In a time-symmetric form, classical relativistic formulations involve not only a reference time but also relatively advanced and retarded times. This work examines a simple model problem having this structure with the equations of motion having closed-form solutions. The complexity of more realistic models is avoided while the general features are retained. The determinism and casuality of the theory are clearly demonstrated. Some problems associated with the quantization of power series representations using conventional procedures are illustrated and discussed.
In a time-symmetric form, classical relativistic formulations involve not only a reference time but also relatively advanced and retarded times. This work examines a simple model problem having this structure with the equations of motion having closed-form solutions. The complexity of more realistic models is avoided while the general features are retained. The determinism and casuality of the theory are clearly demonstrated. Some problems associated with the quantization of power series representations using conventional procedures are illustrated and discussed.
A model problem consisting of two isolated particles with a mutual interaction depending upon retarded and advanced positions and constrained to one-dimensional motion is analyzed herein. This system has a number of features in common with a previously considered one-particle double-delay model problem. The equations of motion have closed-form solutions and the systems are both deterministic and causal. In both cases, standard low-order approximations to the exact problem, in general, have spurious solutions, which must be recognized and removed in order to extract the physically meaningful parts and to proceed reasonably with quantization. In addition, the two-particle system has specific characteristics. The center-of-momentum motion separates out. The dependence of the solutions on the mass ratio of the two particles can be examined. Further, not only does the relative motion have the ordinary solution, in which the two particles move out of phase, it can also have an extraordinary solution, in which they move in phase with each other. Finally, the total generalized linear momentum and Hamiltonian can be evaluated and seen to be constants of the motion.
A model problem, initially proposed by Feynman and Hibbs, is re-examined in this work. It is shown that the equation of motion is deterministic, in spite of its double time delay character, and has chaoticlike solutions. It is further shown that this model is amenable to the methods of higher than first-order Lagrangians, yielding a series of approximations to the initial model. Thus, for each approximation, a complete st of canonical variables can be defined, the Hamiltonian can be written down in canonical form and the conventional quantization rules applied, leading to exact quantizations. Extended sets of basis vectors follow along with the definition of physical state vectors, which are seen to contain all of the classical information. Finally, these results are used to give a possible explanation of the electron-position resonances seen in heavy-ion collision experiments that is consistent with the negative results found in low-energy electron scattering as well as other heavy-ion collision experiments.PACS Nos.: 03.20.+i, 03.65.-w, 05.45.+6
The problem of time in quantum gravity is often presented as a consequence of applying quantum theory to general relativity. These notes adopt a broader perspective. We first examine how time enters classical mechanics, relativity, quantum theory, and quantum field theory, distinguishing three principal aspects: causal order, temporal duration, and the present. This analysis shows that the conceptual difficulties associated with time do not arise only when gravity is quantised. Tensions between causal structure, clocks, observables, and measurement are already present in the theories from which quantum gravity is constructed. We then survey responses to the problem of time across the principal approaches to quantum gravity. These include canonical and Wheeler-DeWitt quantisation, path-integral and histories formulations, perturbative and background-dependent approaches, causal sets, and twistor theory. The programmes are compared according to whether they retain an external time, render time problematic through quantisation, or incorporate causal or temporal structure into their foundations. Particular emphasis is placed on questions arising from quantum foundations and relativistic quantum information. We conclude by examining linearised gravity and weak-gravity quantum systems. This regime shows that the problem of time is not confined to Planck-scale physics, but arises already in attempts to provide a predictive and operationally meaningful quantum description of spacetime near flat geometry.
Relativistic quantum mechanics was developed to reconcile the principles of quantum mechanics with Einstein’s theory of relativity. Despite its success in describing high-energy particles, the theory continues to face unresolved conceptual and mathematical difficulties, particularly in relation to the nature of time, causality, and relativistic consistency. In recent years, the idea of extending space-time into the complex domain has emerged as a useful and potentially meaningful approach to these problems. In this work, a theoretical synthesis is presented to examine the role of complex space-time in relativistic quantum mechanics. The study discusses how complex time arises naturally in relativistic wave equations, quantum field theory, and path integral formulations. Established methods such as Wick rotation, complex coordinate transformations, and non-Hermitian but parity–time (PT) symmetric Hamiltonians are reviewed to illustrate how complex space-time contributes to mathematical stability and clearer physical interpretation (Dirac, 1928; Wick, 1954; Bender & Boettcher, 1998). The relevance of complex space-time is further explored in the context of relativistic particle dynamics and quantum cosmology, where it helps address, issues related to vacuum structure, time evolution, and causality (Feynman & Hibbs, 1965; Hartle & Hawking, 1983). The paper suggests that complex space-time should be viewed not only as a computational technique, but as a framework that may reflect deeper features of relativistic quantum phenomena.
Usually the only difference between relativistic quantization and standard one is that the Lagrangian of the system under consideration should be Lorentz invariant. The standard approaches are logically incomplete and produce solutions with unpleasant properties: negative-energy, superluminal propagation etc. We propose a two-projections scheme of (special) relativistic quantization. The first projection defines the quantization procedure (e.g. the Berezin-Toeplitz quantization). The second projection defines a casual structure of the relativistic system (e.g. the operator of multiplication by the characteristic function of the future cone). The two-projections quantization introduces in a natural way the existence of three types of relativistic particles (with 0, 21, and 1 spins). Keywords: Quantization, relativity, spin, Dirac equation, Klein-Gordon equation, electron, Segal-Bargmann space, Berezin-Toeplitz quantization. AMSMSC Primary: 81P10, 83A05; Secondary: 81R30, 81S99, 81V45
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.