I propose that spinor fields exist as vectors in a complex dimensional space that has it’s own metric tensor is conformally invariant. I then propose one method to project from this complex space to the real space-time and translate equations as such.
To begin, we write the standard equation for the invariant length of a vector, making note that conjugate indices are represented with a bar.
While this is invariant under a coordinate change, we must also require it be invariant under a conjugation. This is only true if our metric is Hermitian. To obtain the covariant derivative of our metric, we look at the differential of this length.
We must make note that while our vector is either a function of regular indices or their conjugates, the metric is a function of both.
W is the complex dimensional affine connection. Plugging these in gives
Now we make a Weyl assumption about the differential of the length
This then allows us to finally write an equation for the covariant derivative of our metric
This is quite different than what we see in space-time, with the difference coming primarily because of the Cauchy-Reimann equations for complex functions. If you add this equation to its own complex conjugate, you can write a version that is just in relation to the derivative of the real part of z, which is easy to see is the same form as we have in space-time. Because this equation is so simple, it has a very simple solution for the affine connection.
Let’s assume there exist functions that relate the complex coordinates z (and their conjugates) to the space-time coordinates x. For the time being, we won’t write what explicitly those functions are. Here is what we can say about them regardless
Let’s write some symbols we see here to help us
Using what we know, we can write relationships between these symbols
This then allows us to write the derivative of our spinor metric in space-time coordinates as it relates to complex coordinates
Lets create some definitions to simplify this further
Then we have
To complete our picture, we need a method to construct the space-time metric tensor from the complex one. In this regard we propose the following
Since the complex metric is Hermitian, the space-time metric will be real-valued. Let’s now take the covariant derivative of this in space-time
This last part is done so that space-time is not conformally invariant as proposed by Weyl (the phis cancel). This is important because we know space-time allows for vectors of fixed length, and that the length of a vector in space-time is related directly to it’s reference frame’s proper time.
Using this, along with the relationship between e and it’s inverse we can write an equation for the space-time affine connection
It’s plain to see that this is real-valued, but also that it contains torsion. Using this, however, we can derive the curvature tensor as it relates to our spinor elements
Using the definition
This hints at a Lagrangian that would involve the square as we can clearly see Electromagnetic elements.