AI Advances Quantum Physics

Introduction

This article presents a new formulation of quantum mechanics, made by AI, that is natively compatible with relativity. It is part of a series of articles in the physics menu. The central object is the Hermitian subspace $\mathbb{M}_+$ of the biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$. Its elements are both the states and the observables of the theory. This is a structural feature that standard quantum mechanics does not have: in the standard formalism, states and observables are different kinds of objects (functionals on one side, operators on the other), while in the biquaternion framework they are elements of the same four-dimensional real subspace.

For the case of a single qubit, the framework reproduces the standard formalism: the Born rule, the Bloch sphere, and the projective measurement rule follow from the algebra, and the unitary dynamics has a natural algebraic formulation as rotor conjugation by a one-parameter group generated by a Hamiltonian. The purpose of this article is to develop this reformulation carefully, to state what is established and what is interpretation, and to flag the open questions.

The article is organized as follows. First the Hermitian subspace $\mathbb{M}_+$ and the two forms that survive on it (the Hermitian form, which is the trace pairing up to a factor of $2$, and the bilinear form $B$ whose quadratic form is the biquaternion norm) are recalled. Then the pure states are identified with the idempotents of $\mathbb{M}_+$, and the mixed states with the elements of the Bloch ball. Then the observables are identified with the general Hermitian elements, and the Born rule is derived from the trace formula. Then the dynamics is described, in the Schrödinger and Heisenberg pictures, in terms of rotor conjugation. Then measurement is described as an algebraic operation. Then the purity, entropy, and fidelity of states are expressed in the framework. Then the symmetries are discussed. Then the ten-point checklist is applied. Then the compatibility of the framework with relativity is developed in a section of its own. The article closes with what is structural and what is new, and with open questions.

Note on the isomorphism convention. Throughout this article, the isomorphism $\mathbb{B} \cong M_2(\mathbb{C})$ is taken to be the one that satisfies $e_j e_k = \sum_l \epsilon_{jkl}\, e_l$ for $j,k \in \{1,2,3\}$ with $j \neq k$ (the standard quaternion convention used in the algebraic articles of this series). Explicitly, the quaternion units map to

$$ e_0 \mapsto I_2, \quad e_1 \mapsto -i\sigma_1, \quad e_2 \mapsto -i\sigma_2, \quad e_3 \mapsto -i\sigma_3, $$

where $\sigma_1, \sigma_2, \sigma_3$ are the Pauli matrices and $i$ is the standard imaginary unit of $\mathbb{C} \subset M_2(\mathbb{C})$ (the image of the scalar imaginary of $\mathbb{B}$ under the isomorphism). Under this convention, a Hermitian element $\tilde{H} = h_0 e_0 + i\mathbf{h}$ maps to $h_0 I + \mathbf{h}\cdot\boldsymbol{\sigma}$, and the idempotent $\tilde\Pi_+(\hat{\mu})$ maps to the standard spin-up projector along $\hat{\mu}$.

The Hermitian Subspace

An element of $\mathbb{M}_+$ has the form

$$ \tilde{H} = h_0\, e_0 + i h_1\, e_1 + i h_2\, e_2 + i h_3\, e_3, \qquad h_0, h_1, h_2, h_3 \in \mathbb{R}. $$

The real scalar part is $h_0 e_0$, and the imaginary vector part is $i(h_1 e_1 + h_2 e_2 + h_3 e_3)$. In compact form,

$$ \tilde{H} = h_0\, e_0 + i\,\mathbf{h}, \qquad \mathbf{h} = (h_1, h_2, h_3) \in \mathbb{R}^3. $$

The subspace $\mathbb{M}_+$ is a real vector space of dimension 4. Its elements are Hermitian (fixed under the Hermitian conjugation ${}^{*}$).

Trace. Writing $\mathrm{Sc}(\tilde{Q})$ for the scalar part of $\tilde{Q}$, the trace of an element of $\mathbb{M}_+$ is twice its scalar part:

$$ \mathrm{Tr}(\tilde{H}) = 2\,\mathrm{Sc}(\tilde{H}) = 2 h_0. $$

Biquaternion norm. The biquaternion norm of $\mathbb{B}$ restricts to a real quadratic form of signature $(1,3)$ on $\mathbb{M}_+$:

$$ N(\tilde{H}) = \tilde{H}\tilde{H}^{\natural} = (h_0^2 - |\mathbf{h}|^2) e_0. $$

This is the form whose vanishing defines the light cone in $\mathbb{M}_+$; the nonzero elements on the cone are the zero divisors of $\mathbb{B}$ that lie in $\mathbb{M}_+$.

The Forms on $\mathbb{M}_+$

The algebra carries three forms, one for each nontrivial conjugation; on $\mathbb{M}_+$ two of them coincide, so two remain. They are the Hermitian form and the bilinear form $B$, and the one that drops out is the Krein form $K$, which agrees with $B$ on this subspace because the coefficient conjugation acts as the natural sign on a Hermitian element. The biquaternion norm is not a third bilinear form but the quadratic form of $B$.

The trace pairing. The trace defines a symmetric bilinear form on $\mathbb{M}_+\times\mathbb{M}_+$:

$$ \mathrm{Tr}(\tilde{H}\tilde{K}) = 2(h_0 k_0 + \mathbf{h}\cdot\mathbf{k}). $$

This is the Euclidean pairing on $\mathbb{R}^4$, up to a factor of 2. It is positive-definite, of signature $(4,0)$. It is the analogue of the Hilbert–Schmidt inner product on the space of operators, and it is the canonical Hermitian form of the algebra in real coordinates, $(\tilde{H}, \tilde{K}) = \mathrm{Sc}(\tilde{H}^{*}\tilde{K}) = h_0 k_0 + \mathbf{h}\cdot\mathbf{k}$, up to the factor of $2$ carried by the trace convention.

The biquaternion norm. The biquaternion norm is the quadratic form

$$ N(\tilde{H}) = h_0^2 - |\mathbf{h}|^2, $$

of signature $(1,3)$. It is the Lorentzian quadratic form whose future light cone defines the state space, and it is the diagonal of the bilinear form $B$ below, $N(\tilde{H}) = B(\tilde{H},\tilde{H})$. It is a quadratic form; it is not a third bilinear form.

