Why Spacetime Is Four-Dimensional and Matter Is Dirac

1. Introduction

1.1. Two Facts Usually Assumed

Two facts about the world are ordinarily written down rather than accounted for. The first is that the spacetime metric is four-dimensional with Lorentzian signature—one time direction and three of space, the stage $(1,3)$ on which all of physics is set. The second is that matter obeys the Dirac equation: among the most precisely confirmed laws in physics, yet one that enters the theory as a postulate. Both are foundational, both are typically assumed, and both have resisted explanation from anything more primitive. This paper obtains both from a single definition—that of an observer-bearing spacetime—and addresses them together rather than singly.

• On the dimension of spacetime.

The dimensionality of space has been traced to the stability of orbits and atoms—bound states exist only for an inverse-square law, hence three spatial dimensions [1,2]—and the full signature to anthropic selection, alternative signatures yielding worlds in which no observer could arise or predict its surroundings well enough to persist [3]. These accounts explain a foundational fact by appeal to phenomena vastly more derived: the stability of planetary orbits, the existence of bound atoms, the viability of living observers. Each presupposes essentially all of physics, chemistry, and biology, and each is built upon the very dimensionality it is invoked to explain. The objection is not ours alone. Callender [4] argues that such “proofs” of the tridimensionality of space are answers in search of a question—that they presuppose the detailed physical laws whose arena is at issue, and so cannot ground the dimension in anything more basic than itself. The stability arguments are, on inspection, partly circular in just this way: the field equation whose solutions are required to be stable already encodes the three-dimensional case, so that the conclusion is built into the premise [5]. To account for the cellar by the roof inverts the order of explanation.

A foundational fact ought to be explained, if at all, by a foundational criterion. The most primitive demand one can make of a spacetime is that it be able to bear an observer at all. We mean “observer” here in a strictly algebraic sense—the existence of a frame in which the probability current is positive-definite and conserved—and not in the biological or predictive sense of an organism that models its surroundings. This is what frees the criterion from the circularity above: positive-definiteness of a probability current is prior to dynamics, prior to kinematics, prior even to the dimension of spacetime, and presupposes no atoms, no chemistry, and no life. We make it precise below as the observer-bearing condition. The observer-bearing spacetimes turn out to be exactly those of signature $0+1$, $1+1$, $2+1$, $3+1$—the dimensional ceiling and the single time direction fixed together—by one algebraic step that invokes nothing about orbits, chemistry, or survival. Where Tegmark asks which spacetimes are hospitable to an observer who can forecast and persist, we ask the prior question, in the algebraic sense: which spacetimes admit a positive-definite, conserved probability current at all.

• On the Dirac equation.

The Dirac equation has been reached before by several routes; each leaves its origin unexplained in a different way. Dirac’s own derivation finds the first-order operator whose square is the Klein–Gordon operator [16]—the construction by which the equation was discovered, and a definitive answer to what the equation is. It was not meant to answer why the dynamics should take that form, and it is the latter question we ask.

Frieden’s Extreme Physical Information (EPI) [6] aims to derive the equations of physics, the Dirac equation among them, by extremizing a Fisher-information functional. Whether EPI is a genuine derivational principle or a formalism adjusted to reach equations already known has been disputed. Streater [7] lists EPI among “lost causes” in physics, on the ground that its central construction does not deliver the explanatory work it claims; the more specific concern is that the “bound information” term in the method is chosen with the target equation in view, so that the result is built in rather than derived.

The quantum-cellular-automaton derivation of D’Ariano and Perinotti [8] is more ambitious and entirely successful on its own terms: the Dirac equation emerges in the large-scale limit, without invoking relativity, from a minimal automaton constrained by unitarity, locality, homogeneity, and isotropy, each constraint narrowing the admissible evolutions until Dirac is the survivor. What this route does not do—and does not set out to do—is proceed in the manner by which physical law is forged from measurement. The one method we possess for building a theory from experiment in a reproducible way—Jaynes’s reconstruction of statistical mechanics [13]—is to commit to nothing beyond what a measured constraint requires: maximal non-commitment, subject to observation. The automaton’s axioms are not of this kind; they are mathematical conditions that corner the equation, rather than a measured constraint imposed on an otherwise non-committal flow. The derivation arrives at Dirac by a sequence of admissibility conditions, not by an inference of Dirac in the way thermodynamics is inferred from the laboratory.

This is the gap we mean to fill. We build the dynamics in the one manner known to yield physical law from measurement: a maximally non-committal evolution—one that creates and destroys no information—subject to a single measured constraint. Applied to the conserved current of an observer-bearing spacetime, that prescription yields the Dirac equation directly: first-order, with no list of cornering axioms. The equation arises as statistical mechanics arises—as the least-committal flow consistent with what is observed (the explicit constraint).

• One definition, two facts.

