This article grew out of a test of GPT-5.5 Pro’s ability to develop the core of a possible quantum-relativistic theory. The Theory of the Minimum Quantum Interval is a speculative and preliminary proposal: its original hypotheses should be read as theoretical material to be discussed and developed, distinct from the established results of special relativity, quantum mechanics, and quantum field theory.
Preface
Special relativity and quantum mechanics are two of the great pillars of modern physics. The first changed the way we understand space and time, showing that their measurements depend on the observer, while the spacetime interval preserves a deeper value. The second changed the way we understand microscopic phenomena, introducing states, probabilities, observables, and processes that escape classical intuition.
The theory introduced here explores a new idea of quantization where relativistic geometry meets the quantum description of physical processes. It does not claim, at least in its preliminary form, that space is a grid or that time advances in ticks. It proposes something subtler: when a quantum process is described through physically resolved events, the invariant separation between two distinct events may be null, or it must exceed a universal minimum threshold, denoted by ℓ₀. In other words, the mathematical continuum remains available as a language, but the physics of distinguishable events might not occupy the whole continuum.
To understand this idea, one must clarify the most important expression: physically resolved events.
A physically resolved event is not simply “a point in space and time.” A point is a mathematical idea: we can imagine it as infinitely small, without extension, perfectly localized. Physics, however, never truly works with pure points. It works with signals, interactions, measurements, records, collisions, emissions, absorptions, and traces left by something. A physically resolved event is an event that, at least in principle, can be distinguished from another event within a physical process.
For example: a photon absorbed by a detector, a particle leaving a track in a cloud chamber, an atom emitting radiation, or an interaction producing a measurable signal. In all these cases we are not speaking of an abstract point, but of something that enters the physical history of the system. The event is “resolved” because the theory, or an ideal measuring apparatus, can treat it as distinct from other events.
This distinction is decisive. Mathematical spacetime can contain infinitely many points, arbitrarily close to one another. But it is not obvious that nature allows every pair of extremely close points to become two truly distinguishable physical events. The theory of the minimum quantum interval begins precisely from this possibility: perhaps the continuum exists as a map, as a language, as a mathematical background; but the physical events that can be separated, counted, and related do not fill that continuum without limit.
A simple analogy may help. A digital photograph shows a continuous scene: a face, a landscape, a shade of light. But the image is made of pixels. If two details are smaller than the resolution of the camera, they do not appear as two separate details: they become a single spot, a single piece of information. The theory proposed here does not say that spacetime is literally made of pixels. It says something more cautious: perhaps quantum processes also have a threshold below which two events can no longer be physically distinguished as separate events.
The difference is subtle but important. To say that space is a grid would mean imagining elementary bricks arranged side by side. This theory does not begin from that picture. It does not say that there is a minimum cell of space, nor a minimum ticking of time. Instead, it says that when we compare two real physical events within a quantum process, what matters is a quantity deeper than ordinary spatial distance: the spacetime interval, that combination of space and time which relativity regards as the same for all observers.
This is where ℓ₀ enters. This new constant would be a minimum threshold for non-null intervals between physically resolved events. If two events are separated exactly by a null interval, that is, if they belong to the light-cone sector, the theory admits them. If instead their separation is different from zero, then it cannot be arbitrarily small: it must be at least equal to ℓ₀.
In intuitive terms, the theory imagines a kind of “forbidden zone” around almost-null intervals. The null value remains possible. Large intervals remain possible. But non-null intervals that are too small would not correspond to pairs of physically distinguishable events. They would be too close, not in ordinary space, but in the deep structure of spacetime.
This point makes the proposal different from a common idea of “minimum length.” Often, when one hears about a minimum length, one thinks of a spatial distance below which nothing can go. Here the hypothesis is different: it does not concern simply “how far apart two points are in space,” but “how separated two events are in spacetime.” This is a fundamental difference, because in relativity space and time are not absolutely separable. Two observers may disagree about the spatial distance or the temporal duration between two events, but they must agree about the invariant interval.
The theory therefore tries to be compatible with relativistic intuition: if a minimum threshold exists, that threshold should not depend on the observer’s point of view. It should not be a length that changes when the reference frame changes. It should be tied to something all observers recognize as the same. For this reason the threshold is applied to the invariant interval.
The expression “physically resolved events” also helps avoid another difficulty. If we said that all points of spacetime must satisfy the threshold ℓ₀, then the mathematical continuum would immediately be excluded. But the theory does not necessarily want to abolish the continuum. It wants to distinguish between the mathematical map and what can become physically real within a process. The map may be continuous; resolved events might instead obey a more selective rule.
In this sense, a physically resolved event is an event with enough physical identity to be named, localized, correlated, or measured. It is not an ideal dot. It is a node in the fabric of a process. It is something that can appear in a quantum story as “this happened here,” “this interacted with that,” “this signal was recorded,” or “this observable played a role in the description.”
The theory of the minimum quantum interval therefore asks a simple question: if two nodes of this fabric are truly distinct, how close can they be in the structure of spacetime? To answer, one must recall a fundamental image from relativity: the light cone. Intuitively, the light cone indicates the set of events that can be connected by a light signal. It is the natural boundary between what can be reached by light, what can be connected by a causal relation slower than light, and what remains separated by a spacelike interval.
The answer hypothesized by the theory is this: two physically resolved events may lie exactly on the light cone, or they must be separated by at least ℓ₀. Exactly null events therefore remain admitted. Events with sufficiently large separation also remain admitted. The excluded zone is the small intermediate interval: the region in which two events would be almost on the light cone without being truly on it, and with an invariant separation below the minimum threshold.
This is the underlying picture: the mathematical world may be continuous like a line drawn with infinite precision; the physical world of quantum processes might instead have a deeper resolution threshold. Not a visible grid, not a ticking time, not a space made of little cubes, but a rule about distinguishable events. Continuity remains the language. Physical resolution might have a limit.
Theory of the Minimum Quantum Interval
Modern physics does not describe only objects, forces, and trajectories. It also describes the limits within which it makes sense to speak of objects, forces, and trajectories. Special relativity showed that space and time are not independent containers, but aspects of a single structure, spacetime. Quantum mechanics showed that physical quantities do not always possess definite values before measurement and that physical states obey principles of superposition, interference, and probability. Quantum field theory then unified these two aspects in a description whose fundamental objects are fields, states, local observables, and correlations.
The proposal developed here introduces a further hypothesis: in quantum processes there is a minimum threshold for non-null invariant spacetime intervals between physically resolved events. This threshold is denoted by \( \style{font-size:1em;}{\ell_0} \). It has dimensions of length and is treated as a universal constant. The theory does not claim that spatial coordinates are discrete, nor that time flows in elementary units. The hypothesis is more specific: two physically resolved events belonging to the same quantum process cannot be separated by an arbitrarily small non-null invariant interval. They may be separated by a null interval, or by an interval at least equal to \( \style{font-size:1em;}{\ell_0} \).
The compact form of the axiom is:
\( \style{font-size:1em;}{\sigma(e_1,e_2)\in \{0\}\cup[\ell_0,+\infty)} \)
Here \( \style{font-size:1em;}{\sigma(e_1,e_2)} \) is not an ordinary spatial distance. It is the positive value associated with the Minkowski interval between two events. The choice to work with \( \style{font-size:1em;}{\sigma} \) is essential because the theory aims to preserve compatibility with special relativity: what is limited is not a coordinate seen by a particular observer, but an invariant quantity.
The theory can be described in one sentence: the physics of quantum processes is not articulated in resolved events separated by non-null invariant intervals smaller than a fundamental length. Geometric spacetime remains continuous as the mathematical reference structure; the restriction concerns instead what is admitted as a distinguishable physical event within a process.
Central Idea
The central notion of the theory is not the point, but the physically resolved event. A geometric point is an ideal element of the spacetime continuum. It can be indicated by coordinates, used in a calculation, and appear as a mathematical limit. A physically resolved event, by contrast, is a unit of physical description: something that can enter a correlation, a record, an observable, or a structure of amplitudes.
This distinction is the first step in avoiding a misunderstanding. If a minimum length were imposed directly on every pair of geometric points, continuous spacetime would immediately become incompatible with the axiom, because in a continuum there are always pairs of points arbitrarily close to one another. The theory does not take this path. It admits the continuum as a mathematical tool, but holds that not every separation in the continuum corresponds to a pair of physically resolved events belonging to the same process.
The proposal is therefore relational. An event acquires meaning within a process. A separation acquires physical meaning when it connects resolved events. The axiom of the minimum interval does not concern geometry taken in the abstract, but the physics of admitted configurations.
A quantum process can be schematized as a structure composed of events, observables, amplitudes, and constraints:
\( \style{font-size:1em;}{P=\{E(P),\mathcal{O}(P),A(P),\Pi_{\ell_0}\}} \)
In this formula, E(P) is the set of physically resolved events of the process P, \( \style{font-size:1em;}{\mathcal{O}(P)} \) is the set of observables associated with the process, A(P) represents correlation amplitudes, and \( \style{font-size:1em;}{\Pi_{\ell_0}} \) denotes the geometric constraint linked to the minimum interval. This is not yet a definitive definition of a quantum process, but a guiding structure: it states which ingredients the theory must make precise.
