The relation between quantum theory and special relativity is peculiar. On the one hand it is close and essential. Steven Weinberg [1], for example, takes the position that the standard model is an inevitable consequence of the marriage of the two theories. On the other, the action at a distance implied by the famous paper by Einstein, Podolsky, and Rosen [2] is certainly contrary to the implications of special relativity in spite of the fact that it does not allow transmission of information. The setting of quantum theory is a Hilbert spaceH and a path inH whose development in time is governed by a one parameter group of unitary transformations while the setting for special relativity can be thought of as Minkowski space with the time development not absolute, but dependent on the inertial system of the observer. The problem is to show how these two very different viewpoints can be reconciled in a way that includes an explanation of the "action at a distance" conundrum.
The relation between quantum theory and special relativity is peculiar. On the one hand it is close and essential. Steven Weinberg [1], for example, takes the position that the standard model is an inevitable consequence of the marriage of the two theories. On the other, the action at a distance implied by the famous paper by Einstein, Podolsky, and Rosen [2] is certainly contrary to the implications of special relativity in spite of the fact that it does not allow transmission of information. The setting of quantum theory is a Hilbert spaceH and a path inH whose development in time is governed by a one parameter group of unitary transformations while the setting for special relativity can be thought of as Minkowski space with the time development not absolute, but dependent on the inertial system of the observer. The problem is to show how these two very different viewpoints can be reconciled in a way that includes an explanation of the "action at a distance" conundrum.
In 1935 Einstein, Podolsky and Rosen (EPR) pointed out that Quantum Mechanics apparently implied some mysterious, instantaneous action at a distance. This paradox is supposed to be related to the probabilistic nature of the theory, but since deterministic alternatives involving "Hidden Variables" hardly agree with the experiments, the scientific community is now accepting this ``quantum nonlocality" as if it were a reality. However, I have argued recently that Quantum Electrodynamics is free from the EPR paradox, due to an indetermination on the number of the unobserved "soft photons" that can be present in any step of any experiment. Here, I will provide a more general proof, based on an approach to the "problem of measurement" that implies the full reconciliation of Quantum Field Theory with Special Relativity. I will then conclude with some considerations on the interpretation of the Quantum Theory itself.
Demonstrations of quantum entanglement which confirm the violation of Bell's inequality indicate that under certain conditions action at a distance is possible. This consequence seems to contradict the relativistic principle of causality, which asserts that an effect never precedes its cause, in any reference frame. By analyzing a numerical example of Bell's experiment with entangled pairs of photons, we show how observers in two inertial reference frames can disagree about the causality relation between two events. One observer claims that event 1 is the cause of event 2, while the other claims that event 1 is the result of event 2. The solution we suggest to the paradox is that in entangled systems, one can find pairs of "entangled events" which have symmetrical causality relations. Each of the events can serve as a cause or as an effect, depending on the frame of reference in which they are observed.
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.
In Mundy [a] I offered an axiomatic analysis of the physical content of the kinematics of special relativity which suggests that, contrary to common belief, there is no incompatibility between special relativity and spacelike (faster-than-light) causation. An anonymous referee pointed out that this conclusion might have some bearing on problems in the interpretation of quantum mechanics such as the Einstein-Podolsky-Rosen problem, since one line of solution to these problems involves the postulation of spacelike causal processes. The present note will develop this idea.
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.