Register to reply

Poincaré and Euclidean groups

by Rasalhague
Tags: euclidean, groups, poincare
Share this thread:
Rasalhague
#1
Dec7-09, 03:28 AM
P: 1,402
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_b...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?
Phys.Org News Partner Science news on Phys.org
Law changed to allow 'unlocking' cellphones
Microsoft sues Samsung alleging contract breach
Best evidence yet for coronal heating theory detected by NASA sounding rocket
George Jones
#2
Dec7-09, 06:02 AM
Mentor
George Jones's Avatar
P: 6,233
Quote Quote by Rasalhague View Post
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_b...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.
Quote Quote by Rasalhague View Post
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.
Rasalhague
#3
Dec7-09, 08:15 AM
P: 1,402
Thanks, George!

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


Register to reply

Related Discussions
Groups, Normalizer, Abstract Algebra, Dihedral Groups...help? Calculus & Beyond Homework 12
Lorentz and Poincare groups Special & General Relativity 1
Euclidean and Non Euclidean Space? Differential Geometry 1
Centers of groups and products of groups Calculus & Beyond Homework 1
Wallpaper Groups, Free Groups, and Trees Introductory Physics Homework 13