AI Physics Review

Introduction

This article is a digest of the biquaternion physics corpus, focused on developing quantum physics in a way natively compatible with relativity. In particular, it can be used by an artificial reader to assess the project. It is written to stand alone: it defines the algebra it uses, it cites no source in support of its claims, and every claim it makes about the algebra can be rechecked from the definitions below. In practice, the text can be copied whole into an AI chat bot, which will produce a first assessment of the theory from it.

The project is a research corpus of about 400+ physics articles, beside 300+ in the biquaternion chapter of the mathematics corpus. The two halves have different status. The mathematics is established: it proves theorems from the definitions, and they hold whatever one thinks of the physical interpretation. The physics is a research programme: its algebra is that same established algebra, and the open question is whether reading the two sectors as relativity and as quantum physics is right. Most of the corpus is known physics restated, which a shared algebra makes expected. What is the corpus's own — the sesqualgebras, the twelve non-associative biquaternion structures, the derivation of the form of the Born rule, the interpretation of the Wick rotation, and the thermal formalism recovered without it — is collected in the result sections below.

This digest gives the central affirmation of the physics corpus, the reasons that affirmation is structurally plausible, the established results the framework rests on or supplies, the new results the corpus adds, and the limits.

The central affirmation. One algebra, the biquaternions, already contains the formal structures of both special relativity and quantum physics. Relativity lives in one of its two complementary four-dimensional real subspaces, called here the material sector $\mathbb{M}_-$: the four-vectors, the invariant interval, the Lorentz group. Quantum physics lives in the other, the informational sector $\mathbb{M}_+$: states, observables, probabilities, measurement. Multiplication by the imaginary unit exchanges the two. The aim is therefore not to quantise relativity or to relativise quantum mechanics. It is to develop quantum physics natively, in a setting where the relativistic structure is already present. The two theories are not two languages to be reconciled but two interpretations of one algebraic object.

The Algebra and the Two Sectors

The algebra

The setting is $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, the complexification of the quaternions. An element is

$$ \tilde{Q} = Q_te_t + Q_x\vec e_x + Q_y\vec e_y + Q_z\vec e_z, $$

with complex coefficients, $e_t = 1$, $\vec e_x^2 = \vec e_y^2 = \vec e_z^2 = -e_t$, $\vec e_x \vec e_y = \vec e_z$ and its cyclic permutations, and $\vec e_\mu \vec e_\nu = -\vec e_\nu \vec e_\mu$ for $\mu \neq \nu$. Two facts carry the whole framework.

• The scalar imaginary $i$ is central. It satisfies $i^2 = -1$ and commutes with every element; it is not one of the units $\vec e_\mu$. There is one algebra here, and its two roles — the imaginary unit of the coefficients and the quaternion units of the directions — cannot be exchanged.
• As a complex algebra, $\mathbb{B}$ is the $2\times 2$ complex matrices. One isomorphism is fixed once and for all, the corpus's matrix element representation $\mathsf{M}_2$: $\mathsf{M}_2(e_t)=I_2$ and $\mathsf{M}_2(\vec e_x)=\begin{pmatrix}0&-i\\-i&0\end{pmatrix}$, $\mathsf{M}_2(\vec e_y)=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, $\mathsf{M}_2(\vec e_z)=\begin{pmatrix}-i&0\\0&i\end{pmatrix}$ — that is, $-i$ times the Pauli matrices — extended over $\mathbb{C}$ by $\mathsf{M}_2(i\tilde{Q})=i\,\mathsf{M}_2(\tilde{Q})$. Consequently $\mathbb{B}$ is associative, complex, and not a division algebra: it has zero divisors. $\mathsf{M}_2$ carries the algebra's structures across exactly: $\mathsf{M}_2(\tilde Q_1\tilde Q_2)=\mathsf{M}_2(\tilde Q_1)\mathsf{M}_2(\tilde Q_2)$, $\mathsf{M}_2(\tilde Q^{*})=\mathsf{M}_2(\tilde Q)^{\dagger}$, $\det\mathsf{M}_2(\tilde Q)=\langle\tilde Q\rangle_{\natural}$ and $\mathrm{Tr}\,\mathsf{M}_2(\tilde Q)=2Q_t$. Two useful corollaries: every identity below is an identity about matrices, and this is not quaternionic quantum mechanics — no one-sided or non-commutative-scalar modification enters, so no quaternionic deviation from ordinary quantum mechanics is to be expected.

Each coefficient splits once and for all as $Q_\mu = q_\mu + iq'_\mu$ with $q_\mu, q'_\mu$ real. The prime marks the slot that carries the $i$: it is a bookkeeping rule and not a physical label. Which of the four parameters is a time is decided by the sector, not by the prime.

The two sectors

Take the Hermitian conjugation, the coefficient-wise complex conjugate with the three vector coefficients negated:

$$ \tilde{Q}^{*} = \bar{Q}_te_t - \bar{Q}_x\vec e_x - \bar{Q}_y\vec e_y - \bar{Q}_z\vec e_z . $$

The two sectors are its two fixed spaces:

$$ \mathbb{M}_+ = \{\tilde{Q} : \tilde{Q}^{*} = \tilde{Q}\} = \{q_te_t + i(q'_x\vec e_x+q'_y\vec e_y+q'_z\vec e_z)\}, \qquad \mathbb{M}_- = \{\tilde{Q} : \tilde{Q}^{*} = -\tilde{Q}\} = \{iq'_te_t + q_x\vec e_x + q_y\vec e_y + q_z\vec e_z\}. $$

So $\mathbb{M}_-$ is imaginary scalar, real vector and $\mathbb{M}_+$ is real scalar, imaginary vector, and on the physical coordinates the same two spaces read

$$ \mathbb{M}_- = \{ict'\,e_t + x\,\vec e_x + y\,\vec e_y + z\,\vec e_z\}, \qquad \mathbb{M}_+ = \{ct\,e_t + i(x'\vec e_x + y'\vec e_y + z'\vec e_z)\}, $$

with $c$ the speed of light and $t',x,y,z$ and $t,x',y',z'$ all real. The material sector is the Minkowski four-vector; the informational sector is its counterpart with the time real and the three space directions imaginary. The two coordinate quadruples are independent, which is why the algebra carries the times $ict'$ and $ct$ and the spaces $x\,\vec e_x+y\,\vec e_y+z\,\vec e_z$ and $i(x'\,\vec e_x+y'\,\vec e_y+z'\,\vec e_z)$ without identifying them.

