#spinors

[min-vs-physics]

[23/7/24] no update except physics hateposting.

[isomorphismes]

It's easy to write down explicit equations for many Calabi-Yau 3-folds (those with SU(3) holonomy), but very difficult to explicitly cast a manifold with holonomy metric G₂ or Spin(7). Robert Bryant wrote out an explicit Spin(7) and an explicit G₂ manifold in 1987: it took 49 simultaneous nonlinear PDE’s to exhibit the symmetry.

Jacob Gross

[eka-mark]

I’m currently reading David Hestenes’s book Space-Time Algebra, in which he argues that Clifford algebras are undervalued as a suitable language for mathematical physics. It’s coordinate-free, unlike the traditional physicist language, and gets more directly at the physically relevant math than the conventional mathematical approach via tensors.

Instead of working with the usual matrix representations of the Pauli and Dirac algebras, they are given in terms of generators and relations, and shown to be instances of (nested) real Clifford algebras. The nesting manifests as follows: taking even subalgebras at each step, we pass from the Dirac algebra to the Pauli algebra to the quaternions to the complex numbers to the real numbers.

As the Clifford algebras of Minkowski spacetime and Euclidean 3-space respectively, the Dirac and Pauli algebras are directly related to the geometry of spacetime and space, in a way that is prior to the concept of spin.

This treatment allows for a purely real formulation of the Dirac spinor calculus, where spinors appear as minimal left ideals in real Clifford algebras.

It’s pretty interesting so far, but I’m not yet convinced any of this is a very big deal. A friend recommended it to me, saying it helped him understand the Dirac theory and the Yang-Mills equation, so we’ll see if I have an “ah-hah” moment in the spinors chapter.

[isomorphismes]

José Figueroa–O'Farrill, Spin Geometry

[isomorphismes]

Consider now a system of n units ι₁,ι₂,...,ιₙ such that the multiplication of any two of them is polar; that is, ιᵣ ιₛ = −ιₛιᵣ.

William Kingdon Clifford, 1878

via José Figueroa O'Farrill

[isomorphismes]

Thus, the existence of a spinor structure appears, on physical grounds, to be a reasonable condition to impose on any cosmological model in general relativity.

Robert Geroch

Chicago, 1969

via José Figueroa O'Farrill, spin geometry

[isomorphismes]

Parallel transport around a closed loop can change a vector. This effect is used to quantify the curvature of space, via the Riemann curvature tensor.

Greg Egan

[isomorphismes]

Lochlainn O’Raifeartaigh, group structure of gauge theories