We describe a new form of retrocausality, which is found in the behavior of a class of causal set theories, called energetic causal sets (ECS). These are discrete sets of events, connected by causal relations. They have three orders: (1) a birth order, which is the order in which events are generated; this is a total order which is the true causal order, (2) a dynamical partial order, which prescribes the flows of energy and momentum among events, (3) an emergent causal order, which is defined by the geometry of an emergent Minkowski spacetime, in which the events of the causal sets are embedded. However, the embedding of the events in the emergent Minkowski spacetime may preserve neither the true causal order in (1) nor correspond completely with the microscopic partial order in (2). We call this disordered causality, and we here demonstrate its occurrence in specific ECS models. This is the second in a series of papers centered around the question: should we accept violations of causality as a lesser price to pay in order to keep realist formulations of quantum theory? We begin to address this in the first paper [E. Cohen, M. Cort\^es, A. C. Elitzur, and L. Smolin, preceding paper, Realism and causality. I. Pilot wave and retrocausal models as possible facilitators, Phys. Rev. D 102, 124027 (2020).] and continue here by giving an explicit example of an ECS model in the classical regime, in which causality is disordered.
We describe a new form of retrocausality, which is found in the behavior of a class of causal set theories, called energetic causal sets (ECS). These are discrete sets of events, connected by causal relations. They have three orders: (1) a birth order, which is the order in which events are generated; this is a total order which is the true causal order, (2) a dynamical partial order, which prescribes the flows of energy and momentum among events, (3) an emergent causal order, which is defined by the geometry of an emergent Minkowski spacetime, in which the events of the causal sets are embedded. However, the embedding of the events in the emergent Minkowski spacetime may preserve neither the true causal order in (1) nor correspond completely with the microscopic partial order in (2). We call this disordered causality, and we here demonstrate its occurrence in specific ECS models. This is the second in a series of papers centered around the question: should we accept violations of causality as a lesser price to pay in order to keep realist formulations of quantum theory? We begin to address this in the first paper [E. Cohen, M. Cort^es, A. C. Elitzur, and L. Smolin, preceding paper, Realism and causality. I. Pilot wave and retrocausal models as possible facilitators, Phys. Rev. D 102, 124027 (2020).] and continue here by giving an explicit example of an ECS model in the classical regime, in which causality is disordered.
Of all basic principles of classical physics, realism should arguably be the last to be given up when seeking a better interpretation of quantum mechanics. We examine the de Broglie-Bohm pilot wave theory as a well-developed example of a realistic theory. We present three challenges to a naive reading of pilot wave theory, each based on a system of several entangled particles. With the help of a coarse graining of pilot wave theory into a discrete system, we show how these challenges can be answered. However, this comes with a cost. In the description of individual systems, particles appear to scatter off empty branches of the wave function as if they were particles and conversely travel through particles as if they were waves. More generally, the ``particles'' of pilot wave theory are led by the guidance equation to move in ways no classical particle would, involving apparent violations of the principles of inertia and momentum conservation. We next argue that the aforementioned cost can be avoided within a retrocausal model. In the proposed version of the pilot wave theory, the particle is guided by a combination of advanced and retarded waves. The resulting account for quantum physics seems to have greater heuristic power, demands less damage to intuition, and moreover provides some general hints regarding spacetime and causality. This is the first of two papers. In the second [E. Cohen, M. Cort\^es, A. C. Elitzur, and L. Smolin, Realism and causality. II. Retrocausality in energetic causal sets, Phys. Rev. D 102, 124028 (2020).], we show that, in the context of an explicit model, retrocausality, with respect to an effective, emergent spacetime metric, can coexist with a strict irreversibility of causal processes.
We propose an approach to quantum theory based on the energetic causal sets, introduced in [M. Cort\^es and L. Smolin, arXiv:1307.6167]. Fundamental processes are causal sets for which the events carry momentum and energy, which are transmitted along causal links and conserved at each event. Fundamentally there are amplitudes for such causal processes, but no space-time. An embedding of the causal processes in an emergent space-time arises only at the semiclassical level. Hence, fundamentally there are no commutation relations, no uncertainty principle and, indeed, no $\ensuremath{\hbar}$. All that remains of quantum theory is the relationship between the absolute value squared of complex amplitudes and probabilities. Consequently, we find that neither locality nor nonlocality is a primary concept; only causality exists at the fundamental level.
The idea that events are equipped with a partial causal order is central to our understanding of physics in the tested regimes: given two pointlike events A and B , either A is in the causal past of B , B is in the causal past of A , or A and B are space-like separated. Operationally, the meaning of these order relations corresponds to constraints on the possible correlations between experiments performed in the vicinities of the respective events: if A is in the causal past of B , an experimenter at A could signal to an experimenter at B but not the other way around, while if A and B are space-like separated, no signaling is possible in either direction. In the context of a concrete physical theory, the correlations compatible with a given causal configuration may obey further constraints. For instance, space-like correlations in quantum mechanics arise from local measurements on joint quantum states, while time-like correlations are established via quantum channels. Similarly to other variables, however, the causal order of a set of events could be random, and little is understood about the constraints that causality implies in this case. A main difficulty concerns the fact that the order of events can now generally depend on the operations performed at the locations of these events, since, for instance, an operation at A could influence the order in which B and C occur in A ’s future. So far, no formal theory of causality compatible with such dynamical causal order has been developed. Apart from being of fundamental interest in the context of inferring causal relations, such a theory is imperative for understanding recent suggestions that the causal order of events in quantum mechanics can be indefinite. Here, we develop such a theory in the general multipartite case. Starting from a background-independent definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal correlations. For a fixed number of settings and outcomes for each party, these correlations form a polytope whose facets define causal inequalities. The case of quantum correlations in this paradigm is captured by the process matrix formalism. We investigate the link between causality and the closely related notion of causal separability of quantum processes, which we here define rigorously in analogy with the link between Bell locality and separability of quantum states. We show that causality and causal separability are not equivalent in general by giving an example of a physically admissible tripartite quantum process that is causal but not causally separable. We also show that there are causally separable quantum processes that become non-causal if extended by supplying the parties with entangled ancillas. This motivates the concepts of extensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant under extension. We characterize the class of ECS quantum processes in the tripartite case via simple conditions on the form of the process matrix. We show that the processes realizable by classically controlled quantum circuits are ECS and conjecture that the reverse also holds.
