3D rotation group

Its representations are important in physics, where they give rise to the elementary particles of integer spin.

This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length (see the law of cosines):

See classical group for a treatment of this more general approach, where SO(3) appears as a special case.

This orthonormality condition can be expressed in the form where RT denotes the transpose of R and I is the 3 × 3 identity matrix.

The group of all 3 × 3 orthogonal matrices is denoted O(3), and consists of all proper and improper rotations.

[3] Every finite subgroup is isomorphic to either an element of one of two countably infinite families of planar isometries: the cyclic groups

For example, counterclockwise rotation about the positive z-axis by angle φ is given by Given a unit vector n in

Then Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that In the next section, this representation of rotations is used to identify SO(3) topologically with three-dimensional real projective space.

Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle.

In physics applications, the non-triviality (more than one element) of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spin–statistics theorem.

The map from S3 onto SO(3) that identifies antipodal points of S3 is a surjective homomorphism of Lie groups, with kernel {±1}.

In this section, we give two different constructions of a two-to-one and surjective homomorphism of SU(2) onto SO(3).

The Möbius transformations can be represented by matrices since a common factor of α, β, γ, δ cancels.

For the same reason, the matrix is not uniquely defined since multiplication by −I has no effect on either the determinant or the Möbius transformation.

In terms of Euler angles[nb 1] one finds for a general rotation one has[6] For the converse, consider a general matrix Make the substitutions With the substitutions, Π(gα, β) assumes the form of the right hand side (RHS) of (2), which corresponds under Πu to a matrix on the form of the RHS of (1) with the same φ, θ, ψ.

It is clear from the explicit form in terms of Euler angles that the map just described is a smooth, 2:1 and surjective group homomorphism.

In Lie algebra representations, the group SO(3) is compact and simple of rank 1, and so it has a single independent Casimir element, a quadratic invariant function of the three generators which commutes with all of them.

The Killing form for the rotation group is just the Kronecker delta, and so this Casimir invariant is simply the sum of the squares of the generators,

That is, the eigenvalues of this Casimir operator are where j is integer or half-integer, and referred to as the spin or angular momentum.

By taking Kronecker products of D1/2 with itself repeatedly, one may construct all higher irreducible representations Dj.

The level of difficulty of proof depends on how a matrix group Lie algebra is defined.

Hall (2003) defines the Lie algebra as the set of matrices in which case it is trivial.

That this gives a one-parameter subgroup follows directly from properties of the exponential map.

This follows from the fact that every R ∈ SO(3), since every rotation leaves an axis fixed (Euler's rotation theorem), and is conjugate to a block diagonal matrix of the form such that A = BDB−1, and that together with the fact that 𝖘𝖔(3) is closed under the adjoint action of SO(3), meaning that BθLzB−1 ∈ 𝖘𝖔(3).

It is mathematically impossible to supply a straightforward formula for such a basis as a function of u, because its existence would violate the hairy ball theorem; but direct exponentiation is possible, and yields where

Then, the logarithm of R is given by[9] This is manifest by inspection of the mixed symmetry form of Rodrigues' formula, where the first and last term on the right-hand side are symmetric.

The general case is given by the more elaborate BCH formula, a series expansion of nested Lie brackets.

Under the hat-isomorphism, It is worthwhile to write this composite rotation generator as to emphasize that this is a Lie algebra identity.

The same explicit formula thus follows in a simpler way through Pauli matrices, cf.

This is manifestly of the same format as above, with so that For uniform normalization of the generators in the Lie algebra involved, express the Pauli matrices in terms of t-matrices, σ → 2i t, so that To verify then these are the same coefficients as above, compute the ratios of the coefficients, Finally, γ = γ' given the identity d = sin 2c'.