Quantum Physics in Biquaternionic Form

Introduction

The companion article, Quantum Physics: Foundations and Structure, presented standard quantum physics in its conventional form: states are density matrices on a complex Hilbert space, observables are Hermitian operators, dynamics is unitary, and measurement is projective. It closed with a checklist of ten structural requirements that any comprehensive quantum theory must satisfy.

This article presents quantum physics in the biquaternion framework. 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 the states and the observables of the theory. A material element is an element of $\mathbb{M}_-$; it is what is observed. Its observable is the Hermitian representative $iA\in\mathbb{M}_+$, whose eigenvalues are the real readings. The states and the representatives therefore all live in $\mathbb{M}_+$, and the object observed lives in $\mathbb{M}_-$. This is a structural feature that standard quantum physics does not have. In its algebraic formulation the states and the observables are of different kinds — a normalized positive functional against an operator — and even in the density-matrix formulation, where both are Hermitian matrices, they are told apart by a trace and a positivity. In the biquaternion framework both are elements of the same four-dimensional real subspace, the state being the positive element normalised by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$ and the observable a general element of the 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 probability form there and restricts to the plain form on the sector, and the general quaternionic bilinear form $\langle\cdot,\cdot\rangle_{\natural}$ 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 Hermitian form. 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.

The Hermitian Subspace

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

$$ \tilde{H} = h_t\, e_t + i h_x\, \vec e_x + i h_y\, \vec e_y + i h_z\, \vec e_z, \qquad h_t, h_x, h_y, h_z \in \mathbb{R}. $$

The real scalar part is $h_t e_t$, and the imaginary vector part is $i(h_x \vec e_x + h_y \vec e_y + h_z \vec e_z)$. In compact form,

$$ \tilde{H} = h_t\, e_t + i\,\vec h, \qquad \vec h = (h_x, h_y, h_z) \in \mathbb{R}^3. $$

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

Hermitian form and Hermitian norm. The Hermitian form pairs two elements by the scalar part of the first with the Hermitian conjugate of the second, and its diagonal is the Hermitian norm:

$$ \langle\tilde Q_1,\tilde Q_2\rangle_{*}=\mathrm{Sc}(\tilde Q_1\tilde Q_2^{*}),\qquad \langle\tilde Q\rangle_{*}=\langle\tilde Q,\tilde Q\rangle_{*}=\sum_\mu\lvert Q_\mu\rvert^{2}. $$

On $\mathbb{M}_+$ both arguments are Hermitian, so $\tilde Q_k^{*}=\tilde Q_k$ and the form equals the plain form $\langle\tilde H_1,\tilde H_2\rangle=\mathrm{Sc}(\tilde H_1\tilde H_2)$; it is positive definite of signature $(4,0)$ there and on the whole algebra. The unit has $\langle e_t\rangle_{*}=1$. The scalar part is the form against the unit, $\mathrm{Sc}(\tilde H)=\langle\tilde H,e_t\rangle_{*}$, and the trace is its reading in the $2\times2$ image, $\operatorname{Tr}(\tilde H)=2\langle\tilde H,e_t\rangle_{*}=2h_t$.

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

$$ \langle\tilde{H}\rangle_{\natural} = \langle\tilde{H},\tilde{H}\rangle_{\natural} = \tilde{H}\tilde{H}^{\natural} = (h_t^2 - |\vec h|^2) e_t. $$

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 four forms, the scalar parts of the four general products of The Four General Products and Their Relations over the Biquaternion Algebra; on $\mathbb{M}_+$ they pair off and coincide, so two remain. They are the Hermitian form, which here equals the general plain bilinear form, and the bilinear form $\langle\cdot,\cdot\rangle_{\natural}$ (the general quaternionic bilinear form), which here equals the general quaternionic sesquilinear form $\langle\cdot,\cdot\rangle_{\natural*}$ because the coefficient conjugation acts as the natural sign on a Hermitian element. The biquaternion norm is not a separate bilinear form but the quadratic form of $\langle\cdot,\cdot\rangle_{\natural}$.

The plain form. For two Hermitian elements $\tilde H_1 = h_{1t}\,e_t + i\vec h_1$ and $\tilde H_2 = h_{2t}\,e_t + i\vec h_2$, the plain form is

$$ \langle\tilde{H}_1,\tilde{H}_2\rangle = \mathrm{Sc}(\tilde{H}_1\tilde{H}_2) = h_{1t} h_{2t} + \vec h_1\cdot\vec h_2. $$

It is symmetric and positive definite, of signature $(4,0)$: the Euclidean pairing on $\mathbb{R}^4$. It is the analogue of the Hilbert–Schmidt inner product on the space of operators, and it coincides on $\mathbb{M}_+$ with the canonical Hermitian form of the algebra, $\langle\tilde{H}_1, \tilde{H}_2\rangle_{*} = \mathrm{Sc}(\tilde{H}_1\tilde{H}_2^{*}) = h_{1t} h_{2t} + \vec h_1\cdot\vec h_2$, because $\tilde H_2^{*}=\tilde H_2$ there. In the $2\times2$ image it reads $\operatorname{Tr}(\tilde{H}_1\tilde{H}_2) = 2\langle\tilde{H}_1,\tilde{H}_2\rangle_{*}$.

