What Is the Difference Between Albenian and Nonalbenian in Quantum Field Theory?

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Discussion Overview

The discussion centers on the distinction between abelian and nonabelian groups in the context of quantum field theory (QFT), exploring their definitions, applications, and relevance to gauge transformations in physical theories.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants explain that an abelian group is defined by the property that the group operation is commutative (ab = ba), while nonabelian groups do not satisfy this property.
  • It is noted that groups are important in field theory because they reflect symmetries in nature, particularly through gauge transformations.
  • Examples are provided, such as the U(1) gauge group of electromagnetism being abelian and SU(3) in quantum chromodynamics (QCD) being nonabelian.
  • A question is raised about whether operators can be considered elements of a group, with a response indicating that elements of a set can form a group if they meet the necessary axioms.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of abelian and nonabelian groups and their significance in QFT, but the discussion includes questions and clarifications that suggest some uncertainty regarding the role of operators in group theory.

Contextual Notes

The discussion does not resolve the implications of operators being elements of a group, nor does it clarify the full extent of how these concepts apply in various QFT scenarios.

captain
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i am have trouble understanding the difference between these two things and why is nonalbenian used in qft.
 
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When we talk about abelian groups, it means :

let a, b be elements of a group G. A group is abelian <--> ab = ba.

Groups are constructed and utilized in field theory primarily b/c we notice the laws of nature satisfy certain symmetries called gauge transformations. For instance the U(1) gauge group of electromagnetism is an abelian group and its the simplest example of relevance in the standard model.

Harder to deal with are nonabelian groups where the above property is not satisfied, and there are several examples (like QCD, where the gauge group is SU(3) -color).
 
Haelfix said:
When we talk about abelian groups, it means :

let a, b be elements of a group G. A group is abelian <--> ab = ba.

Groups are constructed and utilized in field theory primarily b/c we notice the laws of nature satisfy certain symmetries called gauge transformations. For instance the U(1) gauge group of electromagnetism is an abelian group and its the simplest example of relevance in the standard model.

Harder to deal with are nonabelian groups where the above property is not satisfied, and there are several examples (like QCD, where the gauge group is SU(3) -color).

can operators be elements of a group?
 
captain said:
can operators be elements of a group?

The elements of a set with some associated binary operation can be elements of a group if they satisfy the axioms of a group.
 

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