Understanding Spin in Trace Dynamics Using Division Algebras

1. Motivation: Where Is the “Spin Angle” in Trace Dynamics?

In ordinary classical mechanics, angular momentum is the canonical momentum conjugate to an angle, and the conserved total angular momentum arises from rotational symmetry via Noether’s theorem. In relativistic field theory, intrinsic spin is encoded in the Lorentz generators $J^{\mu \nu }$ and the Pauli–Lubanski vector. In trace dynamics (TD), the fundamental degrees of freedom are matrix-valued (more generally operator-valued) configuration variables, and the action is the trace of a polynomial in these variables and their velocities. Canonical momenta are defined by trace derivatives. The question “what is spin?” must therefore be answered canonically: spin should be the canonical momentum conjugate to an appropriate angular (orientation) variable.

A key subtlety is that in TD, classical spacetime is not fundamental; it is emergent in regimes where many degrees of freedom spontaneously localise. Consequently, “spin” cannot be assumed to be an internal angular momentum in pre-existing physical space. One must identify the configuration space that carries the relevant rotational structure. The central proposal of this paper is:

The correct “spin angle” in TD is an orientation variable on the (local) frame bundle, i.e. a group element in a spin group $\mathrm{Spin}(p,q)$ (or an appropriate subgroup), not an angle in physical space.

This is conceptually the same move that distinguishes (i) the position of a rigid body from (ii) its orientation: orientation lives on a Lie group manifold (e.g. $\mathrm{SO}\left(3\right)$ or $\mathrm{SU}\left(2\right)$), not in physical 3-space. In our context, the division-algebra scaffold [1] suggests a natural hierarchy of such groups:

• a pre-localisation 6D stage with signature $(3,3)$ and $\mathrm{Spin}(3,3)$ symmetry;
• emergent 4D Lorentzian leaves with $\mathrm{Spin}(3,1)$ symmetry (and spatial $\mathrm{SU}\left(2\right)$ rotation subgroup);
• additional internal $U\left(1\right)$ phases associated with ${\mathrm{Spin}}^c$ structures on ${\mathbb{CP}}^2$-type fibres.

We now turn to the TD canonical formalism and the emergent quantum commutators.

2. Trace Dynamics Essentials Needed for Spin

2.1. Trace Lagrangians and Trace Derivatives

In TD, one postulates matrix-valued (Grassmann-valued) dynamical variables $\left\{q_r\left(\tau \right)\right\}$ and a trace Lagrangian

$$\mathcal{S}=\int d\tau \mathrm{Tr}L\left(q_r,{\dot{q}}_r\right),$$

(1)

with $\tau$ an intrinsic evolution parameter (often identified with Connes time in noncommutative-geometry-inspired constructions). The canonical momentum is defined by the trace derivative

$$p_r:={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta {\dot{q}}_r}},$$

(2)

where the trace derivative is defined so that variations $\delta q_r$ are cyclically permuted inside the trace to a canonical location, with Grassmann sign rules.

2.2. Bosonic vs. Fermionic Variables and the Adler–Millard Charge

TD distinguishes bosonic (even Grassmann grade) and fermionic (odd grade) variables. A central structural fact is the existence of the Adler–Millard conserved charge $\tilde{C}$ arising from invariance of $\mathrm{Tr}L$ under global unitary transformations. Schematically,

$$\tilde{C}=\sum _{r\in B}[q_r,p_r]-\sum _{r\in F}\{q_r,p_r\}.$$

(3)

At statistical equilibrium, equipartition of $\tilde{C}$ yields emergent canonical commutation/anticommutation relations for the (self-adjoint parts of) coarse-grained variables, reproducing quantum theory [2].

2.3. Why This Matters for Spin

If an angular variable $\Theta$ (bosonic) is introduced in TD and its conjugate momentum is defined canonically as

$$p_{\Theta }:={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta \dot{\Theta }}},$$

(4)

then the emergent relation $[\Theta ,p_{\Theta }]=i\hslash$ implies that $p_{\Theta }$ acts as the generator of translations in $\Theta$ and is quantised when $\Theta$ is periodic. This is precisely the logic used in earlier work to motivate spin quantisation from a TD “phase” variable [3]. The present paper upgrades that phase to a group-valued orientation variable.

3. From Abelian Phase $\theta$ to Nonabelian Orientation U

3.1. Review: Phase-Amplitude Variables in Complexified TD

A particularly useful TD rewriting is to package certain degrees of freedom into complexified variables $Q_e$ with a decomposition into amplitude and phase. In one implementation,

$$Q_{eB}=R_B{e}^{i{\theta }_B},Q_{eF}=R_F{e}^{i\eta {\theta }_F},$$

(5)

with $R_B,{\theta }_B$ bosonic self-adjoint matrices, $R_F,{\theta }_F$ fermionic self-adjoint matrices, and $\eta$ a real Grassmann number inserted so that the fermionic exponential truncates (because ${\eta }^2=0$) [3]. One then defines

$$p_{B\theta }={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta {\dot{\theta }}_B}},p_{F\theta }={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta {\dot{\theta }}_F}}.$$

(6)

At equilibrium, $[{\theta }_B,p_{B\theta }]=i\hslash$ yields integer eigenvalues for $p_{B\theta }$ under ${\theta }_B\sim {\theta }_B+2\pi I$, and an argument based on the $2\pi$ sign flip for fermions motivates half-integer spin [3].

