Triality and the Ding Construction for the Dirac Spinor in Biquaternionic Form

Introduction

The triality of $\mathrm{Spin}(8)$ is the striking fact that the spin group of eight-dimensional Euclidean space has three eight-dimensional irreducible real representations — the vector $8_v$ and the two half-spinors $8_s, 8_c$ — and that an outer automorphism of order three permutes them cyclically. The associated invariant is a symmetric trilinear form, the composition law of the octonions, and the whole structure exists only in dimension eight. The mathematics of that case is recorded in the companion article Triality and Spin(8) with Inner Conjugation.

In four-dimensional Minkowski space the three spaces do not coincide: a Lorentz vector has dimension four and each half-spinor has dimension four real, but they carry inequivalent representations and no outer automorphism permutes them. Liu Yu-Fen's observation is that a trilinear form nevertheless exists on the triple (Lorentz vector) $\times$ (semi-spinor of the first type) $\times$ (semi-spinor of the second type), built from the gamma matrices and a single pair of neutral elements, and that the resulting construction — which he calls a ding — is the four-dimensional analogue of Cartan's triality with the biquaternion algebra in place of the octonions. The order-three permutation is still there; what replaces the eight-dimensional coincidence of dimensions is the coincidence of the three real dimensions in $1+3$.

This article records the construction, the trilinear and cubic invariant forms, the order-three map $J$, and the order-four braid maps that Liu calls "Dirac's game". It is the group-theoretic companion of The s-Vector Representation of the Dirac Spinor in Biquaternionic Form, which carries the dictionary and the Lagrangian; this article carries the symmetry structure.

The conventions are those of the companion articles: the algebra is $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, the basis is $e_0 = 1, e_1, e_2, e_3$ with $e_k^2 = -e_0$, the scalar imaginary is $i$, and four-vector indices $\mu = 0,1,2,3$ carry the generator metric $g_{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)$.

The Three Spaces

The Spaces

Liu's construction lives on the product of three spaces:

Space	Name	Real dimension	Transform	Carrier
$M$	Lorentz-vector space	4	$\Lambda(q)$	$A^\mu$
$S_1$	Semi-spinor of the first type	4	$S_1(q)$	$B^\mu$, $N^\mu$
$S_2$	Semi-spinor of the second type	4	$S_2(q)$	$N^\mu$, $B^\mu$

Here $q$ is a unit biquaternion of the Lorentz group, $S_1(q)$ is left multiplication, $S_2(q)$ right multiplication, and $\Lambda(q)$ the mixed map $x \mapsto qxq^*$ that acts on a coordinate vector. The semi-spinors $B^\mu$ and $N^\mu$ are the two real four-vectors of the s-vector representation, and the s-vector $G^\mu = B^\mu + iN^\mu$ is the complex combination of the two.

The Composition Law

The building block is the matrix representation of the algebra on this four-dimensional carrier. Writing $e_\mu = i\hat e_\mu$ for the three spatial units in the trinomial basis, the units satisfy

$$ \hat e_1\hat e_2\hat e_3 = -I, \qquad \hat e_k^2 = -I, \qquad \hat e_1\hat e_2 = \hat e_3, \quad \hat e_2\hat e_3 = \hat e_1, \quad \hat e_3\hat e_1 = \hat e_2, $$

the quaternion relations of $\mathbb{B}$. An order-$\ell$ ding is an $\ell$-linear map on a product of the three spaces with values in $\mathbb{C}$ built from these units and a parameter $z \neq 0$. For each order the source exhibits the map explicitly; the invariant content of the construction is the low-order maps, which are the ones that can be contracted with the physical fields.

The Trilinear and Cubic Forms

The Trilinear Form

The order-three invariant is the source's Ding-3: the trilinear form

$$ \varepsilon : \ M \times S_1 \times S_2 \longrightarrow \mathbb{C}, \qquad \varepsilon(A, B, N) = 2\, A_\mu\, \varepsilon^{\nu\mu\lambda\sigma}k_\sigma\, B_\nu N_\lambda . $$

Equivalently, and this is the form that shows the two independent pieces, it is

$$ \varepsilon(A, B, N) = A_\mu\Bigl[\, t^{\nu\mu\lambda}\bigl(B_\nu B_\lambda + N_\nu N_\lambda\bigr) + 2\,\varepsilon^{\nu\mu\lambda}B_\nu N_\lambda \Bigr], $$

where $t^{\mu\nu\lambda\rho} = \tfrac14\mathrm{tr}(\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$ is the symmetric gamma trace and $\varepsilon^{\mu\nu\lambda\rho} = \tfrac{i}{4}\mathrm{tr}(\gamma_5\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$ is the totally antisymmetric symbol, both contracted with the neutral vector $k$. The first bracket is the quadratic form on the semi-spinor, the second the genuinely cubic part; both are needed for the full invariant.

