Hermitian Modules over the Biquaternion Algebra with Hermitian Adjoint

Introduction

The biquaternion algebra $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ is the algebra of operators of the framework, and its two distinguished sectors carry the two sides of the physical dictionary: the Hermitian subspace $\mathbb{M}_+$ is the informational sector, the operators of a two-state system, and the anti-Hermitian subspace $\mathbb{M}_-$ is the material sector, Minkowski space (The Hermitian Subspace $\mathbb{M}_+$ as the Informational Sector, The Anti-Hermitian Subspace $\mathbb{M}_-$ as the Material Sector). Both readings rest on the module theory: $\mathbb{B}\cong M_2(\mathbb{C})$ has one simple module $S\cong\mathbb{C}^2$, the spinor module, and the framework's linear structure is that module together with the identification (Modules over the Biquaternion Algebra, The Spinor Module in Biquaternionic Form and Its Lorentz Action).

This article asks the metric question of that module: when does the spinor module carry a positive definite inner product that the algebra acts on by operators with adjoints? The answer is the axiom

$$ (\tilde R\cdot s,t)=\bigl(s,\tilde R^{*}\cdot t\bigr), $$

which says exactly that the action is a $*$-representation, and it is the condition that makes the informational sector the observables, the generators skew-adjoint, and the Dirac-type operator self-adjoint. The mathematical statement, its proof and its uniqueness are in Hermitian Modules over the Biquaternion Algebra with Hermitian Adjoint of the mathematical series, cited here; what this article adds is the physical reading, the two traps, and the operator that the axiom produces.

The Module Axiom as the Physicality of the Inner Product

Definition (the Hermitian form of the module). On the spinor module $S$ a Hermitian form is a complex sesquilinear form, linear in the second argument and conjugate-linear in the first; the axiom of a Hermitian Clifford module is $(\tilde R\cdot s,t)=(s,\tilde R^{*}\cdot t)$ for every element $\tilde R$ and every pair of spinors.

Physical reading. The form is the inner product of the internal Hilbert space, and the axiom is the statement that it is the right inner product: with it the action of the algebra becomes a $*$-representation, so every operator of the algebra has an adjoint and the framework's linear algebra is Hilbert-space linear algebra. Without the axiom one has a vector space with an action; with it one has the operator algebra of a quantum system. The form is not an extra choice: on the irreducible module it is unique up to a positive scalar (Schur, Hermitian Modules over the Biquaternion Algebra with Hermitian Adjoint), so the internal inner product is canonical and the Born-rule normalisation is a choice of unit and not a choice of structure.

Proposition (the sector criterion). The axiom is equivalent to the two statements that every element of the informational sector acts by a self-adjoint operator and every element of the material sector acts by a skew-adjoint one:

$$ (\tilde R\cdot s,t)=+(s,\tilde R\cdot t)\ (\tilde R\in\mathbb{M}_+), \qquad (\tilde R\cdot s,t)=-(s,\tilde R\cdot t)\ (\tilde R\in\mathbb{M}_-). $$

Physical reading. The informational sector is the observables: Hermitian operators, real expectation values. The material sector is the generators: skew-adjoint operators, purely imaginary spectrum, unitary one-parameter groups $e^{ta}$. The two sectors of the algebra are the two roles an operator can play, and the axiom is what fixes which sector plays which role.

The Two Traps

The first trap: the interval form is not the internal form. The form of this article is the scalar form of the dagger,

$$ (\tilde R,\tilde T)=\mathrm{Sc}(\tilde R^{*}\tilde T)=\sum_{\mu=0}^{3}R_\mu^{*}T_\mu , $$

positive definite on the whole eight-dimensional real algebra. The biquaternion norm $N(\tilde{Q})=\sum_\mu Q_\mu^{2}$, complex and indefinite, is a different form on the same algebra, and its restriction to the material sector is the Minkowski interval (Biquaternion Norm and Invertibility, The Clifford Structure of the Biquaternion Algebra). Positivity lives on the internal space, the signature $(1,3)$ on the material space, and no statement of one form transfers to the other. Writing the interval where the inner product belongs is the error that turns the state space into a cone and the symmetry group into the Lorentz group (Two-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint, Hermitian Forms over the Biquaternion Algebra and the Unitary Witt Group with Hermitian Adjoint).

The second trap: the naive spinor form can be isotropic, but not here. The general theory warns that the restriction of the scalar form to a minimal left ideal can be totally isotropic — of Gram matrix zero — in the split signatures, so that the naive construction does not give a spinor inner product (Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint). In the biquaternion algebra this failure cannot occur: the scalar form of the dagger is positive definite, so every subspace inherits a positive definite restriction, and the standard idempotents are self-adjoint, $\tilde\Pi_1^{*}=\tilde\Pi_1$, which is exactly the condition the general construction asks for. Concretely, on the ideal $S=\mathbb{C}\{\tilde\Pi_1,\tilde T\}$ the Gram matrix is $\tfrac12 I_2$ in that basis, positive definite: the spinor inner product exists and needs no correction. The framework is the resolved case, not the exceptional one.

The Spinor Module and Its Compact Symmetry

Theorem (the slice acts by unitaries). The unitary slice $U=\{\tilde A:\tilde A^{*}\tilde A=e_0\}=U(2)$ acts on the spinor module by unitary operators, $(\tilde A\cdot s,\tilde A\cdot t)=(s,t)$, so $\tilde A\mapsto\rho(\tilde A)|_S$ is a unitary representation of $U(2)$ on the internal Hilbert space.

Physical reading. The internal symmetry group of the spinor module is the compact group $U(2)$, and it is the group of unitaries of the internal space: the internal rotations, the internal phase, and nothing else. Its Lie algebra is the material sector, $\mathbb{M}_-\cong u(2)$, whose elements are the skew-adjoint generators $e_1,e_2,e_3$ and $ie_0$; the anticommutation and commutation relations of those generators are the internal Clifford relations (One-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint).