The biquaternion norm. The biquaternion norm is the quadratic form

$$ \langle\tilde{H}\rangle_{\natural} = h_t^2 - |\vec h|^2, $$

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 general quaternionic bilinear form $\langle\cdot,\cdot\rangle_{\natural}$ below, $\langle\tilde{H}\rangle_{\natural} = \langle\tilde{H},\tilde{H}\rangle_{\natural}$. 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 $\tilde{H}_1\tilde{H}_2^{\natural}$:

$$ \langle\tilde{H}_1,\tilde{H}_2\rangle_{\natural}\,e_t = \tfrac{1}{2}\left(\tilde{H}_1\tilde{H}_2^{\natural} + \tilde{H}_2\tilde{H}_1^{\natural}\right). $$

Computing directly, using $\tilde{H}_1^{\natural} = h_{1t} e_t - i\vec h_1$ and the quaternion product of pure vector parts,

$$ \tilde{H}_1\tilde{H}_2^{\natural} = (h_{1t} h_{2t} - \vec h_1\cdot\vec h_2)\,e_t + i(h_{2t}\vec h_1 - h_{1t}\vec h_2) + \vec h_1\times\vec h_2, $$

$$ \tilde{H}_2\tilde{H}_1^{\natural} = (h_{1t} h_{2t} - \vec h_1\cdot\vec h_2)\,e_t - i(h_{2t}\vec h_1 - h_{1t}\vec h_2) - \vec h_1\times\vec h_2. $$

The sum is scalar:

$$ \boxed{\;\langle\tilde{H}_1,\tilde{H}_2\rangle_{\natural} = h_{1t} h_{2t} - \vec h_1\cdot\vec h_2\;}. $$

This is the Lorentzian bilinear form, of signature $(1,3)$, and the biquaternion norm is its diagonal, $\langle\tilde{H}_1,\tilde{H}_1\rangle_{\natural} = \langle\tilde{H}_1\rangle_{\natural}$. On $\mathbb{M}_+$ it is also the Krein form $\langle\tilde{H}_1,\tilde{H}_2\rangle_{\natural*} = \mathrm{Sc}(\tilde{H}_1^{\natural}\tilde{H}_2^{*})$ of the companion article: for Hermitian elements $\tilde{H}_1^{\natural} = \bar{\tilde{H}}_1$ and $\tilde{H}_2^{*} = \tilde{H}_2$, so the form reads $\mathrm{Sc}(\bar{\tilde{H}}_1\tilde{H}_2) = h_{1t} h_{2t} - \vec h_1\cdot\vec h_2$ and agrees with $\langle\cdot,\cdot\rangle_{\natural}$ on $\mathbb{M}_+$ even though the two differ on the full algebra. That is why $\mathbb{M}_+$ carries two forms and not three.

The antisymmetric combination contains the vector part:

$$ \tfrac{1}{2}\left(\tilde{H}_1\tilde{H}_2^{\natural} - \tilde{H}_2\tilde{H}_1^{\natural}\right) = i(h_{2t}\vec h_1 - h_{1t}\vec h_2) + \vec h_1\times\vec h_2. $$

This is not a symmetric bilinear form; it is the antisymmetric part of the product $\tilde H_1\tilde H_2^\natural$, and it plays a role in the commutator structure of the algebra.

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

Form	Expression	Signature
Hermitian form (the plain form on $\mathbb{M}_+$)	$\langle\tilde{H}_1,\tilde{H}_2\rangle_{} = \mathrm{Sc}(\tilde{H}_1\tilde{H}_2^{}) = h_{1t} h_{2t} + \vec h_1\cdot\vec h_2 = \tfrac{1}{2}\mathrm{Tr}(\tilde{H}_1\tilde{H}_2)$	$(4,0)$, Euclidean
Bilinear form $\langle\cdot,\cdot\rangle_{\natural}$ (equal to the Krein form $\langle\cdot,\cdot\rangle_{\natural*}$ on $\mathbb{M}_+$)	$\langle\tilde{H}_1,\tilde{H}_2\rangle_{\natural} = h_{1t} h_{2t} - \vec h_1\cdot\vec h_2$	$(1,3)$, Lorentzian

The Hermitian form is the natural inner product on the space of operators, and on $\mathbb{M}_+$ its restriction is the plain form; the Born rule and the unitary evolution use the Hermitian form there. The bilinear form $\langle\cdot,\cdot\rangle_{\natural}$ is the one whose quadratic form is the biquaternion norm, $\langle\tilde{H},\tilde{H}\rangle_{\natural} = \langle\tilde{H}\rangle_{\natural}$, 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:

$$ \tilde\Pi_{1,2}(\vec u) = \tfrac{1}{2}\left(e_t \pm i\,\vec u\right), $$

where $\vec u = u_x \vec e_x + u_y \vec e_y + u_z \vec e_z$ is a unit pure real quaternion, $|\vec u| = 1$.

Properties of the idempotents:

Hermitian: $\tilde\Pi_{1,2}^{*} = \tilde\Pi_{1,2}$.
Idempotent: $\tilde\Pi_{1,2}^2 = \tilde\Pi_{1,2}$.
Trace one: $\mathrm{Tr}(\tilde\Pi_{1,2}) = 1$.
Complementary: $\tilde\Pi_1 + \tilde\Pi_2 = e_t$, and $\tilde\Pi_1 \tilde\Pi_2 = 0 = \tilde\Pi_2 \tilde\Pi_1$.
Zero divisors: $\langle\tilde\Pi_{1,2}\rangle_{\natural} = 0$.

The parametrization is by the unit sphere $S^2 \subset \mathbb{R}^3$: the direction $\vec u$ determines the idempotent $\tilde\Pi_1(\vec u)$ uniquely, and the complementary idempotent $\tilde\Pi_2(\vec u) = \tilde\Pi_1(-\vec u)$ corresponds to the opposite direction.

