What is the difference between SO(3,1) and SO(1,3)?

[Coin]

Maybe this is kind of a dumb question but... in a lot of places I see lie groups with names like SO(4,1) or O(1,3) or GL(2,3) referred to. I know what it means when you talk about, say, SO(n)-- that would be the rotation group in n dimensions, or the special orthogonal nxn matrices. But what does it mean when you add the comma and the second number? That seems to be a common notation, but I can't find a clear explanation of it.

Even more confusing, it seems like some people will nonchalantly swap the order of the two numbers, such that one source will be talking about SO(3,1) and another will be talking about SO(1,3) but they appear to really be talking about the same group! What does the transposition of the numbers mean?

(This wikipedia page describes in part a notation where you could have, for example, GL(3, R), where the ",R" provides a group that the matrix members are to be pulled from. But this is clearly not what is meant when the second number is an integer...!)

 

[mjsd]

Such language is used widely in advanced physics. SO(3,1) is the algebra of the Lorentz group which is isomorphic to SU(2) x SU(2). You have 6 generators: three for SO(3) like rotation with communtation relation

$$\left[J_i , J_j\right] = i \epsilon_{ijk} J_k\quad;\quad i,j,k = 1,2,3$$

and three for Lorentz boost (one in each direction in space) with algebra

$$\left[J_i , K_j\right] = i \epsilon_{ijk} K_k$$


$$\left[K_i , K_j\right] = -i \epsilon_{ijk} J_k$$

The last relation tells you that two boosts give a rotation. These guys act on the four spacetime coordinates to produce a Lorentz transformation.

I've never seen ppl use SO(1,3) before, but I guess it means the same thing as SO(3,1), the extra 1 is to remind us that rotation is actually in the 4-dim spacetime coordinate and not just 3-dim space as in SO(3). Since $$SO(3,1) \cong SU(2) \otimes SU(2)$$, you can study its represention using knowledge on SU(2). But then again, I could be wrong about SO(1,3).

 

[tacman]

The notation of SO(n,m) or SO(m,n) refers to the special orthogonal group in n+m dimensions, with n being the number of positive eigenvalues and m being the number of negative eigenvalues. In other words, it represents the group of rotations in n dimensions and reflections in m dimensions.

The difference between SO(3,1) and SO(1,3) lies in the signature of the underlying metric. SO(3,1) refers to the special orthogonal group in 3+1 dimensions, where the underlying metric has 3 positive eigenvalues and 1 negative eigenvalue. This group is commonly used in the context of special relativity, where the fourth dimension is considered to be time.

On the other hand, SO(1,3) refers to the special orthogonal group in 1+3 dimensions, where the underlying metric has 1 positive eigenvalue and 3 negative eigenvalues. This group is commonly used in the context of general relativity, where the fourth dimension is considered to be a spatial dimension.

The transposition of the numbers, as in SO(3,1) and SO(1,3), does not change the group itself. It simply reflects the different conventions used in different fields of mathematics. For example, in physics, the convention is to write the time dimension first, while in mathematics, the convention is to write the spatial dimensions first.

In summary, the notation SO(n,m) or SO(m,n) represents the special orthogonal group in n+m dimensions, with n being the number of positive eigenvalues and m being the number of negative eigenvalues. The difference between SO(3,1) and SO(1,3) lies in the signature of the underlying metric, with the former being used in special relativity and the latter in general relativity. The transposition of the numbers does not change the group itself, but simply reflects different conventions used in different fields.