The polarization of the biquaternion norm. The bilinear form associated with the biquaternion norm is obtained by symmetrizing the product $\tilde{H}\tilde{K}^{\natural}$:

$$ B(\tilde{H}, \tilde{K}) = \tfrac{1}{2}\left(\tilde{H}\tilde{K}^{\natural} + \tilde{K}\tilde{H}^{\natural}\right). $$

Computing directly, using $\tilde{H}^{\natural} = h_0 e_0 - i\mathbf{h}$ and the quaternion product of pure vector parts,

$$ \tilde{H}\tilde{K}^{\natural} = (h_0 k_0 - \mathbf{h}\cdot\mathbf{k})\,e_0 + i(k_0\mathbf{h} - h_0\mathbf{k}) + \mathbf{h}\times\mathbf{k}, $$

$$ \tilde{K}\tilde{H}^{\natural} = (h_0 k_0 - \mathbf{h}\cdot\mathbf{k})\,e_0 - i(k_0\mathbf{h} - h_0\mathbf{k}) - \mathbf{h}\times\mathbf{k}. $$

The sum is scalar:

$$ \boxed{\;B(\tilde{H}, \tilde{K}) = (h_0 k_0 - \mathbf{h}\cdot\mathbf{k})\,e_0.\;} $$

This is the Lorentzian bilinear form, of signature $(1,3)$, and the biquaternion norm is its diagonal, $B(\tilde{H}, \tilde{H}) = N(\tilde{H})$. On $\mathbb{M}_+$ it is also the Krein form $K(\tilde{H}, \tilde{K}) = \mathrm{Sc}(\bar{\tilde{H}}\tilde{K})$ of the companion article: for a Hermitian element the coefficient conjugation $\bar{\cdot}$ acts as the quaternion conjugation ${}^{\natural}$, so $K$ and $B$ agree on $\mathbb{M}_+$ even though they differ on the full algebra. That is why $\mathbb{M}_+$ carries two forms and not three.

The antisymmetric combination contains the vector part:

$$ \tfrac{1}{2}\left(\tilde{H}\tilde{K}^{\natural} - \tilde{K}\tilde{H}^{\natural}\right) = i(k_0\mathbf{h} - h_0\mathbf{k}) + \mathbf{h}\times\mathbf{k}. $$

This is not a symmetric bilinear form; it is the antisymmetric part of the full product, and it plays a role in the commutator structure of the algebra.

Summary of the two forms. On $\mathbb{M}_+$ two of the three forms of the algebra remain distinct, one Euclidean and one Lorentzian:

Form	Expression	Signature
Hermitian form (trace pairing $/2$)	$\mathrm{Tr}(\tilde{H}\tilde{K}) = 2(h_0 k_0 + \mathbf{h}\cdot\mathbf{k}) = 2\,\mathrm{Sc}(\tilde{H}^{*}\tilde{K})$	$(4,0)$, Euclidean
Bilinear form $B$ (the Krein form on $\mathbb{M}_+$)	$B(\tilde{H}, \tilde{K}) = h_0 k_0 - \mathbf{h}\cdot\mathbf{k}$	$(1,3)$, Lorentzian

The Hermitian form is the natural inner product on the space of operators, the one the Born rule and the unitary evolution use. The bilinear form $B$ is the one whose quadratic form is the biquaternion norm, $B(\tilde{H},\tilde{H}) = N(\tilde{H})$, and whose future light cone defines the state space. The antisymmetric combination displayed above is not a third form.

States

Pure States

A pure state of a qubit is a rank-one projection operator. In the biquaternion framework, the rank-one projections in $\mathbb{M}_+$ are the idempotents:

$$ \tilde\Pi_\pm(\hat{\mu}) = \tfrac{1}{2}\left(e_0 \pm i\,\hat{\mu}\right), $$

where $\hat{\mu} = \mu_1 e_1 + \mu_2 e_2 + \mu_3 e_3$ is a unit pure real quaternion, $|\hat{\mu}| = 1$.

Properties of the idempotents:

Hermitian: $\tilde\Pi_\pm^{*} = \tilde\Pi_\pm$.
Idempotent: $\tilde\Pi_\pm^2 = \tilde\Pi_\pm$.
Trace one: $\mathrm{Tr}(\tilde\Pi_\pm) = 1$.
Complementary: $\tilde\Pi_+ + \tilde\Pi_- = e_0$, and $\tilde\Pi_+ \tilde\Pi_- = 0 = \tilde\Pi_- \tilde\Pi_+$.
Zero divisors: $N(\tilde\Pi_\pm) = 0$.

The parametrization is by the unit sphere $S^2 \subset \mathbb{R}^3$: the direction $\hat{\mu}$ determines the idempotent $\tilde\Pi_+(\hat{\mu})$ uniquely, and the complementary idempotent $\tilde\Pi_-(\hat{\mu}) = \tilde\Pi_+(-\hat{\mu})$ corresponds to the opposite direction.

Identification with standard states. Under the isomorphism of this article, the idempotent $\tilde\Pi_+(\hat{\mu})$ maps to the standard pure state

$$ |\hat{\mu}+\rangle\langle\hat{\mu}+|, $$

i.e., the spin-up state along direction $\hat{\mu}$. The set of pure states is the Bloch sphere $S^2$.

Mixed States

A mixed state of a qubit is a positive Hermitian operator of trace one. In $\mathbb{M}_+$, such an element has the form

$$ \tilde{\rho} = \tfrac{1}{2}\left(e_0 + i\,\mathbf{r}\right), \qquad \mathbf{r} = (r_1, r_2, r_3) \in \mathbb{R}^3. $$

The positivity condition is the condition that the eigenvalues of the corresponding density matrix are non-negative:

$$ |\mathbf{r}|^2 = r_1^2 + r_2^2 + r_3^2 \leq 1. $$

The eigenvalues of $\tilde{\rho}$ are $\lambda_\pm = (1 \pm |\mathbf{r}|)/2$, which are non-negative if and only if $|\mathbf{r}| \leq 1$.

The state is pure if and only if $|\mathbf{r}| = 1$, i.e., if and only if $\tilde{\rho}$ is idempotent. It is mixed if $|\mathbf{r}| < 1$. The maximally mixed state has $\mathbf{r} = 0$, i.e., $\tilde{\rho} = \tfrac{1}{2} e_0$.

The Bloch Ball

The set of all states (pure and mixed) is the Bloch ball of radius one in $\mathbb{R}^3$:

$$ B^3 = \{\mathbf{r} \in \mathbb{R}^3 : |\mathbf{r}| \leq 1\}. $$

The pure states form the boundary sphere $|\mathbf{r}| = 1$; the maximally mixed state is at the center.

A structural observation. The positivity condition $|\mathbf{r}| \leq 1$ is equivalent to the condition $N(\tilde{\rho}) \geq 0$ on the biquaternion norm of $\mathbb{B}$:

$$ N(\tilde{\rho}) = \tfrac{1}{4}\left(1 - |\mathbf{r}|^2\right) e_0. $$

So the Bloch ball is the intersection of the affine hyperplane $\{\mathrm{Sc} = \tfrac{1}{2}\}$ with the future light cone of the biquaternion norm. The pure states form the boundary of the cone, which is also the set of zero divisors of $\mathbb{M}_+$ at trace one.