Each sector is four-real-dimensional, they meet only at $0$, together they span the algebra, and multiplication by $i$ exchanges them:

$$ i\mathbb{M}_- = \mathbb{M}_+, \qquad i\mathbb{M}_+ = \mathbb{M}_- . $$

The names are the framework's: $\mathbb{M}_-$ material, $\mathbb{M}_+$ informational. Note the labelling, because it is the opposite of what a reader expects: the anti-Hermitian subspace is the material one. The next two sections give the reason.

Material element and observable. An element of $\mathbb{M}_-$ is a material element; it is what is observed, a four-vector or a field. Its observable is the Hermitian representative $iA\in\mathbb{M}_+$, whose eigenvalues are the real readings. The two are one quantity in two writings, and the central $i$ is the exchange between them.

Two forms, two jobs

The convention lists four forms. Two of them carry the physics, and they answer two different questions. The Hermitian form answers a probability question, and it is the only one of the four that is positive definite on the whole algebra. The quaternionic form answers a metric question on $\mathbb{M}_-$. They are stated below in separate subsections. Neither borrows the other's normalisation, and neither uses the trace. The plain form is not a third job: it is the restriction of the Hermitian form to $\mathbb{M}_+$, and it is the reading in which the sector computations below are written. The Krein form is the partner of the quaternionic form and is not needed here.

The Hermitian form: the probability weight

The form. The Hermitian form applies the identity to the first slot and the Hermitian conjugation to the second,

$$ \langle\tilde Q_1,\tilde Q_2\rangle_{*}=\mathrm{Sc}(\tilde Q_1\tilde Q_2^{*})=\sum_\mu Q_{1\mu}\bar Q_{2\mu}=Q_{1t}\bar Q_{2t}+Q_{1x}\bar Q_{2x}+Q_{1y}\bar Q_{2y}+Q_{1z}\bar Q_{2z}. $$

It is positive definite on the whole algebra, $\langle\tilde Q,\tilde Q\rangle_{*}=\sum_\mu\lvert Q_\mu\rvert^{2}$. The star is the Hermitian conjugation and is not decoration: without it the form is indefinite, as the plain form is on $\mathbb{M}_-$.

The scalar part. $\mathrm{Sc}$ is the coefficient of $e_t$. It is defined on $\mathbb{B}$ before any matrix is chosen, and it turns the product into a number. No representation enters.

The restriction to $\mathbb{M}_+$. An element of the sector is $\tilde H=h_te_t+i\vec h$ with $h_t,\vec h$ real, so $\tilde H^{*}=\tilde H$. The Hermitian form therefore equals the plain form there,

$$ \langle\tilde H_1,\tilde H_2\rangle_{*}=\mathrm{Sc}(\tilde H_1\tilde H_2)=h_{1t}h_{2t}+\vec h_1\cdot\vec h_2 , $$

the three minus signs of the bilinear expansion being cancelled by $i^2=-1$ from the imaginary coefficients. The plain form is the restriction, not a second pairing.

The two signatures. The Hermitian form has signature $(4,0)$ on the whole algebra. The plain form has signature $(4,0)$ on $\mathbb{M}_+$ and $(0,4)$ on $\mathbb{M}_-$; it is definite on each sector and indefinite on the algebra.

Which of the four norms. The convention names four algebraic norms: the plain norm $\langle\tilde Q\rangle=\mathrm{Sc}(\tilde Q^2)$, the quaternionic norm $\langle\tilde Q\rangle_\natural=\tilde Q\tilde Q^\natural$, called the biquaternion norm elsewhere in the corpus, the Hermitian norm $\langle\tilde Q\rangle_*=\sum_\mu\lvert Q_\mu\rvert^2$, and the Krein norm $\langle\tilde Q\rangle_{\natural*}=\lvert Q_t\rvert^2-\lvert Q_x\rvert^2-\lvert Q_y\rvert^2-\lvert Q_z\rvert^2$. The weight above is none of them: it is the normalised Hermitian form, a pairing, not a quadratic form. On $\mathbb{M}_+$ the plain and Hermitian norms coincide at $h_t^2+\lvert\vec h\rvert^2$, and the quaternionic and Krein norms coincide at $h_t^2-\lvert\vec h\rvert^2$; only the first pair is positive definite. The normalisation $\mathrm{Sc}(\tilde{Ϛ})=\tfrac12$ is linear and is not a norm condition.

