What is U(1) Lie Group? | Abelian Lie Group | Unit Circle

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In summary, U(1) is a 1 parameter Abelian Lie group that is isomorphic to rotations about a unit circle. It is the 1x1 matrices where x*=x^-1. A U(1) gauge theory is invariant to a local U(1) gauge transformation and is labeled as such because it allows for the establishment of theories that are invariant under SU(2) and SU(3) local gauge transformations, which correspond to the weak and strong force physics. U(1) is identical to SO(2).
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Matterwave
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I keep hearing statements like "XXX theory is based on U(1)" or some such, but I haven't heard what this group actually is. If U(N) are the NxN unitary matrices, then U(1) are the 1x1 matrices such that x*=x^-1. So, I just want to confirm then, that U(1) is simply the 1 parameter Abelian Lie group given by [itex]e^{i\theta}[/itex]? This is simply the unit circle right, and should be isomorphic to rotations about a unit circle?
 
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  • #2
Matterwave said:
U(1) … is simply the unit circle right, and should be isomorphic to rotations about a unit circle?

Yup! :smile:
 
  • #3
Matterwave said:
I keep hearing statements like "XXX theory is based on U(1)" or some such, but I haven't heard what this group actually is. If U(N) are the NxN unitary matrices, then U(1) are the 1x1 matrices such that x*=x^-1. So, I just want to confirm then, that U(1) is simply the 1 parameter Abelian Lie group given by [itex]e^{i\theta}[/itex]? This is simply the unit circle right, and should be isomorphic to rotations about a unit circle?

Yes, it's not very impressive. Though when we say a U(1) gauge theory we mean the physics is invariant to a LOCAL U(1) gauge transformation (i.e exp(i theta(x)), not just a global one (i.e. exp(i* constant)). However, the reason we label such a simple thing is because we also then establish theories which are invariant under SU(2) and SU(3) local gauge transformations which gives us the weak and strong force physics.
 
  • #4
Ok, last question. Is U(1) identical (homomorphic or diffeomorphic?) to SO(2) then?
 

What is U(1) Lie Group?

U(1) Lie Group, also known as the unitary group of degree 1, is a mathematical concept in group theory that represents the collection of all 1x1 complex matrices with determinant equal to 1. It is a subgroup of the general linear group GL(1, C) and is isomorphic to the unit circle in the complex plane.

What is an Abelian Lie Group?

An Abelian Lie Group is a type of Lie Group in which the group operation is commutative. This means that the order in which the group elements are multiplied does not affect the result. The U(1) Lie Group is an example of an Abelian Lie Group.

What does "Unit Circle" refer to in the context of U(1) Lie Group?

In the context of U(1) Lie Group, "Unit Circle" refers to the geometric representation of the group. The unit circle is a circle in the complex plane with radius 1, centered at the origin. Each point on the unit circle corresponds to a complex number with absolute value 1, which can be represented as a 1x1 complex matrix in the U(1) Lie Group.

What are the defining properties of U(1) Lie Group?

The defining properties of U(1) Lie Group include:

  • The group elements are 1x1 complex matrices with determinant equal to 1
  • The group operation is matrix multiplication
  • It is a subgroup of the general linear group GL(1, C)
  • It is isomorphic to the unit circle in the complex plane
  • It is an Abelian Lie Group

How is U(1) Lie Group used in science?

U(1) Lie Group has many applications in science, particularly in quantum mechanics and particle physics. In quantum mechanics, the U(1) symmetry is related to the conservation of electric charge. In particle physics, the U(1) gauge symmetry is used in the Standard Model to describe the electromagnetic interaction. U(1) Lie Group is also used in other areas of mathematics, such as differential geometry and representation theory.

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