Monthly Archives: January 2008

PI Lecture on Density Matrices

Kea kindly pointed out to me that the Perimeter Institute just put on the web a lecture on density matrices and the foundations of quantum mechanics. The lecturer is Christopher Fuchs, and the duration is an hour and a quarter. As a promoter of density matrix theory, I thought I would discuss it here.

The lecture begins with the contribution of the foundations of quantum mechanics to the descent into insanity of John Forbes Nash some 50 years ago.¬†Fuchs shared my view that quantum probabilities are not “non commutative generalizations” of classical probabilities, but his analysis is based on the assumption that quantum states should be written entirely in probability form. In this I disagree.

For those who don’t know what this is about, quantum states are usually represented by “state vectors” which are vectors in a Hilbert space. One can manipulate these state vectors by applying operators to them. That is, an “operator” is a linear transformation that maps the various state vectors to new state vectors. When one operates on a state vector, therefore one gets a new vector. To compute a probability, one takes two vectors and uses the inner product of the Hilbert space to multiply them. This gives a complex number. The probabiliy is the square of the absolute value of that complex number.
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Col. Raigoza’s Calculus Class

The internet contains a lot of information that is difficult to search for. One’s memory for how to spell a name fails with age and even if you find a piece of information that is important to you, if you don’t record it, it is possible for it to sink like one of those unverified island sightings during the age of discovery, and leave you tacking back and forth over the same waters muttering, “I know I saw that link here somewhere.”

Such was the case for me, with regard to the obituary notice of my high school calculus teacher, Juan Raigoza. Due to global pioneering’s recent whining about calculus, I have looked again, and found again, and now can write the post describing him and his class. Where was the link hiding? That I cannot answer, but the sadness I again feel at reading of his death suggests that perhaps my typing fingers held the telescope to a blind eye.
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Lulu Print-on-demand books

On January 4th I found out about Lulu.com, a “print on demand” book publisher. They are the modern equivalent of a vanity printer, whatever you send, they will print, so long as they don’t think it will get them in trouble, in both hard back and paper back in a variety of sizes. Their prices are quite attractive. There are no fees. Your first book costs the same fairly low price as the last, assuming you do not put together a bulk order to get them even cheaper.

Since Lulu is a vanity press, of course it has plenty of crackpot physicists selling books. But it also has some quite good material, and it is easy to distinguish. (My New Year’s Resolution is to be more professional in my physics so that could make my work easier or more difficult to distinguish, depending on your point of view.) For example, they have a tempting copy of Euclid in Greek, with translation (which you must preview to appreciate, and a series of books by Benjamin Crowell such as Vibrations and Waves which can be previewed at its website and is delightfully done.

As any regular reader knows, my physics subject is density matrix (or density operator) formalism, implemented with Clifford algebra (or Geometric algebra). I started writing a book on the subject in 2006 but then didn’t add anything to it until quite recently. Part of the reason for not working was that I didn’t see any easy way of getting it published. Finding Lulu convinced me to begin work again. Just to see what their books look like, I uploaded my LaTeX formatted copy, along with some cover art, and printed a copy. The cost for a hardbound 8.5″ x 11″ book with 175 pages was $24, with shipping. I ordered it January 4, and it arrived a few days ago, on January 21:
Lulu book cover
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Consistent Histories and Density Operator Formalism

I’ve been busy recently, mostly trying to sell an ethanol plant, and I realize that the plan was to talk about how a particle interaction could cause a force like gravity. However, I also made the New Year’s Resolution to be more professional in my physics and that would be a rather scary post. So instead, partly due to a posting of Lubos Motl, I’m going to rewrite the formalism of the Consistent Histories interpretation of quantum mechanics into my favorite formalism, that of pure density operators. This will be added to the first chapter of my book on density operator applications to elementary particles using Clifford / geometric algebra / calculus.

We will begin with a quick review of the consistent histories formalism, largely lifted from the Wikipedia article. We will then detail how the components of that formalism (which uses projection operators and a mixed density matrix) can be rewritten in terms of pure density matrices. Finally, time and space allowing, we will discuss what these things have to do with Margaret Hawton’s photon position operator in this language.
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