This is a clean geometric characterization: states are the trace-one elements of the future light cone of $\mathbb{M}_+$; pure states are the trace-one elements on the cone itself.

Observables

General Observables

A general observable is a Hermitian element of $\mathbb{M}_+$:

$$ \tilde{H} = h_0\, e_0 + i\,\mathbf{h}, \qquad \mathbf{h} = (h_1, h_2, h_3) \in \mathbb{R}^3. $$

The scalar part $h_0 e_0$ is the trace part: $\mathrm{Tr}(\tilde{H}) = 2 h_0$. The imaginary vector part $i\mathbf{h}$ is the traceless part.

The spectral decomposition of $\tilde{H}$ is

$$ \tilde{H} = \lambda_+ \tilde\Pi_+(\hat{\mathbf{h}}) + \lambda_- \tilde\Pi_-(\hat{\mathbf{h}}), $$

with

$$ \hat{\mathbf{h}} = \frac{\mathbf{h}}{|\mathbf{h}|}, \qquad \lambda_\pm = h_0 \pm |\mathbf{h}|. $$

The eigenvalues are $\lambda_\pm$, and the eigenvectors are the idempotents $\tilde\Pi_\pm(\hat{\mathbf{h}})$. So every Hermitian element of $\mathbb{M}_+$ has a spectral decomposition in terms of idempotents.

Traceless observables. The traceless Hermitian elements $i\mathbf{h}$ (with $h_0 = 0$) are the analogs of the traceless observables of the qubit (the Pauli matrices). They have eigenvalues $\pm|\mathbf{h}|$ and eigenvectors $\tilde\Pi_\pm(\hat{\mathbf{h}})$.

Compatibility

Two observables $\tilde{H}, \tilde{K} \in \mathbb{M}_+$ are compatible (simultaneously measurable) if and only if their corresponding matrices commute. In terms of the vector parts, this holds if and only if $\mathbf{h}$ and $\mathbf{k}$ are parallel.

In the general case (non-parallel $\mathbf{h}$, $\mathbf{k}$), the observables are incompatible, and their commutator is

$$ [\tilde{H}, \tilde{K}] = -2\left(\mathbf{h} \times \mathbf{k}\right), $$

where $\mathbf{h}\times\mathbf{k}$ is the ordinary cross product.

Derivation. We compute

$$ \tilde{H}\tilde{K} = (h_0 k_0 + \mathbf{h}\cdot\mathbf{k})e_0 + i(h_0\mathbf{k} + k_0\mathbf{h}) - \mathbf{h}\times\mathbf{k}, $$

$$ \tilde{K}\tilde{H} = (h_0 k_0 + \mathbf{h}\cdot\mathbf{k})e_0 + i(h_0\mathbf{k} + k_0\mathbf{h}) + \mathbf{h}\times\mathbf{k}, $$

$$ [\tilde{H}, \tilde{K}] = -2(\mathbf{h}\times\mathbf{k}). $$

The result is a pure real quaternion (vector part only, with real coefficients): it lies in $\mathbb{M}_- \cap \mathbb{H}_{\mathbb{B}}$, where $\mathbb{H}_{\mathbb{B}} = \operatorname{span}_{\mathbb{R}}\{e_0, e_1, e_2, e_3\}$ is the real quaternion subspace, and this intersection is the real span of $e_1, e_2, e_3$. This is the biquaternion form of the canonical commutation relations of the qubit.

The Born Rule

The Born rule gives the probability of an outcome of a measurement. In the biquaternion framework, it is a consequence of the trace formula.

Expectation Values

For an observable $\tilde{H} = h_0 e_0 + i\mathbf{h}$ and a state $\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$, the expectation value is

$$ \langle \tilde{H}\rangle_{\tilde{\rho}} = \mathrm{Tr}(\tilde{\rho}\tilde{H}) = h_0 + \mathbf{r}\cdot\mathbf{h}. $$

Derivation. We compute

$$ \tilde{\rho}\tilde{H} = \tfrac{1}{2}(e_0 + i\mathbf{r})(h_0 e_0 + i\mathbf{h}) = \tfrac{1}{2}\left(h_0 e_0 + i\mathbf{h} + i h_0 \mathbf{r} + (i\mathbf{r})(i\mathbf{h})\right). $$

For pure quaternions, $(i\mathbf{r})(i\mathbf{h}) = -\mathbf{r}\mathbf{h} = \mathbf{r}\cdot\mathbf{h} - \mathbf{r}\times\mathbf{h}$. Taking the scalar part,

$$ \mathrm{Sc}(\tilde{\rho}\tilde{H}) = \tfrac{1}{2}(h_0 + \mathbf{r}\cdot\mathbf{h}), $$

so $\mathrm{Tr}(\tilde{\rho}\tilde{H}) = 2\mathrm{Sc}(\tilde{\rho}\tilde{H}) = h_0 + \mathbf{r}\cdot\mathbf{h}$.

This is exactly the standard formula $\mathrm{Tr}(\rho H) = h_0 + \mathbf{r}\cdot\mathbf{h}$ for a qubit. So the Born rule is a direct consequence of the trace pairing on $\mathbb{M}_+$. It is not an independent postulate.

Probabilities

For a projective measurement of an observable $\tilde{H} = h_0 e_0 + i\mathbf{h}$ in a state $\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$, the two outcomes are $\lambda_\pm = h_0 \pm |\mathbf{h}|$, with probabilities

$$ p_\pm = \tfrac{1}{2}\left(1 \pm \hat{\mathbf{h}}\cdot\mathbf{r}\right), $$

where $\hat{\mathbf{h}} = \mathbf{h}/|\mathbf{h}|$. The probabilities depend only on the angle between the measurement direction $\hat{\mathbf{h}}$ and the Bloch vector $\mathbf{r}$.

The derivation is direct: $p_+ = \mathrm{Tr}(\tilde\Pi_+(\hat{\mathbf{h}})\tilde{\rho}) = \tfrac{1}{2}(1 + \hat{\mathbf{h}}\cdot\mathbf{r})$, using the trace formula.

