How Are SU(2) and U(1) Representations Combined in Particle Physics?

In summary, the conversation discusses the representation of SU(2)_L\otimes U(1)_Y and the different possibilities for left and right lepton doublets and singlets. It also mentions the math operation of tensor product and suggests looking into Young tableux and group theory for further understanding.
  • #1
lalo_u
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Well, i´m trying to understand this:

I´ve got a representation of [itex]SU(2)_L\otimes U(1)_Y[/itex] such that the left lepton doublets can be represented as (2, -1) and lepton singlets rights as (1, -2).

Then I can be left antiparticles bilinear representations as [itex](2,1)\times(2,1)[/itex] or [itex](1,-2)\times(1,-2)[/itex].

I wonder why, in the first case the possibilities are [itex](1,2)+(3,2)[/itex], and in the second case is [itex](1,-4)[/itex].

What kind of math operation has been done here?
 
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  • #2
Look up "Young tableux". You can also look up chapter 70 in the Quantum Field theory book by Srednikci , free draft here: http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
Is also covered in any "group theory for physics" book.

In the case of ##(2,1) \otimes (2,1)## we have that the U(1) representations just add like normal numbers so 1+1 = 2. The tensor product of two SU(2) doublets becomes ##2 \otimes 2 = 3 \oplus 1## in other words one SU(2) triplet and one SU(2) singlet representation
 
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Related to How Are SU(2) and U(1) Representations Combined in Particle Physics?

1. What is the representation of SU(2)xU(1)?

The representation of SU(2)xU(1) is a mathematical framework used to describe the fundamental forces and particles in the Standard Model of particle physics. It combines the SU(2) gauge group, which describes the weak nuclear force, with the U(1) gauge group, which describes the electromagnetic force.

2. How is SU(2)xU(1) related to the Standard Model?

SU(2)xU(1) is a crucial component of the Standard Model, which is the most widely accepted theory for describing the fundamental particles and forces in the universe. The SU(2)xU(1) representation helps to explain how the weak nuclear force and the electromagnetic force are unified at high energies.

3. What are the different types of representations in SU(2)xU(1)?

There are several types of representations in SU(2)xU(1), including the fundamental representation, the adjoint representation, and the singlet representation. These different representations correspond to different types of particles and their interactions with the fundamental forces.

4. How does SU(2)xU(1) affect particle interactions?

The SU(2)xU(1) representation describes the fundamental forces and particles in the universe, so it affects all particle interactions. The SU(2) gauge group, for example, is responsible for the weak nuclear force and its interactions with particles, while the U(1) gauge group is responsible for the electromagnetic force.

5. What is the significance of SU(2)xU(1) in modern physics?

SU(2)xU(1) is a fundamental part of the Standard Model, which is the most successful and widely accepted theory for describing the fundamental particles and forces in the universe. Its representation allows us to understand and predict the behavior of particles and their interactions at the smallest scales, helping to advance our understanding of the universe and its origins.

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