Identification with standard states. Under the series isomorphism $\mathsf{M}_2:\mathbb{B}\to M_2(\mathbb{C})$ fixed by $\vec e_k\mapsto-i\sigma_k$ (Conventions in the Biquaternion Universe), the idempotent $\tilde\Pi_1(\vec u)$ maps to the standard pure state

$$ |\vec u+\rangle_{*}\langle\vec u+|, $$

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

Mixed States

A mixed state of a qubit is a positive Hermitian matrix normalised to trace one. In $\mathbb{M}_+$, the corresponding element is normalised by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12\langle e_t\rangle_{*}$ and has the form

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

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

$$ |\vec r|^2 = r_1^2 + r_2^2 + r_3^2 \leq 1. $$

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

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

The Bloch Ball

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

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

The pure states form the boundary sphere $|\vec r| = 1$; the maximally mixed state is at the centre.

A structural observation. The positivity condition $|\vec r| \leq 1$ is equivalent to the condition $\langle\tilde{Ϛ}\rangle_{\natural} \geq 0$ on the biquaternion norm of $\mathbb{B}$:

$$ \boxed{\;\langle\tilde{Ϛ}\rangle_{\natural} = \tfrac{1}{4}\left(1 - |\vec r|^2\right) e_t\;}. $$

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 the normalisation $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$.

This is a clean geometric characterization: states are the positive elements of the future light cone of $\mathbb{M}_+$ normalised by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$; pure states are the normalised elements on the cone itself.

Observables

General Observables

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

$$ \tilde{H} = h_t\, e_t + i\,\vec h, \qquad \vec h = (h_x, h_y, h_z) \in \mathbb{R}^3. $$

It is the representative $iA$ of a material element $A\in\mathbb{M}_-$; the material element is what is observed, and the eigenvalues of $iA$ are the readings.

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

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

$$ \tilde{H} = \lambda_+ \tilde\Pi_1(\vec u_h) + \lambda_- \tilde\Pi_2(\vec u_h), $$

with

$$ \vec u_h = \frac{\vec h}{|\vec h|}, \qquad \lambda_\pm = h_t \pm |\vec h|. $$

The eigenvalues are $\lambda_\pm$, and the eigenvectors are the idempotents $\tilde\Pi_{1,2}(\vec u_h)$. So every Hermitian element of $\mathbb{M}_+$ has a spectral decomposition in terms of idempotents.

Traceless observables. The traceless Hermitian elements $i\vec h$ (with $h_t = 0$) are the analogs of the traceless observables of the qubit (the Pauli matrices). They have eigenvalues $\pm|\vec h|$ and eigenvectors $\tilde\Pi_{1,2}(\vec u_h)$.

Compatibility

Two observables $\tilde H_1, \tilde H_2 \in \mathbb{M}_+$ are compatible (simultaneously measurable) if and only if their corresponding matrices commute. In terms of the vector parts, this holds if and only if $\vec h_1$ and $\vec h_2$ are parallel or one of them vanishes.

In the general case (non-parallel, nonzero $\vec h_1$ and $\vec h_2$), the observables are incompatible, and their commutator is

$$ [\tilde H_1, \tilde H_2] = -2\left(\vec h_1 \times \vec h_2\right), $$

where $\vec h_1\times\vec h_2$ is the ordinary cross product.

Derivation. We compute

$$ \tilde H_1\tilde H_2 = (h_{1t}h_{2t} + \vec h_1\cdot\vec h_2)e_t + i(h_{1t}\vec h_2 + h_{2t}\vec h_1) - \vec h_1\times\vec h_2, $$

$$ \tilde H_2\tilde H_1 = (h_{1t}h_{2t} + \vec h_1\cdot\vec h_2)e_t + i(h_{1t}\vec h_2 + h_{2t}\vec h_1) + \vec h_1\times\vec h_2, $$

$$ [\tilde H_1, \tilde H_2] = -2(\vec h_1\times\vec h_2). $$

The result is a pure real quaternion (vector part only, with real coefficients): it lies in $\mathbb{M}_- \cap \mathbb{H}_{\mathbb{B}}$, where $\mathbb{H}_{\mathbb{B}} = \operatorname{span}_{\mathbb{R}}\{e_t, \vec e_x, \vec e_y, \vec e_z\}$ is the real quaternion subspace, and this intersection is the real span of $\vec e_x, \vec e_y, \vec e_z$. This is the biquaternion form of the $\mathfrak{su}(2)$ angular-momentum 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 Hermitian form, the positive-definite form $\langle\tilde Q_1,\tilde Q_2\rangle_{*}=\mathrm{Sc}(\tilde Q_1\tilde Q_2^{*})$ of the whole algebra. Every probability formula below is written with the star. On $\mathbb{M}_+$ the arguments are Hermitian, so the conjugation acts as the identity on them, but it is not omitted: it names the form that is positive definite on the whole algebra. The trace is its matrix reading; in this section it is used only at the end, where normalisation is stated.

