Explaining Electroweak Theory Decomposition to a Beginner

[noahcharris]

I have come across physicists representing electroweak theory as some kind of decomposition (i.e. U(1)xSU(2)). I was wondering if someone could explain this 'crossing' to me a little further. Fair warning, my understanding of group/gauge theory is v rudimentary at this point.

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[Stephen Tashi]

I don't know about the physics, but the operation "X" probably denotes the direct product of groups. Can you understand the explanation at http://en.wikipedia.org/wiki/Direct_product_of_groups ?

 

[homeomorphic]

The physics idea can be thought of as a generalization of electromagnetism. You can describe how electrons interact with electromagnetic fields by re-interpreting the vector potential in terms of the theory of connections on principal bundles, in this case U(1)-bundles, and that can be generalized to groups besides U(1), which leads to non-Abelian gauge theory. It turns out that the electromagnetic tensor can be thought of as a kind of curvature. If you study differential geometry, you learn that you can push around vectors on a surface by parallel transporting them, and that curvature measures the path-dependency of where the vectors end up if you push them along from point a to point b. With principal bundles, this is generalized to things other than vectors, like group elements. The Yang-Mills Lagrangian of the standard model is built out of this sort of generalized curvature. This is all fine for mathematicians, like myself, but the actual physics version generally makes our heads explode because you have to quantize the theory and it gets kind of ridiculous. In a lot of the math side of gauge theory, you end up just using the classical field theory and study the critical points of the action to come up with really weird mathematical facts.

I have no idea why U(1) x SU(2) turns out to be appropriate for electroweak forces, but I can give the somewhat trite answer that it's justified by experiment (and I think that may even turn out to be the "official explanation" by physicists to some degree--I'm not sure how deep of a justification there is beyond that it works).

The cross product is a fairly simple construction where you just take one group and put it in the first slot and the second group in the second slot and just think of it as a group where each slot acts the same way it normally does. It's interesting that such a simple construction, involving two very basic groups, like U(1) and SU(2) (aka a circle and the unit quaternions, respectively) would turn out to be the key to describing two of the fundamental forces of nature. Another interesting point from a physics point of view is that, contrary to what you might expect, the U(1) in the product turns out not to correspond to the U(1) of the electomagnetic force--it's actually a different copy of U(1) that lives inside the product. Or at least that's what physicists tell me.

 

[blue_raver22]

Electroweak theory is a fundamental theory in physics that explains the unification of two of the four fundamental forces in nature: the electromagnetic force and the weak nuclear force. This theory is represented mathematically using a decomposition called U(1)xSU(2). To understand this, let's break it down step by step.

First, U(1) stands for the unitary group of dimension 1, which is a mathematical concept used to describe symmetries in nature. In this case, U(1) represents the symmetry of the electromagnetic force.

Next, SU(2) stands for the special unitary group of dimension 2, which also describes symmetries in nature. In this case, SU(2) represents the symmetry of the weak nuclear force.

So, when we combine U(1) and SU(2) using the "x" notation, it represents the unification of these two forces into one theory, the electroweak theory.

To understand this "crossing" or combination of the two forces, it is important to understand the concept of gauge symmetry. Gauge symmetry is a mathematical concept that describes how particles interact with each other. In the electroweak theory, the U(1) and SU(2) symmetries are combined into a single gauge symmetry, known as the electroweak gauge symmetry.

This gauge symmetry is crucial in understanding the behavior of particles and their interactions with each other. The electroweak theory explains how the electromagnetic and weak nuclear forces are different manifestations of a single unified force at high energies.

In summary, the decomposition U(1)xSU(2) represents the unification of the electromagnetic and weak nuclear forces into a single theory, the electroweak theory. This is achieved by combining the symmetries of these two forces into a single gauge symmetry, which is crucial in understanding the interactions of particles at high energies. Hopefully, this explanation has helped you understand the concept of electroweak theory decomposition a little better.