Measurement Update

After a measurement of $\tilde{H}$ with outcome $+$, the state is updated by the von Neumann–Lüders rule:

$$ \tilde{\rho}' = \frac{\tilde\Pi_+(\hat{\mathbf{h}})\,\tilde{\rho}\,\tilde\Pi_+(\hat{\mathbf{h}})}{\mathrm{Tr}(\tilde\Pi_+(\hat{\mathbf{h}})\tilde{\rho})}. $$

Since $\tilde\Pi_+$ is idempotent, and since $\tilde\Pi_+\tilde{\rho}\tilde\Pi_+$ has rank at most one (as a product of projectors of rank one), the sandwich is proportional to $\tilde\Pi_+$:

$$ \tilde\Pi_+\tilde{\rho}\tilde\Pi_+ = p_+ \tilde\Pi_+, $$

where $p_+ = \mathrm{Tr}(\tilde\Pi_+\tilde{\rho})$. Hence the post-measurement state is simply

$$ \tilde{\rho}' = \tilde\Pi_+(\hat{\mathbf{h}}). $$

So the state collapses to the idempotent associated with the outcome. This is the standard collapse, expressed as a composition of algebraic operations.

Dynamics

The Schrödinger Picture

Time evolution of a pure state. A pure state is an idempotent $\tilde{P} = \tfrac{1}{2}(e_0 + i\hat{\mu})$. It evolves in time by conjugation with a one-parameter group of unitary elements:

$$ \tilde\Pi(t) = \tilde{U}(t)\,\tilde\Pi(0)\,\tilde{U}(t)^{*}, $$

where $\tilde{U}(t)$ is a unitary biquaternion, $\tilde{U}\tilde{U}^{*} = e_0$, satisfying the Schrödinger equation

$$ i\hbar\, \frac{d}{dt}\tilde{U}(t) = \tilde{H}\,\tilde{U}(t), \qquad \tilde{U}(0) = e_0, $$

with $\tilde{H} = h_0 e_0 + i\mathbf{h} \in \mathbb{M}_+$ a Hermitian element (the Hamiltonian).

The solution. We have $-i\tilde{H} = -ih_0 e_0 + \mathbf{h}$. The scalar part and the vector part commute, so

$$ \tilde{U}(t) = \exp(-i\tilde{H}t/\hbar) = e^{-ih_0 t/\hbar}\left(\cos(|\mathbf{h}|t/\hbar)\,e_0 + \sin(|\mathbf{h}|t/\hbar)\,\hat{\mathbf{h}}\right), $$

where $\hat{\mathbf{h}} = \mathbf{h}/|\mathbf{h}|$ (for $\mathbf{h} \neq 0$). The exponential of the vector part uses $\hat{\mathbf{h}}^2 = -e_0$, which gives the trigonometric form.

The element $\tilde{U}(t)$ is unitary: $\tilde{U}(t)\tilde{U}(t)^{*} = e_0$. It is not Hermitian in general: writing $\theta = |\mathbf{h}|t/\hbar$, $\tilde{U}(t)$ is Hermitian exactly when $\tilde{U}(t)^2 = e_0$, which requires either $\sin\theta = 0$ together with $h_0t/\hbar \in \pi\mathbb{Z}$, or $\cos\theta = 0$ together with $h_0t/\hbar \in \tfrac{\pi}{2} + \pi\mathbb{Z}$. For $h_0 = 0$ only the first case survives, at the isolated times $t = n\pi\hbar/|\mathbf{h}|$; for $h_0 \neq 0$ the two conditions must hold simultaneously, so the Hermitian times are more restricted than this. It is a general element of the unitary group $U(2) \subset \mathbb{B}$.

The trace part of the Hamiltonian does not affect the state. Since $e^{-ih_0 t/\hbar}$ is a central complex scalar, and since the conjugation

$$ \tilde{U}(t)\,\tilde{P}\,\tilde{U}(t)^{*} = e^{-ih_0 t/\hbar}R\,\tilde{P}\,e^{+ih_0 t/\hbar}R^{*} = R\,\tilde{P}\,R^{*} $$

(with $R = \cos(|\mathbf{h}|t/\hbar)e_0 + \sin(|\mathbf{h}|t/\hbar)\hat{\mathbf{h}}$ a unit real quaternion), the phase $e^{-ih_0 t/\hbar}$ cancels out. So only the traceless part $\mathbf{h}$ of the Hamiltonian affects the state evolution. This is the standard result: the trace part of the Hamiltonian contributes only to a global, unobservable phase.

Time evolution of a mixed state. A general state $\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$ evolves by the same conjugation:

$$ \tilde{\rho}(t) = \tilde{U}(t)\,\tilde{\rho}(0)\,\tilde{U}(t)^{*}. $$

This is the von Neumann equation, expressed as a rotor conjugation. In differential form, it is

$$ i\hbar\, \frac{d}{dt}\tilde{\rho}(t) = [\tilde{H}, \tilde{\rho}(t)], $$

with the commutator defined in the biquaternion algebra.

Interpretation. The unitary $\tilde{U}(t)$ is an inner automorphism of the algebra: conjugation by a unitary biquaternion preserves multiplication. Its action on $\mathbb{M}_+$ keeps the scalar part fixed and rotates the imaginary vector part, giving the standard precession of a spin-1/2 in a magnetic field, expressed as the biquaternion rotation.

The Heisenberg Picture

In the Heisenberg picture, the state is fixed and the observables evolve:

$$ \tilde{H}_{\mathrm{H}}(t) = \tilde{U}(t)^{*}\,\tilde{H}\,\tilde{U}(t). $$

The evolution equation is the Heisenberg equation

$$ i\hbar\, \frac{d}{dt}\tilde{H}_{\mathrm{H}}(t) = [\tilde{H}_{\mathrm{H}}(t), \tilde{H}_0], $$

where $\tilde{H}_0$ is the Hamiltonian.

Derivation. We compute the time derivative of $\tilde{H}_{\mathrm{H}}(t) = \tilde{U}^{*}\tilde{H}\tilde{U}$, using the Schrödinger equation $i\hbar\,\dot{\tilde{U}} = \tilde{H}_0\tilde{U}$ and its adjoint $-i\hbar\,\dot{\tilde{U}}^{*} = \tilde{U}^{*}\tilde{H}_0$:

$$ i\hbar\, \frac{d}{dt}\tilde{H}_{\mathrm{H}} = i\hbar\left(\dot{\tilde{U}}^{*}\tilde{H}\tilde{U} + \tilde{U}^{*}\tilde{H}\dot{\tilde{U}}\right) = -\tilde{U}^{*}\tilde{H}_0\tilde{H}\tilde{U} + \tilde{U}^{*}\tilde{H}\tilde{H}_0\tilde{U} = \tilde{U}^{*}[\tilde{H}, \tilde{H}_0]\tilde{U}. $$

Since $\tilde{U}(t) = \exp(-i\tilde{H}_0 t/\hbar)$ is a function of $\tilde{H}_0$, it commutes with $\tilde{H}_0$. This is what makes the last expression equal to $[\tilde{H}_{\mathrm{H}}, \tilde{U}^{*}\tilde{H}_0\tilde{U}] = [\tilde{H}_{\mathrm{H}}, \tilde{H}_0]$. So

$$ i\hbar\, \frac{d}{dt}\tilde{H}_{\mathrm{H}}(t) = [\tilde{H}_{\mathrm{H}}(t), \tilde{H}_0]. $$

The two pictures are related by a unitary transformation and give the same physical predictions.