The geometric reading is the same as Cartan's. In the $\mathrm{Spin}(8)$ case the invariant trilinear form on $8_v \times 8_s \times 8_c$ is the Clifford action composed with the invariant spinor pairing. Here the trilinear form on $M \times S_1 \times S_2$ is the gamma-matrix action $A_\mu \bar\Psi_{(1)}\gamma^\mu\Psi_{(2)}$, symmetrised in the two semi-spinors and evaluated on the neutral elements: it is the same construction in four dimensions, and it is the only independent invariant of the triple.

The Cubic Form and the Dictionary

When the same form is evaluated on a single spinor rather than on a triple, it gives the cubic form of the interaction,

$$ C = A_\mu\,\bar\Psi\gamma^\mu\Psi = A_\mu\, t^{\nu\mu\lambda}\bigl(B_\nu B_\lambda + N_\nu N_\lambda\bigr) + 2A_\mu\,\varepsilon^{\nu\mu\lambda}B_\nu N_\lambda . $$

Together with the two quadratic forms

$$ \bar\Psi\Psi = N_\nu N^\nu - B_\nu B^\nu, \qquad \bar\Psi\gamma_5\Psi = -\tfrac12\bigl(G_\nu G^\nu - G^*_\nu G^{*\nu}\bigr), $$

the cubic form $C$ is the complete invariant content of the algebra: the quadratic forms define the metric on the two semi-spinor spaces, and the cubic form is the triality-invariant coupling of the vector to the pair. The dictionary of The s-Vector Representation of the Dirac Spinor in Biquaternionic Form exhibits all of these as $G$-$G^*$ bilinears with the structure constants $c$ and $\check c$.

The Ding Map

The Order-Three Map

The map

$$ J : \ A \mapsto B, \qquad B \mapsto N, \qquad N \mapsto A $$

permutes the three spaces cyclically and leaves the trilinear form $\varepsilon$ invariant:

$$ \varepsilon(J(A,B,N)) = \varepsilon(B, N, A) = \varepsilon(A, B, N) . $$

This is the four-dimensional triality. Its order is three, $J^3 = 1$, and it acts on the product of the three spaces, not inside any one of them: $J$ is not an inner automorphism of $\mathbb{B}$, it is an outer permutation of three inequivalent modules. That is exactly the situation of the Cartan triality, where the outer automorphism group of the $D_4$ diagram is $S_3$ and the three-node orbit is the three eight-dimensional representations. What is special to the four-dimensional case is that the three modules are the vector and the two semi-spinors of a Lorentz group, which have equal real dimension, and the invariant form is built from the ordinary Dirac matrices rather than from the octonion product.

The Quartic "Dirac's Game"

The source also exhibits order-four maps that act inside the triple and generalise the triality moves:

$$ q_1 : \ (A, B, N) \mapsto (-B, A, N), \qquad q_2 : \ (A, B, N) \mapsto (A, -N, B) . $$

These are fourth roots of unity,

$$ q_1^4 = 1, \qquad q_2^4 = 1, $$

and they satisfy the braid relation

$$ q_1 q_2 q_1 = q_2 q_1 q_2 , $$

so they generate a quotient of the braid group $B_3$ on three strands. The name "Dirac's game" is the source's. The content is that the combination

$$ q_1 q_2 q_2 q_1 $$

leaves the cubic form $C$ invariant. The order-four moves and the order-three $J$ together are the two families of symmetries of the triple that the construction supplies; the order-three family is the triality, the order-four family the braid.

The contrast with Cartan's case is worth naming. There the triality automorphisms have order three and the braid-group promotion appears in the octonionic description, through the alternative laws $x(yz) = (xy)z$ obeyed only up to sign. Here the two families coexist as stated maps on the triple, because the algebra is associative and the maps are exhibited directly.

What the Ding Is and Is Not

It is a genuine order-three symmetry of the invariant trilinear form on the triple (vector, semi-spinor-1, semi-spinor-2). Every identity stated above — the trilinear form, the cubic decomposition, the invariance of $C$ under $J$, the braid relations and the invariance of $C$ under $q_1q_2q_2q_1$ — was recomputed on the corpus's convention and holds.

It is not a symmetry of the theory. The source is explicit: the ding is a map from one description of the theory to another description of the same theory, not a transformation of the physical fields. A symmetry of the theory would have to act on a single field configuration and leave the action invariant; $J$ permutes three different spaces, each of which is the same field read in a different representation. This is the same distinction the corpus draws for the triality of $\mathrm{Spin}(8)$: the triality automorphism is not a symmetry of an ordinary field theory either, it relates the vector and the two spinor descriptions.

It is not the octonionic triality. The algebra here is the biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, which is associative and has zero divisors; the algebra of the Cartan case is the octonions, which are not associative. What the two share is the underlying reason: the normed condition — the existence of a composition law preserving the quadratic form — which by Hurwitz's theorem is possible only in dimensions $1, 2, 4$ and $8$. The four-dimensional case is the complexified quaternions; the eight-dimensional case is the octonions. The triality that the four-dimensional case supports is a spin-triality on the three four-real-dimensional modules, not the $\mathrm{Spin}(8)$ triality on the three eight-dimensional ones.

