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Nova Science Publishers eBooks

Space-time geometry and quantum events

Ignazio Francesco Licata

Published Jan 1, 2014• 10 citations
OpenAlex

Abstract

Preface Space-Time Geometry & Quantum Events The Transaction as a Quantum Concept Stochastic Foundation of Quantum Mechanics & the Origin of Particle Spin Fractal Space-Time as an Underlying Structure of the Standard Model Relativity of Scales, Fractal Space & Quantum Potentials Three Possible Implications of Space-Time Discreteness Quantum Computing Space-Time Metric Gauge Fields in Deformed Special Relativity Non-Local Granular Space-Time Foam as an Ultimate Arena at the Planck Scale Structural Approach to the Elementary Particle Theory Forming Physical Fields & Pseudometric & Metric Manifolds. Noncommutativity & Discrete Structures in Classical & Quantum Physics Index.

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PROSPECTIVE The concern of this paper will be different from that of most studies of quantum gravity. There will be no discussion of the problem of quantizing the gravitational field equations, hence nothing about the modifications which quantization requires of space-time structure at distances of the order of the Planck length (∼ 10 −33 cm). The emphasis will be rather upon the implications of quantum mechanics for certain general properties of events and processes which occur in the theatre of space-time, specifically with non-locality and nonlinearity . Quantum mechanics is undoubtedly a non-local theory when it treats correlated spatially separated systems. When one examines closely the character of this non-locality, however, one does not find reasons for modifying the causal structure of space-time as described by special or general relativity theory, but rather reasons for refining the concept of an event. The occurrence of definite outcomes of measurements implies that there are processes in nature governed by nonlinear laws. Nonlinearity is very peculiar from the standpoint of quantum mechanics, since it is contrary to the linearity of the time-dependent Schrödinger equation, which is commonly assumed to govern the dynamics of any isolated physical system. Hence it is necessary to inquire whether a rational treatment of the occurrence of outcomes is possible without some modifications of current quantum mechanics. Whatever decision is reached on this question, however, it does not appear to bear directly on space-time structure, but rather on the general character of processes.

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The possibility that quantum mechanics is foundationally the same as classical theories in explaining phenomena in space and time is postulated. Such a view is motivated by interpreting the experimental violation of Bell inequalities as resulting from questions of geometry and algebraic representation of variables, and thereby the structure of space, rather than realism or locality. While time remains Euclidean in the proposed new structure, space is described by Projective geometry. A dual geometry facilitates description of a physically real quantum particle trajectory. Implications for the physical basis of Bohmian mechanics is briefly examined, and found that the hidden variables pilot-wave model is local. Conceptually, the consequence of this proposal is that quantum mechanics has common ground with relativity as ultimately geometrical. This permits the derivation of physically meaningful quantum Lorentz transformations. Departure from classical notions of measurability is discussed.

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A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. Time and space are treated on exactly the same footing, namely as quantum mechanical observables on a Hilbert space. The theory is not based upon Lagrangian or Hamiltonian mechanics and cannot be formulated within the framework of unitary dynamics. As a most important aspect, the theory breaks with the concept of a continuously flowing time in favour of a discrete jump process in spacetime. All physical statements are formulated in terms of detector events rather than particle states. The physical object under consideration is a spinless particle in empty space. Yet the theory also accounts for particle-antiparticle pair creation and annihilation, and is therefore not a single-particle theory in the strict sense. The Maxwell equations are derived as a straightforward consequence of some fundamental commutation relations. In the non-relativistic limit, and in the limit of infinitely small time uncertainties, one obtains the familiar Schr\\"odinger equation of a spinless particle exposed to an external electromagnetic field.

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The Geometry of Pure Quantum States: Kähler Structures, Fubini–Study Geodesics, and Covariant Tracking Mechanics

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This paper establishes the foundational differential-geometric architecture for the coordinate-independent tracking of pure quantum states within closed physical systems, validated via explicit computational modeling of a driven two-level system. While standard quantum mechanics relies on the linear algebraic structure of complex Hilbert spaces, the unique identification of physical states requires factoring out non-zero complex scaling factors, positioning the true arena of quantum dynamics as the projective Hilbert space P(H). We analyze this space as a compact complex manifold possessing an intrinsic Kähler structure that fuses a Riemannian metric tensor, a symplectic form, and a complex structure into a unified geometric framework. Through this manifold lens, we systematically derive the Fubini–Study metric tensor gF S and its corresponding connection geometries. By evaluating the evolution of state vectors under a time-dependent driving Hamiltonian, we show that the natural propagation of a closed quantum system traces out a geodesic path on the projective manifold. This geometric tracking defines a localized vector gauge field—the Berry connection 1-form—which gives rise to an antisymmetric curvature tensor that modifies the effective classical mechanics of the parameter space. Furthermore, we state and prove three core theorems establishing the geometric constraints of pure state transport: the Holonomy Theorem for cyclic evolutions, the Quantum Cramér–Rao Bound, and the Mandelstam–Tamm Quantum Speed Limit. To bridge theory with physical architectures, we implement an analytical, parameter-shift algorithmic pipeline to extract these metrics natively. Numerical grid analysis confirms the exact structural scaling relation gQF IM = 4 · gF S down to machine precision (1.24 × 10−15). Finally, by tracking the metric volume form det(gij ) alongside the inverse metric components, we visually map out and isolate the topological caustic singularities (θ → 0, π) where the embedding map drops rank. This computational diagnostic provides a robust framework to identify and structurally bypass non-immersive control bottlenecks during the initial Hamiltonian design phase.

