Half-integral multiple

Question

To me, the description "half-integral" would only include the ratios 1/2, 3/2, 5/2... This corresponds to the usage in physics, where fermions are said to have "half-integral spin (projections)".

However, on http://en.wikipedia.org/wiki/Root system,

the usage in condition (4) seems to include 1/2, 1, 3/2, 2, ...

Please comment whether I am reading the wikipedia page correctly. 1/2, 1, 3/2, 2,... are indeed literally "half-integral", so maybe the physics usage should better be "sesqui-integral". That means "one and a half", but we can generalize it to "whole integer and a half".

If only this were the only problem in understanding root systems...

Usenet · Accepted Answer

[nq:1]To me, the description "half-integral" would only include the ratios 1/2, 3/2, 5/2... This corresponds to the usage in physics, ... can generalize it to "whole integer and a half". If only this were the only problem in understanding root systems...[/nq]
The terms "half-integral", "half-integer" are indeed commonly used in physics in the meaning of "half an odd integer". You must know: physicists are generally less formal at terminology than mathematicians.
Wikipedia - Spin in physics: http://en.wikipedia.org/wiki/Spin %28physics%29 This article uses the term "half-integer" three times, the first time in the strict sense and the second and third times in the usual physical meaning of "integer plus 1/2".

See also Google >"half-integer" (about 142,000 hits) and http://en.wikipedia.org/wiki/Half-integer
If I were to write your Wikipedia article I would make... (1) a remark (perhaps edited as a footnote) stating the common meaning of "half-integer" among physicists;
(2) a mention of the representation theory of the 3D rotation group SO(3, R), where AFAIK a similar informal term "half-integral representation" is sometimes used for the 2:1 representations by complex matrices of SO(3, R).
The 2:1 representations of SO(3, R) are even-dimensional complex representations. The simplest of them is SU(2), the standard tool in physics to describe spin properties of particles.
The 1:1 representations of SO(3, R) are odd-dimensional real representations. They are easily realizable by means of spherical functions. The simplest of these is SO(3, R) itself.

Good luck: Johan E. Mebius

Usenet · Answer

[nq:1]To me, the description "half-integral" would only include the ratios 1/2, 3/2, 5/2... This corresponds to the usage in physics, ... "half-integral spin (projections)". However, on system, the usage in condition (4) seems to include 1/2, 1, 3/2, 2, ...[/nq]
There's no way to tell what the author means by the term in whatever you're trying to read. But yes, if you asked me what the term "should" mean, I'd also include the integers. Any integer is half of an integer.
[nq:1]Please comment whether I am reading the wikipedia page correctly. 1/2, 1, 3/2, 2,... are indeed literally "half-integral", so maybe ... can generalize it to "whole integer and a half". If only this were the only problem in understanding root systems...[/nq]
David C. Ullrich
"Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)

Half-integral multiple

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