What Is the Euclidean Analog of the Poincaré Group?

In summary, the Poincaré group consists of all continuous transformations in Minkowski space that preserve the inner product, which includes translations and the restricted Lorentz group. However, it excludes the discontinuous transformations of spatial reflection and time reversal. The full Poincare group has four connected components, with the connected component containing the identity being referred to as the Poincare group. This group is equivalent to the Euclidean group in Euclidean space, consisting of translations and the full Lorentz group. It is also known as the restricted Poincare group.
  • #1
Rasalhague
1,387
2
Benjamin Crowell writes here, "The discontinuous transformations of spatial reflection and time reversal are not included in the definition of the Poincaré group, although they do preserve inner products."

http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html

So, if I've understood this, the Poincaré group consists of all continuous transformations in Minkowski space that preserve the inner product: (1) translations and (2) the restricted Lorentz group (proper orthochronous Lorentz transformations in Minkowski space, i.e. rotations and boosts).

Is there a name for the corresponding set of transformations in Euclidean space? I gather the Euclidean group consists of (1) translations and (2) the orthogonal group (rotations and rotoreflections). It has a subgroup [tex]E^{+}\left ( n \right )[/tex] consisting of translations and the special orthogonal group (rotations). This [tex]E^{+}\left ( n \right )[/tex] is to Euclidean space what the Poincaré group is to Minkowski space, isn't it?

Is there a name or conventional symbol for the set of transformations in Minkowski space corresponding to the Euclidean group (translations together with the full Lorentz group), and does it form a group too?
 
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  • #2
Rasalhague said:
Benjamin Crowell writes here, "The discontinuous transformations of spatial reflection and time reversal are not included in the definition of the Poincaré group, although they do preserve inner products."

http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html

So, if I've understood this, the Poincaré group consists of all continuous transformations in Minkowski space that preserve the inner product: (1) translations and (2) the restricted Lorentz group (proper orthochronous Lorentz transformations in Minkowski space, i.e. rotations and boosts).

This exclusion in non-standard. Similar to the Lorentz group, the full Poincare group has four connected components. Benjamin Crowell defines the "Poincare group" to be the connected component of the Poincare group that contains the identity.
Rasalhague said:
Is there a name or conventional symbol for the set of transformations in Minkowski space corresponding to the Euclidean group (translations together with the full Lorentz group), and does it form a group too?

Yes, it's called the Poincare group.
 
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  • #3
Thanks, George!

I see that what Benjamin Crowell calls "the Poincaré group" is sometimes referred to as "the restricted Poincaré group".
 

Related to What Is the Euclidean Analog of the Poincaré Group?

What are the Poincaré and Euclidean groups?

The Poincaré and Euclidean groups are two types of mathematical groups that describe the symmetries of space. The Poincaré group is a group of transformations that preserve the structure of spacetime, while the Euclidean group is a group of transformations that preserve the geometrical properties of space.

What is the difference between the Poincaré and Euclidean groups?

The main difference between the Poincaré and Euclidean groups is the type of space they describe. The Poincaré group is used in the theory of relativity to describe the symmetries of four-dimensional spacetime, while the Euclidean group is used in classical geometry to describe the symmetries of three-dimensional space.

How are the Poincaré and Euclidean groups related?

The Poincaré group is a subgroup of the Euclidean group, meaning that all transformations in the Poincaré group are also transformations in the Euclidean group. This is because in special relativity, the four-dimensional spacetime can be viewed as a three-dimensional space with an additional time dimension.

What are some real-world applications of the Poincaré and Euclidean groups?

The Poincaré and Euclidean groups have many applications in physics and engineering. For example, the Poincaré group is used in the study of particle physics and the behavior of subatomic particles. The Euclidean group is used in computer graphics to describe the transformation of objects in 3D space.

How do the Poincaré and Euclidean groups contribute to our understanding of the universe?

The Poincaré group plays a crucial role in the theory of special relativity, which has revolutionized our understanding of space and time. The Euclidean group helps us understand the symmetries of 3D space, which is essential in many areas of physics and engineering. Together, these groups provide a powerful tool for studying the structure and behavior of the universe.

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