Coloured Dirac Spinor in Gauge Theories

• Coloured Dirac spinor is a fermionic field with an explicit SU(3) colour degree that generalizes the conventional Dirac formalism for quantum chromodynamics.
• It employs algebraic, quaternionic, and geometric constructions with Z3 grading and extended gamma matrices to seamlessly encode spin and colour degrees of freedom.
• Its coupling with SU(3) gauge fields leads to composite colour-singlet states, providing an algebraic foundation for quark confinement in the Standard Model.

A coloured Dirac spinor is a fermionic field endowed with an explicit SU(3) colour degree of freedom, generalizing the standard Dirac spinor formalism to accommodate quantum chromodynamics (QCD) and related models of elementary particles. The construction of such spinors varies across algebraic, geometric, and quaternionic frameworks, but consistently aims to encode the triplet colour structure intrinsic to the strong interaction and enable a rigorous treatment of gauge invariance, confinement, and symmetry embedding.

1. Algebraic Structure of Coloured Spinors

In the Kerner–Lukierski approach (Kerner et al., 2019), each quark colour (red, green, blue) and its antiparticle is represented by a Pauli doublet, leading to the introduction of a 12-component master spinor: $$\Psi = \begin{pmatrix} \phi^+ \ \phi^- \ \chi^+ \ \chi^- \ \psi^+ \ \psi^- \end{pmatrix} \equiv (\phi^+, \phi^-, \chi^+, \chi^-, \psi^+, \psi^-)^T \in \mathbb{C}^{12}$$ where $\phi, \chi, \psi \in \mathbb{C}^2$. The algebraic enhancement employs $Z_3$ grading, with the group $Z_3 = \{1, j, j^2\}$ cyclically permuting colour and acting nontrivially on mass parameters. Essential “colour matrices” such as $B = \text{diag}(1, j, j^2)$, $Q_3$, and $Q_2 = BQ_3$ operate in the colour space, and extended “gamma-matrices” ($\Gamma^\mu$, $12\times12$) replace the conventional Dirac $\gamma^\mu$, entangling spin and colour. The resulting algebra is not the Clifford algebra ($\{\gamma^\mu, \gamma^\nu\}$), but a $Z_3 \times Z_2$-graded generalization: $$(\Gamma^\mu p_\mu)^6 = (p^6 - |\mathbf{p}|^6)\mathbf{1}_{12}$$ This algebraic structure supports a direct connection between colour and space-time symmetry, foundational for the scheme's subsequent physical interpretations.

2. Interaction with Gauge Fields and Confinement

The coloured Dirac equation in this framework takes the form: $\Gamma^\mu p_\mu \Psi = m \Psi$ When coupled to an SU(3) gauge field $A^a_\mu$, the construction minimally augments the momentum via

$p_\mu \rightarrow \Pi_\mu = p_\mu - g A^a_\mu I_a$

Here, $I_a$ are SU(3) generators in the colour basis, which are in fact realized by combinations of the colour phase matrices ($B, Q_2, Q_3$), as these exhaust SU(3) alongside two other diagonal traceless generators. Consequently, the full coupled system is: $$[\Gamma^\mu \Pi_\mu - m]\Psi = 0,\qquad \Pi_\mu = p_\mu - gA^a_\mu I_a$$ The cyclic grading ensures that colour rotations entwine with space-time transformations, rendering single-quark wavefunctions non-propagating due to complex energy roots: $(p^0)^6 - |\mathbf{p}|^6 = m^6$ Only cubic, colour-singlet composites ($\Psi_{baryon} \sim \Psi_r \Psi_g \Psi_b$) yield real dispersion relations, encoding an algebraic form of quark confinement.

3. Generalized Lorentz Symmetry and Z₃–Graded Extensions

Kerner–Lukierski's framework further generalizes Lorentz symmetry via a $Z_3$-graded extension. The Lorentz algebra decomposes into three six-parameter sectors: $$\mathfrak{L} = L^{(0)} \oplus L^{(1)} \oplus L^{(2)},\qquad [L^{(r)},\,L^{(s)}] \subset L^{(r+s\mod 3)}$$ $L^{(0)}$ is the standard Lorentz algebra, while $L^{(1)}$, $L^{(2)}$ are “complex-conjugate” sectors generated from triple commutators amongst $\Gamma^\mu$ matrices. The full symmetry requires inclusion of all $24$ colour-Dirac multiplets—incorporating colour, chirality, and particle/antiparticle gradings—with the master representation being $144$-dimensional.

