Poincaré and Euclidean groups


by Rasalhague
Tags: euclidean, groups, poincare
Rasalhague
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#1
Dec7-09, 03:28 AM
P: 1,400
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?
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George Jones
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Dec7-09, 06:02 AM
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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
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#3
Dec7-09, 08:15 AM
P: 1,400
Thanks, George!

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


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