The concept of number is fundamental to the formulation of any physical theory. We give a heuristic motivation for the reformulation of Quantum Mechanics in terms of non-standard real numbers called Quantum Real Numbers. The standard axioms of quantum mechanics are re-interpreted. Our aim is to show that, when formulated in the language of quantum real numbers, the laws of quantum mechanics appear more natural, less counterintuitive than when they are presented in terms of standard numbers.
The concept of number is fundamental to the formulation of any physical theory. We give a heuristic motivation for the reformulation of Quantum Mechanics in terms of non-standard real numbers called Quantum Real Numbers. The standard axioms of quantum mechanics are re-interpreted. Our aim is to show that, when formulated in the language of quantum real numbers, the laws of quantum mechanics appear more natural, less counterintuitive than when they are presented in terms of standard numbers.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
In this paper, we will see that we can reformulate the purely classical probability theory, using similar language to the one used in quantum mechanics. This leads us to reformulate quantum mechanics itself using this different way of understanding probability theory, which in turn will yield a new interpretation of quantum mechanics. In this reformulation, we still prove the existence of none classical phenomena of quantum mechanics, such as quantum superposition, quantum entanglement, the uncertainty principle and the collapse of the wave packet. But, here, we provide a different interpretation of these phenomena of how it is used to be understood in quantum physics. The advantages of this formulation and interpretation are that it induces the same experimental results, solves the measurement problem, and reduces the number of axioms in quantum mechanics. Besides, it suggests that we can use new types of Q-bits which are more easily to manipulate.
Quantum theory has created a fundamental change in our ideas as relativity theory had done before. But while relativity theory is essentially classical in its conceptions—it is in fact the culmination of classical physics—quantum mechanics is founded on some new principles. The quantum laws are very different in character from the laws we find in classical theory. In particular, quantum mechanics is a strictly statistical type of theory : from this arises the different conception of causality that marks the new outlook of modern physics. Classical theory describes the so-called macroscopic phenomena, that is, things and events which are more or less of the same size and kind as we find them in ordinary life; and so a description can be given in terms of concepts close to the ones we use in ordinary language. The language of quantum mechanics is based upon a duality principle ; and most of our ordinary language is equally dualistic.
The theory of quantum mechanics is examined using non-standard real numbers, called quantum real numbers (qr-numbers), that are constructed from standard Hilbert space entities. Our goal is to resolve some of the paradoxical features of the standard theory by providing the physical attributes of quantum systems with numerical values that are Dedekind real numbers in the topos of sheaves on the state space of the quantum system. The measured standard real number values of a physical attribute are then obtained as constant qr-number approximations to variable qr-numbers. Considered as attributes, the spatial locations of massive quantum particles form non-classical spatial continua in which a single particle can have a quantum trajectory which passes through two classically separated slits and the two particles in the Bohm-Bell experiment stay close to each other in quantum space so that Einstein locality is retained.
This work presents an elegant and conceptually clean geometric bridge between Pauli algebra and the topology of the Hopf fibration. It shows that the well-known Bell singlet correlation E(a, b) = −a·b is not an additional mystery of quantum theory, but is already fully encoded in the standard spinor geometry S³ → S² and in the non-commutative structure of SU(2). The main result can be understood as a representation statement: the Bell correlation admits a natural formulation as the real part of a U(1) holonomy on the Hopf bundle, namely E(a, b) = Re exp(i(π − θ)), where θ is the angle between measurement directions on the Bloch sphere. This holonomy arises from the Berry–Pancharatnam phase defined on the projective space CP¹ ≃ S², and depends only on the associated oriented solid angle. Importantly, this work does not introduce new physical laws and does not modify quantum mechanics. Instead, it demonstrates that the standard result E = −a·b already admits a rigorous geometric interpretation in terms of connections, curvature, and parallel transport. In this sense, the contribution is not a new theory, but a precise geometric reformulation of existing quantum structure. A key clarification concerns interpretation. The non-abelian SU(2) structure appearing here is not a hidden variable in the sense of Bell’s theorem. It describes the geometry of the quantum state space S³, not a set of pre-existing measurement outcomes. The reduction to U(1) reflects the fact that projective measurements access only equivalence classes on S², discarding the global phase. Accordingly, the violation of Bell inequalities arises from non-commutativity of observables, not from hidden parameters or signalling. The value of the work lies in structural unification. It brings together three standard but often separately presented elements: (i) the algebraic form of entangled correlations, (ii) the geometry of spinor space S³ and the Bloch sphere S², and (iii) the holonomy/phase nature of quantum quantities. In this formulation, Bell correlations appear not only as probabilistic objects, but as global geometric features of the underlying state space. This perspective is particularly useful for geometric quantum mechanics, quantum information theory, and pedagogy, where it provides a clear and rigorous insight into the origin of the cosine correlation law. At the same time, the scope is deliberately modest: the construction is fully consistent with standard quantum theory and does not attempt to extend it. The main risk lies in interpretational overreach. While the geometric picture is compelling, it should not be mistaken for an alternative to quantum mechanics or for a hidden-variable completion. Rather, it reveals an internal geometric layer that is already present in the theory. In summary, this work represents a high-quality incremental advance in quantum foundations: not a paradigm shift, but a compact and illuminating reconstruction of a fundamental result in the language of topology and geometry.