me, as I'd like to give credit where it is due. It's so simple that I can't possibly have been the first to see it.

As I write this, the website is only a few months old. I have so much more to do here. If you want an email when important additions are made, put a comment into the Wiki.

This is one of several educational websites that I've recently started, to see more about them, click the "about" button.