Analytic Conformal Geometry in Complex Spaces

In a previous article, I began exploring concepts for a conformal geometry in complex dimensional spaces. One assumption that was not considered was making the space analytic.

Analytic spaces are those for which the complex dimensional derivatives are independent of internal angle of the complex dimension. Because complex numbers act as planes (2-d spaces), we want our derivatives to be the same regardless of the angle in which we approach with our limits. The end result of this restriction are the Cauchy-Reimann equations.

There are several ways to think about the result of these equations, but the way we will take for our analysis is to treat complex dimensions and their complex conjugates as separate and distinct variables. In the syntax of tensors, this means marking some indices with a bar to indicate that they vary in the conjugate coordinates instead of the regular ones. We will also use a bar to indicate conjugation of scalars. This then allows us to write the Cauchy-Reimann conditions:

(1)

For the remainder of this article we will use pipe notation for partial derivatives. It is possible to combine regular and conjugate indices into a mixed tensor. One just needs to remember that derivatives will only effect the portion of the tensor that matches.

Metric Tensor

The metric for complex spaces can take multiple forms. For our consideration, let us assume the metric to be a mixed tensor that is Hermitian (Kahler).

(2)
(3)

This type of metric has some interesting properties, especially as it pertains to raising and lowering indices. If you apply the metric to a vector, you get the conjugate of its contra-variant form. So, in order to completely raise an index you must use both the metric and complex conjugation. This is an exact parallel to the properties of left and right handed spinors and charge conjugation. We can thusly think of covariant vs contra-variant complex vectors as analogous to matter and anti-matter.

(4)

As with real spaces, we have an affine connection that allows us to define covariant and contra-variant derivatives. Again we must take note that the contra-variant derivative of a covariant tensor is zero per the Cauchy-Reimann conditions and the metric as defined:

(5)

Conformal

As was proposed by Weyl for general relativity, we will allow the length of a vector to change under transplant. This was non-physical in space-time but in our space there is no such restriction. With that in mind, we can write out the differential of equation 3:

Which gives:

(6)

Unlike in real spaces, the affine connection appears here only once. This comes as a result of the analytic nature of our space. Equation 6 is easily solved for the affine connection:

(7)

With this affine connection, we can calculate the curvature tensor for our space:

(8)

What is amazing here is that the metric does not contribute. High level / broad strokes, this is due to our space being analytic which forces angles to remain unchanged as we move through the space. Curvature then is only due to length variations of vectors. We can contract the curvature tensor to produce it’s Ricci version:

(9)

Gauge Invariance

Since our theory is conformal and tensorial, its invariant under coordinate changes as well as a scale imposed on our metric. The vector phi reacts in such as way as to make our affine connection invariant under gauge transformations. This also means the curvature tensor is gauge invariant.

Lagrangian

The Lagrangian of our theory must meet the following criteria to be physical:

  1. Real-valued (invariant under conjugation)
  2. Scalar (invariant under coordinate change)
  3. Conformal (invariant under gauge transformation)

For this we have a simple candidate:

(10)

Where k is an unknown real constant, and g is the determinant of the metric. This assumes 4 dimensions in complex space. If we were dealing with only 2 dimensions we would not need the square root.

Space-time

Before we use our Lagrangian, let’s have a look at how our ideas translate to space-time. To do this, we assume a new set of coordinates (real-valued) and look how how we might right L in terms of those new coordinates. For now, we will not assume any specific mapping between the coordinate system, only that such a mapping is differentiable.

Here we will introduce the following:

(11)

Using this basic definition we have the identities:

(12)

Using this, we can construct the derivative of our metric relative to the space-time coorindates:

(13)

We can simplify this by using some definitions:

(14)

Or:

(15)

It is simple to show that the affine connection term defined in 14 is gauge invariant. It also trivial to show that a phase shift in the complex coordinates results in a shift in the affine term above exactly mimicking a gauge shift in the field A. We of course have chosen the format of A to exactly mimic the electromagnetic field.

Going further we define:

(16)

Resulting in:

(17)

Which we can combine to produce:

(18)

The g here is the determinant of the complex metric tensor. If we assume:

(19)

Then we can write our Lagrangian entirely in terms of the space-time elements:

(20)

Where e is the determinant of the inverse e from above and g is now the determinant of the metric tensor. If we assume the value e to be unity in length, we can arrive at a value of k:

(21)

If we associate the H term with curvature of space-time, we would be forced into a relationship between the values in h with the metric tensor, which itself if related to the complex metric via 19. This would mean the following association:

(22)

Where lp is the Planck length, and alpha is the fine structure constant.

Quick Review

Armed only with a complex dimensional space that is analytic and conformal, we have shown that we can reproduce the Lagrangian of space-time and electromagnetism if we assume a connection between the two space’s metrics via equation 19, where the values of e are an as yet unknown set of relationships between the complex coordinates and the space-time coordinates.

When we translate our covariant derivative from complex space to space-time we get the covariant derivative of QED plus an additional term which we can associate with gravity if we assume the equation 22.

In most cases, we would have zero space-time curvature, which means H is zero.

Also interesting to note that if we assume our complex space metric to be flat (constant), and we derive our space-time metric from H using 22 (or our value of H from our metric), we can derive a set of values for e.

Theoretical Physicist, Entrepreneur