Friday, 20 June 2014

Where did the isospin sign convention come from?

Perhaps a reader may be able to help with this conundrum, which came to my attention on Tuesday following a seminar at Surrey from Mike Bentley, from York, which was all about isospin.

In 1932, Werner Heisenberg introduced the concept of isospin [1].  At least, that's what we call it these days, though it was Wigner, in a 1937 paper [2], who first referred to Heisenberg's idea as isotopic spin, which we've since shortened to isospin.

Heisenberg's idea was that protons and neutrons are really very similar objects - both about the same mass, and having a close link via beta decay in which a neutron can turn into a proton and an electron.  Together neutrons and protons constitute atomic nuclei, and can be termed nucleons.  Heisenberg wondered if it would be possible to conceive a theory where one dealt with just nucleons, but had some way of distinguishing them as either protons or neutrons.  He said (excuse my translation)

Each particle in the nucleus would be characterised in five dimensions:  The three spatial coordinates (x,y,z), the spin in the z-direction, and through a fifth number, ρξ, for which the values of +1 and -1 are possible.  ρξ = +1 would mean that the particle were a neutron, ρξ = -1 that it were a proton.

The whole concept can just be considered a mathematical convenience;  now one can write equations in a higher-dimensional space, but without having to have a notation with 'p' and 'n' subscripts everywhere for proton and neutron states.  However, it also helps notate an apparent underlying symmetry;  that protons and neutrons nearly behave as mirror particles.  The purpose of my post is not about anything as deep as that, but rather about the choice of +1 for neutrons and -1 for protons.   It is just an arbitrary choice, but it's the one originally made by Heisenberg, and repeated by Wigner shortly after.

If I look more or less in any modern textbook, or the Wikipedia article on isospin, one finds the opposite sign definition.  Here I quote from the textbook Nuclear and Particle Physics, by Burcham and Jobes, which I bought while an Undergradute (so you may argue it is not "modern"):

Heisenberg introduced an internal degree of freedom, the isospin I, in complete analogy with the ordinary intrinsic spin s.  The two orientations of the isospin I (I=½) in a notional isospin space, namely I= +½ and I= -½, would correspond with the proton and neutron respectively

The factor of ½ difference I understand, but where did the sign flip come from?  Anyone know?  In Mike's talk on Tuesday, he used the Heisenberg convention, and this is the norm for nuclear physicists, but particle physicists use the opposite sign, as the textbook does.

[1] Über den Bau der Atomkerne. I., W. Heisenberg, Zeitschrift für Physik 77, 1 (1932)

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