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Minkowski Space in Group Theory


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#21 studiot

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Posted 31 August 2016 - 10:11 AM

 

 

If that is the case , how many types of transformation are there in Minkowski space and how many in Euclidean (that is the same as "Galilean" ,isn't it ?) space?

 

You are running ahead of yourself.

 

Much of modern theory in this area revolves around the the difference between linear (euclidian) and affine spaces and transformations.

 

This distinction is often not clearly drawn.

 

Mordred mentioned rotations.

There are also what are known as 'forbidden' rotations' we are important for non symmetrical (handed or chiral) objects.

 

Operators (mathematically) are a particular way of expressing relationships (equations and so on) for convenience of mathematical processing.


Edited by studiot, 31 August 2016 - 10:15 AM.

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#22 ydoaPs

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Posted 31 August 2016 - 05:45 PM

A niggle here.
 
It is not dimensionally true to say you can subtract distance (or its square) from time (or its square).
 
Time has dimension T
The constant, c, has dimensions LT-1
 
When multiplied together the result has dimension, L


That's why the c is there.
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#23 studiot

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Posted 31 August 2016 - 07:21 PM

 

 

(invariant spacetime distance)2 = (distance through time in a given frame)2 - (distance through space in the same frame)2.

 

Forgive me but I can't see a c anywhere.


Edited by studiot, 31 August 2016 - 08:02 PM.

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#24 MigL

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Posted 31 August 2016 - 07:47 PM

The Wiki entry for 'Poincare Group' is actually very understandable, Geordief.

( at my level of understanding, beginner level )

It explains the ten degrees of freedom  of Minkowsky space-time, which comprise the isometries ( the property of invariant intervals between events ), and relates it to the ten parameters of the Galilean Group and its absolute space-time.


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#25 geordief

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Posted 31 August 2016 - 10:49 PM

The Wiki entry for 'Poincare Group' is actually very understandable, Geordief.

( at my level of understanding, beginner level )

It explains the ten degrees of freedom  of Minkowsky space-time, which comprise the isometries ( the property of invariant intervals between events ), and relates it to the ten parameters of the Galilean Group and its absolute space-time.

Thanks . It is very hard going for me. Maybe if I keep coming back to it bits of it will sink in.. hopefully the important bits  

 

Thanks also studiot  for the "Geometry by Transformations" suggestion. I see it is quite reasonably priced on Amazon. 


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#26 Mordred

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Posted 1 September 2016 - 05:19 AM

Learning groups can be daunting. One guide is study each mentioned group. Study how symmetry is established in each group.

This article may help it covers the Poincare group with key details into the Lorentz group (SO)1.3 Hint one is time dimension 3 is spatial dimensions...

O means orthogonal
S is special.

This is included in the article. The lorentz group is needed to understand the Poincare group. So study both...
Put simply a rotation is the spatial components but doesn't involve time. A boost involves the spacetime translations. The Lorentz group has three rotations and 3 boosts. Each is listed in the article.
This article also shows spin statistics aspects of the Poincare group.

Representations of the Symmetry Group of Spacetime

http://www.google.ca...4mg7ARF7VO4hXKw

Unfortunately it doesn't cover parity well.

PS the wiki Lorentz group page is fairly decent.

oh almost forgot you will need to familiarize yourself with Bra-ket notation. The article uses it. group theory often use this notation and others.

https://en.m.wikiped...–ket_notation

YdoaPs, Migl, Studiot and wtf have all provided several key details. So study carefully what each is saying. Keep at it.

You will be amazed how interconnected particle physics, quantum physics, thermodynamics and GR truly is. All described by geometry under the

SO(3)×SO(2)×U(1). standard model particles with no supersymmetric particles. (I don't believe Pati-Salam SO (6) which deals with left and right handedness falls under this but some aspects do.) Particularly in U(1) and SO (2) however Im not positive on Pati-Salam I studied it under the SO (10) MSM and MSSM.
lol I'm fairly confident I lost the audience on that last comment... Just thinking to myself as to group representation under SO (1.3) for left/right hand chirality ( Grr thanks Studiot)

 
Mordred mentioned rotations.
There are also what are known as 'forbidden' rotations' we are important for non symmetrical (handed or chiral) objects.
 
.

if I recall correct Pati-Salam SO(4) included SO (3) left and SO (3) right. but not positive lol. Thankfully "Pati-Salam A la SO(10) on arxiv will tell me.

For others reading an allowable rotation or boost must fall within a symmetry group. As Studiots comment indicates some rotations or boosts may fall into different groups or sub groups. Adding a rotation, boost or translation not already described under a group, changes to a new group...
key note, study symmetry relations in just the spatial components, before tackling the boosts due to time components this is key to understanding the Minkowskii metric, or any other GR metric.

aka line element to ds^2 in regards to Pythagoras theory

Edited by Mordred, 1 September 2016 - 07:28 AM.

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#27 geordief

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Posted 1 September 2016 - 09:42 AM

Learning groups can be daunting. One guide is study each mentioned group. Study how symmetry is established in each group.

This article may help it covers the Poincare group with key details into the Lorentz group (SO)1.3 Hint one is time dimension 3 is spatial dimensions...

O means orthogonal
S is special.

This is included in the article. The lorentz group is needed to understand the Poincare group. So study both...
Put simply a rotation is the spatial components but doesn't involve time. A boost involves the spacetime translations. The Lorentz group has three rotations and 3 boosts. Each is listed in the article.
This article also shows spin statistics aspects of the Poincare group.

Representations of the Symmetry Group of Spacetime

http://www.google.ca...4mg7ARF7VO4hXKw

Unfortunately it doesn't cover parity well.

PS the wiki Lorentz group page is fairly decent.

oh almost forgot you will need to familiarize yourself with Bra-ket notation. The article uses it. group theory often use this notation and others.

https://en.m.wikiped...–ket_notation

YdoaPs, Migl, Studiot and wtf have all provided several key details. So study carefully what each is saying. Keep at it.

You will be amazed how interconnected particle physics, quantum physics, thermodynamics and GR truly is. All described by geometry under the

SO(3)×SO(2)×U(1). standard model particles with no supersymmetric particles. (I don't believe Pati-Salam SO (6) which deals with left and right handedness falls under this but some aspects do.) Particularly in U(1) and SO (2) however Im not positive on Pati-Salam I studied it under the SO (10) MSM and MSSM.
lol I'm fairly confident I lost the audience on that last comment... Just thinking to myself as to group representation under SO (1.3) for left/right hand chirality ( Grr thanks Studiot)
 

Plenty of new territory there. Thanks.


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