3.2. Limitation of a Purely Abelian Phase

The abelian phase $\theta$ captures one component of a more general structure:

• physically, spin is not a single generator but a representation of the rotation/Lorentz algebra;
• relativistically, intrinsic spin is naturally a bivector $S^{\mu \nu }$ (or equivalently an $\mathfrak{so}(3,1)$ generator), not a scalar angle-momentum pair;
• topologically, the relevant configuration space is $\mathrm{SU}\left(2\right)$ (for spinors) or $\mathrm{SO}\left(3\right)$ (for vectors), not just $S^1$.

Thus, we seek a definition that (i) reduces to the $\theta$-construction in a 1-parameter limit, but (ii) naturally produces the full Lorentz-covariant spin tensor and its Casimirs.

3.3. Nonabelian Polar Decomposition

The natural upgrade is the matrix polar decomposition:

$$Q_e=RU,R:={\left(Q_e{Q}_e^{†}\right)}^{1/2},U^{†}U=I.$$

(7)

In ordinary matrix analysis, U is unitary. In our context, U is to be restricted to the image of a spin group in a chosen representation. The core idea is to interpret $U\left(\tau \right)$ as an orientation variable.

4. Canonical Definition of Spin as Momentum on a Spin-Group Manifold

4.1. Orientation Variable and Angular Velocity

Let $G=\mathrm{Spin}(p,q)$ (or a subgroup such as $\mathrm{SU}\left(2\right)$) and let $U\left(\tau \right)\in G$ be a group-valued TD configuration variable. Define the (right-invariant) angular velocity (Maurer–Cartan form)

$$\Omega \left(\tau \right):=U^{-1}\dot{U}\in \mathfrak{g}=\mathfrak{spin}(p,q).$$

(8)

The object $\Omega$ is Lie-algebra valued and transforms by conjugation under left multiplication $U\mapsto gU$.

4.2. Spin Tensor as Canonical Momentum

Definition 1.

(Spin in trace dynamics). Let the trace Lagrangian depend on U only through $\Omega =U^{-1}\dot{U}$ (this is the group-manifold analogue of rotational invariance). Theintrinsic spin tensoris defined as the Lie-algebra-valued canonical momentum

$$S:={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta \Omega }}\in {\mathfrak{g}}^{*}\simeq \mathfrak{g}.$$

(9)

Equivalently, choosing a basis $\left\{{\Sigma }^{AB}\right\}$ of $\mathfrak{spin}(p,q)$ and writing $\Omega =\frac{1}{2}{\Omega }_{AB}{\Sigma }^{AB}$, one defines

$$S^{AB}:={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta {\Omega }_{AB}}}.$$

(10)

Remark 1.

The definition is canonical: it is “momentum conjugate to an angle”, with the “angle” now understood as a point on a Lie group manifold. This is precisely how one defines body-fixed angular momentum for a rigid body, except now implemented within TD trace-derivative calculus.

4.3. Reduction to the Abelian Phase $\theta$

If $U\left(\tau \right)$ lies in a 1-parameter subgroup,

$$U\left(\tau \right)=exp\left(\theta \left(\tau \right)T\right),$$

(11)

for fixed generator $T\in \mathfrak{g}$, then $\Omega =\dot{\theta }T$ and

$$p_{\theta }:={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta \dot{\theta }}}={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta \Omega }}T=\mathrm{Tr}\left(ST\right),$$

(12)

so the earlier $\theta$-momentum is recovered as a component of S.

4.4. Noether Theorem and Conservation of Total Angular Momentum

Assume TD also contains translation-like variables $X^A\left(\tau \right)$ with momenta $P_A\left(\tau \right)$ in an emergent regime where spacetime notions are meaningful. Then define orbital and total generators

$$L^{AB}:=X^A{P}^B-X^B{P}^A,J^{AB}:=L^{AB}+S^{AB}.$$

(13)

Invariance of the trace action under global G transformations implies conservation of $J^{AB}$ (in the TD sense: an operator/matrix constant of motion).

5. Quantisation of Spin in Trace Dynamics

Spin quantisation in this framework has two logically distinct inputs:

1.

an emergent quantum commutator algebra for the coarse-grained canonical variables (from Adler–Millard equipartition);

2.

the topology of the true orientation manifold (e.g. $\mathrm{SU}\left(2\right)$ vs $\mathrm{SO}\left(3\right)$), which controls whether $2\pi$ loops are trivial or not.

5.1. Emergent Lie Algebra of Spin

At equilibrium, the TD canonical structure implies that the self-adjoint parts of $(U,\Omega ,S)$ satisfy the quantum commutator algebra corresponding to $\mathfrak{spin}(p,q)$. Concretely, restricting to the spatial rotation subgroup $\mathrm{SU}\left(2\right)$, one obtains operators $S_i$ with

$$[S_i,S_j]=i\hslash {\epsilon }_{ijk}{S}_k,S^2=s(s+1){\hslash }^2,$$

(14)

so that spin labels are the usual $\{0,\frac{1}{2},1,\frac{3}{2},\dots \}$, emerging from representation theory.

