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…
So what has 24 years of business experience taught me? Well, that list would be rather long and maybe some day I will write a book or three about it all, but I can try to summarize some important points here and share them for anyone who may care to read.
The true purpose of any business is to service a market. You might hear a lot of advice about the importance of mission statements, finding your company “why”, etc. In the end, the “why”, the reason you exist, is to provide a service to a specified market.
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. …
Weyl introduced an extension of Einstein’s General Theory of Relativity in a paper he published in 1918. In this paper, Weyl proposed removing the restriction that vectors maintain their length while undergoing parallel transport. We can write this mathematically in the following way:
where g is the symmetric metric tensor and zeta is an arbitrary vector. If we take the total derivative of this expression we get:
I have been listening to a lot of self help books lately. I know what you’re thinking… oh boy another middle aged man in a mid-life crisis. Sure, perhaps that’s true, but it certainly isn’t uncommon. And lord knows the literary world is filled with gurus who claim to have it all figured out.
But for me, as I drove into this topic, I decided to listen to a wide variety of opinions so that I might perhaps find a path of my own. I wanted to pick what made sense, discard the rest, and help me put into words…
By: Jason Blood
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.
The Pauli matrices form a vector basis. Any 2x2 matrix can be written as a linear combination…
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.
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):
Misunderstanding the Market
It is possible to write an equation for 2-spinors that properly yields the Klein-Gordon equation when squared AND contains elements for a rest energy. Here is the solution:
The sigma are the Pauli matrices with the zero element just being the identity. We can use some basic math to derive the following from equation 1:
I will introduce some calculus for working with tensors in complex spaces. The syntax for such spaces is the same as standard tensor notation but with one important and major distinction. We will use the over-bar to indicate elements that are in the complex conjugate space.
To begin, we start with a metric as we do in real spaces:
Since we are working in a complex space, we will also make Cauchy-Riemann requirements of any tensor in the space. Namely, that there are no mixing between regular complex indices and their complex conjugates. Explicitly: