Rotor (mathematics)

A rotor is an object in geometric algebra (or more generally Clifford algebra) that rotates any blade or general multivector about the origin.[1] They are normally motivated by considering an even number of reflections, which generate rotations (see also the Cartan–Dieudonné theorem).

This article is about the object in geometric algebra. For the vector operator, see curl (mathematics).

The term originated with William Kingdon Clifford,[2] in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre).[3] Hestenes[4] defined a rotor to be any element ${\displaystyle R}$ of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies ${\displaystyle {\tilde {R}}R=1}$, where ${\displaystyle {\tilde {R}}}$ is the "reverse" of ${\displaystyle R}$—that is, the product of the same vectors, but in reverse order.

Generation using reflections

General formulation

α > θ/2

α < θ/2

Rotation of a vector a through angle θ, as a double reflection along two unit vectors n and m, separated by angle θ/2 (not just θ). Each prime on a indicates a reflection. The plane of the diagram is the plane of rotation.

Reflections along a vector in geometric algebra may be represented as (minus) sandwiching a multivector M between a non-null vector v perpendicular to the hyperplane of reflection and that vector's inverse v⁻¹:

${\displaystyle -vMv^{-1}}$

and are of even grade. Under a rotation generated by the rotor R, a general multivector M will transform double-sidedly as

${\displaystyle RMR^{-1}.}$

Restricted alternative formulation

For a Euclidean space, it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of a unit (i.e. normalized) multivector:

${\displaystyle -vMv,\quad v^{2}=1,}$

forming rotors that are automatically normalised:

${\displaystyle R{\tilde {R}}={\tilde {R}}R=1.}$

The derived rotor action is then expressed as a sandwich product with the reverse:

${\displaystyle RM{\tilde {R}}}$

For a reflection for which the associated vector squares to a negative scalar, as may be the case with a pseudo-Euclidean space, such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortcoming.

Rotations of multivectors and spinors

However, though as multivectors rotors also transform double-sidedly, rotors can be combined and form a group, and so multiple rotors compose single-sidedly. The alternative formulation above is not self-normalizing and motivates the definition of spinor in geometric algebra as an object that transforms single-sidedly – i.e. spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product.

Homogeneous representation algebras

In homogeneous representation algebras such as conformal geometric algebra, a rotor in the representation space corresponds to a rotation about an arbitrary point, a translation or possibly another transformation in the base space.

See also

• Double rotation
• Lie group
• Euler's formula
• Generator (mathematics)

References

1. Doran, Chris; Lasenby, Anthony (2007). Geometric Algebra for Physicists. Cambridge, England: Cambridge University Press. p. 592. ISBN 9780521715959.
2. Clifford, William Kingdon (1878). "Applications of Grassmann's Extensive Algebra". American Journal of Mathematics. 1 (4): 353. doi:10.2307/2369379. JSTOR 2369379.
3. Grassmann, Hermann (1862). Die Ausdehnugslehre (second ed.). Berlin: T. C. F. Enslin. p. 400.
4. Hestenes, David (1987). Clifford algebra to geometric calculus (paperback ed.). Dordrecht, Holland: D. Reidel. p. 105. Hestenes uses the notation ${\displaystyle R^{\dagger }}$ for the reverse.