4. Quaternionic and Geometric Realizations

An alternative construction utilizes quaternionic Dirac matrices (Welch, 2008). Here, the coloured Dirac spinor arises in $\mathbb{H}^4$ with imaginary units $i, j, k$ corresponding to red, green, blue. Quasi-particles possess fractional electric charges ($+2/3, -1/3$), derived by imposing Q-covariance and analysing charge assignments under composite formation: $Q(u^r) = Q(u^g) = +2/3,\qquad Q(d^b) = -1/3$ Colour neutrality is enforced by the invariance of total quaternionic charge under permutations of $\{i, j, k\}$, again supporting the SU(3)-singlet rule for observable particles.

A geometric realization on the graded manifold $\mathbb{C}^{\wedge 6}$ (Pfeifer, 2021) employs six anticommuting coordinates $\theta_{a\alpha}$ with R-index $a=1,2,3$ (colour) and spinor index $\alpha=1,2$. The emergent spinors, via Taylor expansion of fields on $\mathbb{R}^{1,3} \times \mathbb{C}^{\wedge 6}$, naturally acquire SU(3) colour charges. Local SU(3) symmetry arises from decomposition of GL(3,$\mathbb{R}$), with the coloured Dirac equation given by: $$(i\gamma^\mu D_\mu - m)\Psi(x) = 0,\quad D_\mu = \partial_\mu + i g T^A A^A_\mu$$ Here, $\Psi^a(x),\,a=1,2,3$ is the “coloured” spinor indexed by the fundamental SU(3).

5. Embedding in the Standard Model Gauge Group and Families

The Kerner–Lukierski construction demonstrates that imposition of full Lorentz covariance, combined with particle–antiparticle and weak isospin gradings, forces the enlargement to $24$ Dirac multiplets. Identifying the further gradings yields three generations tied to flavour doublets $(u,d), (c,s), (t,b)$, culminating in a “144-component master-spinor” that exhaustively encodes colour, weak isospin, electric charge, and three-family structure. Minimal coupling extends naturally to the full SM gauge group SU(3)$_c$ × SU(2)$_L$ × U(1)$_Y$, including gluons, weak bosons, and electromagnetism (Kerner et al., 2019).

6. Conserved Currents and Physical Implications

The quaternionic formalism yields three distinct conserved “colour currents” stemming from the conjugations under quaternionic imaginary generators. These realize colour charge conservation analogous to the SU(3) structure. In the geometric ($\mathbb{C}^{\wedge 6}$) setting, constraint equations on commutators and anticommutators ensure correct transformation properties for colour and Lorentz indices. The emergence of colour confinement is seen not as a dynamical consequence but as an algebraic and geometric inevitability in these generalized Dirac frameworks.

7. Summary Table: Coloured Dirac Spinor in Major Constructions

Framework	Spinor Structure	Colour Embedding
Z₃–graded (Kerner–Lukierski)	12-components, extended Γ^μ matrices	Matrix generators B, Q₂, Q₃; Z₃ grading
Quaternionic (Welch)	4-components over ℍ, γ^μ ∈ ℍ⁴ˣ⁴	i → red, j → green, k → blue
ℂ∧6 Geometry (Pfeifer)	Weyl spinors indexed by a=1,2,3	R-index maps to SU(3) fundamental

Each approach rigorously realizes the SU(3) colour triplet at the algebraic or geometric level, guarantees colour current conservation, and manifests the confinement property either by propagation analysis (Kerner–Lukierski) or Q-covariance (Welch), or through emergent SU(3) singlet selection rules (Pfeifer).

Coloured Dirac spinors thus provide a unified formalism for describing gauge-charged quark fields, enabling the explicit encoding of gauge symmetries, family structure, and confinement within and beyond conventional field-theoretic constructions.