The normalisation. The idempotents along a unit vector $\vec u$ are $\tilde\Pi_\pm=\tfrac12(e_t\pm i\vec u)$. They satisfy $\tilde\Pi_++\tilde\Pi_-=e_t$ and $\mathrm{Sc}(\tilde\Pi_\pm)=\tfrac12$. A state is a convex combination of the two idempotents along its axis, so

$$ \mathrm{Sc}(\tilde{Ϛ})=\tfrac12,\qquad\text{that is}\qquad \langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12. $$

The weight. With $\tilde{Ϛ}=\tfrac12(e_t+i\vec r)$ and $\tilde H=h_te_t+i\vec h$,

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

The star is the Hermitian conjugation; it is inert here because $\tilde{Ϛ}$ and $\tilde H$ are Hermitian, and it is kept because $\langle\cdot,\cdot\rangle_{*}$ is the form that is positive definite on the whole algebra. For $\tilde H=\tilde\Pi(\vec u)$ this is $\tfrac12(1+\vec r\cdot\vec u)$, which lies in $[0,1]$ when $|\vec r|\le1$, and the weights of the two outcomes $\tilde\Pi$ and $e_t-\tilde\Pi$ sum to one. This is the Born rule on the sector, and it is complete before any matrix is chosen.

The quaternionic form: the interval

The form. The quaternionic form applies the quaternion conjugation to the first slot,

$$ \langle\tilde Q_1,\tilde Q_2\rangle_\natural=\mathrm{Sc}(\tilde Q_1^\natural\tilde Q_2)=Q_{1t}Q_{2t}+Q_{1x}Q_{2x}+Q_{1y}Q_{2y}+Q_{1z}Q_{2z}. $$

The norm. Its diagonal is the biquaternion norm,

$$ \langle\tilde Q\rangle_\natural=\langle\tilde Q,\tilde Q\rangle_\natural=\tilde Q\tilde Q^\natural=Q_t^2+Q_x^2+Q_y^2+Q_z^2. $$

The interval. On $\mathbb{M}_-$, with $\tilde R=ict'e_t+x\vec e_x+y\vec e_y+z\vec e_z$,

$$ \langle\tilde R\rangle_\natural=-c^2t'^2+x^2+y^2+z^2, $$

of signature $(3,1)$.

What the norm does. It multiplies: $\langle\tilde Q_1\tilde Q_2\rangle_\natural=\langle\tilde Q_1\rangle_\natural\langle\tilde Q_2\rangle_\natural$. Its zero set is the light cone, and its nonzero null elements are the zero divisors. The elements of norm one are $SL(2,\mathbb{C})$, the double cover of the Lorentz group. In the $2\times2$ image it reads as a determinant, $\det\mathsf{M}_2(\tilde Q)=\langle\tilde Q\rangle_\natural$.

What it does not do. It is indefinite, so it is not a probability. It vanishes on the pure states, so it cannot normalise a state. It has no unit to normalise against and no weight.

The trace: a reading in the $2\times2$ image

The trace is not part of either construction. It is what the Hermitian form looks like once the algebra is written as $2\times2$ matrices. It is declared here, after the physics, and it is used only to restate results already obtained.

The declaration. Fix $\mathsf{M}_2:\mathbb{B}\to M_2(\mathbb{C})$ by $\mathsf{M}_2(e_t)=I_2$ and $\mathsf{M}_2(\vec e_k)=-i\sigma_k$, and define $\mathrm{Tr}:=\mathrm{tr}\circ\mathsf{M}_2$. Since $\mathsf{M}_2(e_t)=I_2$,

$$ \mathrm{Tr}(\tilde Q)=2\,\mathrm{Sc}(\tilde Q),\qquad \mathrm{Tr}(e_t)=2,\qquad \mathrm{Tr}(\tilde Q_1\tilde Q_2^{*})=2\langle\tilde Q_1,\tilde Q_2\rangle_{*}. $$

The factor $2$ is the dimension of the image. It is not a property of the form.

The state. $\mathrm{Tr}(\tilde{Ϛ})=2\cdot\tfrac12=1$. This is the statement "trace one". It restates $\mathrm{Sc}(\tilde{Ϛ})=\tfrac12$; it is not a new condition.

"The trace pairing". That phrase, used throughout the corpus, means $\mathrm{Tr}(\tilde Q_1\tilde Q_2^{*})=2\langle\tilde Q_1,\tilde Q_2\rangle_{*}$: the Hermitian form read through the trace. It is not a fifth form. The star is the Hermitian conjugation and it is not optional in the definition of the pairing: it is what makes the pairing positive definite on the whole algebra. On $\mathbb{M}_+$, where both arguments are Hermitian, $\tilde Q_2^{*}=\tilde Q_2$ and the star acts as the identity, but it is written in every probability formula: it names the form that is positive definite on the whole algebra, and the plain form is the restriction to the sector.

The pair. For any two elements,

$$ \mathrm{Tr}(\tilde Q_1\tilde Q_2^{*})=2\langle\tilde Q_1,\tilde Q_2\rangle_{*}=2\,\mathrm{Sc}(\tilde Q_1\tilde Q_2^{*}). $$

This is the trace reading of the Hermitian form, not a fifth form. It is used only at normalisation; the Born weight itself is stated in the quantum section as the normalised Hermitian form, and its trace reading is recorded there, last.

What the trace is not. It is not defined on $\mathbb{B}$ before a representation is chosen; $\mathrm{Sc}$ is. Its factor is the dimension of the representation, while $\mathrm{Sc}$ and the Hermitian form do not depend on it. The matrix trace of a positive product is the Hilbert–Schmidt inner product, $\operatorname{Tr}(\tilde\varrho\tilde H^{*})=\operatorname{Tr}(\tilde\varrho^{1/2}\tilde H\tilde\varrho^{1/2})\ge0$ for $\tilde\varrho,\tilde H\ge0$; that non-negativity is the reason the star belongs in the probability reading.

