Inspired by Tommaso’s regular column The Quote of The Week, I was amused by a comment on Carl Brannen’s not-so-busy blog:

lots of us 10 year olds really do want to learn physics… Al

It brings to mind the old adage about science progressing one death at a time. If you write more, Carl, perhaps more 10 year olds will join the club. Actually, I must confess, you do write a fair amount, such as this recent comment on Clifford’s post about Eureka moments:

My most recent such moment was when I realized that any non Hermitian projection operators in the Pauli algebra can be written in a unique way as a real multiple of a product of two Hermitian projection operators.

That’s nice.

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## Kea said,

February 20, 2007 @ 12:40 am

Carl, does that mean you can split idempotents? This is sounding more and more like a kind of Cauchy completion one does for categories (by going to a category where idempotents split).

## Mahndisa S. Rigmaiden said,

February 20, 2007 @ 1:28 am

02 19 07

Splitting idempotents eh? Well consider writing any operator as the sum of a negative part and a positive part.

Hmmm, like when we were learning about spherical harmonics, the derivation in Sakurai involved splitting up an operator into 2 parts.

I know that any Unitary operator can be expressed as a linear transformation on a Hermitian operator E.G. U= H + i(dependent variable like time or position) And U naturally has an exponetial representation.

Hmm so what a Hermitian operator generates a non Hermitian operator that can be split into two parts. This is one answer to the question. Let us find some others…

## Carl Brannen said,

February 20, 2007 @ 1:55 am

Kea, the way I see it now, the Hermitian PIs are very special creatures, the more general non Hermitian PIs are kinder and gentler to deal with in certain ways. But each non Hermitian PI can be written as a product of two Hermitian PIs, so one can apply the theorems of Hermitian PIs to the non Hermitian case.

The difference is sort of like how complex numbers are a generalization of real numbers, and one can write a complex number from two real numbers (in a number of ways, the one being sort of similar to this is r and theta). The complex numbers can solve problems that the real numbers cannot.

I signed up for sci.physics.foundations. It’s moderated, but allows weirder posts than usual. Too bad it’s in text format.

There are some interesting articles there by Jay Yablon. One suggests that electrons, muons and taus are composite based on magnetic moments.

One other thing. I’ve got the Java program to convert 3×3 matrices of products of NHPIs into 3×3 matrices of complex numbers. The test is like the diagram chasing you category people like so much. Take a 3×3 matrix of Pauli NHPIs, convert it to complex, square it, and compare with the same 3×3 Pauli matrix, squared, and then converted.

The next task is to write the reverse conversion, to take a complex matrix into a 3×3 matrix of NHPIs. Since the PIs among the complex matrices are known, this gives an exact solution for the NHPIs. That code is written but it was late last night and I haven’t got it to work yet. Probably tonight.

## CarlBrannen said,

February 20, 2007 @ 2:02 am

Mahndisa; the splitting is by multiplication. In particle physics, one represents consecutive events by this sort of multiplication. It’s a causality sort of thing in classical theory of propagators. Another way of writing it is

G(x,x”) = \int G(x,x’) G(x’,x”) dx’

See the multiplication in the above? Physically, a Hermitian PI is an example of a Stern-Gerlach filter. The product of two of them means putting the output of one filter into the input of another. The theorem says that any two Hermitian PIs can be written as a NHPI with a real loss of amplitude.