# Who developed this equation, Pauli?

1. Jun 22, 2012

### PhilDSP

The earliest instance I've seen to this equation, involving the Pauli matrices, is from a 1967 paper by J. M. Levy-Leblond. But it seems general and useful enough that it must have been discovered earlier:

$(\sigma \cdot A)(\sigma \cdot B) = A \cdot B + i \sigma \cdot (A \times B)$

Last edited: Jun 22, 2012
2. Jun 22, 2012

### stevendaryl

Staff Emeritus
Actually, it turns out that this equation is mathematically the same as the equations of geometric algebra, developed by Clifford in the late 1800s, which in turn is a generalization of Hamilton's quaternions.

Pauli introduced spin matrices in 1925, but I'm not sure if he knew about geometric algebras, or whether it was a case of rediscovery.

3. Jun 22, 2012

### tiny-tim

Hi PhilDSP!
is 1846 early enough for you?

see http://www.emis.ams.org/classics/Hamilton/OnQuat.pdf for a reprint of william rowan hamilton's original papers

at page 19 of the .pdf (page 16 of the book) …

21. The fundamental rules of multiplication in this calculus give, in the recent notation, for the scalar and vector parts of the product of any two vectors, the expressions …​
… the formula is given, albeit in different notation, essentially AB = S.AB + V.AB = -A.B + iA×B

(i think this comes originally from The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series), vol. xxix (1846), pp. 26-31, but that volume doesn't seem to be on google e-books yet)

4. Jun 22, 2012

### PhilDSP

Thanks stevendaryl and tiny-tim!

Yes, I have *lots* of material on quaternions and quaternion algebra. But nothing from the original papers by Hamilton so the link provided will be interesting. Levy-Leblond used that equation (with Pauli matrices) to give a derivation of Pauli's linearization of the Schrödinger equation but I'm not sure if Pauli did also.

P.S. According to Wikipedia, Pauli's material is (1927) Zur Quantenmechanik des magnetischen Elektrons Zeitschrift für Physik (43) 601-623. Since I'd have to struggle to understand the paper in German, I'm hesitant to seek it out if it doesn't answer the question posed.

Last edited: Jun 22, 2012