One operation, three uses

A single formula acts on the algebra,

$$ \Gamma_{\tilde{Q}_1}(\tilde{Q}_2) = \tilde{Q}_1\tilde{Q}_2\tilde{Q}_1^{*}, $$

and it preserves each sector and preserves the biquaternion norm whenever $\langle\tilde{Q}_1\rangle_{\natural} = 1$. The same formula is the Lorentz transformation, the evolution of a quantum state and the update of a measurement, the three being distinguished only by which normalisation the acting element is given:

So the reversible and the irreversible are one algebraic operation under two normalisations, and measurement differs from evolution by a property of the acting element — it is not invertible — rather than by a separate axiom. That is the framework's answer to the question of why measurement should be different in kind, and the answer is a theorem about the algebra.

The interlock

One more structural fact is used repeatedly. For two elements of the same sector,

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

The commutator always lands in the material sector and the anticommutator in the informational one. On $\mathbb{M}_-$ the commutator is explicit and minimal,

$$ [\tilde{Q}_1,\tilde{Q}_2] = 2\,\vec q\times\vec r \qquad (\tilde{Q}_1 = iq'_te_t+\vec q,\ \tilde{Q}_2 = ir'_te_t+\vec r), $$

twice the spatial cross product, with the time direction absent: the time axis is central for the bracket. Hence $\mathbb{M}_-$ is closed under the commutator and is a Lie algebra, $\mathfrak{u}(2) = \mathbb{R}ie_t\oplus\mathfrak{su}(2)$, of dimension four; $\mathbb{M}_+$ is closed under the anticommutator and is a Jordan algebra, not a Lie algebra. Neither sector is closed under the product, so both the rotor group and the state module live in the full algebra.

Why the Material Sector Carries Relativity

Five reasons, in increasing order of strength.

1. It is a four-dimensional real space with a real form of the right signature. On $\mathbb{M}_-$, with $\tilde{R} = ict'\,e_t+x\,\vec e_x+y\,\vec e_y+z\,\vec e_z$, the biquaternion norm reads $$ \boxed{\;\langle\tilde{R}\rangle_{\natural} = -c^2{t'}^2+x^2+y^2+z^2\;}, $$ a real form of signature $(3,1)$. Position, velocity, momentum, energy–momentum, four-current and four-potential are all elements of this one subspace, so the kinematical content of relativity — that these are one kind of object — is a statement about the algebra.

2. The Lorentzian signature is derived, not postulated. The biquaternion norm is $Q_t^2+Q_x^2+Q_y^2+Q_z^2$, in which all four directions enter on an equal footing: there is no minus sign in the form itself and none is needed. A minus appears only when a real coordinate is put on a direction whose coefficient carries an $i$, which is exactly the $ict'$ convention, and then it comes from $i^2 = -1$ alone. Nothing about the metric has to be assumed at the start, not even the choice of which direction is timelike. A direction is timelike because its coefficient is imaginary. That is a labelling of the coordinate, not a commitment of the form.

3. The invariance group comes for free, and it is the Lorentz group. A form that multiplies always brings its own isometry group: the elements of unit norm. Those are the biquaternions with $\langle\tilde{U}\rangle_{\natural} = 1$, they form $SL(2,\mathbb{C})$, and they act on $\mathbb{M}_-$ by $\Gamma$ above. That action is the double cover of the proper orthochronous Lorentz group. Lorentz invariance is therefore not an axiom added to the algebra; it is the invariance group of the algebra's own norm. This is the strongest single reason to call $\mathbb{M}_-$ the relativistic sector, because it is the one place where the framework produces a physical symmetry group rather than naming one.

4. The causal structure is an algebraic locus. $\langle\tilde{Q}\rangle_{\natural} = 0$ is the light cone. Its nonzero elements are precisely the elements with no inverse — $3ie_t+\vec e_x+2\vec e_y+2\vec e_z$ has $\langle\tilde{Q}\rangle_{\natural}=-9+1+4+4=0$, so it is a zero divisor. So the light cone, massless propagation and the failure of division are one fact seen three ways.

5. Rotations are the bracket. $[\tilde{Q}_1,\tilde{Q}_2] = 2\,\vec q\times\vec r$, so angular momentum and the rotation generators are the commutator of the sector, and the Lie algebra is $\mathfrak{u}(2)$.

What the material sector does not supply. No scale, no preferred time direction, no Hamiltonian and no equation of motion. The kinematics of relativity is in the algebra; its dynamics comes from outside.

Why the Informational Sector Carries Quantum Physics

Seven reasons. The first is decisive.

1. Only Hermitian elements can be observables or states, and the Hermitian elements are exactly $\mathbb{M}_+$. Under the fixed isomorphism, $\mathsf{M}_2(\mathbb{M}_+)$ is the space of Hermitian $2\times2$ matrices and $\mathsf{M}_2(\mathbb{M}_-)$ is the space of skew-Hermitian ones. Hermitian elements have real eigenvalues, a spectral decomposition in terms of projections, and a positive cone; skew-Hermitian elements have purely imaginary eigenvalues and no positive cone. The reading of a physical quantity must be a real number, and a state must be positive, so the observable — the Hermitian representative $iA$ of the material element $A\in\mathbb{M}_-$ that is the object observed, a four-vector or a field — and the state must both be elements of $\mathbb{M}_+$; the eigenvalues of $iA$ are the readings. Neither a real spectrum nor a positive cone survives on the material element itself. This single fact fixes the assignment of the sectors. It also explains the unfamiliar labelling. The material sector is anti-Hermitian because it carries the four-vectors, whose business is to multiply and so to give isometries. The informational sector is Hermitian because it carries the observables — the representatives $iA$ — and the states, whose business is to have real values.