Reversibility

The evolution generated by a Hermitian Hamiltonian is unitary: the element $\tilde{U}(t)$ satisfies $\tilde{U}(t)\tilde{U}(t)^{*} = e_0$. The evolution is therefore reversible: the state at any later time can be recovered from an earlier time by applying $\tilde{U}(t)^{*}$.

Reversibility is one of the structural features of the biquaternion formulation. The unitary elements form the group $U(2) \subset \mathbb{B}$, and their conjugation action on $\mathbb{M}_+$ preserves the trace pairing and the biquaternion norm.

Measurement as an Algebraic Operation

In the standard framework, the measurement postulate is stated separately from the unitary evolution. In the biquaternion framework, both are consequences of the same algebraic structure.

Projective measurement. A projective measurement of $\tilde{H} = h_0 e_0 + i\mathbf{h}$ is specified by the spectral decomposition $\tilde{H} = \lambda_+ \tilde\Pi_+ + \lambda_- \tilde\Pi_-$, with $\tilde\Pi_\pm = \tfrac{1}{2}(e_0 \pm i\hat{\mathbf{h}})$. The outcome $+$ has:

Probability $p_+ = \mathrm{Tr}(\tilde\Pi_+\tilde{\rho})$, from the trace formula.
Post-measurement state $\tilde{\rho}' = \tilde\Pi_+$, from the sandwich operation.

Generalized measurement. A general measurement is described by a set of Kraus operators $\{\tilde{K}_i\} \subset \mathbb{B}$ satisfying $\sum_i \tilde{K}_i^{*}\tilde{K}_i = e_0$. The action on a state is

$$ \tilde{\rho} \;\longmapsto\; \sum_i \tilde{K}_i\,\tilde{\rho}\,\tilde{K}_i^{*}, $$

and the probability of outcome $i$ is $p_i = \mathrm{Tr}(\tilde{K}_i^{*} \tilde{K}_i \tilde{\rho})$. The Kraus operators are elements of $\mathbb{B}$, and the framework inherits the full POVM formalism in biquaternion language.

A structural point. The reversible–irreversible distinction is algebraic in the biquaternion framework. Unitary elements $\tilde{U}$ preserve the biquaternion norm of $\mathbb{B}$ and generate reversible evolution. Idempotent elements $\tilde{P}$ do not preserve the biquaternion norm (they satisfy $\tilde{P}^2 = \tilde{P}$ but $\tilde{P}\tilde{P}^{*} = \tilde{P} \neq e_0$) and generate irreversible projection. The two kinds of process are distinguished by a property of the acting element, not by a separate postulate.

Purity, Entropy, and Fidelity

Purity

The purity of a state $\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$ is

$$ \mathrm{Tr}(\tilde{\rho}^2) = \tfrac{1}{2}\left(1 + |\mathbf{r}|^2\right), $$

which ranges from $\tfrac{1}{2}$ (maximally mixed, $\mathbf{r} = 0$) to $1$ (pure, $|\mathbf{r}| = 1$).

Deviation from Idempotency

The deviation of $\tilde{\rho}$ from being idempotent is

$$ \tilde{\rho}^2 - \tilde{\rho} = \tfrac{1}{4}\left(|\mathbf{r}|^2 - 1\right)e_0. $$

So the mixedness of a state is captured by the scalar part of $\tilde{\rho}^2 - \tilde{\rho}$. Purity is the condition $\tilde{\rho}^2 = \tilde{\rho}$: the pure states are the fixed points of the squaring map.

Spectral Decomposition and Entropy

The spectral decomposition of $\tilde{\rho}$ is

$$ \tilde{\rho} = \lambda_+ \tilde\Pi_+(\hat{\mathbf{r}}) + \lambda_- \tilde\Pi_-(\hat{\mathbf{r}}), $$

with $\hat{\mathbf{r}} = \mathbf{r}/|\mathbf{r}|$ and $\lambda_\pm = (1 \pm |\mathbf{r}|)/2$. The von Neumann entropy is

$$ S(\tilde{\rho}) = -\mathrm{Tr}(\tilde{\rho}\log\tilde{\rho}) = -\lambda_+ \log \lambda_+ - \lambda_- \log \lambda_-, $$

which depends only on $|\mathbf{r}|$ and vanishes for pure states.

Fidelity

For two pure states $\tilde{P} = \tfrac{1}{2}(e_0 + i\hat{\mu})$ and $\tilde{Q} = \tfrac{1}{2}(e_0 + i\hat{\nu})$, the transition probability (or fidelity squared) is

$$ \mathrm{Tr}(\tilde{P}\tilde{Q}) = \tfrac{1}{2}\left(1 + \hat{\mu}\cdot\hat{\nu}\right). $$

This is the standard formula $|\langle\hat{\mu}|\hat{\nu}\rangle|^2 = \cos^2(\theta/2)$, where $\theta$ is the angle between $\hat{\mu}$ and $\hat{\nu}$ on the Bloch sphere.

For mixed states, the fidelity is given by the Uhlmann formula, which in the biquaternion framework is expressed in terms of the spectral decompositions.

Symmetries

The Unitary Group

The unitary group $U(2)$ acts on $\mathbb{M}_+$ by conjugation:

$$ \tilde{\rho} \;\longmapsto\; \tilde{U}\,\tilde{\rho}\,\tilde{U}^{*}, \qquad \tilde{U} \in U(2). $$

This action preserves the Hermitian property and the trace, so it maps states to states and observables to observables.

Unit Quaternions and Spin Rotations

The unit quaternions (elements of $\mathbb{H}_{\mathbb{B}}$ with unit quaternion norm) form the subgroup $SU(2) \subset U(2)$. Their conjugation action on $\mathbb{M}_+$ preserves the scalar part and rotates the imaginary vector part, giving the rotation group $SO(3)$ acting on the Bloch sphere. This is the standard spin-1/2 rotation group, expressed in the biquaternion framework.

The Lie Algebra

The Lie algebra of the unitary group is the space of anti-Hermitian elements, $\mathbb{M}_-$, with the commutator bracket:

$$ [\tilde{Q}, \tilde{Y}] = \tilde{Q}\tilde{Y} - \tilde{Y}\tilde{Q}, \qquad \tilde{Q}, \tilde{Y} \in \mathbb{M}_-. $$

This is a Lie algebra structure on $\mathbb{M}_-$ (which is not a subalgebra of $\mathbb{B}$, but is a Lie algebra under the commutator). The exponential map sends $\mathbb{M}_-$ to the group of unitary biquaternions.

