# Group Theory and Physics .

• eljose79
In summary: I was only speaking of symmetry and how group theory is not enough to understand it.In summary, the conversation discusses the use of group theory in physics, specifically in relation to quantum field theory and the determination of the Lagrangian of a theory. The first question asks if calculating the invariants of a Lie Group can help determine the Lagrangian, to which the second answer responds that while symmetry is important, it is not the only factor in determining a theory. The conversation also touches on the possibility of unifying gravity and the standard model through the use of group theory, with the caveat that group theory alone is not enough to answer all questions about symmetry in physics.

#### eljose79

Group Theory and Physics...

Suppose we have a quantum field theory with a defined Lie Group of n-parameters, then if we calculated the invariants of the Lie Group...could we then determine the Lagrangian of the theory?.

That is my opinion i think that given a group for a theory we could know all about the physics...and when it comes to gravity and standard model..could they be unified by setting the unified theory group AxB where A would be the group for standard model and B the group for gravity (considered both of them as gauge theory), where "x" means direct product of the two groups.

The first answer if yes, assuming that "the Lagrangian of the theory" translate to "The Lagrangian of the corresponding Yang-Mills theory".

Originally posted by eljose79
Suppose we have a quantum field theory with a defined Lie Group of n-parameters, then if we calculated the invariants of the Lie Group...could we then determine the Lagrangian of the theory?.

That is my opinion i think that given a group for a theory we could know all about the physics...and when it comes to gravity and standard model..could they be unified by setting the unified theory group AxB where A would be the group for standard model and B the group for gravity (considered both of them as gauge theory), where "x" means direct product of the two groups.

The first question reflects indeed the usual methods used to characterize states, the second is not true. The symmetry gives you valuable information, but not all. An example is given by the Poincaré group, where under certain circumstances there are other invariants which cannot be recovered from the group symmetry, but using distributions. Also discrete symmetries cannot usually be found from the Lie group. And then symmetry breakings can evaporate the efforts. Group theory is a powerful tool, but it does certainly do not answer to all questions.

The above two answers are not inconsistent, because it is not clear from the question whethere a gauge group was meant or not. I don't believe the Poincare group is the gauge group of a physical theory.

Who has told that Poincaré is the gauge group? Indeed the special affine groups SA(n,R) where proposed to play the role of gauge groups (more concretely n=4).

## 1. What is group theory and how is it used in physics?

Group theory is a branch of mathematics that studies the properties of groups, which are mathematical objects that represent symmetries or transformations. In physics, group theory is used to describe the symmetries and transformations that occur in physical systems, such as rotations, reflections, and translations. It also helps to classify and understand the fundamental interactions and particles in the universe.

## 2. What is the importance of symmetry in physics and how does group theory relate to it?

Symmetry is a fundamental concept in physics, as it describes the invariance of physical laws under certain transformations. Group theory provides a mathematical framework for understanding and analyzing symmetries in physical systems. It allows physicists to identify and classify different types of symmetries, which can then be used to make predictions and solve problems in physics.

## 3. Can you explain the concept of Lie groups and their significance in physics?

Lie groups are a type of group that describes continuous symmetries, such as rotations and translations. In physics, they are used to study the symmetries of physical systems that involve continuous variables, such as space and time. Lie groups are important in physics because they provide a powerful tool for analyzing and predicting the behavior of physical systems governed by continuous symmetries.

## 4. How does group theory play a role in the Standard Model of particle physics?

The Standard Model of particle physics is a theory that describes the fundamental particles and interactions in the universe. Group theory is essential in this model, as it is used to classify and organize the particles and their interactions according to their symmetries. In particular, the symmetries of the group known as the "SU(3) gauge group" play a crucial role in the Standard Model.

## 5. Are there any real-world applications of group theory in physics?

Yes, group theory has numerous real-world applications in physics. For example, it is used in condensed matter physics to study the symmetries of crystals and their electronic properties. It is also used in quantum mechanics to study the symmetries of atoms and molecules, and in cosmology to understand the symmetries of the universe. Additionally, group theory is used in various engineering fields, such as materials science and signal processing, to study and design complex systems with symmetries.