5.2. Topology: $\mathrm{SU}\left(2\right)$ Double Cover and $2\pi$ Sign

The same formalism explains the difference between integer and half-integer spin:

• If the physical orientation variable effectively lives on $\mathrm{SO}\left(3\right)$ (or bosonic observables are insensitive to the ${\mathbb{Z}}_2$ centre), then only integer representations survive.
• If the relevant degrees of freedom live on $\mathrm{SU}\left(2\right)$ and are sensitive to the nontrivial central element $-I$, then half-integer representations appear. A $2\pi$ loop corresponds to $-I$, which acts as a sign flip on spinors but trivially on vectors.

This is the nonabelian generalisation of the earlier TD observation that a fermionic quantity changes sign under a $2\pi$ shift while a bosonic one does not [3].

5.3. Quantisation as Periodicity on Group Manifolds

For a 1-parameter subgroup with $U\left(\theta \right)=exp\left(\theta T\right)$, periodicity $\theta \sim \theta +2\pi$ implies a quantisation condition for $p_{\theta }$ (and hence for the component $\mathrm{Tr}\left(ST\right)$). In the full group case, the quantisation is encoded in the unitary irreducible representations of G and the spectrum of its Casimirs.

6. Spin–Statistics Connection in TD Language

In TD, statistics is not imposed but is linked to Grassmann grading:

• bosonic variables (even grade) commute, leading to symmetric multi-particle states;
• fermionic variables (odd grade) anticommute, leading to antisymmetric states.

At equilibrium, bosonic variables satisfy commutators while fermionic variables satisfy anticommutators, consistent with the structure of the Adler–Millard charge. Earlier work showed how, once spin is defined canonically via an angle-like variable, integer vs half-integer spin follows in tandem with Bose vs Fermi exchange properties [3].

In the present framework, the connection is tightened:

1.

Spin arises as the representation label of the rotation/spin group acting on the relevant TD degrees of freedom (through the canonical momentum S on G).

2.

Statistics arises from Grassmann parity (commutation vs anticommutation) of those degrees of freedom.

3.

The tie is that fermionic degrees of freedom naturally realise spinorial representations of $\mathrm{SU}\left(2\right)$ (sensitive to the ${\mathbb{Z}}_2$ centre), while bosonic degrees of freedom naturally realise tensorial representations (insensitive to the centre). The same structural distinction (odd vs even Grassmann grade) therefore controls both exchange symmetry and the allowed topology of orientation.

7. Pauli–Lubanski Vector, Casimirs, and the Mass Operator

7.1. 4D Pauli–Lubanski Vector

In an emergent 4D Lorentzian regime, define energy–momentum $P^{\mu }$ and total Lorentz generators $J^{\mu \nu }=L^{\mu \nu }+S^{\mu \nu }$. The Pauli–Lubanski vector is

$$W^{\mu }:={\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \rho \sigma }{P}_{\nu }{J}_{\rho \sigma }.$$

(15)

In the rest frame $P^{\mu }=(m,$0),

$$W^0=0,W^i=m{S}^i,S^i:={\displaystyle \frac{1}{2}}{\epsilon }^{ijk}{S}_{jk}.$$

(16)

Hence the intrinsic spin measured in the rest frame is the spatial dual of the intrinsic Lorentz generator $S^{\mu \nu }$ obtained canonically from Definition 1.

7.2. Mass as Translation Casimir vs. “Square-Root Mass” Charges

For Poincaré symmetry, the mass operator is the translation Casimir

$$P^2:=P_{\mu }{P}^{\mu }=m^2,M:=\sqrt{P^2}.$$

(17)

Separately, in division-algebra-based geometric unification programmes, one may encounter dimensionless charges proportional to square roots of Yukawa couplings or mass ratios. For example, one can define a dimensionless “dark” charge

$$Q_{\mathrm{dem}}\left(f\right):=\sqrt{y_f\left(\mu \right)}=\sqrt{m_f\left(\mu \right)/v_M},$$

(18)

where $v_M$ is a reference mass scale [1]. This is not the same object as the Poincaré Casimir $M=\sqrt{P^2}$; it is a square root of a ratio that can arise from geometric normalisations of internal $U\left(1\right)$ factors. Any identification between the two must be made via the dynamical relation $m=y{v}_M$, not by redefining the Casimir.

8. Division-Algebra Scaffold: 6D $(\mathbf{3},\mathbf{3})$ Geometry and Spin

8.1. Split-Bioctonionic Base and $\mathrm{Spin}(3,3)$

A concrete 6D pre-localisation stage can be built from split bioctonions by selecting quaternionic subalgebras ${\mathbb{H}}_L,{\mathbb{H}}_R\subset \mathbb{O}$ and defining

$$M_6:=\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R,$$

(19)

with a metric of signature $(3,3)$[1]. The maximal compact subgroup contains $\mathrm{Spin}\left(4\right)\simeq \mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R$, acting by conjugation on the imaginary quaternion triples. This provides a natural stage on which a 6D spin group $\mathrm{Spin}(3,3)$ acts.

