Biquaternion

In Mathematical, a biquaternion (or quaternion complexes ) is an element of the Algèbre of the quaternions on the complex numbers. The concept of a biquaternion was mentioned the first time by William Rowan Hamilton with the nineteenth century. William Kingdon Clifford used the same name in connection with a different algebra.

Definition

Either

{1, I, J, K} \,

, the base for the Quaternion S (realities), and or

u, v, W, X \,

of the complex numbers, then

q = U 1 + v I + W J + X K \,

is a biquaternion . The complex scalars are supposed to commutate with the vectors of the base of the quaternions (c.a.d. vj = jv ). While operating judiciously with the addition and the multiplication, in agreement with the group of the quaternions, this collection forms a algebra with 4 dimensions on the complex numbers. The algebra of biquaternions is associative, but not commutative.

The algebra of biquaternions can be regarded as a produces tensorial $\ mathbb {C} \ otimes \ mathbb {H} \,$ where $\ mathbb {C}$ is the body complex numbers and $\ mathbb {H} \,$ is the algebra of the real quaternions.

Place in the theory of the rings

Linear representation

Note that the matric product

\ begin {pmatrix} I & 0 \ \ 0 & - I \ end {pmatrix} \ begin {pmatrix} 0 & 1 \ \ - 1 & 0 \ end {pmatrix}

\ begin {pmatrix} 0 & I \ \ I & 0 \ end {pmatrix}

where each one of these matrices has a square equal to negative of the Matrice identity. When the matric product is interpreted like

i~j = K \,

, one then obtains to a Sous-groupe group of the matrices which is isomorphous with the Groupe of quaternions. Consequently,

\ begin {pmatrix} u+iv & w+ix \ \ - w+ix & u-iv \ end {pmatrix}

represent the biquaternion Q .

Being given a matrix 2x2 complexes unspecified, it exists complex values U , v , W and X to turn it in this form, i.e. the Anneau of the matrices is isomorphous with the ring biquaternions.

Alternative complex plan

Let us suppose that we take W purely imaginary,

w = b~ \ iota \,

, where where

\ iota~ \ iota = - 1 \,

. (Here, one uses

\ iota \,

in the place of I for the imaginary complex to distinguish it from quaternion I). Now, when R = W J , then its square is

r~r = (W J) (W J) = (W W) (J J) = B B (- 1) (- 1) = b^2 \,

. In particular, when B = 1 or - 1, then

r^2 = + 1 \,

. This development shows that biquaternions them are a source of " engines algébriques" as R which high with the cross-section gives +1. Then

{has + b~ \ iota~j: has, B \ in \ mathbb {R}} \,

is a Sous-anneau biquaternions isomorphous with the ring of the split complex numbers.

Application in relativistic physics

The equation of Dirac allows a modeling of the change of Spin of the electron and the introduction of the Positron by a new theory of the orbital kinetic Moment

Presentation of the group of Lorentz

Biquaternions

\ iota~k = \ sigma_1 \,

\ iota~j = \ sigma_2 \,

and

-~ \ iota~i = \ sigma_3 \,

were used by Alexander MacFarlane and later, in their matric form by Wolfgang Pauli. They were known under the name of matrices of Pauli. They are each one the square of the matrix identity and consequently the under-plan

{has + b~ \ sigma; has, B \ in \ mathbb {R}}

generated by one of them in the ring of biquaternions is isomorphous with the ring of the split complex numbers. Consequently, a matrix of Pauli

\ sigma \,

generates a Groupe with a parameter

{U: U = 
exp (\ sigma has), has \ in \ mathbb {R}}

whose actions on the under-plan are hyperbolic rotations. The Groupe of Lorentz is a Groupe of Dregs to six parameters, three parameters (c.a.d. sub-groups generated by the matrices of Pauli) are associated with hyperbolic rotations, sometimes called " boosts". The three other parameters correspond to ordinary rotations in space, a structure of the real quaternions known under the name Quaternions and rotations space. The usual sight by a quadratic Forme of this presentation is that

u^2 + v^2 + w^2 + x^2 = q~q^* \,

is preserved by the orthogonal Groupe on biquaternions when he is seen like

\ mathbb {C} ^4 \,

. When U is real and v, W and X are the imaginary pure ones, then one obtains the subspace

M = \ mathbb {R} ^4 \,

which is appropriate to model the space time.

See too

Biquaternions de Clifford
Octonions conical (isomorphism)

References

Cornelius Lanczos (1949) The Variational Principles off Mechanics , University off Toronto Close, pp. 304-12.
Silberstein, L. (May 1912) Quaternionic form off relativity , Philosophy Magazine , series 6, 23 : 790-809.
Silberstein, L. (1914) .
Synge, J.L. (1972) Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices Communications off the Dublin Institute for Advanced Studies, series has, #21, 67 pages.
Kilmister, C.W. (1994) Eddington' S search for has fundamental theory , Cambridge University Close 0-521-37165-1, pages 121,122,179,180.

Random links: