Real Spinors and Reality Conditions on the Biquaternion Algebra with Hermitian Adjoint

Introduction

A spinor is called real when it equals its own conjugate, and whether such a condition can be imposed is not a matter of taste: it is decided by the reality type of the spinor module, and the type is a function of the signature of the quadratic form modulo eight. The internal spinor space of the biquaternion framework is the defining module $S=\mathbb{B}\tilde\Pi_1\cong\mathbb{C}^2$ of $\mathbb{B}\cong M_2(\mathbb{C})$, of real dimension four, and the question of this article is which reality conditions it admits. The answer is none: the internal space is of complex type, so there is no internal Majorana partner; what the algebra supplies instead is the conjugate module, which is the algebraic source of the two-sided spinor structure of the applications. The mathematical companion is Real Spinors and Reality Conditions on the Biquaternion Algebra with Hermitian Adjoint; the general classification is Real Spinors and Reality Conditions with Inner Conjugation; the low-dimensional types and the general statements are taken from them and are cited, not re-derived.

The physical identifications used here are the corpus's. The internal space with its dagger form, positive definite and Hermitian, is Hermitian Modules over the Biquaternion Algebra with Hermitian Adjoint; the internal spinor, the two-sided Lorentz action and the doubling of the module are The Spinor Module in Biquaternionic Form and Its Lorentz Action; the identification of the coefficient conjugation $\bar{\cdot}$ with the charge-conjugation real structure, its fixed points the real quaternions, its exchange of $S$ with $\bar S$ and the Majorana condition on the doubled module are The Neutrino and Majorana Fermions in Biquaternionic Form; the antiunitary structures on the module and the symmetry classes are Kramers Degeneracy and Antiunitary Symmetry in Biquaternionic Form. The conventions are those of Conventions in the Biquaternion Universe: $\Phi(e_0)=I_2$, $\Phi(e_k)=-i\sigma_k$, $\Phi(i)=iI_2$, and the Clifford generators of $\mathrm{Cl}_{3,0}$ are $\gamma_k=ie_k$, with $\Phi(\gamma_k)=\sigma_k$.

The article's three statements are physical. First, the internal space is of complex type, and the reason is not an accident of dimension: the complex structure of the internal space — the $i$ of the internal quantum mechanics — is the volume element of $\mathrm{Cl}_{3,0}$, itself an element of the algebra, so any antilinear map commuting with the internal Clifford action is annihilated by it. Second, what replaces a reality structure is the conjugate module, and it is exactly the charge conjugation of the corpus: $\bar{\cdot}$ is the conjugate-linear automorphism that ties $S$ to its conjugate $\bar S$, the two being inequivalent as complex modules, and the Majorana condition, when it exists, is a condition on the doubled module and not on the internal one. Third, the ambient four-dimensional module does carry a real structure, and which of its two normalisations one uses decides whether the spinors are called Majorana; the corpus uses the charge-conjugation normalisation, and in that normalisation signature $(1,3)$ admits Majorana and admits no Majorana–Weyl spinor.

The Three Types, Translated

Definition (reality condition on a spinor). A reality condition on a complex spinor module is an antilinear map $J$ with $J^2=\pm1$ commuting with the Clifford action. A spinor $\psi$ is conjugate to $\psi^{c}=J(\psi)$; the condition $\psi^{c}=\psi$ is a Majorana condition and defines the fixed space of a real structure, while $J^2=-1$ is a quaternionic structure and a spinor obeying $\psi^{c}=\psi$ cannot exist for it.

The physical translation of the trichotomy is standard and is collected once.

$d\bmod8$	type of the module	reality structure	spinor
$0,1,2$	real	$J^2=+1$	Majorana
$3,7$	complex	none	Weyl, a conjugate pair
$4,5,6$	quaternionic	$J^2=-1$	symplectic

The physical reading of the three columns is: real spinors are their own antiparticles; complex spinors come in conjugate pairs with no self-conjugate member; quaternionic spinors obey a symplectic reality condition, and the quaternionic structure is the one that enforces Kramers degeneracy. The last row is the third row of the symmetry-class table of Kramers Degeneracy and Antiunitary Symmetry in Biquaternionic Form: $J^2=-1$ is the quaternionic class with its enforced double degeneracy, $J^2=+1$ the orthogonal class with none.

The Internal Space Is of Complex Type

Theorem (no internal reality structure). The internal space $S$, with the Clifford action $c(\gamma_k)=\sigma_k$ of $\mathrm{Cl}_{3,0}$, carries no antilinear $J$ with $J^2=\pm1$ commuting with the action. The internal spinor is therefore of complex type: there is no internal Majorana condition, and no internal reality structure of either sign.

Proof. The volume element $\omega=\gamma_1\gamma_2\gamma_3$ of $\mathrm{Cl}_{3,0}$ is central, satisfies $\omega^2=-1$, and acts on $S$ by the scalar $i$, $c(\omega)=iI$; in the algebra it is the internal imaginary unit, $\omega\mapsto i$. An antilinear map commuting with the generators commutes with every product, hence with $c(\omega)$, so $J(is)=J(c(\omega)s)=c(\omega)J(s)=iJ(s)$, while antilinearity gives $J(is)=-iJ(s)$; therefore $J=0$. The general sign table assigns $d=3$ the complex type, in agreement.

Physical reading (the internal $i$ is not a spectator). The physical content of the proof is worth separating from the computation. In the general theory the complex structure of the spinor module is external to the Clifford algebra: it is the $i$ of the field of scalars, and the Clifford action is real-linear with respect to it, which leaves room for an antilinear symmetry to exist or not. In the biquaternion algebra the internal complex structure is internal to the algebra: the $i$ that the internal quantum mechanics uses is the volume element of $\mathrm{Cl}_{3,0}$, and the internal Clifford action therefore knows about it. An antilinear internal symmetry would have to commute with the internal $i$, and no antilinear map does. This is the reason the internal type is complex and not merely "real in disguise".

Corollary (no internal Majorana, no internal Weyl). On $S$ there is no Majorana spinor, because that would be the fixed space of a real structure and there is none. There is also no Weyl spinor on $S$ alone: the internal chirality operator is $c(\omega)=iI$, of the single eigenvalue $i$, and it does not split the internal space. The two chiral halves, and with them the Weyl pair, appear only on the complexification, that is on the four-dimensional module of the ambient algebra, which is the subject of the next two sections. The internal spinor is therefore not a Majorana spinor and not a Weyl spinor; it is the two-component object whose conjugate lives in the conjugate module, and it becomes a Weyl or a Majorana spinor only when it is embedded in the four-dimensional module.

Charge Conjugation Is the Conjugate Module

Since there is no internal reality structure, the antilinear structure the framework actually uses is the conjugation of the algebra.

The operation. Among the conjugations of $\mathbb{B}$ the coefficient conjugation $\bar{\cdot}$ is the only conjugate-linear automorphism, the quaternion conjugation being $\mathbb{C}$-linear and the Hermitian and anti-Hermitian conjugations being order-reversing; its fixed points are the real quaternions $\mathbb{H}_{\mathbb{B}}$ (The Biquaternion Involution Lattice: Hermitian, Anti-Hermitian and Reversal). Charge conjugation must be a module map compatible with left multiplication, and only an automorphism can be that, so charge conjugation is tied to $\bar{\cdot}$ and not to the order-reversing involutions; the real structure itself lives on the doubled module $\Delta=S\oplus\bar S$, where it is conjugate-linear, satisfies $\mathcal{C}^2=1$ and exchanges the two summands (The Neutrino and Majorana Fermions in Biquaternionic Form).

Proposition (the conjugate module is what makes the spinor two-sided). The conjugate representation $\overline{c}(v)=\kappa\,c(v)\,\kappa^{-1}$, with $\kappa$ entrywise conjugation, is carried by the same complex space $S$ and differs from $c$ on the generators, so it makes $S$ into a second module $\bar S$. The two modules are isomorphic as real representations — the coordinate conjugation $s\mapsto\bar s$ intertwines them — and inequivalent as complex modules, which is exactly the complex type; in particular no $\mathbb{C}$-linear map takes $S$ to $\bar S$. The antisymmetric form $\varepsilon=i\sigma_2=\Phi(-e_2)$ is not such an intertwiner: it realises the coefficient conjugation $\bar{\cdot}$ on the module, $\varepsilon\,\overline{c}(v)\,\varepsilon^{-1}=c(\ast(v))$, and the conjugate pair of simple modules appears as two distinct factors after complexification (Real Spinors and Reality Conditions on the Biquaternion Algebra with Hermitian Adjoint).

Physical reading (why the internal spinor has a conjugate but not a partner). The two statements are the module-level form of the physical situation, and they are worth keeping apart. The internal spinor has a charge conjugate: the conjugate module $\bar S$ exists as a genuinely different complex module, $\bar{\cdot}$ relates the two, and the conjugate spinor is the second component of the doubled object the applications use. It has no Majorana partner: a Majorana spinor would be a fixed point of an internal real structure, and the internal space is of complex type. The doubling is therefore not a self-conjugacy condition inside $S$; it is the passage from $S$ to $S\oplus\bar S$, which is exactly the passage from the algebra's regular module $\mathbb{B}\cong S\oplus S$ to the Dirac module $\Delta=S\oplus\bar S$ (The Neutrino and Majorana Fermions in Biquaternionic Form). The conjugate module is thus the algebraic source of the internal two-sidedness, and it supplies a conjugate without supplying a Majorana condition.

Majorana in the Ambient Spacetime, and the Normalisation

The biquaternion algebra is the even part of a Minkowski Clifford algebra, $\mathbb{B}\cong\mathrm{Cl}^{+}_{3,1}\cong\mathrm{Cl}^{+}_{1,3}$ (The Clifford Structure of the Biquaternion Algebra), and the four-dimensional module is where the Majorana condition lives.

Theorem (the ambient types). In the commuting normalisation of the reality structure the four-dimensional module is real, and Majorana spinors exist, for $\mathrm{Cl}_{3,1}\cong M_4(\mathbb{R})$ with $d=2$, and quaternionic for $\mathrm{Cl}_{1,3}\cong M_2(\mathbb{H})$ with $d=-2\equiv6$. Since both have $\omega^2=-1$, composition with the volume element flips the sign of $J^2$, and in the charge-conjugation normalisation — the one in which a charge conjugation anticommutes with the Clifford generators — the roles are exchanged and it is $(1,3)$ that carries the real structure.

Physical reading (the normalisation is the whole of the apparent disagreement). The corpus works in signature $(1,3)$ with the metric $g=\mathrm{diag}(+,-,-,-)$ and writes the charge conjugation as $\psi^{c}=K\psi^{*}$ with $K=C\gamma^{0T}$ and $KK^{*}=I_4$; the relation $KK^{*}=I_4$ says $J^2=+1$, a real structure, and the fixed space is four-real-dimensional, so Majorana spinors exist in $(1,3)$ while Majorana–Weyl spinors do not, because the charge conjugation exchanges the two chiral halves (The Neutrino and Majorana Fermions in Biquaternionic Form). The same module is quaternionic in the commuting normalisation. The two statements are not in conflict; they are the two normalisations of one structure, exchanged by the volume element, and they agree in the classes $d\equiv0,1,4,5$ (Real Spinors and Reality Conditions with Inner Conjugation). The physical rule that follows is the one the article exists to state: a claim about Majorana spinors is a claim about a normalisation, and the two normalisations must not be compared without saying which is in force. For the internal space the question does not arise, since no antilinear structure exists there at all.

The Quaternionic Reading, Kramers and the Phase

The internal space does carry a quaternionic structure, but against a different Clifford action, and the distinction is the trap of the subject.

Proposition (the same basis against the negative definite form). The basis elements $e_k$ satisfy $e_k^2=-e_0$, the relations of $\mathrm{Cl}_{0,3}$, and against the action $c(e_k)=-i\sigma_k$ the module carries the quaternionic structure $J(s)=\sigma_2\bar s$ with $J^2=-1$. The two readings differ by the internal imaginary unit, $\gamma_k=ie_k$ and $c(\gamma_k)=i\,c(e_k)$: a map commuting with $c(\gamma_k)$ necessarily anticommutes with $c(e_k)$, because an antilinear map reverses $i$. So $\mathrm{Cl}_{3,0}$ gives the complex type and $\mathrm{Cl}_{0,3}$ the quaternionic type, on the same matrices and the same basis.

Physical reading (which internal structure is a symmetry). This is the physical content of the corpus's warning against silently switching between the generators $e_k$ and $\gamma_k=ie_k$ (The Clifford Structure of the Biquaternion Algebra). The quaternionic structure of the second reading is not an internal symmetry of the first: it anticommutes with the internal Clifford action, so it is not a symmetry of the internal Dirac operator. The structure that is available physically is the one Kramers Degeneracy and Antiunitary Symmetry in Biquaternionic Form constructs on the module, $J=UK$ with $U$ an imaginary unit of the algebra and $J^2=UU^{*}=-e_0$; the algebra supplies the unit $U$, and the conjugation $K$ is the module's, which is the corpus's discipline that an algebra involution is not yet a module conjugation. The relation between the two is the sign: $J^2=-e_0$ is the quaternionic class, the class that enforces the Kramers double degeneracy, while $J^2=+e_0$ is the orthogonal class and enforces none.

The phase. In the internal space the central unit is the phase, and the conjugate module is blind to it up to sign: the coefficient conjugation acts on a central unit of phase $\omega$ as $\omega^{*}=\bar\omega$, so the charge-conjugation defect of a phase is the unit $\bar\omega^2e_0$. Charge conjugation therefore commutes with every internal rotation and is detected exactly by the global phase, which is the operator-level statement of the corpus's real-structure reading (Mixed Inner Conjugation on the Biquaternion Algebra with Hermitian Adjoint).

Worked Examples

The internal $i$ is the volume element. With $\Phi(e_k)=-i\sigma_k$ and $\Phi(i)=iI_2$ one has $\Phi(\gamma_k)=\sigma_k$, $\Phi(\gamma_1\gamma_2\gamma_3)=\sigma_1\sigma_2\sigma_3=iI_2=\Phi(i)$, and $\omega^2=-e_0$. The internal imaginary unit is thus the volume element, and the no-go argument of the theorem is the statement $J(i\psi)=iJ(\psi)$ from the Clifford action against $J(i\psi)=-iJ(\psi)$ from antilinearity.

An internal Majorana candidate fails. Writing a general antilinear map on the internal space as $J(s)=T\bar s$, the internal Majorana condition would be the fixed space of such a $J$ commuting with $c(\gamma_k)=\sigma_k$. The commutation forces $T=aI$ and the commutation with the volume element forces $a=0$: the fixed space is empty and the candidate fails, which is the theorem in one line.

The quaternionic structure of the negative definite reading. For $J(s)=\sigma_2\bar s$ one computes $J^2=-I$; this is a genuinely quaternionic structure, of $J^2=-1$, and it commutes with $c(e_k)=-i\sigma_k$ but not with $c(\gamma_k)=\sigma_k$, where $\sigma_2\sigma_1=-\sigma_1\sigma_2$ breaks the relation at $k=1$. The example shows that the two readings give different types on the same space.

Charge conjugation on the doubled module. On $\Delta=S\oplus\bar S$ the charge conjugation exchanges the two summands and satisfies $\mathcal{C}^2=1$ in the charge-conjugation normalisation, the fixed space being parametrised by one half alone and being of real dimension four; the module-level computation is that of The Neutrino and Majorana Fermions in Biquaternionic Form, and the internal contribution is that the two summands are the two inequivalent complex modules $S$ and $\bar S$, related by the coefficient conjugation $\bar{\cdot}$ and by the antilinear coordinate conjugation of Real Spinors and Reality Conditions on the Biquaternion Algebra with Hermitian Adjoint.

Honest Limits

Four limits belong to the article. First, no claim about dynamics. The article states which reality conditions the modules admit and what the corpus reads into them; it does not claim that any particular internal dynamics conserves a charge conjugation or a time reversal, and the corpus's own caution on that point is kept. Second, the normalisation. The internal type is complex in either normalisation, since no antilinear structure exists to normalise; the statements about Majorana spinors in $(1,3)$ are statements in the charge-conjugation normalisation and are cited from the physics article rather than re-derived. Third, the algebra and the module. The three conjugate-linear involutions $\bar{\cdot}$, ${}^{*}$ and ${}^{\flat}$ of the algebra are not reality conditions on the module, and the equation "internal Majorana spinor $=$ fixed point of $\flat$" is the error the article exists to prevent; the module conjugation must be built on the module. Fourth, the competing labelling. The complex type of the internal space is a statement about the $\mathrm{Cl}_{3,0}$ reading with generators $\gamma_k=ie_k$; the $e_k$ reading gives the quaternionic type, and a computation that mixes the two gives a wrong sign for $J^2$. The corpus's spin-geometry article fixes the first reading, and the physics spinor module writes the same action in the $e_k$ basis; the two are related by the internal imaginary unit and must not be interchanged silently.

Summary

The internal spinor space $S\cong\mathbb{C}^2$ of the biquaternion algebra is of complex type: there is no antilinear map with $J^2=\pm1$ commuting with the internal Clifford action, so there is no internal Majorana spinor and no internal Weyl splitting. The reason is structural and is the biquaternion-specific content of the general classification: the internal imaginary unit is the volume element of $\mathrm{Cl}_{3,0}$, an element of the algebra, and an antilinear map commuting with the internal Clifford action must commute with it and is annihilated by its own antilinearity. What the algebra supplies instead is the conjugate module: the coefficient conjugation $\bar{\cdot}$ is the conjugate-linear automorphism that ties $S$ to its conjugate $\bar S$, the two being inequivalent as complex modules, and the real structure $\mathcal{C}$ lives on the doubled module, so the internal spinor acquires a charge conjugate without a Majorana partner. The conjugate module is the algebraic source of the two-sided spinor structure, the passage from $\mathbb{B}\cong S\oplus S$ to $\Delta=S\oplus\bar S$ being exactly the passage to the Dirac module. The Majorana condition itself lives on the ambient four-dimensional module, where the type is real in the charge-conjugation normalisation for signature $(1,3)$ — real structure, four real components, no Majorana–Weyl — and quaternionic in the commuting normalisation, the two being exchanged by the volume element. Finally, against the negative definite reading of the same basis, with generators $e_k$, the module carries the quaternionic structure $J(s)=\sigma_2\bar s$, $J^2=-1$, which is the quaternionic class of the symmetry table and the structure behind Kramers degeneracy; the two readings differ by the internal imaginary unit, which is the competing labelling of the algebra.

Summary of Notation

Symbol	Meaning
$S=\mathbb{B}\tilde\Pi_1\cong\mathbb{C}^2$	Internal spinor space, the defining module
$\gamma_k=ie_k$, $c(\gamma_k)=\sigma_k$	Generators of the internal $\mathrm{Cl}_{3,0}$ action
$\omega=\gamma_1\gamma_2\gamma_3\mapsto i$	Internal imaginary unit; the volume element; $c(\omega)=iI$
$J$, $\psi^{c}=J\psi$	Reality structure and the conjugate spinor
$\psi^{c}=\psi$	Majorana condition; fixed space of a real structure, $J^2=+1$
$J^2=-1$	Quaternionic structure; symplectic class; Kramers degeneracy
$\bar{\cdot}$	Coefficient conjugation; underlies charge conjugation, fixed points $\mathbb{H}_{\mathbb{B}}$
$\kappa$, $\varepsilon=i\sigma_2$	Entrywise conjugation and the antisymmetric form; the conjugate module $\bar S$
$\Delta=S\oplus\bar S$	Dirac module; carries the Majorana condition
$\mathrm{Cl}_{3,1}\cong M_4(\mathbb{R})$, $\mathrm{Cl}_{1,3}\cong M_2(\mathbb{H})$	Ambient forms; real and quaternionic in the commuting normalisation
Commuting, charge-conjugation	The two normalisations; exchanged by $\omega$, agreeing for $d\equiv0,1,4,5$