Spinor Metric

Suppose spinors have their own metric and affine connection that differs from tensors. Let us explore the possibilities. For spinors, we are going to use matrices instead of indices to stay in line with the typical semantics.

(1)

We will also make the following assumption about the metric matrix s in order for our lengths to be real numbers (and the metric will later become an operator):

(2)

We will also create a covariant derivative:

(3)

Using the assumption that if our spinor is “constant”, then its length will also be constant everywhere. We then can arrive at a relationship between the affine connection W and the metric matrix (metricity condition).

(4)

This has a general solution:

(5)

Phi is an unknown of the theory. Using 3, we can also construct a curvature matrix in the usual way:

This has a simple solution for the affine connection W.

(6)

Plugging in (5) and with some minor arithmetic, we can show the solution for the curvature matrix in terms of the metric s:

(7)

Knowing the Lagrangian for the electromagnetic field, (7) gives us a direct clue to a Lagrangian for our theory:

(8)

(9)

Setting (9) to zero, and using the relationship below, we get:

Now, we can break up C into it’s two parts, recognizing that the metric tensor g is a function of the spinor metric s. This allows us to separate out the part of our equation that deals with F on one side, and the rest on the other:

The left side becomes:

(10)

We can immediately recognize the very first part. It is the stress-energy tensor of the electromagnetic field:

(11)

Notice that the left hand side has a real AND imaginary component. This means we will be left with two equations. Let’s right out the right hand side now:

We can now equate the real and imaginary parts to yield our two equations:

(12)

Let’s define a new tensor:

(13)

Putting this into 12 gives:

(14)

The last term is just a value “at the boundary”. You can see that from the derivation found here:

Einstein-Hilbert action

This last value is zero IF we assume a vanishing boundary term. If the term does NOT vanish, then we would have a candidate for the cosmological constant (in principle). We can thus reproduce Einstein’s equation exactly by the proper introduction of constants:

(15)

Using (7) we can write this as a relationship between the spinor metric matrix s and the Ricci tensor:

(16)

The solution presented in (15) along with the Lagrangian in (8) is quite a miracle. We have reproduced Einstein’s field equations by assuming spinors have their own metric and affine connection, in effect their own geometry, and have demonstrated that doing so leads to not only curved space-time, but electromagnetism as a geometric effect in the spinor space.

It is worth noting that our choice of associating the unknown field Phi in (7) with the electromagnetic field is no accident, and could be extended to a more generic case. Doing so yields a much more complicated version of the stress energy tensor but the relationship of (14) is maintained.

Coordinate Changes

Sticking to our matrix formalism, we can explore coordinate changes in the spinor space. A generic coordinate change looks as follows:

(17)

This means the spinor metric transforms as:

(18)

Let us now introduce a series of matrices that allow us to project a vector from a spinor:

(19)

In order for the vector to transform as expected under a shift in our spinor, we must have the following relationship:

(20)

Now comes the fun part. We will also require that covariant derivatives match. Namely:

Which gives us:

(21)