What is So(3): Definition and 55 Discussions

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space



{\displaystyle \mathbb {R} ^{3}}
under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e. handedness of space). Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition. Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.
Rotations are linear transformations of



{\displaystyle \mathbb {R} ^{3}}
and can therefore be represented by matrices once a basis of



{\displaystyle \mathbb {R} ^{3}}
has been chosen. Specifically, if we choose an orthonormal basis of



{\displaystyle \mathbb {R} ^{3}}
, every rotation is described by an orthogonal 3 × 3 matrix (i.e. a 3 × 3 matrix with real entries which, when multiplied by its transpose, results in the identity matrix) with determinant 1. The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3).
The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its representations are important in physics, where they give rise to the elementary particles of integer spin.

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  1. DanCoimbra

    I Physical reality of nontrivial loops in SO(3)

    I will ask a mathematical and a physical-cum-philosophical question pertaining to the fact that SO(3) is not simply connected. Context Classical rotations in three spatial dimensions are represented by the group SO(3), whose elements represent 3D rotations. Having said that, note that classical...
  2. cianfa72

    I Is the Belt Trick Possible with Continuous Deformation in 3D Rotation Space?

    Hi, in the following video at 15:15 the twist of ##4\pi## along the ##x## red axis is "untwisted" through a continuous deformation of the path on the sphere 3D rotations space. My concern is the following: keeping fixed the orientation in space of the start and the end of the belt, it seems...
  3. T

    I SO(3) -- What is the advantage of knowing something is in a group?

    Good Morning! I know that Rotation matrices are members of the SO(3) group. I can prove some useful properties about it: The inverse is the transpose; Closure properties; However, what is the advantage of asserting that a rotation matrix is a member of the SO(3) group, when all I really need...
  4. L

    A SO(3) group, Heisenberg Hamiltonian

    We have commutation relation ##[J_j,J_k]=i \epsilon_{jkl}J_l## satisfied for ##2x2##, ##3x3##, ##4x4## matrices. Are in all dimensions these matrices generate ##SO(3)## group? I am confused because I think that maybe for ##4x4## matrices they will generate ##SO(4)## group. For instance for...
  5. J

    Model with SU(2) gauge symmetry and SO(3) global symmetry

    1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##. The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...
  6. sergiokapone

    I Rodrigues' rotation formula from SO(3) comutator properties

    Is any way to get Rodrigues' rotation formula from matrix exponential \begin{equation} e^{i\phi (\star\vec{n}) } = e^{i\phi (\vec{n}\cdot\hat{\vec{S}}) } = \hat{I} + (\star\vec{n})\sin\phi + (\star\vec{n})^2( 1 - \cos\phi ). \end{equation} using SO(3) groups comutators properties ONLY...
  7. S

    I What is the Dimension of SO(3) with Constraint det(O) = 1?

    The group ##\rm{O(3)}## is the group of orthogonal ##3 \times 3## matrices with nine elements and dimension three which is constrained by the condition, $$a_{ik}a_{kj} = \delta_{ij}$$ where ##a_{ik}## are elements of the matrix ##\rm{A} \in O(3)##. This condition gives six constraints (can be...
  8. sophiatev

    I The SO(3) group in Group Theory

    In Griffith's Introduction to Elementary Particles, he provides a very cursory introduction to group theory at the start of chapter four, which discusses symmetries. He introduces SO(n) as "the group of real, orthogonal, n x n matrices of determinant 1 is SO(n); SO(n) may be thought of as the...
  9. M

    I Relationship between a Lie group such as So(3) and its Lie algebra

    I am just starting a QM course. I hope these are reasonable questions. I have been given my first assignment. I can answer the questions so far but I do not really understand what's going on. These questions are all about so(N) groups, Pauli matrices, Lie brackets, generators and their Lie...
  10. T

    A Question about the irreducible representation of a rank 2 tensor under SO(3)

    When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...
  11. K

    I Weinberg gives relations of SO(3)

    (Weinberg QFT, Vol 1, page 68) He considers Mass-Positive-Definite, in which case the Little Group is SO(3). He then gives the relations Is it difficult to derive these relations? I'm asking this mainly because I haven't seen them anywhere other than in Weinberg's book. Also, I'm finding...
  12. N

    I Computation of the left invariant vector field for SO(3)

    I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward. I have been looking at these notes: https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example...
  13. S

    I How Do SU(2) and SO(3) Relate to Spinors and Vectors in Physics?

