Introduction

The term “Symmetric Gauge theory” is suppose to be an oxymoron, and not to mention, a technical impossibility!

First Lesson: Gravity and Operator Calculus (the stuff quantum theories are built from) hate each other!

Actually as the story goes, Gauge theory is as “antisymmetric” as you can get, as it is based on an antisymmetric field strength tensor, known as the Yang-Mills tensor. Gravity on the other hand is expressed with the full symmetric Riemann tensor.

First, let’s explain what is meant by (anti)symmetric here! Well use simple example, say I wish to multily two numbers, x and y. I can say that the product of x and y is commutative,

xy = yx

this is a symmetric operation. But if we have,

xy = -yx

this an antisymmetic operation. These rules generalize to tensors.

All this leads to the fact that Gravity is very different from the other forces. One key aspect that creates this difference lies in the subject known as geometrodynamics, which in my opinion sounds like the description of the motion of a GMC economy car! But in reality, this literally means that space-time geometry is dynamical, so this thing we like to call Gravity is not a simple force as it were with the case of the the other three, but the actual change in the curvature of space-time itself!

Second Lesson: The first three (gauge) forces of nature use space-time as a “static” background, while gravity represents changes in the background at point to point.

Field Strength Tensors and Contractions

Let’s look at the field strength tensors mentioned above in more detail,

The Yang-Mills tensor is,



First note that with the formula for the hermiticity condition gives the ‘allowed’ rank of the tensors, (for the ambitious reader, this relates to the constrain coming from unitarity, i.e., the fact that the Yang-Mills tensor is based on the symmetry of the SU(n) group)



Of course excluding the special case of , for which we get the abelian group U(1) for Electromagnetism, Also this field is invariant under Lorentz transformation and possess space-time symmetry,



From purely mathematical considerations, the U(1) group is the 1-dimensional group over the complex numbers and forms a 2-d normed division algebra over the reals. Taking the appropriate numeric assignement and going from Greek to Latin indices gives the electric and magnetic fields,



At this point I should interject that Electrodynamics is a beautiful and elegant theory in the history of physics (not to mention successful as all hell in everyday life).

Considering the more generalized cases when    , gives the massive vector bosons and gluons, given by the gauge groups SU(2) and SU(3), respectively. These forces, unlike electromagnetism are strictly internal (quantum) symmetries.

All of the gauge theory assumes a metric of unity, .

That’s all good if you live in flatland. If you introduce space-time curvature, the metric is no long a constant, but becomes dynamical (i.e. it can deviate below unity due to curvature). The metric becomes a solution to a differential equation for 2nd order curvature. This is given by the two contractions of the Riemann tensor,



And forms the Einstein-Hilbert field equation,



Note that the general structure of the Riemann tensor is very much like the Yang-mills. But if we express the Christoffel symbol (connection coefficient) in terms of the metric,



It is obvious that the apparent similarity in the structure of the equations is a superficial one at most.

Everything in GR is about the metric! I can here Jan Brady now, “Metric, metric, metric!”

Enter Symmetric Gauge Theory – Go with the best, use Maxwell.

Maxwell’s theory along with the weak and strong forces are antisymmetic in nature, as mentioned above. This has a direct correspondence with a simple but profound experimental fact,

For an antisymmetric field theory (tensor), like charges always repel. This is a first blow to the use of a  rank-1 field theory for gravity, as with gravity, like charges always “universally” attract.

Turns out that at the classical level (and perhaps the quatnum) this difficulty can be overcome by addressing a simple accounting issue with the way Nature may choose to multiply quantities (to quote The Stand-Up Physicist). But to use this practice, we are forced to abandon the use of tensors (like the ones above), as tensors can’t perform this feat.

Question is, “how does one do field theory without tensors?” the short answer lies in a mathematical field know as a quaternion. The quaternions are a 4-dimensional division algebra over the reals with the “built-in” definition of a norm, this comes in handy when desribing an inertial frame and a Lorentz invariant scalar (i.e. normally taken as the product to tw0 4-vectors, but our case the product to two quaternions). In 4-dimensions, quaternions share all the important properties of tensors.

The re-write is actually much easier than one might think, we start by defining the U(1) group as a quaternion over it’s own absolute value,



The group SU(2) is called the group of Unit Quaternions, it is obtained by taking the exponential of the same quaternion minus its conjugate,



We arrive at the fully quaternion by simply taking the product of 1st and 2nd result,



The Lie algebra of the SU(3) group is different from that of SU(2), we get the correct multiplication table by using a second exponential of a second quaterion minus it’s conjugate, but this time we “double conjugate” the first quaternion minus it’s conjugate to give SU(3),



One important comment is in order about quaternions vs. tensors. Tensors can be generalized to higher dimensions, where as quaternions (via their unique structure) makes them restricted to 4-dimensions. In a certain sense, we are ‘guided’ via accident or destiny to use this formalism to describe how fields in nature behave together.

So at this point we’ve got Special Relativity and Gauge theory nailed down pretty good. But as mentioned above, a symmetric theory is need to handle gravity.

Hypercomplex Numbers and the “Symmetrization of Maxwell”

Quaternion multiplication is as follows,

ijk = -1

ij = k, ji = -k …

On the other hand, multiplication with hypercomplex numbers is,

ij = k, but ji = k also!

jk = kj = -i

ki = ik = -j, end result

i^2 = j^2 = -k^2 = -1

Then choose the so-called “California representation,” so everyone multiplies out to just positive 1.

So is this a potential route to the quantization of gravity (exclusively in 4-D, BTW, the stuff that can be tested!)? I say, hells yea.

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