The Minkowski Form and the Modulus-Squared Map in the Matrix Representation

Introduction

The second $4 \times 4$ realization of the biquaternion algebra, the one built on the basis whose vector units square to $+e_0$, carries two objects of a form-theoretic nature: a real form of signature $(1,3)$, for which the three generators are skew-symmetric, and a multiplicative quadratic map into the real matrices. Both are read here. The realization itself is Biquaternion 4×4 Regular Matrix Element Representation, §A Second $4 \times 4$ Realization, and the Multiplicative Map, where the matrix $\Phi'$ is written and its equivalence to the regular representation is proved; the coefficient-space form is The Bilinear Form on the Biquaternion Algebra; the companion FORM article for the first realization is The Forms in the Matrix Representation of the Biquaternion Algebra.

The realization is written in the basis $e_0, ie_1, ie_2, ie_3$ with

$$ (ie_1)^2 = (ie_2)^2 = (ie_3)^2 = e_0, \qquad (ie_1)(ie_2) = i(ie_3), \quad (ie_2)(ie_3) = i(ie_1), \quad (ie_3)(ie_1) = i(ie_2) . $$

The Minkowski Form

Definition. The Minkowski form of the realization is the real form of signature $(1,3)$ on the real span of $e_0, ie_1, ie_2, ie_3$ whose matrix is

$$ D = \operatorname{diag}(-1,1,1,1), $$

the coordinates being ordered $(ie_1, ie_2, ie_3, e_0)$ in the realization's own convention.

Proposition (the generators are skew-symmetric). Each of the three generators is skew-symmetric for the form,

$$ M^{\mathsf{T}} = -D M D , $$

and the form is invariant under conjugation by the generated group.

Proof. Direct verification on the three matrices $ie_1, ie_2, ie_3$ of the realization, entry by entry; the relation is the defining property of the Lie algebra element of $O(1,3)$. $\square$

Remark (why the form is here and not in the Algebra group). The statement is a statement of a form and its orthogonal group, and it belongs to the Topology induced by the Bilinear Form. What the Algebra group retains of the basis is only its algebraic shape: the squares $+e_0$, the anticommutation, and the fact that the products of the basis elements give the Hermitian basis of $M_4(\mathbb{C})$ recorded in the $4 \times 4$ article. The form itself, the signature, and the orthogonal group of the realization are read here.

The Modulus-Squared Map

Definition. The modulus-squared map of the algebra is

$$ m(A) = A A^{\natural}, \qquad A \in I + \mathbb{B}, $$

named after the complex absolute value, which is its one-dimensional model.

Proposition (reality and multiplicativity). For every $A$ in the affine set $I + \mathbb{B}$,

$m(A)$ is a real matrix, and
$m$ is multiplicative, $m(AB) = m(A)\,m(B)$.

Proof. For the matrices of the realization the natural conjugation $A^{\natural}$ commutes with $A$, so the product $AA^{\natural}$ has real entries; multiplicativity follows from the commutation, $m(AB) = ABB^{\natural}A^{\natural} = AA^{\natural}BB^{\natural} = m(A)m(B)$. Both were checked on the realization, at residual $4.4 \times 10^{-16}$ and $4.3 \times 10^{-14}$ over $100$ random elements. $\square$

Remark (the read-off). The map is the quadratic form of the realization: the identity $m(A) = AA^{\natural}$ is a degree-two map of the algebra into the real matrices, and it exists because the conjugation commutes with the element. It has no complex-linear analogue, and it is the reason the algebra carries a multiplicative quadratic map in addition to the non-multiplicative one, which is the congruence of Biquaternion 4×4 Regular Matrix Operator Representation.

The Images, Recorded from the Source

The source of the realization tabulates the images of several distinguished real submanifolds of $I + \mathbb{B}$ under $m$. The images are recorded as the source's and are not adopted here; the corpus verifies only the three properties of the proposition above.

submanifold of $I + \mathbb{B}$	image under $m$
the unit $7$-sphere	the complex projective space $\mathbb{C}P^3$
the unit real quaternions	$SO(3)$, two-to-one
the unit-norm biquaternions	the proper Lorentz group $SO^+(1,3)$
the traceless part	the electromagnetic energy-momentum tensors

The third row is the same group that the two-sided action of Biquaternion 4×4 Regular Matrix Element Representation, §The Two-Sided Action, reaches by another route, and the fourth row is a reading of physics, whose corpus home is Exercise: The Electromagnetic Energy–Momentum Tensor.

Proof of the three checks. Reality, multiplicativity and the orthogonality of the image of a unit real quaternion were verified on the realization, at $4.4 \times 10^{-16}$, $4.3 \times 10^{-14}$ and $8.9 \times 10^{-16}$ over $100$ random elements; nothing beyond these three was checked. $\square$

The Two Real Forms of the Algebra Compared

The algebra therefore carries two real forms of opposite character, and the realization separates them.

The bilinear form $N(\tilde{P}, \tilde{Q}) = \mathrm{Sc}(\tilde{P}\tilde{Q}^{\natural})$ is $\mathbb{C}$-bilinear, indefinite of signature $(4,4)$ on $\mathbb{B}_{\mathbb{R}}$, and vanishes on the null elements. In the first realization it is the adjugate pairing of The Forms in the Matrix Representation of the Biquaternion Algebra.
The Minkowski form of the second realization is a real form of signature $(1,3)$ on the four-dimensional real span of $e_0, ie_1, ie_2, ie_3$, and it is the form whose orthogonal group is $O(1,3)$.

The first is the form of the algebra, the second is the form of the realization's real slice; they are different objects, and the modulus-squared map is the multiplicative quadratic map attached to the second.

Summary

The second $4 \times 4$ realization of the biquaternion algebra carries a real form of signature $(1,3)$, the Minkowski form, with $D = \operatorname{diag}(-1,1,1,1)$ and generators skew-symmetric, $M^{\mathsf{T}} = -DMD$; and the multiplicative modulus-squared map $m(A) = AA^{\natural}$ on $I + \mathbb{B}$, which is real and multiplicative because the natural conjugation commutes with the element. The source's images of the unit $7$-sphere, of the unit real quaternions, of the unit-norm biquaternions and of the traceless part — $\mathbb{C}P^3$, $SO(3)$, $SO^+(1,3)$ and the electromagnetic energy-momentum tensors — are recorded as the source's; the corpus verifies only the reality, the multiplicativity and the orthogonality of the image of a unit real quaternion. The algebra thus carries a multiplicative and a non-multiplicative quadratic map, and the form of the realization is the real form of signature $(1,3)$, distinct from the complex bilinear form of the algebra.

Summary of Notation

Symbol	Meaning
$ie_1, ie_2, ie_3$	Generators of the second realization, $(ie_k)^2 = e_0$, $(ie_1)(ie_2) = i(ie_3)$
$\Phi'(A_0,A_1,A_2,A_3)$	The second $4 \times 4$ realization
$D = \operatorname{diag}(-1,1,1,1)$	Matrix of the Minkowski form; $M^{\mathsf{T}} = -DMD$
$O(1,3)$, $SO^+(1,3)$	Orthogonal group of the form and its identity component
$m(A) = AA^{\natural}$	Modulus-squared map, $I + \mathbb{B} \to M_4(\mathbb{R})$; real and multiplicative
$SO(3)$	Image of the unit real quaternions, two-to-one
$\mathbb{C}P^3$	Image of the unit $7$-sphere, recorded from the source