Taylor expanding to first order in " ? " gives the infinitesimal rotation operator:
3.
The space of rotations is isomorphic with the set of rotation operators and the set of orthonormal matrices with determinant + 1.
4.
The relationship between the angular momentum operator and the rotation operators is the same as the relationship between lie algebras and lie groups in mathematics.
5.
Likewise, exponentiating the representations of the generators gives the representations of the boost and rotation operators, under which a particle's spinor field transforms.
6.
The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below.
7.
Though all three movements can be represented by a rotation operator with constant coefficients in some frame, they cannot be represented by these operators all at the same time.
8.
In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.
9.
Exponentiating the generators gives the boost and rotation operators which combine into the general Lorentz transformation, under which the spacetime coordinates transform from one rest frame to another boosted and / or rotating frame.
10.
From this and the infinitesimal rotation operator and its Hermitian conjugate, and ignoring second order term in ( \ delta \ theta ) ^ 2, one can derive the commutation relation with the rotation generator:
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