Quantum Gravity

10 min readOct 29, 2019

By: Jason Blood

Introduction

In this paper, I propose a new method to quantize gravitation. This is accomplished through a generalization of the Pauli matrices and the connection between spinors and vectors. It follows some of the same systems proposed by spin connections, making use of tetrads, but adds components that take complex conjugation into account. This latter part leads to some interesting results. We also take the stance that the metric is “derived” from the spinor elements and not the other way around.

Pauli Matrices and the Metric

The Pauli matrices form a vector basis. Any 2x2 matrix can be written as a linear combination of them, and if the combination elements are all real valued, the matrix will be Hermitian. Because of the Pauli matrices great utility, we will opt to use the Weyl convention for spinors. This means that instead of a single 4-component spinor, we will use two 2-component spinors, marking them as left-handed and right-handed. If we use a carat to indicate the adjoint of a matrix, the Dirac matrices in this Weyl convention become:

This then allows us to write the relationship for the metric tensor as

We propose that this equation is in general true, and so as a general case the Pauli Matrices vary over space-time. Because the base set form a vector basis, the general versions can be written as linear combinations:

We want to make note that while the capital latin indices run from 0 to 3, they are NOT tensor elements. “Eta” above is a static matrix with the same values found in the Minkowski metric, but it does NOT change when we change coordinate systems. All static elements will be indexed using capital letters.

Affine Connections

As with vectors, we propose that spinors also have their own affine connection (covariant derivative).

It should be noted that the spinor affine connection W is a set of four 2x2 matrices. The symbols L and R indicate left and right handed 2-spinors accordingly.

The Pauli Matrices not only give us the metric when we “square” them, but they also allow us to project a vector from any spinor.

Now let’s work a little magic. Let’s assume that the covariant derivatives match.

Which, after some simple math, gives

(1)

This is really one equation as the second can be derived from the first. To reduce this further, we can break out the tetrads as they relate to the Pauli spin matrices. First let’s define:

Then (1) reduces to:

(2)

Where we define the lambdas as follows:

The values can be found in Appendix A. We can simplify this further by creating condensed notation:

Metricity

We now propose a metricity condition on (4):

From this it is easy to see that we get the usual metricity condition for the metric tensor. If we assume that the vector affine connection is symmetrical in its lower indices (torsion free), then it can be determined entirely by the metric, which itself is determined from the tetrads. This means that (5) allows us to determine the values of the spinor affine connection entirely from the tetrads as well, with one very important exception.

If you review the values of lambda- in the appendix, you will see that equation (5) does not contain any reference to the zero element of the imaginary component b. This means that b0 is an unknown of our theory.

We will not solve for the other elements of a and b here, those values can be found in Appendix B.

Curvature

From (5) we can write an equation for the vector affine connection

This allows us to write the Ricci tensor in terms of omega

(6)

Using (6) we can also determine the scalar R

If we define our Lagrangian in the usual way, as it relates to R, we get exactly the Tetradic Palatini action. You can read about that further here: https://en.wikipedia.org/wiki/Tetradic_Palatini_action

Using the spinor affine connection W, we can also construct a spinor equivalent to R, which we will label C:

(7)

The values of C are 2x2 matrices. This means we can write them as a linear combination of the Pauli matrices:

Where

(8)

Equation (4) along with our usual definition of the electromagnetic field allows us to write the 0 elements of A and B precisely

(10)

We can also rewrite (7) using the a and b fields (though this is messy):

Using (9) we can simplify this further

(10)

We can contract further to get the scalar R

(11)

Lagrangian

Instead of relying on the vector curvature R to define our Lagrangian, we will instead propose something based on C.

We can expand this further to break out the A and B terms

(13)

Varying this relative to the tetrads and setting it to zero, we have:

The left hand side of this equation will give us the electro-magnetic stress-energy tensor. To give us the energy element from the Einstein equation, we need to multiply both sides by a constant:

The e inside bars above is the determinant of a matrix formed from the tetrads (equivalent to the square root of the determinant of the metric). The constant on the right can be rewritten as:

Where l is the Planck length and alpha is the fine structure constant. To solve (15) further we need to calculate the variation of the element M. M contains only tetrads terms and is quite complicated.

Spinor Metric

The spinor affine connection implies the existence of a geometry where the spinors act as vectors. If this is the case, the affine connection would have an associated metric.

If we allow the length of spinor to change under parallel transport (conformal), via Weyl, we can give an equation relating the affine connection W andthe spinor metric s:

We can break this up into components using:

The values for epsilon are found in Appendix A.

So our unknowns now change from tetrads and the electromagnetic field to the elements of the spinor metric (4 real values), the Weyl length vector (4 real unknowns), and the components of the electromagnetic field.

Let’s equate (19) to our version containing a’s and b’s. To do that, we first make use the Pauli matrices as a vector basis.