From the analysis of the measurement process we make the hypothesis that we have to add to the quantum state psi a label z and a special function alpha in order to describe completely the preparation of a (pure) quantum system . Given such a terne every observable A receives a univocal value . Therefore all physical quantities of the system are well defined independently from the measuring process. Making the unknown parameter z to vary in its space we get the probabilities prescribed by quantum mechanics.We show also that every evolution of quantum states can be extended to an evolution of the ternes.
From the analysis of the measurement process we make the hypothesis that we have to add to the quantum state psi a label z and a special function alpha in order to describe completely the preparation of a (pure) quantum system . Given such a terne every observable A receives a univocal value . Therefore all physical quantities of the system are well defined independently from the measuring process. Making the unknown parameter z to vary in its space we get the probabilities prescribed by quantum mechanics.We show also that every evolution of quantum states can be extended to an evolution of the ternes.
Every quantum physical system can be considered the ''shadow'' of a special kind of classical system. The system proposed here is classical mainly because each observable function has a well precise value on each state of the system: an hypothetical observer able to prepare the system exactly in an assigned state and able to build a measuring apparatus perfectly corresponding to a required observable gets always the same real value. The same system considered instead by an unexpert observer, affected by the ignorance of a hidden variable, is described by a statistical theory giving exactly and without exception the states, the observables, the dynamics and the probabilities prescribed for the usual quantum system.
The Question of the Completeness of Quantum Mechanics When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory. Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level. The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.
In this article we define and build the one method of description of stochastic evolution of a physical quantum system.For each quantum state U E ω ∈ we construct the probability measure ω µ in the space ( ) , U P S , where U P is the space of the pure states of the quantum system, S the Borel σ -algebra in U P .Farther, for any Hermit's positive element with norm 1 u = , in the C * -algebra of observables U , we define the probability measure u µ on the set of states U E .If strongly conti- nuous group { } t α of * automorphisms on U describes the evolution of struc- ture of observables, according to this, we have a picture of evolution of distribution of states of quantum system relatively to each observable u .
Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state η where η does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [Qsτ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement ∀Qsτ [U(Qsτ) defined⇒E(H,U(Qsτ),ψQsτ)],where U(Qsτ) is the probability measure U associates with the infinite instruction string Qsτ and ψQsτ is the outcome sequence obtained by carrying out Qsτ. E(H, U(Qsτ),ψQsτ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(Qsτ) are true for ψQsτ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(Qsτ) corresponds to a nontrivial product probability measure and U H-agrees with experiment, then the outcome sequence ψQsτ is H-random. H-randomness is defined here in terms of the statement E(H, P, φ). Another property of a valid physical theory, which is defined here, is that, for some Qsτ, U(Qsτ) must be determinable on much of AH from ψQsτ. Sufficient conditions for this property to hold are given. AH is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, φ) are given. Among other things, it is proved that, if E(H, P, φ) holds and P is a nontrivial probability measure on A, then φ is not definable in H.
We develop the observer function as a context-dependent process connecting continuous preparations to distinguishable records. The central mathematical result remains conditional: separated asymptotically stable outcomes produce a discrete basin quotient on the regular domain, while finite registration requires additional preparation and settling conditions. We extend that foundation with an integrated switched receiver whose preparation probabilities, fidelity, sufficient registration time, and disturbance margin follow from one specified law. A thermal version derives the outcome kernel and finite-horizon retention probabilities from an overdamped diffusion. Acceptance sensitivity and a declared predictive test expose the dependence on readout resolution and preparation. This integrated error budget belongs to one maintained-source receiver class; it does not validate a universal observer law. The detailed state space remains continuous. Scope. We establish conditional mathematical mechanisms and one physically interpretable receiver class. The binary source target is declared in advance. The calculations do not constitute experimental validation, a derivation of fundamental quantization, a quantum measurement theory, or a universal law of consciousness. Candidate definitions complete structures left open in the source manuscripts. Standard mathematical results and methods are organized into explicit proofs and reproducible model calculations; they are not claimed as new mathematical discoveries. The contribution is a conditional construction and its quantitative limits within that declared model. Universal physical scope and a fundamental-continuum ontology remain research hypotheses. In simple terms, the paper asks: How can a continuous system produce clear, discrete records (like “this” versus “that”) without the underlying state itself becoming discrete? It treats observation as a process that starts with a continuous preparation (a starting condition that can vary smoothly) and ends with a distinguishable record. The main mathematical claim is conditional and limited: - If a dynamical system has a few well-separated, asymptotically stable outcomes (think of two valleys a ball can roll into and stay in), then the continuous state space can be divided into open basins that lead to those different outcomes. The *labels* of those basins form a discrete structure (a discrete quotient), even though every individual trajectory stays continuous and the full state never jumps. - Getting a usable record in finite time, and knowing how reliable it is, requires extra conditions: the preparation has to be good enough, the system has to settle fast enough, and you need margins against disturbances. Attraction alone is not enough. The paper then builds one concrete example of this a “switched receiver.” Imagine a simple overdamped device (like a particle in a double-well potential) that is temporarily coupled to a source (the thing being observed), then disconnected so it can settle. From a single specified law the paper derives: the probabilities that a random preparation ends up correct, wrong, or still unresolved by a deadline, a sufficient time and disturbance margin that guarantees correct registration for every allowed preparation (in the deterministic, bounded-forcing case), and, when thermal noise is added, an outcome probability kernel plus retention probabilities over finite horizons. It shows where weaker claims fail. For example, two states that both get the same accepted label can still respond very differently to a later probe, so the label alone is not always a sufficient description. A wider acceptance window improves the numbers, but only because you relaxed the criterion, not because the dynamics improved. And there are simple counterexamples where a discrete label persists without the system settling to an attracting equilibrium. It does not derive fundamental quantization, a general quantum measurement theory, a universal observer law, or anything about consciousness. The detailed state remains continuous. The results hold inside one declared model class (a maintained-source cubic receiver). Extending them to freely evolving sources, higher dimensions, or other physical systems requires new derivations. Universality stays a research hypothesis. In short: the paper supplies a precise, limited mathematical mechanism by which continuous dynamics can generate discrete records under stated conditions, works out the numbers for one realistic receiver class (including noise and retention), and carefully lists what remains open. It is a conditional construction with quantitative limits, not a universal derivation. Explainer video Watch on YouTube (Roots of Reality): https://youtu.be/HJ9AYugKj_0 The video, its thumbnail and description are included in this record.