Expectation Values

For an observable $\tilde{H} = h_t e_t + i\vec h$ and a state $\tilde{Ϛ} = \tfrac{1}{2}(e_t + i\vec r)$, the state is normalised by the form against the unit,

$$ \langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12\langle e_t\rangle_{*},\qquad \langle e_t\rangle_{*}=1, $$

and the expectation value is the normalised Hermitian form,

$$ \langle \tilde{H}\rangle_{\tilde{Ϛ},*} = \frac{\langle\tilde{Ϛ},\tilde{H}\rangle_{*}}{\langle\tilde{Ϛ},e_t\rangle_{*}} =\frac{\mathrm{Sc}(\tilde{Ϛ}\tilde H^{*})}{\mathrm{Sc}(\tilde{Ϛ})} = h_t + \vec r\cdot\vec h. $$

Derivation. We compute

$$ \tilde{Ϛ}\tilde{H}^{*} = \tfrac{1}{2}(e_t + i\vec r)(h_t e_t + i\vec h) = \tfrac{1}{2}\left(h_t e_t + i\vec h + i h_t \vec r + (i\vec r)(i\vec h)\right), $$

where $\tilde H^{*}=\tilde H$ and $\tilde{Ϛ}^{*}=\tilde{Ϛ}$ have been used. For pure quaternions, $(i\vec r)(i\vec h) = -\vec r\vec h = \vec r\cdot\vec h - \vec r\times\vec h$. Taking the scalar part,

$$ \mathrm{Sc}(\tilde{Ϛ}\tilde{H}^{*}) = \tfrac{1}{2}(h_t + \vec r\cdot\vec h), $$

so $\langle\tilde{Ϛ},\tilde{H}\rangle_{*} = \tfrac12(h_t + \vec r\cdot\vec h)$, and dividing by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$ gives the expectation value $h_t + \vec r\cdot\vec h$. The star is present in every step; it acts as the identity because both arguments are Hermitian.

On $\mathbb{M}_+$ the star is inert, so the same number is $\mathrm{Sc}(\tilde{Ϛ}\tilde H)/\mathrm{Sc}(\tilde{Ϛ})$, the plain scalar. So the Born rule follows from the Hermitian form once the states and observables are placed in $\mathbb{M}_+$. It is not an independent postulate. The density-matrix form of the statement is recorded with the other normalisation statements at the end of the section.

Probabilities

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

$$ \boxed{\;p_\pm = \tfrac{1}{2}\left(1 \pm \vec u_h\cdot\vec r\right)\;}, $$

where $\vec u_h = \vec h/|\vec h|$. The probabilities depend only on the angle between the measurement direction $\vec u_h$ and the Bloch vector $\vec r$.

The derivation is direct: $p_+ = \langle\tilde\Pi_1(\vec u_h),\tilde{Ϛ}\rangle_{*}/\langle\tilde{Ϛ},e_t\rangle_{*} = \mathrm{Sc}(\tilde\Pi_1(\vec u_h)\tilde{Ϛ}^{*})/\mathrm{Sc}(\tilde{Ϛ}) = 2\,\mathrm{Sc}(\tilde\Pi_1(\vec u_h)\tilde{Ϛ}^{*})$, and this is $\tfrac{1}{2}(1 + \vec u_h\cdot\vec r)$.

Measurement Update

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

$$ \tilde{Ϛ}_{\mathrm{out}} = \frac{\tilde\Pi_1(\vec u_h)\,\tilde{Ϛ}\,\tilde\Pi_1(\vec u_h)}{2\,\mathrm{Sc}(\tilde\Pi_1(\vec u_h)\tilde{Ϛ}^{*})}. $$

Since $\tilde\Pi_1$ is idempotent, the sandwich $\tilde\Pi_1\tilde{Ϛ}\tilde\Pi_1$ is Hermitian, has its range in the line of $\tilde\Pi_1$, and vanishes on the kernel of $\tilde\Pi_1$; it is therefore a real multiple of $\tilde\Pi_1$:

$$ \tilde\Pi_1\tilde{Ϛ}\tilde\Pi_1 = p_+ \tilde\Pi_1, $$

where $p_+ = 2\,\mathrm{Sc}(\tilde\Pi_1\tilde{Ϛ}^{*})$. Hence the post-measurement state is simply

$$ \boxed{\;\tilde{Ϛ}_{\mathrm{out}} = \tilde\Pi_1(\vec u_h)\;}. $$

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

Normalisation and the Trace Reading

Every formula above is complete without a trace. The trace is the reading of the normalisation in the $2\times2$ image: $\operatorname{Tr}(e_t)=2$, so a normalised state has

$$ \operatorname{Tr}(\tilde{Ϛ})=2\,\mathrm{Sc}(\tilde{Ϛ})=1, \qquad \operatorname{Tr}(\tilde{Ϛ}\tilde H^{*})=2\,\mathrm{Sc}(\tilde{Ϛ}\tilde H^{*})=2\,\mathrm{Sc}(\tilde{Ϛ}\tilde H), $$

and the weight reads

$$ p=\frac{\operatorname{Tr}(\tilde{Ϛ}\tilde H^{*})}{\operatorname{Tr}(\tilde{Ϛ})}. $$

The factor $2$ is the dimension of the image and cancels in the ratio; it is not part of the form. Written this way the Born rule has the standard density-matrix form $\mathrm{Tr}(\rho H)=h_t+\vec r\cdot\vec h$. On $\mathbb{M}_+$ both factors are Hermitian, so the star in the numerator is inert and $\operatorname{Tr}(\tilde{Ϛ}\tilde H^{*})=\operatorname{Tr}(\tilde{Ϛ}\tilde H)$. The unstarred trace is real, because $(AB)^{\dagger}=BA$ for Hermitian $A,B$ gives $\overline{\mathrm{tr}(AB)}=\mathrm{tr}(AB)$; it is non-negative for $0\le\tilde H\le e_t$, because $\mathrm{tr}(AB)=\mathrm{tr}(A^{1/2}BA^{1/2})\ge0$. Hence $p=\operatorname{Tr}(\tilde{Ϛ}\tilde H)/\operatorname{Tr}(\tilde{Ϛ})\in[0,1]$.Written this way the Born rule has the standard density-matrix form $\mathrm{Tr}(\rho H)=h_t+\vec r\cdot\vec h$.

