Talk:Clifford module

Initial comments

This has become a useful page for explicit information. It would be good also to have a statement of the basic Atiyah-Bott-Shapiro result on Clifford modules.

Charles Matthews 19:30, 16 Jul 2004 (UTC)

See also

"A generalized FFT for Clifford algebras" ([1] 2005, preprint [2] 2004) and references. The paper gives detailed constructions for both naive and fast versions of both the forward and inverse real matrix representations for all signatures of real universal Clifford algebras.

Also see my discussion item Talk:Spinor#Mathematics_vs_Physics. Leopardi 07:16, 14 February 2006 (UTC)

Where are the Spinors?

Elements of Clifford modules are usually called spinors, or more precisely, pinors. This page needs a link to something about spinors. Maybe it should even be merged in. John Baez (talk) 03:12, 7 October 2009 (UTC)

Real Clifford algebra R(3,1)

The section on the real Clifford algebra R_3,1 says: "The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix. The signature is (+++−). For the signatures (+−−−) and (−−−+) often used in physics, 4×4 complex matrices or 8×8 real matrices are needed." This does not seem to make sense - 4x4 complex matrix = 32 real coefficients, 8x8 real matrix = 64 real coefficients. Also the article "Classification of Clifford algebras" lists (+---) as a 2x2 quaternion matrix, and (-+++) as a real 4x4 matrix, though perhaps the context is different. This section is very sketchy, it seems to convey no substantive information. Someone who knows more about this should revise and expand this section.Enon (talk) 21:26, 1 July 2012 (UTC)

What is written is correct. While a general 8x8 matrix has 64 independent components, not all of this freedom is used when representing an arbitrary member of R(3,1) -- instead a member of R(3,1) is represented by a linear combination of sixteen particular 8x8 matrices which represent the full linear basis of the algebra. It is found to be necessary to represent each of these basis elements as an 8x8 matrix, in order for their matrix representations to have the correct products to match the multiplication group of the basis elements, and the correct signatures. Jheald (talk) 09:14, 2 July 2012 (UTC)

Thanks, Jheald. That would be good information to add, though I think a bit more context and working through the reasoning and implications is needed for this section to convey much to anyone who isn't already familiar with the topic. This way of using the Pauli matrices seems quite a different approach than that used in Geometric Algebra. (For other readers, Jheald is a frequent contributor to the geometric algebra article. GA has methods for avoiding using the Pauli matrices directly, although it is also a style of using Clifford algebras.) I also hope somebody will address John Baez's long-standing request for more in the article on the connections to (IIRC) spinors, spin and pin groups.Enon (talk) 17:25, 2 July 2012 (UTC)