What unites the two results is that neither is reached by positing the physics it recovers. We make one definition—that of an observer-bearing spacetime—and then enumerate the spacetimes that satisfy it. The enumeration is the first result: the observer-bearing spacetimes are exactly $0+1,1+1,2+1,3+1$. It presupposes nothing about quantum mechanics; it asks which probability structures a spacetime can carry observably at all, and classifies them. The admissible structures range from a single classical weight in the lowest arena—one number evolving in time, no phase, no interference—through the richer structures available as the dimension grows, to the four-component spinor of the maximal arena. Classical and quantum probability are both members of this family, not assumptions placed into it. The second result is that the information-preserving dynamics of the maximal member is the Dirac equation. The four-dimensional Lorentzian arena and the Dirac equation—ordinarily independent assumptions laid atop a dynamics already in place—are here two findings within a single classification of what an observer-bearing spacetime can be. We do not derive the world from nothing; we define what it is for a spacetime to bear an observer, list those that do, and find our world’s arena and law as the maximal one.

1.2. The Observer-Bearing Condition

Let the probability be carried by a multivector amplitude $\psi$, an element of the even subalgebra of the Clifford algebra $C\ell \left(V\right)$ of the tangent space V of spacetime. We use a multivector amplitude, rather than the usual complex one, because the observer-bearing criterion below has force only when the amplitude is built from the geometry itself: in $3+1$ dimensions the two representations are isomorphic, so nothing is lost, but in other dimensions and signatures they diverge, and it is that divergence that makes the criterion discriminating. We define what it is for a spacetime to bear an observer in terms of two requirements on this amplitude:

(P1) 

a positive-definite scalar density $\rho ={\langle \tilde{\psi }e\psi e\rangle }_0\ge 0$ along some unit basis vectore, and

(P2) 

a conserved grade-1 current $J=\tilde{\psi }e\psi$ that is a pure vector (no higher-grade parts).

(P1) is the Born requirement together with the demand that there be a frame to read it: some direction must assign the configuration a real, non-negative density. (P2) is the requirement that probability flow be a conserved four-vector current admitting a continuity equation—the minimal condition for a relativistic probability balance.

Definition 1 

(Observer-bearing spacetime). A spacetime isobserver-bearingif its multivector amplitude admits, in some frame, both (P1) and (P2): there is a unitvectore for which the canonical density $\rho ={\langle \tilde{\psi }e\psi e\rangle }_0$—formed by the algebra’s own geometric product, not by a freely chosen inner product—is real and positive-definite over all ψ, and for which the current $J=\tilde{\psi }e\psi$ is a pure grade-1 vector. The “observer” is algebraic, not physical: it is a direction e along which a positive, conserved probability can be read, not an organism that models its surroundings. The reference element e is a grade-1 vector because that is the operational content of such a direction—a lab axis, a worldline tangent. We donotrequire e to be timelike; that the qualifying direction is forced to be the one of opposite square—i.e. timelike—is a consequence (Theorem 1), not an assumption. A spacetime that bears no such direction in any frame is, in principle, unmeasurable.

1.3. Results (Summary)

Theorem 1 

(Enumeration of observer-bearing spacetimes). A spacetime is observer-bearing if and only if its metric is Lorentzian with exactly one time dimension and total dimension $n\le 4$. The observer-bearing signatures are exactly $(1,0)/(0,1)$, $(1,1)$, $(2,1)/(1,2)$, $(3,1)/(1,3)$; all Euclidean $(n,0)$, all split (e.g. $2+2$), all multi-time, and all $n\ge 5$ signatures are not observer-bearing.

Theorem 2 

(Dynamics of the maximal observer-bearing spacetime). On the maximal observer-bearing spacetime, $3+1$, the information-preserving transport of the conserved current of Theorem 1—the flux·force flow of vanishing net entropy production, with scalar entropy-production density, with ${\partial }_{\mu }\psi$ made a genuine entropic force by the addition of the required conjugate constraint—is the Dirac equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$. The coupled-channel matrix form and the unitarity of the evolution are supplied by the flux·force structure; the scalarity constraint admits the grade-1 (Dirac) coupling; the constraint that guarantees an entropic force turns out to have the form of the mass term, with the mass m as its Lagrange multiplier.

The first theorem enumerates the observer-bearing spacetimes and, with them, the conserved current each carries; the second transports that current on the maximal one and finds the Dirac equation. The two stages share the current itself: the grade-1 current whose existence makes a spacetime observer-bearing is the flux whose information-preserving transport is the dynamics.

2. The Multivector Amplitude

Let V be the tangent space with metric g of signature $(p,q)$, $n=p+q$, and let $C\ell \left(V\right)$ be its Clifford algebra, defined by $e_{\mu }{e}_{\nu }+e_{\nu }{e}_{\mu }=2g_{\mu \nu }$. The even subalgebra$C{\ell }^{+}$, spanned by products of an even number of basis vectors, is closed under the geometric product (even · even = even), and hence is closed under both superposition (addition) and rotor action (multiplication by elements R with $R\tilde{R}=1$, acting as $\psi \mapsto R\psi$). It has dimension $2^{n-1}$. Reversion, written $\tilde{\psi }$, reverses the order of vector factors in each term.