The new hypothesis enters in the way E(P) may be configured. Not every set of events is physically admissible. For every pair of resolved events in the same process, the invariant separation must belong to the admitted spectrum.
Established Results and the Theory’s Own Hypotheses
It is useful from the start to distinguish what belongs to established physics from what is introduced by the theory.
It belongs to established physics that, in special relativity, the Minkowski interval between two events is invariant under Lorentz transformations. It belongs to established physics that quantum mechanics uses states in Hilbert spaces, complex amplitudes, operators, and probabilistic rules. It also belongs to established physics, especially in quantum field theory, that local objects must be handled with care, often as operator-valued distributions integrated against test functions.
What does not belong to established physics, however, is the axiom according to which non-null invariant intervals between physically resolved events must have a minimum value \( \style{font-size:1em;}{\ell_0} \). This is the specific hypothesis of the theory. It is not proved, not deduced from a known experiment, and must not be presented as an established fact. It is a proposed principle, to be developed and tested.
The theory must therefore satisfy two conditions. The first is compatibility: it must reduce to known theories in regimes where those theories have already been verified. The second is fertility: it must produce, at least in principle, calculable consequences that distinguish it from ordinary physics.
The first condition is expressed by the continuum limit:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}K_{\ell_0}=K_0} \)
The second condition instead requires that, for finite \( \style{font-size:1em;}{\ell_0} \), there exist controlled corrections to amplitudes, propagators, or correlation functions:
\( \style{font-size:1em;}{A_{\ell_0}(f_2,f_1)\neq A_0(f_2,f_1)} \)
This inequality should not be read as a general prediction already proved. It indicates the theoretical place where the new scale might manifest itself: in high-resolution amplitudes, especially near the band of forbidden intervals.
Reference Geometry
The first formulation of the theory is set in Minkowski spacetime. This choice does not claim to include gravity immediately. It is a methodological choice: before generalizing to curved spacetimes, one must understand whether the axiom can be made coherent in the simplest relativistic framework.
A geometric event is described by four coordinates:
\( \style{font-size:1em;}{x^\mu=(ct,x,y,z)} \)
The chosen metric has positive time signature:
\( \style{font-size:1em;}{\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)} \)
The quadratic separation between two geometric events is:
\( \style{font-size:1em;}{\Delta s^2=\eta_{\mu\nu}\Delta x^\mu\Delta x^\nu} \)
In ordinary coordinates:
\( \style{font-size:1em;}{\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2} \)
Setting:
\( \style{font-size:1em;}{\Delta r^2=\Delta x^2+\Delta y^2+\Delta z^2} \)
one can write:
\( \style{font-size:1em;}{\Delta s^2=c^2\Delta t^2-\Delta r^2} \)
The theory does not directly use \( \style{font-size:1em;}{\Delta s^2} \) as a positive quantity, because it can be positive, negative, or null. It introduces instead:
\( \style{font-size:1em;}{\sigma(e_1,e_2)=\sqrt{\left|\eta_{\mu\nu}\Delta x^\mu\Delta x^\nu\right|}} \)
or, in coordinates:
\( \style{font-size:1em;}{\sigma(e_1,e_2)=\sqrt{\left|c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2\right|}} \)
The quantity \( \style{font-size:1em;}{\sigma} \) has dimensions of length. It treats timelike and spacelike separations uniformly, taking the positive value of the interval. Null intervals are those for which:
\( \style{font-size:1em;}{\eta_{\mu\nu}\Delta x^\mu\Delta x^\nu=0} \)
and therefore:
\( \style{font-size:1em;}{\sigma(e_1,e_2)=0} \)
The distinction between infinitesimal intervals and finite intervals is important. The theory does not formulate its principle as a constraint on the differential element ds. It formulates it on finite separations between physically resolved events. In this sense, an intuitive notation such as “ds greater than a constant” is less precise. The correct formulation is a condition on the admitted spectrum of \( \style{font-size:1em;}{\sigma} \) for pairs of physical events.
The Axiom of the Minimum Quantum Interval
Let P be a quantum process. Let \( \style{font-size:1em;}{e_1} \) and \( \style{font-size:1em;}{e_2} \) be two distinct physically resolved events belonging to E(P). The fundamental axiom states:
\( \style{font-size:1em;}{\sigma(e_1,e_2)\in \{0\}\cup[\ell_0,+\infty)} \)
This formula contains two admissions and one exclusion. Null intervals are admitted:
\( \style{font-size:1em;}{\sigma(e_1,e_2)=0} \)
Non-null intervals at least equal to \( \style{font-size:1em;}{\ell_0} \) are admitted:
\( \style{font-size:1em;}{\sigma(e_1,e_2)\geq\ell_0} \)
Non-null intervals smaller than \( \style{font-size:1em;}{\ell_0} \) are excluded:
\( \style{font-size:1em;}{0\lt\sigma(e_1,e_2)\lt\ell_0} \)
The forbidden band can be denoted by:
\( \style{font-size:1em;}{\mathcal{F}_{\ell_0}=\{(e_1,e_2)\mid 0\lt\sigma(e_1,e_2)\lt\ell_0\}} \)
The admitted domain is:
\( \style{font-size:1em;}{\Gamma_{\ell_0}=\{(e_1,e_2)\mid \sigma(e_1,e_2)=0\ \text{or}\ \sigma(e_1,e_2)\geq\ell_0\}} \)
The axiom can be read as a spectral condition:
\( \style{font-size:1em;}{\mathrm{Spec}(\sigma)_{\mathrm{phys}}\subseteq\{0\}\cup[\ell_0,+\infty)} \)
This notation must be interpreted cautiously. It does not necessarily mean that a rigorously defined distance operator with that spectrum already exists. It means that, in the physical space of the theory, the invariant separations between resolved events belong to that domain. A more advanced mathematical formulation may indeed define an operator, or a family of operators, associated with intervals, but the axiomatic core can already be expressed as a restriction on configurations.
The explicit presence of the zero value is essential. If the axiom were simply:
\( \style{font-size:1em;}{\sigma(e_1,e_2)\geq\ell_0} \)
it would also eliminate the null sector. This would be in tension with the structure of the light cone, with the propagation of light signals, and with the role of null intervals in special relativity. The theory avoids this difficulty by isolating the null sector as an admitted singular class.
The resulting geometry is peculiar: the light cone remains included; around the light cone there appears a forbidden band of non-null intervals; beyond that band, admitted physical intervals begin again. The theory therefore introduces a gap not around coincidence in the naive sense, but around the null sector in the space of invariant intervals.
Physical Meaning of \( \style{font-size:1em;}{\ell_0} \)
The constant \( \style{font-size:1em;}{\ell_0} \) is the new fundamental scale proposed by the theory. It has dimensions of length and is treated as a universal scalar. Its value is not fixed a priori. It could be connected to the Planck scale, but the theory must not assume this identification without further arguments. In a cautious version, \( \style{font-size:1em;}{\ell_0} \) is a new constant, to be estimated or experimentally constrained.
It is not an absolute minimum spatial distance. It does not say that two spatial coordinates cannot differ by less than \( \style{font-size:1em;}{\ell_0} \). It does not say that time is composed of minimum intervals. It does not replace continuous spacetime with a regular grid. Its meaning is more precise: \( \style{font-size:1em;}{\ell_0} \) is the minimum admitted value for the non-null invariant separation between physically resolved events of the same process.
The ratio between \( \style{font-size:1em;}{\ell_0} \) and the physical scale of the process is controlled by a dimensionless parameter. If \( \style{font-size:1em;}{L_\Psi} \) denotes the characteristic spacetime scale over which a state, a profile, or a correlation function varies significantly, one defines:
\( \style{font-size:1em;}{\varepsilon_{\ell}=\frac{\ell_0}{L_\Psi}} \)
The ordinary regime is:
\( \style{font-size:1em;}{\varepsilon_{\ell}\ll1} \)
When this condition is satisfied, the new scale is much smaller than the physical resolution of the process. In this regime, the effects of \( \style{font-size:1em;}{\ell_0} \) must become negligible.
The regime in which the theory may differ from ordinary physics is instead the one in which:
\( \style{font-size:1em;}{L_\Psi\sim\ell_0} \)
In this case, the structure of the forbidden band can no longer be ignored. Amplitudes between resolved events, propagators, and correlation functions could receive corrections. Such corrections, however, must be calculated in a specific model: the axiom alone is not enough to provide numbers.
Physically Resolved Events
A physically resolved event is described as a localized but non-punctual structure. In a first formulation, each event e is associated with an effective four-dimensional position, a resolution profile, and an observable content.
The effective position is:
\( \style{font-size:1em;}{x_e^\mu=(ct_e,x_e,y_e,z_e)} \)
The resolution profile is a test function:
\( \style{font-size:1em;}{f_e\in\Phi} \)
The function \( \style{font-size:1em;}{f_e} \) is concentrated around the spacetime region associated with the event. It represents the resolution with which the event enters the process. A very concentrated profile approximates a point event; an extended profile represents an event distributed over a finite region.
