# 3D phase space of point particle and spinors.

1. Jun 18, 2009

### Spinnor

Can we make a connection?

Consider the phase space of a point particle in R^3. Six numbers are required, three for position and three for velocity.

Now consider an isotropic vector, X, in C^3 with X*X = 0.
X = (x1,x2,x3), X*X = (x1*x1 + x2*x2 + x3*x3),
x1 = c1 + i*c2, x1*x1 = (c1*c1 + c2*c2 +2*i*c1*c2)

From:

http://www.sjsu.edu/faculty/watkins/spinor.htm

"It can be shown that the set of isotropic vectors in C^3 form a two dimensional surface. This two dimensional surface can be parametrized by two coordinates, z0 and z1 where

z0 = [(x1-ix2)/2]1/2
z1 = i[(x1+ix2)/2]1/2.

The complex two dimensional vector Z=(z0, z1) Cartan calls a spinor. But a spinor is not just a two dimensional complex vector; it is a representation of an isotropic three dimensional complex vector."

Let the real part of X represent the position of a point particle and let the imaginary part of X represent the velocity of the same particle. If we require X*X = 0 how does that restrict the phase space path of a particle?

Thank you for any help.