Quantum mechanics (QM) provides a variety of ideas that can assist in developing Artificial Intelligence for healthcare, and opens the possibility of developing a unified system of Best Practice for inference that will embrace both QM and classical inference. Of particular interest is inference in the hyperbolic-complex plane, the counterpart of the normal i-complex plane of basic QM. There are two reasons. First, QM appears to rotate from i-complex Hilbert space to hyperbolic-complex descriptions when observations are made on wave functions as particles, yielding classical results, and classical laws of probability manipulation (e.g. the law of composition of probabilities) then hold, whereas in the i-complex plane they do not. Second, i-complex Hilbert space is not the whole story in physics. Hyperbolic complex planes arise in extension from the Dirac-Clifford calculus to particle physics, in relativistic correction thereby, and in regard to spinors and twisters. Generalization of these forms resemble grammatical constructions and promote the idea that probability-weighted algebraic elements can be used to hold dimensions of syntactic and semantic meaning. It is also starting to look as though when a solution is reached by an inference system in the hyperbolic-complex, the hyperbolic-imaginary values disappear, while conversely hyperbolic-imaginary values are associated with the un-queried state of a system and goal seeking behavior.
The aim of the present paper is to introduce and to discuss the most basic fundamental concepts of quantum physics by means of a simple and pedagogical example. An appreciable part of its content presents original results. We start with the Euclidean plane which is certainly a paradigmatic example of a Hilbert space. The pure states form the unit circle (actually a half of it), the mixed states form the unit disk (actually a half of it), and rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed and open systems. The set of pure states or a set of mixed states solve the identity and they are used for understanding the concept of integral quantization of functions on the unit circle and to give a semi-classical portrait of quantum observables. Interesting probabilistic aspects are developed. Since the tensor product of two planes, their direct sum, their cartesian product, are isomorphic (2 is the unique solution to x^x= x\times x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), we describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU
(2). We explain the most general form of the Hamiltonian in the real plane by considering the integral quantization of a magnetic-like interaction potential viewed as a classical observable on the unit
2-sphere. Finally, we present an example of quantum measurement with pointer states lying also in the Euclidean plane.
This work reformulates quantum mechanics on the real plane
(R2), establishing an algebraic isomorphism to the complex field
(C) and demonstrating that the complex structure inherent to the Schrödinger equation can be fully captured by coupled real-valued dynamics. By constructing a field
(R2,+,⋅) isomorphic to
C, the real counterpart of the Schrödinger equation is derived, yielding a system of coupled real partial differential equations. This system reveals a symplectic structure where the symplectic matrix
J geometrically represents the imaginary unit
i, acting as a phase-space rotation operator. Time-independent solutions decouple into identical real Helmholtz equations, preserving spectral equivalence. The reformulation provides geometric insights into foundational aspects: the geometric representation of
i, the emergence of probability conservation from real-component interactions, and the encoding of quantum phase relationships through the multiplicative structure of the
R2 field. This perspective reinforces connections to classical Hamiltonian mechanics and geometric quantization, suggesting that complex numbers are structural, emerging from field operations on the real plane, rather than fundamental. Furthermore, quantum computation elements like qubits and gates are shown to have equivalent representations as real symplectic and special orthogonal matrices, maintaining unitary equivalence. We introduce a framework for quantum computation within a real vector space and provide a generalized protocol for generating entangled Bell states. These results offer a new perspective on the necessity of complex numbers in quantum theory, providing a framework with potential computational advantages for hybrid quantum-classical systems and enhanced geometric interpretability.
Quantum computing is no longer a far-off science fiction fantasy; it is an aggressively developing area that has the potential to change all areas of life, including medicine and cryptography. With the increasing availability of quantum computers, the shortage of resources to connect theoretical ideas and practice has taken a central position. This book is an attempt to bridge this gap. The journey of the fundamentals and real-world applications of quantum computing this book offers is an obligatory process on the way to exploring this fast-growing sector. We started by establishing the mathematical basis that we needed to know about quantum mechanics, including complex numbers and matrices and the density matrix. This understanding would be the foundational basis upon which the qubit, the basic unit of quantum information and the potent concepts of superposition and entanglement that provides quantum systems with their inherent advantage, can be studied. We have now equipped the reader with the essentials to build and control the quantum states by an exhaustive description of the quantum gates, and circuits. The book finishes off with a real-world discussion on groundbreaking quantum algorithms such as those of Shor and Grover, and how quantum computation provides fundamentally more efficient solutions to complex problems. Lastly, looking into the future, we pointed out the radical possibilities of this technology in various industries, including smart healthcare and cryptography, machine learning and financial analysis. Looking ahead, this book is more practice than theory, and it is meant to help fill the gap between theory and practice. Through these ideas in Python, readers can not only now see the quantum revolution but also have a role to play in its future. The understanding of these concepts and algorithms can provide a critical footing for contribution to the next big advance in the field of computational science.
Optical science and engineering have been playing key enabling roles behind the continuous advancement of physics since ancient times. We have to pick up that role again. Recent publications by physicists Smolin [1], “Trouble with Physics”, 2007 and Hossenfelder [2], “Lost in Math”, 2018, are deeply troubling for the future of Physics and our credibility as the gatekeeper of the enabling knowledge of Optical Physics. We will explain how the various erroneous interpretations related to Relativity (e.g. time dilation), to Quantum Mechanics (Superposition Principle) and Astrophysics (Holographic Universe) arise due to the (i) use of non-causal Fourier modes (existing in all space) as tools of nature’s reality and (ii) neglect of Non-Interaction of Waves (NIW). NIW indicates that simple superposition of Fourier modes, in the absence of interacting materials, cannot generate spatio-temporal re-grouping of wave energies. System engineering thinking will be underscored that can constrain our math models to stick to realities. We will present examples on how understanding NIW in optical phenomena can help us bring out causal interpretations and causal models in some of the key branches of Physics.