    Hello! I want to make sure I understand the relation between this and rotation (mainly between SU(2) and SO(3), but also in general). Also, I am a physics major, so I apologize if my statements are not very rigorous, but I want to make sure I understand the basic underlying concepts. So SU(2) is...
  14. S

    I Understanding SU(2) and SO(3) Representations

    Hello! I am reading some representation theory and I am a bit confused about some stuff. I read that SU(2) is the double covering of SO(3), so to each matrix in SO(3) corresponds one in SU(2). I am not sure I understand this. So if we have a 3D representation of SU(2), the 3D object it acts on...
  15. JTC

    I Left translate back to I in SO(3)

    Hello I am hoping someone can explain a sentence to me. Unfortunately, I do not even recall where I read it. I wrote it down years ago and long since lost the source. (Now I think some of it is making sense, but I don't remember the source.) Consider R(t) as an orthogonal rotation matrix...
  16. M

    I Constructing left invariant vector fields on SO(3)

    hello every one can one please construct for me left invariant vector field of so(3) rotational algebra using Euler angles ( coordinates ) by using the push-forward of left invariant vector field ? iv'e been searching for a method for over a month , but i did not find any well defined method...
  17. S

    I SO(3) rotation of eigenvectors

    Consider the eigenvectors ##(0, 1)## and ##(1, 0)## for the quantum system described the magnetic field ##\vec{B} = (0,0,B)##. Say I now rotate the magnetic field as ##\vec{B} = (B\sin\theta\cos\phi,B\sin\theta\sin\phi,B\cos\theta)##. Then the eigenvectors are supposed to change as...
  18. S

    I Rotation Matrix for Vector v=(a,b,c) by Angle θ | Efficient Computation Method

    Hello! I need to find the rotation matrix around a given vector v=(a,b,c), by and angle ##\theta##. I can find an orthonormal basis of the plane perpendicular to v but how can I compute the matrix from this? I think I can do it by brute force, rewriting the orthonormal basis rotated by...
  19. M

    Isomorphism between so(3) and su(2)

    Homework Statement How do I use the commutation relations of su(2) and so(3) to construct a Lie-algebra isomorphism between these two algebras? Homework Equations The commutation relations are [ta, tb] = i epsilonabc tc, the ts being the basis elements of the algebras. They basically have the...
  20. S

    Conserved Noether current under SO(3) symmetry of some Lagrangian

    Homework Statement Verify that the Lagrangian density ##\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi_{a}\partial^{\mu}\phi_{a}-\frac{1}{2}m^{2}\phi_{a}\phi_{a}## for a triplet of real fields ##\phi_{a} (a=1,2,3)## is invariant under the infinitesimal ##SO(3)## rotation by ##\theta##, i.e...
  21. G

    I What is the rank of the SU(2)xSU(2) algebra?

    I am reading in my group theory book the well known commutation relations of the Lie algebra of SO(3), i.e. [J,J]=i\epsilon J. What I don't understand is the statement that "from the relations we can infer that the algebra has rank 1". Any ideas?
  22. I

    Scalar as one dimensional representation of SO(3)

    Hi to all the readers of the forum. I cannot figure out the following thing. I know that a representation of a group G on a vector spaceV s a homomorphism from G to GL(V). I know that a scalar (in Galileian Physics) is something that is invariant under rotation. How can I reconcile this...
  23. L

    Su(2), so(3) and their representations

    I try to understand the statement "Every representation of SO(3) is also a representation of SU(2)". Does that mean that all the matrices of an integer-spin rep of SU(2) are identical to the matrices of the corresponding spin rep of SO(3)? Say, the j=1 rep of SU(2) has three 3x3 matrices, so...
  24. M

    SO(N) adjoint rep. under SO(3) subgroup

    Hi. I'm having trouble figuring out how SO(N) adjoint rep. transforms under a SO(3) subgroup. Unlike SU(N), SO(N) fundamental N gives \begin{equation} N \otimes N = 1 \oplus A \oplus S \end{equation} So the \begin{equation} S \end{equation} part really bothers. Can you give a help?
  25. mef51