Dynamics

The Schrödinger Picture

Time evolution of a pure state. A pure state is an idempotent $\tilde\Pi = \tfrac{1}{2}(e_t + i\vec u)$. 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, $\tilde{U}\tilde{U}^{*} = e_t$, satisfying the Schrödinger equation

$$ i\hbar\, \frac{d}{dt'}\tilde{U}(t') = \tilde{H}\,\tilde{U}(t'), \qquad \tilde{U}(0) = e_t, $$

with $\tilde{H} = h_t e_t + i\vec h \in \mathbb{M}_+$ a Hermitian element (the Hamiltonian).

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

$$ \boxed{\;\tilde{U}(t') = \exp(-i\tilde{H}t'/\hbar) = \exp(-ih_t t'/\hbar)\left(\cos(|\vec h|t'/\hbar)\,e_t + \sin(|\vec h|t'/\hbar)\,\vec u_h\right)\;}, $$

where $\vec u_h = \vec h/|\vec h|$ (for $\vec h \neq 0$). The exponential of the vector part uses $\vec u_h^2 = -e_t$, which gives the trigonometric form.

The element $\tilde{U}(t')$ is unitary: $\tilde{U}(t')\tilde{U}(t')^{*} = e_t$. It is not Hermitian in general: writing $\theta = |\vec h|t'/\hbar$, $\tilde{U}(t')$ is Hermitian exactly when $\tilde{U}(t')^2 = e_t$, which requires either $\sin\theta = 0$ together with $h_tt'/\hbar \in \pi\mathbb{Z}$, or $\cos\theta = 0$ together with $h_tt'/\hbar \in \tfrac{\pi}{2} + \pi\mathbb{Z}$. For $h_t = 0$ only the first case survives, at the isolated times $t' = n\pi\hbar/|\vec h|$. For $h_t \neq 0$ and $\vec h \neq 0$, apart from $t'=0$, the first case gives Hermitian times exactly when $h_t/|\vec h|$ is rational, and the second contributes in addition only when that ratio, in lowest terms, is a ratio of odd integers. It is an element of the unitary group $\mathrm{U}(2) \subset \mathbb{B}$.

The trace part of the Hamiltonian does not affect the state. Since $\exp(-ih_t t'/\hbar)$ is a central complex scalar, and since the conjugation

$$ \tilde{U}(t')\,\tilde\Pi\,\tilde{U}(t')^{*} = \exp(-ih_t t'/\hbar)\tilde u\,\tilde\Pi\,\exp(+ih_t t'/\hbar)\tilde u^{*} = \tilde u\,\tilde\Pi\,\tilde u^{*} $$

(with $\tilde u = \cos(|\vec h|t'/\hbar)e_t + \sin(|\vec h|t'/\hbar)\vec u_h$ a unit real quaternion), the phase $\exp(-ih_t t'/\hbar)$ cancels out. So only the traceless part $\vec h$ 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 $\tilde{Ϛ} = \tfrac{1}{2}(e_t + i\vec r)$ evolves by the same conjugation:

$$ \tilde{Ϛ}(t') = \tilde{U}(t')\,\tilde{Ϛ}(0)\,\tilde{U}(t')^{*}. $$

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

$$ i\hbar\, \frac{d}{dt'}\tilde{Ϛ}(t') = [\tilde{H}, \tilde{Ϛ}(t')], $$

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 $i\hbar\,d\tilde{U}/dt' = \tilde{H}_0\tilde{U}$ and its adjoint $-i\hbar\,\left(d\tilde{U}/dt'\right)^{*} = \tilde{U}^{*}\tilde{H}_0$:

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

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 $\tilde{U}(t')\tilde{U}(t')^{*} = e_t$. 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 $\mathrm{U}(2) \subset \mathbb{B}$, and their conjugation action on $\mathbb{M}_+$ preserves the Hermitian form 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 structure, the Hermitian form on $\mathbb{B}$, which restricts to the plain form on $\mathbb{M}_+$.

Projective measurement. A projective measurement of $\tilde{H} = h_t e_t + i\vec h$ is specified by the spectral decomposition $\tilde{H} = \lambda_+ \tilde\Pi_1 + \lambda_- \tilde\Pi_2$, with $\tilde\Pi_{1,2} = \tfrac{1}{2}(e_t \pm i\vec u_h)$. The outcome $+$ has:

Probability $p_+=\langle\tilde\Pi_1,\tilde{Ϛ}\rangle_{*}/\langle\tilde{Ϛ},e_t\rangle_{*}=\mathrm{Sc}(\tilde\Pi_1\tilde{Ϛ}^{*})/\mathrm{Sc}(\tilde{Ϛ})$, whose trace reading at normalisation is $\operatorname{Tr}(\tilde\Pi_1\tilde{Ϛ}^{*})=\operatorname{Tr}(\tilde\Pi_1\tilde{Ϛ}) = 2\langle\tilde\Pi_1,\tilde{Ϛ}\rangle_{*}$.
Post-measurement state $\tilde{Ϛ}_{\mathrm{out}} = \tilde\Pi_1$, from the sandwich operation.