The Dirac delta can be used as an ideal limit:
\( \style{font-size:1em;}{\delta_{x_e}(f)=f(x_e)} \)
but in the theory it is not assumed as a primitive physical event. A physically resolved event is closer to a regular profile than to a perfectly punctual distribution. This choice is coherent with the role of \( \style{font-size:1em;}{\ell_0} \): if there is a minimum scale for physically resolved non-null intervals, point localization must be treated as a mathematical limit, not as elementary physical reality.
The event can therefore be represented schematically as:
\( \style{font-size:1em;}{e=(x_e^\mu,f_e,\hat{O}_e)} \)
where \( \style{font-size:1em;}{\hat{O}_e} \) is the observable associated with the event. This formula is not a definitive definition, but a useful parametrization. It separates three aspects: where the event is localized, with what resolution it is described, and what observable content it carries.
Two events may be geometrically close but not physically distinguishable. The theory expresses this idea by requiring that, if two events are distinct and resolved, then their non-null separation must exceed the threshold \( \style{font-size:1em;}{\ell_0} \). If the separation falls inside the forbidden band, the correct description is not that of two distinct events, but of a single unresolved structure or of a nonphysical configuration.
Configurations of Events
A process may contain more than two events. A configuration of n events is:
\( \style{font-size:1em;}{\mathcal{E}_n=(e_1,e_2,\ldots,e_n)} \)
The configuration is physically admissible if every distinct pair satisfies the axiom:
\( \style{font-size:1em;}{\sigma(e_i,e_j)\in \{0\}\cup[\ell_0,+\infty)} \)
for every pair with distinct indices:
\( \style{font-size:1em;}{i\neq j} \)
The configuration is forbidden if there exists at least one pair such that:
\( \style{font-size:1em;}{0\lt\sigma(e_i,e_j)\lt\ell_0} \)
The set of admissible configurations of a process P can be written as:
\( \style{font-size:1em;}{\mathcal{C}_{\ell_0}(P)=\{\mathcal{E}_n\mid \sigma(e_i,e_j)\in \{0\}\cup[\ell_0,+\infty)\}} \)
The physical space of the theory must be supported on such configurations:
\( \style{font-size:1em;}{\mathrm{supp}(\Psi_{\mathrm{phys}})\subseteq\mathcal{C}_{\ell_0}(P)} \)
This formula is one of the most important in the whole framework. It says that the physical state does not live on all imaginable configurations of the continuum, but only on those compatible with the minimum interval. Forbidden configurations remain describable as geometric constructions, but they do not belong to the physical domain.
For extended regions, the notion of separation must be handled carefully. If two profiles \( \style{font-size:1em;}{f_1} \) and \( \style{font-size:1em;}{f_2} \) have extended supports, there is not necessarily a single value of \( \style{font-size:1em;}{\sigma} \) between them. There are many pairs of points, some admitted and others possibly forbidden. In these cases the constraint must be expressed on the support of the kernel or on the effective configurations, not through a single pointwise distance.
The stronger condition is:
\( \style{font-size:1em;}{\mathrm{supp}(f_2)\times\mathrm{supp}(f_1)\subseteq\mathcal{F}_{\ell_0}} \)
In this case, if the propagator satisfies the geometric constraint, the amplitude must vanish:
\( \style{font-size:1em;}{A_{\ell_0}(f_2,f_1)=0} \)
When, instead, the supports cross both admitted and forbidden regions, one must integrate the filtered propagator and evaluate the effective contribution.
Space of States
The natural mathematical structure for the theory is a Gelfand triple, or rigged Hilbert space:
\( \style{font-size:1em;}{\Phi\subset H\subset\Phi’} \)
The space H is the kinematical Hilbert space. It preserves the probabilistic core of quantum mechanics: inner product, norm, superposition, and probabilistic interpretation.
The space \( \style{font-size:1em;}{\Phi} \) is a dense subspace of regular states. It contains test functions, localized profiles, and states sufficiently well behaved to define smeared observables, amplitudes, and distributional limits.
The space \( \style{font-size:1em;}{\Phi} \)‘ is the distributional dual. It contains generalized states, Dirac deltas, propagators, and objects that do not necessarily belong to H but are indispensable in the physical formalism.
A kinematical state can be written as:
\( \style{font-size:1em;}{\Psi\in\Phi’} \)
but not every kinematical state is physical. To become physical, a state must satisfy at least two constraints: a dynamical constraint and a geometric constraint. The dynamical constraint selects the solutions admitted by the law of motion. The geometric constraint selects the configurations compatible with \( \style{font-size:1em;}{\ell_0} \).
In preliminary form:
\( \style{font-size:1em;}{H_{\mathrm{phys}}=\{\Psi\in\Phi’\mid \hat{C}\Psi=0,\ \Pi_{\ell_0}\Psi=\Psi\}} \)
When one works with the free constraint \( \style{font-size:1em;}{\hat{C}_0} \), the form becomes:
\( \style{font-size:1em;}{H_{\mathrm{phys}}=\mathrm{ker}(\hat{C}_0)\cap\mathrm{Im}(\Pi_{\ell_0})} \)
This formula is compact, but it hides several technical problems. One must define the domain of \( \style{font-size:1em;}{\hat{C}_0} \), the domain of \( \style{font-size:1em;}{\Pi_{\ell_0}} \), the physical inner product, and the relation between the geometric and dynamical constraints. If the two constraints do not commute, the construction of the physical space cannot be obtained simply by applying two projectors in succession. In that case, the physical intersection of the conditions must be defined directly.
A strong condition, but not an automatically true one, would be:
\( \style{font-size:1em;}{[\Pi_{\ell_0},\hat{C}_0]=0} \)
If this relation holds, the geometric and dynamical constraints are compatible in a simple way. If it does not hold, the theory must specify which of the two constraints has operational priority, or how the overall physical projector is constructed.
The Geometric Projector
The geometric projector \( \style{font-size:1em;}{\Pi_{\ell_0}} \) represents the restriction imposed by the axiom of the minimum interval. It eliminates the components of the state supported on forbidden configurations.
The physical-state condition is:
\( \style{font-size:1em;}{\Pi_{\ell_0}\Psi=\Psi} \)
If \( \style{font-size:1em;}{\Pi_{\ell_0}} \) is truly a projector, it must satisfy:
\( \style{font-size:1em;}{\Pi_{\ell_0}^2=\Pi_{\ell_0}} \)
If it is an orthogonal projector with respect to the kinematical inner product, one must also have:
\( \style{font-size:1em;}{\Pi_{\ell_0}^{\dagger}=\Pi_{\ell_0}} \)
Its image is the space of geometrically admissible configurations:
\( \style{font-size:1em;}{\mathrm{Im}(\Pi_{\ell_0})=\{\Psi\mid\mathrm{supp}(\Psi)\subseteq\mathcal{C}_{\ell_0}(P)\}} \)
This definition is conceptually clear, but mathematically incomplete. One needs a measure on the space of configurations. One needs a rigorous definition of support for distributional states. One needs a rule for events described by extended profiles. One also needs a distinction between a true sharp projector and a regular filter.
A regular filter is not necessarily idempotent. If one uses a function \( \style{font-size:1em;}{F_{\ell_0,\delta}} \) to damp certain separations, in general:
\( \style{font-size:1em;}{F_{\ell_0,\delta}^2\neq F_{\ell_0,\delta}} \)
This means that a smooth filter is not a geometric projector in the strict sense. It may be used as an effective model or regularization, but it must be distinguished from the pure axiom. The theory must therefore keep two ideas separate: the axiomatic constraint, which defines the physical domain, and the operational filter, which is used to construct regular propagators.
Dynamical Constraint
The theory aims to be four-dimensional and covariant. For this reason it does not take the Schrödinger equation, which describes evolution with respect to an external time, as its fundamental form. At the fundamental level, the physical state is selected by a constraint:
\( \style{font-size:1em;}{\hat{C}\Psi=0} \)
In the free scalar sector, the natural constraint is the Klein-Gordon one. Setting:
\( \style{font-size:1em;}{\mu=\frac{mc}{\hbar}} \)
one defines:
\( \style{font-size:1em;}{\hat{C}_0=\partial^\mu\partial_\mu+\mu^2} \)
or:
\( \style{font-size:1em;}{\hat{C}_0=\partial^\mu\partial_\mu+\left(\frac{mc}{\hbar}\right)^2} \)
The d’Alembertian operator is:
\( \style{font-size:1em;}{\partial^\mu\partial_\mu=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial z^2}} \)
Free states satisfy:
\( \style{font-size:1em;}{\hat{C}_0\Psi=0} \)
This choice is cautious. The theory does not immediately modify the free relativistic dispersion relation. The new scale first enters the domain of physical configurations and the support of amplitudes. At a later stage, direct deformations of the dynamical operator may be studied, but they introduce risks: new poles, instabilities, violations of unitarity, or problems of spectral positivity.