The even subalgebra is the largest subspace of $C\ell \left(V\right)$ closed under both operations a quantum amplitude requires—superposition and rotation. It is in this precise sense the most general amplitude space, and we adopt it as the home of the multivector amplitude $\psi$.

Throughout, the canonical density along a unit vector e is written ${\langle \tilde{\psi }e\psi e\rangle }_0$ and the current $J=\tilde{\psi }e\psi$, with reversion on the leading factor; we use this ordering verbatim in every signature below.

3. Theorem 1: Enumerating the Observer-Bearing Spacetimes

3.1. The Question

We ask a purely kinematic question: in which spacetimes is the multivector amplitude observer-bearing—admitting a positive directional density in some frame and a conserved vector current? This is prior to any dynamics; it concerns only what can be observed. Its answer is a complete list of observer-bearing spacetimes, and with each comes the amplitude’s Clifford algebra, its phase structure, and the conserved current the dynamical stage will transport.

3.2. Why a Multivector Amplitude—and Why It Is More Demanding Than a Spinor

Conventional relativistic quantum mechanics represents the state as a column spinor: an element of an abstract complex vector space ${\mathbb{C}}^k$ carrying a representation of the spin group. Such a representation exists in every dimension and every signature—Clifford algebras admit spinor representations whatever the metric, and one may write a Dirac-type equation on a seven-dimensional or a $(2,3)$-signature spacetime without obstruction. The column spinor is a bookkeeping device: its components are complex numbers carrying no intrinsic geometric meaning, and the matrices ${\gamma }^{\mu }$ act on them from outside. Nothing in that construction cares how many dimensions there are, or how many are timelike. The abstract spinor is dimensionally promiscuous because it lives outside spacetime.

We make a stronger demand, in the spirit of Hestenes’ geometric algebra [9,10]: that the amplitude be a multivector livinginspacetime, not an abstract column attached to it. Concretely, $\psi$ is an element of the Clifford algebra of the tangent space itself—a sum of a scalar, an oriented plane (bivector), an oriented volume (pseudoscalar), and so on, each piece a genuine geometric quantity in the spacetime where the physics happens. The “internal” spin degrees of freedom are not indices on an external column; they are the higher-grade parts of a single spacetime multivector. The basis vectors $e_{\mu }$ are not external matrices but elements of spacetime itself, acting by the geometric product.

This identification—amplitude = spacetime multivector—is what costs us our generality, and that is exactly the point. Once the amplitude is required to be a multivector built from the spacetime metric, two quantities that were free in the abstract-spinor picture become fixed by the metric and dimension themselves: its density must be a genuine non-negative scalar of that geometry, read along a direction (P1), and its current must be a genuine grade-1 vector of that geometry (P2). These are now computed from the multivector by the geometric product, not freely posited bilinears. In the abstract picture one may declare ${\psi }^{†}\psi$ positive by choosing the inner product; here $\rho ={\langle \tilde{\psi }e\psi e\rangle }_0$ is fixed by the algebra and either is or is not positive depending on the signature. In the abstract picture the current is a vector because one contracts with ${\gamma }^{\mu }$ by hand; here $J=\tilde{\psi }e\psi$ is whatever grade the geometric product makes it, and only in the right dimensions is it purely grade-1.

The status of this representational choice is therefore the engine of the enumeration. In $3+1$ the multivector amplitude is isomorphic to the complex column spinor, so the choice costs nothing observationally; it is in the other signatures, where the two diverge, that the multivector amplitude has content the column spinor lacks. The column spinor, representable everywhere, excludes nothing and so says nothing about which spacetimes can bear an observer. The multivector amplitude, being built from the metric, is observer-bearing only in a bounded family of signatures—and that family, computed below, is exactly the one observed.

3.3. Positive-Definiteness Implies Exactly One Time Dimension

For $\psi ={\sum }_k{\langle \psi \rangle }_k$, the scalar density ${\langle \tilde{\psi }e\psi e\rangle }_0$ along a unit vector e is a signed sum of squares of the component coefficients. The sign attached to each grade-k blade is the product of the reversion sign of grade k, the sign accrued in commuting e through the blade, and the metric factors $g_{\mu \mu }$ of the basis vectors composing the blade. Crucially, this density depends on the chosen e only through $e^2=\pm 1$ and the signature of the remaining $n-1$ directions relative to it; all directions of a given square are equivalent under the orthogonal symmetry of the metric. The observer-bearing criterion—that some vector e make the form definite—therefore needs to be checked only on one representative of each square class.

Lemma 1. 

There exists a unit vector e for which ${\langle \tilde{\psi }e\psi e\rangle }_0$ is definite (a single overall sign) if and only if exactly one basis vector has signature opposite to the rest—i.e. signature $(1,n-1)$ or $(n-1,1)$. When it holds, the qualifying e is the odd-signature direction, which is therefore timelike (up to overall sign). For $(n,0)$ and $(0,n)$ all directions share one square and the form is indefinite for every vector e; for split signatures such as $2+2$ both square classes give indefinite forms. Established exhaustively for $n\le 4$ in Section 3.5; $n\ge 5$ is excluded independently by Lemma 2.