2. The sector is the observable algebra of a two-state system, exactly. $\mathsf{M}_2(\mathbb{M}_+)$ is the Hermitian $2\times2$ matrices; an element is $$ \boxed{\;\mathsf{M}_2(\tilde{H}) = h_tI_2 + \vec h\cdot\boldsymbol{\sigma}, \qquad \tilde{H} = h_te_t + i(h_x\vec e_x+h_y\vec e_y+h_z\vec e_z)\;}, $$ with real $h_t,\vec h$, and its spectrum is $h_t+\lvert\vec h\rvert$ and $h_t-\lvert\vec h\rvert$. This is not an analogy with quantum mechanics: it is the operator algebra of one qubit.

3. The probability weight is the normalised Hermitian form, and on $\mathbb{M}_+$ it is the Born rule. The form is $\langle\tilde Q_1,\tilde Q_2\rangle_{*}=\mathrm{Sc}(\tilde Q_1\tilde Q_2^{*})$; on $\mathbb{M}_+$ it coincides with the plain form $\langle\tilde Q_1,\tilde Q_2\rangle=\mathrm{Sc}(\tilde Q_1\tilde Q_2)$, which is its restriction, but the weight below is written with the star on the sector and on the algebra alike; the scalar part is the coefficient of $e_t$ and needs no matrix. On $\mathbb{M}_+$ the form is real and positive definite, of signature $(4,0)$, and it does not multiply — which is exactly what a probability needs. The unit satisfies $\langle e_t\rangle_{*}=1$; a state is normalised against it by $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$, that is $\mathrm{Sc}(\tilde{Ϛ})=\tfrac12$; and the weight of an outcome with effect $\tilde{H}$ is $$ \boxed{\;p=\frac{\langle\tilde{Ϛ},\tilde{H}\rangle_{*}}{\langle\tilde{Ϛ},e_t\rangle_{*}} =\frac{\mathrm{Sc}(\tilde{Ϛ}\tilde H^{*})}{\mathrm{Sc}(\tilde{Ϛ})} =h_t+\vec r\cdot\vec h\;}. $$ In the $2\times2$ image, where $\mathrm{Tr}:=\mathrm{tr}\circ\mathsf{M}_2$ as declared above, $\mathrm{Tr}=2\,\mathrm{Sc}$ and $\mathrm{Tr}(e_t)=2$, so $\mathrm{Tr}(\tilde{Ϛ}\tilde H^{*})=2\,\mathrm{Sc}(\tilde{Ϛ}\tilde H^{*})=h_t+\vec r\cdot\vec h$ and the weight reads $p=\mathrm{Tr}(\tilde{Ϛ}\tilde H^{*})/\mathrm{Tr}(\tilde{Ϛ})$. This is the trace formula, a reading of the pairing and not one of the four algebraic norms. On $\mathbb{M}_+$ the star in the numerator is inert, since $\tilde H^{*}=\tilde H$, so the same number is $\operatorname{Tr}(\tilde{Ϛ}\tilde H)/\operatorname{Tr}(\tilde{Ϛ})$. The star is what makes the pairing positive definite on the whole algebra. On the sector the unstarred trace is real, because $(AB)^{\dagger}=BA$ for Hermitian $A,B$ makes $\mathrm{tr}(AB)$ equal to its conjugate; it is non-negative for $0\le\tilde H\le e_t$, because $\mathrm{tr}(AB)=\mathrm{tr}(A^{1/2}BA^{1/2})\ge0$. The weight is therefore $p=\operatorname{Tr}(\tilde{Ϛ}\tilde H)/\operatorname{Tr}(\tilde{Ϛ})\in[0,1]$. Note what this says: probability is here a pairing of two elements with the positive-definite form, not a product of weights. That is exactly what the quantum side needs, and it is why the framework can claim that the difference between its two halves is the difference between a form that multiplies and a form that does not.

4. The state space is not added: it is the positive cone of the sector. The pure states are the algebra's own idempotents, $$ \tilde{\Pi} = \tfrac12(e_t \pm i\vec u_{\mathrm r}), \qquad \tilde{\Pi}^2 = \tilde{\Pi}, \qquad \tilde{\Pi}^{*} = \tilde{\Pi}, \qquad \langle\tilde{\Pi},e_t\rangle_{*} = \tfrac12, \qquad \langle\tilde{\Pi}\rangle_{\natural} = 0, $$ parametrised by the unit sphere — the Bloch sphere — and the mixed states are the positive elements with $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$, that is $\mathrm{Sc}(\tilde{Ϛ})=\tfrac12$ $$ \tilde{Ϛ} = \tfrac12(e_t + i\vec r), \qquad \lvert\vec r\rvert \le 1, $$ with $\det\mathsf{M}_2(\tilde{Ϛ}) = (1-\lvert\vec r\rvert^2)/4 \ge 0$ as the positivity condition: the Bloch ball, exactly the convex set of states of a qubit. Nothing about the state space has to be put in by hand, and the pure states sit on the same null cone that the material sector calls the light cone, since $\langle\tilde{\Pi}\rangle_{\natural} = 0$.

5. The non-commutativity has exactly the quantum pattern. The product of two Hermitian elements is Hermitian exactly when the two commute. The commutator of two Hermitian elements is anti-Hermitian, hence an element of $\mathbb{M}_-$. Therefore $i$ times a commutator is again Hermitian, hence again an observable. That is the pattern of the quantum commutator and of the generator of a dynamics. Equivalently, $\mathbb{M}_+$ closes under the anticommutator and not under the commutator: it is a Jordan algebra, as an algebra of observables is, while the Lie algebra sits in the other sector.