8.2. Leaf Selection: $\mathrm{SO}(3,3)\to \mathrm{SO}(3,1)\times \mathrm{SO}\left(2\right)$

A key geometric mechanism is the selection of an oriented negative 2-plane $N\subset T_x{M}_6$ with orthogonal complement $W=N^{\perp }$ of signature $(3,1)$. The stabiliser is

$${\mathrm{Stab}}_{\mathrm{SO}(3,3)}\left(N\right)\simeq \mathrm{SO}(3,1)\times \mathrm{SO}\left(2\right),$$

(20)

and the Lie algebra decomposes as

$$\mathfrak{so}(3,3)=\mathfrak{so}\left(W\right)\oplus \mathfrak{so}\left(N\right)\oplus (W\wedge N),$$

(21)

where $W\wedge N$ are “mixing” generators that rotate leaf directions into normal directions [1]. This splitting is conceptually important for spin: it tells us how a 6D intrinsic spin bivector $S^{AB}$ decomposes into

• a 4D Lorentz spin $S^{\mu \nu }$ on the leaf ($\mathfrak{so}\left(W\right)$ part),
• a normal $\mathfrak{so}\left(2\right)$ generator (an internal $U\left(1\right)$-like quantum number),
• mixed components $S^{\mu \widehat{a}}$ associated with the broken coset $(W\wedge N)$ (typically heavy/decoupled after localisation).

8.3. 6D Pauli–Lubanski Generalisation

In d dimensions the Pauli–Lubanski object is a $(d-3)$-form. In $d=6$ one may define a 3-form

$$W_{ABC}:={\displaystyle \frac{1}{2}}{\epsilon }_{ABCDEF}{P}^D{J}^{EF}.$$

(22)

Its invariants play the role of spin Casimirs in 6D. Under reduction to a 4D leaf using the projector formalism described in [1], components reduce to the familiar 4D $W^{\mu }$ plus additional terms involving the normal directions.

8.4. Octonionic Split and Internal Fibres as ${\mathbb{CP}}^2$ Tangents

A division-algebra realisation of the geometric $\mathrm{SU}\left(3\right)$ branching uses the octonionic split

$$\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon ,\Im \mathbb{O}=\Im \mathbb{H}\oplus \mathbb{H}\epsilon ,$$

(23)

where $\epsilon \perp \mathbb{H}$ and ${\epsilon }^2=-1$[1]. Under $\mathrm{SU}\left(3\right)\to \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$ one has

$$\mathbf{8}\to {\mathbf{3}}_0\oplus {\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}\oplus {\mathbf{1}}_0,$$

(24)

and the real 4-space underlying ${\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}$ can be identified with $T{\mathbb{CP}}^2$ at a point. The associated $\mathrm{U}\left(1\right)$ acts as the canonical ${\mathrm{Spin}}^c$ line connection on ${\mathbb{CP}}^2$-type fibres [1].

8.5. Interpretation for Spin Angles

This division-algebra scaffold yields three distinct, conceptually separate “angular” structures:

1.

Lorentz/spin orientation:$U\left(\tau \right)\in \mathrm{Spin}(3,1)$ (or $\mathrm{Spin}(3,3)$ pre-localisation), whose conjugate momentum is intrinsic spin S.

2.

Spatial rotations:$U\left(\tau \right)\in \mathrm{SU}\left(2\right)\subset \mathrm{Spin}(3,1)$, giving the familiar three-component spin algebra.

3.

Geometric fibre phase: an internal $\mathrm{U}\left(1\right)$ acting on $T{\mathbb{CP}}^2$, required by ${\mathrm{Spin}}^c$ structure. This provides a separate angle whose conjugate momentum is an internal charge-like quantum number.

A recurring pitfall is to conflate (1) and (3). In the present view, the earlier TD phase variable $\theta$ is best interpreted as either (i) a restriction of (1) to a 1-parameter subgroup, or (ii) an effective abelianisation of a more general nonabelian orientation dynamics. Any further identification with the ${\mathrm{Spin}}^c$$\mathrm{U}\left(1\right)$ should be made only after an explicit bundle-level matching of the relevant connections.

9. Understanding Spin Space

9.1. What We Mean by “Spin Space”

In conventional relativistic quantum theory, spin is encoded in the representation theory of the rotation/Lorentz group: orbital angular momentum acts on (spatial) coordinates, whereas intrinsic spin acts on an additional index carried by the field (spinor, vector, tensor, etc.). In our framework, the question “where does this additional space come from?” has a concrete answer: it is geometric and it is already present in the $E_8\times \omega E_8$ scaffold through the two extra $SU\left(3\right)$ factors that we interpret as geometric structure groups [1].

We will use the term spin space in three closely related (but distinct) senses:

1.

Spin state space: the representation space on which $SU\left(2\right)$ acts as a doublet, i.e. a complex two-dimensional space $\cong {\mathbb{C}}^2$ (for spin-$\frac{1}{2}$) or its tensorial descendants (for integer spin).

2.

Spin orientation space: the group manifold $SU\left(2\right)\simeq S^3$ of local frame orientations, coordinatised by three angles (e.g. Euler angles).

3.

Spin generator space: the Lie algebra $\mathfrak{su}\left(2\right)\cong {\mathbb{R}}^3$ of infinitesimal rotations, i.e. the three generator directions corresponding to the three Pauli matrices.