So the subspace $\mathbb{M}_+$ carries the states and observables; the subspace $\mathbb{M}_-$ carries the generators of the symmetry group. The two subspaces are complementary in the Hermitian decomposition of $\mathbb{B}$.

Application of the Ten-Point Checklist

The companion article listed ten structural requirements for a comprehensive quantum theory. We now assess the biquaternion framework against each.

1. State space. The states are the positive trace-one elements of $\mathbb{M}_+$, i.e., the Bloch ball. This is a canonical object of the algebra, not a postulate.

2. Observables. The observables are the Hermitian elements of $\mathbb{M}_+$. The spectral decomposition is in terms of idempotents.

3. Born rule. The Born rule is the trace formula $\mathrm{Tr}(\tilde{\rho}\tilde{H})$. It is a consequence of the trace pairing on $\mathbb{M}_+$, not an independent postulate.

4. Dynamics. The dynamics is generated by unitary biquaternions via rotor conjugation, with the Schrödinger equation for the unitary and the von Neumann equation for the state. This matches the standard formalism.

5. Measurement rule. Projective measurement and state update are expressed as algebraic operations: probability from the trace formula, state update from the sandwich operation. The reversible–irreversible distinction is algebraic (unitary vs. idempotent).

6. Composition rule. The tensor product of two qubits corresponds to $\mathbb{B}\otimes_\mathbb{C}\mathbb{B} \cong M_4(\mathbb{C})$ (tensor product over $\mathbb{C}$, treating $\mathbb{B}$ as a $\mathbb{C}$-algebra), and the partial trace is defined in terms of the standard trace on the tensor product. This is a natural extension of the framework.

7. Symmetry group. The unitary group $U(2)$ acts on $\mathbb{M}_+$ by conjugation, with $SU(2)$ generating the spin rotations and $SO(3)$ acting on the Bloch sphere.

8. Relativistic extension. The biquaternion algebra contains the proper orthochronous Lorentz group through its double cover $SL(2,\mathbb{C})$, realized as the group of biquaternions of unit norm, and the material sector $\mathbb{M}_-$ carries the four-vectors. The algebraic structure required for the Lorentz group and the spinor representation is present in the algebra. The quantum formalism presented here and the relativistic structure share the same algebraic home, which makes the framework naturally suited to a relativistic quantum theory of spinor fields. The classical precursor of such a theory — the biquaternion Dirac equation — is treated in the companion article, and the quantization of the spinor field is a natural direction for future work.

9. Classical limit. The classical limit of the qubit formalism is the classical spin, obtained when the state is in a coherent state (a pure state) and the dynamics is a classical precession. The general classical limit of quantum mechanics requires the machinery of decoherence and the correspondence principle, which are not developed in this article.

10. Interpretation. The biquaternion framework admits several interpretive readings. The most natural is algebraic: the states and observables of a quantum system are elements of the Hermitian subspace $\mathbb{M}_+$, and the quantum structure is a consequence of the algebra. This is a structural interpretation, not a metaphysical one, and it is compatible with several of the standard interpretations (Copenhagen, relational, informational) without committing to any of them.

Compatibility with Relativity

Item 8 of the checklist states the compatibility with relativity as a requirement to be met. It deserves a section of its own, because the compatibility is stronger than the coexistence of two structures inside one algebra: the relativistic and the quantum objects are made from the same elements, and their transformation groups are two subgroups of the same invertible elements, acting by one and the same conjugation formula.

One conjugation action, two groups. The symmetry transformations of this article and of its relativistic companions all act by conjugation,

$$ \tilde{Q} \;\longmapsto\; \tilde{\Lambda}\,\tilde{Q}\,\tilde{\Lambda}^{*}, \qquad \tilde{\Lambda} \in \mathbb{B} \ \ \text{invertible}, $$

and the two theories differ only in the subgroup of the invertible elements over which $\tilde{\Lambda}$ ranges. The quantum operations of the Schrödinger picture are the unitary case, $\tilde{\Lambda}\tilde{\Lambda}^{*} = e_0$; these elements form the group $U(2)$. The proper orthochronous Lorentz transformations of $\mathbb{M}_-$ are the case of unit norm, $\tilde{\Lambda}\tilde{\Lambda}^{\natural} = e_0$; these elements form $SL(2,\mathbb{C})$, the double cover of that group. The two normalization conditions are genuinely different — neither group contains the other — yet both act by the same formula, on the same algebra, and their intersection is the rotation group:

$$ SU(2) \subset U(2), \qquad SU(2) \subset SL(2,\mathbb{C}), \qquad U(2) \cap SL(2,\mathbb{C}) = SU(2). $$

The intersection is the set of unit real quaternions. Its conjugation action rotates the Bloch vector of a state in $\mathbb{M}_+$ and the spatial three-vector of a four-vector in $\mathbb{M}_-$ by the same $SO(3)$ rotation, with the same rotor. The spin rotation of the qubit and the spatial rotation of the material sector are therefore one algebraic operation, not two that happen to agree. This is the double cover $SU(2) \to SO(3)$ of the companion articles, read inside the single algebra $\mathbb{B}$.

What distinguishes the two groups. The two groups are subgroups of the same invertible elements, acting by the same formula, and what separates them is the algebraic character of the transformation. For a unitary rotor the conjugation is multiplicative,

$$ \tilde{U}\left(\tilde{Q}\tilde{Y}\right)\tilde{U}^{*} = \left(\tilde{U}\tilde{Q}\tilde{U}^{*}\right)\left(\tilde{U}\tilde{Y}\tilde{U}^{*}\right), \qquad \tilde{U} \in U(2), $$

so it is an automorphism of $\mathbb{B}$; conjugation by a non-unitary element has no such property, and is only a congruence. The difference is visible in the quadratic forms each conjugation leaves invariant.

The trace pairing $\mathrm{Tr}(\tilde{H}\tilde{K})$, the Euclidean Hilbert–Schmidt form on $\mathbb{M}_+$ on which the Born rule rests, is preserved by exactly the unitary group:

$$ \mathrm{Tr}\!\left(\tilde{\Lambda}\tilde{H}\tilde{\Lambda}^{*}\,\tilde{\Lambda}\tilde{K}\tilde{\Lambda}^{*}\right) = \mathrm{Tr}(\tilde{H}\tilde{K}) \ \ \text{for all Hermitian } \tilde{H},\tilde{K} \quad \Longleftrightarrow \quad \tilde{\Lambda} \in U(2), $$

since the left-hand side is $\mathrm{Tr}\big((\tilde{\Lambda}^{*}\tilde{\Lambda})\tilde{K}(\tilde{\Lambda}^{*}\tilde{\Lambda})\tilde{H}\big)$, and the condition on $\tilde{\Lambda}^{*}\tilde{\Lambda}$ is the unitarity condition. A boost fails it.