A general deformation could have the form:
\( \style{font-size:1em;}{\hat{C}_{\ell_0}=\hat{C}_0+\ell_0^2\hat{D}_2+O(\ell_0^4)} \)
where \( \style{font-size:1em;}{\hat{D}_2} \) must be constructed covariantly and with the correct dimensions. A formal example is:
\( \style{font-size:1em;}{\hat{D}_2=a_2(\partial^\mu\partial_\mu)^2+a_1\mu^2\partial^\mu\partial_\mu+a_0\mu^4} \)
with dimensionless coefficients:
\( \style{font-size:1em;}{a_0,a_1,a_2\in\mathbb{R}} \)
This possibility nevertheless remains secondary to the minimal version of the theory. The minimal version preserves \( \style{font-size:1em;}{\hat{C}_0} \) and introduces \( \style{font-size:1em;}{\ell_0} \) as a geometric constraint on states and amplitudes.
Physical Projector and Inner Product
The physical projector must select the states satisfying both the dynamical constraint and the geometric constraint. Its image is:
\( \style{font-size:1em;}{\mathrm{Im}(\Pi_{\mathrm{phys}})=\mathrm{ker}(\hat{C}_0)\cap\mathrm{Im}(\Pi_{\ell_0})} \)
A symbolic notation is:
\( \style{font-size:1em;}{\Pi_{\mathrm{phys}}=\Pi_{\mathrm{ker}(\hat{C}_0)\cap\mathrm{Im}(\Pi_{\ell_0})}} \)
This form is more cautious than the simple multiplication of projectors, because in general two projectors do not commute. If there exists a dynamical projector \( \style{font-size:1em;}{\Pi_C} \) associated with the constraint, one may write formally:
\( \style{font-size:1em;}{\Pi_C=\delta(\hat{C}_0)} \)
or, in an integral representation:
\( \style{font-size:1em;}{\Pi_C=\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\lambda\,e^{i\lambda\hat{C}_0}} \)
The parameter \( \style{font-size:1em;}{\lambda} \) is not physical time. It is an auxiliary averaging parameter over the constraint.
The physical inner product may be proposed as:
\( \style{font-size:1em;}{\langle\Phi\mid\Psi\rangle_{\mathrm{phys}}=\langle\Phi\mid\Pi_{\mathrm{phys}}\mid\Psi\rangle_{\mathrm{kin}}} \)
This formula is natural, but not sufficient. One must verify that the inner product is positive, well defined, and compatible with the physical observables. In the relativistic case, defining the inner product is often delicate, especially for Klein-Gordon-type constraints. The theory must therefore avoid presenting this formula as a final solution. It is a starting point.
Probabilities are defined through projections onto physical alternatives. If \( \style{font-size:1em;}{\Pi_\alpha} \) is the projector associated with an alternative \( \style{font-size:1em;}{\alpha} \), then:
\( \style{font-size:1em;}{P(\alpha)=\frac{|\Pi_\alpha\Psi_{\mathrm{phys}}|_{\mathrm{phys}}^2}{\sum_\beta|\Pi_\beta\Psi_{\mathrm{phys}}|_{\mathrm{phys}}^2}} \)
The alternatives must satisfy the geometric constraint:
\( \style{font-size:1em;}{\Pi_\alpha=\Pi_{\ell_0}\Pi_\alpha\Pi_{\ell_0}} \)
This condition prevents a physically admitted measurement from selecting forbidden configurations.
Localized Observables
In a relativistic theory, observables should not be defined only on a spatial slice at fixed time. They must be associable with regions of spacetime. Since pointlike objects are distributional, the theory uses smeared observables.
Given a test function \( \style{font-size:1em;}{f_R} \) concentrated in a region R, one defines:
\( \style{font-size:1em;}{\hat{O}(f_R)=\int_M f_R(x)\hat{O}(x)d^4x} \)
Here M is Minkowski spacetime and \( \style{font-size:1em;}{\hat{O}(x)} \) is a local object, to be treated in the distributional sense. The physical observable is not directly \( \style{font-size:1em;}{\hat{O}(x)} \), but its version integrated against \( \style{font-size:1em;}{f_R} \).
For a physically resolved event e:
\( \style{font-size:1em;}{\hat{O}(e)=\hat{O}(f_e)} \)
A physical observable must preserve the space of physical states:
\( \style{font-size:1em;}{\hat{O}_{\mathrm{phys}}H_{\mathrm{phys}}\subseteq H_{\mathrm{phys}}} \)
It must be compatible with the geometric constraint:
\( \style{font-size:1em;}{\Pi_{\ell_0}\hat{O}_{\mathrm{phys}}\Pi_{\ell_0}=\hat{O}_{\mathrm{phys}}} \)
and, on the physical space, it must be compatible with the dynamical constraint:
\( \style{font-size:1em;}{[\hat{O}_{\mathrm{phys}},\hat{C}_0]\Psi_{\mathrm{phys}}=0} \)
These conditions may be too strong in some contexts, but they express the fundamental requirement: a physical observable must not transform an admitted state into a forbidden state.
Correlations between localized observables take the form:
\( \style{font-size:1em;}{G_{\ell_0}(f_2,f_1)=\langle\Psi_{\mathrm{phys}}\mid\hat{O}(f_2)\hat{O}(f_1)\mid\Psi_{\mathrm{phys}}\rangle_{\mathrm{phys}}} \)
These correlation functions are one of the main places where the theory might produce predictions. If the supports of \( \style{font-size:1em;}{f_1} \) and \( \style{font-size:1em;}{f_2} \) explore the forbidden band, \( \style{font-size:1em;}{G_{\ell_0}} \) may differ from the ordinary correlation \( \style{font-size:1em;}{G_0} \).
Relational Amplitudes
The dynamics can be formulated through amplitudes between events or regions. For two physically resolved events \( \style{font-size:1em;}{e_1} \) and \( \style{font-size:1em;}{e_2} \), with profiles \( \style{font-size:1em;}{f_{e_1}} \) and \( \style{font-size:1em;}{f_{e_2}} \), a natural form is:
\( \style{font-size:1em;}{A_{\ell_0}(e_2,e_1)=\langle f_{e_2}\mid\Pi_{\mathrm{phys}}\mid f_{e_1}\rangle_{\mathrm{kin}}} \)
This formula says that the physical amplitude is obtained by projecting the kinematical profiles onto the physical space. The role of external time disappears from the fundamental form; what matters is the physical correlation between events or regions compatible with the constraints.
For regions \( \style{font-size:1em;}{R_1} \) and \( \style{font-size:1em;}{R_2} \):
\( \style{font-size:1em;}{A_{\ell_0}(R_2,R_1)=\langle f_{R_2}\mid\Pi_{\mathrm{phys}}\mid f_{R_1}\rangle_{\mathrm{kin}}} \)
When two events fall into the forbidden band, the physical amplitude must vanish:
\( \style{font-size:1em;}{A_{\ell_0}(e_2,e_1)=0} \)
if:
\( \style{font-size:1em;}{0\lt\sigma(e_2,e_1)\lt\ell_0} \)
Conversely, the amplitude may be nonzero when:
\( \style{font-size:1em;}{\sigma(e_2,e_1)=0} \)
or:
\( \style{font-size:1em;}{\sigma(e_2,e_1)\geq\ell_0} \)
This rule must not be confused with a classical dynamics of propagation. It does not say that a signal travels from \( \style{font-size:1em;}{e_1} \) to \( \style{font-size:1em;}{e_2} \). It says that the correlation amplitude between two resolved events is defined only on the admitted geometric domain.
Composition of amplitudes requires a measure on the physical space. If the physical projector is idempotent:
\( \style{font-size:1em;}{\Pi_{\mathrm{phys}}^2=\Pi_{\mathrm{phys}}} \)
then one can write a symbolic composition rule:
\( \style{font-size:1em;}{A_{\ell_0}(e_3,e_1)=\int_{\Gamma_{\ell_0}}d\mu_{\mathrm{phys}}(e_2)A_{\ell_0}(e_3,e_2)A_{\ell_0}(e_2,e_1)} \)
The integration is over the admitted physical domain. This formula remains formal until the measure \( \style{font-size:1em;}{d\mu_{\mathrm{phys}}} \) and the nature of the space of intermediate events are specified.
Physical Propagator
The propagator is the object that connects the abstract formulation to concrete calculations. In the ordinary free sector, the propagator \( \style{font-size:1em;}{K_0} \) satisfies:
\( \style{font-size:1em;}{\hat{C}_0K_0=\delta^{(4)}} \)
The theory of the minimum interval requires a propagator compatible with the admitted domain. We define:
\( \style{font-size:1em;}{\Gamma_{\ell_0}=\{(x,y)\mid \sigma(x,y)=0\ \text{or}\ \sigma(x,y)\geq\ell_0\}} \)
and:
\( \style{font-size:1em;}{\mathcal{F}_{\ell_0}=\{(x,y)\mid 0\lt\sigma(x,y)\lt\ell_0\}} \)
The physical propagator must satisfy a support condition:
\( \style{font-size:1em;}{\mathrm{supp}(K_{\ell_0})\subseteq\Gamma_{\ell_0}} \)
This condition is the dynamical translation of the axiom. It does not deny that, in the continuum, there are points x and y with forbidden separation. It says that the physical propagator between resolved events must not have support on those pairs.
