Penrose and Weinstein on Spinors

A brief exposition on the central role of Spinors in modern Cosmology and modern physics, or proof of the importance of mathematical physics.

What follows is a summary of a constructive discussion between Penrose and Weinstein on the role of Spinors in theoretical physics.

1. Dirac recognised the importance of Spinors in theoretical physics.
2. Spinors were previously developed by mathematicians such as Élie Cartan.
3. Weinstein thinks of Spinors as mathematical entities that remain very mysterious like the Monolith in 2001.
4. Penrose has a geometric understanding of Spinors through the Dirac Belt trick which show that Spinors are the square root of geometry.
5. Penrose points out that the number of Spinors grows exponentially at a rate of \(\propto 2^N\) where \(N\) is the dimension of the vector space, which allows us to identify them with even-graded oriented surfaces or what most physicists call ‘multivectors’. Three-dimensional geometric algebra generates \(2^3 = 8\) ‘linearly independent’ oriented surfaces, of which \(3 \choose 2\) are Spinors.

Finally, Weinstein points out that without the validity of the Dirac Belt trick in the Universe the Pauli Exclusion Principle wouldn’t function and therefore we wouldn’t have a periodic table.