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

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

and the probability of outcome $i$ is $p_i=\mathrm{Sc}(\tilde{K}_i^{*}\tilde{K}_i\tilde{Ϛ})/\mathrm{Sc}(\tilde{Ϛ})$, whose trace reading at normalisation is $\operatorname{Tr}(\tilde{K}_i^{*} \tilde{K}_i \tilde{Ϛ})$. 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 $\tilde\Pi$ do not preserve the biquaternion norm (they satisfy $\tilde\Pi^2 = \tilde\Pi$ but $\langle\tilde\Pi\rangle_{\natural} = \tilde\Pi\tilde\Pi^{\natural} = 0$) 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 $\tilde{Ϛ} = \tfrac{1}{2}(e_t + i\vec r)$ is

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

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

Deviation from Idempotency

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

$$ \tilde{Ϛ}^2 - \tilde{Ϛ} = \tfrac{1}{4}\left(|\vec r|^2 - 1\right)e_t. $$

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

Spectral Decomposition and Entropy

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

$$ \tilde{Ϛ} = \lambda_+ \tilde\Pi_1(\vec u_r) + \lambda_- \tilde\Pi_2(\vec u_r), $$

with $\vec u_r = \vec r/|\vec r|$ and $\lambda_\pm = (1 \pm |\vec r|)/2$. The von Neumann entropy is

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

which depends only on $|\vec r|$ and vanishes for pure states.

Fidelity

For two pure states $\tilde\Pi_{\mathrm p} = \tfrac{1}{2}(e_t + i\vec u_{\mathrm p})$ and $\tilde\Pi_{\mathrm q} = \tfrac{1}{2}(e_t + i\vec u_{\mathrm q})$, the transition probability (or fidelity squared) is

$$ \mathrm{Tr}(\tilde\Pi_{\mathrm p}\tilde\Pi_{\mathrm q}^{*}) = \tfrac{1}{2}\left(1 + \vec u_{\mathrm p}\cdot\vec u_{\mathrm q}\right). $$

This is the standard formula $|\langle\vec u_{\mathrm p}|\vec u_{\mathrm q}\rangle_{*}|^2 = \cos^2(\theta/2)$, where $\theta$ is the angle between $\vec u_{\mathrm p}$ and $\vec u_{\mathrm q}$ 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 $\mathrm{U}(2)$ acts on $\mathbb{M}_+$ by conjugation:

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

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 $SU(2) \subset \mathrm{U}(2)$. 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:

$$ [\tilde{Q}_1, \tilde{Q}_2] = \tilde{Q}_1\tilde{Q}_2 - \tilde{Q}_2\tilde{Q}_1, \qquad \tilde{Q}_1, \tilde{Q}_2 \in \mathbb{M}_-. $$

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 elements of $\mathbb{M}_+$ normalised by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$ (trace one in the $2\times2$ image), 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. Each Hermitian element is the representative $iA$ of a material element $A\in\mathbb{M}_-$, the object observed; the readings are the eigenvalues of $iA$.

3. Born rule. The Born rule is the normalised Hermitian form $\langle\tilde{Ϛ},\tilde{H}\rangle_{*}/\langle\tilde{Ϛ},e_t\rangle_{*}$. It is a consequence of the Hermitian form, positive definite on the whole algebra and equal to the plain form on $\mathbb{M}_+$, not an independent postulate; the trace formula $\operatorname{Tr}(\tilde{Ϛ}\tilde H^{*})=2\langle\tilde{Ϛ},\tilde H\rangle_{*}=2\,\mathrm{Sc}(\tilde{Ϛ}\tilde H^{*})$ is its $2\times2$ reading.

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 normalised Hermitian form, 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 $\mathrm{U}(2)$ 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 space carries the structures the relativistic side reads — the bilinear ones and the spinor module — and the Lorentz group sits in the invertible elements. The quantum formalism presented here and the relativistic structure share the same biquaternion space, 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 physics requires the machinery of decoherence and the correspondence principle, which are not developed in this article.

10. Interpretation. The biquaternion framework admits several interpretations. 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 the two theories on one biquaternion space: 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,

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

and the two theories differ only in the subgroup of the invertible elements over which the conjugating element ranges. The quantum operations of the Schrödinger picture are the unitary case, $\tilde U\tilde U^{*} = e_t$; these elements form the group $\mathrm{U}(2)$. The proper orthochronous Lorentz transformations of $\mathbb{M}_-$ are the case of unit norm, written $\tilde U_{\mathrm L}$, with $\tilde U_{\mathrm L}\tilde U_{\mathrm L}^{\natural} = e_t$; 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 biquaternion space, and their intersection is the rotation group:

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

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 biquaternion space $\mathbb{B}$.

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,

$$ \tilde{U}\left(\tilde{Q}_1\tilde{Q}_2\right)\tilde{U}^{*} = \left(\tilde{U}\tilde{Q}_1\tilde{U}^{*}\right)\left(\tilde{U}\tilde{Q}_2\tilde{U}^{*}\right), \qquad \tilde{U} \in \mathrm{U}(2), $$

so it is an automorphism of $\mathbb{B}$; the map $\tilde Q\mapsto\tilde U\tilde Q\tilde U^{*}$ for 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 Hermitian form $\langle\tilde{H}_1,\tilde{H}_2\rangle_{*}$, equivalently $\mathrm{Tr}(\tilde{H}_1\tilde{H}_2^{*})=2\langle\tilde{H}_1,\tilde{H}_2\rangle_{*}$ in the $2\times2$ image, is the Euclidean Hilbert–Schmidt form on $\mathbb{M}_+$ on which the Born rule rests; it is preserved by exactly the unitary group:

$$ \mathrm{Tr}\!\left(\tilde U\tilde{H}_1\tilde U^{*}\,\tilde U\tilde{H}_2\tilde U^{*}\right) = \mathrm{Tr}(\tilde{H}_1\tilde{H}_2) \ \ \text{for all Hermitian } \tilde{H}_1,\tilde{H}_2 \quad \Longleftrightarrow \quad \tilde U \in \mathrm{U}(2), $$

since the left-hand side is $\mathrm{Tr}\big((\tilde U^{*}\tilde U)\tilde{H}_2(\tilde U^{*}\tilde U)\tilde{H}_1\big)$; setting $S=\tilde U^{*}\tilde U$ and taking $\tilde{H}_2=e_t$ gives $S^2=e_t$, and $S$ positive then forces $S=e_t$, which is the unitarity condition. A boost fails it.