For two test functions, the amplitude is:
\( \style{font-size:1em;}{A_{\ell_0}(f_2,f_1)=\int_M d^4x\,\int_M d^4y\overline{f_2(x)}K_{\ell_0}(x,y)f_1(y)} \)
In compact form:
\( \style{font-size:1em;}{A_{\ell_0}(f_2,f_1)=\langle f_2\mid K_{\ell_0}\mid f_1\rangle} \)
The ordinary limit requires:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}K_{\ell_0}=K_0} \)
in the sense of distributions:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}\langle f_2\mid K_{\ell_0}\mid f_1\rangle=\langle f_2\mid K_0\mid f_1\rangle} \)
This is an essential condition. If it is not satisfied, the theory does not recover ordinary physics.
Decomposition of the Propagator
Since the null sector must remain admitted, the ordinary propagator is decomposed into two parts: one component associated with the null sector and one component outside the null sector.
\( \style{font-size:1em;}{K_0=K_{\mathrm{null}}+K_{\mathrm{off}}} \)
The part \( \style{font-size:1em;}{K_{\mathrm{null}}} \) represents the contribution supported on the light cone or associated with the null singularity. The part \( \style{font-size:1em;}{K_{\mathrm{off}}} \) represents the contribution outside the null sector. This decomposition is operational and depends on the chosen prescription: Feynman, retarded, advanced, or another Green function may have different properties. The theory must specify the choice in each concrete application.
An effective model of filtered propagator is:
\( \style{font-size:1em;}{K_{\ell_0,\delta}=K_{\mathrm{null}}+F_{\ell_0,\delta}(\sigma)K_{\mathrm{off}}} \)
The filter \( \style{font-size:1em;}{F_{\ell_0,\delta}} \) suppresses the part outside the null sector in the forbidden band. The parameter \( \style{font-size:1em;}{\delta} \) controls the smoothness of the transition and is technical, not a new fundamental constant.
A filter choice can be built from a smooth function S(u) such that:
\( \style{font-size:1em;}{S(u)=0\quad\text{for}\quad u\leq0} \)
\( \style{font-size:1em;}{S(u)=1\quad\text{for}\quad u\geq1} \)
\( \style{font-size:1em;}{0\lt S(u)\lt1\quad\text{for}\quad 0\lt u\lt1} \)
The filter is:
\( \style{font-size:1em;}{F_{\ell_0,\delta}(\sigma)=S\left(\frac{\sigma-\ell_0}{\delta\ell_0}\right)} \)
From this definition it follows that:
\( \style{font-size:1em;}{F_{\ell_0,\delta}(\sigma)=0\quad\text{for}\quad\sigma\leq\ell_0} \)
and:
\( \style{font-size:1em;}{F_{\ell_0,\delta}(\sigma)=1\quad\text{for}\quad\sigma\geq(1+\delta)\ell_0} \)
This choice also cancels the boundary \( \style{font-size:1em;}{\sigma} \)=\( \style{font-size:1em;}{\ell_0} \), which the axiom instead admits. This must be interpreted correctly: the axiom states which separations are possible; the dynamics may nevertheless assign zero amplitude to a possible separation. If one also wants to respect the boundary dynamically, a different filter may be chosen. The theory must therefore distinguish between admitted domain and dynamical weight.
The continuum limit is:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}K_{\ell_0,\delta}=K_0} \)
provided that the filter tends to one on fixed non-null separations and that the null sector remains preserved.
Dynamical Effect of the Filter
For finite \( \style{font-size:1em;}{\ell_0} \), the filtered propagator is not in general the ordinary Green function of the free constraint. Applying \( \style{font-size:1em;}{\hat{C}_0} \) gives:
\( \style{font-size:1em;}{\hat{C}_0K_{\ell_0,\delta}=\delta^{(4)}+\mathcal{Q}_{\ell_0,\delta}} \)
The term \( \style{font-size:1em;}{\mathcal{Q}_{\ell_0,\delta}} \) arises from the action of the differential operator on the filter:
\( \style{font-size:1em;}{\mathcal{Q}_{\ell_0,\delta}=\hat{C}_0\left(F_{\ell_0,\delta}K_{\mathrm{off}}\right)-F_{\ell_0,\delta}\hat{C}_0K_{\mathrm{off}}} \)
This term is localized in the region where the filter varies:
\( \style{font-size:1em;}{\ell_0\leq\sigma\leq(1+\delta)\ell_0} \)
This point is decisive. The theory must choose between two interpretations.
The first interpretation is effective: \( \style{font-size:1em;}{K_{\ell_0,\delta}} \) is a modified propagator obtained by imposing a geometric restriction on amplitudes. In this case \( \style{font-size:1em;}{\mathcal{Q}_{\ell_0,\delta}} \) is an effective correction due to the filter.
The second interpretation is fundamental: there exists a new dynamical operator \( \style{font-size:1em;}{\hat{C}_{\ell_0,\delta}} \) such that:
\( \style{font-size:1em;}{\hat{C}_{\ell_0,\delta}K_{\ell_0,\delta}=\delta^{(4)}} \)
This second path is more ambitious but also more difficult. It requires constructing a coherent dynamical operator, controlling its spectrum, verifying unitarity, and guaranteeing the absence of unwanted degrees of freedom.
The cautious formulation regards \( \style{font-size:1em;}{K_{\ell_0,\delta}} \) as an initial effective model. The complete theory will then have to clarify whether the filter is only a rule for selecting amplitudes or whether it is the sign of a modified fundamental dynamics.
Continuum Limit
The continuum limit is the main test of compatibility with known physics. It can be expressed in several ways.
The admitted spectrum tends to the ordinary spectrum:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}\left(\{0\}\cup[\ell_0,+\infty)\right)=[0,+\infty)} \)
The forbidden band disappears:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}\mathcal{F}_{\ell_0}=\emptyset} \)
The propagator becomes ordinary again:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}K_{\ell_0,\delta}=K_0} \)
The amplitudes become ordinary again:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}A_{\ell_0,\delta}(f_2,f_1)=A_0(f_2,f_1)} \)
The nonrelativistic corrections must vanish:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}\mathcal{R}_{\ell_0}[\psi]=0} \)
For a process with characteristic scale \( \style{font-size:1em;}{L_\Psi} \), a general estimate of the corrections is:
\( \style{font-size:1em;}{\mathcal{R}_{\ell_0}[\psi]=O\left[\left(\frac{\ell_0}{L_\Psi}\right)^p\right]} \)
with:
\( \style{font-size:1em;}{p\gt0} \)
The value of p is not fixed by the axiom. It depends on the filter, the regularity of the profiles, the type of propagator, and the dynamical model. A complete theory will have to calculate it in specific cases.
The continuum limit should not be understood only as the mathematical limit \( \style{font-size:1em;}{\ell_0\to0} \). Physically, it also corresponds to the regime in which \( \style{font-size:1em;}{\ell_0} \) is much smaller than every scale resolved by the process. In that case, the new structure remains hidden.
Recovery of Special Relativity
Special relativity is recovered because the basic geometry remains Minkowskian and the axiom is built from a Lorentz-invariant quantity. Under Lorentz transformations, the quadratic interval remains invariant:
\( \style{font-size:1em;}{\eta_{\mu\nu}\Delta x’^\mu\Delta x’^\nu=\eta_{\mu\nu}\Delta x^\mu\Delta x^\nu} \)
Consequently:
\( \style{font-size:1em;}{\sigma’(e_1,e_2)=\sigma(e_1,e_2)} \)
If \( \style{font-size:1em;}{\ell_0} \) is a universal scalar, then the condition:
\( \style{font-size:1em;}{\sigma(e_1,e_2)\in \{0\}\cup[\ell_0,+\infty)} \)
has the same value for all inertial observers. No privileged reference frame is introduced at the kinematical level.
In the free sector, a plane-wave solution of the constraint \( \style{font-size:1em;}{\hat{C}_0} \) is:
\( \style{font-size:1em;}{\Psi(t,\mathbf{x})=e^{-iEt/\hbar}e^{i\mathbf{p}\cdot\mathbf{x}/\hbar}} \)
Applying the constraint:
\( \style{font-size:1em;}{\hat{C}_0\Psi=0} \)
one obtains the relativistic dispersion relation:
\( \style{font-size:1em;}{E^2=p^2c^2+m^2c^4} \)
This relation belongs to established physics. In the minimal version of the theory, it is not modified. The novelties appear instead in the physical domain of configurations and in the support of amplitudes.
This choice is important because it reduces the risk of conflict with already very precise experiments. Directly deforming relativistic dispersion can produce observable consequences that are strongly constrained. The theory of the minimum interval, in its cautious version, introduces the new scale in a subtler way: not as an immediate modification of free kinematics, but as a restriction on physically resolved correlations.
Recovery of Ordinary Quantum Mechanics
Ordinary quantum mechanics must emerge when a laboratory time can be chosen and when the regime is nonrelativistic. In the fundamental theory, time is not a privileged external parameter; it is a coordinate of spacetime. However, in ordinary experiments, clocks, apparatuses, and reference systems exist that allow the state to be described as a function of laboratory time.
