Recently I’ve come across a few papers from China (e.g. Xiang-Yao Wu et al., arXiv:1212.4028v1 14 Dec 2012) that make the following statement:
...any quantity which transforms linearly under Lorentz transformations is a spinor.
It’s my understanding that e.g. a 4-momentum vector also transforms linearly under a Lorentz transformation.
Is the first statement simply false, or should one take it to be true in the sense that a 4-vector is capable of being written in spinor notation?
Perhaps the first statement might be a confusion between Lorentz transformations and spin matrices? In the chapter on spinors in Misner, Thorne and Wheeler’s Gravitation (p. 1148) they show that while a vector transforms under a spin matrix (aka rotation operator / quaternion / spinor transformation) as:
$$X \to X' = RXR^*,$$
a quantity that transforms as
$$ξ \to Rξ'$$
is known as a spinor.
After further web-searching, I’ve come across references making statements that seem to throw some light on the issue
Andrew Steane’s recent and very readable (to such tyros as myself) “An introduction to spinors” (http://arxiv.org/abs/1312.3824 13 Dec 2013), in which he writes (p.1, 2nd para): “… One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed.” (But see (3), here below).
I’ve also now traced back the original quotation, repeated word-for-word, through a number of earlier papers (Chinese and Russian) to: V. V. Varlamov arXiv:math-ph/0310051v1 (2003), in which he cites – as do all the later papers – one of the earliest writers on spinors, B. L. van der Waerden, Nachr. d. Ces. d. Wiss. Gottingen, 100 (1929). Varlamov also wrote a densely mathematical and well-referenced paper “Clifford Algebras and Lorentz Group” (math-ph/0108022, 2001), which inclines me to give more credence to the original statement, even though it was parroted by a number of later authors.
However, it appears that Dirac himself suggested an even more general entity than the spinor: “A new kind of quantity with components which transform linearly under Lorentz transformations must be introduced, and I call it an expansor. It is rather more general than a tensor or a spinor in that the number of its components is infinite, but enumerable.” P. 1205, section 1946:1 DEVELOPMENTS IN QUANTUM ELECTRODYNAMICS (p. 21 of the section). The Collected Works of P. A. M. Dirac: 1924-48: 1924-1948 By (author) P. A. M. Dirac, Volume editor Richard Henry Dalitz. Cambridge University Press, 26 Oct 1995 - Science - 1310 pages.
Further, on p. 1163 of the above, Dirac notes that “the present theory of expansors applies, of course, only to integral spins, but probably it will be possible to set up a corresponding theory of two-valued representations of the Lorentz group, which will apply to half odd integral spins.”
These two-valued entities were subsequently supplied by Harish-Chandra, who called them “expinors” (“Infinite Irreducible Representations of the Lorentz Group”, Published 1 May 1947 doi:10.1098/rspa.1947.0047Proc. R. Soc. Lond. A 1 May 1947 vol. 189 no. 1018 372-401).
I haven’t access to van der Waerden’s classic, so cannot check whether he made the claim referenced by Varlamov (but have little reason to doubt him).
So what can I conclude?
The subsequent work by Dirac and Harish-Chandra seems to invalidate the van der Waerden claim, as later cited by Varlamov, at least for infinite dimensional IRs of the Lorentz group.
So perhaps the queried definition does apply fairly generally, but not universally?
If so, it would be good to have an expert clarify the distinction.
PS Gratitude to Qmechanics for tidying up the original posting.