The Causal Set hypothesis for Quantum Gravity asserts that the smooth Lorentzian spacetime manifold of General Relativity is only an approximation to a fundamental microscopic discrete structure: a locally-finite partially-ordered set. The points of this set are akin to the point-events of the spacetime manifold, and the order-relations among these points are akin to the causal-relations among these point-events. The local-finiteness condition implies a fundamental discreteness and provides an important [conformal] factor needed for the continuum approximations. In this dissertation, we address two aspects toward the formulation of a quantum dynamics for causal sets. The first aspect concerns the dynamics of a zero-mass classical scalar field on a background causal set, Poisson-sampled from an Alexandrov interval in (1 + 1)-dimensional Minkowski spacetime. Our numerical simulations suggest that actions can be defined for such a scalar field, expressed solely in terms of the causal set and the scalar field, which approximates the corresponding classical action on the continuum Minkowski spacetime. This result may help us describe matter and gauge fields in a local and Lorentz-invariant way on a causal set, a discrete structure now being considered as the arena for spacetime physics. More importantly, it may help us formulate an action for the causal set itself, which would be used in a sum-over-histories approach to describe the quantum dynamics of causal sets. The second aspect concerns Quantum Measure Theory, a new approach to quantum dynamics inspired by the sum-over-histories approach, which is better suited to formulating the dynamics of causal sets and of other alternative structures for spacetime. To better understand the mathematical structure underlying this new approach, we derive some algebraic identities involving the sum-rules of the Quantum Measure and its generalizations. We conclude with a discussion on possible applications and open problems concerning these aspects.
We propose an approach to quantum theory based on the energetic causal sets, introduced in [1]. Fundamental processes are causal sets whose events carry momentum and energy, which are transmitted along causal links and conserved at each event. Fundamentally there are amplitudes for such causal processes, but no space-time. An embedding of the causal processes in an emergent space-time arises only at the semiclassical level. Hence, fundamentally there are no commutation relations, no uncertainty principle and, indeed, no ~. All that remains of quantum theory is the relationship between the absolute value squared of complex amplitudes and probabilities. Consequently, we find that neither locality, nor non locality, are primary concepts, only causality exists at the fundamental level.
Beyond Causality: Temporal Geometry, Informational Consistency, and the Foundations of Physical Time What if the deepest mystery of time is not whether the past can be reached, but whether causality itself is more fundamental than the physical laws through which we describe it? For more than a century, modern physics has transformed our understanding of space, time, matter, and information. General relativity revealed that temporal order is inseparable from spacetime geometry. Quantum theory challenged classical intuitions about measurement, correlations, and physical reality. Information theory provided a mathematical language for describing what can be known, transmitted, and recovered. Yet a profound question remains: must every physically consistent universe possess a globally ordered sequence of causes and effects? Beyond Causality investigates this question through an extended mathematical and conceptual study of temporal geometry, informational dependence, retrocausal feedback, and global consistency. Organized into six interconnected parts, this 430-page research monograph develops a framework for examining causal structures beyond conventional temporal ordering. It brings together established results from general relativity, quantum information, causal modeling, and fixed-point theory, while exploring additional mathematical constructions designed to clarify the conditions under which temporally nonstandard information networks can remain internally consistent. Particular attention is devoted to distinguishing mathematical self-consistency from physical realizability, statistical correlation from controllable communication, and unconventional temporal descriptions from genuine violations of operational causality. Among its principal investigations are the geometric and energetic constraints associated with localized chronology-violating spacetime models, the existence and stability of globally consistent informational feedback networks, the behavior of finite stochastic systems under intervention, and the relationship between quantum causal structures and hypothetical retrocausal channels. The work also extends the theoretical research program introduced in Eon Reverser: Toward Temporal Bubble Radio Communication, examining what additional mathematical, physical, and experimental conditions would be required before communication with an earlier receiver record could be considered a meaningful scientific possibility. Analytical derivations, formal propositions, computational examples, and reproducibility materials accompany the discussion. The proposed constructions are explicitly distinguished from established physical results, and no experimental evidence for backward-in-time communication is claimed. The central ambition of this monograph is neither to declare that causality can be violated nor to assume that temporal paradoxes admit a universal resolution. It is to ask a more precise question: what mathematical structures, consistency principles, and observable constraints would a physical theory have to satisfy if causal order were not universally fixed? The possibility that time, information, and causality may possess a deeper relationship than our present formulations reveal remains an open intellectual frontier. Exploring that frontier requires imagination disciplined by mathematics, speculation constrained by physical principles, and theoretical proposals that remain accountable to observation. Beyond Causality is offered as a contribution to that exploration: an attempt to extend the questions we can rigorously formulate, identify the limits of existing models, and provide a foundation for further investigation into the nature of physical time. Keywords: Causality; Temporal Geometry; General Relativity; Retrocausality; Closed Timelike Curves; Quantum Causal Structures; Information Theory; Global Consistency; Fixed-Point Theory; Chronology Protection; Advanced Electromagnetic Response; Eon Reverser.