One starts from the free constraint:
\( \style{font-size:1em;}{\left(\partial^\mu\partial_\mu+\left(\frac{mc}{\hbar}\right)^2\right)\Psi=0} \)
One separates the oscillation due to rest energy:
\( \style{font-size:1em;}{\Psi(t,\mathbf{x})=e^{-imc^2t/\hbar}\psi(t,\mathbf{x})} \)
Substituting into the constraint and isolating the dominant terms gives:
\( \style{font-size:1em;}{i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi+\frac{\hbar^2}{2mc^2}\frac{\partial^2\psi}{\partial t^2}} \)
In the nonrelativistic regime, the second time derivative is negligible relative to the main terms. The free Schrödinger equation remains:
\( \style{font-size:1em;}{i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi} \)
With an effective potential:
\( \style{font-size:1em;}{i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V\right)\psi} \)
In the theory of the minimum interval, the most general form includes corrections:
\( \style{font-size:1em;}{i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V\right)\psi+\mathcal{R}_{\ell_0}[\psi]} \)
The recovery of ordinary quantum mechanics requires:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}\mathcal{R}_{\ell_0}[\psi]=0} \)
and, for states varying on scales large compared with \( \style{font-size:1em;}{\ell_0} \):
\( \style{font-size:1em;}{\mathcal{R}_{\ell_0}[\psi]\to0\quad\text{when}\quad\frac{\ell_0}{L_\Psi}\to0} \)
Thus the Schrödinger equation is not denied. It is reinterpreted as the effective limit of covariant dynamics when a laboratory time exists and when the new scale is unresolved.
Causality and Microcausality
The theory preserves the null sector:
\( \style{font-size:1em;}{\sigma=0} \)
This choice preserves the light cone as a fundamental structure. However, preserving the light cone is not enough to guarantee full quantum causality. In a relativistic quantum theory one must avoid the possibility of controllable superluminal signals.
The theory admits correlations between spacelike-separated events. This is not problematic in itself: ordinary quantum mechanics also admits nonclassical correlations between spatially separated systems. The crucial point is that such correlations must not become channels of communication faster than light.
In a formulation close to QFT, microcausality is required. For observables localized in spacelike-separated regions, one imposes:
\( \style{font-size:1em;}{[\hat{O}(f_1),\hat{O}(f_2)]=0} \)
when the supports of \( \style{font-size:1em;}{f_1} \) and \( \style{font-size:1em;}{f_2} \) are spacelike separated and belong to causally independent regions. For fermionic fields, the condition must be modified with anticommutators, but the logic remains: physically separated observables must not allow operational superluminal influence.
The theory of the minimum interval must therefore add a compatibility condition:
\( \style{font-size:1em;}{[\hat{O}_{\mathrm{phys}}(f_1),\hat{O}_{\mathrm{phys}}(f_2)]\Psi_{\mathrm{phys}}=0} \)
for appropriate regions. This condition does not automatically follow from the axiom on \( \style{font-size:1em;}{\sigma} \). It must be assumed as an additional requirement or derived from the complete construction of the propagator and the observables.
The presence of the forbidden band near the light cone could modify some high-resolution correlations, but it must not alter the operational principle according to which no observer can use such correlations to transmit superluminal signals. This is one of the most important checks that a complete mathematical version of the theory will have to pass.
Null Sector
The null sector is the most delicate and most original part of the theory. Null intervals satisfy:
\( \style{font-size:1em;}{c^2\Delta t^2-\Delta r^2=0} \)
and therefore:
\( \style{font-size:1em;}{\sigma=0} \)
The theory explicitly admits them. Almost-null intervals, however, may be forbidden if they have nonzero \( \style{font-size:1em;}{\sigma} \) but are smaller than \( \style{font-size:1em;}{\ell_0} \).
This creates a discontinuous structure in the space of physically resolved intervals: the zero value is admitted, then there is a gap, then values from \( \style{font-size:1em;}{\ell_0} \) upward are admitted. Formally:
\( \style{font-size:1em;}{\{0\}\cup[\ell_0,+\infty)} \)
This choice distinguishes the theory from many ideas of “minimum length” that replace the null interval with a nonzero minimum length. Here the light cone is not erased. It is preserved as a singular class.
The consequence is that the theory does not automatically regularize all singularities on the light cone. If \( \style{font-size:1em;}{K_{\mathrm{null}}} \) is preserved, any distributional singularities associated with the null sector remain present. This is a limitation, but also a choice of relativistic coherence. The theory cannot claim, in its minimal form, to eliminate all ultraviolet divergences of QFT. It may suggest a new regularization for some correlations outside the null sector, but the treatment of null singularities requires further hypotheses.
If one wished to make the theory more regularizing, one could introduce a minimum profile also for the null sector. But that would be a different and stronger extension. The version presented here preserves the null sector.
Interactions
The minimal version works with the free constraint. To include interactions, one can deform the constraint:
\( \style{font-size:1em;}{\hat{C}_{\mathrm{int}}=\hat{C}_0+\hat{U}} \)
where \( \style{font-size:1em;}{\hat{U}} \) represents an interaction term. Since the theory is four-dimensional, \( \style{font-size:1em;}{\hat{U}} \) should be built from observables localized on spacetime regions, not only from instantaneous potentials.
A schematic form is:
\( \style{font-size:1em;}{\hat{U}=\hat{U}(f_R)} \)
The physical condition becomes:
\( \style{font-size:1em;}{\hat{C}_{\mathrm{int}}\Psi=0} \)
together with the geometric constraint:
\( \style{font-size:1em;}{\Pi_{\ell_0}\Psi=\Psi} \)
Interactions must also respect the admitted domain. This means that the interaction must not generate physical configurations supported entirely in the forbidden band. A symbolic condition is:
\( \style{font-size:1em;}{\Pi_{\ell_0}\hat{U}\Pi_{\ell_0}=\hat{U}_{\mathrm{phys}}} \)
The construction of interactions is an open part of the theory. In particular, one must understand how to treat local QFT interactions, which are often defined through products of fields at the same point and require renormalization. The presence of resolution profiles and of a scale \( \style{font-size:1em;}{\ell_0} \) could provide a new regulatory scheme, but this is not automatic. It must be shown through explicit calculations.
One possibility is that \( \style{font-size:1em;}{\ell_0} \) produces covariant form factors in amplitudes. In momentum space one could have an effective form:
\( \style{font-size:1em;}{\widetilde{K}_{\ell_0}(k)=F_{\mathrm{eff}}(\ell_0^2 k^2)\widetilde{K}_0(k)} \)
with:
\( \style{font-size:1em;}{\lim_{\ell_0^2|k^2|\to0}F_{\mathrm{eff}}(\ell_0^2 k^2)=1} \)
This, however, is an effective representation, not the fundamental principle. The fundamental principle remains the constraint on intervals between resolved events.
Relation to Special Relativity
The theory takes from special relativity the centrality of the Minkowski interval. It does not introduce an absolute Euclidean distance, does not select a privileged reference frame, and does not modify the role of the speed of light in the minimal sector.
Special relativity classifies intervals as timelike, spacelike, and null. The theory of the minimum interval adds a second classification: among non-null intervals, only those whose invariant modulus is at least equal to \( \style{font-size:1em;}{\ell_0} \) are physically resolved.
An admitted timelike interval satisfies:
\( \style{font-size:1em;}{c^2\Delta t^2-\Delta r^2\geq\ell_0^2} \)
An admitted spacelike interval satisfies:
\( \style{font-size:1em;}{\Delta r^2-c^2\Delta t^2\geq\ell_0^2} \)
A null interval satisfies:
\( \style{font-size:1em;}{c^2\Delta t^2-\Delta r^2=0} \)
The forbidden band can be written as:
\( \style{font-size:1em;}{0\lt\left|c^2\Delta t^2-\Delta r^2\right|\lt\ell_0^2} \)
This formula clarifies the geometry: the forbidden zone is a region around the light cone, but it excludes the cone itself. It does not coincide with a small ball around the origin of spacetime; it depends on the invariant interval.
For almost simultaneous events, with \( \style{font-size:1em;}{\Delta} \) t very small:
\( \style{font-size:1em;}{\sigma\simeq\Delta r} \)
Therefore spacelike-separated events are admitted if:
\( \style{font-size:1em;}{\Delta r\geq\ell_0} \)
For almost-lightlike events, with:
\( \style{font-size:1em;}{c^2\Delta t^2\simeq\Delta r^2} \)
the quantity \( \style{font-size:1em;}{\sigma} \) can be very small. If it is nonzero and smaller than \( \style{font-size:1em;}{\ell_0} \), the pair is forbidden as a pair of resolved events.
This is an important difference from a simple minimum spatial distance. The theory does not limit only what is “close in space”; it limits what is “too close to the light cone” in invariant terms, except for the cone itself.
Relation to Quantum Mechanics
The theory preserves the fundamental principles of quantum mechanics: states, superposition, amplitudes, observables, and probabilities. A state can be a linear combination of states:
\( \style{font-size:1em;}{\Psi=a\Psi_1+b\Psi_2} \)
with:
\( \style{font-size:1em;}{a,b\in\mathbb{C}} \)
The probabilistic rule remains based on norms and squared moduli. The differences with respect to ordinary quantum mechanics do not concern the principle of superposition, but the physical domain of events and configurations.