    [Differential Geometry] Matrix of Differential Equations in SO(3)

    Homework Statement Suppose that ##s \to A(s) \subset \mathbb{M}_{33}(\mathbb{R})## is smooth and that ##A(s)## is antisymmetric for all ##s##. If ##Q_0 \in SO(3)##, show that the unique solution (which you may assume exists) to $$\dot{Q}(s) = A(s)Q(s), \quad Q(0) = Q_0$$ satisfies ##Q(s) \in...
  26. H

    Connection between SU(2) and SO(3)

    I am somewhat confused with the connection between the two groups. In the text I'm reading (An Introduction to Tensors and Group theory for physicists N. Jeevanjee), there is a chapter quite early on (in the group theory part) which outlines a homomorphism from SU(2) to SO(3), however I find...
  27. S

    Transforming Representations of SO(3) to Act on Vectors?

    Hi Everybody! I am working on QFT and learning representation theory from Coleman's lecture notes. Just the necessary stuff to go to the Dirac equation. To my question: From the generators of SO(3) I get through exponentiation an element of SO(3), this holds naturally for any Lie group...
  28. BWV

    Commutator SO(3) - what am I doing wrong?

    reading that the commutator of rotations on two orthogonal axes is i * the rotation matrix for the third axis but if I commute this \begin{pmatrix}\mathrm{cos}\left( \theta\right) & -\mathrm{sin}\left( \theta\right) & 0\cr \mathrm{sin}\left( \theta\right) & \mathrm{cos}\left(...
  29. A

    How Do You Draw a Doublet in a Young Diagram for SO(3)?

    Hello all! Clebsch-Gordon Coefficients tell us that in an SO(3) representation 3 x 2 = 4 + 2 (x/+ is the tensor product/sum). In practicing my Young Diagrams I tried to re-create this calcuation, but can't seem to figure out how to draw a doublet. I would appreciate some advice! (The triplet...
  30. C

    Noether current for SO(3) rotation

    This is a problem from my theoretical physics course. We were given a solution sheet, but it doesn't go into a lot of detail, so I was hoping for some clarification on how some of the answers are derived. Homework Statement For the Lagrangian L=1/2(∂μ∅T∂μ∅-m2∅T∅) derive the Noether...
  31. conquest

    Finding Hamiltonian as Legendre transform on SO(3)

    Homework Statement We need to find the Hamiltonian that corresponds to a given Lagrangian by finding the Legendre transform. The system is a rigid body pinned down in some point. This means the motion is described essentialy by SO(3). So the Lagrangian is given in terms of these matrices and...
  32. D

    Left group actions involving SO(3) and the 2-sphere

    the group of proper orthogonal transformations SO(3) acts transitively on the 2-sphere S2. show that the isotropy group of any vector r is isomorphic to SO(2) and find a bijective correspondence between the factor space SO(3)/SO(2) and the 2-sphere such that SO(3) has identical left actions...
  33. L

    Can SO(3) be used for Poincare spacetime symmetry in the standard model?

    I'm a layman trying to understand the symmetries used in the std model. I understand that U(1), SU(2), & SU(3) are incorporated in the Lagrangians for internal symmetries. I've read that SO(3) is also used in the std model for Poincare spacetime symmetry. Is that true and if so, how is it...
  34. M

    SU(2) as representation of SO(3)

    The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))? Is a representation R of some group G a group too? If so, is it true that G is representation of R?
  35. L

    Solving SO(3) Irreps: Find Eigenvectors & Eigenvalues of X_3

    Hi, I just wondered if someone could check my understanding is correct on this topic. I understand that to find the irreps of a group we can find the irreps of the associated Lie algebra, i.e. in the case of SO(3) find irreducible matrices satisfying the comm relations...
  36. N

    Irreducible representation of so(3)

    Hi guys, I have a question which is very fundamental to representation theory. What I am wondering is that whether a first rank cartesian representation of so(3) is irreducible. As I understand first rank cartesian representation of so(3) can be parametrized in terms of the Euler angles. That...
  37. W

    SO(3) as a quotient group of SU(2)?