The key point is that all three appear naturally and canonically once we identify the internal 4-real-dimensional fibre and its $SU\left(2\right)\times U\left(1\right)$ action.

9.2. From the Geometric $SU\left(3\right)$ to a Canonical Internal Fibre

On each side $X=L,R$ of $E_8\times \omega E_8$, the maximal chain

$$E_8\supset E_6\times SU{\left(3\right)}_{\mathrm{geom}}$$

provides an extra $SU{\left(3\right)}_{\mathrm{geom},X}$ which we interpret as geometry rather than a gauged force [1]. Choosing the standard embedding $SU\left(2\right)\subset SU\left(3\right)$ with a complementary $U\left(1\right)$, the adjoint branches as

$$\mathbf{8}⟶{\mathbf{3}}_0\oplus {\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}\oplus {\mathbf{1}}_0.$$

(25)

The interpretation proposed in [1] is:

• the ${\mathbf{3}}_0$ supplies the three spatial directions associated with an imaginary quaternion triple;
• the ${\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}$ supplies a real 4-dimensional internal fibre at each point, naturally identified with the tangent of ${\mathbb{CP}}^2$;
• the two $U\left(1\right)$ factors play the role of ${\mathrm{Spin}}^c$ line connections on these fibres.

Thus, the additional “spin space” is not an ad hoc postulate: it is the canonical internal fibre carried by the same geometric $SU\left(3\right)$ which also supplies the three external spatial directions in (25).

9.3. Octonions Make the Split Explicit: $\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon$

The geometric meaning of (25) becomes transparent using the octonionic split [1]:

$$\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon ,\Im \mathbb{O}=\Im \mathbb{H}\oplus \mathbb{H}\epsilon ,$$

(26)

where $\mathbb{H}\subset \mathbb{O}$ is a chosen quaternionic subalgebra and $\epsilon \in \Im \mathbb{O}$ is orthogonal to $\mathbb{H}$ with ${\epsilon }^2=-1$. The split (26) realises the group-theoretic branching as follows:

$${\mathbf{3}}_0\leftrightarrow \Im \mathbb{H},{({\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1})}_{\mathbb{R}}\leftrightarrow \mathbb{H}\epsilon .$$

(27)

The real 4-space $\mathbb{H}\epsilon$ is the internal fibre, denoted $F_4$ (and $F_{4,X}$ on side $X=L,R$).

To make the doublet structure manifest, one chooses a unit imaginary quaternion $u\in \Im \mathbb{H}$ and defines an intrinsic complex structure J on $\mathbb{H}\epsilon$ by [1]

$$J\left(a\epsilon \right):=\left(ua\right)\epsilon ,J^2=-{\mathrm{Id}}_{\mathbb{H}\epsilon }.$$

(28)

Then $(\mathbb{H}\epsilon ,J)$ becomes a complex vector space of complex dimension 2, hence isomorphic to ${\mathbb{C}}^2$:

$$(\mathbb{H}\epsilon ,J)\cong {\mathbb{C}}^2.$$

(29)

This is precisely the spin-$\frac{1}{2}$ state space.

The accompanying $U\left(1\right)$ acts as phases generated by J:

$$e^{\theta J}:v\mapsto cos\theta v+sin\theta Jv,v\in \mathbb{H}\epsilon ,$$

(30)

which is multiplication by $e^{i\theta }$ in the complex structure (29). In the geometric interpretation, this $U\left(1\right)$ is the ${\mathrm{Spin}}^c$ line on the ${\mathbb{CP}}^2$ fibre [1].

9.4. How $SU\left(2\right)$ Produces Three Spin Directions from a Doublet Fibre

There is a standard but crucial representation-theoretic point:

The same Lie algebra $\mathfrak{su}\left(2\right)$ has (at least) two fundamental geometric incarnations: (i) the adjoint (vector) representation on a real 3-space, and (ii) the fundamental (spinor) representation on a complex 2-space.

In our octonionic split, these appear simultaneously:

• Vector side (external):$\Im \mathbb{H}$ is a real 3-space carrying the adjoint action. Concretely, for a unit quaternion $q\in SU\left(2\right)$, the adjoint action is

  $$x\mapsto qx{q}^{-1},x\in \Im \mathbb{H},$$

  (31)

  which realises the covering $SU\left(2\right)\to SO\left(3\right)$.
• Spinor side (internal):$(\mathbb{H}\epsilon ,J)\cong {\mathbb{C}}^2$ carries the fundamental (spin-$\frac{1}{2}$) action. Equivalently, after choosing an identification $(\mathbb{H}\epsilon ,J)\cong {\mathbb{C}}^2$, each $q\in SU\left(2\right)$ acts by a $2\times 2$ special unitary matrix on the column vector $v\in {\mathbb{C}}^2$.

The three spin directions are then simply the three basis directions in the Lie algebra

$$\mathfrak{su}\left(2\right)\cong {\mathbb{R}}^3,$$

(32)

i.e. the three generators (Pauli matrices) acting on the doublet. Thus, the fibre does not add three extra coordinates; instead it adds a canonical carrier space for the same three-generator rotation algebra, now in its spinorial representation.