Worked example, $n=4$, signature $(1,3)$, taking $e={\gamma }_0$ the timelike direction (${\gamma }_0^2=+1$), writing $\psi =a+F+bI$ (scalar, bivector with six components $F_{ij}$, pseudoscalar):

$${\langle \tilde{\psi }{\gamma }_0\psi {\gamma }_0\rangle }_0=a^2+b^2+\sum _{i<j}{F}_{ij}^2\left(\mathbf{positive}\mathbf{definite}\right).$$

(1)

By contrast, the Euclidean $4+0$ case has no timelike direction; taking any spacelike vector $e=e_1$ (all four directions equivalent),

$${\langle \tilde{\psi }{e}_1\psi e_1\rangle }_0=a^2-b^2-F_{01}^2-F_{02}^2-F_{03}^2+F_{12}^2+F_{13}^2+F_{23}^2,$$

(2)

which contains terms of both signs and is indefinite; by the symmetry above the same holds for $e_2,e_3,e_4$, so no vector yields a positive density. The discriminating, basis-independent fact is the presence of terms of both signs—genuine indefiniteness—for every direction. Euclidean signature is thus not observer-bearing, not by lacking a timelike vector by fiat, but by admitting no positive directional density in any frame.

One might ask whether some non-directional bilinear could rescue a Euclidean signature. It can, algebraically: because $\mathrm{Spin}\left(4\right)$ is compact, the even subalgebra of $C\ell (4,0)$ carries a positive-definite invariant form built from a general (non-vector) reference element. This form is excluded here on physical, not algebraic, grounds—it is not a density along any direction, hence not a quantity an observer reads off in a frame (Definition 1). The restriction to a grade-1 reference vector is exactly the content of bearing an observer; with it, no Euclidean or split signature is observer-bearing, by the explicit sign computations above.

3.4. Pure-Vector Current Implies $n\le 4$

The even subalgebra carries grades $\{0,2,4,\dots \}$. For $n=4$ it is exactly $\{\mathrm{scalar},\mathrm{bivector},\mathrm{pseudoscalar}\}=1+6+1=8$, so $\psi =a+F+bI$ exhausts it, and the bilinear $J=\tilde{\psi }e\psi$ (with e the qualifying vector) is a pure grade-1 vector—the spacetime-algebra Dirac current, satisfying $J^2={\rho }^2\ge 0$ and future-pointing timelike. (That J is pure grade-1 for a generic even multivector, not only for a constrained Dirac spinor, is verified by direct computation in $C{\ell }_{1,3}$.)

Lemma 2. 

For $n\ge 5$ the even subalgebra acquires an independent grade-4 sector. At $n=5$, for instance, $C{\ell }^{+}$ is $1+10+5=16$-dimensional, so $\psi =a+F+Q$ with Q a five-component grade-4 object (notthe pseudoscalar, which at $n=5$ is the odd grade-5 element and lies outside $C{\ell }^{+}$). The bilinear $J=\tilde{\psi }e\psi$ then acquires grade-3 and grade-5 parts for any vector e; the current is no longer a pure vector, and (P2) fails. The failure is monotonic in n—higher even grades accumulate as n grows—so no spacetime of dimension $n\ge 5$ is observer-bearing. (Explicit grade-3/grade-5 contamination at $n=5$: Appendix A.)

3.5. The Complete List

Two requirements act together: some vector must make the density definite (Lemma 1) and the current must be a pure grade-1 vector (Lemma 2). Dimensions $n\ge 5$ fail the current test outright, so the density need only be checked for $n\le 4$—a finite list, settled by computing ${\langle \tilde{\psi }e\psi e\rangle }_0$ on a representative of each square class in every signature. We tabulate the density along the most favorable direction (timelike where one exists, otherwise spacelike):

n	Signature	${\langle \tilde{\psi }e\psi e\rangle }_0$ (best representative)	definite?
1	$1+0$	$a^2$	yes
1	$0+1$	$-a^2$	yes (global sign)
2	$1+1$	$a^2+b^2$	yes
2	$2+0$	$a^2-b^2$	no
2	$0+2$	$-a^2+b^2$	no
3	$2+1,1+2$	$a^2+F_{01}^2+F_{02}^2+F_{12}^2$	yes
3	$3+0$	$a^2-F_{01}^2-F_{02}^2-F_{12}^2$	no
3	$0+3$	$-a^2+F_{01}^2+F_{02}^2+F_{12}^2$	no
4	$3+1,1+3$	$a^2+b^2+{\sum }_{i<j}{F}_{ij}^2$	yes
4	$2+2$	$a^2-b^2$, bivectors split $3+/3-$	no
4	$4+0,0+4$	see Eq. (2); terms of both signs	no