6. Interference has a home. A state can be represented as the left ideal $\mathbb{B}\tilde{\Pi}$ with basis $\{\tilde{\Pi}, \tilde\Upsilon\}$. Here $\tilde{\Pi} = \tfrac12(e_t+i\vec e_z)$ is idempotent and $\tilde\Upsilon = \tfrac12(i\vec e_x+\vec e_y)$ is nilpotent, $\tilde\Upsilon^2 = 0$, with $\tilde\Upsilon^{*}\neq\tilde\Upsilon$: a partner to the state and not itself a state. On that basis the positive form has Gram matrix $\tfrac12I_2$. So the two amplitudes are the two coefficients of the element, and the amplitudes add while the weights do not.

7. What the sector does not supply. The complex phase lives in the state module and not in the sector. $\mathbb{M}_+$ is a four-real-dimensional subspace, $\mathbb{R}e_t\oplus i\mathbb{R}\vec e_x\oplus i\mathbb{R}\vec e_y\oplus i\mathbb{R}\vec e_z$ — one real direction and three imaginary ones. It is not an algebra under the ambient product, only a Jordan algebra under the anticommutator; the complex structure acts through the centre. So the $i$ of quantum mechanics is carried by the module. The canonical commutation relation is not in the algebra and must be imported. And only one qubit is built: a boost of a trace-one element exits the trace-one slice, so the relativistic state space does not exist yet.

Established Results Rewritten with Biquaternions

Nothing in this section is new. It holds the two bodies of established material the framework stands on, one in physics and one in mathematics. The section name is the author's: the physics block is the results whose subject is a physical object and to which the algebra adds something; the mathematics block is established algebra that the framework rests on rather than rewrites.

In physics

This is what the algebra supplies. Two rows are the biquaternion rewriting proper: Maxwell's four collapse into the one equation $\tilde{\nabla}\tilde{F} = -\sqrt{\mu}\,\tilde{J}$, and the Klein–Gordon and Dirac pair is the one factorisation $\tilde{\nabla}\tilde{\nabla}^{\natural} = \Box$. The standard results the framework merely carries are not repeated here: the hydrogen spectrum, the Casimir force and the Unruh temperature, linearized gravity, and the qubit results — the observables, the Bloch ball, the Born rule, evolution and measurement. Since $\mathbb{B}\cong M_2(\mathbb{C})$, the corpus calls those the same mathematics in different notation — hydrogen transcribed, the Unruh effect imported, linearized gravity not derived, the Casimir vacuum on an imported module — and the ones the article needs are stated where they do work, in the sections above. The ten rows below are what remains: the two biquaternion equations, and the eight results the algebra supplies in its own objects, one negative and one a correction.

In mathematics

Every row is a theorem about $\mathbb{B}$ and its subspaces alone, provable from the multiplication table, independent of any physics. Established, and load-bearing: everything the framework does rests on these. The rows whose subject is a physical object sit in the physics block above, and they rest on these.

New Results

Everything in this section is the corpus's own, and the section is where what is new is collected. It is of two kinds, one in physics and one in mathematics. The mathematical part is proved from the definitions and needs no experiment. The physical part is new as statements about the physics; it rests on the identification of the two sectors, and where the algebra does not prove a statement, the statement is proposed rather than proved.

In physics

These are new as statements about the physics. The first states the identification of the two sectors, and all but the comparison with the earlier quaternionic programme rest on it; that comparison rests on the algebra rather than on the identification.

In mathematics

These are constructions and theorems proved from the definitions, new to the standard literature. They stand on their own: they make no physical claim, and a reader can check them as mathematics. The sizes are given because they are the measure of how far each is developed, and every one is read off the mathematics menu, chapter by chapter, rather than asserted. The biquaternion chapter holds 302 articles, of which 216 are distributed over the twelve biquaternion structures — fifteen for each of the eight symmetric and antisymmetric bands, and 19, 20, 24 and 33 for the four general products. Beside it stand the general chapters the biquaternion articles draw on, and they are not counted in that 302. The four sesqualgebra chapters hold 138: Sesqualgebras 40, Sesqualgebras with a degree-2 form 49, Topology on Sesqualgebras 24, Topology on Sesqualgebras with a degree-2 form 25. The two chapters on algebras with a degree-2 form hold 90: Algebras with a degree-2 form 54 and Topology on Algebras with a degree-2 form 36. The six chapters together hold 228. Inside the biquaternion chapter the forms live in two places. The Form band of each of the twelve biquaternion structures holds 31 articles in all. The 13 of Products, Forms and Algebraic Norms own the four general products, the four forms and the four algebraic norms. A further 14 articles compare the biquaternions with the neighbouring number systems. The claim that these objects are absent from the standard literature is the corpus's claim; I could not test it from inside the corpus, and the test is a literature check rather than a computation.

None of the new results in physics is a measurement prediction, and the corpus does not claim otherwise. Their content is of four kinds: an identification that fixes the objects of a theory, a derivation that turns a postulate into a theorem about the algebra, an explanation of a pattern already observed in practice, and a comparison that fixes the framework's relation to an earlier programme. The new mathematics is of a different kind again: it is proved, so it stands or falls with its proofs and with the literature check, and no experiment bears on it.

What the Framework Does Not Do

The project keeps its own list of proven obstructions. These are the ones that change what a reader can expect.

