Understanding Minkowski Space Compactification

[ritzo]

Hi. I'm reading about the compactification of Minkowski Space, and there is a subject that is keeping me awake. They say that the group of conformal transformations is isomorphic to the group of pseudoorthogonal transformations with determinant equal to 1. I don't know how this happen and it would of great help if someone could tell me why this happen.

 

[strangerep]

Hi. I'm reading about the compactification of Minkowski Space, and there is a subject that is keeping me awake. They say that the group of conformal transformations is isomorphic to the group of pseudoorthogonal transformations with determinant equal to 1. I don't know how this happen and it would of great help if someone could tell me why this happen.

More accurately, the group SO(4,2) is a double cover of the conformal group of Minkowski space. The Lie algebras of these two groups coincide, as can be seen by direct evaluation.

 

[ritzo]

First of all, thank you so much for answer me. Second, I'm sorry, but I'm not familiar with Lie algebra. I don't know if it is possible to solve this in a more elemental way or maybe you can tell me about a book where I can find this. Again, thanks.

 

[Bill_K]

Consider a 3-space with orthogonal coordinates T, X, S and signature + - -. Consider the rotations in this 3-space, which form the group SO(2,1), and their actions on the unit hyperboloid T² - X² - S² = 1.

Parametrize points on this hyperboloid with coordinates x, s whose relation to T, X, S is

T - X = e^x
T + X = s² e^x + e^-x
S² = s² e^2x

(By substitution, one can verify that for any values of x and s we have T² - X² - S² ≡ 1, which lies on the hyperboloid.)

Now consider the action of SO(2,1) on x and s. One of the rotations, a boost in the T,X plane, amounts to a translation in x and a rescaling of s:

x → x + c
s → s e^-c

Another, a rotation about the null direction T - X, is a translation of s,

s → s + c

Ok, now generalize this to any number of dimensions. Replace the original S² everywhere by Ʃ ε_i S_i² where ε_i = ±1. (This is the line element in an n-dimensional space with signature determined by the choice of ε_i's.) We are now looking at the unit hyperboloid T² - X² - Ʃ ε_i S_i² = 1. The parameters x and s_i

T - X = e^x
T + X = Ʃ s_i² e^x + e^-x
Ʃ S_i² = Ʃ s_i² e^2x

and the transformations

x → x + c
s_i → s_i e^-c

and

s_i → s_i + c_i

Thus we have a rescaling of the s_i's and a translation of the s_i's, which, together with rotations among the s_i's, produce the conformal group in n dimensions.

This shows explicitly that the transformations in the rotation group SO(n+1, m+1) map to those in the conformal group C(n,m).

 

[strangerep]

[...] I'm not familiar with Lie algebra. I don't know if it is possible to solve this in a more elemental way or maybe you can tell me about a book where I can find this.

Hmm. You said you were "reading about the compactification of Minkowski Space", but not knowing anything about Lie group theory is a serious handicap for most subjects in physics.

You could try the book by Greiner & Muller: "QM - Symmetries". It contains a lot of introductory material on Lie groups (though of course it's not about compactification of Minkowski Space).

This shows explicitly that the transformations in the rotation group SO(n+1, m+1) map to those in the conformal group C(n,m).

Nice explanation. I hope the OP can appreciate it eventually. (BTW, since it's been ages since I studied this stuff closely, where does the double-cover arise?)

 

[Jim Kata]

Check this out. https://www.physicsforums.com/showthread.php?t=172461 specifically look at post number 6 in this thread. I think this is the answer you're looking for.

 

[ritzo]

Ok, let me be completely honest with you guys. I'm reading the book "Conformal differential geometry and its generalizations", of Akivis. In this book the author don't use anything what you are using. He starts with the euclidean case (n dimensions) and then he generalize to the pseudo euclidean case. Since he want to study the conformal aspects of this space, he introduces the point at infinity, son R^n (euclidean space) becomes C^n (conformal space). Next, he use "polyspherical coordinates", which are points in a (n+1)-projective space, P^(n+1), and gets an inner product which helps him to identify every point in C^n with a point on a quadric in P^(n+1).This identification is knows as "Darboux mapping". He then says that every conformal transformation is equivalent, up to a factor c, with a linear transformation that leaves the quadric invariant. This linear transformations is of order n+2. Since, if n is odd, you can take an appropriate value of c to make 1 the determinat of the linear transformation, he concludes that the conformal group is isomorphic to SO(2+n,1)

And I am like... "What?", Why?. If the linear transformations are of order 2+n, why it is isomorphic to a matrix of SO(2+n,1)?

So, please you guys. I don't know anything about Lie algebra and I have weeks thinking on this. Thank you.

 

[Bill_K]

He then says that every conformal transformation is equivalent, up to a factor c, with a linear transformation that leaves the quadric invariant. This linear transformations is of order n+2.

This quadric is the "unit hyperboloid" that I referred to above in #4.

So, please you guys. I don't know anything about Lie algebra and I have weeks thinking on this. Thank you.

I didn't mention anything about Lie algebras.