For each signature the entry shows the density along the most favorable vector; by the symmetry of Lemma 1 no other direction does better. The form is definite—some frame can read it—if and only if exactly one basis vector has signature opposite to the others, namely $(1,n-1)$ or $(n-1,1)$. (In the indefinite rows the explicit per-term signs depend on basis ordering and conjugation convention; the convention-independent and sufficient fact is that terms of both signs are present for every direction.) Imposing both requirements leaves exactly the observer-bearing family

n	Signature	density definite?	current	observer-bearing?	fails by
1	$1+0,0+1$	yes	grade-1		—
2	$1+1$	yes	grade-1		—
2	$2+0,0+2$	no	—	×	no readable density
3	$2+1,1+2$	yes	grade-1		—
3	$3+0,0+3$	no	—	×	no readable density
4	$3+1,1+3$	yes	grade-1	(max)	—
4	$2+2$	no	—	×	no readable density
4	$4+0,0+4$	no	—	×	no readable density
$\ge 5$	any	—	grades 3,5,...	×	current not a vector

The observer-bearing spacetimes are exactly $0+1$, $1+1$, $2+1$, $3+1$: every dimension up to four, each with a single time direction, and nothing beyond. The list is a sharp exclusion: no spacetime above four dimensions, and none of multi-time, Euclidean, or split signature, can bear an observer; every signature within the family can. This matches observation: measurements are made at $0+1$ (a single degree of freedom), $1+1$, $2+1$, and $3+1$, and never outside it; the lower members appear as effective descriptions of subsystems—quantum wires $(1+1)$, planar electron systems and anyons $(2+1)$—within the $3+1$ ambient world. The maximal observer-bearing spacetime is $3+1$.

3.6. The Ladder of Observer-Bearing Spacetimes

The enumeration is not merely a count; the kind of probability an observer-bearing spacetime carries changes along it, set by the even-subalgebra dimension $2^{n-1}=1,2,4,8$ for $n=1,2,3,4$:

• $0+1$: a single real number. The lowest rung makes plain what “classical” means here: there is only time, and a single real weight evolving along it—a time-ordered series, the bare succession of outcomes when a coin is flipped. There is no space for the probability to inhabit, hence no geometry to support a phase and no interference; time is the only thing conserved.
• $1+1$: two real components, a two-component object with left- and right-moving sectors.
• $2+1$: a complex two-spinor; a $\mathbb{C}$ phase and an $SU\left(2\right)$ rotor group first appear—a qubit.
• $3+1$: a four-component Dirac spinor; full spin-$\frac{1}{2}$, the maximal case.

This is the content of the classification, and it is worth stating plainly what it shows: classical probability and quantum probability are both members of the observer-bearing family, sitting at different rungs of one ladder. Interference and the complex phase appear only higher up, once the dimension affords a geometry in which the amplitude can rotate. The quantum character is therefore not assumed and then propagated; it is what an observer-bearing spacetime acquires once it is large enough to carry a phase. The construction commits to no quantum-ness in advance—it enumerates what can be observed, finding the classical time-series at the bottom rung and the Dirac spinor at the top. We do not claim the ladder forces the complex structure, only that it becomes available, rung by rung, as the dimension climbs to its ceiling.

4. Theorem 2: The Dirac Equation as Information-Preserving Flow

4.1. Evolving the Maximal Arena’s Current

The first stage delivered, for the maximal observer-bearing spacetime $3+1$, a finished object: the multivector amplitude $\psi$, its Clifford algebra, its phase, and its conserved current J. Nothing further about the amplitude is decided here; the second stage takes these as given and asks a single new question—how must the conserved current be transported if the flow neither creates nor destroys information? The novelty of this stage is entirely in the thermodynamics of that transport; the amplitude’s features enter only as fixed inputs from Theorem 1.

We work in the conventional column (matrix) representation. In $3+1$ this is equivalent, by Hestenes’ isomorphism $C{\ell }_{1,3}^{+}\cong C{\ell }_{3,0}$ [9,10], to the multivector amplitude of Theorem 1, under $\tilde{\psi }\leftrightarrow \overline{\psi }={\psi }^{†}{\gamma }^0$ and $J=\tilde{\psi }{\gamma }_0\psi \leftrightarrow J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$, with ${\gamma }_0$ the timelike direction the kinematic stage singled out; we use the column form only because the kinetic bilinear is most transparent there.

4.2. What Evolves, and What Costs Information

The object is the amplitude $\psi$ and its conserved current $J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$. To make “committing to nothing” precise we need a measure of how much a candidate evolution injects into, or extracts from, the information content of the probability. That measure is the net rate of entropy change along the flow. An evolution that produces entropy is forgetting—coarse-graining, mixing, losing information about the state. One that consumes entropy is manufacturing information from nowhere—asserting more than the data warrant. The maximally non-committal evolution does neither: it transports the probability while producing no net entropy. This is the dynamical analogue of the maximum-entropy principle of Jaynes [13]: where maximum entropy fixes a state by adding nothing beyond the constraints, information preservation fixes a flow by adding nothing beyond conservation of the probability. The two are analogues, not identical operations—one is an extremum of a state functional, the other the vanishing of a production rate along a flow—but they share the same structure: a least-committal selection subject to a measured constraint.