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Introduction: Conceptual issues in quantum field theory Part I. Philosophers' Interests in Quantum Field Theory: 1. Why are we philosophers interested in quantum field theory 2. Quantum field theory and the philosopher Part II. Three Approaches to the Foundations of Quantum Field Theory: 3. The usefulness of a general theory of quantized fields 4. Effective field theory in condensed matter physics 5. The triumph and limitations of quantum field theory 6. Comments Discussions Part III. Does Quantum Field Theory Need a Foundation: 7. Does quantum field theory need a foundation? Part IV. Mathematics, Statistical Mechanics and Quantum Field Theory: 8. Renormalization group theory: its basis and formulation in statistical physics 9. Where does quantum field theory fit into the big picture? 10. The unreasonable effectiveness of quantum field theory 11. Comment: the quantum field theory of physics and of mathematics Part V. Quantum Field Theory and Spacetime: Introduction 12. Quantum field theory and spacetime: formalism and reality 13. Quantum field theory of geometry 14. 'Localization' in quantum field theory: how much of QFT is compatible with what we know about spacetime 15. Comments VI. 16. What is quantum field theory and what did we think it was? 17. Comments Discussions Part VII.Renormalization Group: 18. What is fundamental physics? A renormalization group perspective 19. Renormalization group: an interesting yet puzzling idea Part VIII. Non-Abelian Gauge Theory: 20. Gauge fields, gravity and Bohm's theory 21. Is the Aharonov-Bohm effect local? Discussions Part IX. The Ontology of Particles or Fields: 22. The ineliminable classical face of quantum field theory 23. The logic of quanta 24. Do Feynman diagrams endorse a particle ontology? 25. On the ontology of QFT Part X. Panel Discussion.

