Symmetries in particle physics

In summary, when discussing symmetries in particle physics, SU(N) and SO(N) are commonly used. SU(3) is used to describe the three colors of quarks because the orthogonal group, SO(3), would not accurately represent nature. Similarly, SU(2) is used to describe quark doublets because the kinetic terms in the Lagrangian are invariant under global SU(N) transformations. To introduce a local gauge symmetry, minimal coupling to the corresponding gauge field is necessary.
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We often use SO(N) and SU(N) to describe symmetries in particle physics. I am not clear which one to choose when I try to discuss a symmetry. For example, why do we use SU(3) but not SO(3) to describe the symmetry of the three colors of quarks? Similarly, why do we use SU(2) but not SO(2) to describe a quark doublet, for instance, the (u d) doublet?
 
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The obvious answer would be ”because the orthogonal groups would not describe nature”.

On a more technical note, the quark fields allow complex rephasing and the kinetic terms in the Lagrangian are invariant under the global SU(N) transformations. Promoting the global symmetry to a local gauge symmetry forces you to introduce the minimal coupling to the corresponding gauge field.
 
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Related to Symmetries in particle physics

1. What are symmetries in particle physics?

Symmetries in particle physics refer to the inherent properties of particles and their interactions that remain unchanged under certain transformations. These transformations can be rotations, translations, or other mathematical operations. Symmetries play a crucial role in understanding the fundamental forces and building theories in particle physics.

2. Why are symmetries important in particle physics?

Symmetries are important in particle physics because they provide a framework for understanding the underlying laws of nature. By identifying and studying symmetries, scientists can make predictions about the behavior of particles and their interactions. Additionally, symmetries allow for the development of elegant and concise theories that can accurately describe complex phenomena.

3. What is the relationship between symmetries and conservation laws?

Symmetries and conservation laws are closely related in particle physics. Conservation laws state that certain quantities, such as energy or momentum, remain constant in a closed system. Symmetries dictate that the laws of physics are the same in all directions or under certain transformations. These symmetries often lead to conservation laws, as they dictate that certain quantities cannot change in a physical system.

4. How do scientists use symmetries to search for new particles?

Scientists use symmetries to guide their search for new particles. The Standard Model of particle physics is based on symmetries, and any deviations from these symmetries can indicate the presence of new particles or interactions. By analyzing the behavior of known particles and their symmetries, scientists can make predictions about the properties of undiscovered particles, which can then be searched for in experiments.

5. Can symmetries be broken in particle physics?

Yes, symmetries can be broken in particle physics. This occurs when the laws of physics do not exhibit the same symmetries as the mathematical equations that describe them. For example, the weak nuclear force does not exhibit the same symmetries as the electromagnetic force, indicating that there is a breaking of symmetry. The study of broken symmetries can provide insight into the fundamental laws of nature and the origins of mass in particles.

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