So before writing any equation we model the local entropy balance of the flowing, conserved current, in the sense of irreversible thermodynamics—a conserved quantity redistributed against a gradient. We do not answer by fiat; the balance framework fixes the form.

4.3. The Flux·Force Form: What the Structure Supplies

In the thermodynamics of irreversible processes, the local rate of entropy production of a redistributed conserved quantity has a single universal structure, a sum of flux times force [14,15]:

$$\sigma =\sum _a{J}^a·X_a,$$

(3)

where the flux $J^a$ is the conserved current being transported and the force$X_a$ is the gradient of its conjugate variable. Applied to our object, the flux is the conserved current of Theorem 1, the force is the gradient of the amplitude, ${\partial }_{\mu }\psi$, and the entropy-production density is

$$\sigma \sim \overline{\psi }{\Gamma }^{\mu }{\partial }_{\mu }\psi +\left(\mathrm{conjugate}\right),$$

(4)

with ${\Gamma }^{\mu }$ the coupling between flux and force on the internal (spinor) space.

Three features of the evolution are supplied by this structure, and it is worth separating them:

1.

The coupled-channel matrix form. A flux·force density with internal degrees of freedom is a sum over directions $\mu$ of channel-mixing couplings ${\Gamma }^{\mu }$ contracted with the gradient: ${\sum }_{\mu }\overline{\psi }{\Gamma }^{\mu }{\partial }_{\mu }\psi$. The coupling ${\Gamma }^{\mu }$ carries a spacetime index and mixes the components of $\psi$—it is a matrix, not a scalar multiplier. This multi-component, matrix-contracted, summed-over-directions form is the gross skeleton of the Dirac operator, and it is supplied by the flux·force structure itself.

2.

First order. The flux is the conserved current, algebraic in $\psi$ (it carries no derivative), while the force is a first gradient. The density is therefore bilinear—flux × force—hence linear in the gradient, and its Euler–Lagrange equation is first-order. This is the structural difference from an information content such as Fisher’s, which is quadratic in the gradient and yields a second-order equation; a balance carries one derivative, an information measure two.

3.

Scalarity of the entropy density (a requirement, resolved in §Section 4.5). Entropy is a scalar state function, so its production rate $\sigma$ is a scalar density. The coupling ${\Gamma }^{\mu }$ must therefore be such that the bilinear is a scalar. This requirement is issued by the entropy form; it does not by itself fix the coupling, and its resolution is deferred to the next subsection.

What the flux·force structure does not supply is the specific algebra of the ${\Gamma }^{\mu }$: the coefficient of a generic balance is merely a coupling matrix, and the scalarity requirement above only constrains it. The admissible couplings are determined within the algebra inherited from Theorem 1 (§Section 4.5); the reality of the Dirac coupling (the i) comes from the reversible root (§Section 4.6).

4.4. The Entropic-Force Constraint, Which Turns Out to Be the Mass Term

The flux·force reading carries one further demand. For ${\partial }_{\mu }\psi$ to count as a genuine thermodynamic force—not merely a gradient we have chosen to call one—it must be conjugate to the flux through a scalar pairing of the field with itself. We therefore add to the functional the term that guarantees this conjugacy. In the inherited algebra the scalar pairing of $\psi$ with $\overline{\psi }$ is a grade-0 bilinear, of which ${\langle \overline{\psi }\psi \rangle }_0$ is the representative, and the added term is

$$-2m{\langle \overline{\psi }\psi \rangle }_0.$$

(5)

We add this term for a purely structural reason—to make the transport genuinely entropic, by furnishing the conjugation the force requires—and we then observe that it has exactly the form of the relativistic mass term. The mass term is thus not a separate dynamical postulate; it is the constraint the entropic-force reading demands, recognized after the fact. Its coefficient m is the Lagrange multiplier conjugate to that constraint—precisely as, in the canonical ensemble, temperature is the multiplier conjugate to the energy constraint:

Jaynes maximum entropy	this construction
entropy functional $S\left[p\right]$	transport functional $\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -\mathrm{c}.\mathrm{c}.$
constraint $\langle E\rangle$	force-conjugacy pairing ${\langle \overline{\psi }\psi \rangle }_0$
multiplier $\beta$ (temperature)	multiplier m (mass)

The mass therefore enters the construction in a definite structural role: it is the multiplier of the constraint that renders the transport entropic. Just as the canonical distribution is what entropy maximization gives subject to the energy constraint, the Dirac equation is what information-preserving transport gives subject to the force-conjugacy constraint, with the mass as the conjugate multiplier.

4.5. Scalarity, Issued by Entropy, Resolved by the Inherited Algebra

The previous subsection established a requirement of thermodynamic origin: because the entropy-production density is a scalar, the coupling ${\Gamma }^{\mu }$ must make $\overline{\psi }{\Gamma }^{\mu }{\partial }_{\mu }\psi$ a scalar. We stress the division of labor. The requirement that the coupling be scalar-producing is issued by the entropy form—the statement that $\sigma$ is a scalar density. The resolution of that requirement is supplied by the algebra inherited from Theorem 1.