This answers the conceptual question:

Spin has three components because the internal orientation group is $SU\left(2\right)$, whose Lie algebra has three independent generators. The doublet fibre is the state space on which these generators act.

9.5. Matching the Three Spin Directions with the Three Orbital Directions

Orbital angular momentum exists because physical space has three spatial directions, hence a spatial rotation group with three generators. In our $E_8\times \omega E_8$ scaffold, the relevant three spatial directions arise from the ${\mathbf{3}}_0$ in (25), realised octonionically as $\Im \mathbb{H}$[1]. The same $SU\left(2\right)$ that rotates $\Im \mathbb{H}$ also acts on the internal fibre $(\mathbb{H}\epsilon ,J)\cong {\mathbb{C}}^2$. Therefore the identification

$$\Im \mathbb{H}\cong \mathfrak{su}\left(2\right)\cong “\mathrm{generator}\mathrm{space}”$$

provides a clean geometric explanation of why there are “three spin directions” corresponding to the “three orbital directions”: they are the same three $\mathfrak{su}\left(2\right)$ directions, seen in two representations:

$$\mathrm{orbital}:\Im \mathbb{H}(\mathrm{adjoint}/\mathrm{vector}),\mathrm{spin}:(\mathbb{H}\epsilon ,J)\cong {\mathbb{C}}^2(\mathrm{fundamental}/\mathrm{spinor}).$$

(33)

In this sense, the octonionic split (26) gives a concrete “division-algebra picture” of the familiar quantum statement:

Orbital angular momentum acts on spatial vectors; spin acts on spinors; both are governed by the same rotation algebra.

9.6. Fermions and Bosons: Does the Same Spin Space Work for Both?

Yes, with a precise interpretation.

• Fermions: a spin-$\frac{1}{2}$ field is naturally valued in the doublet fibre $(\mathbb{H}\epsilon ,J)\cong {\mathbb{C}}^2$. The $SU\left(2\right)$ generators act directly on this fibre, so the internal fibre is the spin state space.
• Bosons: integer-spin objects arise as tensorial representations (including the adjoint) of the same $SU\left(2\right)$. Concretely, the triplet $\Im \mathbb{H}$ realises spin-1 (vector) behaviour, and higher integer spins arise from symmetric tensor powers. Hence the same geometric $SU\left(2\right)$ supports both fermionic and bosonic spins; the difference is the representation carried by the field/degree of freedom.

This is also consistent with the trace-dynamical viewpoint where Grassmann grading distinguishes fermionic from bosonic degrees of freedom, while the group action selects which spin representations are realised dynamically [3].

9.7. Spin Orientation Space and the “Three Internal Angles”

The discussion above identifies the state space for spin. To connect to a canonical definition of spin as momentum conjugate to angles (as required in trace dynamics), one uses the fact that $SU\left(2\right)$ itself is a 3-parameter manifold ($S^3$). Locally, write an $SU\left(2\right)$ frame/orientation as

$$U\left(\tau \right)\in SU\left(2\right),U\left(\tau \right)=U\left(\varphi \right(\tau ),\vartheta (\tau ),\psi (\tau \left)\right),$$

with three angles $(\varphi ,\vartheta ,\psi )$ (e.g. Euler angles). The corresponding angular velocity is the Maurer–Cartan form

$$\Omega \left(\tau \right):=U^{-1}\dot{U}\in \mathfrak{su}\left(2\right),\Omega ={\Omega }^a{T}_a,a=1,2,3.$$

In trace dynamics, the canonical spin components are then the momenta conjugate to these $SU\left(2\right)$ orientation velocities:

$$S_a:={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta {\Omega }^a}},a=1,2,3,$$

which realises the three internal spin directions as canonical momenta on the $SU\left(2\right)$ orientation manifold. (The earlier single angle $\theta$ corresponds to restricting $U\left(\tau \right)$ to a one-parameter subgroup.)

9.8. Two Sides and Chirality: A Cautious Geometric Remark

On each side $X=L,R$ there is a distinct geometric $SU{\left(3\right)}_{\mathrm{geom},X}$, hence a distinct quaternionic subalgebra ${\mathbb{H}}_X$ and a distinct internal fibre $F_{4,X}\cong {\mathbb{H}}_X{\epsilon }_X$. Simultaneously, the split-bioctonionic base construction yields two embedded Lorentzian 4D leaves ${\Sigma }_L$ and ${\Sigma }_R$ inside a $(3,3)$ bulk [1]. This naturally suggests an association between left/right geometric sectors and left/right chiral sectors; however, a strict identification of “the two fibres” with “the two 4D chiralities” requires an explicit analysis of: (i) the relevant Clifford modules on each leaf, and (ii) how the ${\mathrm{Spin}}^c$ twisting by the fibre $U\left(1\right)$ is implemented for chiral zero-modes. The present section therefore emphasises the robust, representation-theoretic conclusion: each side supplies a canonical doublet fibre supporting spinorial $SU\left(2\right)$ action, and hence a canonical geometric spin space.

9.9. Why a Point Particle Can Have Spin: Motion Outside Spacetime

A recurrent puzzle in standard presentations is: how can a point particle have angular momentum? In our framework the resolution is geometric and does not require spatial extension. The key is that a “particle” (more precisely, an STM atom in the $E_8\times \omega E_8$ scaffold) is not described solely by a spacetime trajectory; it also carries internal fibre degrees of freedom attached to each base point.