Ordinary quantum mechanics often uses time as an external parameter. The theory of the minimum interval instead seeks a covariant formulation in which time is an internal coordinate of spacetime. The Schrödinger equation emerges as a limit, not as a fundamental principle.
The theory admits correlations between events belonging to the same quantum process, even when those events are separated by spacelike intervals. This possibility follows from the fact that the geometric constraint does not require a classical trajectory between events; it requires only that their invariant separation belong to the physically admitted domain.
The initial task of the theory is therefore more circumscribed: to define which configurations of events are compatible with the minimum interval and to study how this restriction may affect amplitudes, propagators, and correlation functions.
A physical correlation may be described, in general form, as a relation between observables localized on profiles or regions of spacetime:
\( \style{font-size:1em;}{G_{\ell_0}(f_2,f_1)=\langle\Psi_{\mathrm{phys}}|\hat{O}(f_2)\hat{O}(f_1)|\Psi_{\mathrm{phys}}\rangle_{\mathrm{phys}}} \)
The geometric constraint remains the condition selecting the physical domain of events:
\( \style{font-size:1em;}{\Pi_{\ell_0}\Psi_{\mathrm{phys}}=\Psi_{\mathrm{phys}}} \)
This setting leaves open the future study of complex correlations, without assigning to the theory the premature task of explaining specific quantum phenomena already described by the ordinary formalism.
Relation to Quantum Field Theory
The theory recalls QFT in three main respects. The first is the use of observables localized in spacetime. The second is the role of propagators and correlation functions. The third is the need to treat local fields as distributions.
In ordinary QFT, a point field \( \style{font-size:1em;}{\hat{\phi}(x)} \) is often a distributional object. A well-defined observable requires smearing:
\( \style{font-size:1em;}{\hat{\phi}(f)=\int_M f(x)\hat{\phi}(x)d^4x} \)
The theory of the minimum interval adopts a similar logic. Physically resolved events are associated with profiles \( \style{font-size:1em;}{f_e} \), and amplitudes are calculated by integrating propagators against test functions.
The novelty lies in the support constraint on the physical propagator:
\( \style{font-size:1em;}{\mathrm{supp}(K_{\ell_0})\subseteq\Gamma_{\ell_0}} \)
This condition is not standard in ordinary QFT. It is the new element that attempts to incorporate \( \style{font-size:1em;}{\ell_0} \) into the domain of correlations.
The theory must, however, face the typical problems of QFT: renormalization, local causality, spectral positivity, unitarity, and the definition of composite observables. The simple insertion of a filter does not automatically guarantee quantum consistency. In particular, a nonlocal filter in spacetime can modify the analytic structure of propagators. One must verify that it does not introduce violations of causality or nonphysical states.
Relation to Minimum Lengths and Spacetime Granularity
The idea of a minimum length is not new. It has been discussed in various contexts: quantum gravity, string theory, the generalized uncertainty principle, causal sets, noncommutative models, zero-point length, covariant discretizations, and approaches to spacetime granularity.
The theory of the minimum quantum interval reworks this theme, but it does not coincide with a simple discretization of space. It does not say that there is a grid of points separated by \( \style{font-size:1em;}{\ell_0} \). It does not say that all measurable distances have a discrete spectrum. It does not say that the null interval is replaced by a minimum length.
Its specific form is:
\( \style{font-size:1em;}{\sigma\in \{0\}\cup[\ell_0,+\infty)} \)
This structure is more selective. It preserves the null sector and introduces the gap only on physically resolved non-null intervals. It also applies the constraint to the events of a quantum process, not to every pair of geometric points.
This combination is the most original part of the proposal. The general idea of a minimum length has precedents. The specific idea of a gap in the spectrum of non-null invariant intervals, with the null sector preserved and application to resolved events, is the distinctive formulation of the project.
Originality of the Theory
The originality of the theory does not consist in generically introducing a minimum scale. That theme is already widely present in theoretical physics. The original part lies in the combination of five choices.
The first is that the limited quantity is the invariant interval \( \style{font-size:1em;}{\sigma} \), not a spatial or temporal coordinate.
The second is that the null value remains admitted:
\( \style{font-size:1em;}{\sigma=0} \)
The third is that only the band is excluded:
\( \style{font-size:1em;}{0\lt\sigma\lt\ell_0} \)
The fourth is that the constraint applies to physically resolved events of a process, not to the geometric continuum itself.
The fifth is that the theory attempts to incorporate the constraint into a physical space defined by the intersection between covariant dynamics and geometric projection:
\( \style{font-size:1em;}{H_{\mathrm{phys}}=\mathrm{ker}(\hat{C}_0)\cap\mathrm{Im}(\Pi_{\ell_0})} \)
These elements produce an autonomous theoretical structure. It is not enough to say that the theory “postulates a minimum length”; one must specify that it postulates a gap in the spectrum of non-null invariant intervals between resolved events.
Necessary Missing Parts
To become a complete physical theory, the framework must still develop several parts.
The first is the rigorous definition of \( \style{font-size:1em;}{\Pi_{\ell_0}} \). One must specify its domain, the measure on configuration space, the way it acts on distributional states, and its compatibility with the dynamical constraint.
The second is the physical inner product. The formula:
\( \style{font-size:1em;}{\langle\Phi\mid\Psi\rangle_{\mathrm{phys}}=\langle\Phi\mid\Pi_{\mathrm{phys}}\mid\Psi\rangle_{\mathrm{kin}}} \)
is plausible but not sufficient. Positivity and normalization must be proved.
The third is microcausality. One must verify:
\( \style{font-size:1em;}{[\hat{O}(f_1),\hat{O}(f_2)]=0} \)
for appropriate spacelike regions.
The fourth is unitarity, or its equivalent in a constrained theory. If a laboratory time emerges, one must have:
\( \style{font-size:1em;}{\frac{d}{dt}|\psi(t)|^2=0} \)
The fifth is spectral positivity. The filter must not introduce negative-norm states, nonphysical energies, or unwanted dynamical poles.
The sixth is the construction of a fundamental propagator. One must decide whether \( \style{font-size:1em;}{K_{\ell_0,\delta}} \) is only an effective model or the Green function of a new operator:
\( \style{font-size:1em;}{\hat{C}_{\ell_0,\delta}K_{\ell_0,\delta}=\delta^{(4)}} \)
The seventh is the production of quantitative predictions. A physical theory must arrive at formulas comparable with observations, even if the effects are extremely small.
Possible Predictions
The first predictions should concern the propagator near the forbidden band. The difference between the modified propagator and the ordinary propagator is:
\( \style{font-size:1em;}{\Delta K_{\ell_0,\delta}=K_{\ell_0,\delta}-K_0} \)
Using the decomposition:
\( \style{font-size:1em;}{K_{\ell_0,\delta}=K_{\mathrm{null}}+F_{\ell_0,\delta}K_{\mathrm{off}}} \)
and:
\( \style{font-size:1em;}{K_0=K_{\mathrm{null}}+K_{\mathrm{off}}} \)
one obtains:
\( \style{font-size:1em;}{\Delta K_{\ell_0,\delta}=\left(F_{\ell_0,\delta}-1\right)K_{\mathrm{off}}} \)
This difference is concentrated where the filter deviates from one, that is, near the threshold \( \style{font-size:1em;}{\ell_0} \). For test profiles:
\( \style{font-size:1em;}{\Delta A_{\ell_0,\delta}(f_2,f_1)=\langle f_2\mid\Delta K_{\ell_0,\delta}\mid f_1\rangle} \)
A first prediction could be an amplitude suppression for almost-null but non-null pairs of events. Another could concern corrections to two-point functions at extremely high resolution. Still another could be a form factor in high-energy scattering processes, if the momentum-space model is well defined.
However, without a value of \( \style{font-size:1em;}{\ell_0} \) and without a precise choice of filter or dynamical operator, these remain research directions, not definitive predictions.
Compact Formulation of the Theory
The theory can be summarized in a set of fundamental relations.
