The multiparticle spacetime algebra (MSTA) is an extension of Dirac theory to a multiparticle setting, which was first studied by Doran, Gull and Lasenby. The geometric interpretation of this algebra, which it inherits from its one-particle factors, possesses a number of physically compelling features, including simple derivations of the Pauli exclusion principle and other nonlocal effects in quantum physics. Of particular importance here is the fact that all the operations needed in the quantum (statistical) mechanics of spin 1/2 particles can be carried out in the “even subalgebra” of the MSTA. This enables us to “lift” existing results in quantum information theory regarding entanglement, decoherence and the quantum / classical transition to spacetime. The full power of the MSTA and its geometric interpretation can then be used to obtain new insights into these foundational issues in quantum theory. A system of spin 1/2 particles located at fixed positions in space, and interacting with an external magnetic field and/or with one another via their intrinsic magnetic dipoles provides a simple paradigm for the study of these issues. This paradigm can further be easily realized and studied in the laboratory by nuclear magnetic resonance spectroscopy.
The multiparticle spacetime algebra (MSTA) is an extension of Dirac theory to a multiparticle setting, which was first studied by Doran, Gull and Lasenby. The geometric interpretation of this algebra, which it inherits from its one-particle factors, possesses a number of physically compelling features, including simple derivations of the Pauli exclusion principle and other nonlocal effects in quantum physics. Of particular importance here is the fact that all the operations needed in the quantum (statistical) mechanics of spin 1/2 particles can be carried out in the “even subalgebra” of the MSTA. This enables us to “lift” existing results in quantum information theory regarding entanglement, decoherence and the quantum / classical transition to spacetime. The full power of the MSTA and its geometric interpretation can then be used to obtain new insights into these foundational issues in quantum theory. A system of spin 1/2 particles located at fixed positions in space, and interacting with an external magnetic field and/or with one another via their intrinsic magnetic dipoles provides a simple paradigm for the study of these issues. This paradigm can further be easily realized and studied in the laboratory by nuclear magnetic resonance spectroscopy.
The multi particle spacetime algebra (MSTA) is an extension of Dirac theory to a multi particle setting, which was first studied by Doran, Gull and Lasenby. The geometric interpretation of this algebra, which it inherits from its one-particle factors, possesses a number of physically com pelling features, including simple derivations of the Pauli exclusion princi ple and other nonlocal effects in quantum physics. Of particular importance here is the fact that all the operations needed in the quantum (statistical) mechanics of spin 1/2 particles can be carried out in the even subalge bra of the MSTA. This enables us to lift existing results in quantum information theory regarding entanglement, de coherence and the quantum / classical transition to spacetime. The full power of the MSTA and its geometric interpretation can then be used to obtain new insights into these foundational issues in quantum theory. A system of spin 1/2 particles lo cated at fixed positions in space, and interacting with an external magnetic field and/or with one another via their intrinsic magnetic dipoles provides a simple paradigm for the study of these issues. This paradigm can fur ther be easily realized and studied in the laboratory by nuclear magnetic resonance spectroscopy.
In this chapter we study the application of geometric algebra to both nonrelativistic and relativistic quantum mechanics. We concentrate on the quantum theory of spin-1/2 particles, whose dynamics is described by the Pauli and Dirac equations. For interactions where spin and relativity are not important the dynamics reduces to that of the Schrödinger equation. There are many good textbooks describing this topic and we will make no attempt to cover it here. We assume, furthermore, that most readers have a basic understanding of quantum mechanics, and are familiar with the concepts of states and operators. Both the Pauli and Dirac matrices arise naturally as representations of the geometric algebras of space and spacetime. It is no surprise, then, that much of quantum theory finds a natural expression within geometric algebra. To achieve this, however, one must reconsider the standard interpretation of the quantum spin operators. Like much discussion of the interpretation of quantum theory, certain issues raised here are controversial. There is no question about the validity of our algebraic approach, however, and little doubt about its advantages. Whether the algebraic simplifications obtained here are indicative of a deeper structure embedded in quantum mechanics is an open question. In this chapter we only consider the quantum theory of single particles in background fields. Multiparticle systems are considered in the following chapter.
Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin’s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.
This work presents a focused theoretical extension of the Time-Delay Field (TDF) framework, previously introduced in “A Theory of Everything via the Five-Dimensional Delay Field Model” (Masarrat-Bakhsh, 2025). The present paper applies the TDF approach to the physical interpretation of spin, offering a complementary perspective to the standard Dirac formulation. Within the TDF framework, spacetime is endowed with intrinsic temporal memory, modeled through an effective time-delay field arising from a higher-dimensional (5D) structure. Spin is reinterpreted not as a purely algebraic or intrinsic quantum number, but as an emergent dynamical feature associated with delayed phase responses in non-Markovian field evolution. The characteristic spin-1/2 behavior and 4π rotational periodicity are shown to arise naturally from this delay-induced phase structure. The standard Dirac theory and its successful predictions are fully recovered in the limit of vanishing delay, ensuring consistency with established relativistic quantum mechanics. This work does not propose a replacement of the Dirac framework, but rather provides a causal reinterpretation that may offer deeper insight into the origin of spin and spin–orbit coupling. The paper is intended as a conceptual and foundational contribution, suitable for discussions in theoretical physics, quantum foundations, and spacetime-based interpretations of quantum phenomena.
The algebraic approach to physical theories provides a general framework encompassing both classical and quantum mechanics. Accordingly, by looking at the behaviour of the relevant algebras of observables one can investigate structural and conceptual differences between the theories. Interesting foundational questions can be formulated algebraically and their answers are then given in a mathematically compelling way. My dissertation focuses on some philosophical issues concerning entanglement and quantum information as they arise in the algebraic. These two concepts are connected in that one can exploit the non-local character of quantum theory to construct protocols of information theory which are not realized in the classical world. I first introduce the basic concepts of the algebraic formalism, by reviewing von Neumann's work on the mathematical foundations of quantum theories. After presenting the reasons why von Neumann abandoned the standard Hilbert space formulation in favour of the algebraic approach, I show how his axiomatic program remained a mathematical "utopia" in mathematical physics. The Bayesian interpretation of quantum mechanics is grounded in information-theoretical considerations. I take on some specific problems concerning the extension of Bayesian statistical inference in infinite dimensional Hilbert space. I demonstrate that the failure of a stability condition, formulated as a rationality constraint for classical Bayesian conditional probabilities, does not undermine the Bayesian interpretation of quantum probabilities. I then provide a solution to the problem of Bayesian noncommutative statistical inference in general probability spaces. Furthermore, I propose a derivation of the a priori probability state in quantum mechanics from symmetry considerations. Finally, Algebraic Quantum Field Theory offers a rigorous axiomatization of quantum field theory, namely the synthesis of quantum mechanics and special relativity. In such a framework one can raise the question of whether or not quantum correlations are made stronger by adding relativistic constraints. I argue that entanglement is more robust in the relativistic context than in ordinary quantum theory. In particular, I show how to generalize the claim that entanglement across space-like separated regions of Minkowski spacetime would persist, no matter how one acts locally.