What is the use of infinite-dimensional representation of group

In summary, an infinite-dimensional representation of a group is a mathematical structure where the elements of a group are represented by infinite-dimensional matrices or linear transformations. These representations have important applications in mathematics, physics, and other scientific fields, allowing for the study of continuous symmetries and providing a more general and flexible representation of group elements. However, they can be more challenging to analyze compared to finite-dimensional representations and not all groups have infinite-dimensional representations available. In physics, infinite-dimensional representations of groups are used to study the symmetries of physical systems, such as particles, wave functions, and space-time, and have applications in areas like fluid dynamics and statistical mechanics.
  • #1
liucl78
3
0
What is the use of infinite-dimensional representation of lie group?
Now, I know Hilbert space is infinite-dimensional, and physical states must be in Hilbert space.
However, for massive fields, the transformation group is SO(3), its unitary representation is finite.
For massless fields, the transformation group is non-compact ISO(2), but only the finite representation of ISO(2), namely SO(2) representation, has physical meanings.

I want to know where the infinite-dimensional representation is used in physics?

thanks
 
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  • #2
It provides the most general possible representation of what you are trying to do.
 

Related to What is the use of infinite-dimensional representation of group

1. What is an infinite-dimensional representation of a group?

An infinite-dimensional representation of a group refers to a mathematical structure in which the elements of a group are represented by infinite-dimensional matrices or linear transformations. This representation allows for the study of groups with infinitely many elements or continuous groups, such as Lie groups.

2. Why are infinite-dimensional representations of groups important?

Infinite-dimensional representations of groups have a wide range of applications in mathematics, physics, and other scientific fields. They allow for the study of continuous symmetries, which are essential in understanding many physical phenomena, such as quantum mechanics and relativity. They also have important applications in harmonic analysis, differential geometry, and representation theory.

3. How are infinite-dimensional representations of groups different from finite-dimensional representations?

The main difference between infinite-dimensional and finite-dimensional representations is the size of the matrices or operators used to represent the elements of the group. In finite-dimensional representations, the matrices have a fixed size, while in infinite-dimensional representations, the matrices have an infinite number of rows and columns. This allows for a more general and flexible representation of the group elements.

4. Are there any limitations to using infinite-dimensional representations of groups?

One limitation of using infinite-dimensional representations is that they can be more difficult to study and analyze compared to finite-dimensional representations. Additionally, not all groups have infinite-dimensional representations, and those that do may have multiple inequivalent representations.

5. How are infinite-dimensional representations of groups used in physics?

Infinite-dimensional representations of groups are widely used in physics to study the symmetries of physical systems. For example, they are used in quantum field theory to describe the symmetries of particles, in quantum mechanics to study the symmetries of wave functions, and in general relativity to describe the symmetries of space-time. They also have applications in other areas of physics, such as fluid dynamics and statistical mechanics.

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