Lemma 3 

(Admissible couplings). The scalarity requirement on $\overline{\psi }{\Gamma }^{\mu }{\partial }_{\mu }\psi$ does not single out a unique coupling. Solving the constraint within the inherited algebra, the admissible couplings are of two kinds:

1. 

the grade-1 vector ${\gamma }^{\mu }$: since the geometric product of a grade-1 element with the grade-1 force ${\partial }_{\mu }\psi$ contains a grade-0 part, ${\gamma }^{\mu }$ scalarizes and yields the Dirac coupling (a purely higher-grade left coupling produces no scalar and is excluded); and

2. 

right-multiplications $\psi \mapsto \psi N$ by the even subalgebra $C{\ell }_{1,3}^{+}\cong M_2\left(\mathbb{C}\right)$, which commute with the Lorentz (left-rotor) action and therefore scalarize automatically; in the reversible (density-preserving) sector these are the generators of $\mathfrak{u}\left(2\right)=\mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$.

Scalarity thus admits, alongside the Dirac vector coupling, an internal $\mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ sector. (Grade arithmetic for the vector case: Appendix B.)

• Remark (why this is physics, not a defect).

The appearance of an $\mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ sector alongside the Dirac coupling might seem foreign to a relativistic wave equation. It is not a defect of the method. These right-acting structures are precisely the geometric-algebra ingredients that Hestenes and successors identify with the electroweak gauge group [10,11,12]; the scalarity constraint is thus carving at a real joint—its solutions are the Dirac coupling together with the algebraic seed of electroweak gauge structure, not an artifact. The full electroweak realization—in particular the chiral action that distinguishes it from a vectorlike $\mathfrak{u}\left(2\right)$—requires complexifying the spacetime algebra and lies outside the present scope. We therefore proceed with the grade-1 (Dirac) sector, noting only that the additional admissible sector is known physics, not a flaw in the construction.

Adopting the grade-1 coupling, the ${\gamma }^{\mu }$ are the grade-1 elements of the inherited algebra, satisfying

$$\frac{1}{2}\{{\gamma }^{\mu },{\gamma }^{\nu }\}=g^{\mu \nu }\mathbf{1}$$

(6)

as the defining relation of the Clifford algebra of Section 2. This is the same Clifford relation Dirac obtained in 1928 [16], there by seeking a first-order operator whose square is the Klein–Gordon operator; here it arises by a different route—an entropy-issued scalar requirement met by the algebra the kinematic stage already fixed.

4.6. The Reversible Sector and the Stationary Statement

The coupling ${\gamma }^{\mu }$ decomposes the flow into a symmetric (Hermitian) and an antisymmetric (anti-Hermitian) part. The symmetric part is the dissipative sector ($\sigma >0$, irreversible relaxation); the antisymmetric part yields $\sigma =0$, transporting the probability reversibly. Our criterion—neither producing nor consuming information—selects the $\sigma =0$ sector, i.e. the anti-Hermitian root ${\gamma }^{\mu }\to i{\gamma }^{\mu }$. Assembling the reversible flux·force action with the entropic-force constraint of §Section 4.4,

$$\Sigma [\psi ,\overline{\psi }]=\int d^{4}x\left[\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -\left({\partial }_{\mu }\overline{\psi }\right){\gamma }^{\mu }\psi -2m\overline{\psi }\psi \right],$$

(7)

and varying $\overline{\psi }$ (with $\psi ,\overline{\psi }$ independent), with the reversible root $i{\gamma }^{\mu }$:

$$\begin{array}{|c|}\hline (i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0.\\ \hline\end{array}$$

(8)

The information-preserving transport of the conserved current of the maximal observer-bearing spacetime, in the grade-1 sector of Lemma 3, is therefore the Dirac equation. On any solution the current is conserved, ${\partial }_{\mu }{J}^{\mu }=0$: the flow transports probability without producing entropy, which is the content of its reversibility.

5. What Is Explained

Both results follow from the single definition of an observer-bearing spacetime (Definition 1), as every physical uniqueness theorem follows from its axioms. Under that definition, the observer-bearing spacetimes are exactly the Lorentzian signatures with one time direction and $n\le 4$; and the Dirac equation is the information-preserving transport of the maximal one’s current.

It is worth stating plainly which part of the construction supplies which feature of the result, since no single ingredient yields the Dirac equation and each contributes a named part:

• The observer-bearing definition (Theorem 1). The complete list of observer-bearing spacetimes; with each, its Clifford algebra, its phase, and its conserved grade-1 current. Classical probability sits at the bottom rung of this list and the Dirac spinor at the top; quantum structure is a member of the family, not an input.
• The flux·force structure (entropy). The coupled-channel matrix form of the evolution operator; its unitarity (via the reversible $\sigma =0$ sector); the first-order character (the balance being linear in the gradient); and the scalar requirement on the coupling.
• The inherited algebra. The admissible couplings: the grade-1 vector (adopted, giving Dirac) and, additionally, an internal $\mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ sector (the electroweak seed, scoped out).
• The entropic-force constraint. The mass term: the scalar pairing ${\langle \overline{\psi }\psi \rangle }_0$ added to make ${\partial }_{\mu }\psi$ a genuine force turns out to be the mass term, with the mass m as its Lagrange multiplier—a structural role, not an independent postulate.