•Mar 1, 2004•25 citations
Preface Space-Time Geometry & Quantum Events The Transaction as a Quantum Concept Stochastic Foundation of Quantum Mechanics & the Origin of Particle Spin Fractal Space-Time as an Underlying Structure of the Standard Model Relativity of Scales, Fractal Space & Quantum Potentials Three Possible Implications of Space-Time Discreteness Quantum Computing Space-Time Metric Gauge Fields in Deformed Special Relativity Non-Local Granular Space-Time Foam as an Ultimate Arena at the Planck Scale Structural Approach to the Elementary Particle Theory Forming Physical Fields & Pseudometric & Metric Manifolds. Noncommutativity & Discrete Structures in Classical & Quantum Physics Index.
PROSPECTIVE The concern of this paper will be different from that of most studies of quantum gravity. There will be no discussion of the problem of quantizing the gravitational field equations, hence nothing about the modifications which quantization requires of space-time structure at distances of the order of the Planck length (∼ 10 −33 cm). The emphasis will be rather upon the implications of quantum mechanics for certain general properties of events and processes which occur in the theatre of space-time, specifically with non-locality and nonlinearity . Quantum mechanics is undoubtedly a non-local theory when it treats correlated spatially separated systems. When one examines closely the character of this non-locality, however, one does not find reasons for modifying the causal structure of space-time as described by special or general relativity theory, but rather reasons for refining the concept of an event. The occurrence of definite outcomes of measurements implies that there are processes in nature governed by nonlinear laws. Nonlinearity is very peculiar from the standpoint of quantum mechanics, since it is contrary to the linearity of the time-dependent Schrödinger equation, which is commonly assumed to govern the dynamics of any isolated physical system. Hence it is necessary to inquire whether a rational treatment of the occurrence of outcomes is possible without some modifications of current quantum mechanics. Whatever decision is reached on this question, however, it does not appear to bear directly on space-time structure, but rather on the general character of processes.
The possibility that quantum mechanics is foundationally the same as classical theories in explaining phenomena in space and time is postulated. Such a view is motivated by interpreting the experimental violation of Bell inequalities as resulting from questions of geometry and algebraic representation of variables, and thereby the structure of space, rather than realism or locality. While time remains Euclidean in the proposed new structure, space is described by Projective geometry. A dual geometry facilitates description of a physically real quantum particle trajectory. Implications for the physical basis of Bohmian mechanics is briefly examined, and found that the hidden variables pilot-wave model is local. Conceptually, the consequence of this proposal is that quantum mechanics has common ground with relativity as ultimately geometrical. This permits the derivation of physically meaningful quantum Lorentz transformations. Departure from classical notions of measurability is discussed.
A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. Time and space are treated on exactly the same footing, namely as quantum mechanical observables on a Hilbert space. The theory is not based upon Lagrangian or Hamiltonian mechanics and cannot be formulated within the framework of unitary dynamics. As a most important aspect, the theory breaks with the concept of a continuously flowing time in favour of a discrete jump process in spacetime. All physical statements are formulated in terms of detector events rather than particle states. The physical object under consideration is a spinless particle in empty space. Yet the theory also accounts for particle-antiparticle pair creation and annihilation, and is therefore not a single-particle theory in the strict sense. The Maxwell equations are derived as a straightforward consequence of some fundamental commutation relations. In the non-relativistic limit, and in the limit of infinitely small time uncertainties, one obtains the familiar Schr\\"odinger equation of a spinless particle exposed to an external electromagnetic field.
This paper establishes the foundational differential-geometric architecture for the coordinate-independent tracking of pure quantum states within closed physical systems, validated via explicit computational modeling of a driven two-level system. While standard quantum mechanics relies on the linear algebraic structure of complex Hilbert spaces, the unique identification of physical states requires factoring out non-zero complex scaling factors, positioning the true arena of quantum dynamics as the projective Hilbert space P(H). We analyze this space as a compact complex manifold possessing an intrinsic Kähler structure that fuses a Riemannian metric tensor, a symplectic form, and a complex structure into a unified geometric framework. Through this manifold lens, we systematically derive the Fubini–Study metric tensor gF S and its corresponding connection geometries. By evaluating the evolution of state vectors under a time-dependent driving Hamiltonian, we show that the natural propagation of a closed quantum system traces out a geodesic path on the projective manifold. This geometric tracking defines a localized vector gauge field—the Berry connection 1-form—which gives rise to an antisymmetric curvature tensor that modifies the effective classical mechanics of the parameter space. Furthermore, we state and prove three core theorems establishing the geometric constraints of pure state transport: the Holonomy Theorem for cyclic evolutions, the Quantum Cramér–Rao Bound, and the Mandelstam–Tamm Quantum Speed Limit. To bridge theory with physical architectures, we implement an analytical, parameter-shift algorithmic pipeline to extract these metrics natively. Numerical grid analysis confirms the exact structural scaling relation gQF IM = 4 · gF S down to machine precision (1.24 × 10−15). Finally, by tracking the metric volume form det(gij ) alongside the inverse metric components, we visually map out and isolate the topological caustic singularities (θ → 0, π) where the embedding map drops rank. This computational diagnostic provides a robust framework to identify and structurally bypass non-immersive control bottlenecks during the initial Hamiltonian design phase.
Introduction: Conceptual issues in quantum field theory Part I. Philosophers' Interests in Quantum Field Theory: 1. Why are we philosophers interested in quantum field theory 2. Quantum field theory and the philosopher Part II. Three Approaches to the Foundations of Quantum Field Theory: 3. The usefulness of a general theory of quantized fields 4. Effective field theory in condensed matter physics 5. The triumph and limitations of quantum field theory 6. Comments Discussions Part III. Does Quantum Field Theory Need a Foundation: 7. Does quantum field theory need a foundation? Part IV. Mathematics, Statistical Mechanics and Quantum Field Theory: 8. Renormalization group theory: its basis and formulation in statistical physics 9. Where does quantum field theory fit into the big picture? 10. The unreasonable effectiveness of quantum field theory 11. Comment: the quantum field theory of physics and of mathematics Part V. Quantum Field Theory and Spacetime: Introduction 12. Quantum field theory and spacetime: formalism and reality 13. Quantum field theory of geometry 14. 'Localization' in quantum field theory: how much of QFT is compatible with what we know about spacetime 15. Comments VI. 16. What is quantum field theory and what did we think it was? 17. Comments Discussions Part VII.Renormalization Group: 18. What is fundamental physics? A renormalization group perspective 19. Renormalization group: an interesting yet puzzling idea Part VIII. Non-Abelian Gauge Theory: 20. Gauge fields, gravity and Bohm's theory 21. Is the Aharonov-Bohm effect local? Discussions Part IX. The Ontology of Particles or Fields: 22. The ineliminable classical face of quantum field theory 23. The logic of quanta 24. Do Feynman diagrams endorse a particle ontology? 25. On the ontology of QFT Part X. Panel Discussion.