The biquaternion norm $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural}$, and with it the interval on $\mathbb{M}_-$, is invariant under the whole rotor group, by multiplicativity of the biquaternion norm together with $N(\tilde{\Lambda}^{*}) = \overline{N(\tilde{\Lambda})} = 1$:

$$ N\!\left(\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}\right) = N(\tilde{\Lambda})\,N(\tilde{Q})\,N(\tilde{\Lambda}^{*}) = N(\tilde{Q}), \qquad \tilde{\Lambda} \in SL(2,\mathbb{C}). $$

This form does not, however, separate the two groups, and the point is worth stating: since $|N(\tilde{U})| = |\det\tilde{U}| = 1$ for a unitary element, the unitary group preserves the biquaternion norm as well. In general $N$ is invariant under conjugation by every element with $|N| = 1$, of which the unitary elements and the unit-norm rotors are the two cases that occur here. The Lorentzian form is thus common to the quantum and the relativistic transformations. What separates them is the Euclidean trace pairing, which only the quantum operations preserve, together with the automorphism property, which only they possess. Neither group contains the other, and their common subgroup is the rotation group $SU(2)$.

A boost is not a quantum operation. A pure boost has a Hermitian rotor, and therefore lies in $\mathbb{M}_+$ itself, the subspace that carries the states and observables:

$$ \tilde{\Lambda} = \cosh\frac{\psi}{2}\,e_0 + i\sinh\frac{\psi}{2}\,\hat{\mathbf{u}}, \qquad N(\tilde{\Lambda}) = \cosh^2\frac{\psi}{2} - \sinh^2\frac{\psi}{2} = 1 . $$

It lies on the norm-one hyperboloid of $\mathbb{M}_+$, whereas the states lie on the trace-one slice of the future light cone, with $N(\tilde{\rho}) = \tfrac{1}{4}(1 - |\mathbf{r}|^2) \geq 0$ and pure states at $N = 0$. Boost conjugation preserves Hermiticity, positivity (a congruence preserves the inertia), and the biquaternion norm, but it preserves neither the trace nor the trace pairing. On the maximally mixed state, for instance,

$$ \tilde{\Lambda}\,\tfrac{1}{2}e_0\,\tilde{\Lambda}^{*} = \tfrac{1}{2}\left(\cosh\psi\,e_0 + i\sinh\psi\,\hat{\mathbf{u}}\right), $$

whose trace is $\cosh\psi$, not $1$. A boost therefore maps the state cone into itself while moving the state off the trace-one slice; the physical state is recovered by renormalizing. This is the algebraic form of a familiar relativistic fact: the Lorentz transformation of a state is not unitary, and the quantity it preserves is the Lorentzian biquaternion norm rather than the Euclidean trace pairing.

The Wigner rotation. The product of two boost rotors is not Hermitian in general — only when the two boosts are collinear. Its polar decomposition,

$$ \tilde{\Lambda}_2\tilde{\Lambda}_1 = \tilde{W}\,\tilde{\Lambda}, \qquad \tilde{W} \in SU(2), $$

separates a unitary factor $\tilde{W}$ — the Thomas–Wigner rotation — from a positive-definite Hermitian boost factor $\tilde{\Lambda}$, both of unit norm. The composition of two relativistic transformations therefore produces a genuine quantum operation, a spin rotation: the relativistic group generates an element of the quantum group. This is where the two structures meet, and it is the algebraic origin of the non-commutativity of the boosts, treated in the companion article on the Lorentz group.

The shared imaginary unit. The $i$ of the Schrödinger equation $i\hbar\,d\tilde{U}/dt = \tilde{H}\tilde{U}$, the $i$ of the phase $e^{-i\tilde{H}t/\hbar}$, and the $i$ of the material coordinate $ict$ are one and the same element of $\mathbb{B}$: the scalar imaginary. The generator of the quantum phase and the origin of the Lorentzian signature of the interval are not independent conventions, but two consequences of the complex structure that defines the material sector. The quantum dynamics of this article runs on the same complex time as the relativistic kinematics of $\mathbb{M}_-$.

What the compatibility does and does not provide. The framework supplies a common algebraic home: the relativistic and quantum objects are made from the same elements, their transformation groups are subgroups of the same invertible elements acting by one conjugation formula, their intersection is the rotation group, and the Wigner rotation has an algebraic locus. It does not supply, by itself, a relativistic quantum dynamics. A Lorentz-covariant state space, microcausality, and the many-particle structure are not contained in the single-qubit formalism above, which is the non-relativistic restriction: the state space is a fixed Bloch ball, and a boost moves a state off it. Those questions belong to the biquaternion Dirac equation and its canonical quantization, treated in the companion articles, where the state space, and not only the transformation group, becomes relativistic.

What Is Structural and What Is New

What is standard QM, reformulated.

The qubit state space (Bloch ball) is recovered.
The Born rule is $\mathrm{Tr}(\tilde{\rho}\tilde{H})$.
The unitary dynamics is rotor conjugation.
The projective measurement rule is the sandwich operation.
The spectral decomposition is in terms of idempotents.
The entropy and fidelity have the standard forms.

What is structurally new or newly visible.

States and observables live in the same subspace. In the standard formalism, states and observables are both operators, but conceptually different (functionals vs. operators). In $\mathbb{M}_+$, both are elements of the same four-dimensional real subspace, paired by the trace.
The state space is canonical. The idempotents of $\mathbb{M}_+$ are canonical objects of the algebra; the Bloch sphere is their parametrization by $S^2$. The Bloch ball is the intersection of the trace-one hyperplane with the future light cone of the biquaternion norm.
The Born rule is algebraic. It is a consequence of the trace pairing, not an independent postulate.
Measurement is algebraic. The Born rule and state update are algebraic operations on $\mathbb{M}_+$. The reversible–irreversible distinction is a property of the acting element.
The positivity condition is a biquaternion-norm condition. The condition $|\mathbf{r}| \leq 1$ on states is equivalent to $N(\tilde{\rho}) \geq 0$. The "physical" constraint on states is a consequence of the quadratic form of the algebra.
The symmetry group and observables share the same algebraic home. The unitary elements of $\mathbb{B}$ act on $\mathbb{M}_+$ by conjugation; their generators lie in $\mathbb{M}_-$.
The framework contains the algebraic structure for relativity. The Lorentz group is contained in the algebra (via its double cover $SL(2,\mathbb{C})$) as the group of biquaternions of unit norm, and the material sector $\mathbb{M}_-$ carries the four-vectors. The biquaternion framework is therefore the natural setting in which the quantum formalism and the relativistic structure coexist, without the incompatibilities that appear when they are formulated separately.