Reference spacetime:
\( \style{font-size:1em;}{\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)} \)
Quadratic interval:
\( \style{font-size:1em;}{\Delta s^2=\eta_{\mu\nu}\Delta x^\mu\Delta x^\nu} \)
Positive separation:
\( \style{font-size:1em;}{\sigma(e_i,e_j)=\sqrt{\left|\eta_{\mu\nu}\Delta x^\mu\Delta x^\nu\right|}} \)
Minimum interval axiom:
\( \style{font-size:1em;}{\sigma(e_i,e_j)\in \{0\}\cup[\ell_0,+\infty)} \)
Forbidden band:
\( \style{font-size:1em;}{\mathcal{F}_{\ell_0}=\{(e_i,e_j)\mid0\lt\sigma(e_i,e_j)\lt\ell_0\}} \)
Admitted domain:
\( \style{font-size:1em;}{\Gamma_{\ell_0}=\{(e_i,e_j)\mid\sigma(e_i,e_j)=0\ \text{or}\ \sigma(e_i,e_j)\geq\ell_0\}} \)
Gelfand triple:
\( \style{font-size:1em;}{\Phi\subset H\subset\Phi’} \)
Resolved event:
\( \style{font-size:1em;}{f_e\in\Phi} \)
Smeared observable:
\( \style{font-size:1em;}{\hat{O}(f_R)=\int_M f_R(x)\hat{O}(x)d^4x} \)
Free dynamical constraint:
\( \style{font-size:1em;}{\hat{C}_0=\partial^\mu\partial_\mu+\left(\frac{mc}{\hbar}\right)^2} \)
Physical space:
\( \style{font-size:1em;}{H_{\mathrm{phys}}=\mathrm{ker}(\hat{C}_0)\cap\mathrm{Im}(\Pi_{\ell_0})} \)
Filtered propagator:
\( \style{font-size:1em;}{K_{\ell_0,\delta}=K_{\mathrm{null}}+F_{\ell_0,\delta}(\sigma)K_{\mathrm{off}}} \)
Amplitude between profiles:
\( \style{font-size:1em;}{A_{\ell_0,\delta}(f_2,f_1)=\langle f_2\mid K_{\ell_0,\delta}\mid f_1\rangle} \)
Continuum limit:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}K_{\ell_0,\delta}=K_0} \)
Recovery of null corrections:
\( \style{font-size:1em;}{\lim_{\ell_0\to0}\mathcal{R}_{\ell_0}[\psi]=0} \)
Conclusion
The Theory of the Minimum Quantum Interval proposes adding a new universal scale \( \style{font-size:1em;}{\ell_0} \) to the structure of physics. This scale does not directly discretize space, does not directly discretize time, and does not erase the geometric continuum. Instead, it limits the non-null invariant separations between physically resolved events of the same quantum process.
Its central axiom is:
\( \style{font-size:1em;}{\sigma(e_1,e_2)\in \{0\}\cup[\ell_0,+\infty)} \)
The strength of the proposal lies in its covariant formulation. Since \( \style{font-size:1em;}{\sigma} \) is built from the Minkowski interval, the constraint can be compatible with Lorentz invariance. The choice to preserve the null sector avoids an immediate break with the role of the light cone. The distinction between geometric points and physically resolved events allows the mathematical continuum to be maintained without renouncing a physical granularity of processes.
The theory takes up already existing tools: special relativity, quantum mechanics, quantum field theory, distributions, test functions, rigged Hilbert spaces, covariant constraints, and propagators. Its originality does not lie in any one of these tools taken separately, but in the way they are organized around a gap in non-null invariant intervals.
The framework remains preliminary. One must rigorously define \( \style{font-size:1em;}{\Pi_{\ell_0}} \), construct the physical inner product, verify microcausality and unitarity, control spectral positivity, clarify whether the filtered propagator is effective or fundamental, and produce quantitative predictions. Only after these steps can the theory be evaluated as a possible physical theory, and not merely as an axiomatic program.
Postscript: Why This Theory and These Hypotheses?
A theory of this kind arises from a very simple tension: on one side we need to preserve the mathematical tools of modern physics, while on the other we suspect that physical reality, especially at extremely small scales, may have a more selective structure than the ideal continuum used in calculations.
The mathematical continuum is an extremely powerful tool. Differential calculus, coordinates, functions, derivatives, field equations, Hilbert spaces, and the language of geometry have made it possible to build some of the most effective theories in the history of science. Abandoning this continuity would mean losing a huge part of the descriptive capacity of physics. The theory of the minimum quantum interval therefore begins from a cautious choice: spacetime remains continuous as a mathematical language, while the physics of distinguishable events is subjected to a new constraint.
The hypothesis takes shape from two intuitions. The first is the idea, attributed to a suggestion by Heisenberg, that there may exist a third universal constant with the dimensions of a length. After the speed of light, which organizes the relativistic structure of spacetime, and after Planck’s constant, which organizes the quantum structure of action, one can imagine a new constant capable of introducing a physical threshold into the description of events. The second intuition comes from relativity: a fundamental length, to be truly universal, should be formulated through a quantity shared by all inertial observers. That quantity is the spacetime interval.
The most naive choice would be to assign the new constant to three-dimensional space, as if there existed an absolute minimum spatial distance. This path immediately raises a difficulty: in relativity, space and time depend on the reference frame. What for one observer is mainly spatial distance may, for another, mix space and time differently. A universal constant applied only to space would therefore risk introducing a privileged reference frame. The theory chooses another path: it applies the threshold to the invariant interval, that is, to the combination of space and time that preserves the same value for all observers.
In this way the new constant, denoted by ℓ₀, does not act on ordinary spatial distance and does not impose a separate minimum time. It acts on the four-dimensional separation between physically resolved events. Time is transformed into length through ct, as in relativity, and is considered together with the three spatial coordinates. The threshold therefore concerns spacetime, not an isolated part of it.
This choice has a strong internal logic. If we want to introduce a form of quantization or fundamental limit, it makes more sense to apply it to a complete relativistic quantity than to a single component. Quantizing space directly or quantizing time directly would force us to explain why one of the two dimensions receives privileged treatment. Applying the constraint to the spacetime interval avoids this asymmetry and keeps the theory within a relativistic sensibility.
The central point is that the theory does not quantize coordinates. The coordinates x, y, z, and t continue to be used as continuous variables. The constraint appears at another level: that of physically resolved events. A physically resolved event is an event that truly enters the description of a physical process. It may be a measurement, a record, an interaction, a transition, an emission, an absorption, or any recognizable node in the history of a system. A geometric point is an ideal entity; a physically resolved event is an entity with operational meaning inside a physical theory.
The theory then states that two distinct events of the same process may have null invariant separation, or a separation at least equal to ℓ₀. The null sector is preserved because it belongs to the structure of the light cone. Photons and, more generally, processes associated with propagation at the speed of light occupy a special role in relativity. Eliminating null intervals would immediately make compatibility with the known causal structure problematic. For this reason the theory admits the zero value and introduces the limit only on non-null intervals.
The consequence is a particular structure. The light cone remains admitted. Sufficiently large intervals remain admitted. The excluded zone is the small intermediate interval: separations different from zero, but smaller than ℓ₀. Intuitively, two events may lie exactly on the luminous boundary, or far enough apart in the structure of spacetime. The theory excludes almost-null events with too small a separation.
This hypothesis also produces an interesting relation between space and time. Since the relativistic interval contains a temporal part and a spatial part with opposite signs, compressing one of the two components too much may force the other to change in order for the constraint to remain satisfied. If two events of the same process are extremely close in space, their temporal separation must be sufficient to keep the interval admitted. If, instead, two events are compressed into an almost instantaneous temporal interval, their spatial distance must avoid the forbidden band.
At this point the problem of localization enters. If the theory speaks of physically resolved events, one must avoid treating them as infinitely precise points. An infinitesimal point is useful in calculations, but a physical event always has a resolution. A detector has finite sensitivity, a measurement has a duration, a track has a width, an interaction is described through a physical region. Even when the formalism uses ideal points, concrete physics passes through profiles, regions, and signals.
For this reason the theory introduces the language of distributions. Distributions make it possible to work with ideally localized objects, such as the Dirac delta, while at the same time allowing the perfect point to be replaced by a test function, that is, by a localized profile. In simple terms, an event can be treated as a small “spot” in spacetime, concentrated enough to be recognizable, but without being reduced to a point with no extension. This choice makes the theory more coherent with the very idea of a minimum interval: if a resolution threshold exists, physical events must be described through resolved profiles, not as infinitely punctual entities.
Distributions therefore serve to hold together two requirements. On one side we preserve the mathematics of the continuum, with functions, integrals, operators, and propagators. On the other side we assign finite resolution to physical events, avoiding the confusion between the geometric point and the real event. This step is decisive because it allows the theory to preserve ordinary mathematical tools while introducing a new physical threshold into the domain of admitted configurations.
The question “why this theory?” then has a precise answer. Because it offers a way to introduce a new universal constant without breaking spacetime into a grid. Because it applies the limit to a relativistically meaningful quantity, the invariant interval. Because it preserves mathematical continuity, indispensable to theoretical physics, and moves the restriction onto the level of physically resolved events. Because it distinguishes the ideal point from the measurable event. Because it allows distributions and localized profiles to be used as a bridge between the mathematical continuum and a possible physical granularity.
The theory remains speculative. It does not arise as an already established experimental result and does not claim to replace consolidated theories. Its initial value lies in the logic of the framework: it takes seriously the idea of a universal length constant, places it in relativistic spacetime, avoids a rigid discretization of coordinates, and proposes a constraint on physically resolved intervals. In this sense, it is a proposal for theoretical organization: simple in principle, cautious in mathematics, open to more technical developments.
The hypotheses are therefore three. There exists a universal scale ℓ₀. This scale concerns the invariant spacetime interval between physically resolved events. Physical events, to be described coherently with that scale, must be treated as localized profiles or distributions, rather than as absolute infinitesimal points.
From these hypotheses the whole framework arises: a continuous spacetime as mathematical background, a physics of resolved events as selected domain, a minimum interval as invariant threshold, and a distributional formalism as the natural language for connecting the mathematical ideal to physical resolution.
