We derive the Hilbert space formalism of quantum mechanics from epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. An epistemic formalism is developed, where concepts like individual and collective knowledge are used, and knowledge may be actual or potential. The physical state
S corresponds to the collective potential knowledge. The state
S is a subset of a state space
S={Z}, such that
S always contains several elements
Z, which correspond to unattainable states of complete potential knowledge of the world. The evolution of
S cannot be determined in terms of the individual evolution of the elements
Z, unlike the evolution of an ensemble in classical phase space. The evolution of
S is described in terms of sequential time
n∈N, which is updated according to
n→n+1 each time potential knowledge changes. In certain experimental contexts
C, there is initial knowledge at time
n that a given series of properties
P,P′,… will be observed within a given time frame, meaning that a series of values
p,p′,… of these properties will become known. At time
n, it is just known that these values belong to predefined, finite sets
{p},{p′},…. In such a context
C, it is possible to define a complex Hilbert space
HC on top of
S, in which the elements are contextual state vectors
SˉC. Born's rule to calculate the probabilities to find the values
p,p′,… is derived as the only generally applicable such rule. Also, we can associate a self-adjoint operator
Pˉ with eigenvalues
{p} to each property
P observed within
C. These operators obey
[Pˉ,Pˉ′]=0 if and only if the precise values of
P and
P′ are simultaneoulsy knowable.
Abstract Chapter 9 begins laying a foundation for quantum statistical mechanics with a discussion of relevant topics in quantum mechanics. The fundamental postulates of quantum mechanics are reviewed, beginning with the definition of a quantum state and the introduction of the concept of a state vector in a Hilbert space. The representation of observables as Hermitian operators on Hilbert space vectors is described; it is shown that the eigenvalues of Hermitian operators are real and, therefore, constitute possible outcomes of measurements made on these observables. The notions compatible and non-compatible observables and implication for the measurement problem are presented. The time-dependent Schrödinger equation is introduced as generating the time-evolution of the state vector, and it is shown how to cast the Schrödinger equation into a coordinate basis to yield the wave equation of quantum mechanics. Coordinate and momentum bases and their properties are desribed. Several analytically solvable quantum systems are presented as illustrative examples. The chapter concludes with a discussion of spin and the treatment identical particles
The purpose of the paper is to study the foundations of the main axioms of Quantum Mechanics. From a general study of the mathematical properties of the models used in Physics to represent systems, we prove that the states of a system can be represented in a Hilbert space, that a self-adjoint operator is associated to any observable, that the result of a measure must be the eigen value of the operator and appear with the usual probability. Furthermore an equivalent of the Wigner's theorem holds, which leads to the Schr{ö}dinger equation. These results are based on well known mathematics, and do not involve any specific hypothesis in Physics. They validate and explain the methods currently used, which are made simpler and safer, and open new developments. In the third edition of this paper developments have been added about the estimation of physical anomalies.
This article examines the scope and limitations of quantum mechanics in its Schrödinger and Dirac formulations. Quantum theory provides an extraordinarily precise description of how physical states evolve, interfere, and produce measurable probabilities. However, like other foundational theories, it presupposes a specific mathematical structure—complex Hilbert spaces, linear unitary dynamics, an external time parameter, and the Born rule—that it does not itself explain. The work argues that quantum mechanics answers the question of how quantum states behave, but not the deeper question of why physical reality admits a state-based, probabilistic description at all. To address this missing level of explanation, the article places quantum mechanics within the broader framework of the Thermodynamic Imperative of Order (TIO) and the Granular Entropic Program (GEP). Within this framework, quantum states are interpreted as emergent, statistically stable macroscopic representations of typical correlations in high-dimensional configuration spaces. Linearity, unitarity, and the Born rule arise as necessary consequences of measure concentration and stability, rather than as fundamental postulates. The article does not propose modifications to quantum mechanics. Instead, it clarifies the domain of validity of the theory and explains why a framework with its structure and remarkable universality must exist if stable quantum phenomena are to be observed at all.
In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution of the probabilities given by the Generalized Born's Rule. In other models, such as the Heisenberg model, the auxiliary equations are quite different, but the Generalized Born's Rule is the same formula (covariance) and gives the same results (invariance). Also some aspects of the Schrodinger model are not found in the isomorphic Heisenberg model, and they therefore do not have any physical significance. One example of this is the infamous collapse of the quantum state. Other quantum phenomena, such as entanglement, are easy to analyze in terms of the Generalized Born's Rule without any reference to the unnecessary concept of collapse. Finally, this leads to the possibility of quantum theory with other sorts of auxiliary equations instead of Schrodinger's equation, and examples of this are given. Throughout this treatise the leit motif is the central importance of quantum probability and most especially of the simplifying role of the time dependent Generalized Born's Rule in quantum theory.
We present the transformation-response framework, an operational reformulation of quantum mechanics. A quantum state is not a Hilbert space object but the catalog of a system's responses to all physical transformations: for each operation
g from the system's local group
G, an interference experiment gives a complex value
χ(g). The collection
{χ(g):g∈G} is the characteristic function and defines the state. The only postulate is that
χ is positive-definite, encoding the requirement that no superposition of transformations yields negative probability. From this single assumption, the entire standard formalism is derived: Hilbert space via GNS construction, Born rule via Bochner theorem, Schrödinger equation from group automorphisms, and especially Feynman path integral as a Trotter limit. The framework is background-independent and time-neutral: time is a coordinate along a one-parameter subgroup of
G. It also reveals a new physical constraint, product order positivity, which may lead to testable predictions. The framework provides a unified, economical, and falsifiable foundation for quantum theory rooted in operational primitives.