Open Questions

The idempotent formalism raises several questions.

1. Extension to many qubits. The formalism above describes a single qubit. The extension to $n$ qubits requires the tensor product $\mathbb{B}^{\otimes_\mathbb{C} n}$ (over $\mathbb{C}$), which is isomorphic to $M_{2^n}(\mathbb{C})$. How do the idempotents, states, and observables generalize? Is the tensor product natural in the biquaternion framework, or does it require additional structure?

2. Second quantization. The formalism is first-quantized. To describe creation and annihilation of particles, one needs a Fock space and operator-valued fields. How does the biquaternion framework extend to this setting?

3. The full dynamics. The dynamics described here is for a qubit with a fixed Hamiltonian. The general dynamics of an open quantum system — with dissipation, decoherence, and feedback — is described by the Lindblad equation, which extends the von Neumann equation with additional terms. How does the Lindblad equation read in the biquaternion framework?

4. Relativistic quantum theory. The biquaternion algebra contains the Lorentz group and the spinor representation, and the biquaternion Dirac equation describes a relativistic spin-1/2 field classically. The quantization of this field, with a Lorentz-covariant state space and a consistent many-particle structure, is the natural next step. The relation between the single-qubit quantum formalism presented here and the relativistic spinor-field theory remains to be worked out in detail.

5. The interpretation of the informational sector. The companion article on $\mathbb{M}_+$ proposes an informational interpretation of the Hermitian subspace. Under this reading, the idempotents are "pure information states" and the Hermitian elements are "informational observables". The Born rule is the "probability of extracting information". Whether this leads to new structure is an open question.

6. The measurement problem. The idempotent formalism makes the Born rule and the state update algebraic, but it does not explain why one idempotent is selected in a given measurement. This is the same measurement problem as in standard quantum mechanics, in different notation.

7. Empirical content. The reformulation reproduces standard quantum mechanics for a single qubit. Does it predict anything new, or does it only reinterpret? If it predicts new physics, at what scale does the deviation from standard QM appear?

8. The path integral. The standard path integral formulation of quantum mechanics is not obviously expressible in the biquaternion framework. Is there a natural path integral in the biquaternion algebra, with a natural measure inherited from the algebra?

Summary

Quantum mechanics in the biquaternion framework is built on the Hermitian subspace $\mathbb{M}_+$ of the biquaternion algebra. The states are the positive trace-one elements of $\mathbb{M}_+$ (the Bloch ball), the pure states are the idempotents (the Bloch sphere), and the observables are the general Hermitian elements. The Born rule is the trace pairing $\mathrm{Tr}(\tilde{\rho}\tilde{H})$, the dynamics is rotor conjugation by unitary biquaternions, and measurement is the sandwich operation $\tilde{\rho}\mapsto\tilde{P}\tilde{\rho}\tilde{P}$.

The framework reproduces the standard qubit formalism for the states, observables, Born rule, dynamics, and measurement, with the structural advantages that states and observables live in the same space, the Born rule is a consequence of the trace pairing, measurement is an algebraic operation, and the positivity condition on states is a biquaternion-norm condition.

The biquaternion algebra contains the algebraic structure required for relativity — the Lorentz group (via its double cover $SL(2,\mathbb{C})$), the spinor representation, and the material sector of four-vectors — and the quantum formalism presented here and the relativistic structure coexist within the same algebra. The full relativistic quantum theory of spinor fields is the natural continuation of the framework: the classical precursor is the biquaternion Dirac equation treated in the companion article, and the quantization of the spinor field is a natural direction for future work.

The extension to many qubits, the second-quantized version, the connection to quantum field theory, and the question of empirical content are open.

Summary of Notation

Symbol	Meaning
$\mathbb{B}$	Biquaternion algebra
$\mathbb{M}_+$	Hermitian subspace (states, observables)
$\mathbb{M}_-$	Anti-Hermitian subspace (generators)
$\mathbb{H}_{\mathbb{B}}$	Real quaternion subspace, $\operatorname{span}_{\mathbb{R}}\{e_0, e_1, e_2, e_3\}$ (unit elements: $SU(2)$)
$\tilde{H} = h_0 e_0 + i\mathbf{h}$	Hermitian element (observable)
$\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$	State
$\tilde\Pi_\pm(\hat{\mu}) = \tfrac{1}{2}(e_0 \pm i\hat{\mu})$	Idempotent (pure state)
$\mathrm{Tr}(\tilde{H}) = 2 h_0$	Trace
$\mathrm{Tr}(\tilde{\rho}\tilde{H}) = h_0 + \mathbf{r}\cdot\mathbf{h}$	Born rule
$N(\tilde{H}) = \tilde{H}\tilde{H}^{\natural} = h_0^2 - \|\mathbf{h}\|^2$	Biquaternion norm, signature $(1,3)$
$B(\tilde{H}, \tilde{K}) = h_0 k_0 - \mathbf{h}\cdot\mathbf{k}$	Polarization of the biquaternion norm
$[\tilde{H}, \tilde{K}] = -2(\mathbf{h}\times\mathbf{k})$	Commutator
$\tilde{\rho}\mapsto\tilde{P}\tilde{\rho}\tilde{P}$	Projective measurement
$\tilde{U}(t) = e^{-ih_0t/\hbar}(\cos(\|\mathbf{h}\|t/\hbar)e_0 + \sin(\|\mathbf{h}\|t/\hbar)\hat{\mathbf{h}})$	Unitary evolution
$\tilde{\rho}(t) = \tilde{U}(t)\tilde{\rho}\tilde{U}(t)^{*}$	Von Neumann evolution
$\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$, $\tilde{\Lambda}$ invertible	Conjugation; quantum if $\tilde{\Lambda}\tilde{\Lambda}^{*} = e_0$ (group $U(2)$), Lorentz if $\tilde{\Lambda}\tilde{\Lambda}^{\natural} = e_0$ (group $SL(2,\mathbb{C})$)
$\mathrm{Tr}(\tilde{\rho}^2) = \tfrac{1}{2}(1+\|\mathbf{r}\|^2)$	Purity
$\tilde{\rho}^2 - \tilde{\rho} = \tfrac{1}{4}(\|\mathbf{r}\|^2-1)e_0$	Deviation from idempotency