Remark (why the Lorentz group is not the internal symmetry). The norm-one slice $\mathbb{B}^{\times}_1=SL(2,\mathbb{C})$, the spin group of the material sector, is not a group of unitary operators of the internal form: only its compact part $SU(2)$ is. There is no positive definite form preserved by the spinor representation of the Lorentz group — the Lorentz-invariant bilinear pairing of spinors is the antisymmetric $\varepsilon$, not an inner product — and this is the operator-level reason the material symmetry is a symmetry of the interval and the internal symmetry is a symmetry of the inner product. The two groups act on different structures of the same algebra, and the framework's unitary group of the module is $U(2)$.

The Dirac Element of the Module

Definition. The Dirac element of the module is the sum of the left multiplications by the generators,

$$ D_{\mathrm{alg}}=\sum_{k=1}^{3}L_{e_k}, $$

which preserves the left ideals and so acts on the spinor module.

Theorem (square, adjoint and gap). On the spinor module the generators act by $c(e_k)=-i\sigma_k$, and

$$ D_S=-i\bigl(\sigma_1+\sigma_2+\sigma_3\bigr), \qquad D_S^{*}=-D_S, \qquad D_S^{2}=-3\,\mathrm{id}, \qquad D_S^{*}D_S=3\,\mathrm{id}>0 . $$

The real operator $iD_S$ is self-adjoint with spectrum $\{\pm\sqrt3\}$ and orthonormal eigenspinors, so the element is invertible, with no zero mode, and a gap of $\sqrt3$.

Physical reading. The operator is the internal Clifford element: the sum of the three internal complex structures, an internal "Dirac-type" operator rather than the spacetime Dirac operator. Its two structural properties are the ones the axiom guarantees: it is skew-adjoint, and its Hermitian square is positive. Positivity of the square is the statement that the internal Hilbert space has no negative-norm direction, and invertibility is the statement that the framework has no internal zero mode at this level: the internal Clifford element has a spectral gap. The spacetime Dirac operator $\sum_\mu e_\mu\partial_\mu$, its square and its analysis are a different operator on a different carrier and are in The Dirac Equation in Biquaternionic Form and The Spinor Module in Biquaternionic Form and Its Lorentz Action; the present one is the algebraic element without derivatives.

Corollary (formal self-adjointness of the Dirac operator with derivatives). For the operator $D=\sum_k\rho(e_k)\partial_k$ with $\partial_k^{*}=-\partial_k$ the two sign flips cancel and $D^{*}=D$: the Dirac operator of a Hermitian Clifford module is formally self-adjoint, and its square is the positive operator $\lVert D\cdot\rVert^{2}$. This is the property that makes the framework's Dirac dynamics unitary at the level of the inner product, and its analysis belongs to the analysis of the algebra.

Summary

The spinor module of the biquaternion algebra carries a canonical positive definite inner product, the scalar form of the dagger, and the algebra acts on it by a $*$-representation: the axiom $(\tilde R\cdot s,t)=(s,\tilde R^{*}\cdot t)$ is equivalent to the informational sector acting by observables and the material sector by generators. The form is unique up to a positive scalar on the irreducible module, so the internal inner product is not a choice. The two traps are that the interval form $N$, complex and indefinite, must never be used as the internal form, and that the general warning of an isotropic spinor form has no instance here, because the scalar form is positive definite and the standard idempotents are self-adjoint: on the minimal left ideal the Gram matrix is $\tfrac12 I_2$. The compact group $U(2)$ acts by unitaries on the module, while the Lorentz spin group $SL(2,\mathbb{C})$ does not — only $SU(2)$ does — because the material symmetry preserves the interval and the internal symmetry preserves the inner product. The Dirac element $D_{\mathrm{alg}}=\sum_kL_{e_k}$ acts as $-i(\sigma_1+\sigma_2+\sigma_3)$, is skew-adjoint, has square $-3\,\mathrm{id}$ and positive Hermitian square $3\,\mathrm{id}$; the real operator $iD_{\mathrm{alg}}$ is a self-adjoint internal observable with spectrum $\{\pm\sqrt3\}$, and the module has no zero mode.

Summary of Notation

Symbol	Meaning
$S$	The spinor module, $\cong\mathbb{C}^2$; the internal Hilbert space
$(\tilde R\cdot s,t)=(s,\tilde R^{*}\cdot t)$	The axiom; the action is a $*$-representation
$\mathbb{M}_+$ self-adjoint, $\mathbb{M}_-$ skew	Observables and generators
$(\tilde R,\tilde T)=\mathrm{Sc}(\tilde R^{}\tilde T)=\sum_\mu R_\mu^{}T_\mu$	The internal form; positive definite
$N(\tilde{Q})=\sum_\mu Q_\mu^{2}$	The interval; indefinite; never the internal form
Gram $=\tfrac12 I_2$ on $S=\mathbb{C}\{\tilde\Pi_1,\tilde T\}$	The spinor inner product; not isotropic
$U=U(2)$	The internal symmetry; unitary on $S$
$SL(2,\mathbb{C})$, $SU(2)$	Material spin group; only the compact part is unitary on $S$
$D_{\mathrm{alg}}=\sum_kL_{e_k}=-i(\sigma_1+\sigma_2+\sigma_3)$	Dirac element of the module
$D_{\mathrm{alg}}^{2}=-3\,\mathrm{id}$, $D_{\mathrm{alg}}^{*}D_{\mathrm{alg}}=3\,\mathrm{id}$	Clifford square and positive Hermitian square; gap $\sqrt3$