- #1
jobinjosen
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What are the properties of SO(4) group? , How this acts as a rotator in 4 dimensions?, What are the elements of Rotation matrix in a specific dimension among four dimensions?
Properties and Elements of SO(4) Group in 4 Dimensions
- Thread starter jobinjosen
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In summary, the SO(4) group is a group of rotations in four dimensions. It acts as a rotator by having a two dimensional "axis" of rotation. The generators of rotation are components of Angular momentum (Lx, Ly, Lz) and components of Laplace Runge Lenz (LRL) vector (Ax, Ay, Az), which creates an additional symmetry.
- #2
jostpuur
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I've been waiting for some kind of answer for this post too. I cannot answer the OP, but I'll throw more questions 
When a rotation is carried out in three dimensions, there is an axis of rotation, that is a one dimensional subspace of the three dimensional space, and the rotation is in fact just a two dimensional rotation in the orthogonal complement of this axis. In analogy with this I might guess, that in four dimensions the one dimensional axis is replaced by a two dimensional subspace, that is then some kind of "axis" of rotation. Is this correct?
In analogy with SO(3), I might guess that [itex]SO(4)=\textrm{exp}(\mathfrak{so}(4))[/itex], where [itex]\mathfrak{so}(4)[/itex] consists of those 4x4 matrices that are antisymmetric (satisty [itex]X^T=-X[/itex]). However, these matrices depend only on 6 real variables, which is not enough to define two four dimensional vectors that would span the "axis space", so it seems I'm guessing something wrong.
When a rotation is carried out in three dimensions, there is an axis of rotation, that is a one dimensional subspace of the three dimensional space, and the rotation is in fact just a two dimensional rotation in the orthogonal complement of this axis. In analogy with this I might guess, that in four dimensions the one dimensional axis is replaced by a two dimensional subspace, that is then some kind of "axis" of rotation. Is this correct?
In analogy with SO(3), I might guess that [itex]SO(4)=\textrm{exp}(\mathfrak{so}(4))[/itex], where [itex]\mathfrak{so}(4)[/itex] consists of those 4x4 matrices that are antisymmetric (satisty [itex]X^T=-X[/itex]). However, these matrices depend only on 6 real variables, which is not enough to define two four dimensional vectors that would span the "axis space", so it seems I'm guessing something wrong.
- #3
George Jones
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jostpuur said:
This is true of SO(n) and so(n).
https://www.physicsforums.com/showpost.php?p=1110359&postcount=20 may be of interest to both you and jobinjosen.
- #4
jobinjosen
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Here are some more points regarding SO(4) group.
In SO(3) rotations, generator of rotation are components of Angular momentum (Lx, Ly, Lz) for rotation w.r.t corresponding axis.
Now, In SO(4), what are the generators of rotation?
They are components of Angular momentum (Lx, Ly, Lz) and components of Laplace Runge Lenz (LRL) vector (Ax, Ay, Az). Constancy of this LRL vector creates aditional symmetry. Am I correct?
In SO(3) rotations, generator of rotation are components of Angular momentum (Lx, Ly, Lz) for rotation w.r.t corresponding axis.
Now, In SO(4), what are the generators of rotation?
They are components of Angular momentum (Lx, Ly, Lz) and components of Laplace Runge Lenz (LRL) vector (Ax, Ay, Az). Constancy of this LRL vector creates aditional symmetry. Am I correct?
Related to Properties and Elements of SO(4) Group in 4 Dimensions
1. What is the SO(4) group in 4 dimensions?
The SO(4) group is a mathematical group that represents the symmetries of a four-dimensional space. It is a special orthogonal group, meaning that its elements are matrices that preserve the length of vectors and the angles between them. In simpler terms, it describes the rotations and reflections in four-dimensional space.
2. What are the properties of the SO(4) group?
The SO(4) group has several properties that make it unique. Some of these include closure, meaning that the composition of two elements in the group is also an element of the group, and associativity, meaning that the order in which operations are performed does not affect the final result. Additionally, the group has an identity element, which is the identity matrix, and each element has an inverse element.
3. How is the SO(4) group related to physics?
The SO(4) group has many applications in physics, particularly in quantum mechanics and particle physics. It is used to describe the symmetries of physical systems and can be used to predict the behavior of particles and their interactions. It is also important in the study of spacetime and the theory of relativity.
4. What are the elements of the SO(4) group?
The elements of the SO(4) group are 4x4 matrices with real numbers as entries. These matrices have a determinant of 1, and their inverse is equal to their transpose. The elements can be represented by four rotations and six reflections, which can be visualized in four-dimensional space.
5. How is the SO(4) group different from other groups?
The SO(4) group is unique in that it is a special orthogonal group in four dimensions. It differs from other groups in its properties and the way it is used in mathematics and physics. It is also larger than the more well-known SO(3) group, which represents rotations in three-dimensional space.
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