• No scale. The algebra fixes no length, mass, coupling or dimensionless ratio. Every scale is an input.
• No preferred time and no Hamiltonian. Time is one coordinate of an element; the algebra fixes no arrow for it, no Hamiltonian and no basis.
• No single norm is both multiplicative and positive. The positive definite norm exists: the Euclidean norm $\lVert\tilde Q\rVert_E=\sqrt{\mathrm{Sc}(\tilde Q\tilde Q^{*})}$, the Frobenius norm under $\mathbb{B}\cong M_2(\mathbb{C})$, real and vanishing only at the origin. It is not multiplicative. The norm that multiplies, the biquaternion norm $\langle\tilde Q\rangle_{\natural}=\tilde Q\tilde Q^{\natural}$, is the interval and is indefinite, vanishing on nonzero elements. The metric and its isometry group use the multiplicative norm; the topology and the Hilbert space use the Euclidean one.
• No canonical state module. It depends on a chosen idempotent: different choices give isomorphic but distinct submodules, so the split into amplitudes is canonical only up to unitary equivalence.
• The biquaternion norm's invariance group exceeds the symmetry of states. The group preserving it is $\mathrm{U}(1)\cdot SL(2,\mathbb{C})$, strictly larger than the unitary group $\mathrm{U}(2)$ that acts on states: the framework has more symmetry than the physics uses.
• No commutation relation in the algebra. The canonical commutator must be imported.
• No complex structure in the sector. $\mathbb{M}_+$ is a real Jordan algebra, closed under the anticommutator and not under the product; the complex structure acts through the centre, and the phase is carried by the module.
• Multiplication is blind to chirality. Every operation the algebra supplies is a multiplication, and a multiplication acts alike on the two chiral components of a Dirac module. Chirality, the discrete symmetries, and the split of positive from negative energy are external.
• Three of the four brackets fail the Jacobi identity. The plain bracket closes and gives the Lie algebra, but the conjugation-carrying brackets, which the quantum pairings need, do not: they generate a higher structure instead.
• The internal group stops at $\mathrm{U}(2)$. The algebra's own continuous symmetry is the central phase $\mathrm{U}(1)$ together with the quaternion triple $SU(2)$, that is $\mathrm{U}(2) = (SU(2)\times \mathrm{U}(1))/\mathbb{Z}_2$, the internal group of one qubit. $SU(3)$ cannot be generated: the sector is four-real-dimensional, and the algebra's modules have even complex dimension, so no three-dimensional colour multiplet is closed under it. The fermion generations and the strong-interaction octet lie outside.
• The formalism is single-qubit. The algebra carries one qubit's worth of structure: no internal tensor factor and no selection rule for measurement. There is no stable classical sector under a generic evolution — classicality must come from decoherence or a chosen context. Curvature cannot reside in the algebra and needs an inserted tetrad, the field equations are transcribed rather than derived, and gravity is a research agenda.

The empirical position. The project's own assessment, and the one a reader should adopt until it revises itself, is negative: the framework is currently empirically equivalent to standard physics. It makes no prediction that distinguishes it. Empirical equivalence is not the absence of new content: what is new here is mathematics, which needs no experiment, and interpretations that reorganise or explain known results. What the framework lacks is a new prediction, and no experiment can confirm or refute an interpretation. Each candidate signature has been examined and found unavailable, and every reason is structural rather than accidental. Modified dispersion needs a scale the algebra does not have. Birefringence and a second light cone are foreclosed, because there is only one norm. A coupling of the material to the informational sector is unspecified — no field, no action, no constant — and this is named as the framework's largest gap. Deviations of the quaternionic kind cannot appear, since the algebra is complex and associative. Derived dimensionless relations, the one class that would need no new scale, are absent.

The criterion to apply is the project's own. For any number the framework reproduces, ask which of its structures produced that number. If the answer is a structure it shares with standard physics, the number is not evidence for the framework's addition. The shared structures are the identification of $\mathbb{M}_-$ with Minkowski space, the isomorphism with $2\times2$ matrices, and every inserted constant.