    we know there is a two to one homomorphism from SU(2) to SO(3) suppose u is an element in SU(2) then u and -u map into the same element in SO(3) the question is, maybe SO(3) is a quotient group of SU(2)? with respect to the subgroup {I,-I}?
  38. W

    Why so(3) is not isomorphic to su(2)?

    it is generally known that there is a two-to-one automorphism from su(2) to so(3) but consider the problem in this way: all elements in so(3) are of the form (up to a unitary transform of the basis) R(\alpha,\beta.\gamma)=e^{-i\alpha F_z} e^{-i\beta F_y} e^{-i \gamma F_z} where F_x...
  39. P

    Difference between SO(3) and so(3)

    Hi, What is the difference between lie group SO(3) and lie algebra so(3)? I just got the idea in terms of terminology that one represents the group and the other stands for algebra. But can anyone please provide details as to what exactly is the difference? Regards, Priyanshu
  40. L

    SO(3) Special Othorgonal Group

    For the special orthogonal group SO(3), with G-set R^3, and the usual G-action, we choose x in R^3 not equal to 0. Then the stabilizer of x (set of all the transformations in SO(3) that doesn't change x) is all the rotations about the axis produced by x (and -x). Can someone explain why the...
  41. B

    Exploring the Isomorphism Between SU(2) and SO(3) Groups

    I have not seen why SU(2) and SO(3) groups are isomorphic?
  42. M

    Does a faithful action of SO(3) imply a metric on R^3?

    I think that the usual action of SO(3) on R^3 (defined by matrix multiplication) is faithful, because to non-identity rotations belong non-identity transformations.If we don't have originally a norm on R^3, but do have a faithful action of SO(3) on it, then we can try to define a norm by taking...
  43. C

    Möbius transformations and SO(3)

    Hi, I was given the following problem, and i couldn't solve it yet: Give a bijection between the elements of SO(3) and the fractional linear transformations of the form \varphi_{z,w}\,(u)=\frac{zu+w}{-\bar wu+\bar z}, where u\in \mathbb C\cup \{\infty\};\, z,w\in \mathbb C. Any ideas...
  44. R

    The groups O(3), SO(3) and SU(2)

    Homework Statement How can irreducible representations of O(3) and SO(3) be determined from the irreducible representations of SU(2)? The Attempt at a Solution I believe there is a two-one homomorphic mapping from SU(2) to SO(3); is that enough for some shared representations? If I had...
  45. N

    What Are the Generators of the 4D Irreducible Representation of SO(3)?

    Homework Statement Find the generators of the four dimensional irreducible representation of SO(3), such that J_3 is diagonal. The Attempt at a Solution I know how to get the rest if I know J_3, by using ladder operators. But what is J_3? For a 3d representation it's diagonal with 1,0-1, in 4d...
  46. T

    Understandig Representation of SO(3) Group

    Hi, I'm very new on Group Theory, and lacking of easy to understand document on it. I can't get Representation of SO(3) Groups. Is there anyone can tell me useful information about it? Thanks, Tore Han
  47. Jim Kata

    How Is the Center of SU(2) Related to the Fundamental Group of SO(3)?

    there's a surjective homomorphism from a : SU(2) --> SO(3) The kernel of this homomorphism is the center of SU(2) which is Z/2Z. Now the fundamental group of SO(3) is Z/2Z. This is a general thing. The simplest version of my question is how is the center of SU(2) related to the...
  48. C

    Structure constants of su(2) and so(3)

    SU(2) and SO(3) "have the same Lie algebra". While I understand that their corresponding lie algebras su(3) and so(2) have the same commutator relations \mbox{SO(3)}: \left[ \tau^i, \tau^j\right] = \iota \varepsilon_{ijk} \tau^k \mbox{SU(2)}: \left[ \frac{\sigma^i}{2}...
  49. M

    Visualizing left action of SO(3) on itself

    The SO(3) group is topologically a 3-dimensional ball of radius \pi, if the opposite points on its surface are identified with each other. (the name of it is 3-dimensional projective space). The center of the ball represents the unit element e of the group. An arbitrary point g in the ball...
  50. X

    Showing SO(3) Subset is Projective Plane Diffeomorphic

    I was wondering if anyone can help me to show that the subset of SO(3) contaning all matrices A with det(A+id)=0 is a submanifold diffeomorphic to real projective plane. Thanks.