Internal fibre at each spacetime point. On each side $X=L,R$, the geometric subgroup $SU{\left(3\right)}_{\mathrm{geom},X}$ branches as

$$\mathrm{Adj}SU\left(3\right)⟶3_0\oplus 2_{+1}\oplus 2_{-1}\oplus 1_0,$$

(34)

and the realification of $2_{+1}\oplus 2_{-1}$ furnishes a canonical real rank-4 internal fibre $F_{4,X}$ naturally identified with $T{\mathbb{CP}}^2$ at each point [1]. Octonionically this is realised by

$$\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon ,\Im \mathbb{O}=\Im \mathbb{H}\oplus \mathbb{H}\epsilon ,$$

(35)

with the identifications

$$3_0\leftrightarrow \Im \mathbb{H},{(2_{+1}\oplus 2_{-1})}_{\mathbb{R}}\leftrightarrow \mathbb{H}\epsilon \equiv F_4.$$

(36)

Choosing a unit imaginary quaternion $u\in \Im \mathbb{H}$ defines an intrinsic complex structure on the real 4-space $F_4=\mathbb{H}\epsilon$ by

$$J\left(a\epsilon \right):=\left(ua\right)\epsilon ,J^2=-{\mathrm{Id}}_{F_4},$$

(37)

so that $(F_4,J)\cong {\mathbb{C}}^2$ carries an $SU\left(2\right)$-doublet structure and the accompanying $U\left(1\right)$ acts as phases $e^{\theta J}$ [1]. In other words: a point of spacetime is decorated by an internal space on which rotations/phases can act.

“Point particle” kinematics with fibre motion. Accordingly, the configuration of an elementary object is not merely $x^{\mu }\left(\tau \right)$ (a point on a 4D leaf) but a pair

$$\left(x^{\mu }\left(\tau \right),\xi \left(\tau \right)\right),\xi \left(\tau \right)\in F_4\mathrm{or}\mathrm{more}\mathrm{generally}\mathrm{in}\mathrm{an}\mathrm{associated}\mathrm{bundle}.$$

(38)

The internal symmetry group contains an $SU\left(2\right)$ acting on the doublet fibre $(F_4,J)\cong {\mathbb{C}}^2$, so it is natural to introduce an internal orientation variable

$$U\left(\tau \right)\in SU\left(2\right)\cong S^3,$$

(39)

which rotates the fibre state $\xi$ as $\xi \left(\tau \right)\mapsto U\left(\tau \right)\xi \left(\tau \right)$ (after choosing a local trivialisation).

The crucial observation is now immediate:

$${\dot{x}}^{\mu }\left(\tau \right)=0(\mathrm{no}\mathrm{orbital}\mathrm{motion}\mathrm{in}\mathrm{spacetime})\mathrm{is}\mathrm{compatible}\mathrm{with}\dot{U}\left(\tau \right)\ne 0(\mathrm{nontrivial}\mathrm{internal}\mathrm{motion}).$$

(40)

Thus a spacetime point can carry a nontrivial internal dynamics (a trajectory on the group manifold $SU\left(2\right)$), and this is precisely the sense in which a point particle can have spin:

spin is generated by motion in the internal fibre/orientation space attached at each spacetime point, not by spatial extension of the object.

Canonical spin as momentum conjugate to internal angles. To make this statement canonical (in the precise Noether–Hamiltonian sense), one introduces the Lie-algebra valued angular velocity

$$\Omega \left(\tau \right):=U{\left(\tau \right)}^{-1}\dot{U}\left(\tau \right)\in \mathfrak{su}\left(2\right),\Omega ={\Omega }^a{T}_a,a=1,2,3,$$

(41)

and defines the intrinsic spin components as the canonical momenta conjugate to these internal rotation rates:

$$S_a:={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta {\Omega }^a}},a=1,2,3.$$

(42)

Because $SU\left(2\right)$ is three-dimensional as a manifold, it has three local angle coordinates (e.g. Euler angles), and (42) provides three spin directions as canonical momenta. This reproduces the familiar fact that spin has three components while the fibre state space is a doublet: the doublet is the representation space on which the three generators act.

The octonionic split (35) makes the external/internal parallelism transparent: orbital rotations act in the adjoint/vector representation on $\Im \mathbb{H}$ (the $3_0$ piece), while intrinsic spin acts in the spinorial/fundamental representation on $(\mathbb{H}\epsilon ,J)\cong {\mathbb{C}}^2$ (the $2_{\pm 1}$ piece) [1]. The symmetry is therefore not “two copies of spacetime”; it is the same $\mathfrak{su}\left(2\right)$ algebra appearing in inequivalent representations on the external (vector) and internal (spinor) sectors.

