We shall argue in this paper that a central piece of modern physics does not really belong to physics at all but to elementary probability theory. Given a joint probability distribution J on a set of random variables containing x and y, define a link between x and y to be the condition x=y on J. Define the {\it state} D of a link x=y as the joint probability distribution matrix on x and y without the link. The two core laws of quantum mechanics are the Born probability rule, and the unitary dynamical law whose best known form is the Schrodinger's equation. Von Neumann formulated these two laws in the language of Hilbert space as prob(P) = trace(PD) and D'T = TD respectively, where P is a projection, D and D' are (von Neumann) density matrices, and T is a unitary transformation. We'll see that if we regard link states as density matrices, the algebraic forms of these two core laws occur as completely general theorems about links. When we extend probability theory by allowing cases to count negatively, we find that the Hilbert space framework of quantum mechanics proper emerges from the assumption that all D's are symmetrical in rows and columns. On the other hand, Markovian systems emerge when we assume that one of every linked variable pair has a uniform probability distribution. By representing quantum and Markovian structure in this way, we see clearly both how they differ, and also how they can coexist in natural harmony with each other, as they must in quantum measurement, which we'll examine in some detail. Looking beyond quantum mechanics, we see how both structures have their special places in a much larger continuum of formal systems that we have yet to look for in nature.
We shall argue in this paper that a central piece of modern physics does not really belong to physics at all but to elementary probability theory. Given a joint probability distribution J on a set of random variables containing x and y, define a link between x and y to be the condition x=y on J. Define the {\it state} D of a link x=y as the joint probability distribution matrix on x and y without the link. The two core laws of quantum mechanics are the Born probability rule, and the unitary dynamical law whose best known form is the Schrodinger's equation. Von Neumann formulated these two laws in the language of Hilbert space as prob(P) = trace(PD) and D'T = TD respectively, where P is a projection, D and D' are (von Neumann) density matrices, and T is a unitary transformation. We'll see that if we regard link states as density matrices, the algebraic forms of these two core laws occur as completely general theorems about links. When we extend probability theory by allowing cases to count negatively, we find that the Hilbert space framework of quantum mechanics proper emerges from the assumption that all D's are symmetrical in rows and columns. On the other hand, Markovian systems emerge when we assume that one of every linked variable pair has a uniform probability distribution. By representing quantum and Markovian structure in this way, we see clearly both how they differ, and also how they can coexist in natural harmony with each other, as they must in quantum measurement, which we'll examine in some detail. Looking beyond quantum mechanics, we see how both structures have their special places in a much larger continuum of formal systems that we have yet to look for in nature.
Quantum theory is the soul of contemporary physics. It was discovered in an adventurous way, under the urge to solve the puzzles posed by atomic spectra and blackbody radiation. But after its invention, it immediately became clear that it was not just a theory of specific physical systems: it was rather a new language of universal applicability. Already in 1928, the theory had received solid mathematical foundations by Hilbert, von Neumann, and Nordheim, and this work was brought to completion in the monumental work of von Neumann, in the form that we still study nowadays. The theory is extraordinarily successful, and its predictions have been confirmed to an astonishing level of precision in a large spectrum of experiments. However, almost 90 years after von Neumann's book, quantum theory remains mysterious. Its mathematical formulation – based on Hilbert spaces and self-adjoint operators – is far from having an intuitive interpretation. The association of physical systems to Hilbert spaces whose unit vectors represent pure states, the representation of transformations by unitary operators and of observables by self-adjoint operators – all such postulates look artificial and ad hoc . A slightly more operational approach is provided by the C * -algebraic formulation of quantum theory – still, this formulation relies on the assumption that observables form an algebra , where the physical meanings of the multiplication and the sum are far from clear. In short, the postulates of quantum theory impose mathematical structures without providing any simple reason for this choice: the mathematics of Hilbert spaces is adopted as a magic blackbox that “works well” at producing experimental predictions. However, in a satisfactory axiomatization of a physical theory the mathematical structures should emerge as a consequence of postulates that have a direct physical interpretation. By this we mean postulates referring, e.g., to primitive notions like physical system, measurement, or process, rather than notions like, e.g., Hilbert space, C * -algebra, unit vector, or selfadjoint operator. The crucial question thus remains unanswered: why quantum theory? Which are the principles at the basis of the theory? A case that is often invoked in contrast is that of Special Relativity theory, which directly follows from the simple understandable principle of relativity.
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.
As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.
It is shown that quantum mechanics is, like thermodynamics, a phenomenological theory i.e., not a causal theory, ( not because it is a statistical theory - statistical theories with caused probability distributions can be regarded as causal) but because pure states, i.e., probability distributions of measurement values, cannot inhere in elementary particles and therefore cannot change when their world tubes intersect and hence they cannot be regarded as interacting causally. By a causal theory is meant a theory that specifies the changes in time of the states of causally interacting entities in its domain. The areas in quantum mechanics in which causal interactions are relevant include, though not explicitly, measurement and therefore the Born rule, and, explicitly, the unitary Schrodinger time development of states. The Born rule probabilities are shown to to refer not to conjoint superpositions of eigenstates but to classical mixtures of mutually exclusive eigenvalues and the Schrodinger time development of states is shown to refer to the time development of the states of non-causally interacting elementary particles and hence cannot be regarded as as a causal time development equation, appearances to the contrary notwithstanding. The recognition that quantum mechanics is not a causal theory but a phenomenological theory like thermodynamics does not affect the way it is employed to calculate an predict and hence preserves its empirical success but it does allow a typically simple phenomenological theory explanation of entanglement and other apparently non-local phenomena.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.
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.