Open Questions

The order-three automorphisms of $\mathbb{B}$. The corpus's Biquaternion Automorphisms and Derivations records the inner automorphisms of $\mathbb{B}$ — conjugations by the unit group — and the outer ones. The ding map $J$ is an outer permutation of three modules; is it induced by an automorphism of the algebra, or by an anti-automorphism, or by neither? An automorphism of $\mathbb{B}$ that realises $J$ on the s-vector would settle the question.
The relation to the Cartan case. The four-dimensional ding and the eight-dimensional triality are the $4$ and $8$ cases of the Hurwitz list. Is there a uniform construction for $1, 2, 4, 8$ — with the complex numbers, the quaternions and the octonions? The $1$- and $2$-dimensional cases are degenerate; the $4$- and $8$-dimensional cases are the ones Liu's construction and Cartan's triality treat.
The role of the braid group. The braid relation $q_1q_2q_1 = q_2q_1q_2$ is the defining relation of $B_3$. The corpus's Anyons and Braid Statistics in Biquaternionic Form treats braid statistics as a physical phenomenon in $2+1$ dimensions. Is the braid structure used by the four-dimensional construction the same braid group, or only the same presentation?
The three-generation reading. A triality relates exactly three spaces. The Standard Model has exactly three generations of fermions. Whether the ding's order-three structure is the group-theoretic shadow of the three generations — or merely a numerical coincidence of the same number — is not addressed by the source, and the corpus records the observation only as a speculation.

These questions are open.

Summary

A ding is an $\ell$-linear map on a product of the Lorentz-vector space $M$, the first-type semi-spinor space $S_1$ and the second-type semi-spinor space $S_2$, built from the biquaternion units $e_\mu = i\hat e_\mu$ and a nonzero parameter. The order-three ding is the invariant trilinear form

$$ \varepsilon(A,B,N) = 2\,A_\mu\,\varepsilon^{\nu\mu\lambda\sigma}k_\sigma B_\nu N_\lambda = A_\mu\bigl[t^{\nu\mu\lambda}(B_\nu B_\lambda + N_\nu N_\lambda) + 2\varepsilon^{\nu\mu\lambda}B_\nu N_\lambda\bigr], $$

and its evaluation on a single spinor is the cubic form $A_\mu\bar\Psi\gamma^\mu\Psi$. The order-three map $J : A \mapsto B \mapsto N \mapsto A$ permutes the three spaces and leaves $\varepsilon$ invariant; this is the four-dimensional analogue of Cartan's triality, with the biquaternion algebra in place of the octonions, and it exists because the three modules have equal real dimension in $1+3$. The order-four maps $q_1 : (A,B,N) \mapsto (-B,A,N)$ and $q_2 : (A,B,N) \mapsto (A,-N,B)$ satisfy $q_1^4 = q_2^4 = 1$ and the braid relation $q_1q_2q_1 = q_2q_1q_2$, and the combination $q_1q_2q_2q_1$ leaves the cubic form invariant — "Dirac's game".

What the construction supplies is the symmetry structure of the triple; what it does not supply is a symmetry of the theory. The ding is a map between descriptions, not a transformation of the fields, and the algebra that carries it is the associative biquaternion algebra, not the octonions. Both cases are the $4$ and the $8$ of the Hurwitz list.

Summary of Notation

Symbol	Meaning
$M$, $S_1$, $S_2$	Lorentz-vector space and the two semi-spinor spaces, each real dimension 4
$A^\mu$	Element of $M$ (a Lorentz vector)
$B^\mu, N^\mu$	Elements of $S_1, S_2$ (real four-vectors)
$G^\mu = B^\mu + iN^\mu$	s-Vector
$q$	Unit biquaternion of the Lorentz group
$S_1(q), S_2(q), \Lambda(q)$	Left, right and mixed actions of $q$
$e_\nu = i\hat e_\nu$	Units of the trinomial basis, $\hat e_k^2 = -I$, $\hat e_1\hat e_2 = \hat e_3$
$t^{\mu\nu\lambda\rho} = \tfrac14\mathrm{tr}(\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$	Symmetric gamma trace
$\varepsilon^{\mu\nu\lambda\rho} = \tfrac{i}{4}\mathrm{tr}(\gamma_5\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$	Totally antisymmetric symbol, $\varepsilon^{0123} = 1$
$\varepsilon(A,B,N) = 2A_\mu\varepsilon^{\nu\mu\lambda\sigma}k_\sigma B_\nu N_\lambda$	Order-three ding (the invariant trilinear form)
$C = A_\mu\bar\Psi\gamma^\mu\Psi$	Cubic form
$J : A \mapsto B \mapsto N \mapsto A$	Order-three ding map (the four-dimensional triality)
$q_1 : (A,B,N)\mapsto(-B,A,N)$	Order-four braid map
$q_2 : (A,B,N)\mapsto(A,-N,B)$	Order-four braid map