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

$$ \langle\tilde U_{\mathrm L}\tilde{Q}\tilde U_{\mathrm L}^{*}\rangle_{\natural} = \langle\tilde U_{\mathrm L}\rangle_{\natural}\,\langle\tilde{Q}\rangle_{\natural}\,\langle\tilde U_{\mathrm L}^{*}\rangle_{\natural} = \langle\tilde{Q}\rangle_{\natural}, \qquad \tilde U_{\mathrm L} \in SL(2,\mathbb{C}). $$

This form does not, however, separate the two groups, and the point is worth stating: since $|\langle\tilde{U}\rangle_{\natural}| = |\det\tilde{U}| = 1$ for a unitary element, the unitary group preserves the biquaternion norm as well. In general $\langle\tilde Q\rangle_{\natural}$ is invariant under conjugation by every element $\tilde U$ with $|\langle\tilde U\rangle_{\natural}| = 1$, 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 the two full groups is the Euclidean plain form: $\mathrm{U}(2)$ preserves it, while a non-unitary Lorentz element does not. The automorphism property points the same way: it holds for the unitary elements and fails for the non-unitary Lorentz elements. Neither group contains the other, and their common subgroup is the rotation group $SU(2)$.

One space, several structures. The two forms that define the groups are the forms of two different structures of the twelve, read over the one space (The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space). The indefinite form is the general quaternionic bilinear form $\langle\cdot,\cdot\rangle_{\natural}$, the form of the structure $\mathrm{GQA}$, an algebra over $\mathbb{C}$, whose diagonal is the interval. The positive form is the general plain sesquilinear form (the Hermitian form), the form of the structure $\mathrm{GPS}$, a sesqualgebra over $(\mathbb{C},\bar{\cdot})$, the pairing on which the Born rule rests. The Lorentz group is defined by the bilinear structure and the unitary group by the sesquilinear one. That one space carries both is the whole of the compatibility, and it is why the two theories can be written side by side without being one theory. On $\mathbb{M}_+$ the four forms collapse pairwise — the Hermitian form coincides with the general plain bilinear form, and $\langle\cdot,\cdot\rangle_{\natural}$ with $\langle\cdot,\cdot\rangle_{\natural*}$ (§The Forms on $\mathbb{M}_+$, above) — so the distinction is a statement about the full space, not about the Hermitian sector.

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:

$$ \tilde U_{\mathrm b} = \cosh\frac{\psi}{2}\,e_t + i\sinh\frac{\psi}{2}\,\vec u_{\mathrm b}, \qquad \langle\tilde U_{\mathrm b}\rangle_{\natural} = \cosh^2\frac{\psi}{2} - \sinh^2\frac{\psi}{2} = 1 . $$

It lies on the norm-one hyperboloid of $\mathbb{M}_+$, whereas the states lie on the state slice of the future light cone, with $\langle\tilde{Ϛ}\rangle_{\natural} = \tfrac{1}{4}(1 - |\vec r|^2) \geq 0$ and pure states at $\langle\tilde{Ϛ}\rangle_{\natural} = 0$. Boost conjugation preserves Hermiticity, positivity (a congruence preserves the inertia), and the biquaternion norm, but it preserves neither the normalisation $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$ nor the plain form. On the maximally mixed state, for instance,

$$ \tilde U_{\mathrm b}\,\tfrac{1}{2}e_t\,\tilde U_{\mathrm b}^{*} = \tfrac{1}{2}\left(\cosh\psi\,e_t + i\sinh\psi\,\vec u_{\mathrm b}\right), $$

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

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

$$ \tilde U_{\mathrm b,2}\tilde U_{\mathrm b,1} = \tilde{W}\,\tilde U_{\mathrm b}, \qquad \tilde{W} \in SU(2), $$

separates a unitary factor $\tilde{W}$ — the Thomas–Wigner rotation — from a positive-definite Hermitian boost factor $\tilde U_{\mathrm b}$, 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 $\exp(-i\tilde{H}t'/\hbar)$, 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 biquaternion space: 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 the normalised Hermitian form $\langle\tilde{Ϛ},\tilde{H}\rangle_{*}/\langle\tilde{Ϛ},e_t\rangle_{*}$, whose trace reading is $\operatorname{Tr}(\tilde{Ϛ}\tilde H^{*})=\operatorname{Tr}(\tilde{Ϛ}\tilde H)=2\langle\tilde{Ϛ},\tilde H\rangle_{*}=2\,\mathrm{Sc}(\tilde{Ϛ}\tilde H^{*})$.
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 the two are of different kinds — a normalized positive functional on one side, an operator on the other — or, in the density-matrix formulation, Hermitian matrices told apart by a trace and a positivity. In $\mathbb{M}_+$, both are elements of the same four-dimensional real subspace: the state is the positive element normalised by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$, the observable a general element, so the two families are nested rather than disjoint and a state is itself an observable. The positivity is the sector's norm condition $\langle\tilde{Ϛ}\rangle_{\natural} \geq 0$ on the state slice. The object observed, however, lives one factor of $i$ away, in $\mathbb{M}_-$; it is a material element, and its observable is the Hermitian representative whose eigenvalues are the readings.
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 normalisation hyperplane $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$ with the future light cone of the biquaternion norm.
The Born rule is algebraic. It is a consequence of the Hermitian form, 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 $|\vec r| \leq 1$ on states is equivalent to $\langle\tilde{Ϛ}\rangle_{\natural} \geq 0$. The "physical" constraint on states is a consequence of the quadratic form of the algebra.
The symmetry group and observables share the same biquaternion space. 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 interpretation, 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 physics, in different notation.