9.10. Why Spin Has No Classical Analogue (In the Relevant Sense)

It is important to state carefully what “no classical analogue” means here. Classical mechanics certainly has angular momentum for extended bodies, and one can introduce classical “internal orientation” variables for a rigid rotor. What is special about quantum spin is:

1.

a point particle in ordinary classical mechanics has no intrinsic orientation degrees of freedom;

2.

half-integer representations of rotations (spinors) are intrinsically tied to the double cover $SU\left(2\right)\to SO\left(3\right)$ and to quantum operator kinematics;

3.

in our trace-dynamical setting, the fundamental spin variables arise from noncommutative and (for fermions) Grassmann-graded configuration variables that are not obtainable by quantising an ordinary classical point-particle theory.

Trace-dynamical origin of the spin variable. In the Planck-scale matrix dynamics, one introduces amplitude–phase variables for the fundamental degrees of freedom, for instance

$$Q_B^e\equiv R_B{e}^{i{\theta }_B},Q_F^e\equiv R_F{e}^{i\eta {\theta }_F},$$

(43)

and defines bosonic/fermionic spin angular momenta as the canonical momenta conjugate to the phase velocities:

$$p_{B\theta }:={\displaystyle \frac{\delta L}{\delta {\dot{\theta }}_B}},p_{F\theta }:={\displaystyle \frac{\delta L}{\delta {\dot{\theta }}_F}}.$$

(44)

A central point is that the fermionic configuration variable ${\theta }_F$ has no analogue in ordinary quantum mechanics derived by quantising classical theories; it is a genuinely Planck-scale trace-dynamical degree of freedom [3]. This provides a precise sense in which spin (especially fermion spin) is not the quantisation of a classical spacetime angle for a point particle: the underlying configuration variable is not present in classical point-particle dynamics to begin with.

Quantisation and the loss of a classical limit. At low energies, equipartition of the Adler–Millard charge yields the emergent canonical (anti)commutators for coarse-grained self-adjoint parts, including

$$[{\theta }_B,p_{B\theta }]=i\hslash ,\{{\theta }_F,p_{F\theta }\}=i\hslash ,$$

(45)

and one obtains discrete spin eigenvalues (integer for bosonic spin variable; half-integer for fermions via the spinorial $2\pi$ sign structure) [3]. Discreteness of eigenvalues is already enough to explain why there is no continuous classical limit in which a point particle carries an arbitrarily small intrinsic angular momentum: the intrinsic generator is not a c-number parameter but an operator with a quantised spectrum.

Geometric interpretation of “no classical analogue”. From the division-algebra point of view, the internal fibre $(F_4,J)\cong {\mathbb{C}}^2$ is the natural carrier space for the spinorial action of $SU\left(2\right)$, whereas $\Im \mathbb{H}$ carries the vectorial action. A purely classical point particle moving in spacetime samples only the vectorial sector. Quantum spin exists because the fundamental degrees of freedom also live in the fibre sector and can undergo internal $SU\left(2\right)$ evolution even when spacetime position is fixed. The absence of a classical analogue is therefore not mysterious: it is the absence (in classical point-particle mechanics) of the fibre/orientation degrees of freedom and of the operator/Grassmann structure that makes their conjugate momenta quantised.

In summary: a point particle can carry spin because it can move on the internal orientation manifold ($SU\left(2\right)$ and its associated fibre) while remaining pointlike in spacetime; and spin has no classical analogue (in the relevant point-particle sense) because its canonical “angle” variables are internal, noncommutative, and (for fermions) Grassmann-graded, yielding quantised generators already at the kinematical level.

10. Discussion and Open Problems

10.1. Summary of the Proposal

We have proposed a canonical definition of spin in TD:

$$\begin{array}{|c|}\hline \mathrm{Spin}\mathrm{is}S={\displaystyle \frac{\delta \mathrm{Tr}L}{\delta \left(U^{-1}\dot{U}\right)}}\in \mathfrak{spin}(p,q),\\ \hline\end{array}$$

(46)

with $U\left(\tau \right)\in \mathrm{Spin}(p,q)$ an orientation variable. This definition:

• reduces to the previously studied phase-momentum spin in a 1-parameter subgroup limit;
• naturally yields Lorentz-covariant spin $S^{\mu \nu }$ and Pauli–Lubanski invariants;
• makes the quantisation of spin a consequence of TD emergent commutators and group topology.

10.2. Open Problems

A non-exhaustive list of technical questions that deserve careful analysis:

1.

Constrained TD variation on group manifolds: a systematic derivation of Euler–Poincaré equations in TD trace-derivative language, including Grassmann-valued group variables where relevant.

2.

Precise emergence of $\mathrm{Spin}(3,1)$ from $\mathrm{Spin}(3,3)$: an explicit dynamical model of how the leaf projector and Higgs-like breaking selects the physical Lorentz group and decouples mixed generators.

3.

Spin–statistics beyond kinematics: the extent to which microcausality assumptions of local QFT are replaced by TD structural inputs (Grassmann parity + topology) when spacetime is emergent.

4.

Role of internal ${\mathrm{Spin}}^c$ phases: whether any component of spin-like TD angles should be identified with $\mathrm{U}\left(1\right)$ connections on ${\mathbb{CP}}^2$ fibres, and if so how this impacts family structure and charge quantisation.

5.

6D Casimirs and reduction: a detailed mapping between 6D $\mathrm{Spin}(3,3)$ Casimirs (including the 3-form Pauli–Lubanski object) and the 4D $(m,s)$ labels on a leaf.