We state the scope honestly:

(a) The dimensional result is a classification of observer-bearing spacetimes: nothing above $3+1$, and no multi-time, Euclidean, or split signature, can bear an observer; every signature within the family $0+1,\dots ,3+1$ can. It bounds what is observable, making no claim about an underlying ontology, which may be of any dimension. The maximal observer-bearing slice is $3+1$; the lower members appear as effective descriptions of subsystems within it. The result is conditional on the representational choice—the multivector amplitude, with a directional, grade-1 density (Definition 1)—which §Section 3.2 motivates and grants is a choice, not a necessity. In $3+1$ it agrees with the complex amplitude; its content lies in the other signatures, where the two diverge.

(b) The Dirac result is the information-preserving transport of the maximal arena’s current. Entropy supplies the form of the evolution operator (coupled-channel, first-order), its unitarity, and the scalar requirement on the coupling; the inherited algebra furnishes the admissible couplings, of which we adopt the grade-1 (Dirac) sector; and the constraint added to guarantee an entropic force turns out to be the mass term, with the mass as its conjugate multiplier—the same role temperature plays in the canonical ensemble. The Clifford relation coincides with the one Dirac obtained in 1928 [16], reached here by a thermodynamic route rather than by factorizing Klein–Gordon. The additional $\mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ sector admitted by scalarity is the geometric-algebra seed of electroweak structure (§Section 4.5), recorded but not developed.

(c) The definition itself—that a spacetime is observer-bearing when its multivector amplitude admits a readable density and a conserved vector current—is posited, not derived. We do not eliminate postulates; we replace dynamical postulates about the world with a single definition of what it is for a spacetime to bear an observer, and obtain both the admissible arenas and the law of motion from it.

6. Discussion

The construction realizes, in two stages, a single discipline: define what it is for a spacetime to bear an observer, enumerate the spacetimes that do, then fix the transport of the maximal one’s conserved current by what conservation requires. The observer-bearing condition is the condition of possibility for measurement—logically prior to any dynamics—and yields the list of admissible arenas together with the conserved current each carries. The information-preservation requirement, applied to that current, yields its evolution: the flux·force structure supplies the coupled-channel, first-order form and, in its reversible sector, the unitarity; scalarity fixes the admissible couplings within the inherited algebra; and the constraint that makes ${\partial }_{\mu }\psi$ a genuine entropic force turns out to be the mass term, with the mass as its conjugate multiplier.

What is gained is a route to two facts usually met with a postulate. Why is the spacetime metric Lorentzian, with one time dimension and no more than four dimensions? Because exactly these metrics can bear an observer—a frame assigning a positive directional density and a vector current. A spacetime that can bear no observer in any frame is unmeasurable in principle; no other signature can, and none of dimension five or higher can. Why does matter obey the Dirac equation? Because the Dirac equation is the information-preserving transport of the maximal arena’s current—its operator form, unitarity, and scalar coupling-requirement fixed by the entropy structure, its coupling the grade-1 sector of the inherited algebra, and its mass term the very constraint that makes the transport entropic, with the mass as the conjugate multiplier.

The two facts are usually assumed separately, atop a dynamics already in place. Here they are two findings within one classification: define the observer-bearing spacetimes, list them, and the world’s arena is the maximal entry while the world’s law is that entry’s information-preserving dynamics. The same classification contains classical probability at its lowest rung—a single weight in time, the bare succession of outcomes, with no geometry to permit interference; quantum structure is not assumed but found, partway up a ladder whose top is the Dirac spinor. We do not assume the world is quantum; we ask what a spacetime can observably carry, and find the classical and the quantum together in the answer.

What is not gained, and we do not claim it, is the inevitability of the observer-bearing definition itself. Every reconstruction bottoms out in some primitive; ours bottoms out in that definition, whose warrant is the empirical content it carries that its alternatives do not. The next “why”—why an amplitude, why this notion of observation—recurses beyond this paper. But the primitive we rest on is an epistemic object, a statement about what can be measured, rather than a dynamical postulate about the world; and from it the dimension of spacetime and the form of the law of relativistic matter are not assumed but constrained.

No datasets were generated or analysed during the current study; all results are analytic and contained within the manuscript.

Symbolic verifications of the grade-by-grade signature and scalarity computations (Section 3 and Section 4.5) were performed with a computer-algebra system and are available from the author on reasonable request.

The author used a large language model (Anthropic’s Claude) as an aid during the development and drafting of this manuscript, for assistance with exposition, symbolic algebra checks, and editorial structuring. All scientific content, claims, derivations, and conclusions were conceived, verified, and approved by the author, who takes full responsibility for the work. The AI tool is not, and does not qualify as, an author.

The author declares no conflict of interest.