7. Empirical content. The reformulation reproduces standard quantum physics 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 physics 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 physics in the biquaternion framework is built on the Hermitian subspace $\mathbb{M}_+$ of the biquaternion algebra. The states are the positive elements of $\mathbb{M}_+$ normalised by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$ (trace one in the $2\times2$ image), i.e., 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 normalised Hermitian form $\langle\tilde{Ϛ},\tilde{H}\rangle_{*}/\langle\tilde{Ϛ},e_t\rangle_{*}$, the dynamics is rotor conjugation by unitary biquaternions, and measurement is the sandwich operation $\tilde{Ϛ}\mapsto\tilde\Pi\tilde{Ϛ}\tilde\Pi$.

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 Hermitian form, measurement is an algebraic operation, and the positivity condition on states is a biquaternion-norm condition.

The biquaternion space carries the structures the relativistic side reads — the bilinear ones, the spinor module, and the material sector of four-vectors, with the Lorentz group in the invertible elements through the double cover $SL(2,\mathbb{C})$ — and the quantum formalism presented here and the relativistic structure coexist within the same biquaternion space. 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, $\operatorname{span}_{\mathbb{R}}\{e_t, \vec e_x, \vec e_y, \vec e_z\}$ (unit elements: $SU(2)$)
$\tilde{H} = h_t e_t + i\vec h$	Hermitian element (observable)
$\tilde{Ϛ} = \tfrac{1}{2}(e_t + i\vec r)$	State
$\tilde\Pi_{1,2}(\vec u) = \tfrac{1}{2}(e_t \pm i\vec u)$	Idempotent (pure state)
$\langle\tilde{Ϛ},\tilde{H}\rangle_{} = \tfrac12(h_t + \vec r\cdot\vec h)$, $\langle\tilde{Ϛ},e_t\rangle_{}=\tfrac12$	Hermitian form on a state
$\langle\tilde{Ϛ},\tilde{H}\rangle_{}/\langle\tilde{Ϛ},e_t\rangle_{} = h_t + \vec r\cdot\vec h$	Born rule, the normalised Hermitian form
$\mathrm{Tr}(\tilde{H}) = 2 h_t$, $\mathrm{Tr}(\tilde{Ϛ}\tilde H^{*}) = h_t + \vec r\cdot\vec h$	Trace, the $2\times2$ reading of the Hermitian form
$\langle\tilde{H}\rangle_{\natural} = \langle\tilde{H},\tilde{H}\rangle_{\natural} = \tilde{H}\tilde{H}^{\natural} = h_t^2 - \|\vec h\|^2$	Biquaternion norm, signature $(1,3)$
$\langle\tilde{H}_1,\tilde{H}_2\rangle_{\natural} = h_{1t} h_{2t} - \vec h_1\cdot\vec h_2$	Polarization of the biquaternion norm
$[\tilde{H}_1, \tilde{H}_2] = -2(\vec h_1\times\vec h_2)$	Commutator
$\tilde{Ϛ}\mapsto\tilde\Pi\tilde{Ϛ}\tilde\Pi$	Projective measurement
$\tilde{U}(t') = \exp(-ih_tt'/\hbar)(\cos(\|\vec h\|t'/\hbar)e_t + \sin(\|\vec h\|t'/\hbar)\vec u_h)$	Unitary evolution
$\tilde{Ϛ}(t') = \tilde{U}(t')\tilde{Ϛ}\tilde{U}(t')^{*}$	Von Neumann evolution
$\tilde{Q}\mapsto\tilde U\tilde{Q}\tilde U^{*}$, $\tilde U$ invertible	Conjugation; quantum if $\tilde U\tilde U^{*} = e_t$ (group $\mathrm{U}(2)$), Lorentz if $\tilde U_{\mathrm L}\tilde U_{\mathrm L}^{\natural} = e_t$ (group $SL(2,\mathbb{C})$)
$\mathrm{Tr}(\tilde{Ϛ}^2) = \tfrac{1}{2}(1+\|\vec r\|^2)$	Purity
$\tilde{Ϛ}^2 - \tilde{Ϛ} = \tfrac{1}{4}(\|\vec r\|^2-1)e_t$	Deviation from idempotency
$\langle\tilde{Q}_1,\tilde{Q}_2\rangle_{*}$	the general plain sesquilinear form, $\mathrm{Sc}(\tilde{Q}_1\tilde{Q}_2^{})$; on the diagonal $\langle\tilde{Q}_2,\tilde{Q}_2\rangle_{}=\sum_\mu\lvert Q_{2\mu}\rvert^{2}$
$\langle\tilde{Q}_1,\tilde{Q}_2\rangle_{*}$	the general plain bilinear form, the scalar part of the general plain bilinear product, $\mathrm{Sc}(\tilde{Q}_1\tilde{Q}_2)$