Common Misreadings

1. The material sector is the anti-Hermitian one. The more familiar convention would take the Hermitian subspace as the physical one. Here it is the other way round, and the reason is stated above: the four-vectors must multiply, the readings must be real.
2. The prime marks the $i$, not the sector. The primed parameter is the one attached to $ie_\mu$: a rule about the coefficients and not a label of a subspace. The crossing it produces is the point. In $\mathbb{M}_-$ the $i$ sits on the scalar, so the material time is the primed scalar slot, $ict' = iq'_t$; in $\mathbb{M}_+$ it sits on the vector, so the informational positions are the primed slots, $i(x'\,\vec e_x+y'\,\vec e_y+z'\,\vec e_z)$. That one crossing is what makes the two sectors mirror images.
3. The material time enters as an imaginary coefficient and the informational time as a real one. $ict'$ against $ct$. The minus of the metric comes from $i^2 = -1$ in $ict'$ and from nowhere else.
4. Two forms, two jobs, never conflated. The Hermitian form $\langle\cdot,\cdot\rangle_{*}$ answers the probability question: it is real and positive definite on the whole algebra, and it does not multiply. On $\mathbb{M}_+$ it equals the plain form $\langle\cdot,\cdot\rangle$, which is the reading used for the sector computations, while the Born weight keeps the star. The quaternionic form $\langle\cdot,\cdot\rangle_{\natural}$ answers the interval question on $\mathbb{M}_-$: it is indefinite, it multiplies, and its zero set is the light cone. In particular $\langle\tilde{Q}\rangle_{\natural} = 0$ does not mean $\tilde{Q} = 0$. A state is normalised by the linear condition $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$, equivalently $\mathrm{Sc}(\tilde{Ϛ})=\tfrac12$, never by the quadratic norm.
5. The interval is the biquaternion norm, not the square. $\langle\tilde{Q}\rangle_{\natural} = \tilde{Q}\tilde{Q}^{\natural}$; the bare $\tilde{Q}^2$ is not a scalar and is not the interval.
6. Neither sector is a subalgebra. Neither is closed under the product: a product of two elements of one sector generally has components in both. That is why the rotor group and the state module live in the full algebra.
7. The electromagnetic field strength is in neither sector. Unlike the four-potential it is not a four-vector: its scalar part vanishes, and its vector part is mixed real and imaginary, the imaginary half carrying the electric field and the real half the magnetic field. The four-potential, not the field, is a material element.
8. The two sector-exchange operations are not the same map. The exchange of the two sectors is multiplication by $i$, of order four and invertible. The Wick rotation is a continuation of the time coordinate. The two agree on the time axis and differ away from it. And being invertible and of order four, the exchange carries no arrow of time.
9. Agreement with standard physics is not confirmation. The project's own criterion, stated above: for a number the framework reproduces, ask which of its structures produced it. Where the structure is shared, the agreement tests the dictionary and not the framework.
10. A form that multiplies and a form that does not are two different tools. A metric is the first; a probability is the second. Reading a probability off the multiplicative form, or a metric off the definite one, is the error that the framework exists to avoid.
11. "The framework is only a rewriting of standard physics" is half right. Most of the corpus is indeed standard physics restated, and that part is evidence about the language and not about the world. The word to reject is only: the new results above are not rewritings.
12. "Observable" has two senses, and only one is Hermitian. The word names the Hermitian representative $iA\in\mathbb{M}_+$, whose eigenvalues are the readings. The object observed is the material element $A\in\mathbb{M}_-$, a four-vector or a field. Calling the material element an observable — reading it in the classical sense of a quantity that simply has a value — is a change of dictionary, not of algebra: the two writings differ by the central $i$ and generate the same numbers.
13. The trace is a reading, not an intrinsic operation. $\mathrm{Sc}$ is defined on $\mathbb{B}$ without a representation: it is the coefficient of $e_t$. $\mathrm{Tr}$ is defined only once a matrix representation is chosen. In $\mathsf{M}_2$ the identity $\mathrm{Tr}=2\,\mathrm{Sc}$ holds because $\mathsf{M}_2(e_t)=I_2$, and the factor is the dimension of the image. Normalisation in the algebra is $\langle\tilde{Ϛ},e_t\rangle_{*}=\tfrac12$, that is $\mathrm{Sc}(\tilde{Ϛ})=\tfrac12$; "trace one" is the name that condition takes in the $2\times2$ image.

Summary

The framework's central affirmation is that the biquaternion algebra already contains both special relativity and quantum physics, as a matter of algebra. They live in two complementary real four-dimensional subspaces, joined by multiplication by the imaginary unit. Quantum physics can therefore be developed natively in a setting where the relativistic structure is already present. The strongest argument for the first half is that the biquaternion norm is multiplicative, and real of signature $(3,1)$ on $\mathbb{M}_-$. Its isometry group is the unit-norm biquaternions, $SL(2,\mathbb{C})$, acting by $\tilde{Q}\mapsto\tilde{U}\tilde{Q}\tilde{U}^{*}$. That group is the Lorentz group, and the light cone is the biquaternion norm's zero set. The strongest argument for the second half is that the Hermitian elements are exactly $\mathbb{M}_+$. The observables — the Hermitian representatives of the material elements — and the states of a two-state system are therefore elements of that sector and of no other. Real spectra and a positive cone exist there. The pure states are the sector's idempotents, the mixed states its positive normalised cone, and the Born weight its normalised Hermitian form. The one form $\Gamma$ serves as Lorentz transformation, evolution and measurement under three normalisations.

The two halves are separated by the two ways a form can behave: a form that multiplies gives a group of isometries, and a form that does not gives a pairing and a probability. One algebra carries both.

Most of the corpus restates known physics in the biquaternion language, which confirms the dictionary and not the framework. What it adds beside that comes in two kinds. In mathematics: the sesqualgebras, the systematic construction of the non-associative complexoid, quaternioid and biquaternion structures, and the Bochner theorem for the biquaternion transform, which fixes the form it needs — the Hermitian form $\langle\tilde Q_1,\tilde Q_2\rangle_{*}$ — and the roots for which the transform of a positive measure is positive definite, namely the degenerate ones. All of these are proved from the definitions, so that they stand on their own and no experiment bears on them. In physics: the derivation of the form of the Born rule from the normalised Hermitian form, read through the trace, the interpretation of the Wick rotation as a transfer to the elliptic quaternion subspace, with its classification of applications, and the Matsubara thermal formalism obtained without the rotation, inside the material sector, where the positivity is carried by the Hermitian form and is the central-root case of the Bochner theorem, together with the cutoff of the renormalization group obtained the same way, on the Hermitian form of a material element rather than on a rotated norm. The physics results rest on the identification of the two sectors; they are new to the literature as statements about the physics, and they are proposed rather than proved from the algebra, except the thermal rewriting and the regulator, whose algebraic step is the identity between the Hermitian form and the rotated norm. Beside them stands the comparison with the quaternionic quantum theory, where the algebraic core — the tensor obstruction, the forced centrality of the unit, and the trace obstruction to the commutation relation — is proved, and the reading it carries is that the biquaternion framework is that programme's corrected successor: the same algebra and geometry, with the quaternions moved from the scalar field to the elements. Neither kind is a prediction.

The limits are as sharp as the results. The algebra contains no scale, no time direction, no Hamiltonian, no positive form that multiplies and no commutation relation — the commutator of two Hermitian elements has no scalar part, so no pair in $\mathbb{M}_+$ can satisfy one. Its state module is not canonical, multiplication is blind to chirality, and the internal group reaches $\mathrm{U}(2)$ and not $SU(3)$. The quantum formalism is single-qubit. And the framework is currently empirically equivalent to standard physics, with no distinguishing prediction.

Summary of Notation