The pages previously dedicated to explaining what quaternions are have, without clearly stating it, dealt with the historical version of these numbers.
They have a particularity: their components are real numbers.
Some works discussing the subject consider quaternions with complex components.
Is it possible to bring this concept to life? If so: under what conditions and how? That is what I am exploring here. The subject turns out to be much more complicated than intuition suggests.
This is a good question to which several answers can be given.
First, for the natural and simple reason that mathematics loves to try to generalize and extrapolate concepts that work on a given set to larger sets encompassing the previous one.
Secondly, because real quaternions induce questioning whose answers depend on the domain of belonging of their components.
There is a relatively common illustration of these questions: the characterization of the nullity of the square of a quaternion with real components. this square is generally equal to:
q2
= [q0, < h, q >]2
= [(q0)2 - < q, q >, 2.q0. <h, q >]
For it to be zero, it is necessary and sufficient that its real part and its imaginary part are simultaneously zero; which translates to:
(q0)2 - < q, q > = 0(R)
2. q0. q = 0
The nullity of the imaginary part is achieved in only two cases.
First possibility: this quaternion is a pure real number. It amounts to saying that its vector part is null:
q = 0
It follows that the quaternion whose square is studied and assumed to be null is equal to:
q = q0 ∈ R
The square of this real number can only be zero if the real part of the quaternion under consideration is zero.
{ q ∈ R et q^2 = 0(R) } ⇒ q = 0(R) = 0(H)
Second possibility, this quaternion is purely imaginary:
q ∈ Im(H).
It amounts to the same thing as saying that its real part is zero:
q0 = 0(R).
The quaternion q whose square is studied can therefore only be null if the Euclidean scalar product of its vector part q by itself is zero.
This is the exact point where the issue of the domain to which the components of this quaternion belong appears.
Indeed, from the moment these components are real numbers, the square of the studied quaternion can only be null if its vector part is also null.
{ q0 = 0(R), q ∈ E(3, R), q2 = 0(H) }
⇓
< q, q > = 0(R)
⇓
q = 0
⇓
q = 0(H)
The conclusion remains identical to the one resulting from the first possibility, even though the logical path leading to it differs.
The square of the zero quaternion with real components is equal to itself; it is zero.
Conversely, if the square of a quaternion with real components is zero, then that quaternion was the zero quaternion.
The lemma no longer applies to quaternions whose components belong to the commutative field of complex numbers C. Why?
Answer
Assuming a priori that this type of quaternions exists (which is not certain), they are intuitively written as:
q = a + b.i + c.j + d.k a, b, c, d ∈ C
The relations characterizing the nullity of the square of such a quaternion remain formally identical to those written above for a classical quaternion (synonym: with real components).
Here again, the two possibilities analyzed previously should be considered. The first leads to the same conclusion. This is not the case for the second because:
{q0 = 0(C), q ∈ E(3, C), q2 = 0(H)}
⇓
< q, q > = 0(C)
⇓
q ∈ SpinE(3, C)
The new domain of definition of the studied quaternion components brings this exploration squarely into the universe of three-dimensional space spinors. The theory was developed in a pioneering way by E. Cartan during the first quarter of the twentieth century [01].
The square of the zero quaternion with complex components is equal to itself: it is zero.
On the other hand, the converse of this statement is false because non-zero quaternions with complex components whose square is the zero quaternion seem to be able to exist.
Is this really the case? Does the intuitive formulation of quaternions with complex components make sense?
The apparent simplicity with which the previous questioning found an answer hides a huge problem: that of a coherent and unified definition of the notion of complex-component quaternions; why?
It is known that quaternions with real components can be interpreted as double complex numbers (bicomplex numbers) since, thanks to their multiplication table, they can always be written as:
q = a + b.i + c.j + d.k = (a + b.i) + j.(c – i.d)
But they could just as easily be written:
q = a + b.i + c.j + d.k = (a + c.j) + k.(d – j.b)
Etc. Therefore, there are multiple ways to decompose a given real-component quaternion into a pair of complex numbers.
The situation worsens when considering (for the moment virtually) the existence of quaternions with complex components. The first fundamental difficulty stems from the fact that the theory of quaternions with real components introduces three distinct pure imaginary numbers (i.e. i, j, k) all of which have a square equal to minus one. Because of this, it theoretically becomes possible to create three types of complex numbers:
z1 = a1 + i.b1 ∈ C(i), a1, b1 ∈ R
z2 = a2 + j.b2 ∈ C(j), a2, b2 ∈ R
z3 = a3 + k.b3 ∈ C(k), a3, b3 ∈ R
The question of "how to construct quaternions with complex components?" arises and must therefore be studied in this context.
As I have already pointed out above, an intuitive extrapolation would be to naively write:
q = a + b.i + c.j + d.k a, b, c, d ∈ C
But that would be ignoring the coexistence of the three images of the set of complex numbers: C(i), C(j) and C(k)! To write that the components a, b, c and d are complex numbers therefore means nothing sufficiently precise. Writing that directly and naively extrapolates the formalism of quaternions with real components makes no sense.
As a consequence, the reason that justified considering quaternions with complex components is weakened (see above the question of the characterization of quaternions whose square is zero.
It is therefore necessary to find another way to introduce complex numbers into a discussion initially focused on the study of the historical version of quaternions.
One choice among infinitely many others consists in considering the set of real and linear combinations of these three types of complex numbers:
q = c1. z1 + c2. z2 + c3. z3 c1, c2, c3 ∈ R
TAny combination of this type is a quaternion with real components; indeed, it is equal to:
q = (c1. a1 + c2. a2 + c3. a3) + i. c1. b1 + j. c2. b2 + k. c3. b3
This procedure tacitly implies three vectors of a three-dimensional real vector space:
a : (a1, a2, a3) ∈ E(3, R)
b : (b1, b2, b3) ∈ E(3, R)
c : (c1, c2, c3) ∈ E(3, R)
... and the resulting real quaternion can be written concisely as:
q = (< c, a >, < h, c • b >)
Where "•" designates the operation of sliding one vector over the other.
Knowing a triplet of complex numbers, each of which is of a different type, is equivalent to knowing three vectors a, b, c of a three-dimensional real vector space E(3, R).
Each linear combination of this kind of triplets always makes it possible to construct a quaternion with real components.
Strictly speaking, the construction in question does not define quaternions with complex components, but it establishes a correspondence between quaternions with real components and triplets of complex numbers.
(z1, z2, z3) ∈ C(i) x C(j) x C(k) ⊂ E(3, C)
↓↑
(a, b, c) ∈ E^3(3, R)
⇓
∃ q
=
c1. z1 + c2. z2 + c3. z3
=
(< c, a >, < h, c • b >)
∈ H
The real quaternion resulting from a linear combination of the kind set forth above is a pure imaginary when the vectors c and a are perpendicular:
c ⊥ a ⇔ < c, a > = 0
⇓
q
=
c1. z1 + c2. z2 + c3. z3
=
(0, < h, c • b >)
∈ Im(H)
It is the subject discussed earlier on this page that sparked the interest in quaternions with complex components.
As the tri-complex numbers constructed in the previous section are only a particular subset of classical quaternions, the lemma applies to them.
Only the zero quaternion has a zero square and the discussion on the zero square is closed.
But it indirectly opens another one, raising the question of counting the elements (a, b, c) of the vector space e^3(3, r) whose representation is the classical zero quaternion.
Now, an element of this type of quaternion set is zero whenever these three vectors are linked by the two relations:
c • b = 0 et < c, a > = 0
The first contains the eight situations:
c = 0 ∀ b
c : (0, 0, c3) b : (b1, b2, 0)
c : (0, c2, 0) b : (b1, 0, b3)
c : (c1, 0, 0) b : (0, b2, b3)
c : (0, c2, c3) b : (b1, 0, 0)
c : (c1, 0, c3) b : (0, b2, 0)
c : (c1, c2, 0) b : (0, 0, b3)
∀ c b = 0
They share the common feature of inducing the orthogonality of the arguments of the pair of vectors (c, b):
c • b = 0 ⇒ < c, b > = 0
The concomitance with the second orthogonality relation between c and a as well as the way in which the orthogonality of the pair (c, b) is achieved induce in particular the existence of a real number φ such that:
∃ φ ∈ R : b = φ. a
Finally, the sought-after vector triplets have a very specific formalism:
(a, φ. a, c) : c ⊥ a, φ ∈ R
It is possible to deduce the complex number triplets associated with the standard null quaternion.